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REFORMING MATHEMATICS TEACHING:
EXAMINING THE RELATIONSHIP BETWEEN INSTRUCTIONAL POLICY
AND A TEACHER'S OPPORTUNITIES TO LEARN

By

Pamela Kaye Geist

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Teacher Education
1998

ABSTRACT

REFORMING MATHEMATICS TEACHING:
EXAMINING THE RELATIONSHIP BETWEEN INSTRUCTIONAL POLICY
AND A TEACHER'S OPPORTUNITIES TO LEARN

By

Pamela Kaye Geist

This dissertation offers a portrait of one teacher's efforts to attend to
state-initiated reform in mathematics education. The portrait illuminates the
complexity involved for the individual teacher in the context of a strong
systemic effort by the state to change mathematics instruction in public
schooling. It illuminates that there are a variety of different components to a
teacher's practice and much for the teacher to think about and do to revise
and compose in the course of efforts to improve instruction. Grounded in
current frameworks for viewing relations between policy and practice, teacher
learning is used as a lens for investigating a teacher's responses to policy.

One teacher's thinking and teaching is traced over time, examining the
impact of the teacher's learning on responses to two different strands of
proposals: reforming teaching of basic computational skills and introducing
discrete mathematics into elementary mathematics teaching. Three questions
guide the investigation: What mediates the teacher's interpretations and
learning in the context of efforts to reform? What does the teacher learn?
And, how does learning across time impact a teacher's interpretations and
enactment of policy? Data collection includes interviews and observations of
teaching across a three year period in two different teaching environments:

elementary school teaching in a predominantly white, middle class school

district; and teaching other teachers in state-sponsored professional
development activities. Data also includes observations of and interviews
with others involved in the teacher's professional development activities.

The case studies illuminate the ways in which an elementary teacher
encounters, interprets, transforms and implements the ideas of reform. They
portray how specific proposals elevate in importance, evolve and take root in
a teacher’s thinking, and resurface in the teacher's thought and practice again
and again. Results include a description of what the teacher learns, the
influences on her learning, and an appraisal of the impact of learning on the
teacher's understanding and enactment of the recommendations offered in
three central reform documents.

The cross-case analysis uncovers aspects of practice that make it
particularly difficult for the teacher to learn and enact instructional policy. It
shows, for example, that the mathematics framework and other levers
developed by the state are not the only influences on a teacher's practice and
that many of the influences are not always coordinated or recognized. Results
illuminate that what the teacher brings and musters as resources and learning
will have a great deal to do with whether reform presses forward at the level
of a teacher's practice. At the same time, findings reveal that the teacher, in
trying to locate and develop learning opportunities, encounters a set of weak
links between efforts to reform and opportunities to learn about reform-based
teaching. In essence, findings point to the importance of understanding better
the relationship between the individual teacher's needs as a learner and the
environments of instruction offered teachers, and working to create more

meaningful links between the two.

Copyright by
Pamela Kaye Geist
1998

Dedication

To my children, Evan and Jenna.

ACKNOWLEDGMENTS

This dissertation has traveled with me across the country three
different times as my family moved from one location to another. It has also
sat for long periods as I gave birth and began my responsibilities parenting
two wonderful children. Across each of these events, the ideas in this work
have continually consumed and captured my intellect in ways I had never
before imagined possible. It has truly been a remarkable intellectual and
deeply emotional experience. And through the process, I have grown more
in awe of the teacher focusing this study and the many others on whom I
relied to make this possible. I turn to those individuals now.

With deepest respect and gratitude I want to acknowledge Sandy Wise
for providing the opportunity to learn from her efforts to respond to the
reforms in California. I am indebted to the remarkable way that she opened
her mind, her classroom, and her heart as she shared the many promising
and the most difficult aspects of what it was like for an exemplary teacher to
learn to teach differently. I am forever grateful to Sandy for helping me to
understand with greater depth and clarity the complexities of teaching and
the difficulties involved in changing practice.

There are also many faculty members at Michigan State University that
I wish to acknowledge. I am particularly thankful to Deborah Ball as the chair
of my graduate committee and director of this dissertation. I wish to thank
Deborah for the wonderful support and guidance she has graciously given
through every stage and aspect of this work. I am deeply appreciative of her

unending patience, kindness, and keen insight into what this work could be.

vi

It truly would not have taken a final form without the continued
commitment and wisdom she gave to the process. Not only has my work
benefited greatly, but equally important, I have benefited as a person by
knowing her.

I also wish to acknowledge the other members of my graduate
committee. David Cohen, Suzanne Wilson, Helen Featherstone and
Penelope Peterson each have remarkable vision and are wonderfully kind
peOple. I am truly fortunate to have worked with each one of you. David was
the first to suggest that I could work from a distance on the EPPS project. In a
sense, his belief in me and continual support made all of this possible. By
way of conference calls, meetings by speaker phone, and rare occasions
meeting in person, he helped me to define the focus of this study and work
through some of the more difficult conceptual issues. I am truly grateful for
his genuine and warm way of asking very hard questions and gently pressing
me in promising directions.

I am grateful to Suzanne for her amazing insight and uncanny ability
to take the long-winded, jumbled thoughts I offered and return them in the
form of crystal clear ideas and new directions for focusing this work. At times
when I felt very stuck, I turned to Suzanne. She helped me to understand
some of the finer aspects of doing qualitative research, especially the analytic
work involved.

I am also thankful to Helen and Penelope, who each joined my
committee in the latter stages of the work. They each gave generously of their
time in order to read and provide feedback on earlier drafts. The perspective

that each one brought to this study helped me immensely in shaping the final

form.

vii

I also wish to acknowledge the support and friendship of Dr. Lyn
Taylor, a mathematics education professor at University of Colorado in
Denver. Lyn was instrumental in supporting my connections to projects and
professors at Michigan State University while I lived in Denver. She
generously gave her time and resources to support my efforts. I am very
fortunate to have met Lyn, she is one of the kindest and most thoughtful
individuals I have ever known.

There are two fellow graduate students to whom I wish to give special
thanks. Steve Smith and Janine Remillard have continued to support my
work in the form of a long-distance study group. By way of conference calls,
we offered each other the opportunity for colleagues to examine and
challenge ideas and writing, within the contexts of our three dissertations. I
am grateful for their support of friendship and thoughtful reading and
responses to the many, many memos and sketches of chapters throughout
this work. It has been a very special privilege to have Steve and Janine with
me as I progressed along the many promising paths and muddled my way
through the not so promising directions this work has taken me.

In addition, I wish to thank Jean Beland and Tena Harrington for
working with me long distance, helping me edit and format the document,
making sure it was prepared properly and completed by deadline.

There are many others I wish to acknowledge who had much to do
with my progress. My parents, Gaines and Mary, are two wonderfully kind
people who taught me that I could accomplish whatever I set my heart to.
They instilled an attitude of never give up, upon which I relied often,
especially during the writing stages. My family members, Randy, Kevin and
Kim, are three of the most wonderful people I know. We are very different

from one another and yet, we respect each other for the gifts we each have.

viii

Each one of you have been there for me, especially when I needed to see a
bigger picture than what the university or this work could offer.

In addition, I want to give special thanks to my husband, Tom. Tom
has supported my efforts in ways no other could. He gave me the space I
needed to think through difficult aspects of this work, without demand or
interruption. To do so, he took care of our children, taking them on many
long hikes and other outings that opened up substantial, uninterrupted
writing time for me as well as fond memories for Tom and our children.
This allowed me to do what was needed and with peace of mind. I am
especially grateful to Tom for the many hours he gave me in the last few
months of writing. There were many days that he took from his own busy
and very demanding work schedule to free up time for me to make progress.
After the children were tucked into bed, he would work late into the night.

Finally, I want to acknowledge my children, Evan and Jenna. They
have taught me more in the short time that I have had the pleasure of their
company than I could ever return. I have been truly blessed, especially in that
they arrived one year apart and in the middle of this project. Evan traveled
with me many times to collect data in California. He was a real trooper and
sat very still in his seat during many hours of travel, observations, and
interviews. Jenna came along during the writing phases. She sat with me for
long hours at the computer. A fond memory for me will be the way that she
held my little finger with her entire hand while I read and revised text. I am
very grateful to Evan and Jenna for their patience with me. Somehow I think
they understood what I needed. And in watching their growth and learning, I

am continually reminded of the importance of doing this work. .

ix

TABLE OF CONTENTS

List of Tables ............................................................................... xvi

List of Figures ............................................................................. xvii

Chapter 1 Instructional Policies, Teacher Learning, And

Reforming Mathematics Teaching ................................................. 1
Introduction ........................................................................... 1
Relations Among Policy, Teacher Learning, and Teaching Practice ........... 4

A Cognitive Frame .......................................................... 6
The Pedagogy of Policy .................................................... 12

Studying Teacher Learning In the Context of External Efforts to Reform ..... 14

Research on Teacher Learning ............................................. 16
The Role of Prevailing Professional Development Activities .......... 20
Focusing on a Single Teacher .............................................. 22

The Mathematics Education Reforms .3 ............................................. 25
What Are These Reforms? ................................................. 25
Implications For Teacher Learning ........................................ 27
Looking Ahead ....................................................................... 30
Chapter 2 Methodology and Research Procedures .......................... 32
Introduction ........................................................................... 32
Methodological Overview ........................................................... 33
Focusing The Study ......................................................... 36
Teacher Selection ................................................... 36

The Relationship Between Teacher Learning

and Teaching Practice ............................................ 37

Opportunities to Learn ............................................. 38

Distilling the Reform Vision ...................................... 40

Data Collection ............................................................... 47

Observation Data ................................................... 47

Interview Data ...................................................... 49

Other Data Sources ................................................. 50

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

Analysis of Observation and Interview Data .................... 51

Cross-Case Analyses .............................................. 53

Chapter 3 The Ideas of Policy ..................................................... 54
Introduction ........................................................................... 54
What is the Policy? A Document Analysis Approach ........................... 55
Reforming Basic Computational Skills Instruction ...................... 58

A Variety of New Computational Skills ......................... 62

Conceptual Grasp of Underlying Mathematical Ideas .......... 70

Contextualize Computational Skills .............................. 74

An Overall View of Teaching

Computational Skills Differently ......................................... 75
Introducing Discrete Mathematics ......................................... 83
What is Discrete Mathematics? .................................... 85
Why Discrete Mathematics Should Be Introduced ............. 92
How Discrete Mathematics Could Be Introduced .............. 93
Advice to Teachers ................................................. 96
Other Resources on Discrete Mathematics ................................ 99
Revisiting The Question -- What is Discrete Mathematics? . . .100
Revisiting The Question -- Why Introduce
Discrete Mathematics? ........................................... 107
Should Discrete Mathematics Be Introduced
at the Elementary Level? ......................................... 108

xi

Chapter 4 Learning To Teach Basic Computational
Skills Differently ....................................................................... 112
Introduction ........................................................................... 1 12

A Teacher's Responses to Proposals to Teach

Computational Skills Differently ................................................. l 13
A Single Mathematics Lesson .............................................. l 13
Mathematics Teaching on a Wider Scale .................................. 1 19

The Next School Year: Responding to

Problems In Practice ...................................................... 129

An Automaticity Lesson .................................................... 133
Reinterpreting the State‘s Goals ................................................... 139
Factors Affecting the Teacher's Interpretation ........................... 140

The Practice of Teaching .......................................... 141

Doctoral Studies .................................................... 143

State Mandated Testing ............................................ 145

Conversations With Other Educators ............................ 145

Reform Documents ................................................. 147

A Teacher's Prior Knowledge and Beliefs ...................... 148

An Appraisal of a Teacher's Responses to Changing

Teaching of Computational Skills ................................................ 150

A Variety of New Computational Skills .................................. 151

Contextualize Computational Skills ....................................... 152

Conceptual Grasp of Underlying Ideas ................................... 157

Chapter 5 Learning To Teach An Unfamiliar Mathematics ............... 166
Introduction ........................................................................... 1 67

A Teacher Leader's Responses to Proposals to Introduce Discrete
Mathematics Into the Elementary School Mathematics Curriculum .......... 167

xii

Learning About Discrete Mathematics .................................... 170

The Reform Documents ........................................... 171
Conversations With Other Educators ............................ 174
Curriculum Materials ............................................... 180
Teacher Workshops ................................................ 185
Teaching Combinatorial Counting For Understanding ................. 194
The Mathematics This Teacher Focused On ..................... 195
Combinatorial Mathematics ....................................... 197

An Appraisal of A Teacher Leader's Teaching
the Mathematics Of Counting ............................................ 200

Chapter 6 Re-Forming Mathematics Teaching

And The Teacher's Opportunities to Learn ................................... 205
Introduction ........................................................................... 205
A Pedagogical Framework .......................................................... 205

A Curriculum of Reform ................................................... 206
Introducing New Content ......................................... 207
Teaching Familiar Mathematics in New Ways .................. 208
Introducing New Pedagogical Practices ......................... 21 l

The Teacher as Learner ..................................................... 213
What Does The Teacher Bring? ................................... 213
Prior Knowledge as Prerequisite Knowledge ................... 217
Prior Knowledge and Framing a Course of Action ............ 219

A Teacher's Opportunities to Learn (Links Between

Policy and Teaching Practice) ............................................ 221
The Big Ideas ....................................................... 227
Connections to A Teacher's Prior Knowledge and Beliefs. . . .236
The Teacher's Views of What the

Opportunities To Learn Offered ................................ 246

xiii

 

The Qualities And Conditions Of A Teacher's

Opportunities To Learn About Reform-Based Teaching ...................... 247
Mismatch in Content ............................................... 247
Few Connections to the Teacher's Prior Knowledge and
Teaching Experience ............................................. 248
Lack of Continuity Across Learning Experiences .............. 249
Few Opportunities to Connect Learning to
Teaching Practice ................................................. 250
Absence of Purposeful Teaching ................................. 250
Little Common Learning Community ............................ 251
A Search for Certainty ............................................. 252

Chapter 7 Linking Instructional Policies To Professional Development
Opportunities For Teachers .......................................................... 255
Introduction ........................................................................... 25 5

Responding to Reform Initiatives: Focusing at the Level

of the Individual Teacher .......................................................... 260
The Dynamic Nature of the Learning Process ........................... 261
Factors Affecting Teacher Learning ....................................... 264

Analysis of the Impact of Changes on Student Learning ...... 265

Learning Contexts Not Intended to Press Policy Objectives. .265
The Teacher's Personal Stance Toward Learning .............. 266
Topic-Specific Differences in Teachers' Prior Knowledge

and Teaching Experience ........................................ 267

The Non-Systematic Nature of Teachers' Opportunities to Learn ..... 268

Creating Stronger Learning Links Between Instructional Policies

and Teachers’ Opportunities to Learn ............................................ 272
Attending to the Nonsystematic Nature of Teachers'

Professional Development Opportunities ............................... 275

A Combination of Freedom and Control ........................ 276

The Important Role and Preparation of Teacher Leaders ...... 280

xiv

Proposals to Improve Teachers' Opportunities to Learn ................ 283

Learning to Analyze the Impact of Change ...................... 283
Fostering Beliefs About Improving

Students' Opportunities to Learn ............................... 285
Fostering A Stance of Critique And Inquiry .................... 287

Subject-Matter Focused, Topic-Specific
Opportunities to Learn ........................................... 289
Change At the Level of The Smallest Unit ........................................ 292
Appendix A .................................................................................... 298
References ...................................................................................... 307

XV

LIST OF TABLES

Table 1. What Sandy Brings ................................................................................ 214

Table 2. Central Reform Ideas and the Central Ideas of Sandy's
Opportunities to Learn About Computational Skill Instruction .............. 228

Table 3. Central Reform Ideas and the Central Ideas of Sandy's
Opportunities to Learn About Introducing Discrete Mathematics ........... 232

Table 4. Central Ideas of Sandy's Opportunities to Learn About
Reforming Teaching of Computational
Skills and What Sandy Brings .......................................................................... 238

Table 5. What Sandy Brings and the Central Ideas of Sandy's

Opportunities to Learning About Introducing
Discrete Mathematics .......................................................................................... 241

xvi

LIST OF FIGURES

Figure 1. Learning Template Basic Conputational Skills Instruction ....... 52
Figure 2. Basic Computational Skills ................................................................ 63
Figure 3. Teaching Computational Skills ........................................................ 76
Figure 4. Traditional Forms of Computational Skills Instruction ............. 79
Figure 5. Reforming Computational Skills Instruction ............................... 80
Figure 6. Sandy Wise - Master Plan 1991-92 School Year ............................. 125
Figure 7. Ven Diagram ......................................................................................... 210

xvii

CHAPTER 1

INSTRUCTIONAL POLICIES, TEACHER
LEARNING, AND REFORMING MATHEMATICS TEACHING

Introduction

Lessons from the past paint a bleak picture. American public schooling
has a poor track record involving many failed attempts to introduce more
ambitious curricula and instructional practices into public schooling (Cohen &
Neufeld, 1981; Cuban, 1984; Tyack 82: Cuban, 1995) Instructional policies
historically have had little overall success. Our history teaches us that reforming
school instruction is a difficult task and that teachers will not have an easy time
of it (Cuban, 1984; Powell, Farrar, & Cohen, 1985; Sarason, 1982, 1990).

Yet, despite the lessons from our past, the last decade has been
characterized by an explosion of new and ambitious policies aimed at reforming
mathematics teaching and learning (California State Department of Education,
1985, 1992; National Council of Teachers of Mathematics, 1989, 1991; National
Research Council, 1990). The State Department of Education in California was
no exception launching massive efforts to reform mathematics teaching in
widely-touted documents such as the Mathematics Framework for California

u 1i 1 (1992), and It’s Elementagg (1993). Yet, even as reform was
underway, there was much evidence to suggest that few instructional reform
efforts ever make it past the classroom door (Cuban, 1990; Sarason, 1990).

Modern day efforts to reform are ambitious. They aim to alter public

education by challenging extant views of knowledge and enduring patterns of

classroom teaching and learning (NCTM, 1989, 1991 ; CSDE, 1992). Yet, as visions

of more ambitious educational experiences for US. students, they present
substantial problems of comprehension, interpretation, and enactment for the
classroom teacher (Cohen, 1989; Cuban, 1984). An increasing number of
educators and researchers are trying to understand what these problems entail
and are giving serious attention to the role of the teacher in enacting reform
(Cohen 8: Ball, 1990; Cohen et al., 1990; Elmore 8: McLaughlin, 1988; Elmore et
al., 1996; Featherstone 8: Smith, 1996; Little, 1993; Schifter 8: Fosnot, 1993;
Shulrnan, 1983; Simon 8: Schifter, 1991 ; Wilson et al., 1996).

This dissertation is situated along that line of research. It explores how a
single teacher in California responds to state-initiated efforts to reform
mathematics teaching and learning in public schooling. The chapters ahead
portray what happens when a teacher is completely willing to learn and change
her teaching as the state proposed. The case studies that follow paint a portrait
of the kind of teacher policymakers would tend to admire. Sandy Wisel, a third
grade teacher in the Forest Glen school district, suspended any disbelief she may
have had concerning the state’s plans to reform. She was endlessly energetic in
her efforts to learn about the new policy and change her teaching accordingly.
Sandy was quite adventurous in her attempts to construct a teaching practice
that offered students significantly different opportunities to learn mathematics.

What we stand to learn from Sandy’s efforts at reform is invaluable.
Unless and until we understand how teachers like Sandy interpret and learn
from the waves of reform that come their way, we are likely to continue to fly
blind with little overall success at changing what goes on inside classrooms. Part
of what is at stake is getting past the common assumption that policy, if well-

developed, can easily be put into practice by willing teachers. This study

 

1As I promised confidentiality, Sandy Wise, the school and district are all pseudonyms.

challenges the belief that traditional instruction is somehow rooted in teachers’
bad habits or unwillingness and inabilityA even if policymakers were able to
promote better solutions to curricular and instructional problems (see Cremin,
1961; Kaestle, 1972, 1983; Katz, 1987; Slavin, 1989) and could construct better
policies to promote such changes, there would remain the problem of educating
teachers about any efforts to reform.

Inattention to what is involved for the practitioner would only produce
more wheel-spinning and for many an accumulation of despair. These risks are
not worth taking. Given our history of failed attempts to create more ambitious
teaching and learning in US. classrooms, we no longer can afford to ignore what
is involved for the teacher. Currently, we know very little about how the teacher
encounters, responds, and enacts new policy. We have few studies that portray
what teachers make of proposed instructional changes, the difficulties they
encounter, or how teachers come to understand policy and change their teaching.
If we ever hope to improve education in the US, these issues must be given
much more attention.

This study sheds light on several of these issues. It provides policymakers
and teacher educators a rear-view mirror. It offers a way to continually check in
with what is happening once a policy is in place. At times, the mirror reflects
back the experiences of the classroom teacher as she introduces changes in her
practice. At other times it reflects how policy ideas were encountered by the
teacher, what ideas received attention, what was understood about those ideas,
and how policy ideas were formulated into changes in teaching. The mirror also
illuminates the role and influence of the teacher’s professional development
opportunities in learning about and from new policy. Ultimately, this study
offers an explanation for how the teacher, despite much new learning and great

effort, can produce a practice that resembles little of what is imagined in a policy.

Relations Among Policy,
Teacher Learning, and Teaching Practice

What would it take for policy ideas to successfully influence teaching
practice? Policymakers have considered this question mostly on the basis of how
they might persuade educators to adopt and implement policy ideas. The issues
involved often have more to do with providing appropriate incentives that
convince irnplementors that a new policy is worthwhile. From this vantage
point, adopting and implementing policy often involves more a process of telling
educators what the policy is and offering the necessary incentives. Some
incentives served more as enticements, such as merit pay or elevated
professional roles. Others served more to punish implementors. Teacher
evaluations, new rules, and standards often played out as negative incentives.
No matter how incentives were perceived, relations between policy and practice
were understood in terms of the incentives offered and the effects of those
incentives.

Conventional policy-practice research offered different ideas and
explanations for the ways in which policy did or did not find its way into
teaching practice. Much of the earlier analyses focused on quantitative
measures, such as students’ standardized test scores, to determine the effects of
policy. These measures usually resulted in the claim that new policy had little or
no effect on practice. Often, the results led to increased regulation in educational
programs. The ”failure” to implement policy more often was attributed to
teachers’ and administrators’ bad habits and resistance to change (Cuban, 1984;
Fullan, 1982; Sarason, 1990; Tyack 8: Cuban, 1995). What we learned from these
scholars was that when teachers were not directly a part of the development of
new policy and their ownership was not ensured, top-down imposition led to

changes that did not last (Elmore, 1983; Sarason, 1982).

Policy analysts then began studying relations between the bottom and the
top of the educational system. Some argued that policy implementation
involved a process of transmitting new ideas about teaching and learning as
envisioned in policy instruments into teaching practice. Fullan, for example,
argued that, "Implementation consists of putting into practice an idea, program
or set of activities and structures new to the people attempting or expected to
change" (Fullan, 1992, p. 65). This view fostered the notion that policy could be
placed into practice, in a rather straightforward fashion, by way of teachers’ use
of new texts, tests, or other structures that embodied the policy vision.
Implementation was thus understood in terms of those structures and the effects
they had on teaching practice.

Michael Lipsky (1980), offered yet another account in Street level
Mm. He explained relations between policy and practice as more a
matter involving individual decision-making. He suggested that employees
attempt to solve problems and exercise discretion in an effort to meet clients
needs. He argued that formal bureaucratic controls may actually compromise
quality as a result. Studies such as this one, viewed relations between policy and
practice to involve the individual’s need to cope with the uncertainties and
pressures that come along with policy changes and an examination of the
structures that promote those changes (Lipsky, 1980).

Others, such as McLaughlin, perceived policy implementation to involve a
mutually adaptive process, one where policy was transformed to meet the needs
of the educator and practice was transformed to fit the goals of policy
(McLaughlin, 1976). From this View, both the transformation of project goals and
the changes personnel made to incorporate those goals were key to

understanding policy implementation. Here, both policy ideas and teaching

practice underwent change. McLaughlin argued that often these transformations
were rooted in individual sense making and learning.

This study begins at a fundamentally different point than those mentioned
thus far. The teacher in this study was already persuaded about the policy. She
needed no new incentives. She already was convinced that the intellectually
ambitious goals of California’s reform efforts were not only essential but
necessary for improving mathematics education for all students in the state.
There was little, if any, resistance on her part or her administrators. Similar to
other studies, I considered the individual to be a crucial link between policy and
practice. I assumed like others that teacher thinking and interpretation were key
factors influencing policy implementation. But, I took that logic one step further.
I assumed that policy changes ultimately would depend on teacher learning and
so examined any efforts at reform by exploring what and how the teacher
learned and in relationship to changed teaching.

From this perspective, this dissertation explores relations between policy
and practice from the vantage point of teacher learning and the effects of teacher
learning on teaching practice. Findings grow out of a longitudinal investigation
of a single teacher's learning and change in relationship to a state-initiated policy
to reform mathematics education. It investigates how the teacher responded to
policy proposals over time and across teaching environments. It examines what
and how the teacher learned to understand and enact the proposed changes.

And it looks carefully at the factors that mediated teacher learning.

A Cegnitive Frame

The origins of the framework underlying this work can be traced back to
policy-practice research of the early eighties. Researchers were beginning to

focus on what was happening with policy between the points of conception and

enactment. Policy analysts began looking in local settings and asking what
teachers and administrators were making of new policy. They began to realize
that the individual had much influence over how policy ideas actually played
out in practice. Elmore (1983), explained the phenomenon as the ”the power of
the bottom over the top” arguing that local meanings of policy were more useful
for understanding whether and how policies were adopted and institutionalized.
Studies such as the RAND Change Agent (1975) investigation illuminated further
how top-down policies constrained practice and often did not help teachers
construct the kind of teaching imagined. These studies suggested that local
leadership and motivation were critical to policy success (Darling-Hammond 8:
Berry, 1988; Darling-Hammond 8: Wise, 1981; Fullan, 1982; Sarason, 1982).

They also made another significant contribution. Studies such as these
suggested that teachers’ and administrators" opportunities for continual
learning, experimentation, and decision making throughout the implementation
process had a significant impact on whether policies came alive in classrooms
(McLaughlin, 1990, 1993). These findings set the stage for examining policy-
practice relations at the local level.

Later studies would focus on district, school, classroom, and teacher level
investigations of school reform. Studies of teachers’ thinking and learning, in
particular, were recognized as key to providing better information about the
outcomes of policy in practice (Cohen 8: Ball, 1990; Fullan, 1992). Researchers
argued that studies focusing on teacher learning and change could improve
policy making itself, as they provided deeper insights into aspects of teaching
practice that made policy implementation particularly difficult.

Califomia’s efforts to reform mathematics education in the mid-eighties
set the stage for what later would become groundbreaking research on relations

of policy and practice. Embodied in California’s efforts were assumptions about

what it would take to make the proposed changes possible. Teacher learning,
although acknowledged, was not portrayed as a critical factor in the early stages
of the implementation process. Instead, policymakers aimed their influence in
other directions. New goals and standards for practice and aligning those
standards with assessments, texts, and other curricular links were considered
central. The instruments of policy (curriculum frameworks and guides, tests,
texts and related professional development) proposed new conceptions of
student learning, new images of good teaching, and a commitment to serving the
needs of a diverse student population (CSDE, 1992; NCTM, 1989, 1991).

Policymakers in California began pressing practices toward these goals.
In 1985, the State Department of Education introduced a new mathematics
curriculum framework and corresponding curriculum guidelines. State officials
had consulted mathematicians, mathematics educators, and teachers in the
development of these documents. Although such frameworks had been issued
to local educators since the 1960’s, the 1985 version played a very different role
than earlier frameworks. State officials used the new document to press
publishers to revise both the content and pedagogical suggestions in their texts,
arguing that texts should conform to the goals of the new framework.

For the first time publishers were exhorted to emphasize understanding
mathematical ideas and problem solving rather than rote-memorization of facts
and algorithms. They were told if they did not make major changes in their
books, the texts would not be considered for adoption. In 1986, the state upheld
its position and rejected every text publishers submitted during the first round of
subsequent textbook adoptions.

Officials in Sacramento further used the framework to launch major
revisions in the state’s achievement testing program. Policymakers believed

these instruments would promote further the ideas of the policy. Instructional

alignment also played a key role. Education agencies recast curriculum
guidelines to send the same clear messages to teachers about reform. By 1992,
there were many elements of the reform in place. The framework, testing
program, instructional alignment, and newly adopted texts, all seemed to carry
the same message. Systemic alignment was, for the most part, accomplished.

A clear cut shift in ideas about what it would take to promote the new
policy took place near the end of the decade. Scholars and reformers alike began
to recognize the need to support teachers’ understanding of the policy and
shifted the focus on what it would take to teachers’ professional learning (Cohen
et al., 1990; Little, 1993; Sparks 8: Loucks-Horsley, 1990). Professional
development became the center of the reform movement. Teachers began
participating in workshops and institutes, each one aiming to educate teachers
about the policy and support teachers’ efforts to change their practice.

Research studying the effects of the 1985 mathematics framework
concluded that the policy did influence teachers’ instructional strategies (Cohen 8:
Ball, 1990a; Cohen 8: Spillane, 1992; Smith, 1991). Yet, they argued that teachers
often viewed the policy more as an add on, something to be done over and above
modal practice or made more compatible with prevailing practices (Cohen 8: Ball,
1990a; Wilson 8: Corbett, 1990). Preliminary research reports argued that the
policy, much like past efforts (see Powell, Farrar 8: Cohen, 1985; Sarason, 1982),
had failed to appreciate the teacher’s role in implementation. In particular, the
policy failed to acknowledge teachers need to learn (Cohen 8: Ball, 1990).

By 1990, researchers were arguing that the effects of educational policies
and programs depended chiefly on what teachers would make of them (Ball, 1990;
Elmore 8: McLaughlin, 1988). Teacher case studies of California’s efforts published

inE a ' nalEval ati nand P 1i Anal sis (Fall, 1990) and the Elementagg
ihgollgml (1992), illustrated how teachers constructed different meanings

from policy (Cohen, 1990; Peterson, 1990; Putnam et al., 1992; Wilson, 1990). They
suggested that the different constructions depended on a number of factors,
including teachers' knowledge and beliefs about teaching and the subject matter.
Cohen and Ball (1990, p. 238) pointed out that, "instructional policies are filtered
through teachers' knowledge and beliefs about academic subjects and through
their established practices."

Over the course of the next seven years, the Education Policy and Practice
Study (EPPS), would continue to draw much attention to teachers’ professional
learning and the relations among teacher learning, policy and practice. EPPS
researchers began investigating the course of instructional reform in mathematics
and reading in and across three states, California, Michigan and South Carolina.
This dissertation grows out of that research.2

I became involved in the EPPS study as a third year graduate student.
Prior to that, most of my work was situated in professional development schools,
partnerships formed between faculty at local school sites and Michigan State
University. My work in these schools surprised and frustrated me. I observed
that some teachers worked very hard to understand the mathematics education
reforms. Some went to great lengths to involve themselves in learning
opportunities that would help to understand the proposed recommendations in
new policy and formulate changes into their teaching. Yet, even though some
teachers learned a great deal, overall, students’ learning experiences seemed to

change little. Consequently, I wanted to understand more about why.

 

2This study was done as part of the Educational Policy and Practice Study at Michigan State
University. It was supported in part from grants from the National Science Foundation: Pew
Charitable Trust: Carnegie Corporation of New York: the Consortium for Policy Research in
Education, which is funded by a grant from the US. Department of Education, Office of

Educational Research and Improvement (Grant No. OERI-G-008690011): and Michigan State
University.

10

My interest in the EPPS project grew in relationship to my growing
despair with what I had observed in working in professional development
schools. Even as there was much professional development, there was little
change in teaching. I aimed to look more closely at what mediated teacher
learning, what the teacher was learning, and how learning informed the kinds of
changes teachers made in their teaching. I imagined that this work would be
useful for guiding future efforts to provide professional development around
policy. The EPPS project provided an opportunity to study these issues and
within a community of researchers asking similar questions about teacher
learning and improving teaching.

Early on in the project researchers noticed that what teachers were
learning influenced their interpretations of the policy and what should change in
their teaching. Yet, they were unclear as to what teachers were actually learning
or how they were learning it. In response, EPPS researchers launched an
investigation into teacher learning and the relationship of that learning to policy
implementation to see better what impacted teachers’ interpretations of the
policy and changing their teaching.

As researchers interviewed teachers and observed their practices, they
noticed how teachers unpacked the ideas of the policy and how teacher learning
impacted determinations of what should change. Teaching and learning became
a metaphor for viewing and understanding how policy played out in teaching
practice. This metaphor represented a new and different account of the interplay
between policy and practice. EPPS researchers claimed that the new educational
policies, because they called for dramatic departures from what was currently
practiced and understood, required extensive and profound learning on the part
of teachers (Cohen 8: Barnes, 1993a). They posited that all change would depend

on learning and began examining efforts to reform on that basis.

11

The Pedagggy Qf Pglicy
Cohen and Barnes (1993a, 1993b), in two essays, developed a descriptive

framework for examining the complex and dynamic nature of the learning
process within relations between educational policies and teaching practice. The
ideas they put forth paralleled relations among teachers, students, and subject
matter in classrooms. Policymakers were viewed much like educators, offering
new ideas about teaching, learning, and reforming teaching. Policy instruments
embodied those ideas and represented a curriculum of reform. Policy learners
were those responsible to implement the proposed changes. Implementation
involved learning on the part of teachers, administrators, teacher educators,
curriculum specialists, school boards, and many others (Wilson et al., 1996).

Applying this frame to three recent episodes in US. school reform, Cohen
and Barnes concluded that the ”pedagogy of policy” had been didactic and
inconsistent at best. Policymakers mostly told teachers what to do and little had
been done to educate teachers about what the new proposals would mean for
classroom teaching (1993a). Having critiqued the traditional pedagogy of policy,
Cohen and Barnes (1993b), asked what an educative frame might mean for the
design and implementation of future instructional policies that press for
ambitious classroom teaching. They concluded that policy design and
implementation processes (e.g., developing instructional frameworks, selecting
curricular materials, redesigning tests) should include rich opportunities for
teachers and other educators to learn (Cohen 8: Barnes, 1993b).

The notion that policy design and implementation was educative in
nature became central to all future EPPS work. Researchers identified a number
of factors that conceivably would affect the quality of teachers’ learning

experiences surrounding policy. These factors included experiences with the

12

ideas of policy (e.g., subject matter frameworks), the resources learners brought
to bear in their learning experiences (e. g., content knowledge, beliefs about
learning), and the discourse patterns between policymakers and practitioners.

Several studies examined relations among these factors and teacher
learning from policy. For example, Jennings (1995) examined teacher learning
from policy by exploring ways in which teachers' prior experiences and beliefs
shaped their learning of new policy. Grant, Peterson, and Shojgreen-Downer
(1996), underscored that teacher learning from policy occurs within multiple and
embedded contexts that are frequently misaligned, presenting challenges for
teachers’ construction of ambitious pedagogy. Investigations also focused on
district and school administrators' learning from policy. For instance, Spillane
(1993), examined local administrators' knowledge and beliefs and how personal
histories with the ideas shaped learning experiences surrounding state-
sponsored reading policy. Peterson, Prawat, and Grant (1994) reported on a
district where administrators responded to declining fiscal resources, changing
demographics, and new curriculum policies, by "learning."

Once EPPS researchers had developed some ideas about the influences on
teacher learning and change, they also wondered whether and how teachers’
understanding of policy changed over time. This study takes up these questions
and investigates teacher learning and change over the course of two years. It
focuses on how the teacher’s understanding of policy changes over time and
examines what mediates those changes.

This study extends earlier EPPS work in another way as well. Previous
EPPS studies already had established that there were differences in teachers’
responses to reading and mathematics reforms arguing that subject-matter
makes a difference in reforming teaching (Ball 8: Cohen, 1995b; Grant, 1995;

Spillane, 1996). Researchers suggested that educators were better—positioned to

13

learn and respond to new instructional policy in reading than in mathematics
(Ball 8: Cohen, 1995b). This study examines subject-matter aspects of changing
teaching by focusing specifically on mathematics and investigating teacher
learning and change across two different reform-targeted subject specific content
areas: basic computational skills and discrete mathematics. Across topic,
subject-specific investigations of teacher learning and change have rarely been
pursued. Yet, it would make sense to think that changing how multiplication is
taught requires different learning than, for example, introducing combinatorial
counting ideas into teaching. My purpose was to explore whether and how
specific content within a subject-matter makes a difference in reform and its
enactment.

The next two major sections of this chapter describe how I developed the
two larger bodies of work underlying this dissertation. Unlike most studies of
teacher learning to date, this study investigates the individual teacher's learning
and change across a range of different opportunities to learn about reform-based
teaching. In terms of design, this meant there were no pre-determined occasions
to examine what or how the teacher learned. Consequently, I had to develop a
framework for guiding my investigation of teacher learning and change. I also
had to develop some ideas about the policy itself. This proved quite challenging,
in part because the policy proposed a kind of teaching that was not well-

understood or readily observable in classrooms.

Studying Teacher Learning
In the Context of External Efforts to Reform

In general, we lack good conceptual and theoretical frameworks for
understanding what happens when teachers respond to instructional policies.
We know little about what they learn in the context of efforts to reform or how

their learning impacts the decisions they make to change their teaching. We

14

know even less about what happens once changes in practice are in place.
Problematic is that policymakers often behave as though policy implementation
is virtually over when a policy has been developed and regulations are in place.
In addition, most studies that investigate teacher learning do so in specific
professional development contexts or in relation to particular learning goals
rather than broad-based efforts to reform teaching (Lieberman, 1995; Schifter,
1993; Wood et al., 1991). Typically, these studies make claims about prevailing
forms of teacher professional development and discuss whether the structures or
community were adequate to support learning (Little, 1993; Lord, 1994; Sparks 8:
Loucks-Horsley, 1990). Understanding the impact of learning across a variety of
different contexts on a teacher's interpretations and enactment of policy remains
mostly unexamined.

My purposes required a much more comprehensive investigation into
teacher learning and change. I wanted to understand what happened to a
teacher's ideas and practice over time and in relationship to the variety of
learning opportunities a teacher encountered. This suggested a longitudinal
investigation across many learning contexts. In a sense, I imagined positioning a
searchlight on an individual teacher’s encounters with policy ideas. I imagined
using the searchlight to locate any occasions, circumstances, or instances that the
teacher encountered for making sense of policy proposals. My aim would be to
unpack these occasions, examining what mediated the teacher's learning, what
and how the teacher learned, and whether and how her learning impacted her
interpretations and enactment of policy. Across time, I would explore a series of
learning events trying to understand both what the teacher was offered for
understanding state efforts to reform mathematics education and how the
teacher responded to those occasions. The literature on teacher learning and

change informed how I would examine the teacher's opportunities to learn.

15

Researeh en :1 eaeher Learning

Historically, teacher development considered teaching to be more a
technical craft where teacher training of technical skills was considered central.
This history suggested the likelihood that some professional development would
focus only on the technical aspects of teaching and the ideas and practices
involved on these occasions would require little serious learning (Cohen, 1989).
It would be important to consider whether and what these occasions added to a
teacher’s understanding of the policy.

Other studies indicated the importance of tracing a teacher’s personal
history with teaching, unpacking the lessons of past experience and the impact of
that experience on a teachers’ understanding and implementation of the policy.
There existed sufficient evidence in the literature that suggested that teachers’
previous teaching experiences hinder teachers’ efforts to learn about and change
teaching. Dan Lortie described the "apprenticeship of observation," arguing that
what teachers bring to their teacher learning experiences are more potent than
formal teacher education courses.3 This research indicated that the lessons of
experience are difficult to overturn and that preservice teacher education and
prevailing professional development often promote further didactic teaching of
facts and skills (Lortie, 1975; Featherstone et. al, 1996; Feiman-Nemser, 1983).

More recent research suggested that, just like student learning, teachers
bring to their learning opportunities other knowledge that shapes what and how
they learn (Ball, 1988, 1989; Borko et al., 1992; Brown 8: Borko, 1992; Nelson,
1995; Schifter, 1993). The EPPS research revealed that teachers fill in the gaps of

their understanding of policy in light of the thin guidance they receive creating a

 

3D.C. Lortie. Schoolteacher: A Sociological Study of Teaching. (Chicago: 1975).

16

”mélange” of understanding and practices relating to new policy (Cohen, 1989).
These studies pointed to a range of factors impacting teachers’ understanding of
policy including new learning, the educational context, and teachers’ prior
knowledge and teaching experience. Others pointed toward the importance of
considering the variety of structures and contexts in which teachers learn as well
as the importance of giving attention to other factors influencing teacher learning
such as teachers’ opportunity to converse with other educators about similar
issues and teaching circumstances (Featherstone, Pfeiffer, 8: Smith, 1993;
Featherstone et al., 1993; Heaton, 1994).

These studies indicated the importance of looking at teachers’ learning as
not only shaped by teachers' encounters with the reform (i.e., policy documents),
but also by the prior knowledge, beliefs, and dispositions teachers bring to
learning to improve teaching. They indicate the importance of investigating
teacher learning in both a backward and forward direction giving adequate
attention to teachers’ prior knowledge, skills and dispositions. From this
standpoint, I would construct a view of the teacher’s history of learning about
mathematics, teaching, student learning, and improving teaching and I would
consider the interplay of that history with new learning. This would require
something similar to an archeological dig, an excavation of prior understanding
in relationship to new learning.

There also existed a growing literature that recast teacher learning in ways
that paralleled the kind of learning underlying the mathematics education
reforms. The epistemological position of learning inherent in the NCTM
flandards documents, for example, is a constructivist / socio-constructivist
perspective (NCTM, 1989). From this view, knowledge is considered dynamic
and conditional, very much dependent upon the individual’s sense-making

within intellectual communities.

17

The mathematics education and learning psychology communities
explored and offered theoretical explanations for the nature of the
transformation that happens when teachers change their beliefs, deepen their
knowledge and reinvent their practices. Carpenter, Fennema, Peterson, and their
colleagues suggested that teacher change was a matter of acquiring and using
new knowledge of how children’s mathematical thought evolves and developing
enriched and reorganized conceptual structures of mathematics (Carpenter et al.,
1988; Fennema et al., 1993; and Peterson et al., 1989). In the Cognitively Guided
Instruction project, researchers had been working with teachers on building
conceptual links between research-based models of children’s mathematical
thought and teaching practice (Carpenter et al., 1994). For many teachers, the
process of focusing on students’ mathematical thinking in light of what they
were learning about the mathematics framework, generated the integration of
research based knowledge about teaching and learning into their View of
children’s learning (Fennema et al., 1993). Other work focused on teachers’
encounters with renegotiating the norms of the classroom to foster students’
construction of mathematical concepts. For example Wood and colleagues
suggested that teachers resolve conflicts between their own prior knowledge and
beliefs about learning as they observed students learning in their practices. They
argued that conflict resolution supported by reflection and resolution supported
teacher change (Wood et al., 1991). Similarly, Schifter 8: Fosnot argued that
changes in teachers’ ideas about the nature of learning requires a process of
disequilibration of prior ideas and reconstruction of more powerful ones
(Schifter, 1993; Schifter 8: Fosnot, 1993; Schifter 8: Simon, 1992).

Although all of these researchers focus on the phenomenon of teacher
learning and change, each study emphasized a unique aspect of teacher learning.

Together, they portrayed teachers’ professional learning as a reconstruction

18

process, one marked by points of disequilibrium, discomfort, comfort, conceptual
change, uneven and more settled points of reconstruction.

These characteristics would become guideposts in my investigation of
teacher learning. They would signal a time to pause and explore the conceptual
focus and process of teacher learning. At each point, I would launch both a
forward and backward investigation, examining the interplay of prior
knowledge and teaching experience and new learning. In each instance, I would
track on the various resources that mediated the teacher’s interpretation and
implementation of the policy, examining what was offered and how. Over time,
I would uncover the impact of these resources on the teacher’s interpretations of
the policy and her teaching.

Finally, research on teacher learning also suggested the importance of
looking at subject—matter specific aspects of teacher leaming and change.
Shulrnan pointed to what he termed the ”missing paradigm” in educational
research and argued that increased attention should be given to subject specific
aspects of teaching and learning to teach (Shulrnan, 1987). Stodolsky studied
teachers’ efforts to change their teaching across different subject matters and
argued that the subject makes a difference in teachers’ learning to change their
practice (Stodolsky, 1988). Earlier EPPS studies revealed more about the
unevenness of the challenge for teachers in responding to, for example, the
reading and the mathematics reforms (Ball 8: Cohen, 1995b; Grant, 1995; Spillane,
1996). These studies suggested the importance of looking carefully at the role of
teachers’ subject matter knowledge in learning to teach. I aimed to uncover and
understand the relationship between a teacher's subject matter knowledge and a
teacher's responses to policy. My goal would be to examine a teacher’s subject
matter learning and the impact of that learning on the teacher’s capacity to

understand and enact policy.

19

An important question that arises when thinking about examining the
various aspects of teacher learning identified above involved whether prevailing
professional development opportunities provides a good context for exploring
these issues. Below I suggest the benefits of examining teacher learning in the
context of the prevailing professional development activities that are available to

most teachers.

The Rele Qf Prevailing Prefessional Develepment Activities

There is no shortage of professional development opportunities for
teachers in the United States. Many State Departments of Education, national
and private organizations, and local school staffs, continue to offer teachers
opportunities to learn about improving their teaching. The occasions range from
one-day workshops, longer-term in service on tests and texts, and a host of other
activities involving many other educational organizations. These dominant
forms of professional development are situated in a substantial infrastructure,
one that is available and reaching large numbers of teachers. Yet, most of these
occasions are highly criticized. They tend to offer teachers fragmented
experiences, only bits and pieces of a more complex puzzle (e.g., cooperative
learning, problem solving, new content) and little guidance about how to put the
pieces together. Educators and researchers alike are beginning to understand
that constructing more ambitious teaching and learning opportunities for
students first requires a substantially different approach to teachers’ professional
development (Ball 8: Cohen, 1995a; Wilson et al., 1996; Sykes, 1996; Smylie, 1996).

Given the existing problems, I wrestled with a nagging question: What
would be the point of studying teacher learning and the impact of learning on a
teacher's understanding and enactment of policy in the context of professional

development activities that are already considered highly ineffective for

20

developing teachers? My purpose was not to report any further on the
inadequacies of the existing system. And most research had already dismissed
existing professional development activities arguing that the structures alone do
not create the kinds of opportunities teachers would need to learn.

At the same time that prevailing professional development is problematic,
most state departments were relying to some extent on professional development
activities for educating teachers about efforts to reform. I wondered what
professional development activities teachers relied upon to support their efforts
to respond to policy? And what messages about reform-based teaching were
getting heard?

We know very, very little about how teachers encounter the ideas of
reform. We know that State Boards of Education often will develop a new vision
of teaching and learning as well as multiple levers to support that vision. Yet,
how does the individual teacher encounter those messages? This study
potentially could reveal a great deal about whether and how reform messages
are transformed into opportunities for teachers to understand them. It could also
reveal much about how teachers transform the messages they are offered into
changes for their teaching. We know precious little about what teachers make of
the policy levers they encounter and even less about how teachers make use of
these levers to transform their teaching.

It makes sense to suggest that although current structures fail to support
the ongoing development of teachers, we still could stand to learn a great deal
about whether and how teachers come to understand policy levers through these
structures. For example, are the assumptions about mathematics teaching and
learning that underlie policy offered to teachers in the context of the professional
development opportunities they encounter? If so, what do teachers make of

these assumptions and how do they transform the ideas into changes for their

21

 

teaching? Uncovering whether and how teachers’ opportunities to learn help
them to understand policy could support new ideas for improving teachers'
professional development opportunities, both substantively and structurally, and
with an eye toward supporting efforts to reform (Feiman-Nemser 8: Remillard,

1996). A study of this design could clarify many of these issues.

Focusing en a Single Teacher
The complexity of the data I hoped to gather and the potential of learning

from the teacher focusing this study guided my decision to focus on just one
teacher. Yet, I had to consider carefully what a study of one teacher could teach
us about the relations among policy, teacher learning, and teaching practice.

It was through the EPPS project that I met Sandy Wise.4 Sandy taught in
the Forest Glen School District in the state of California for eight years, the last
four in the same school. Sandy was an exemplary mathematics teacher,
identified as such by other leaders in her district. That is how she came to the
EPPS study. One of our curiosities in the project was to see how teachers that
seemed best positioned to reform their teaching went about their work. Through
numerous interviews and observations I learned that Sandy was committed to
high teaching standards. She was competent mathematically and steeped in
ideas about what these reforms involved for her learning and practice.

In fact, mathematics education reform had played a significant role in
Sandy’s practice from the onset of her career. In the early eighties, Sandy
decided to continue her college education and began working toward an
elementary teaching certificate. She recalled her formal teacher education

experience,

 

4A5 I promised confidentiality, Sandy Wise, the school and district are all pseudonyms.

22

”I was really born on the edge of the reform movement. I was
taught to teach the way instruction is talked about today. I wasn’t
taught with a basal reader, I wasn’t taught with a math textbook. I
was taught through Project AIMS,5 and whole language, and I really
relish in the fact that my professors were far ahead of the game"

(Interview, 5/92).

After receiving her teaching certificate, Sandy began teaching third-grade
in California public schools. She also decided to stay on at the university to
complete her masters degree. By then, Project AIMS had grown into a university
based teacher leadership program sponsoring a masters degree in education.
Sandy began her teacher leadership role as a Project AIMS consultant educating
teachers across the United States in week-long workshops aimed at improving
mathematics and science teaching. As an AIMS consultant, Sandy continued
formal teacher education course work in required seminars.

My initial observations of Sandy’s third-grade teaching revealed that she
was spending nearly two hours of each instructional day teaching mathematics.
Most of this time students were engaged in investigation-type activities
emphasizing problem solving and conceptually-oriented mathematical ideas.

By 1989, Sandy had enrolled in a satellite doctoral program of The
University of Southern California. She specialized in curriculum and instruction.

It was in this setting that she was involved in designing and analyzing

 

5Project AIMS is a grass-roots teacher organization aimed at improving the teaching and learning
of mathematics and science. Project AIMS (Activities Integrating Mathematics and Science) is a
privately funded education foundation aimed at improving science and mathematics education
in our schools. AIMS advocates and provides teachers with hands-on learning activities where
the notion of "learning by doing" is valued. The project has been recognized by Congress
through Project 2061 as an outstanding program aimed at meeting the goals and guidelines for
taking mathematics and science education into the twenty-first century.

23

elementary mathematics curriculum materials and instructional practices. Sandy
focused extensively on NCTM's (1989) Currieulum and Evaluatien Standards For
h l a mati .

Sandy also became a mentor teacher in her district. In this role, she was
involved in teacher education projects in her district. In particular, Sandy offered
state-sponsored professional development opportunities to teachers for learning
about the state's new Mathematies Framewerk Fer Califernia Publie Seheels
(CSDE, 1992). Her ongoing teacher development work later lead to her
promotion to curriculum specialist in her district.

Sandy’s personal history reveals her mindfulness of the mathematics
education reforms. She was both a consumer and provider of professional
development opportunities aimed at improving mathematics teaching and
learning. She had positioned herself to be successful at implementing the state’s
reform agenda. She had ample opportunities to learn, ample learning community,
and ample resources to support her learning.

Sandy’s practice itself served as part of the basis of my decision to study a
single teacher. She was obviously mindful and deeply invested in the state-
initiated reform effort in mathematics education. Studying her practice offered an
opportunity to see how teacher learning about policy was conceived,
conceptualized and constructed for the practicing teacher. It offered the
opportunity to explore relations between policy and practice from the vantage
point of teacher learning.

Sandy’s practice also offered the perspective of a teacher developer. As a
teacher leader in California and across the US, Sandy already was making
determinations of what and how teachers should learn. Her practice as a teacher

leader provides a rather unique opportunity to understand how policy ideas

24

become transformed into opportunities for teachers to learn.6 A study of Sandy’s

practice could potentially address each of the questions focusing this work.

The Mathematics Education Reforms
The other larger piece of work underlying the investigation of Sandy’s
practice involved unpacking the mathematics education reforms. Asa backdrop
for viewing and understanding Sandy’s encounters with the policy to reform her
mathematics teaching, I necessarily had to develop some ideas about what these
reforms proposed for changing mathematics teaching and what the implications
were for teacher learning. To conclude this chapter, I describe the assumptions I
made about the reforms in mathematics education and I explain how I focused

my work within the larger landscape of reform ideas.

What Are These Refgrms?

One impetus for the current reform movement (NCTM, 1989, 1991a,; NRC,
1990) is the widely held belief that American mathematics education is failing. In
most US. elementary classrooms, modal mathematics teaching and learning
emphasizes rules, procedures, memorization, and right answers (Goodlad, 1984;
Stodolsky, 1988). In other words, students seldom are confronted with serious
mathematical problems and are rarely expected to reason about mathematical
ideas. Teachers stand in the front of the room and show students how to do
particular procedures, later assigning practice exercises. Students practice the
procedures asking teachers for help only when they get stuck. Teachers check

students answers and assign more practice when needed.

 

6Sandy functioned in various teacher leader roles in her district and more nationally. Eventually
Sandy left elementary classroom teaching. She earned her Ed.D in 1997 and currently works

with prospective elementary teachers, teaching math methods courses at a satellite location of
USC. '

25

In these classrooms, mathematics is represented as calculation and rote-
memorization. Mathematics is not experienced as ways of thinking about
quantity or space and usually does not involve reasoning. Topics such as
probability, geometry, and discrete mathematics are often not given attention
providing the necessary time to cover the traditional topics of arithmetic
including subtraction with regrouping and long division (NCTM, 1989; Peterson,
1990; Prawat et al., 1992; Putnam et al, 1992; Putnam 8: Geist, 1994).

Current policies paint a very different view of mathematics teaching and
learning. They argue for wide scale changes in both content and pedagogy.
Underlying these arguments are very different assumptions about mathematics,
teaching, and student learning. In general terms, the reforms envision
mathematics as dynamic, more of an ongoing process of construction and
reconstruction, one involving sense-making. From this view, knowing
mathematics is much more than memorizing facts. Instead, knowing
mathematics implies understanding many domains of knowledge and
connections among mathematical ideas.

Students become involved in mathematical reasoning, constructing
plausible arguments, and analyzing arguments for when they do and don’t make
sense. These changes in turn, imply that students should explore more novel
topics than arithmetic, topics like probability, chance, statistics, and discrete
mathematics. Student learning no longer would emphasize memorization.
Instead, students would construct their own ideas in the context of group.

The reforms envision a pedagogy very different from modal teaching.
Teaching would involve facilitating student learning, stimulating meaningful
conversation, fostering conceptual understanding, and guiding analysis of
mathematical arguments. Teachers no longer would act as authorities and givers

of mathematical knowledge. Instead, they would become facilitators, fostering

26

skill development and conceptual understanding in tandem. Pedagogy would
no longer be dominated by telling students what to know and do, but instead
would involve framing opportunities for students to learn, coaching,
orchestrating, and guiding investigations and discussions. (CSDE, 1985, 1992;
NCTM, 1989, 1991; NRC, 1990).

To say these reforms are ambitious is an understatement. They propose to
shift mathematics teaching and learning from mechanical drill and memorization
toward mathematical reasoning and understanding. They expect that students
will learn mathematics in more meaningful ways and to accomplish that end,
reformers have proposed fundamental and wide-reaching changes in the content
and pedagogy of school curricula. At the same time, they are not new. Dewey,
Bruner, and many others have continued to press for more ambitious school
teaching and learning throughout the history of US. education. Yet, despite
much argument, modal practice has persisted. And consequently, there are few

examples available for observing what these reforms entail for teaching practice.

Implicatigns for Teacher Learning

Instructional policy serves many purposes. At its best, it can serve to
educate an entire population about a new trajectory for US. public education.
And as enlightening as it can be, at the same time, it involves visions of uncertain
practices (Ball, 1993, 1996). Instructional policies are not designed, nor can they
provide, the specifics of minute-to-minute practice. Educational policies aim to
sketch broad goals, set new directions and standards, and in some cases provide
glimpses of classroom teaching useful for imagining new aims and purposes.
What they do not provide are programs for teaching practice.

More complicated still, the conceptions of knowledge, teaching, and

learning underlying the policies focusing this study are mostly unfamiliar to

27

teachers. And because much of the picture of teaching practice is undeveloped
and underdetermined in the policies, they become more open to multiple
interpretations of practice and implications for teacher learning (Ball, 1996; Ball 8:
Cohen, 1995b; Cohen, 1989; Shulrnan, 1987).

One of the more complicated aspects of this work involved unpacking the
reform documents for what they implied for the individual teacher who must
work on them. I understood well from my previous work that the proposals in
such documents as California’s mathematics frameworks and NCTM’s standards
documents involved a substantial and difficult departure from modal teaching.
Yet, spelling out more specifically what this involved and what teachers must
learn to understand and enact such changes was very difficult.

Other EPPS research had already concluded that these reforms would
require an enormous agenda for teacher learning, one not well-understood or yet
defined (Ball, 1996). Part of what this study would provide is a more detailed
picture of what was involved for the teacher as a learner of new policy in
California and in relation to the mathematics education reforms more generally.

Consider, for example, the National Council of Teachers of Mathematics’
recommendation that teachers must create classroom environments that promote
logic and evidence as verification for knowing and move away from the notion
of the teacher as the authority for right answers (NCTM, 1991, p. 3). This
proposal represents a multidimensional problem for teachers. To enact such a
proposal, teachers would need to reconsider much of what they currently do.

For example, teachers’ roles, students’ roles, the role of the text, tests, and other
curriculum materials would all need to be reconsidered in light of the proposal to
redirect issues of authority away from the teacher and toward the students’ logic
and evidence. The ways in which students and teachers interact, in particular

the discourse patterns, would necessarily need to change. Teachers would have

28

to invent new ways to propose tasks and problems as well as facilitate students’
learning. New ideas including the notion of accountability and responsibility for
providing evidence of claims would become central. A single policy proposal,
taken seriously, represents a host of issues, questions, changes, and in particular
w for the individual teacher (Heaton, 1994; Lampert, 1990, 1992;
Thompson, 1985). Unpacking policy proposals in terms of what they imply for
teacher learning proved to be a complex and sometimes daunting task.

Because it was clear that the agenda for teacher learning was vast and
because I wanted to learn in as much breath and depth as possible what the
implications were for teachers trying to understand and enact policy proposals, I
narrowed the focus of this study to two strands of policy proposals. I selected
the two strands on the basis of the central role each played in Sandy’s efforts to
respond to the state's agenda during the time-frame of data collection for this
study. Sandy was particularly focused on the proposals to reform her teaching
of basic computational skills and introduce discrete mathematics into her
teaching. In her work with the proposals surrounding these two strands, she
faced a set of issues, questions, and unknowns that would continually draw her
attention and focus her learning throughout the data-collection period.

Although Sandy certame worked on other aspects of reforming her mathematics
teaching during the data-collection period, the strands focusing this work

remained the most prominent in her efforts to reform her teaching.7

 

7For example, I decided not to focus on the notion of conceptually understanding mathematical
ideas - Sandy had previously worked on this idea and it no longer remained a central tenet in
her learning. I also noticed that each time Sandy returned to questions about conceptual
understanding it was in the context of content specific teaching. In other words, conceptual
understanding was enveloped by a bigger question about what the content was and what it
involved for teaching it. This seems less so the case from a policy perspective as conceptual
understanding appears the more wide scale idea that envelopes specific content. The reserve
seemed true from the perspective of the teacher.

29

My decision to focus on these two strands was also based on my goal to
examine the role of teachers’ subject-matter learning on teachers' capacity to
respond to efforts to reform. The two strands I selected represented very
interesting contrasts in terms of subject-matter issues and teachers’ content
knowledge. One strand challenged the mathematics content and instructional
practices that have continued to have a stranglehold on elementary mathematics
(Burns, 1994; Putnam 8: Geist, 1994). The teaching of computational skills is
firmly entrenched in school practices. The strand of proposals to introduce
discrete mathematics represented quite an opposite scenario. The mathematics
content for the most part had never been an explicit piece of the elementary
mathematics curriculum. Most elementary teachers would be unfamiliar with
the mathematics content of discrete mathematics. In contrast, each strand
represented entirely different problems of change for the individual teacher and
vastly different agendas for teacher learning. There seemed a rich Opportunity to
explore the role of teachers’ prior knowledge and teaching experience as well as

teachers' subject-matter learning on reforming teaching:

Looking Ahead

Chapter Two provides a discussion of the methodological issues involved
in conceptualizing, designing and conducting this study. In that chapter, I go
into greater detail as to the logic, benefit, and process of narrowing the focus of
this dissertation to one teacher's learning about two aspects of the larger
landscape of reform.

In Chapter Three I provide a detailed policy/ document analyses of the
recommendations for change surrounding each strand. In addition, I identify the
vastness of what there is for teachers like Sandy to learn if they are to understand

and implement the proposals. This chapter serves as a backdrop in my

30

investigation of Sandy’s encounters with the proposals. Chapter Three is also an
important contribution to better understanding relations between policy and
practice. Not only does it illuminate the gap between policy and practice in
terms of the vastness of the learning required for teachers to understand and
enact policy, it also highlights the great complexity involved in constructing a
View of the recommendations offered and the implications for changed practice.

Chapters Four and Five are case studies of Sandy’s efforts to understand
and enact the proposals regarding each strand. I trace the evolution of her
learning and interpretations of the proposals over time and I examine the
interplay between her personal history with the proposed ideas and practices
and what she was offered by way of new opportunities to learn. Each of these
chapters offer insight into how a teacher learning agenda surrounding policy is
conceptualized and constructed and the effects of learning on changing teaching.
These chapters help us understand relations between policy and practice from
the vantage point of practice.

Chapter Six is a cross—case analyses of the two case studies of Sandy’s
learning and change. It offers a comprehensive View, characterizing the qualities
and conditions of Sandy’s opportunities to learn about the policy proposals. The
analyses in this chapter sheds new light on relations among teacher learning,
educational policy, and teaching practice. It offers new insight into how the US.
professional development system functions to educate teachers about policy.

Chapter Seven draws together my findings across all of the chapters and
the larger literature to argue that the US. professional development system is a
non-system for supporting teacher learning about policy. I then use what this
study teaches us about the relations among teacher learning, policy, and practice,
to suggest several ways in which teachers’ opportunities to learn might be

constructed to support more systematic learning about policy.

31

CHAPTER 2
METHODOLOGY AND RESEARCH PROCEDURES

Introduction

Some might question what can be learned about the relations among
policy, practice, and teachers’ learning from a case study investigation of a single
teacher’s encounters with the mathematics education reform agenda in
California. They might ask what can one teacher teach us? Although this is a
study of one teacher’s interpretations and learning about and from an
instructional policy, it is an investigation of the resources, orientations, views,
content, and learning across several case studies of this teacher’s encounters with
policy ideas over the course of time. The data is robust and portrays in depth
and detail the nature of policy implementation from the perspective of the
individual teacher trying to enact policy. What this study offers is an
understanding of the relations among policy, practice, and teachers’ learning
from the perspective of the individual teacher.

In this study, I assume that learning is the central activity of policy
implementation. I examine a single teacher’s efforts to learn about and interpret
several strands of policy proposals. I use a cross-case analyses approach to
understand one teachers’ learning across two different strands of proposals. One
set of proposals involves the recommendation that elementary teachers introduce
discrete mathematics into their mathematics teaching. The other focuses on ideas
and recommendations for reforming ideas and practices for teaching
computational skills. The data was drawn out of a structure that was bounded

by the occasions of this teacher’s policy learning, her prior knowledge, beliefs

32

and dispositions surrounding the specific proposals on which she focused, and a
policy analyses approach to understanding the mathematics education reforms.

My method for collecting the data involved a process much like the
positioning of three large searchlights, and watching for when they came
together at a common point. One light came from the perspective of the policy,
another from Sandy’s learning and interpretation, and a third, focusing on her
teaching. From these three vantage points, I searched for common ground,
places where conceptual ideas and interpretations of proposed changes
overlapped in meaning. Often, these three lights never arrived on a common
point. In these instances, my analysis focused on explaining why. At other
times, I recognized common points of intersection. When this occurred, I tried to
understand how that happened and what the factors were that supported the
similarities.

The purpose of this chapter is to explain the methodological decisions and
analysis procedures I used to position these lights and search for the common
ground. I begin with a methodological overview highlighting these procedures.
I then explain my methods and analysis procedures for each of the two larger
bodies of work -- launching an empirical investigation into this teachers’ learning
and changes in her practice and defining and understanding the conceptual

territory of this study from the perspective of the policy.

Methodological Overview
This study investigated one teacher’s encounters with the mathematics
education reforms. It traces how a single teacher encountered and defined the
reform agenda, how she made interpretations about the proposals, and how she
Constructed the ideas in her own thought and practice. Because I aimed to

understand teachers’ encounters with policy, there were two larger bodies of

33

work to develop. One involved investigating this teacher’s learning and
interpretation in relation to her encounters with the state’s policy. And the other
involved understanding what these ambitious reforms were that this teacher was
working on and what they required of her learning.

I began with the assumption that teachers’ learning was more an
evolutionary process, one that continually informed and effected teachers’ efforts
to reform their teaching. As I began to understand more about the complexity of
tracing teacher’s learning regarding teachers’ efforts to understand new policy
and improve teaching, I necessarily needed to find ways to track on the evolution
of specific ideas and practices over time and the association of changes in
thought and practice to particular learning. To get the richest data possible that
would reflect something of this process, I realized I would need to identify
reform ideas that were most prominent in this teacher’s work, those that
captured a great deal of her thought process and interest. Further, to capture the
significance of her learning in relation to her understanding of the policy ideas
that would focus this work, I would also need to focus my efforts on
understanding the influences of the most prominent sources for her learning
about those ideas. Thus, my early fieldwork would shape the focus of this study.
I began by exploring what this teacher was learning surrounding the ambitious
reform efforts in the state, how her learning was constructed, and how it effected
her teaching practice and interpretations of the new policy. Decisions I made
during this period shaped the focus and conceptual domain of this study. I
discuss these decisions in more detail in the next section.

Because I wanted to understand teachers’ learning from the perspective of
the individual teacher trying to work on these reforms, I proposed to use case
studies of one teacher’s learning. I assumed that learning was a process of

construction involving various complex factors that influenced learning. A case

34

study approach provided the means to look deeply over time, with detail, and
within the context (Stake, 1978, 1995; Ericson, 1986). By gathering data on this
teacher’s background, following leads, interviewing and observing her work
with the reforms as well as other constituents involved in her learning, I could
gradually take on the perspective of the individual teacher (Cusick, 1983), and
develop a View of her learning and the association of that learning to her ongoing
interpretations of the new policy and changes in her teaching.

As background to my investigation into this teacher’s learning and
change, I proposed a document / policy analysis of the reform agenda in
mathematics education. My purpose was to formulate a View of what the
central ideas of reform involved, what changes were proposed, and what
teachers would need to learn to accomplish those changes. This body of work
would define the conceptual/ analytical territory of this dissertation. Guiding
my work was a question about what reform ideas were seen as central by those
who envisioned them. I asked what changes are proposed and what is suggested
for accomplishing those changes? The purpose of the analysis would be two-
fold. First, it would serve to characterize the nature of the changes and learning
policymakers envisioned. Second, it would serve as a backdrop for describing
and making sense of this teacher's responses to policy.

Methodologically, my goal was to develop several case studies of this
teacher’s learning and the relationship of that learning to understandings of
policy and reformed teaching. I would then look across the cases to learn what I
could about the nature of the teacher's learning experiences and the impact of
learning on the teacher's interpretations and enactment of policy in teaching.
Ultimately, my goal was to provide an analysis of whether and how what was
learned across several cases fostered this teacher’s capacity to understand and

change her teaching in ways imagined in the reform documents.

35

Eggusing the Study

Field methods have the advantage of flexibility. My initial exploration
into the field helped to shape and refine the focus of this study over time. There
were two ways in which this happened. One had to do with the decision to focus
on one teacher’s learning and change. The other had to do with the way in
which I made determinations about the reform ideas that would focus this
dissertation. With each of these decisions I purposefully narrowed the
conceptual territory of this work (Peshkin, 1993). This required that I adapt my
data gathering process to focus on specific ideas and circumstances, ones that
often occurred unexpectedly at different stages of the research . I describe how I

made these decisions below.

Teaeher Seleetjen

Studying one teacher’s learning was a decision I made, in part, out of my
initial observations and interviews with Sandy Wise. It was what she potentially
could teach us and what I could potentially learn from examining her work with
the reforms that guided my decision to focus on case studies of her learning.

Given that my purpose was to advance our understanding of teacher
learning and the relationship of teacher learning to policy and practice, I
necessarily needed to study teachers deeply invested in learning about the
mathematics reforms. The EPPS study provided a wealth of options. In chapter
one I described that Sandy Wise’s emerging leadership role in mathematics
education reform and her View of herself as a learner about reforming her
teaching made her a likely candidate. My initial observations and interviews of
her work with the reforms revealed that the new policy in the state of California

played a central role in her determinations of what should change in her

36

teaching. Because she came to the project identified as an exemplary
mathematics teacher and a teacher developer, she represented a good candidate
for studying a teacher’s learning on two levels: as a learner about reforming her

third grade teaching and as a teacher developer.

The Relatienship Between Teacher Learning and Teaching Praetiee

Because my purpose was to develop ideas about what teachers were
learning about reforming their teaching, how that learning was structured, and
whether what was learned supported an understanding of the proposals in the
framework, I examined several different aspects of Sandy’s practice. When I first
began visiting Sandy’s classroom I observed that she had organized students into
eight cooperative learning groups for solving mathematical problems. She spent
nearly two hours of each school day on activities that ranged from mental
arithmetic lessons to probability and statistics. Sandy’s students were usually
engaged in a process of collecting data and analyzing it. Students talked to each
other for the most part and Sandy seemed to work from the background.

Sandy had already constructed a practice that was significantly different
than modal practice. Students responsibly went about collecting, analyzing and
making sense of mathematical problems, modeling for others what they had
done. There was movement, noise, action, and discussion each day.

In other ways Sandy’s practice reflected traditional patterns. Discourse, in
larger group and small group interactions seemed to focus on getting right
answers. Patterns were direct and linear. Students usually gave short responses
to more directive-type questions asked by Sandy. She often provided the logic
for working through a problem and ultimately pointed students toward a correct

answer or right way to think about the mathematics.

37

Sandy’s teaching practice was also bound up in her views about
developmental instruction. She argued that students need slow steady steps that
can be repeated and practiced. Problems were organized in terms of the number
of steps or number of things to remember for solving them. Sandy made a
decision not to use the state adopted textbook because she thought it did not
provide the incremental or developmental approach to teaching mathematics she
thought necessary.

These early observations and interviews suggested that although Sandy
had constructed a practice that looked quite different than modal practice, at the
same time there were fundamental ideas and practices that ran contrary to the
proposals in the policy. It became clear that tracing her learning and the
connection of that learning to her teaching would involve a data collection
process that reached as far backward and as far forward into her thinking,
learning, and teaching, as possible. The backward process, I imagined would be
much like an archeological dig, requiring much care in the excavation process.
The forward process, I imagined to be more straightforward requiring data
collection in the form of interviews and observations of Sandy’s teaching and her

opportunities to learn about the policy.

Wm

My early work investigating her learning proved the forward process to
be much more difficult than imagined. I noticed that Sandy’s prior learning was
etched by a variety of circumstances and ideas, some not easily accessible at this
point in time. I identified that there were at least eight settings or sites that her
prior learning was associated with and went on in relationship to. They

included:

38

* State and District Professional Development Activities
* Conversations With Other Educators

* Reading Reform Documents

* Project AIMS Workshops

* Mentor Teacher Work

* Doctoral Coursework

* Teaching Practice

* Assessments

I also realized that Sandy was learning very different things in each of
these settings, and that each site had entirely different assumptions and purposes
for her learning. Yet, Sandy drew on each one formulating ideas about the
reforms and her teaching, some contributing directly to her mathematics
teaching. Because I was interested in the conceptual orientations of Sandy’s
learning and how her learning affected her mathematics teaching, it became clear
that I would necessarily need to understand something of what underlies each of
these occasions. I would have to follow not only Sandy’s learning in relation to
these sites for her learning, but I also would need to uncover the histories
underlying theses occasions.

Based on these issues I proposed that focusing on one teachers’ learning
and change across a variety of reform ideas and in relation to the variety of
circumstances for her learning the most promising for understanding the
conceptual orientations to her learning. The case studies could then focus on
what she was learning conceptually and the conceptual orientations that
undergird her ongoing development of ideas and practices. The case study
methodology would provide an appropriate detail for understanding what

underlay this teacher’s learning experiences (Patton, 1990).

39

As the investigation into this teacher’s learning and change progressed, I
noticed how her ideas and understandings of several reform ideas changed,
shifted, and intermingled with other reform ideas. I tracked the development of
several reform ideas backward and forward creating what I called maps of her
learning. I worked to uncover Sandy’s embedded and explicit assumptions
about the reform ideas she was focusing and I tried to determine the embedded
and explicit assumptions within the opportunities to learn she was relying. I
aimed to analyze the conceptual orientations and patterns in her learning and
expected that my analysis would focus on whether there were matches between
what she was learning, the opportunities she was relying, and the assumptions

and ideas in the new policy.

D' ' ' f Vi i n

Making determinations about what reform ideas would focus this work
presented the second set of issues that would further focus this study. My goal
was to look carefully at the manifestation of important policy changes where it
mattered most - in the classroom - and from the joint vantage point of the policy
and the teacher whose teaching it intended to change. Conceptualizing the
policy - in terms of what was proposed - would not only help me to bound the
work in this study but also would serve to extend our ideas and understandings
of what the policy involved for the teacher and teacher learning.

The new mathematics education policy in California provided a rich site
for such work. The ideas of the policy came in the form of descriptions and
vignettes of teaching embodied in documents, newly adopted texts, new
curriculum guides, changes in testing, alignment with national standards and
goals, and various professional developments aimed at fostering the new vision.

Yet, even as the new policy hoped to inform and educate teachers toward

4O

important, essential, and major ideas and changes for their practice, it did not
prescribe specifically what teachers should do. The actual changes teachers
should introduce was less defined.

In an effort to focus this study on central tenets of the new policy, ideas
that would matter to policymakers and teachers alike, I initially developed a set
of questions and a criteria to help me identify a defensible set of reform ideas on
which to focus. I asked what did the new policy recommend teachers change?
What was recommended for accomplishing those changes? I developed the

following criteria to begin.

1. the idea was seen as a key element of the new reform
2. the idea is likely to be encountered and introduced into teaching
practice by the teacher in this study

3. the idea is substantially different from modal practice

The first criterion assured that each idea identified was seen as an important in
the new policy. The second assured the contrast between policy ideas and the
teacher’s conceptualization and construction of the idea could occur. And the
third insured that the idea was substantially enough that it might be a focal point
for judging the successful implementation of new policy. In other words, the
more adventurous the idea, the more appealing the idea would be for study.

I began by analyzing NCTM’s Prefessional Standards fer Teaching
Mathematics (1991), NCTM’s Currigulum And Evaluatien Standards fer
Teaching Mathematies (1989), and the Califemia Mathematies Framewerk (1985,

1992). Woven through each document were ideas and recommendations that

centered around three cornerstones of teaching: 1) mathematics content (what

41

mathematics is and what it means to know mathematics) 2) pedagogy for
teaching mathematics 3) how students learn mathematics.
Using the criteria above, I nominated the following as a preliminary list of

reform ideas for focusing this work.

Mathematics

-mathematics is not a body of isolated facts and procedures,
but connected in its ideas and applications

-knowing mathematics is a process of reasoning where
students support claims with evidence

-doing mathematics is conjecturing, inventing, and problem
solving rather than mechanistic answer finding

-broadening conceptions of the mathematical content for
elementary school teaching

Mathematics Pedagegy

-students construct their own understandings

-students must be active learners in the classroom

Learning

-teachers must become facilitators of student inquiry

-classrooms must become mathematical communities

42

My aim was not to provide an exhaustive list of the reform agenda but to
articulate a set of ideas that policymakers, and others might regard as central
tenets of the reform. This set of ideas served to focus my initial investigation into
Sandy’s learning and change. I continued to revise and refine this list as the
study proceeded. Eventually, I used the subset of these ideas for exploring
Sandy’s work in depth. This narrowed the focus of this study.

Conceptualizing what teachers would need to learn to respond to the
proposed changes was much more of a challenge. I searched the documents for
what they suggested teachers would need to learn. This process involved
unpacking the commentary around specific reform ideas. For example, each
document asserted teachers would necessarily need to learn new mathematics.
Probability and statistics and discrete mathematics were mentioned. Although
much was said about learning familiar mathematics in new ways, there were few
explicit statements of what would be involved. For instance, the conceptual
underpinnings of familiar procedural steps or connections among mathematical
ideas were mentioned. Yet, the specifics were left unaddressed.

My nominations below were drawn from my reading of the documents,
my own understanding of mathematics, my work with prospective teacher’s, my
early observations and interviews with teachers in the EPPS study. I used this
framework as an initial lens for examining Sandy’s learning and change. Again,
my purpose was not to formulate an exhaustive list of the possibilities but more

to capture a list of central tenets for teacher’s learning.

Mathematiee
-the conceptual underpinnings of mathematical ideas such as number,
number operations, computation, place value, measurement, fractions,

decimals, geometric shapes and ideas, estimation

43

 

 

-connections between mathematical ideas such as numbers, number
operations, fractions, etc.

-conceptual foundations of algorithms,

-broader conceptions of the ideas taught across grade levels

-new content such as probability / statistics ideas and discrete
mathematics

-new conception of mathematical knowledge such as how
mathematical ideas change over time, are open to debate and are
flexibly understood

-mathematical modes of reasoning, different problem solving

strategies, structures of valid arguments

Mathematics Pedagegy

-new roles for students and teachers in terms of authority for knowing
and responsibility for doing the work

~how to facilitate students’ sense making of mathematical ideas, where
growth and change in the construction of knowledge become part of
the process

-engage students in communication both orally and in writing with
each other and teachers around mathematics

-structure the social organization of classrooms differently so that
students have the opportunity to collaborate with each other and take
responsibility for their own and each others learning

-structure the environment and tasks so that students confront
mathematical ideas make conjectures and search for ways to solve

problems

-to use new texts and other materials in ways that do not promote
prescriptive learning

-how to engage students in discourse around problem solving that
promotes testing theories, debate, and sense making

-how to manage uncertainty in learning environments

Learning

-constructivist ideas about learning

-implications for constructivist ideas about learning on other aspects of
practice such as assessment

-ongoing informal assessment integrating instruction and assessment
of student understanding

-role of prior knowledge in sense making

-how to structure time for students to explore, investigate, and grapple
with significant mathematical ideas

-to use physical space, materials, and manipulatives in ways that
facilitate learning significant mathematical ideas

—to encourage students to work together to make sense of
mathematical ideas

-to encourage students to take risks, raise questions, formulate

hypotheses about mathematical ideas

This list guided my initial inquiry into Sandy’s learning and change. I

searched for evidence of learning around these ideas and began to investigate

her understanding both in a forward and backward direction. Influencing my

focus was the evidence I uncovered that specific ideas tended to be predominant

in Sandy’s thought. For example, she clearly worked on the idea of learning

45

communities where students work together in groups on mathematical problem
solving tasks. Other ideas Sandy seemed to give little attention. Discourse
patterns, for instance, seemed not to be something she thought much about. The
discourse patterns in her teaching were very traditional. In an interview Sandy
confirmed she had not given the patterns of talk in her classroom much thought.
Because my purpose was to systematically follow the evolution of a subset of
ideas, I selected from the list above leaving others mostly unexamined. My
selections were based, in part, on what I thought I could get substantial data on.
In other words, I had to be convinced that the idea was central in Sandy’s
thought and she was actively pursuing learning about the idea.

My determinations were that Sandy was consumed with ideas and
practices relating to reforming her teaching of basic computational skills
instruction and introducing discrete mathematics into her teaching. These two
strands of ideas were focal in her work during the time of this study. In delving
deeply into Sandy’s learning and change regarding these two strands of reform
ideas, I would create the opportunity to investigate the complexity of the
learning involved as she worked out interpretations of the proposed changes and
introduced the changes into her teaching. I would trade the opportunity to look
across a wider range of reform ideas contrasting how a variety of different ideas
take root or intermingle into Sandy’s thought and practice for detailed views of
two strands of ideas (Goetz 8: LeCompte, 1984).

My determinations to focus on these two strands were also rooted in what
I thought I could learn from them in contrast. As I argued in chapter one, they
represented opposite challenges for reform. One involved uprooting and
changing the content and pedagogy of the mainstay of the elementary
mathematics curriculum. The other involved introducing ideas and practices

Currently not present in the elementary curriculum. And they represented an

46

opportunity to consider the role of different mathematical topics in teachers’

learning from and about policy.

Data gellection

Data focusing on Sandy’s learning and change was collected in the form of
interviews, classroom observations, observations and interviews of the
professional development activities she encountered, various articles and
documents she relied upon in connection with coursework or other learning
experiences, papers she authored, professional development materials, and other
materials such as curriculum materials and curriculum guidelines.

Interview and observation instruments developed recursively as the study
proceeded. I began with interview and observation protocols adapted from the
larger EPPS study3. These protocols were developed and revised during the
course of this study. Each data point and subsequent analysis contributed to
redefining and refining data points on future instruments. The changes
consisted primarily of follow up questions in order to trace the evolution of

Sandy's learning and teaching of computational skills and discrete mathematics.

gzbservatign Data
Observation data were collected across a two year span. Each classroom
observation included two days of observation, across the entire day, followed by

interviews and discussions of her practice. Observations of her professional

 

8The instruments used in this study were developed as part of a longitudinal study of curricular
reform across three states, including California's efforts to reform mathematics teaching (Cohen,
D. K., Peterson, P. L., Wilson, S. M., Ball, D. L., Putnam, R., Prawat, R., Heaton, R., 8: Wiemers, N.
(1990). Effects of state level reform of elementary school mathematics curriculum on classroom
practice (Elementary Subject Center Series No. 25), East Lansing, Michigan State University,
College of Education. The instruments have also been used and tested in subsequent studies,
including QUASAR and From Congress to Classroom.

47

development activities involved three days of observations each of two different
Project AIMS workshops. In each instance Sandy was an instructional leader
facilitating other teacher’s learning. Observations of her district inservice
involved two one day workshops. In total, there were 16 days of observation.

Fieldnotes of each day of observation were written. Portions of each
observations were audio-taped and later transcribed. Extensive handwritten
notes were used to assist in the writing of fieldnotes. My focus in these
observations involved tracing any aspect of Sandy’s work with computational
skills and discrete mathematics in relation to her teaching, interactions with
students, choice and use of curriculum materials, changes and or patterns and
practices in her teaching. Write-ups of each observation included a narrative
summary of the activities, analytic questions to guide later interviews, and
preliminary analyses of Sandy’s work with the specific reform ideas traced.
These notes represented the observations data.

The schedule of observations involving four classroom observations and
three professional development observations formulated the basis of seven sets

of fieldnotes.

Observation F ieldnotes
1292
January, 1992 (Classroom Observation)
May, 1992 (Classroom Observation)
August, 1992 (AIMS Workshop)
323
January, 1993 (Classroom Observation)
August, 1993 (District Inservice)
August, 1993 (AIMS Workshop)

48

October, 1993 (Classroom Observation)

Interview Data
I tracked Sandy’s learning in association with the following activities,

interviewing her and others in relation to those activities.

Mentor Teacher Work

State and District Professional Development Activities
Reading Reform Documents

Project AIMS Workshops

Doctoral Coursework

Professional Development Activities

Teaching Practice

Conversations With Other Educators

Assessments

Interview data was collected across the entire study. Interviews were
audio taped and later transcribed. Following each observation, I conducted a
post observation interview. These interviews tracked on changes or refinements
of reform ideas I noticed or that Sandy identified in her practice. Interviews
conducted around her professional development activities involved Sandy,
teacher participants, and other teacher educators involved in those settings.
These interviews focused on what Sandy and others were learning in relation to
these activities. Protocols were designed to probe what Sandy was learning, how
determinations were made for how to structure that learning, and how that

learning effected her teaching practice and her interpretations of the new policy.

49

Follow up interviews involved tracking specific reform ideas into Sandy’s

practice.

Other Data Segrees

During the course of the study, other data such as course assignments,
papers authored by Sandy, articles she relied upon for learning, interviews
involving other teachers and administrators in the larger project were collected.
In addition, ongoing discussions with researchers involved in the larger EPPS
study contributed to my ideas about what these reforms involved and what

teachers would need to learn to accomplish them.

Data Analysis
DQeument Analysis

Analyzing the reform agenda in mathematics education was not a
straightforward task. In part, this is because the desired changes and what
teachers need to learn to make those changes remains largely undefined. The
policy in California, as embodied by such instruments as reform documents,
frameworks, texts, tests, and professional development opportunities, offers a
vision of what desired teaching would be like. At the same time, none of these
instruments provide prescriptions for teaching practice. Nor do they specify
what teachers would need to learn to accomplish the proposed changes.
Readers, for the most part, are left to figure these things out for themselves.

The significance of this point is important for making sense of the analytic
work in this study. One aspect of the work involved formulating ideas about
what the proposed changes involved for teaching and making determinations of
what teachers would need to learn to accomplish those changes. Guided by my

analyses of three central reform documents, I developed a view of what the

50

proposed changes implied for teaching practice and for teacher learning. My
analysis of the reform agenda and a teacher's construction of the proposals
pertaining to computational skills and discrete mathematics were
interdependent. The development of each informed the other.

In analyzing the text across the reform documents, I conducted a search
for any ideas, recommendations, explanations, proposals or vignettes that gave
definition to or provided insights into introducing discrete mathematics or
reforming computational skill instruction. I cross-examined the documents for
similarities and differences in ideas, recommendations and visions of practice.
From these data I formulated views of what the proposed changes involved for
teaching and what teachers would need to learn to make those changes in their
practice. These findings appear in chapter three of this work.

My analysis of the proposed changes across these two strands of
proposals served as a back drop for making sense of the empirical investigation

into the teacher’s thinking and practice.

An 1 i f rvati n and Int rvi w Data

Analysis of interview and observation data of Sandy’s learning and
change was ongoing throughout this study. Initial visits provided the basis for
formulating hypotheses about what she was learning and how her learning
effected her teaching. Follow up visits provided the opportunity to check out
these hypotheses and look for further evidence of change or evolution in her
thought and practice. Once transcripts and fieldnotes of observations were
complete, I indexed the data on templates focusing on specific reform ideas.
Each template tracked specific reform ideas. Early on, I tracked many ideas and

later resolved to work carefully with two.

51

The templates consisted of a matrix of the relationship between the
various contexts of Sandy’s learning in relationship to and the specifics of what
she was learning. The templates were influenced in part by what I was learning
from the document analysis and in part by what I was noticing in relation to the
sites for learning. These templates evolved and changed over time as Sandy’s
thought and practice evolved and changed. I used several rounds of each
template tracing the development of reform ideas and practices. These templates
served as the basis of my understanding of how ideas were evolving and in

association to what circumstances for her learning.

 

estima- student altern cgncep centext new teehnel
tien genstruet al ri devel. ped

 

 

Project
Aims

 

Course
work

 

Teaching

 

Reading
Policy

 

Assessment

 

Conversa-
tions

 

 

Profes
Dev.

 

 

 

 

 

 

 

 

 

Figure 1. Learning Template
Basic Computational Skills Instruction

Each round of data collection provided evidence of how the particular ideas
were evolving in Sandy's thought and practice. From each template, I developed
analytic memos of Sandy’s learning and change and contrasted these memos

with what I was learning from the policy/ document analysis. Eventually, I

52

pulled together a series of memos to develop descriptive and analytic pictures of
Sandy’s learning and change over time. These descriptive and analytic case
studies appear in chapters four and five of this work. They serve as the basis of

the cross-case analytic work in chapter six.

Cress—Case Analyses

The cross-case analyses involved iterations of comparative analysis,
exploring various themes and returning to the cases to check validity. I
compared the cases on the basis of what Sandy learned substantively, the
qualities and conditions of the learning experiences she encountered, and
Sandy's View of what the learning experiences offered (Miles 8: Huberman,
1984). I then looked to see whether and to what extent the experiences Sandy
encountered offered and represented the ideas and practices proposed in the
reform documents. The analyses revealed patterns across Sandy’s learning
experiences suggesting a lack of coordination and coherence in the educational
experience encountered for learning about reformed-based change. I will argue
that these patterns represent significant aspects of practice that make it
particularly difficult for a teacher, working as Sandy did, to learn and respond to

policy objectives (Kennedy, 1979; Wehlege, 1981).

53

CHAPTER 3

THE IDEAS OF POLICY

Introduction
In the previous chapter I described how I made decisions about the reform
ideas that would focus my investigation of Sandy’s policy learning and
interpretation. Two strands of policy proposals were central in Sandy’s learning
during the course of this study. One focused on reforming computational skills
instruction and the other proposed that discrete mathematics be introduced into
the elementary school mathematics teaching. This chapter is devoted to taking a
close look at what each of these strands involve from a policy perspective.
Throughout this dissertation I use the terms strands, strands of ideas, two
strands, and sets of ideas and proposals to refer to the specific ideas, proposals,
and recommendations made by policymakers for reforming basic computational
skills instruction and introducing discrete mathematics into the elementary
School mathematics curriculum. The word strand means fibers that are twisted
together to form a rope. I use the notion of strands in this study because it ,
appropriately portrays a cluster of reform ideas, proposals, or recommendations
intended to work together to accomplish a desired goal.9 In this case, reforming

basic computational skills instruction and introducing discrete mathematics into

 

9The notion of strands is also used in the California mathematics frameworks to specify particular
content areas considered important to the state’s view of what the mathematics curriculum
should attend. In the 1985 framework seven strands of content were identified. The 1991
framework added an eighth strand called discrete mathematics. This use of strand focuses on
desired mathematical content. My use of the terms strand includes the reference to particular
content but further includes the range of proposals, ideas and recommendations made by policy
makers for reforming basic computational skills instruction and introducing discrete
mathematics into the school mathematics curriculum.

54

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elementary school mathematics teaching are the desired ends. The variety of
ideas, proposed changes, recommendations, and practices offered to accomplish

those ends are what I refer to as strands of the reform.

What is The Policy?
A Document Analysis Approach

Instructional policies communicate ideas about changing teaching. By
way of texts, reform documents, mandated tests, and many other policy levers,
policymakers send messages about what should be changed, why they are
recommended, and what is suggested about how to carry the changes. When
effective, policy creates images for how teaching and learning might be re-
imagined.

Some of the most detailed descriptions of current views on what must
change, why, and how those changes might be accomplished appear as written
documents that have surfaced over the last two decades. In this chapter, I
examine a subset of those documents for the purpose of formulating for the
reader a view of the proposals for changing basic computational skills instruction
and introducing discrete mathematics. My analyses is an effort to identify the
central tenets of these proposals and uncover whatever guidance is offered to
teachers. I use my analyses as a backdrop for understanding and analyzing
Sandy’s learning and the changes she makes in her teaching.

Underlying the analysis in this chapter is the assumption that the visions
and arguments in the reform documents leave much of the picture of change
largely undefined. Reform documents, such as those analyzed in this chapter,
attempt to set standards and describe goals for how teaching and learning must
change. In other words, they are statements of direction. In this endeavor, they
leave much of the very practical how-to of daily teaching unattended. Cohen

and Ball argue that policies are not clear programs for practice (Ball 8: Cohen

55

1995a, 1995b). Policies, by their very nature, offer suggestions, arguments,
glimpses and images of what the proposed changes may involve for teachers and
the education of students. There is much that the policies leave unattended. For
example, the practical moment to moment experience of teaching is not attended
to as well as any careful examination of the relations between mathematics as a
discipline of knowledge and mathematics teaching pedagogy.

The inconclusive nature of policy interacted with the empirical work in
this study. My understanding of the reform rhetoric and the manifestation of
reform ideas in Sandy’s practice intermingled. As I observed Sandy work with
the ideas, my own ideas of what the proposals involved changed and deepened.
The examples and illustrations I provide as I unpack the proposals in this chapter
reveal the interaction.

My analyses makes two significant contributions to our understanding of
the relations between policy and practice. First, it informs the rhetoric of policy
by separating out, unpacking, defining and making sense of the variety of
proposals offered. Another contribution involves uncovering the complexity of
such a task. Searching the reform documents for ideas and recommendations is
not a straightforward process. The nature of policy as visions of uncertain
practice make it difficult to construct a detailed and complete picture of change.
As a result, my analyses is also incomplete and inconclusive.

This is an important insight in itself. Unpacking policy proposals does not
lead to specifics for practice. Instead, the unpacking process further delineates
the uncertainty of practice. Re-imagining and remaking the teaching of basic
computational skills and discrete mathematics by way of policy proposals
involves much speculation and decision making. The speculation and decisions I
make in this chapter to carry this process out illuminate what the individual

teacher is confronted with in trying to understand and respond to any set of

56

proposals. This chapter illustrates the nature of understanding policy and the
ways in which policy interpretations can have multifaceted faces. The analyses I
provide illuminates the problematic nature of formulating a well-grounded
understanding of any one strand of policy proposals. This may be the most
important contribution of this chapter. The individual teacher is faced with
distilling what these reforms are and what they suggest for her practice. Yet, the
complexity and uncertainty involved in such a task may explain much of the
disjuncture between instructional policies and teaching practice.

I turn now to outline the sections that follow. The chapter is organized in
two parts. The first part is an analysis of reformers’ envisioned changes and
recommendations for how basic computational skills might be re-imagined and
remade in teaching practice. My analysis is based on ideas and recommended
changes as communicated by three central mathematics education reform
documents: Currieulum and Evaluatien Standards fer ghee] Mathematies, 1989,
Prefessienal Standards fer Teaehing Mathematies, 1991, and the Ca_life_rn_i_a_
Mamematies Framewerk, 1992.10 The Framewerk (1992), I rely upon in

particular because of the more central role it played in Sandy Wise’s work
surrounding the reforms. The Curriculum Standards (1989), and the Teaehing
Standards (1991), I use because of the significant role each document plays in
establishing, guiding, and implementing mathematics education reform more
nationally as well as more locally within the state of California.

In the second part of the chapter I focus on reformers’ ideas and
recommendations for how discrete mathematics might be introduced into the

existent school mathematics curriculum. My analysis is based on reformers’

 

1° Throughout this chapter the 'ul v ai n tan ar f r h 1M emai

will be referred to as the Cerricelum Standards, the Professienel Standards fer Teaching will be

referred to as the IeeehjngStandges, and the Califernie Mathematies Framewerk will be
referred to as the Framewerk

57

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ideas and recommendations as communicated by the Currieulum Standards
(1989), and the Framework (1992). Once again, I rely upon these documents
because of the central role they play in guiding the mathematics education
community toward reforming mathematics teaching and learning. In addition, I
rely upon the ideas and recommendations advanced in NCTM’s 1993 Yearbook
titled Diserete Mathematies Aeress the Currieulum K—12, an interview with a
mathematician specializing in discrete mathematics, and several textbooks
focusing on discrete mathematics. I made decisions about including these
sources as I realized there was a need for a wider range of ideas and practices
surrounding the introduction of discrete mathematics for making sense of what

was proposed in the two documents.

Referming Basie Cemputatienal Skills Instruetien

It is surprising how little text is directly afforded to the consideration of
teaching and learning basic computational skills. Given that computation has
continued to dominate teaching practice at the elementary grades and therefore
is often at the forefront of teachers minds, one might expect that considerable
direct attention be given to the topic. Helping teachers rethink the place and
importance of basic computational skills in relationship to a student's
mathematical experiences in school seems an essential element for reform. Yet,
of the 215 pages in the Framewerk, there are 6 pages explicitly devoted to
answering the question, "How do computational procedures fit into this
mathematics program?" (p. 54-59). Later in the document a one page discussion
focuses directly on how number facts ought to be taught (p. 73). The Curriculum
Standards, a 256 page text, devotes 4 pages specifically aimed at discussing
whole number computation and 3 pages of text exploring concepts of whole

number operations for elementary grades (p. 41-47 Standards 7 8: 8). Standard 7

58

for grades 5-8 focuses 4 pages of text on ideas about computation and estimation
for the middle grades. Of the 665 pages of text searched, only 20 or so pages or
about 3% of the texts devote direct, explicit attention to questions and issues
related to the teaching and learning of basic computational skills. In one sense,
the documents are relatively silent about what many elementary teachers are
preoccupied with in their practices.

Beyond these sections, recommendations attending to computational
goals are less noticeable. One would have to probe the texts much more deeply
to find more and with little guidance about where to search. Many ideas are
tucked in various places throughout all three texts. For instance, there are
several pages in the Framework (1992), that focus on units of instruction. On
page 110, there are three short paragraphs describing a unit on understanding
arithmetic operations for elementary grades. These paragraphs, although they
are intended to illustrate the idea of units of instruction, also communicate that
an over emphasis on recall and the speedy execution of algorithms actually
interferes with understanding operation concepts.

In another context titled Characteristics ef Empewering Mathematies
Pregrams (p. 40-43), there is text describing specifically how students ought to be
introduced to traditional algorithms. The Teaching Standards, 1991, offers
similar test. For example, there is a paragraph under appraising worthwhile
mathematical tasks (p. 26), that includes a recommendation that computational
skill and the automaticity of those skills ought to be considered when evaluating
whether particular tasks are worthwhile for students. Several examples of
worthwhile tasks that also accentuate computational skills are provided.

Many other examples of ideas and recommendations for reconsidering
basic computational skill exist throughout the texts. For example, a K-4

curriculum standard titled Estimatien fer K-4 (p. 36-37), focuses on the ideas that

59

estimation must be reconsidered as a computing strategy in and of itself.11 There
are also short stories, sample mathematical problems and vignettes located in a
variety of contexts where specific content, general principles and pedagogical
ideas surrounding the teaching and learning of basic computational skill is
considered. Some of the stories in particular are very telling of as they offer
glimpses into what reformers may have imagined students to be doing
differently in their learning of computation. At the same time, one would have
to deduce from these stories the specifics related to computation.

In addition to narrative text there are outlines and tables portraying ideas
about computation. (i.e., p. 20-21 Curriculum Standards, p. 208-210 Framework).
Within these pages there are summaries of the skills that ought to get more and
less attention. For example, thinking strategies for learning basic number facts
should receive more attention whereas isolated treatment of paper-and-pencil
computations should receive decreased attention. There also is a description of
what students would be doing in terms of tasks and emphasis if they were
engaged in understanding arithmetic rather than memorizing routine
algorithms. The descriptions are written across grade levels to provide a vision
of what students' experiences over time might be like (p. 188 Framework). One
would have to search each table and outline to extract the many position
statements related to computational skills.

It seems important to remember that the documents are intended as
statements of direction, offering visions and arguments for what must change. In
providing new visions and images of what mathematics teaching and learning

ought to look like, a kind of teaching much different than modal practice, explicit

 

11 This standard may not be immediately recognized as about computational skill. Only after
doing a preliminary analysis of reformers' views would one realize reformers goal to shift current
views of estimation as a "checking" strategy toward estimation as a computational strategy in
itself. Within this standard estimation is developed as a legitimate computational strategy.

60

and extensive attention to any one topic is perhaps not reasonable. It also seems
sensible to suggest that as reformers imagined a new teaching practice, one
where computation i_s less central, that less attention be afforded to that topic.
Whatever may be the case, the effect is that the policies offer an indirect route
toward an understanding of the ideas and recommendations proposed for
teaching basic computational skills. For the most part, the topic is not
approached directly nor explicitly in the reform documents.

Given that policies are not programs for practice, relative silence or
indirect attention to a topic would not equate with simple approval or
disapproval of a particular kind of practice. At the same time, as I have also
argued, it would be very difficult and highly unlikely for any one teacher to
encounter all of what is proposed for reforming basic computational skills
instruction, even in several passes through the reform documents. Further, so
many ideas and topics are given such meager treatment, that even if particular
ideas are located, the treatment often doesn’t provide much guidance or
stimulate imagination at the level of practice.

Some ideas and recommendations might be overlooked completely
simply because they are embedded in contexts that the reader does not give
attention. Others are not easy to uncover because they are presented in ways
that bring forward other reform ideas leaving computational skill issues in the
background and less noticeable. One would have to come prepared to work
extensively with the documents to flesh out a detailed sketch of reformers' ideas
related to computational skill. This process would involve not only a detailed
search but a sorting of issues and ideas as well as the inclination to consider
multiple interpretations of ideas across an extensive amount of text.

I now take up this task. I ask what are the proposals that policymakers

make to reform teaching of computational skills? How do traditional ideas for

61

developing students’ proficiency with computation compare? What follows is a

detailed sketch of policymakers' proposals as described in the reform documents.
The reform documents articulate a new vision of computational skills

instruction. There are three central tenets described. Content and pedagogy are

interwoven in my descriptions as they are in the reform documents.

A Variety of New Computational Skills
A Conceptual Grasp of What Underlies Basic Computational Skills

Contextualize Basic Computational Skills

A Va ' f N w m utational Skills

A variety of basic computational skills involves introducing a range of
new ways to make computations. Included are mental computations, estimating,
students’ strategies for computing, calculators and computers for more complex
computations, and traditional paper-and-pencil algorithms for more
straightforward kinds of computations. Also recommended are opportunities
for students to evaluate the strengths and weaknesses of skills within various
contexts. In other words, traditional algorithms no longer would dominate
computational skill instruction. Instead, an emphasis on understanding and
using a variety of skills would dominate.

To help the reader see more clearly what these changes involve, the table

below contrasts proposed ideas with traditional ideas and practices.

62

 

 

Traditienal Ideas Proposed Ideas
traditional computing algorithms extensive mental computational skill
basic addition and multiplication facts estimation as a computing strategy
limited mental computational skill non-traditional computing strategies
students’ computing strategies
technology for computing
traditional algorithms

basic addition and multiplication facts

 

 

Figure 2.
Basic Computational Skills

Each of the three documents assert the importance of teaching children a
variety of different ways to compute. The above set of skills are specifically
proposed for use in the service of computation.12 In contrasting the proposed set
of skills with extant practice, there are four new skills proposed that are currently
not emphasized in elementary school curriculums: estimation as a computing
strategy, non-traditional strategies, students’ strategies, and calculator and
computer use. Estimation, a very familiar idea in traditional forms of practice,
plays the role of a checking strategy in the context of computation. New
proposals suggest, "estimation is not solely a means of checking required by

calculation: it may be the appropriate technique in itself," (p. 54 Framework).13

 

12Reformers encourage a broader view of what the basic mathematical skills include. The
mathematics of computation represents only one category of mathematical skill. Traditionally
skills in mathematics referenced only the basic facts and computing algorithms. The new vision
assumes basic mathematical skills to include, for example conjecturing, reasoning, arguing,
proving as well as computational skill.

13 Estimation is given much attention in the reform documents. Within the Cerrieelem
fiandmds for K—4, Standard 5 focuses explicitly on estimation. Standard 7 for grades 5-8 deals
with computation and estimation. At the same time, estimation might not be recognized initially

63

 

Estimation is viewed as a computational strategy itself. Mental computation,
often a part of traditional computational curricula, must be treated with more
emphasis. New proposals suggest, ”both mental computation and estimation
should be ongoing emphases that are integrated throughout all computational
work," (p. 45 Curriculum Standards). Thus, basic addition and multiplication
facts are foundational in both forms of teaching. Yet, later in this chapter, I
explain how instructional practices surrounding basic-fact acquisition are very
different across traditional practices and reform-based teaching.

Most of the discussion in the texts promoting a variety of skills is aimed at
broadening ideas about what computational includes and why computation is
done in relationship to both in-school and out-of—school contexts. The image
created is that computational skill no longer implies turning automatically to
traditional algorithms for the purpose of computing. Instead, there is first a
consideration of what strategy seems most sensible in a given context. For
example, if the context requires an estimate rather than an exact answer, an
estimate would be made. If an estimate is not sufficient, other strategies would
be used based on the circumstances and numbers in the desired computation.
The idea is that traditional algorithms are not automatic and instead belong in a
pool of possible choices. Computational skills are selected based on their merit
in the circumstance (see p. 9, Currieulum Standards, for a diagram of this
decision making process).

What constitutes computational skill would change dramatically if the
proposals in the documents were realized. Traditional paper and pencil
algorithms would no longer be the automatic choice for efficient and accurate

computations. Knowing a variety of skills and understanding contextual

 

by a reader as related to computational skill given the traditional views of estimation by most.
The new policies assume knowledge of estimation a strategy for computing.

64

 

influences for using those skills would represent ”computational skill.”
Statements supporting these claims would include, "Almost all complex
computation today is done by calculators and computers," and, "in many daily
situations, answers are computed mentally or estimates are sufficient, and paper-
and-pencil computations are useful when the computation is reasonably
straightforward" (p. 44 Curriculum Standards). In addition, "frequent use of
calculators, mental computation, and estimation, helps children develop a more
realistic view of computation and enables them to be more flexible in their
selection of computing methods" (p. 45 Curriculum Standards). And further,
students must have the opportunity to, "compare different approaches and
algorithms for obtaining the same results, evaluating strengths and weaknesses.
They must understand why the approach they choose makes sense for the
problem they are solving (p. 56 Framework).

Traditionally context was not an issue in relationship to computational
skills. Isolated practice of procedures and basic number facts have always
dominated computational skill instruction. The proposals in the policies suggest
the opposite ought to be the case. Contextualizing computational skills is

perhaps the most central proposal. In the Curriculum Standards the authors

 

write, "some proficiency with paper-and-pencil computational algorithms is
important, but such knowledge should grow out of the problem situations that
have given rise to the need for such algorithms" (p. 8 Curriculum Standards).
This statement implies that mastery, proficiency, and accuracy in making
calculations must grow out of the problem contexts that make those goals
worthwhile. In contrast, traditional practices valued the goals of speed, accuracy
and mastery as ends in themselves.

Non-traditional computational techniques and students’ strategies can

overlap. They include strategies such as doubling or adding back. Once again,

65

given the particular context, non-traditional computing strategies might be a
better alternative than any other computational skill. For example, given the
context and specific numbers in the problem, if a student had to compute $5.00 -
$3.99, it would make more sense to use a strategy of counting back rather than
the traditional subtraction algorithm. Students would count toward $5.00 by
adding one cent to get $4.00 and then $1 to make $5.00, getting the answer $1.01
as the difference. This strategy might be emphasized as students come up with it
or more explicitly as a non-traditional technique. Either way, the authors argue
that strategies such as these are more often the way computation is done in out-
of-school contexts. They also claim that these strategies are often more efficient
and can be done without paper and pencil. The Framework on p. 54 reads,
"There are many different ways to perform a calculation. Some students will
invent and refine procedures for themselves, and immigrant students may bring
valid alternative procedures. Teachers can encourage inventions and other
alternatives ..." In the Currieulum Standards, "Children need more time to
explore and to invent alternative strategies for computing mentally," (p. 45
Curriculum Standards).

Past practices emphasized learning one computing technique for each of
the four operations of addition, subtraction, multiplication and division. The
new vision implies students would know and have flexible use of a number of
different computational strategies. New teaching practices would be different as
a result. They would provide students an opportunity to select and defend
choices as well as demonstrate alternative strategies, analyzing both strengths
and weaknesses in use. These proposals imply other changes as well. For
example, they imply different discourse patterns and new goals surrounding
instruction of basic computational skills. No longer would executing the same

procedures for getting the right answer be the goal. Instead, the examination of

66

contexts, consideration of various computational techniques, selecting and using
various techniques, and considering the reasonableness of results would be more
central to the learning opportunity. Students would therefore have very
different conversations about computation than traditional practices.

A particular point should be made regarding proposals for the use of
technology for computational purposes. Proposals in the documents
surrounding the use of technology for computational goals tend to muddy the
issue at best. For instance, the new Framewerk (1992) describes calculators as
"electronic pencils" of today, and that, "A reasonable goal is to make calculators
available at all times for in-class activities, homework and tests" (p. 59
Framework). At the same time, reformers caution, "the availability of calculators
does not eliminate the need for students to learn algorithms" (p. 8 Curriculum
Standards), and "Calculators do not replace the need to learn basic facts, to
compute mentally or to do reasonable paper-and-pencil computation" (p. 17
Curriculum Standards). Other issues confound the role of technology further.
One position is, "Technology in the classroom can be a positive force for equity; it
helps break doWn the barriers to mathematical understanding created by
differences in computational proficiency (p. 57 Framework). At the same time,
"Classroom experience indicates that young children take a commonsense View
about calculators and recognize the importance of not relying on them when it is
more apprOpriate to compute in other ways" (p. 19 Curriculum Standards).

Although the proposals clearly recommend the use of calculators and
computers in relationship to computational goals, various interpretations of the
arguments could be made. For example, should calculators replace instruction
of three and four digit addition, subtraction, multiplication and division
algorithms? Should calculators be given to students with low computational

skills for all computing? It would be difficult to draw conclusions about when

67

calculators and computers should be used based on the information in the
reform documents. There is little guidance given for what sorts of computations
should involve the use of technology, what technology helps students
understand and not understand about computation, and nothing mentioned
about what teachers might do when the technology is not available for student
use.

Yet, there does seem to be agreement across the documents about the role
calculators and computers ought to serve in relationship to students’ learning of
computation. Proposals agree that calculators and computers should be used,
”to investigate numerical patterns rather than check paper-and—pencil
calculations," and to ”analyze data rather than perform rapid drill on basic facts"
(p. 56 Framework). These statements suggest that calculators and computers
ought to be used for purposes other than students’ learning of executing
procedures. In other words, they should be used in the process of analyzing and
solving problems rather than for developing proficiency with basic facts and
algorithms.

Very little attention is aimed at examining the traditional computational
curriculum. For instance, there are few statements to look to concerning
recommendations for what to include or exclude from the traditional curriculum.
Take for example, long division, the mainstay of the fourth grade curriculum.
Within each document, there are hints that long division is no longer considered
valuable. In one paragraph, in the context of calculator and computer usage, the
authors raise the question, "How many adults, whether store clerks or -
bookkeepers, still do long division (or even long multiplication) with paper and
pencil?" (p. 57 Framework). In response to the question, the authors explain that
long division and possibly longer multiplication problems are outdated and

rarely used, "except in instances where calculations are done on the backs of

68

envelopes."14 Yet, little guidance is offered in the texts explaining whether long-
division and multiplication algorithms ought to be excluded or to what degree
they should be taught or to what level of complexity.

Because so little is said and because teachers bring variation in prior
knowledge and teaching experience, the documents can be read in almost
entirely opposite ways. For example, the reform documents give little attention
to the kinds of emphasis that should be given to traditional computational
techniques. Yet, they suggest that long division, complex paper-and-pencil
computations and paper-and-pencil fraction computation be given decreased
attention (p. 21 Curriculum Standards, p. 208 Framework). What this implies for
practice is not clear. It could suggest that all that is currently done with division
should still be done, but not to the degree of mastery. It may mean that only
single or double digit divisors be expected to the degree of mastery and more
complex divisions be shown but not emphasized. There are various other
interpretations depending on what one brings to reading the documents. For
example, a fourth grade teacher may bring the View that the traditional long
division algorithm is highly valuable for students and interpret the above
statement to suggest she continue teaching the long division algorithm but
include other strategies as well. Another fourth grade teacher might argue that
the documents suggest all traditional goals for teaching long division be
abandoned. In essence, one interpretation may be to expand curricular goals but

also keep what is there. Another may be to abandon certain goals.

 

14 It is also noted in a footnote that computations using traditional algorithms are often done on
the backs of envelopes even when estimates make more sense. Reformers are suggesting that an
overemphasis on traditional algorithms in school robs children of their ability to make more
sensible computing choices.

69

Coneeptual Grasp ef Underlying Mathematical Ideas

A great deal of attention is given to the notion of understanding
mathematics on a conceptual level. A conceptual grasp of the ideas underlying
computation implies that emphasis be shifted away from mastery of basic facts
and proficiency with algorithms and toward understanding the conceptual
underpinnings of computational strategies and linking those ideas with
procedural steps. The documents provide extensive argument supporting the
idea that the mathematics curriculum must be conceptually oriented rather than
skill oriented. In other words, emphasis must shift away from pure skill
acquisition toward underlying conceptual knowledge and understandings.

This View assumes that problem solving, conceptual understanding and
skill acquisition are intimately tied together instructionally. There is much text to
support this notion. For example, "A strong conceptual framework also provides
anchoring for skill acquisition," and "a strong emphasis on mathematical
concepts and understandings also support the development of problem solving"
(p. 17 Curriculum Standards).

A conceptual orientation to computational skills implies very different
teaching practices than what has been traditionally acceptable. Instead of
devoting instructional time to practicing the execution of pre-determined
procedural steps, students would instead focus on understanding the concepts
underlying the procedures and understanding the relationships between these
concepts. This implies, "Emphasizing mathematical concepts and relationships
means devoting a substantial amount of time to the development of
understandings" (p. 17 Curriculum Standards). And, "it also means relating this
knowledge to the learning of skills by establishing relationships between the

conceptual and procedural aspects of tasks" (p. 17 Curriculum Standards).

70

Instructional practices would be very different. For example, "By
emphasizing underlying concepts, using physical materials to model procedures,
linking the manipulation of materials to the steps of procedures and developing
thinking patterns, teachers can help children master basic facts and algorithms
and understand their usefuhiess and relevance to daily situations" (p. 44
Curriculum Standards). Instead of asking students to memorize the basic facts,
students are asked to look for patterns and understandings in an array of basic
facts useful for deriving the other facts. For example, instead of arraying the 100
basic facts resulting from combining all combinations of the digits 0 through 9
and memorizing each combination, the goal would be placed on recognizing
patterns in the rows and columns. For example the pattern of all zeros in the
zero column and the same number in the ones column. Other patterns such as
the idea that reverse-ordered facts provide the same answer and understanding
why this happens would be more prominent.

A conceptually-oriented curriculum does not emphasize drill or practice
of traditional algorithms. In fact, drill is not given any attention in the reform
documents other than to suggest that speed of processing is no longer a central
goal. Practice is mentioned in several places. Both the Framewerk (1992), and
the Teaehing Standards (1991), suggest some practice on basic number facts and
traditional algorithms ought to be done for achieving proficiency with
computing algorithms. In particular, one statement reads, "Practice designed to
improve speed and accuracy should be used, but only under the right conditions:
that is, practice with a cluster of facts should be used only after children have
developed an efficient way to derive the answers to those facts" (p. 47
Curriculum Standards). Multiple meanings could be associated to how practice
fits in the new proposals. One interpretation might be that traditional practice

exercises for learning basic number facts should be done, but only after learning

71

how to derive the answers using patterns and thinking strategies. This
interpretation implies practice would be used to support proficiency with a
cluster of related facts where relationships among those facts might be realized.
Another reference to the role of practice and when it ought to occur can be found

on page 47 of the Curriculum Standards.

Exploratory experiences in preparation for paper-and-pencil
computation give children the opportunity to develop underlying
concepts related to partitioning numbers, operating on the parts,
and combining the results. Many such experiences can be provided
in the context of using place value materials, computing mentally,
or performing computational estimation. Only after these ideas are
carefully linked to paper—and pencil procedures is it appropriate to
devote time to developing proficiency by providing practice. (p. 47

Curriculum Standards).

What kind of practice is not forthcoming in this text or the others. It seems
practice does play a role in developing proficiency with computation in the new
proposals. Whether it is isolated practice of facts and procedures or practice in
contexts of problems is not clear. What is clear is the idea that conceptually
oriented curricula emphasize the thoughtful use and meaningful development of
the mathematical ideas underlying computational skills and practice would be
done after those understandings are developed.

A clearer picture of what that would look like in practice is offered on p.
26 of the Teaching Standards (1991). The example illustrates how basic
multiplication facts can be developed for a cluster of facts even as there is aim to

contextualize the use of those facts and provide an opportunity to explore the

72

conceptual dimensions of the notion of factors. The paragraph is situated in a
section of the text aimed at setting standards for worthwhile mathematical tasks
and serves as an example of the kind of task reformers' suggest is worthwhile for

developing basic number facts.

Trying to figure out how many ways 36 desks can be arranged in
equal-sized groups - and whether there are more or fewer possible
groupings with 36, 37, 38, 39 or 40 desks - presses students to
produce each number's factors quickly. As they work on this
problem students have concurrent opportunities to practice
multiplication facts and to develop a sense of what factors are.
Further the problem may provoke interesting questions: How
many factors does a number have? Do larger numbers necessarily
have more factors? Is there a number that has more factors than

36? (NCTM, 1991, p. 26)

In this illustration, student's work with a cluster of multiplication facts
exploring the conceptual basis of the notion of factor. The idea is to promote
both the acquisition of a set of facts and a conceptual understanding of the
mathematical concept of factor. There is also aim at problem solving. As a
consequence of engaging in problem solving, students have concurrent
opportunities to explore the idea of factors and practice a cluster of
multiplication facts. Through many similar problems students are provided the
opportunity to develop mastery of basic facts, fluency with calculations,
conceptual understanding and the influence of contextual issues for solving
problems. Keeping in mind that, "the focus of attention has shifted from

proficiency with computational procedures to attempts to make sense of and use

73

addition, subtraction, multiplication and division appropriately" (p. 110

Framework).

Centextualize Cemputatienal Skills

Contextualize basic computational skills refers to the idea that
computation must be developed within the contexts of genuine problems, the
actual settings that give rise to the need for such skills. Proposals in the
documents suggest approaching computational skill instruction using a problem
solving perspective. From this View computational skill would net be developed
in isolation of other mathematical experiences. Instead, computational skill
would be developed within experiences that require the use of those skills. This
implies that proficiency with computation would develop concurrently with
other skills such as problem solving and mathematical reasoning. Reformers
argue that children must come to understand why computation is valued. They
argue, "The purpose of computation is to solve problems," (NCTM, 1989, p. 44).
And, "An awareness that computation is learned and used to attain some goal
develops when problem situations and computations are explicitly linked
throughout all aspects of work with computations" (NCTM, 1989, p. 44).

Linking problem solving and computational skill instruction is the most
central pedagogical change proposed in the policies for instruction of basic
computational skills. A problem solving approach is used to foster how
computational skills are used in the service of doing mathematics. From the
perspective of practice, this change assumes an entirely different orientation to
learning computation. Traditional ideas promote teaching computational skills
in isolation of and prior to any work with problem solving. New proposals
reverse this notion and argue, "a goal is to create contexts that foster skill

development even as students engage in problem solving and reasoning"

74

(NCTM, 1991, p. 26). And, "instead of the expectation that skill in computation
should precede word problems, experience with problems helps develop the
ability to compute" (NCTM, 1989, p. 9). Further, "present strategies for teaching
computation may need to be reversed: knowledge often should emerge from
experience with problems" (NCTM, 1989, p. 9). These statements suggest that as
students work through problems, they can also work on computational skills. In
fact, problems can be designed to promote development with computational
skills. In addition to learning proficiency with computing skills, students will
also learn what computing strategies make sense in different contexts. Through
the experience of solving meaningful problems and considering meaningful

mathematical situations, computational skills can developed.

An derall View ef Teaching Cemputatienal Skills Differently

The table below illuminates contrasts between traditional ideas and
practices for the development of students’ computational proficiency and
proposals articulated in the reform documents. There is very little overlap.
None of the traditional practices are central to the new proposals. Practice with
computing, although it appears to be a point of overlap, is very different from

reformers' perspective when compared to traditional ideas.

75

 

 

Traditional Ideas New Proposals

memorization of basic facts and contextualize use
traditional algorithms
emphasize understanding by linking
practice exercises aimed at conceptual knowledge and procedural
developing accuracy and precision steps

in using basic facts and traditional
algorithms thinking strategies for learning basic
facts

drill in the form of timed tests
aimed at developing speed in practice under certain conditions
executing facts and algorithms

 

 

 

Figure 3.
Teaching Computational Skills

Traditional practices for developing computational skill consist mostly of
memorization and isolated drill and praetiee of routine addition and
multiplication facts and traditional computing algorithms. These practices are
highly criticized in the reform documents as promoting that computational skill
is synonymous with proficiency at executing traditional computing algorithms.
These practices promote the idea that speed, precision, and accuracy is more
important than contextual and conceptual understandings. These practices are
believed to create the belief in children that computation is done purely for the
sake of computing and assumes children will know when and how to apply
computational skills as circumstances arise.

Reformed practices begin by situating all computational goals inside
problem solving experiences. Conceptual goals and contextual understandings
are attended concurrently with computational goals. Some practice of number
combinations that are useful in the problem solving context is recommended.

Practice outside these settings is not recommended.

76

Computational proficiency in reformed practice assumes flexible
understanding of a variety of computational skills. It includes efficiency and
accuracy in computing but extends these aims to include conceptual
understanding and contextual understanding, how the context influences what
computational strategies are used. Those making strides to understand and ’
implement the policy must manage to teach students to compute proficiently and
achieve conceptual understanding of the underlying ideas. Reformed ideas
support and encourage memorization and facility with number facts and
operations even as students engage in mathematical problem solving.
Traditionally, memorizing facts and algorithms went on separate from problem
solving settings. Understanding what this change implies instructionally would
be at the heart of any program to foster teacher learning about changing
computational skills instruction.

Two main arguments underlie these proposals. First, the changes provide
students the opportunity to understand why computation is performed. Second,
these practices foster more efficient use of computation. Statements such as,
"Skills can be acquired in ways that make sense to children and in ways that
result in more effective learning" (p. 17 Curriculum Standards) foster the notion
that a conceptual orientation to learning mathematics can also support
development of skills and with higher degrees of accuracy and less instructional
time involved. The documents make the argument that a conceptual approach to
computation, "result in good achievement, good retention, and a reduction in the
amount of time children need to master computational skills. Furthermore many
of the errors children typically make are less prevalent" (p. 44 Curriculum
Standards). In addition, ”placing computation in a problem solving context
motivates students to learn computational skills and serves as an impetus for the

mastery of paper-and-pencil algorithms" (p. 45 Curriculum Standards).

77

The documents also propose that teachers must, "consider which skills are
essential and why and seek ways to develop essential skills in the contexts in
which they matter" (p. 26 Teaching Standards). There is very little guidance for
making these determinations. The only direct reference is that decreased
emphasis be given to long division and multi-digit multiplication that involves
extensive carrying procedures.

The documents acknowledge conceptual orientation would require much
more time. In other words, students’ computation-related skills would develop
on a different pace than the current curriculum provides for. Several statements
allude to the extent of the work that would be involved to make this happen. For
example, "The approach to computation taken in this standard requires
educators to rethink the traditional scope and sequence decisions," and, "the time
required to build an adequate conceptual base should cause educators to rethink
when children are expected to demonstrate a mastery of complex skills." (p. 44,
p. 17 Curriculum Standards). Taken seriously, remaking computational skills
instruction would involve restructuring and remaking the entire elementary
skill-oriented curriculum. For example, to advance the idea that, "the
presentation of computational procedures can be delayed until students need it
and meaningful examples and motivation can be provided before the algorithm
is presented" (p. 56 Framework), most teachers would necessarily need to
restructure all that they currently do around computation. Traditionally,
instruction begins with the goal of mastery of procedural steps. Instead, new
proposals suggest instruction would begin by focusing on conceptual ideas, the
consideration of contexts, developing a variety of computational strategies, and
using those strategies in different situations. Mastery, speed and accuracy would
be attended only after those understandings are developed. Further arguments

suggest that only after students come to conceptually understand procedures

78

and learn why certain steps make sense, fluency will develop as a result (p. 56
Framework). Taken seriously, the proposals would require a reframing of the
more dominant skill-oriented curriculum and much reconsideration of the scope
and sequence of the entire elementary mathematics curriculum.

The diagrams below highlight extant practice first and then policy
proposals regarding computation. Examining the differences across these
diagrams can begin to illuminate what would be involved to teach computation

differently and what teachers might need to know to accomplish that goal.

 

     

   

Traditional Algorithms

  
   
 
   
 

Mental
Computations

Procedural Orientaion
Problem Solving

l

Computational Proficiency

 

 

 

 

 

Figure 4.
Traditional Forms of Computational Skills Instruction

Extant practice involves two basic approaches to computing: traditional

algorithms and mental computations. Instructionally, each is approached

79

procedurally. Facility typically involves mental and pencil-paper calculations

using mostly traditional algorithms. Speed and accuracy are valued. Problem—

solving is introduced only after students have mastered computational skills.
The diagram below captures ideas for reforming computational skills

instruction. It is far-more complex.

         
    
     
    
   
 

Procedural I . Conceptual
Orientation Orientation

...........
.........
.................
0000
"""""""""""
00000
000000
O .‘|‘

'\ estimation

    

l (MI. ln‘

 

  

student's l l ,
‘algorill \\\ ,' l , mental

  

traditional
algorithms

\
| l‘
llllll
...........
..................
"""""""""""""""""""
............

 

-------------------------------
"""""""""""""""
............

a n
"""""
aaaaa
0' ‘u
,n '-

n' 'n
I o
o n
I \

0..
c..
n...
'1
..............
...........

Conceptual Understanding

Learning In Context

Figure 5.
Reforming Computational Skills Instruction

80

The inner most circles represent a range of new computational skills not included
in traditional forms of practice. For instance, students’ algorithms, technology,
estimation, and other alternative algorithms are new computational skills that
must be taught. Both conceptual and procedural orientations to these skills are
recommended. Conceptual understanding is to be valued over the procedural
orientation emphasized in extant ideas and practices. For example,
understanding underlying ideas such as place value and flexible use of
computational skills are to be valued over speed and accuracy with algorithms.
The darker circle in the diagram highlights the importance of problem-solving
contexts. All computational skills development is to be situated inside and go on
concurrently with problem solving experiences. The goal is for students to learn
to pay attention to problem solving aspects of computational skills, both what
skills to use and how best to do the computation.

Contrasting proposed changes with traditional practice can serve as a
beginning for speculating about what there is for teachers to learn to change their
teaching of computation. For example, teachers would necessarily need to learn
about estimation as a computing strategy as most elementary teachers would not
have experiences thinking about estimation as a computing strategy itself.
Teachers would also need to learn about the issues that arise in using technology
to compute. Many teachers choose not to use calculators because of the
difficulties that arise in using technology with students in classrooms. Further,
teachers would need to understand that students often develop their own
strategies for computing. And they would need to learn the conceptual
grounding of the traditional algorithms and other mental operations already
taught in school.

Examining these diagrams can reveal other issues as well. Notice that

problem solving is treated differently. It only becomes important after students

81

have developed procedural understandings of the traditional algorithms and
routine mental computations in traditional forms of teaching computational
skills. Extant practices treat problem solving contexts as the core activity for
students. Proficiency is not required to work on problems. Contextual
influences of problems are important in terms of what students should learn
about computation. Unpacking the range of issues and considering how teachers
might attend to these issues instructionally would be a basis for determining
what teachers would need to learn.

Yet, uncovering all of what is proposed and the implications of the
proposals for teacher learning would be more complicated still. There are wide-
scale issues that dwell within public schooling. For example, there is a constant
force of the back-to-the basics argument. Historically, we have seen consistent
return to the basics following any proposals to produce more ambitious
instructional goals. Another likely issue would be exploring different
conceptions of balance and what balance implies instructionally. The next
chapter illustrates how a teacher can interpret current policy proposals to imply
a balance between traditional forms of instruction with more conceptual and
problem-solving oriented teaching. My analysis earlier in this chapter suggests a
different interpretation. The term balance often refers to the idea of drill and
practice with facts and algorithms along side more ambitious problem solving
and conceptual orientations to teaching mathematics. Reformed ideas envision
balance to imply basic computational skills instruction going on concurrently
with reasoning and problem solving. Both issues, ”back-to-the-basics" and
”balance” continue to get teachers stuck in formulating new practices. These
issues can act as barriers to long-lasting change.

Another likely issue involves the relationship between procedural and

conceptual knowledge. Policy proposals place a premium on conceptual

82

understanding of the underlying mathematics of computation. Conceptual
knowledge of place value, relations between ones and tens, as well as operation
sense to name only a few are considered much more important in reformed-
based teaching. Yet, the relationship between conceptual knowledge and
proficiency with computation is complicated. One does not guarantee the other.
The relationship is not well-understood. Research offers conflicting results and

most research is not topic-specific.

Intreducing Discrete Mathematics

It is not unusual to find instructional policies that ask teachers to
introduce new and different mathematics into the existent mathematics
curriculum. Sometimes this can even involve mathematics that teachers
themselves have never learned or taught before (Sarason, 1982). The Curriculnm
Standards (1989) argue "it is crucial that all students have experiences with the
concepts and methods of discrete mathematics" (p. 176). And, the Planner/m
(1992), calls attention to the importance of discrete mathematics by introducing a
new strand of mathematics content, suggesting discrete mathematics be taught in
equal balance with other more traditional content areas such as geometry,
algebra, measurement, and number.

Yet, most teachers, especially elementary teachers have few ideas for what
discrete mathematics is. Some have never heard of it before. Most elementary
teachers would be hard pressed to describe discrete mathematics or come up
with examples of the central tOpics, questions or methods. Obviously,
introducing a mathematics that teachers have little ideas about suggests there are
at least two levels of the problem of teachers’ learning. Teachers not only would
have to learn what would be involved for introducing students across grade-

levels to discrete mathematics but, they would have to learn what discrete

83

mathematics is for themselves. Teachers would not be able to help children learn
what they themselves do not understand (Feiman-Nemser, 1983).

The process of uncovering the ideas, proposals, and recommendations
articulated in reform documents for introducing discrete mathematics also did
not lead to programs and specifics for practice. The exact opposite occurred.
Although the goals and directions became more focused, the process did not lead
to specific actions for practice.

My analysis focuses only on the issues, mathematical ideas, and practices I
uncovered as central to the policy proposals conveyed in the reform documents.
Because the Curriculum Standards (1989) and the Framewerk (1992), were
substantially incomplete for providing understandings of the mathematics and
related issues Sandy confronted, I turned to a number of other sources to help me
sketch in a View of what these proposals involved for the individual teacher.

The analyses are informed by the reform documents, my mathematics
background and teaching of secondary and collegiate level mathematics courses,
an interview I conducted with a research mathematician specializing in discrete

mathematics, textbooks of discrete mathematics, and NCTM’s 1991 Yearbook

titled Discrete Mathematics Acress the Curriculum K—12.15

I explore four core issues:

1) What discrete mathematics is.

 

15The 1991 NCTM Yearbook titled Discrete Mathematics Across the Cnrxicnlnm K-12, Margaret
Kenney, editor, is identified in the Framework (1992), as a resource for teachers to consult for
incorporating the study of discrete mathematics into their teaching practice. I separate the
germ (1991a), from the other two sources in my analysis both because it is of a different
nature and distribution than the reform documents and because Sandy Wise did not rely upon
the Yendpeek (1991a) in her efforts to learn about incorporating a study of discrete mathematics
into her teaching practice.

84

2) Why discrete mathematics should be introduced into school
mathematics curriculums.

3) How discrete mathematics should be introduced into the school
curriculum.

4) How teachers might introduce discrete mathematics into their

practice.

What is Discrete Mathematics?

The 1985 mathematics framework identified seven strands of content to
guide teachers' decisions about what to teach across the K-12 school curriculum.
The 1992 framework introduced an eighth strand titled discrete mathematics.
The Framewerk (1992), makes no specific mention of discrete mathematics in the
table of contents, but in a section entitled, What's New in 1992, on page 2, there is
an announcement of a new strand of content, discrete mathematics.

The announcement appears on page 5, of the document as the reader is
reminded of the seven content strands of the previous framework: number,
measurement, geometry, patterns and functions, statistics and probability, logic
and algebra. The announcement reads, "The 1992 document endorses those
strands and adds another, discrete mathematics." The reader is directed to two

very brief paragraphs describing discrete mathematics in a footnote.

This unfamiliar title includes things teachers at all grade
levels have been doing for years. Discrete implies emphasis on
separate (discrete) entities rather than on measures of continuous
quantities - on questions of hew many rather than hew much. In
third grade, for example, Boris can be asked in how many ways he

can dress if he has three shirts and two pairs of pants (six): and in

85

seventh grade students can be asked to design a tournament. This
I discrete mathematics includes topics such as combinatorial

counting principles (how to count permutations and combinations

for probability problems) and discrete structures (such as networks

and trees). A resource for teachers is Discrete Mathematics Across

the Cnrriculum, K-12, the NCTM yearbook for 1991. (p. 5
Framework, 1991)

Two more paragraphs appear later in the document (p. 84), in a sub-

section entitled, Strands. and within a section called Structnre and Centent ef the
Mathematics Pregram.16

The discrete mathematics strand did not appear in the 1985
Framewerk, although some of its ideas appeared under statistics
and probability. Discrete mathematics includes a cluster of related
ideas, such as principles for counting arrangements of discrete
objects (permutations, combinations, selections); other counting
principles (the inclusion / exclusion principle, the pigeonhole
principle): some basic and useful ideas from set theory (unions and
intersections); the study of discrete structures (networks, graphs,
and tree structures); recurrence relations (such as the Fibonacci
relation, Fn = Fn-1 + Fn-2 : and the analysis of algorithms.

Discrete in this context means focusing on discrete and

separate entities rather than on measures of continuous quantities:

 

16A few additional comments related to discrete mathematics can be found in various other
sections of the text. Most of what is said in these other sections pertain to the issue of how

discrete mathematics might fit within the larger K-12 mathematics program. For an example see
p. 107.

86

it does not mean that everything not continuous is to be considered
discrete mathematics. Arithmetic with integers, for example, is
treated under number, not under discrete mathematics. (California

State Department of Education, 1992, p. 84)

These paragraphs represent what is offered in terms of explanatory
statements for understanding what discrete mathematics is. Depending on what
the reader brings to these statements, there is either a wealth of information
about discrete mathematics of little to make sense of. The notion that discrete in
mathematics implies separateness, distinct from another, or elements that are
countable, suggests that discrete mathematics involves, for example, study of sets
of numbers such as { 1, 2, 3}. Combinatorial analysis is the study of the different
ways to count arrangements, combinations, or permutations on a set. For
example, there would be two different arrangements possible for the set of
elements {1,2}, both {1,2} and {2,1}. Extending ways in which to configure
elements of sets opens up the mathematics of combinatorial counting.

A combinatorial counting problem, a sub-topic of discrete mathematics, is
embedded in these paragraphs. It asks how many ways a student can dress if he
has three shirts and two pairs of pants. One could argue that each shirt can be
worn with each of the two pairs of pants, concluding that there are six different
combinations of outfits possible. Although there is no discussion of these ideas
or techniques for soling them in these paragraphs, they involve the central tenets
of combinatorial mathematics mentioned as, "combinatorial counting principles”

and "discrete structures." NCTM‘s Discrete Mathematics Acress the Curriculnm

K-12, is named as a resource.17

 

17As mentioned earlier I provide an analysis of this text later in this work.

87

Other ideas that one could draw out of these paragraphs, such as the
notion of counting arrangements, combinations, permutations, and selections, as
well as set theory (unions and intersections) may be familiar (i.e. new math era or
college coursework) to some elementary teachers. But they are not commonly
part of the elementary mathematics program. The terms discrete structures and
recurrence relations are most likely unfamiliar to most elementary teachers and
therefore would not offer much in the way of understanding what discrete
mathematics involves. The terms, "the inclusion/ exclusion counting principle,
the pigeonhole principle," would probably fall into the unfamiliar category as
well. The phrase, "the analysis of algorithms," may or may not be useful to
elementary teachers trying to understand these paragraphs, depending on their
understanding of what an algorithm is.

It seems that elementary teachers would have to bring fairly well-worked
out ideas about discrete mathematics to recognize and use the examples and
illustrations provided in the documents for formulating a vision of what discrete
mathematics is. Paradoxically, what the reform documents articulate to
introduce discrete mathematics assumes that teachers bring fairly well-worked
out ideas about discrete mathematics. At the same time, discrete mathematics is ,
an unfamiliar content area to most teachers.

The Cnrricnlum Standards (1989), suggests that, "Although discrete
mathematics is a relatively new term, we will consider it simply to be the study
of mathematical properties of sets and systems that have a countable number of
elements" (p. 176). And, "Whereas the physical or material world is most often
modeled by continuous mathematics, that is, the calculus and prerequisite ideas
from algebra, geometry, and trigonometry, the non material world of
information processing requires the use of discrete (discontinuous) mathematics"

(p. 176). Once again, depending on the reader, there is either a wealth of

88

information or a mountain of confusion. Formulating any understanding of
discrete mathematics based on these statements would require a great deal of
assumed mathematical knowledge. For example, ideas about what continuous
and discontinuous quantities are, what underlying systems refers to, the idea of
infinity would be central, and understanding the mathematics of algebra and
calculus would all be required prerequisite knowledge for making sense of the
contrast portrayed. One would have to bring this knowledge to their reading of
the documents in order to understand the ideas.

Most of what is written in the Standards documents is clearly labeled in
sections for secondary teaching practice. It would be easy for an elementary
teacher to dismiss these proposals altogether, arguing that incorporating a study
of discrete mathematics into the elementary curriculum is not a central goal of
reform for elementary grades. Interestingly, Sandy perceives the introduction of
discrete mathematics to be central to elementary mathematics teaching reform.

The Curriculum Standards (1989), direct treatment of discrete
mathematics is geared toward a secondary teacher audience. Much of what is
written is focused toward grades 9-12. Yet, there are many examples and
illustrations throughout each text corresponding to various other topics that are
representative of discrete mathematics. Because these illustrations are not
identified as such, one would not be informed explicitly that they are useful for
understanding something about discrete mathematics. From this standpoint,
they may not be useful for that purpose.

Within Standard 12, (p. 176) the following statement suggests, "finite
graphs (structures consisting of vertices and edges), together with their
associated matrix representations, offer an important addition to the student's
repertoire of representation schemes." Following is an illustration involving a

diagram of a network of one-way streets referred to as a directed graph and a

89

representation of the diagram as a matrix of numbers. Some discussion explains
how matrices can be manipulated to yield various kinds of information about the
graph. The example illustrates how the very familiar ideas (to the secondary
audience) of matrices can also be used to introduce a very non-traditional
combinatorial counting problem. The illustration assumes a basic knowledge of
matrices. This may be problematic for elementary teachers in particular as it
assumes knowledge many elementary teachers may not have.

On p. 177 and p. 178 two other mathematical ideas are mentioned. The
terms, "recurrence relations" or "thinking recursively" and "the development and
analysis of algorithms" are each discussed briefly and illustrated through several
examples. The text uses a fair amount of mathematical symbolism making it
difficult to follow. Most likely the symbolism is unfamiliar to most elementary
teachers, making it highly unlikely for them to discern any information about
discrete mathematics.

A number of topics are mentioned in relationship to the analysis of

algorithms. Recommended is an algorithmic perspective to topics such as,

"the greatest common factor of two integers, the solution of
quadratic equations; approximating roots of polynomial equations;
geometric constructions; the specification of sequence of
transformations mapping one figure onto another, similar one; the
construction of LOGO procedures to produce figures satisfying
certain conditions; determining shortest/ critical paths in finite
graphs; random-number generation to simulate probability
problems; sums of sequences; and solutions of systems of linear

(and nonlinear) equations" (NCTM, 1989, p. 178).

90

The first idea is probably familiar to most elementary teachers because it
falls within the current scope of the elementary curriculum. The ideas of an
algorithmic perspective may or may not be familiar to elementary teachers
depending on the teacher’s mathematics background. The example used to
illustrate the notion of an algorithmic perspective involves evaluating a fourth
degree polynomial equation with general terms of the form ax. The illustration is
complicated and difficult to follow. Ironically, the tone developed in relationship
to the illustration is one of ease. For example, "The development and analysis of
algorithms often add clarity and precision to the student's understanding of
mathematical ideas and provides a context for nurturing careful logical
reasoning," (NCTM, 1989, p. 178).

The Cnrriculum Standards (1989), mentions other ideas as well. Included
are sets and relations; deductive proof, particularly proof by mathematical
induction; the algebra of matrices; recursively defined functions; mathematical
modeling and algebraic structures. Once again, depending on the background
knowledge of the reader, these ideas may or may not clarify a sense for what
discrete mathematics involves. There is no explanation for how these ideas fit
within a study of discrete mathematics or in relation to other mathematics.

It should be pointed out that the documents treat all content strands
similarly. Short descriptive paragraphs, lists of central ideas and topics, and
some illustration serves to introduce a reader to each content strand. Most of the
ideas mentioned in sections describing discrete mathematics would be familiar to
secondary mathematics teachers. Very few would be familiar to elementary
teachers. Thus, developing insights into what discrete mathematics involves
based on reading the documents would be highly'dependent on the

mathematical understandings one brings to their reading. In the case of

91

elementary teachers, it would be very difficult to develop any sense for what

discrete mathematics involves.

Wh Di r t Math matics h uld B Intr duced

The Framewerk (1992), offers nothing in the way of an explanation for
adding an additional strand of content beyond the idea that the Framewerk
(1992) endorses the set of standards recommended by NCTM in the Curriculum
Standerds (1989). The Curriculum Standards (1989), on the other hand, make
several different arguments for including discrete mathematics in the school
mathematics curriculum. The biggest push seems to come out of the connection
discrete mathematics has with computer technology. On page 176, the following

statement supports this conclusion.

"Computer technology, too, wields an ever-increasing influence on
how mathematics is created and used. Computers are essentially
finite, discrete machines, and thus topics from discrete mathematics
are essential to solving problems using computer methods. In light
of these facts, it is crucial that all students have experiences with
the concepts and methods of discrete mathematics" (NCTM, 1989,
p. 176).

Embedded in this argument are a number of related rationales. For instance,
discrete mathematics, because of the connection it has with computer technology,
has potentially more obvious connections to real-world phenomena as well and
thus provides a more direct means for strengthening the connection between
theoretical mathematics and real-world applications, a constant criticism of the

traditional curriculum.

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Further, the Curriculum Standards (1989), point out, "the non material
world of information processing requires the use of discrete (discontinuous)
mathematics" (p. 176). Given that we live in an information processing society,
discrete mathematics offers a means for making societal demands and concerns
more relevant within the school curriculum. A side argument to the issue of
relevance or usefulness also involves the rationale of modernizing the school
mathematics program. Discrete mathematics has been termed "the math for our
time" (p. 1, NCTM Yearbook, 1991a), and there is argument that discrete
mathematics would update or modernize the school math program providing
avenues for, "moving school mathematics toward the 21St century" (p. 1, NCTM
Yearbook, 1991a).

A secondary argument for introducing discrete mathematics, although
less emphasized in the reform documents and perhaps less noticeable, involves
the idea that discrete mathematics provides a setting or opportunity to
incorporate many of the other goals and recommendations of the reform. The
argument posits that besides promoting and capitalizing on the connection to
technology and real-world problems, discrete mathematics potentially fosters
reasoning and critical thinking, primary aims of the reform. Thus, discrete
mathematics creates a desirable setting for emphasizing logical reasoning,
deductive thinking and inductive argument. The documents suggest discrete
mathematics can provide an avenue for school mathematics to move away from
the dominance on manipulating symbols and memorization formulas, the very

thing school mathematics is criticized for.

Hew Discrete Mathematics Ceuld Be Introduced

There are mixed messages concerning how discrete mathematics might be

situated within the larger school mathematics program. The Curriculum

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Standards (1989), proposes that discrete mathematics must be introduced to all
students at the secondary level. Yet, there are no specific instructions for
whether or how to introduce discrete mathematics into the elementary or middle
grades. At the same time, there exists many examples throughout the K-8 set of
standards where discrete mathematics is featured. These examples fall within
contexts such as number sense or numeration and probability and statistics. The
question of how to or whether explicit attention should be given to discrete
mathematics at levels other than grades 9-12, is left ambiguous in the Curriculum
Standards (1989). The NCTM Yearbeek (1991a), proposes explicitly that
attention be given to discrete mathematics across the entire curriculum, stressing
the importance of introducing discrete mathematics at all levels on the basis that
the entire math program can be strengthened as a result. NCTM’s message is
confusing at best.

The Framework (1992), advocates explicitly that attention should be given
to discrete mathematics at all levels of schooling. Yet, the authors argue that the
level of formalism should be adjusted at different levels. For instance the
document explicitly states that formal attention to discrete mathematics in grades
K-5 is net appropriate (p. 107). Recommended instead is that discrete
mathematics be explored informally in a variety of different contexts already
common to the elementary school math program. There is very little discussion
for how this might be done or how any informal explorations might connect to
form a coherent understanding of discrete mathematics or connect to more
formal work in discrete mathematics later. A few examples of activities in which
teachers might engage students for exploring discrete mathematics are included.
Two examples, one at the third grade level and one at the middle grades level
were offered earlier in this text. There is no discussion for how these examples

might be situated in practice or what students might be able to learn from them.

94

The Curricnlum Standards (1989), the Framewerk (1992), and the NCTM
Yearbeek (1991a), all agree that a formal course at the high school level is not
appropriate. Instead, the documents suggest that topics in discrete mathematics
be integrated throughout existing courses. The following statement on p. 176 of
the Curriculum Standards (1989) supports this position, "This standard neither
advocates nor describes a separate course in discrete mathematics at the
secondary school level; rather, it identifies those topics from discrete
mathematics that should be integrated throughout the high school curriculum."
Further, "The depth and formalism of treatment should be consistent with the
level of the courses in which a topic appears." The idea is that topics in discrete
mathematics can be inserted into an already existent mathematics program.

The Framewerk (1992), provides some information about how discrete
mathematics topics might be integrated and organized within the K-5
mathematics curriculum. In a section on page 107, under a sub-section titled
Integrating Strands in the Elementant Crades, the authors write, "because the
strands are intended to be interwoven, students need not have experiences with
each strand separately. In kindergarten through grade five, some strands,
particularly algebra, functions, and discrete mathematics, seldom appear alone
because it is not appropriate to deal with that material formally." Several
examples are given explaining how this might happen. For instance, unifying
ideas such as "how many" or "how much" (counting and measuring) are
recognized as important ideas for children to learn. They represent contexts
where several strands might be interwoven at once, in this case "number and
discrete mathematics" (p. 107). On page 108 the ideas are explained in a little

more detail.

95

"How many? How much? A count tells how many things are in a
specified group. A measure tells how much of a specified attribute
something has. Generally, counts are discrete and measures are
continuous. Both counts and measures identify quantities,

therefore both are identified by unit labels."

Specific examples of activities or content that can be interwoven as
unifying ideas are not provided in this section. Later in the text there is a
description of what a recommended program organization and structure for
elementary grades might look like (CSDE, 1992, p. 109). In examining this
program there is no specific mention of discrete mathematics or any of the ideas
outlined earlier as important within a study of discrete mathematics. What does
appear are a few examples of counting problems under number and numeration
and Venn diagrams for classifying objects and displaying data under statistics.
The illustration is compatible to the idea that discrete mathematics not be treated

formally but introduced in relation to other strands or unifying ideas.

Advice te Teachers

The big message is that much of the content of discrete mathematics is
already largely in the school curriculum. The Framewerk (1992), introduces
discrete mathematics as, "This unfamiliar title includes things teachers at all
grade levels have been doing for years" (p. 2). The implication seems to be that
much of what discrete mathematics involves is already getting attention. What
then would be involved for teachers introducing discrete mathematics into their
practice? The documents offer few details. One possible interpretation might be
that incorporating discrete mathematics into the school curriculum would

involve nothing new and only a reorganization of topics. Another might be that

96

familiar ideas, already getting attention, could be used in the process of
introducing discrete mathematics. There are several illustrations of this point.
Recall the illustration involving matrices that I discussed earlier in this chapter.
At the same time, the issue of how to take familiar mathematics and use it to
introduce discrete mathematics is not explicitly discussed. The documents tend
to leave this question mostly unattended.

One idea mentioned in the documents is the notion ”slant.” The term
”slant” refers to the idea that discrete mathematics should not be taught in ways
that promote more memorization of formulas and pure symbol manipulation,
the exact thing current practices in mathematics teaching is criticized for.
Instead, reformers’ recommend discrete mathematics be represented in ways that
promote logical reasoning and thinking. The following paragraph points to this

emphasis:

In grades K-8, counting typically involves matching the elements of
a set with a finite subset of the natural numbers. But real-world
problems that can be simplified to the form "How many different
subsets of size k can be selected from the members of a set having n
distinct members?" require an entirely different method of
counting. To develop students' ability to solve problems with this
structure, instruction should emphasize combinatorial reasoning as
opposed to the application of analytic formulas for permutations
and combinations. To illustrate this shift in perspective consider a
fundamental identity involving binomial coefficients: nCr = nCn-r.
This identity usually is established by algebraic manipulation of the

formula for combinations. In contrast, a student who reasons

combinatorially may observe that nCr represents the number of

97

ways one can choose an r—element set from and n-element set. For
each r-element set chosen, however, there corresponds a set of n - r
elements not chosen. Thus, the number of ways of choosing an r-
element set is equal to the number of ways of choosing an (n - r)

element set (NCTM, 1989, p. 179).

This paragraph potentially could foster ideas and understandings of
combinatorial mathematics, an area of specialization under the scope of discrete
mathematics. It also represents important ideas about pedagogy, instruction that
emphasizes thinking, understanding, and reasoning rather than memorization of
formulas or symbol manipulation. Yet, the impact of the paragraph may be very
different for different readers. The complexity of the mathematical terms, and
the mathematics may interfere with sense making about discrete mathematics
and how to introduce it into teaching practice. The assumption in the policies
seems to be that most readers would bring a relatively good working
understanding of the mathematics of combinatorial counting. It may be that the
very opposite is true, at least for elementary teachers.

The reform documents offer relatively little for understanding what
discrete mathematics is or how it might be introduced into the school
curriculum. The little that is provided points readers in various directions for
learning more about discrete mathematics. The ideas and recommendations to
this point represent what I uncovered in the reform documents. Because much
was left unattended and unclear, and because I wanted to explain Sandy’s
encounters with these proposals, I turned to other sources to develop further
understandings of the ideas and practices for introducing discrete mathematics
into the school curriculum. The analyses that follow are not based in the reform

documents but instead, are derived from interviews and my reading of various

98

other texts including NCTM’s yearbook entitled Discrete Mathematics Acress the
Curricnlum K-12, (1991a).

Other Reseurces en Discrete Mathematics

Davis and Hersh propose in The Mathematical Experience (1981), that
mathematics is the science of quantity and space and that mathematicians are
involved in studying ideas and relationships related to quantity and space. They
claim that mathematics, both what it is and how it gets done, changes over time
making it impossible to provide a completely adequate View of what
mathematics is, mostly because of the changing nature of the work in progress.
At best, one can develop a sense for the current state of affairs by characterizing
what mathematicians working in the community do, keeping in mind that there
are different points of view about what that is (Barratta-Lorton, 1977; Davis 8:
Hersh, 1981; Kitcher 1984; Kline, 1980; Lakatos, 1976; Tymoczdo , 1986).

Bruce Sagan, a research mathematician, in an interview offered the

following three categories for describing the work that mathematicians do.

algebra / number theory
analysis
geometry / topology. 18

He explained that although the categories do a grave injustice to describing all of
mathematics, they can provide a very general frame for situating the work of

discrete mathematics.

 

18I borrow this category structure from Bruce Sagan, a practicing mathematician and professor of
mathematics at Michigan State University, who helped me think about discrete mathematics
from the perspective of a mathematician working in the field.

99

Revisiting The Qnestion —- What Is Discrete Mathematics?

The first category, algebra / number theory, includes a wide range of
interests around the notion of quantity. Algebra, in this categorical scheme, is
not the mathematics of the typical algebra course one might recall from high
school. Instead, it involves ideas of modern algebra such as group theory.
Examples include the more primitive work of arithmetic, set theory,
combinatorial counting, and more complex ideas of modern algebra involving
studies of groups, rings and fields. Much of discrete mathematics falls into this
category. Discrete mathematics is a specialization involving studying the logical
and algebraic relationships between objects that are discrete or separate from one
another (Bamier 8: Chan, 1989). Objects that are separate or distinct are often
termed discontinuous.

The second category of analysis, focuses on mathematics involving
continuous numbers and regions. The more familiar mathematics many
remember from high school and college algebra and calculus courses resides in
this category. The mathematics is often described as continuous mathematics, as
it involves ideas and relationships of continuous quantities and the notion of
infinity. Even though discrete mathematics falls into the first category, there is
overlap into this category as well. For example, mathematical analysis can be
used to determine the asymptotic behavior of certain combinatorial sequences.

Topology is representative of the third category and focuses on ideas and
ongoing work around the notion of space. There are combinatorial techniques
(one aspect of discrete mathematics) for computing certain homology groups, a

basic goal of topology.19

 

19Bruce Sagan, a research mathematician at Michigan State University helped me to think about
the overlapping nature of all of mathematics and in particular offered examples as illustrations.

100

The examples I mention illustrate that there is overlap across the general
categories and that discrete mathematics resides in all three categories.20 This is
not atypical. Most specializations overlap into other areas of mathematical
study. For example, optimization and graph theory are not considered the
mathematics of number theory. Combinatorial counting typically is. Yet, each of
these topics is considered discrete mathematics. There really are no clear-cut
boundaries in mathematics and discrete mathematics is not an exception.

Interestingly, arithmetic and discrete mathematics overlap. Although
arithmetic is not thought of as discrete mathematics or vice-versa, each involves
the other. The reform documents portray the mathematics of arithmetic and
discrete mathematics as separate entities, entirely set apart, with little, if any,
common ground. In reality each is connected and work in one area is fertile
ground for work in the other.

The above categories, I believe are useful for understanding better the
distinctions drawn between discontinuous and continuous mathematics in the
reform documents. Discrete mathematics is often categorized as ”discontinuous
mathematics” because its focus is on systems of numbers where elements are
distinct, finite or separate, hence the term discontinuous. The counting numbers
and the integers are examples. Analysis, on the other hand, is often described as
”continuous mathematics.” The real and complex numbers underlay the
mathematics of analysis. The integers are considered discrete because each
element in the set is distinct from another element. For example, there are no
numbers between 3 and 4. The real numbers are considered continuous because

between any two real numbers there exists another real number. Numbers with

 

201 do not introduce these terms with intent that all readers understand the specific examples. I
do hope to show that discrete mathematics is a mathematical field of inquiry that is not bound by
particular topics.

101

repeating decimal expansions are often the basis of continuous mathematics.
They are often not the basis of discrete mathematics.

Once again, this is not clear-cut. For example, linear algebra involves both
work with continuous and discrete sets and ideas. Mathematically speaking, the
'notion that there is mathematics that is continuous and discontinuous is
superficial. Yet, because we try to explain the nature of different mathematics we
are bound to generalize the boundaries of ongoing work. The terms
discontinuous and continuous help to do that. At the same time,lto understand
that distinction one would have to bring ideas about the larger landscape of
mathematical study. A fairly typical problem of combinatorial mathematics, a
sub-category of discrete mathematics, might involve, for example, determining
the number of combinations that can be arranged of the numbers 1, 2, 3, and 4 or
the number of unique paths that can be found between two locations on a map.
These problems ask for specific counts. The underlying number system for each
question is finite in terms of the number of elements and the elements themselves
are separate or discrete. In contrast, continuous mathematics such as calculus,
involves more the idea of taking measures as for example finding the area under
a curve. The terms continuous and discontinuous provide information about the
underlying systems of numbers.

The mathematics of discrete mathematics, because it exists on the
boundaries of many other specialization’s, encompasses many familiar
mathematical ideas also associated to other areas of study. For example, set
theory or probability make use of central mathematical ideas also useful to
ongoing work in discrete mathematics. Venn diagrams may be a familiar
example used across many mathematical specialization’s. Venn diagrams
display pictorially relationships between sets of objects or numbers. Yet, Venn

diagrams, as mathematical ideas do not represent the mathematics of discrete

102

mathematics, set theory or probability. They are useful only to the extent they
can display particular principles or ideas central to each mathematical field of
study. This is problematic when characterizing areas of mathematical study by
naming central mathematical ideas. One would have to be knowledgeable about
the field to understand how any particular idea is associated to it. The ideas
mentioned in the documents for describing discrete mathematics overlap into
many different studies. Using only central ideas to describe discrete
mathematics would provide little information if other, more structural
knowledge was not known. For example, combinatorial analysis, in part, focuses
on techniques and theories useful for formulating counts to non-traditional kinds
questions where configurations or different arrangements are of interest.
Arithmetic involves more traditional kinds of counting questions. Each overlap
in their use of Venn diagrams or number operations. They are set apart by the
mathematical structures of the field, for example, by the central questions,
underlying systems of numbers on which questions are examined, and central
techniques used for exploring those questions.

Discrete mathematics can involve numbers that have either infinite
expansions or finite expansions. Numbers such as pi and the square root of 2
have infinite decimal expansions, while the number three does not. In discrete
mathematics there are discrete algebraic systems or field extensions that involve
the square root of 2.21 The terms discontinuous and continuous reference the
nature of the relationships among elements in the sets. At the same time, they
are associated to particular mathematics. There are many mathematical ideas
and techniques associated to each continuous and discontinuous mathematics

that are useful for solving problems or proving theories in discrete mathematics.

 

21This example is provided by Bruce Sagan.

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An example might be the processes of solving systems of equations, often
associated to continuous mathematics such as linear algebra and calculus, to
locate the optimum solution to a problem given a set of constraints. In essence
the boundaries between discontinuous and continuous mathematics are
superficial. School mathematics, because of the way mathematics is segmented
for the purposes of select courses, promotes these boundaries far more than the
mathematics itself. This is not to say there are not trends and patterns in
mathematics, making various fields different from one another. Discrete
mathematics is sometimes described as falling through the cracks or existing on
the edges of many areas of specialization. For example, one mathematician
described discrete mathematics as having roots in set theory, probability,
combinatorics, matrix algebra, graph theory, formal languages, and automa
theory (Barnier 8: Chan, 1989).

The documents suggest that discrete mathematics is concerned with
mathematical questions that ask ”how many” rather than, ”how much.”
Combinatorial mathematics, the mathematics of counting arrangements and
configurations is an example. Yet, counting is only one specialized area of
discrete mathematics. Further, there are different kinds of counting questions.
In an interview, Bruce Sagan identified three different kinds of counting
questions. They included enumerative problems, optimization problems, and
existence problems.

Enumeration questions engage mathematicians in the work of deriving
and proving theories of counting. For example, combinatorial counting
principles, combination and permutation theories, partitions of numbers and
sets, and principles such as the inclusion-exclusion principle are all examples of
ideas included in the broader theory of enumeration. I mention the ideas here to

inform the reader of the many different mathematical ideas involved in the

104

mathematics of counting. Combination and permutation, for example, are
fundamental to the mathematics of counting. Each are highlighted in the reform
documents and ultimately become the focus of Sandy’s work in discrete
mathematics.

Optimization questions involve determining best possible solutions to
problems given particular constraints. Familiar mathematical ideas and
techniques, commonly taught as part of more traditional mathematics courses,
are also useful for working on optimization and enumeration questions. An
example of a common idea might be matrices. Matrices are familiar ideas to
school mathematics, usually appearing in a second algebra course. They are also
useful in combinatorial mathematics. The mathematics of matrices can be used
to determine counts and when combined with analysis can determine the best
solution to a problem. Yet, as a topic itself, matrices are not considered discrete
mathematics. The central questions asked and processes used to solve certain
questions may involve the use of matrices, making the mathematics of matrices
centrally connected to discrete mathematics.

A third category of questions in discrete mathematics involves the idea of
existence. Existence questions ask if there is in fact a solution to a problem.
Design theory, block designs, projective planes, Latin squares and coding theory
are examples of techniques or theories about existence. Once again, I mention
these not because I assume understanding of the ideas but simply to portray the
multiplicity of ideas that cut across the three kinds of counting questions central
to discrete mathematics.

Computer programming and the influence of computer techniques are
also of considerable importance to the ongoing work of discrete mathematics.
Computer programming and new technologies are tied closely to the ongoing

development of the field. The computer has not only provided new avenues for

105

solving and extending the tedious tasks of counting, optimizing, and
determining existence, but computers also have had great influence on the
subject matter of discrete mathematics by way of programming languages and
the applications possible. For example, binary number systems and
programming languages are often considered part of the subject matter of
discrete mathematics and usually are included in most discrete math textbooks.
Computers have also influenced the nature and kind of enumeration,
optimization, and existence questions and their applications to real world
problems as well. In this sense, computer technology has tended to play a kind
of dual role in the ongoing development of the field.

Discrete mathematics is not computer mathematics. Because computers
are highly prominent in the work of discrete mathematics, there is a tendency to
generalize terms. Often, computer related terms and discrete mathematics have
become synonymous. Yet, discrete mathematics is not computer programming
or other related computer work. To the contrary, the mathematics of discrete
mathematics existed far before the influx of computers. Many of the central
ideas have been around for a very long time, long before computer technologies.
At the same time, discrete mathematics would be altered drastically if void of the
influence of computer technology.

My point in trying to delve deeper into what discrete mathematics is and
what underlies discrete mathematics is to illuminate the complexity involved in
sketching even a fairly general View of the central questions, underlying
structures, techniques and ideas used to solve questions in the field of discrete
mathematics. I also have tried to suggest that although the mathematical ideas
such as Venn diagrams, systems of equations, and matrices each are centrally
connected to the ongoing work of discrete mathematics, they themselves are not

discrete mathematics. This would be true of any mathematics.

106

Understanding discrete mathematics implies knowledge of the central
kinds of questions asked, techniques for solving those questions, routines and
patterns mathematicians identify and make use of in that process, as well as
good ideas about the systems that underlay that work and define the boundaries
of the theories and arguments.

Discrete mathematics changes over time, like all mathematics, it is not a
fixed body of knowledge where particular mathematical ideas exist. Rather, it
grows and changes making use of any mathematical idea that advances an
understanding of the central questions asked. In this light, discrete mathematics,
as with any topic in mathematics, involves the familiar and the new, each
shaping the other and progressing the work.

Obviously, describing the nature of discrete mathematics in documents
not even designed for that purpose would be a difficult task. Portraying what
discrete mathematics involves in general categories and descriptions is hard and
can mean many different things to different readers. In chapter five of this work,
the reader will learn how Sandy interpreted the meaning and proposals focusing

on discrete mathematics.

Revisiting flfhe Qnestienu Why Intreduce Discrete Mathematics?

In my interview with Bruce Sagan I encountered a different rationale for
introducing discrete mathematics into the school curriculum not explained in the
documents. He explained that currently there is a big draw toward discrete
mathematics courses in the college curriculum. There is even argument among
mathematicians about introducing college students to discrete mathematics prior
to calculus. The argument stems from concern that many students are, ”weeded
out,” of further mathematical study by way of the traditional calculus sequence.

This limits the number of students continuing beyond elementary level college

107

mathematics. The argument rests on the difficulty students encounter very early
on in calculus with ideas such as delta-epsilon proofs. Proof in this context is a
complex idea often ending the majority of students’ mathematical careers. In
effect, the traditional mathematics sequence of courses acts as a barrier to a wider
range of career choices for students.

Some mathematicians propose that discrete mathematics is a more
promising site for introducing and exploring the notion of mathematical proof
with college-level students. This argument rests on the assumption that students
will find the mathematics of counting with finite sets much easier and therefore,
discrete mathematics may be an easier introduction to the notion of proof. Some
mathematicians claim the earlier proofs in discrete mathematics are much easier
to understand than the delta-epsilon proofs. Removed would be the more
complex ideas of infinity and continuous functions. Finite numbers of objects
and finite outcomes would be easier to understand. Discrete mathematics would
hopefully be more appealing and less threatening to students.

The ongoing arguments about the college curriculum filters into K—12
public schooling as well. If discrete mathematics were to become an alternative
for more and more students, then preliminary work would necessarily begin in
earlier education. The introduction of discrete mathematics into the school
curriculum may be related to the ongoing debates concerning college course
work. Given that mathematicians consider understanding and constructing a
proper mathematical proof one of the most important goals in learning

mathematics, school programs constantly seek avenues for meeting that goal.

h it Ma mati BIntrduedathElmnta Lvl?
I suggested earlier that the documents offer somewhat mixed messages

about whether discrete mathematics ought to be introduced in the elementary

108

grades. In particular, the Curricnlum Standards (1989), focus on discrete
mathematics in sections for grades 9-12. Yet, NCTM, in the 1991 yearbook,
explicitly suggests that K-8 programs should be strengthened by including
discrete mathematics. In chapter three the authors argue that children’s
elementary years provide the foundation for all strands that will be studied in
depth in the upper grades.

Further, Claire Graham writes, ”Discrete mathematics is not a new branch
of mathematics that must be added to the existing curriculum. Rather it is a
collection of topics that most elementary teachers know something about and
almost certainly already teach. These topics include counting techniques, sets,
logic, reasoning, and patterning (iteration an recursion, algorithms, probability,
and networks,” (p. 18). The basis on which the author argues the ideas are
common is not disclosed.

Throughout a 246 page text, there are numerous ideas and suggestions for
introducing discrete mathematics some focusing on the elementary grades.
Sandy did not make use of this text. Later on the reader will learn about Sandy’s
sense making around an example offered in the Curricnlnm Standatds (1989),
aimed at helping secondary teachers think about discrete mathematics in relation
to matrices. Sandy used this example to try to understand a combinatorial
mathematics problem, a topic in discrete mathematics. The illustration she used
involved directed graphs and matrices, ideas Sandy knew very little about.
Unbeknownst to Sandy, in the NCTM yearbook, a similar discussion of these
mathematical ideas appears for elementary teachers (p. 30). Direct attention is
given to how the mathematics of directed graphs can be introduced very early on
in children’s mathematical experiences. Sandy did not access this resource. The
authors offer many examples and illustrations in various sections of the text that

would provide an introduction to discrete mathematics for elementary grades

109

(see p. 42 for a combinatorial counting problem and p. 61 for an example
involving recurrence relations).

In checking the distribution of the yearbook since publishing in 1991, three
printings, totaling 20,390 copies were published. In the progression of sales
across the years, 6000 were sold in the first year, 5000 in the second, 2000 in the
third, 1000 in the fourth, 480 in the fifth, and finally 288 during the 95-96 school
year. There remained 5000 copies in stock. It seems she could have had access.

At the same time, although I did not consider these figures in contrast to
other yearbook sales and therefore do not know if this pattern is typical, it does
seem ironic given NCTM’s position that discrete mathematics should be
introduced into the K-12 curriculum that sales continued to decrease. The
yearbook would seem an essential document for all elementary teachers given
that what is presented in the Curriculum Standards (1989), regarding discrete
mathematics is geared to secondary teachers.

Elementary teachers would have to bring a fairly well-worked sense of the
ideas of discrete mathematics to understand the examples and illustrations.
First, they would have to recognize the examples and illustrations scattered
throughout all the texts as illustrations of discrete mathematics. Most are not
identified as such. From there, teachers would have to bring much in the way of
prerequisite knowledge to see the discrete mathematics in these illustrations.
Paradoxically, what the documents hope teachers to get from reading them, may
also be what teachers would need to bring to understand them.

Further, there remains the problem of locating and formulating an overall
view of what the documents propose. In drawing much attention toward a new
vision of mathematics teaching and learning, basic computational skills
instruction appeared a well-hidden agenda. Rhetorically, it was not a central

reform issue. Recall that only 11 in over 600 pages, in three different documents,

110

devoted direct attention to basic computational skills instruction. Everything
else was scattered in various contexts and forms making it difficult to find and fit
together. Proposals for introducing discrete mathematics were communicated
more directly and openly, yet there were few details and inconsistencies across
the documents. Recall that the Curriculum Standards (1989), proposed discrete
mathematics be included for grades 9-12. At the same time, the NCTM in the
1991 yearbook on discrete mathematics recommended discrete mathematics be
included across the entire curriculum.

In the next two chapters the reader will soon see that the documents were
only one of many resources that the teacher used to understand the proposals for
reforming her teaching of computation and introducing discrete mathematics
into the school mathematics curriculum. The analytic work in this chapter will
hopefully serve as a back-drop for making sense of Sandy's responses to the

proposals she tried to understand and enact in her teaching.

111

CHAPTER 4

LEARNING TO TEACH
BASIC COMPUTATIONAL SKILLS DIFFERENTLY

Introduction

In the first and second chapters of this work I introduced a thoughtful,
enthusiastic teacher. I described Sandy Wise as reform-minded and deeply
committed to teaching mathematics in ways envisioned by policy. I suggested
that she positioned herself to be successful at reforming her teaching. She was
deeply invested and well-supported in her efforts to improve her mathematics
teaching. This chapter tells a story about Sandy’s efforts to change her teaching
of computational skills. In it, I describe how Sandy’s learning about reform—
based teaching evolved and how her learning interacted with how she changed
her teaching. The chapter illuminates how policy fostered new visions of
teaching and how teacher learning fostered new interpretations of policy. The
reader may be surprised at the turn of events.

I begin by examining a particular episode of Sandy’s teaching during the
first year of my observations. The lesson illustrates how Sandy encountered a
puzzling set of circumstances when she changed her teaching of basic
computational skills. As the story develops, the reader will learn about Sandy's
efforts to abandon traditional forms of teaching aimed at the rote-learning of
computational algorithms. From there, I develop a picture of the teaching
practice Sandy created, one she thought attended to conceptual and
computational goals. Toward the end of the story, the reader will learn what

Sandy did to manage the problematic circumstances she encountered. I devote

112

the remainder of the chapter to describing the role and significance of Sandy's

learning on the turn of events.

A Teacher's Responses To
Proposals to Teach Computational Skills Differently

Sandy Wise, a reform minded teacher, was committed to improving her
mathematics teaching. Yet, in that process, an ironic twist emerged. In trying to
teach mathematics more conceptually, Sandy encountered a crossroads in her
teaching of basic computational skills. To illustrate, I focus on an episode of
Sandy’s third-grade teaching during the Spring of the first year of my
observations. I use this lesson to illustrate the tensions that arose in her practice

and Sandy’s interpretations of what was happening.

A Single Mathematics Lessen
Sandy asks the following question: How many chopsticks would we need

if everyone in this room needed them to eat their Chinese food? Sandy explained
that what she really wanted wasn't the answer so much but for students to come
up with strategies for finding the total number of chopsticks needed. The thirty-
two students in Sandy’s classroom seemed familiar with this kind of work.
Immediately they began working in smaller groups of three and four for about
ten minutes and without any direction to do so. Some students talked to each
other while others worked alone even as they sat in groups. Many used paper
and pencil, others relied on calculators. Seemingly unsurprised when Sandy
asked groups to share their strategies aloud, students volunteered a variety of

approaches.

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Joe: We counted for each one at our table and then two for each one at

Sarah's table and two for each one at Tom's table and made a chart, then I

added.

Jen: I added 2 + 2 + 2 + 2 + for everyone.

Adam: We counted each student and there were 32 and you [points to
Sandy] made 33. I then added 33 and 33 because each one gets two

chopsticks.

Sarah: We added 32 and 32 to get 64 and then two more for you.

After each group reported their strategies, Sandy asked one child from each
group to represent the strategy on the chalkboard. She then asked students to
compare the different approaches on the chalkboard. A lively discussion arose.
Several students noticed that Sandy had not been included in the totals. Others
wondered if I, the observer, should be included. Sandy agreed that such
decisions would certainly make a difference in the results, but she did not tell

students what they should do. Jen interrupted:

But, our answers should all be the same.

Sandy: Why is that?

Because we have to get chopsticks for everyone.

Sandy: Okay, I suppose if we were to order our food we would have to let

the restaurant know how many chopsticks we will need. Maybe we
should decide.

Sandy asked students what they thought. One student said that both Sandy and
I should be included because the question indicated that everyone in the room

would be eating food. Sandy asked everyone to indicate if they agreed by vote.

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Students quickly voted in agreement that both Sandy and myself should be
included.

Sandy then asked Joe to come up to the front of the room to explain his
strategy. Joe, pointing to the table he had previously written on the chalkboard,
explained that each name corresponded to the group leader at each table and the

number corresponded to the number of chopsticks needed at that table.

Joe 8
Sarah 16
Tom 12
Jim 18
Jen 10

He then began writing the following addition problem on the board:
8+16+12+18+10=
Joe began adding the numbers in his head. He paired the first two numbers,
added them saying ”24”, and then became stuck. It was at this point that the
lesson began to take a different direction than Sandy had imagined. Instead of
moving the discussion toward comparing strategies as she had hoped, the
discussion began to center on computational work. Sandy, realizing Joe was
stuck, asked the class to help him with the computation. After a few minutes,
several other answers were offered, each different from the other and different
from Joe’s. Sandy then asked each group to work on the computation by helping
each other look for any mistakes they may have made. Sandy walked around the
room trying to help students work on the addition problem Joe had developed.
The lesson, on that day, never found its way back to comparing and contrasting

the strategies on the board (Fieldnotes, 5/92).

115

Later, in an interview, Sandy explained that as she walked around the
room she noticed that many students could not add the numbers correctly. She
explained that whether students were using their own strategies or attempting
the traditional addition algorithm, they made many errors. Sandy reported that
she helped students mostly by demonstrating the traditional addition algorithm
to compute the problem (Interview, 5/92).

This lesson represented only one of many occasions where Sandy found
herself worrying about students’ computational proficiency. The same
circumstances occurred so often that Sandy decided that students’ lack of
computational skill was actually getting in the way of her larger more conceptual
and reform-oriented goals. For instance, in this lesson, Sandy had hoped that
students would explore relationships among the various computational
strategies and that ultimately students would notice connections between
strategies involving addition and those involving multiplication. She hoped
students might see that adding 2 over and over, thirty-two times, was the same
as 32 times 2, and that both strategies give the same result. She also hoped
students would begin to question the efficiency of their strategies and see
benefits to the notion of grouping and multiplication.

Yet, Sandy’s goals in this lesson were never realized. Even though she
returned to the problem the very next day, many students remained stuck in
various computational problems related to the different strategies. Sandy
commented in an interview, ”you know, most of these kids I really think could
tell me that multiplying 32 times two means 32 groups of two, and I think they
could draw me a picture of it, and show me that it also means adding two, 32
times. But, I don’t think they could necessarily add it correctly.” She continued,
”maybe people tended to overreact, and went a little too far with not wanting

kid’s to memorize facts and algorithms anymore” (Interview, 5/92).

116

Sandy no longer stressed traditional algorithms in her teaching as she had
done in previous years. Instead of drilling students on traditional computational
algorithms, she presented the algorithms to students once and only at the end of
more conceptual orientations into addition, subtraction, and multiplication. At
this point in her career, Sandy worried less about students memorizing and
practicing the traditional algorithms and instead emphasized developing
conceptual understanding. Only after many encounters with the circumstances
above did she wonder, ”Maybe in approaching things more conceptually and
through problem solving, I am dropping the basic math skills from my teaching”
(Interview, 5/ 92).

Confronted by this dilemma, Sandy began to wonder about the role and
purpose of basic computational skills in relationship to what she hoped students
learn about mathematics. Sandy found no easy answers. She knew both
conceptual and computational goals were valued. Yet, she pondered, should
she, ”stop in the middle of the lesson and review the traditional algorithm or
more simply try not to introduce problems with more complicated
computational problems ” (Interview, 5/92). The latter seemed impossible given
the various strategies students were coming up with. She also seemed puzzled
about the enormous amount of time students needed when they were left to
themselves to compute an answer. There were no simple solutions.

Sandy searched the reform documents for help. She read, ”students will
invent and refine procedures themselves,” and that ”students should compare
different approaches and algorithms for obtaining the same results, evaluating
the strengths and weaknesses of each” (California State Department of
Education, 1992, p. 56, 57). Further Sandy wanted students to, ”understand why
the approach they choose makes sense for the problem they are solving, and if it

makes sense, they will develop fluency” (p. 56).

117

Sandy focused on the idea that,

”traditional teaching emphasizing practice in manipulating
expressions and practicing algorithms as a precursor to solving
problems ignores the fact that knowledge often emerges from the
problems. This suggests that instead of the expectation that skill in
computation should precede work with problems, experience with

problems helps develop the ability to compute (NCTM, 1989, p. 9).

Sandy tried to create a practice she thought fit with this idea. She began her
lessons with a mathematical problem that offered many different possibilities for
students. Yet, as she developed a more conceptual and problem solving
approach to students' learning about the four operations, she realized there were
significant problems. While problem solving experiences like the chopsticks
problem offered students an opportunity to learn about computation differently
than traditional practices, they also involved doing the computations. Sandy, a
teacher interested in developing students’ conceptual understanding and
computational skills, confronted a problem of what to do when students’
computations didn’t get them the desired answers.

Computational skills and conceptual goals seemed at odds to Sandy. She
wasn’t sure about the relationship between them. She began questioning the role
of speed, accuracy, memorization and practice in learning to compute. And
although she wanted to believe her students would develop fluency with
computation as they engaged in more problem solving and conceptually
oriented work, her teaching experiences began to suggest otherwise.

Over the next year Sandy worked on understanding this relationship

better. The interpretations and decisions she made to change her teaching of

118

basic computational skill the next school year may seem surprising. To

understand how Sandy arrived at the decisions she made, one would first need
to consider Sandy’s practice on a wider scale. In the next section, I explore

I Sandy’s mathematics teaching across the school day and across the 1991-1992

school year. My aim is to provide a picture of how Sandy attended to students’

computational skill learning across the school day and year.

Mathematics Teaching en e Wider Scale

When I first began visiting Sandy's classroom, I was struck in general by
the amount of time she devoted to mathematics instruction on a single day.
Nearly two hours of each school day, usually in 20 to 40 minute chunks, were
devoted to learning mathematics. More modal mathematics lessons occur only
once in a single school day and for much less than two hours. Consider a typical

day. On January 21, 1992, Sandy posted the following schedule for students.

Daily Schedule

8:00 Morning Business

8:20 Mental Math

8:40 Good Morning Meeting
9:00 Blockout

9:20 Fair Shares and Division
9:50 Probability

10:00 Recess

10:30 Language Arts

11:30 Lunch

12:00 Probability

12:45 Class Meeting

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1:15 Oral Language

1:50 Prep to go home

1:55 Dismissed

On this particular day, there were five different time slots for

mathematics, two hours and five minutes. During ”Mental Math”, Sandy
pointed at random to single numerical digits and operation signs hanging by
string from the ceiling. Students were to silently follow the numerical sentence
Sandy created. When called upon, they were to provide the correct answer. This
activity Sandy designed to provide students with practice on mental addition,
subtraction, and multiplication of two and three step problems. The activity
lasted for about 20 minutes, as students computed in their heads problems such
as 4 + 8 x 3 =. Paper-pencil and calculators were not allowed. No attention was
given to the standard order of operations like, ”Do multiplication before
addition.” Instead, Sandy’s goal was for students to work step by step as the
problem was created. So, for example in the problem 4 + 8 x 3, since 4 + 8 was
created first, it would be added first. The multiplication would be done second.
Sandy would point to the 4 hanging from the ceiling, the plus sign, the 8, the
times sign, and the 3. She would then point to the equal sign and wait for
students to raise their hands. Students almost always gave the correct answer.
Sandy would ask students whether everyone agreed with the answer offered.
Students almost always agreed. Overall, there was little discussion. Mostly,
students shook their heads up and down and recited back the steps of the
problem. On a few occasions students disagreed with a given answer. Usually
Sandy would ask other students for the answer, Typically it was correct. There
were no discussions of alternative strategies or how numbers might be combined

or broken down in ways convenient for computing mentally.

120

”Blockout,” was similar to Bingo. Students placed marker chips in a
position on an axis system after Sandy called out a horizontal and vertical
number. The activity was designed to familiarize students with the process of
locating points on an x-y coordinate plane. Sandy thought of the activity as a
prerequisite to the geometry and algebra units she would teach later. Like
Mental Math, Blockout was intended to develop facility, only in this case, with
locating points on a coordinate system. It lasted approximately twenty minutes.

In between Mental Math and Blockout a class meeting was held. A
number of issues arose. Sandy confronted students on their lack of attention
during a lesson involving a guest speaker the previous day. She asked students
to think about what they might have done differently and whether they want to
have speakers in the future. Students also raised concerns. They wondered why
they were not permitted in the gym during lunch anymore. Sandy explained the
construction going on and indicated when the gym might be available again.
The meeting lasted about 20 minutes and did not involve any mathematics
instruction.

”Fair Shares and Division,” the third math lesson of the day, focused on
the idea of division and that of equal-distribution or fair-sharing. The activity,
approximately 30 minutes in length, involved students dividing packs of
lifesavers among 4 and later 8 friends. Students distributed the lifesavers among
friends and kept track of how many each friend received and the number left
over.22 No paper-pencil calculations were made. Instead, Sandy hoped students
would link the notion of fair-sharing to their ideas about division. For example,
she expected students would say I divided 221ifesavers fairly among four

friends, each getting five lifesavers and a remainder of 2.

 

22A closer analysis of this activity appears later in this chapter.

121

”Probability” had two time-slots, one before and one after lunch. Before
lunch students collected data by pulling and replacing a lifesavers from a brown
paper bag. The data was recorded on the chalkboard by placing hash marks
under the letters r (red), 0 (orange), y (yellow), g (green), and w (white). In the
afternoon segment students were asked to make and explain their predictions of
how many of each color lifesaver there was in the bag. Sandy also asked
students whether the data collected supported the actual outcome of colors in the
bag. Both segments took 55 minutes.

Fair Shares and Probability focused on two mathematical ideas, equal
distribution and chance. Calculation was not emphasized. Sandy was much
more interested in how students reasoned about the problems. ”Probability”
also offered students an opportunity to justify their answers by providing
evidence (Fieldnotes, 1 /92). Together the two lessons took one hour and twenty-
five minutes of the two hour and five minutes devoted to mathematics.

During the other parts of the afternoon students were involved in
language arts lessons, lunch, recess and another class meeting. None of these
activities involved mathematics instruction on that day.

Sandy’s practice was much like a patchwork of practices, each element
offering a different contribution to the overall goal of becoming what she called,
”mathematically powerful.” Coherence across lessons on any single day did not
appear to be an important goal to Sandy. The mathematics lessons across the
day were not connected in substantive ways. They did not build or extend
mathematical ideas. Instead, they involved unrelated mathematical goals. With
the exception of the two segments on probability, each lesson on this day had
entirely different aims. Several lessons focused on computational skill, but each
one emphasized a different skill. Other lessons focused on developing

mathematical ideas such as division and chance on this day.

122

Lessons did connect substantively across the days of the week and across
the month. For example, the probability lessons on January 21 were connected to
a lesson on January 22. On the next school day Sandy introduced a very similar
probability problem involving the ideas of prediction and chance. She planned
to extend students’ understanding of prediction and chance across the entire
month of January. A little later in this section, I will show how Sandy’s master
plan was aimed at developing mathematical connections across weeks and
months according to how Sandy believed the lessons extended students’ learning
about specific mathematical ideas.

These data suggest that Sandy’s mathematics teaching was separated
along two lines. Statistically, Mental Math and Blockout, were skill-oriented
activities, making up a little more than 30% of the instructional time on that day.
About half of that time focused on mental calculations involving addition,
subtraction and multiplication. Thus, about 15% of the instructional day on
mathematics focused on basic computational skills. In contrast, one hour and
twenty-five minutes, or approximately 70% of the time was spent on
investigative activities or exploring mathematical ideas. During these activities,
problem solving strategies and conceptual orientations to mathematical ideas
were of central concern whereas computation remained in the background.

Sandy confirmed in an interview that she tried to focus 80% to 90% of the
total instructional time in mathematics on conceptual understanding and
problem solving. She indicated that 10% to 20% was devoted to students
developing computational skills and the proficiency of those skills. Sandy
claimed she emphasized mostly mental arithmetic and mental estimates instead
of paper-pencil calculations. She explained she devoted only one week each on
the traditional algorithms, showing students the steps and requiring they

practice them. The changes represented a significant departure from years past

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where Sandy spent months repeating and reviewing instruction of the traditional
algorithms for computational purposes (Interview 1 /92).

The changes described here had accumulated over several years. At this
stage Sandy was devoting a little more than four weeks of the entire school year
to teaching the procedural steps of the traditional algorithms (Interview, 1 /92).
She explained that most of what she provided for learning basic computational
skills emphasized mental calculations and estimates. My observations on
January 21, 22, May 5 and 6 fit with Sandy’s estimates. I observed no lessons
involving rote-memorization or practice of any traditional algorithms during my
visits that year.

Sandy’s Master Plan also illustrates the two-dimensional nature of her
mathematics teaching. Yet, it suggests there is a third dimension. At the
beginning of each school year Sandy constructed an overall yearly plan for her
mathematics teaching. Below is Sandy’s plan for the 1991-92 school year.23

Sandy explained she developed this plan to use in place of the district's
mandated curriculum guide and textbook. She describes the plan as, "a really
healthy road map, I know exactly where I am going, and I no longer worry
whether I am hitting the skills, the concepts, or the global outcomes, it's all
there,” (Interview, 1 / 92).

”Math” resembles a more traditional third-grade curriculum plan.
Initially, there is a pre-assessment of students' computational skills, a test on the

basic number facts and the traditional addition and subtraction algorithms. Each

 

231 have included only those elements of the yearly plan pertaining to mathematics. Sandy
constructed a master plan with each subject area represented.

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125

month thereafter focuses on a particular aspect of computation: number, place
value, operations and patterns in computations. There is a lengthy section on
regrouping. There is a unit on calculators and a post assessment of student’s
computational skill. This strand of instruction emphasizes some memorization
of the traditional algorithms, some drill and practice on the basic facts and
algorithms, and timed tests. Attention is also given to place value and bases
other than ten (Interview, 1 / 92).

The second row relates more directly to the state’s framework. Each
month corresponds to a content strand identified in the framework. The 1985
Framewerk identified seven strands: number, pattern, logic, functions, statistics
and probability, geometry, measurement and algebra.24 Sandy's plan
corresponds to these but includes an additional strand titled fractions which she
calls a unit. On any given instructional day, there may be as many as three
activities relating directly to the content strand for that month. These activities
are of the investigative, hands-on, learning by doing type. Sandy uses materials
by Project AIMS, Marilyn Burns, Family Math and many others to draw from.

”Problem Solving” was not represented in Sandy's teaching on January 21,
1992. The emphasis here is on developing student’s problem solving strategies.
Activities are intended to offer students a chance to work on developing
strategies such as guess and test, look for a pattern, formulating the question,
look for a counter-example, using a table, drawing a picture, and trying an easier
problem. Sandy selects activities mostly from Project AIMS , Family Math and
Marilyn Burns to provide students these experiences?-5 The months that are left

blank are open as Sandy continues to search for new materials to use.

 

24In the 1991 version an additional strand, Discrete Mathematics, was added. I take up Sandy's
learning of discrete mathematics and her representation of these ideas in practice in the next
chapter.

25Project AIMS plays a significant role in Sandy's ideas and practices related to mathematical

126

Sandy’s master plan organizes her mathematics teaching into three
dimensions. One dimension takes up computational goals, another involves
investigations into mathematical ideas and corresponds to the content strands of
the framework, and the third focuses on problem solving strategies. Lessons
across the school day, week, and month correspond to one or the other of each
dimension. There are connections among ideas within each dimension. For
example, the strand dimension provides experiences that aim to connect division
and the idea of average. Each are linked by the idea of fair-sharing (Interview,
5/92). The computational skill dimension is organized across the year by level of
difficulty of each skill as well as by operation. For instance, three step addition
problems follow two step addition problems.

There are few connections across dimensions. The more conceptual
explorations of division in Sandy’s practice on January 21, for example, were not
linked to any experiences students would have practicing division mentally.
Thus, the dimensions are not connected in substantive ways.26

For Sandy, the three dimensions represent what children need to know
and be able to do mathematically (Interview, 5/92). Instructionally, each
dimension offered students various pieces of mathematical knowledge to be
successful in mathematics. Recall that Sandy devotes somewhere around 20-30%
to the Math dimension and 70-80% to the content strands and problem solving

dimensions. Experiences corresponding to the Math dimension emphasize

 

knowing and doing in the elementary classroom. I discuss the relationship Sandy has with
Project AIMS and her use of AIMS materials in more detail later in this chapter as well as in other
places throughout this work.

26This may have seemed not necessary given Sandy’s view of mathematics. Because she viewed
mathematics as an enterprise in the service of other sciences, she may not see as well the benefit
of understanding ideas deeply. Instead ideas had to be understood well enough to put them to
use for solving problems or making sense of the world. If one could compute and also find
averages, that was enough . I take this up further in the next section involving Sandy’s
knowledge and beliefs about mathematics.

127

practice with mental calculations, mental estimates and routine algorithms as
well as explorations into ideas such as place value, various bases and the four
operations. Skill oriented lessons are taught in small chunks separate from
problem solving and content strand dimensions. In contrast, students’
experiences corresponding to the strand dimension are more open-ended,
involve little if any practice, and invite students to consider mathematical ideas
on a basis other than procedurally. Activities often begin by describing a context
that lead to a problem solving opportunity. Students work in small and large
groups, explorations often continue over several days, and students are often
asked to justify their thinking by providing rationales for their work.

Although the dimensions for the most part are separate, there are a few
activities that were multifaceted in purpose. For example, the chopsticks
problem not only was intended to provide an opportunity for students to
develop and use their own ideas to solve a problem, it also was intended to foster
understandings of the relationship between addition and multiplication. In this
sense, facility problem solving and conceptual understanding were intertwined.

It would be plausible to argue that students’ mathematical experiences in
Sandy’s classroom that year were quite different from modal practice.
Instructional time overall was dominated by the investigations experience.
Experiences with rote-memorization and practice of the traditional algorithms
was much less than most conventionally taught classrooms. And although
students were asked to practice computations, more often the experience
involved making mental calculations and estimates rather than paper-pencil drill
and practice of the traditional algorithms. Sandy’s practice during the first year
of my observations in many ways reflected a very different approach to teaching
and learning mathematics and in particular there were significant differences

surrounding instruction of basic computational skills.

128

Sandy had developed a practice she thought met the challenge to teach for
conceptual understanding and computational proficiency. Yet, as the lesson I
used to open this chapter illustrates, Sandy also realized that the changes she
made placed computational goals in jeopardy. Further, she interpreted students’
lack of facility and skill at computing to interfere with the exploration into more
conceptual, idea-oriented side of mathematics. As Sandy concluded the 1991-
1992 school year, she worried she had misread the state's goals. Sandy

wondered What she was doing wrong (Interview 5/92).

Th Next ch 1 Year:
Respending te Preblems In Practice
One year later, it was very obvious that Sandy had changed her teaching
significantly. Yet, the changes I observed seemed ironic given Sandy’s
commitment to understanding and reforming her teaching in ways characterized
by the policies. In an interview Sandy revealed there was a dramatic shift in her
thinking and practices related to computational skills. And, consequently, she
reinstated traditional teaching practices aimed at the rote-learning of basic
number facts and traditional algorithms, only now, more pervasive and
extensive than ever before. Sandy was now devoting as much as 30-50% of
instructional time on mathematics to developing pencil-paper algorithms for
computing. Her daily schedule reflected these changes by a new title,
”Automaticity/Q7 The following schedule is for January 10, 1993.

 

27The term Automaticity surfaced initially for Sandy in one of her doctoral courses over the
summer. The term itself indicates a significant change in Sandy's thoughts about what ought to
be emphasized in relation to computational goals when comparing one year to the next.

129

Daily Schedule - Year 2

8:00 Morning Business

8:20 Citizenship Discussion

8:50 Problem Solving Strategies

8:55 Polar Pizza Challenge

9:05 Spelling Test

9:15 Automaticity - Mathematics

9:45 Book Awards

9:50 Automaticity — Quick Read

9:55 Recess

10:20 Dear Reports

10:30 Mail Call (Project Aims Mathematics Activity)
10:45 Science

11:30 Lunch

12:25 Continuation Mail Call

12:40 Congratulations to Polar Pizza Party
12:45 Oral Language

1:05 Afternoon Recess

1:15 Zip Around (Speaking and Listening Language Arts)
1:20 A Square Deal (Problem Solving Activity)
1:40 Alleyway (Math Activity Project AIMS)
1:50 Prepare to go Home

1:55 Dismissed

Like the previous year, several time-slots were devoted to mathematics
instruction. On this day, there were eight. ”Problem Solving Strategies,” the first

mathematics activity, involved a brief discussion of how to arrange objects such

130

as marshmallows and hands in a way to make measures more precise. This
discussion was one of a series across several days involving an ongoing problem-
solving activity aimed at understanding measuring.

”Polar Pizza Challenge,” involved a timed multiplication-fact test. The
timer was set and frantically student’s answered as many of the single-digit
multiplication facts on the page. When finished, students clapped to signal to
Sandy that they were finished. Sandy would record their time on the overhead
projector. In turn, the student recorded the time at the top of their page and
turned in the worksheet. Later in the day papers were graded and times were
recorded in Sandy’s grade book. Students waited to hear if their score was a
perfect one. If so, and their recorded time was under two minutes, students were
invited to the Polar Pizza Party. Sandy compiled results and announced in the
afternoon segment, ”Congratulations to Polar Pizza Party,” the names of those
students who succeeded. A round of applause was given to each student
awarded the coupon admitting them to the party. Accumulated coupons could
be used for other rewards such as fifteen minutes of free-time. The two
instructional segments took fifteen minutes that day.

”Automaticity-Mathematics” was next on the schedule. On this day the
lesson involved practicing routine addition and subtraction problems involving
numbers with many zeros. I examine this lesson in more detail later in this
section. For now, it suffices to say that this lesson represented a very traditional,
procedural approach to learning computational algorithms. Eight computational
problems, void of any context, were presented on the overhead projector. Sandy
instructed all students to follow a traditional step-by-step procedure regardless
of the numbers involved.

At 10:30, ”Mail Call” was next. Recall the investigative-type lessons I

described as dominating Sandy’s mathematics instruction the previous year.

131

This lesson, a Project AIMS activity, was of this kind. Students collected and
recorded a set of data involving the number of mail items received over a three
day period at their home. Using bear counters to represent mail items, students
explored the mathematical idea of average by equally distributing the counters
across the three days. Discussion focused on what it would mean to equally
distribute the items, whether the total is different after distribution, and language '
that linked equal distribution to the word average. The lesson, broken into two
segments, one before and one after lunch, totaled 30 minutes.

”Square Deal,” involved students trying to place the digits 0 through 9 at
the vertices of embedded squares so that the numbers of connecting vertices total
20. Sandy illustrated a guess and check method for accomplishing the task and
recommended students search for other strategies as well. This activity took
twenty minutes on this day but would be continued over several days. It was
followed by the introduction into a Project AIMS activity titled ”Alleyway,”
which involved students collecting data on how many puffs of air it took to
move cotton balls across various surfaces such as students' desk tops. Sandy
organized this activity at the end of the school day because of the commotion it
created in the classroom. Sandy planned to make use of the data the next
instructional day focusing on the ideas of mean median and mode.

In sorting through the various lessons on the day, Alleyway and Mail Call
were more hands-on, learning by doing activities focusing on the idea of average.
Together, the two activities totaled 40 minutes. Square Deal, lasting 20 minutes,
typified the problem solving dimension of Sandy’s practice. Combined with the
short problem solving discussion that went on earlier in the day, the lessons
totaled 25 minutes. The remaining activities focused on recognizing and
developing students' progress with basic fact acquisition and traditional

computing algorithms. These lessons made up 45 minutes or approximately 40%

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of the instructional time for mathematics on that day. Sandy confirmed in an
interview that she had decided to increase the time students would spend
focusing on learning traditional computing algorithms. In effect, across the 1992-
1993 academic year, this decision decreased her emphasis on conceptual and
problem solving goals (Interview, 5/ 93).

Perhaps more important to note is the different nature of the experiences
students had for learning about computation than the previous year. The lessons
no longer emphasized mental calculations and estimation. Instead, emphasis
was on memorizing, practicing and reciting basic number facts and traditional
computing algorithms. .

To summarize the picture thus far, Sandy's practice during the second
year of my observations was strikingly different. It reflected mene time on basic
computational skill learning and 1% time on activities emphasizing conceptual
understanding and problem solving. In addition, it reflected mm traditional
computational goals and lgs alternative skills for computing such as student’s
strategies, mental calculations and estimation. I turn now to look more closely at
what Sandy referred to as an automaticity lesson. I describe the lesson on

January 10, 1993 at 9:15 in detail.

An Autematicing Lessen

Automaticity was not a word that students the previous year had
experience with. This year, automaticity carried great meaning and significance
for students. It not only represented a certain kind of activity students would
engage in, but, it also represented great achievement and reward. Students
understood that they would put everything away off their desks, except for a
pencil. They knew speed and accuracy were key factors. And, there was a great

sense of excitement and privilege associated to automaticity lessons.

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Automaticity also was associated to reading instruction. ”Automaticity-
Quick Read,” at 9:50, involved a speed test in reading. Students understood that
automaticity had a great deal to do with speed and accuracy.

Automaticity in mathematics on January 10, 1993, involved a speed test. I
observed the following.

Sandy turns the overhead projector on. The following eight problems are

displayed under the title Automaticity.

1. 1000 3. 10,000 5. 75 7. 497
+3976 - 967;} x4 697

+ 119

2. 100 4. 11,496 6. 20,000 8. 700
x e + 14.697 - 10,201 400
ill!)

Turning on the projector was the signal for students to begin. Students
worked quickly. When the problems were completed, students raised their arms
to let Sandy know they had finished. Sandy recorded each students' time on the
overhead. In turn, the student recorded their time at the top of their paper. The
paper was then turned over and students waited quietly until everyone had
finished. Although desks were arranged in groups of four facing each other,
each student worked silently alone.

Once everyone had completed the exercise, students traded papers and
graded each other's answers. Sandy collected and filed the papers. Later she
recorded the number right and the time for each student in her grade book.
After all papers had been collected Sandy went over every step of each problem
and requested that students recite in unison as she wrote down the appropriate

numbers on the overhead for everyone to follow.

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For example: 3. 10,000
-9,e7S
Sandy placed a slash mark through the first zero at the right and everyone
joined in saying:
"slash, burn it off, regroup, slash, burn it off, regroup,
slash, burn it off regroup, slash burn it off, regroup."

Sandy continued placing slash marks through the rest of the zeros until
she came to the zero in the thousands place. She then placed a 1 above the 0 in
the ones place and a 9 above the other three 0's. She asked, "Have we done any
subtraction?" Students chimed in loudly "no!” Sandy continued, "That's right, all
we have done is regroup to reorganize this number. Remember the value of the
top number is still the same." Students chimed in by signal, "Ten minus three
equal seven, nine minus seven equals two, nine minus six equals three, and nine
minus nine equals zero, zero minus zero equals zero." Sandy wrote each number
as students recited. Then she asked, "Is it reasonable that 10,000 minus
approximately 9,700 is about 300?" Everyone answered, "Yes!"

Sandy’s approach for each of the eight problems is similar. All were
computed using a traditional algorithm. Students recited each step. Problem
five, 75 x 4 , was worked in two different ways. First, the two numbers were
multiplied using the traditional multiplication algorithm. Then, Sandy asked if
there might be an alternative way to work the problem. One student suggested,

"You could add 75 four times." Sandy wrote,

75
75
75
i7_5

Students recited back, "Five plus five plus five plus five is twenty, put down the

zero and carry the two." Then, "Seven plus seven plus seven plus seven is

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twenty-eight, add the two and you get thirty." Sandy wrote the number 300
under the problem. She then asked students to compare both answers, "Does
everyone agree the two ways produce the same answer?" Student's responded in
unison, "Yes!"

An automaticity lesson represented a significant change in Sandy’s
thought and teaching practice. It was not an idea that Sandy returned to when
tensions arose in her practice. Instead, it represented new learning. Sandy
explained that the purpose of the above lesson was to promote automatic
computational skill for the purposes of, ”becoming so proficient that students
can actually do it without thinking," (Interview, 1 / 93). Each and every
automaticity lesson that year was intended to support that goal. Emphasis was
on getting right answers in the least amount of time. The problems were not
situated in any problem solving contexts nor were they connected to any other
dimension of Sandy’s mathematics teaching.

Sandy's practice represented what most reformers would criticize. The
Curriculum Standards (1989), for instance, argues that decreased attention
should be given to complex paper-pencil computations, the isolated treatment of
those algorithms, and rounding for estimating (p. 21). On p. 46, ”it is
inconsistent with the Standards to isolate paper and pencil procedures by
focusing on them for an extended time prior to the introduction of other
computing methods: this traditional practice suggests to children that computing
means using paper-and pencil methods.” And further, ”Instruction should
emphasize meaningful development of these procedures not speed of
processing.” And finally on p. 231, ”it is essential that instructional programs
provide opportunities for students to generate procedures. Such opportunities
should dispel the belief that procedures are predetermined sequences of steps

handed down by some authority.”

136

The reform documents paint a very different picture than Sandy's practice.

Exploratory experiences in preparation for paper and pencil
computation give children the opportunity to develop
underlying concepts related to partitioning number
operations on the parts and combining the results. Many
such experiences can be provided in the context of using
place value materials, computing mentally, or performing
computational estimation. Only after these ideas are
carefully linked to paper and pencil procedures is it
appropriate to devote time to developing proficiency by

providing practice. (NCTM, 1991, p. 47)

The new Framewerk (1992), suggests that, ”Depth is to be valued over pace so
that the presentation of a computational procedure can be delayed until students
need it and meaningful examples and motivation can be provided before the
algorithm is presented. Also to be valued is the critical use of alternative
algorithms.”

Another suggestion focuses on the relationship with the conceptual

underpinnings of procedures. For instance,

”A strong conceptual framework also provides anchoring
for skills acquisition. Skills can be acquired in ways that
make sense to children and in ways that result in more
effective learning.” Further, ”it also means relating this
knowledge to the learning of skills by establishing

relationships between the conceptual and procedural aspects

137

of tasks. The time required to build an adequate conceptual
base should cause educators to rethink when children are
expected to demonstrate a master of complex skills (NCTM,

1989, p. 17).

The reform documents suggest that students opportunity to consistently
make decisions about the most efficient means for computing is of primary
importance. Sandy’s lesson limits students to practicing the traditional
computational procedures. Proposals suggest that computational goals should
be situated in contexts that give rise to computations. Further, they suggest the
numbers in the problems should be realistic given the problem at hand. Sandy’s
practice isolates computational goals. On January 10, 1993, she focused on
practicing computational procedures involving numbers with many zeros,
numbers that students at this age are not likely to encounter. Under the new
guidelines this would not be appropriate unless situations students encountered
required it. The contrasts are striking.

Yet, recall that Sandy’s practice had changed significantly from the
previous year. She left the 1991-1992, school year frustrated. She knew she had
to rethink her ideas about computation. That summer Sandy reconsidered her
teaching and put a new plan into action. She identified a set of computational
skills she believed to be essential for the third grade curriculum. Using the
district’s curriculum guides and the state’s standardized tests as guides, Sandy

formulated the following:

"single digit addition and multiplication facts (0-10)

138

*mental arithmetic of two operations and single digits (3 x 6 + 5)
*traditional algorithms for addition, subtraction, multiplication and
division (multiple digits)

*traditional addition and multiplication algorithms with carrying
*traditional subtraction algorithm with borrowing and regrouping

*reasonableness of answers using rounding and mental estimates

She designed automaticity lessons to foster memorization of computational
procedures and proficiency with those skills. She developed worksheets for
timed-tests and a way to motivate students' speed and accuracy with those skills.
Sandy offered a reward, participation in a pizza party. And although she
continued to incorporate lessons that emphasized mental calculations and
estimation, they occurred less often and were shorter in length (Interview, 5/ 93).
Automaticity now dominated Sandy’s instruction of basic computational skills.
The changes Sandy made presents a puzzle for policymakers. Amid
reforms that take aim at traditional practices, a reform-minded teacher ultimately
emphasizes traditional forms of practice even more. How can this happen? We
have a teacher deeply invested in the reforms and yet becomes committed more
than ever to developing students’ proficiency with traditional computing
algorithms. For most, Sandy decisions would mark a clear turning away from
the visions in policies. Yet, interestingly enough, for Sandy they represent a
closer approximation of what reformers’ may have hoped for. My analysis in the

next section provides some insight into how this could happen.
Reinterpreting the State's Goals

In formulating my analysis of Sandy's learning and the affect of her

learning on her responses to the proposals to change her teaching of

139

computation, I returned to the theoretical framework underlying this
dissertation. The ”policy as pedagogy” frame views the relationship between
instructional policies and teaching practice centrally as matters involving
teaching and learning. Thus, I examined what Sandy learned and the contexts
she used to help her understand what the state suggested she change in
relationship to computational skills teaching. Further, I examined whether and
how her learning experiences impacted the changes she made in her teaching as
well as her interpretations of policy. The analyses illuminates that Sandy’s
learning played a significant role in responding to the state's efforts to reform
mathematics teaching. It suggests that what Sandy learned, in part, fostered the
exact ideas and practices criticized most in the reform documents. And further

that Sandy's learning experiences fostered a new interpretation of policy.

Facters Affecting the Teacher's Intecpretatien

Interestingly, Sandy did not create this mix of ”old” and ”new” practices
because she somehow turned away from policymakers' agenda or gave up on
reforming her teaching. Quite the contrary. Sandy formulated her direction in
an effort to improve her teaching and respond more effectively to the state's
goals. To understand how, I look carefully at what Sandy considered for
changing her teaching of basic computational skills. Sandy took extraordinary
steps to help her understand what she thought would be important for creating
the kind of teaching described in the framework. For example, Sandy called
individuals at the state department and arranged for meetings specifically to
hear their views of what was suggested. She arranged for coursework as part of
a doctoral program to help her understand better issues in mathematics
education. Below, I describe the various contexts and ideas Sandy considered as

she formulated her ideas about computational skills instruction.

140

The Practice ef Teaching

Earlier, I described a number of tensions that arose in Sandy’s practice
during the 91-92 school year. Sandy’s interpretation of those tensions involved a
reconsideration of the traditional computational curriculum and how it fit with
policymakers' ideas for change. For example, she struggled with whether she
should focus attention on the traditional algorithms and if so, how much. She
wondered whether speed and accuracy with computing remained a valued goal.
During that year, Sandy believed her students had developed conceptual
understandings of addition, subtraction, multiplication and division. But she
began to worry whether they were developing an accompanying facility with
computation. Sandy explained, "They hold on to the concept across the year but
they can't do the computations just right," and ”They can explain multiplication
conceptually as repeated addition or grouping of the same number, but they
can't multiply," (Interview, 1 / 93).

Sandy offered this illustration in an interview. A student suggested
during a lesson that 75 X 4 is the same as adding 75 + 75 + 75 + 75. Yet, when he
began adding the numbers to get the answer for comparison, he became stuck.
Sandy argued, "my students can show what 75 x 4 means visually using
diagrams, pictures and manipulatives, but later in the school year when routine
addition and multiplication problems are part of problem solving situations, they
get lost, they can’t do it." She explained further, "The outcome of finding the
right answer to 75 + 75 + 75 + 75 or 75 x 4 just wasn't happening.” She
continued, "I found that most kids could tell me four groups of 75 is the same as
adding 75 four times, but they could not necessarily add it. And, I found that I

was just dropping those basic math skills" (Interview, 1 / 93).

141

Sandy stumbled on a big issue. She was circling around a question about
the relationship between conceptual understanding and procedural knowledge.
By the close of the school year Sandy had concluded that, "understanding
mathematical ideas conceptually did not guarantee fluency with computing
skills across the school year." Sandy argued, "to know what multiplication
means or why the algorithms work does not necessarily mean one can multiply

efficiently and accurately" (Interview, 1 / 93). She explained:

"Suppose later in the year after multiplication has been
taught the kids are involved in a problem situation where
they need to add quite a few numbers together to solve
maybe only part of the problem. It is my goal that they
can see and look for likeness in numbers when adding so
that they can use multiplication. If they do, I know they
have internalized why multiplication is useful to us. So,
at that point they are multiplying for example 6 x 12. If
they can do this accurately and fairly quickly then we can
concentrate on the problem at hand. If not, then we are
stuck. I was finding that my kids knew things like
adding 6 twelve's is the same as multiplying 6 x 12 but
whether they tried to add or multiply to compute, too
many were stuck. Then we had trouble staying in the

problem."

Sandy learned from changing her practice that students will often get
stuck on fairly simple and routine computations inside problem solving

situations. She realized that students’ lack of facility with computing interferes

142

with their progress into the more conceptual and reasoning sides of mathematics.
Sandy’s hunches were confirmed when her students did not fair well on the state
exams that year. Sandy learned that basic computational skills can act as a
barrier to conceptual and problem solving goals. Ultimately, Sandy decided that
a more conceptually focused curriculum does not necessarily support student’s

proficiency with basic computational skills.

Decteral Studies

Sandy learned a great deal over the summer. Because Sandy left the 1991-
1992 school year worried about how computational goals might also be met
within a more conceptually and problem-oriented practice, she searched out
circumstances that might help her think about these issues. The interviews
suggest that automaticity was at the forefront of Sandy’s mind. She described
how she came across the notion of automaticity in one of her doctoral courses.
The course, a curriculum course, was not focused on issues surrounding
computational goals or the mathematics framework, but it did focus on questions
about curriculum in the context of reform. Sandy explained that the instructor
distributed an article by Benjamin Bloom in entitled The Hands and Feet ef
Ce_ni_us, (1986), for the purposes of exploring historical perspectives surrounding
the elementary school curriculum. She described the study to analyze experts in
their fields, and found that in each field, mathematics included, there were
desirable skills that experts learned to the point of mastery. The notion
automaticity referred to the idea that the execution of these skills could be done
without conscious thought so that other more complicated kinds of thinking
could go on as well.

Something important happened. Sandy connected Bloom’s ideas about

automaticity to her observations of students in her classroom. She explained,

143

"I found my students do a phenomenal job with conceptual
analysis but I couldn't figure out why for instance on my
state exams the application of basic skills was not
maintaining. When I saw this article, I said, I've got a
problem here and basically I am not helping my kids master
their skills. When we are through with a unit at the end of
the month and I think they have understood multiplication
and multiplying, we are off to the next unit, never to practice
multiplying again. I said I've got a hole here and I need to
plug it. I decided kids need practice with many basic kinds
of drill and kill, and the bottom line is that they will be
maintaining it at a higher level of success than before"

(Interview, 1 / 93).

Sandy interpreted Bloom’s work to suggest that students must master to

automatic levels the traditional algorithms for computing if they are to be

successful in doing other mathematics. She reasoned, "I have some children that

can really reason and problem solve, they know how to go after a task, work on

concepts, but dog-gone it they can't compute the problem accurately. To me

there is a component missing. The computational abilities also have to be

quality, they have to be automatic" (Interview, 1 /93).

Bloom's work offered Sandy a solution to the tensions she experienced in

her practice. She acted on her learning and incorporated memorization and

practice of basic computing skills arguing that they are, ”not to be re-taught, but

rather to be practiced, for a level of mastery" (Interview, 1 /93).

144

State Mandated Testing
A second idea that Sandy encountered that contributed to Sandy’s

reformation of basic computational skills teaching involved reconsidering her
students’ CAT scores. She examined her students’ tests and wondered why they
did so poorly on the computational portion of the state's standardized test. She
certainly wanted to do whatever she could to improve students’ opportunity to
do well on these tests but she also believed her teaching was evaluated using
these scores. She commented, "I am really being evaluated on those scores. I
know it is not reflective of how I teach. I know that. But, not everybody else
does" (Interview, 1 / 93).

Sandy’s decision to reinstate traditional forms of teaching were influenced
by her beliefs about the role and purpose of the CAT. She believed her principal
evaluated her teaching on students’ scores. She explained, "I tried to make these
changes in my instruction to really see if I could make a difference on these

tests."

Conversatiens With Other Educaters

Sandy also encountered criticisms of her teaching. Parents, as well as
fourth grade teachers at her school, complained that students could not do the
"basics." In an interview Sandy revealed she worried a great deal about such
statements and in part, was influenced by them to reinstate drill and practice on

the traditional algorithms. She explained,

"I got more criticisms a year ago from other teachers about
things like regrouping wasn't solid enough or that
multiplication wasn't strong enough. I even had a few

parents say God we had to work on that all summer long

145

again, and fall came and they said gosh Ms. Wise we
thought you worked them really hard last year but they are

not ready for this year." (Interview 1 /93).

Sandy understood that criticism was pervasive in education. She knew that any
kind of change carried with it much criticism and from multiple directions. Yet,
in her case, the comments others made about her teaching of the basics seemed to
really shake her. She felt badly that other teachers in her school thought she did
not prepare students to enter their classrooms. They complained directly to
Sandy that students from her classes arrived unable to do routine calculations
efficiently and accurately. The principal criticized Sandy about the CAT and that
her students did poorly on the computation portion of the exam.

Although it would be difficult to suggest the impact of any of these
criticisms, Sandy acknowledged that they did play a role in her decision to return
to traditional practices. The comments established some sense of what other
educators thought important (Interview, 1 /93). Sandy also consulted several
individuals at the state department. In conversations with others Sandy
questioned whether memorizing algorithms was still a valued goal. She
explained in an interview,

"I spoke with several people at the state department and my
understanding is the reason we want kids to memorize algorithms

is so that there is fluency, fluency that they can move around in

bigger concepts and the actual computations don't get in the way,

and they verified that was correct. So, it's not memorization for

memorization sake, it's memorization to develop a tool."

146

Sandy’s ideas were confirmed by those she thought to have a good
knowledge of the new framework. From her view, she felt sure that her ideas

were matched well with others whom supported the framework. Sandy argues,

"you know for so long we argued that having a child compute
doesn't necessarily mean they understand or have the ability to
problem solve. Then, we said our goal is to help them understand
and we want them to be able to problem solve but now I wonder,
having a child who understands doesn't mean they can compute.
Now I think like this. You can't throw away the insides to the pie.
Being able to acquire number facts is imperative to being a
mathematician. I don't think it needs to be stressed as an end in
itself. But, it is definitely part of the solution. In order to have a
broader talent and ability to apply and work with the concepts, one
has to acquire a number of different understandings, and
computing is one of them. Now, for a lot of students, that ability
just isn't happening, and I do think we have to find a different
avenue to get there, to get to that big picture. But, for now this is

it." (Interview, 5 /93).

Referm Decuments

Sandy continued to examine reformers’ position in the policies and read
many sections over to establish what was said. Her reading was shaped sharply
by her questions about the traditional computational curriculum. She searched
for ideas concerning the traditional computational algorithms and she searched

for guidance in the form of specific answers, programs for practice. But, Sandy

147

found very little.28 Yet, it seemed the relative silence in the policies combined
with all that she was learning and thinking related to computational skills
convinced her that the traditional computational curriculum taught in very

traditional ways remained valued. Sandy reasoned:

"and that's when the question comes, at what point do you give on
skills acquisition for the depth of the concept? And that's when I
changed my yearly plan, I decided to embrace both more fully. I'm
not convinced you can do one or the other any longer. I definitely
believe in the strands and the unit approach, there's no issue there
and that enlarges the scope of what mathematical empowerment is.
But, for most children, it's still a necessity to practice daily the
skills. I can see that they are still valued in the documents and I can
see how they can interfere with other learning. You have to do it,
and you have to do it everyday. It's like riding a bike, you just have

to keep practicing it everyday until it becomes automatic."

A Teacher's Prier Knewledge and Beliefs

Sandy came to her learning experiences knowing much about the
traditional algorithms and teaching practices for teaching those skills. Her ideas
broadened to include estimation as a computational skill, mental arithmetic and
some ideas about students’ strategies as well. She emphasized these skills more

so during the first year of my observations. During the second year Sandy

 

28In chapter three, I argue the question of what to do about the traditional computational
curriculum is mostly left behind as policymakers focused instead on including new
computational skills and new instructional strategies. In that effort, the policies seemed to mostly
leave unattended such questions as what of the traditional computational curriculum remains
valued or to what levels of complexity.

148

continued to teach estimation and mental arithmetic with less emphasis making
room for more traditional goals in the automaticity lessons.

Sandy’s prior knowledge and beliefs were reinforced by her new learning
about the role of memorization and automaticity in learning basic computational
skills. In combination with an inattention to issues surrounding the traditional
computational curriculum, Sandy comfortably and confidently reformulated a
new interpretation of reformers’ ideas about computational skills instruction. In
effect, Sandy’s learning functioned as a basis for formulating policy.

Sandy’s learning, in and across the various contexts, contributed
significant insights to Sandy's views of improved teaching. The combination and
interaction of ideas that she encountered functioned to change her stance toward
her teaching across the years of my observations. Her learning failed to promote
and sustain new approaches toward computational goals and instead the
combination of factors fostered m learning that resulted in a return to
traditional forms of teaching.

Yet, several questions remain. On the surface, Sandy’s practice during the
1991-1992 school year, seemed more compatible, more promising, and moving
toward policymakers' goals. Her practice the following year seemed less so. Yet,
did Sandy’s ideas and practices the first year of my observations, the changes
and interpretations she made, represent the ideas of reform? If so, there remains
a question about why computational goals were not met that first year. And,
why would Sandy return to the exact practices criticized most as opposed to
other ideas she may have tried? In the next section, I examine Sandy's learning
in light of these questions. I ask whether Sandy's learning, both what and how
she learned, supported her understanding and enactment of the ideas and

practices envisioned in the reform documents.

149

An Appraisal of A Teacher's Responses
To Changing Teaching of Computational Skills

Sandy had developed a practice that prioritized conceptual understanding
and reasoning about mathematical ideas. She used manipulatives to explore the
mathematics of a situation and consistently encouraged students to use
manipulatives in place of traditional procedural algorithms.

The second year of my observations there was evidence of much change.
Sandy emphasized the rote-learning of traditional algorithms much more. This
marked a clear departure from the state's reform agenda. My analysis in the
previous section illuminates that what Sandy was learning ran contrary to
envisioned ideas and practices? In addition, other factors influenced Sandy's
decisions that also were contradictory toward the state's goals. A clear turning
away from policy would be expected under these conditions. After all, all that
she encountered supported the position she took.

Yet, for Sandy, this was not a turn away from policy. She believed the
changes she made represented a closer match to policy. How could she make
this argument? To understand why, I revisit Sandy's practice. My analysis
suggests that Sandy’s ideas and practices across hem years of my observations
were substantially different than envisioned practice. I argue that Sandy’s return
to traditional forms of teaching computation in the second year may not be the
best marker of Sandy's interpretation and enactment of policy. A more adequate
marker may be understanding what had gone wrong in that first year and how
her learning failed to address these issues.

Recall in chapter three, I argued that reforming instruction of basic
computational skills would require a great deal of new learning for teachers. Not

only would the teacher need to learn an entirely new set of computational skills

150

but also required would be learning a new pedagogy for teaching those skills.

My analysis of what was proposed in the reform documents included:

* A Variety of New Computational Skills
" Contextualize Computational Skills

" Conceptual Grasp of Underlying Ideas

In this section, I take each of these and use them as points of intersection for

examining Sandy’s learning and the ideas and practices her learning fostered.

A Varieng ef New Cemputatienal Skills

Overall, Sandy’s views of what the important computational skills are for
children to know went mostly unchallenged. Her practice continued to reflect
traditional algorithms as the most central method for computing. Aside from the
attention she gave mental arithmetic and estimation as a computing strategy
during the 1991-92 school year, her ideas and practices returned to a steadfast
View that the traditional algorithms were the most central computing techniques
for students to learn.

Sandy did not include in either year any emphasis on technology, other
strategies such as doubling, adding back, etc., nor did she focus explicitly on
strategies students’ bring. Even though problem solving experiences dominated
students’ work in mathematics during the 1991-1992 school year, Sandy
continued to recommend traditional algorithms to students for computing. In
the second year, Sandy insisted students use traditional algorithms for all
computations in the automaticity lessons. The purpose of automaticity lessons
was to provide students with practice of efficient and accurate ways to compute

(traditional computing algorithms) to use in investigative problem solving.

151

Centextualize Cemputatienal Skills

Policymakers imagined in fairly general ways how traditional
computational goals might be met. By emphasizing underlying concepts, using
physical materials, linking the manipulation of materials to steps of procedures
and developing thinking patterns, teachers can help children master basic facts
and algorithms and understand their usefulness and relevance to daily situations
(NCTM, 1989, p. 44).

This paragraph suggests that conceptual and procedural understanding
both remain valued. Yet, the recommendation is that both goals can be met
within the same instructional experience. Mathematical reasoning, conceptual
grasp, and mastery of basic facts and algorithms should not be treated
separately. Instead, each goal is attended in problem solving settings, as
students explore conceptual ideas, use manipulatives to explore those ideas and
link those experiences to algorithmic procedures. Instructionally, computational
goals and conceptual understanding happen simultaneously.

Sandy’s practice, both years, for the most part, did not emphasize any
links between ideas such as place value and the procedural steps of the
algorithms.29 Although Sandy made extensive use of manipulatives, these
instances were not linked to the procedural steps of traditional or non-traditional
algorithms for computing. Instead, Sandy's practice reflected a separation
between procedural and conceptual goals. There were no opportunities that I
observed across my Visits that attempted to link the two. For example, I
observed lessons across both years around the idea of average. None focused on

linking the idea of average with the step-by-step procedures for calculating the

 

29This is not to suggest Sandy ought to have focused on algorithms but rather to suggest that
emphasis on connecting algorithms with conceptual ideas was not what Sandy aimed to do.

152

average. Sandy confirmed in interviews that she did not emphasize relationships
between the conceptual work and the procedural work she did with the
algorithms (Interview, 5/93). Missing was any explicit attention to linking
procedures and the conceptual basis of those procedures.

The mathematics education community remains unclear about these
issues. There is not clear evidence to suggest how the two are intertwined.
There exists some evidence to suggest that students’ understanding of
underlying ideas, such as the conceptual underpinnings of algorithms, help
students monitor the success of their computations (Heibert 8: Lefevre, 1986;
Nesher, 1986; Resnick, 1984). In other words, understanding supports effective
use and efficiency with algorithmic steps. Sandy’s practice, because the two are
not linked in instruction, does not offer an explicit opportunity for conceptual
and procedural learning to benefit each other.30

Recall that Sandy's practice during the 1991-92 school year emphasized
investigations into mathematical ideas. In fact, she focused 80 to 90% of
instructional time in mathematics on hands-on learning-by—doing activities.
These lessons did not include any instruction aimed at developing computational
skill. The remaining 10 to 20% of instructional time focused separately on
computational goals. During that year, Sandy had constructed a practice where
little direct attention was aimed at learning basic facts and traditional algorithms
yet, when this did happen it was not linked to conceptual understanding or
mathematical reasoning. This practice contrasts sharply with Sandy’s practice
the following year. Sandy worked on the traditional computing algorithms each
day across the entire school year in relation to what she called automaticity of

skills. Yet, automaticity and conceptual goals were instructed separately.

 

30See for example Pearla Nesher, Are Mathematical Understanding and Algorithmic Performance

Related? In (1986), Be; the Learning et Mathematics 6(3), 2-9.

153

Recall also that during the 91-92 school year the multiplication algorithm
was shown to students in only one week at the end of the multiplication unit.
Thereafter the algorithm only came up indirectly inside problem solving
contexts. In these settings it was not mandatory that students use the traditional
algorithm. They could work the computation any way they thought made sense.
Some students had the use of calculators and others tried to use their own
alternative techniques. Sandy did not give any attention to what techniques
might be the most sensible or proficient during these opportunities. Even though
Sandy emphasized problem solving contexts for learning computation, she gave
no direct attention to any relationships between computation and the context of
the problem. Students were encouraged to multiply when it made sense to them
to do so but there was no explicit attention linking any procedures to the
underlying conceptual basis of the problem.

Sandy’s ideas and practices during the first year of my observations
minimized the importance and study of the traditional computational
curriculum. Instead, Sandy focused on problem solving experiences that

involved computational goals as well. She explained,

"I don't want them to multiply just to go through the steps of
multiplication, that's nonsense, the reason you learn to multiply is
because, well, just yesterday we had a situation where there were a
series of 3 or 4 sevens to add in a problem and the kids had big
trouble with it. This hung up the whole activity. We had to add 7
+ 7 + 7 + 7 + 1. I explained this is a place where multiplication can
help you solve the problem. You have four sets of seven here,
right, what is four groups of 7, they said it was twenty-eight, plus

one, they said twenty-nine. So we put down the nine and carried

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the two. They all went oh, I said remember the goal in
multiplication is to add like things fast. So now, I said remember to

look for likeness in numbers" (Interview, 5/92).

Sandy's learning to that point seemed to shift her thinking away from procedures
and toward situations where multiplication was useful. Her ideas focused more
toward such notions as likeness in numbers as a signal to use multiplication as
opposed to memorizing procedures. Her practice reflected these changes by de-
emphasizing the amount of time students would spend memorizing and
practicing procedures and emphasizing problem solving activities with
computational goals.

At the same time, Sandy did not attend directly to basic fact acquisition or
practice of procedural steps of any algorithms within those problem solving
contexts. Ultimately, she abandoned the notion that problem solving contexts
could successfully provide students an opportunity to learn computation. In
contrast, the following year was significantly different. Automaticity lessons
were equally central with problem solving contexts for learning computation.
During automaticity lessons Sandy used traditional algorithms to work all
problems regardless of the numbers involved. She incorporated catchy phrases
such as ”slash, burn it off,” to represent crossing out a zero and replacing it with
a nine in the traditional subtraction with regrouping procedure. Sandy repeated
practice with problems involving many zeros or more difficult regroupings.
And, estimation was used as a checking procedures rather than a computing
strategy itself.

Recall the automaticity lesson I discussed earlier in the chapter. Sandy

presented students with the following problems to work.

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1. 1000 310,000 5. 75 7. 497

+s97s - 967s x4 697
+ 119

2. 100 4. 11,496 6. 20,000' 8. 700
x 6 + 14,597 - 10.201 400
+600

In looking more closely at these eight problems, only two, problems 4 and 7,
seem to require the use of a traditional algorithm and only if an exact answer is
necessary. The other problems could be done more efficiently with an alternative
algorithm. For example, problems 1, 5, 2, and 8 might more efficiently be done
by bringing to bear an understanding of place value into the calculation and
computing on components of the problem. Computations could be done
mentally, quickly and quite easily. Problems 3 and 6 might be more efficiently
done using a strategy of ”adding back" or "adding up," the idea being adding up
to get to the number 10,000 in problem 3 or 20,000 in problem 6. Problems 2 and
5 might also be done using a strategy of doubling or multiples. For problems 1
and 8, adding only the thousands and hundreds place respectively need be done,
also requiring an understanding of place value to get the correct answer. In
effect, Sandy’s emphasis on traditional algorithms in this lesson promotes
practice and memorization of routine procedures and less the notion of efficiency
or underlying understandings.

Drill and practice under timed conditions emphasizes memorization of
routine procedures and de-emphasizes conceptual and contextual
understandings. For example, understanding place value and how the value of
numbers is influenced by sums, products, differences, and quotients becomes
hidden. Sandy’s practices during the 92—93 school year shifted away from the
exact ideas she hoped to make more central the previous year. Issues of

efficiency, understanding, and contextual aims actually became less central.

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Alternative computing strategies such as doubling, adding back, looking for
blocks of ten, twenty-five, or a hundred, all were disqualified as reasonable
options in automaticity lessons. In effect, Sandy’s decisions limited students’
opportunity to become fluent with computation.

In problem solving situations, students were not provided an opportunity
to size up problem solving situations and determine with insight and intuition
how best to calculate in various contexts. Instead Sandy’s purpose with the
automaticity was for students to make use of it inside the problem solving
contexts. Although she did not monitor students use of traditional algorithms,
she hoped she had provided students with efficient and accurate methods for

doing the necessary computations.

Cenceptual Crasp ef Underlying Ideas

Sandy’s practices often focused on a gimmick for remembering a
particular concept rather than understanding any underlying basis of a
mathematical idea. For example, in one lesson aimed at exploring the idea of
division, Sandy asked students to divide a pack of life-savers among four

children.

Ms. Wise: You see even when Jason was two, he was an expert in math.
You know one thing about a two year old, they always want their fair-
shares. Jason why don't you grab three friends. (Jason points to three

friends as students begin cheering him on.)

Now, Mrs. Jacobson has to figure out how to give the children their life-
savers. (Ms. Wise opens a pack of life-savers placing each one on the

overhead projector and notes there are 11 in the pack. She then begins to

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pass them out to Jason and his friends.) Here's two for you, two for you,

two for you and here's one for you and three left over. Is that all right?
Students: Noll That's not fair.
Ms. Wise: Well, why don't I give these three to three of the children?

Students: Because it's not fair.

Ms. Wise: See you guys knew this already, when you divide things '

among kids, everyone wants their fair-share. Okay, we have four children
and 11 life-savers to divide among them. (At this point Ms. Wise begins to
re-distribute the life-savers one by one and then asks a series of questions.)

So, how many does each child get?

Students: two

Ms. Wise: And, how many are left over?

Students: three

Ms. Wise: Okay, suppose we had another pack of life-savers and eight
kids. I need four more kids. (She lays out twenty-two life-savers on the
overhead inside a symbol that looks like a division bar. In front of the
symbol she draws eight hands.) Now, Ms. Jacobson has to figure out how
to give all of these life-savers to eight children. Does anyone want to

make a prediction on how many life-savers each child will get?

158

Student: Four

Ms. Wise: Why do you say four?

Student: Cause it sounds good.

Ms. Wise: Okay, anyone else want to make a prediction?

Student: Approximately two.

Ms. Wise: Any other guesses?

Student: Only one.

Ms. Wise: Okay, Tommy why don't you distribute the life-savers.

As Tommy distributes the life-savers Ms. Wise waves the wand repeating the

phrase "noit-ca-il-pitlum" over and over. Tommy distributes one life saver to

each child. Ms. Wise asks, "does it look like there is enough for another round?"

Several students answer "yes!" Tommy continues to distribute a second round

of life-savers. When he finishes Ms. Wise asks:

Ms. Wise: Okay how many life-savers did we have to start?

Students: twenty-two

Ms. Wise: And how many children did we have?

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Students: eight

Ms. Wise: And, how many life-savers did each child get?

Students: two

Ms. Wise: And, how many were left over?

Students: Six

Ms. Wise: Raise your hand if you see a pattern.

Matthew: Well, each time we were splitting up the lifesavers, you were

teaching us division.

Ms. Wise: I'm teaching you what?

Matthew: division!

Ms. Wise: No, no that is a sin in third grade. I was teaching you "noit-ca-

il-pitlum."

Matthew: (and several other students at the same time) It's multiplication

backwards!

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Ms. Wise comments, ”see, division is in our blood" and "a two-year-old can do
it." She announces that next week there will be more opportunities to do

division.

Ms. Wise: okay then, division is two things, right gang, now say it with
me, division is two things, it is fair-shares and multiplication backwards.

Ready go.

Students: fair shares and multiplication backwards
Ms. Wise: Ready go.

Students: fair-shares and multiplication backwards!!

Student’s interest and energy level was very high during this lesson. They
were attentive, listening, calling out, and laughing out in what seemed pure
enjoyment. They especially seemed to like the wand and the phrase "noit-ca-il-
pitlum." Just saying it seemed fun.

Sandy aimed for two connections. First, she hOped students understand
that division is an equal-distribution of objects into groups with something
leftover. Sandy drew eight hands in front of a box holding twenty-two life-
savers . Students physically distributed the life-savers from the box into the
eight hands. She wanted students to experience that 22 can be distributed into 8
equal groups of two in each group with six left over. Symbolically, she hoped
students would associate the idea of equal-distribution to the division symbol.
Yet, there was no explicit connection made. There also was no explicit

connection made between the idea of groups and division symbol. Whether

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eight groups of two or two groups of eight represented the same thing did not
come up.

Sandy also wanted students to make a connection between multiplication
and division. Yet, there was no explicit attention given to what this connection
involved. Only the phrase ”noit ca il pitlum” was mentioned. Students seemed
to know the phrase that division is multiplication backwards but, any support
for understanding the idea was not addressed. The opportunity represented
more a memorization rather than an understanding of what the connection
involved. The exploration of the conceptual underpinnings of division were
weak. At best, students were offered an opportunity to equally distribute life-
savers.

In another lesson on the idea of average, Sandy related fair-sharing to
average without making any explicit distinctions to division. In the average
lesson Sandy asked students to keep track of the number of pieces of mail their
household receives over a three day period. Students had to predict what the
"average" number of mail items would be. Sandy explained that finding the
average meant "fair-sharing" or equally distributing the mail across the three day
period. Using colored bear counters, students found averages by distributing
equally the total number of mail items into equal groups. They began using
language such as "my average is four" (Fieldnotes 5/92).

Sandy explained her goal in this lesson was to approach the idea of
average without using the algorithm. She said she wanted students to see that
average was linked to the idea of fair-sharing. Although students could come up
with their averages fairly quickly by distributing the bear-counters equally across
the number of days, they were not asked to think about the underpinnings of the
idea of an average. In other words, students were not involved in an

opportunity to consider the meaning of an average, what an average represents

162

about a set of data or why an average might be useful to know in particular
situations.31 Students did learn a technique for calculating an average, an
alternative to the traditional algorithm. Yet, they were not asked to link those
steps to the underlying meaning of the idea of an average. Thus, on one level
students approached the idea of average in a very different way than traditional
approaches involving memorizing the algorithm. At the same time the lesson
did not ask students to understand conceptually the idea of average nor did it
engage students in connecting procedural steps to the idea of an average.

The lesson on average also makes no distinctions between fair-sharing and
the idea of division. Although Sandy does much more with the idea of average
in later lessons, students are not given any direct opportunity to think about
connections between fair-sharing and the ideas of average and division. Mostly,
students are involved in learning alternative algorithms for computing answers

in the form of distributing manipulatives in particular ways.

36*!6391-31-1-31-31-1-31-2-31-

My analysis of Sandy’s teaching across the three core constructs proposed
in the reform documents suggests that Sandy attended to this agenda only
minimally, never fully integrating the proposals into her teaching. Each year
students' opportunities to learn pit computational skill against conceptual
understanding. Each year, conceptual and skill-oriented learning are instructed
separately. Across the years, one goal dominated, the other received less
attention. In the first year, Sandy's instructional approach rested on the belief

that conceptual understanding would promote computational fluency. At the

 

31Understanding the idea of average conceptually involves much more than what Sandy’s
practice offers. Jan Mokros and Susan Jo Russel in their paper Children’s Concepts of Average
and Representativeness helped me to think about these issues. See TERC Working Paper 4-92,
January, 1992.

163

same time, Sandy's views of what it would mean to understand mathematics
conceptually differed from what was described in the reform documents. The
following year, Sandy's reasoned that students need more opportunities to
practice procedural steps to computational algorithms. In that year, she returned
to traditional forms of teaching aimed specifically at developing proficiency with
basic number facts and algorithms.

Sandy's practice across both years runs in stark contrast with proposals to
reform computational skills instruction. Yet, what Sandy encountered typifies
some of the tensions any teacher would likely encounter when trying to attend to
both procedural and conceptual goals. The complexity involved in learning to
teach mathematics for conceptual understanding has been well-documented
(Ball, 1989; Eisenhart et al, 1993; Heibert 8: Lefevre, 1986) A strong debate about
the relationship between conceptual and procedural knowledge and the
pedagogical practices needed to support each continues (Heibert 8: Weame,
1988; Nesher 1986). There is also debate specifically about the role that basic
computational skills plays in students’ understanding of mathematics (Resnick,
1984). It should be no surprise that Sandy would face a great deal of uncertainty
around these issues.

Even as there is much debate and uncertainty surrounding these issues,
teachers must try out new instructional strategies if they are to respond to the
state's efforts at reform. They must try to change their teaching in the context of
an environment that holds the pervasive perception that teaching is a practice of
certainty. Sandy's approach, under these conditions, suggests that she believed
there were clear-cut answers offered in the reform documents. She formulated
what she thought were the right answers and tried them out. The problems that
arose in her teaching were not viewed so much as tensions, but more things she

needed to fix. Sandy thought of the problems as errors in her own judgment,

164

miscalculations on her part, even personal deficiencies. And her response was to
find the right answers for her teaching.

The uncertainty of teaching practice and enacting policy is what makes
Sandy’s stance toward "solving" the problems that arose in her practice so
significant. If teaching and enacting policy are seen as uncertain crafts, it would
seem natural to encounter a set of tensions, as a natural consequence of any
change. Instead, Sandy encountered a stance of certainty toward changing her
teaching across the professional development opportunities she participated.
This stance of certainty is endemic in education. The perception of teaching as a
practice of certainty is far more common at all levels of the educational system
McDonald, 1992). At the same time, the policy in California painted the practice
of teaching and enacting policy as one of uncertainty. Yet, Sandy encountered
little opportunity to alter her stance. She had few opportunities that suggested,
for example, she would need to unpack the issues underlying her teaching of
computational skills and see them as choices with consequences for students'
learning, rather than right or wrong answers for her teaching. Instead, the
opportunities to learn that Sandy encountered did not function in this capacity.
They failed to suggest she would need to unpack, understand, and manage
underlying tensions. Mostly, the experiences offered more activities to try and in

the form of right answers for teaching.

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CHAPTER 5

LEARNING TO TEACH AN UNFAMILIAR MATHEMATICS

Introduction

The previous chapter investigates Sandy's efforts to change her teaching of
basic computational skills. This chapter focuses on a different strand of content.
Though here, the mathematics is not as familiar as computation. The teaching
context is also different. In 1989 Sandy accepted the position of elementary
curriculum specialist in her district. One of her responsibilities included helping
other teachers learn about the new mathematics framework. Sandy’s role as
curriculum specialist represents a unique opportunity for this study. It offers the
opportunity to investigate the learning experiences of a teacher leader in the
context of external efforts to reform.

As the district’s elementary curriculum specialist, Sandy felt comfortable
with her responsibilities to help other teachers understand the state’s goals.
However, she learned that the new framework proposed a strand of content that
was not recommended in the previous framework. This strand was unfamiliar to
Sandy. In fact, she had never heard of discrete mathematics until now. To
prepare for an upcoming workshop on the new framework, Sandy decided that
she would develop some ideas of what discrete mathematics is and how
elementary teachers might introduce it into their teaching.

The story in this chapter focuses on what Sandy did and the experiences
she encountered for learning about discrete mathematics. It illuminates the

difficulties a teacher encountered in learning an unfamiliar mathematics for

166

herself as well as the problem of teaching other elementary teachers how they
might introduce discrete mathematics into their teaching.

The chapter begins by describing the experiences Sandy encountered for
learning about discrete mathematics. Like most elementary teachers, Sandy had
very little knowledge of what discrete mathematics involved. She ran into
difficulties very early in the learning process. As the story evolves, I explore
Sandy’s learning in and across a variety of settings that she made use of. The
reader will learn about the mathematical ideas she grappled with, what directed
her attention there, and the role several other factors played in promoting or
hindering her evolving understandings. The story ends with an illustration of
the opportunity for learning Sandy offered other elementary teachers for their
learning about discrete mathematics and how to introduce it into their practices.

In the second part of this chapter, I consider the events in this story from a
pedagogical perspective. I examine what and how Sandy learned in relation to
the proposals to introduce discrete mathematics and whether her learning
fostered understandings and changes envisioned in the policy. My analysis
suggests there is an enormous amount of new learning required to understand
and introduce new mathematics into an already bursting elementary
mathematics curriculum. And the problem of teacher learning is a very serious

one having consequences far beyond the work of one teacher.

A Teacher Leader's Responses to
Proposals to Introduce Discrete Mathematics
Into The Elementary School Mathematics Curriculum

Sandy encountered proposals to introduce discrete mathematics as she
prepared for a workshop to teach other teachers in her district. The central
purpose of the workshop was to introduce teachers to the new mathematics

Framewerk (1992). Sandy accepted the position of district elementary

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curriculum specialist two years prior and knew well in advance that the
development of a new framework was underway.

Sandy expressed the importance of introducing discrete mathematics in
our very first interview. She previewed the new Framewerk (1992), in one of her
doctoral courses prior to publication and more widespread distribution. Sandy
explained, "the new document is an extension of the document from '86. That
document emphasized seven strands. A lot of other states modeled after it. In
the new document they added an eighth component called discrete mathematics,
I think it is kind of like math in the real world" (Interview, 1 /92). Sandy had
already begun to think about discrete mathematics and what it might mean for
her teaching of other teachers.

When asked again later about what else was new in the forthcoming
framework, Sandy reiterated, ”the new strand of content was a big change”
(Interview, 1 /92). Although Sandy mentioned other changes and additions to
the new document, the introduction of discrete mathematics preoccupied her
mind. She explained, ”I have been asked to speak to, have a conversation with
groups of teachers from the county to develop an understanding of the new
framework, for the County Office of Education, it's a math consortium group. So
when I was posed with this task, I knew I had to develop the discrete math
strand, which I didn't do formally at that point, and certainly I didn't have it
internalized enough to talk about it" (Interview, 11 /93).

Sandy’s tone and comments indicated mostly guesswork. She proposed,
”1 think it is like math in the real world” (Interview, 1 /92). And, she read

inquisitively,

”it indicates here that it is a study of systems of separate entities, so

it's used in the sciences, maybe with the elements chart. They also

168

indicate finite graphs so maybe it is related to statistical data, but
this is difficult. And, students are supposed to construct, analyze
and compare algorithms. Well, we don't have many algorithms in
the third grade" (California State Department of Education, 1992, p.
149 / Interview 2 / 92).

As Sandy tried to understand the connections in the above paragraph, it seemed
she was grasping at straws. She was hard-pressed to make any sense of the
written words on the page.

The reform documents represented Sandy’s first opportunity to learn
something about discrete mathematics. The Framewerk (1992), provided a
description. Sandy read that discrete mathematics should be introduced into the
K-12 curriculum and that all students should learn the topics and ideas of
discrete mathematics. Sandy decided she would need to learn much more to
introduce the strand of content to the teachers at her upcoming workshop.

Given the enormous reform agenda, it seemed somewhat curious that
Sandy would center so much of her attention on discrete mathematics. Yet, the
pressure of the upcoming workshop and how Sandy organized workshops
around the content strands of the framework seemed to make it good sense.
Sandy perceived her role and responsibilities to include educating other teachers
about the content strands. In fact, Sandy structured her seminars in direct
relationship to the content strands.32 She introduced teachers to instructional

activities she thought fit within each category of content and tried to support

 

32A colleague observed Sandy’s teaching in the context of a one day district sponsored inservice
for third-grade teachers titled Third Grade With Math Manipnlatives. He noted that Sandy's
organization and focus throughout the workshop centered on activities associated to the eight
strands of content identified by the new Framework document. The impression he had was that
the workshop was aimed more at introducing the new framework rather than exploring the use
of manipulatives.

169

teachers in learning to use those activities. This structure, in part, pressed Sandy
to consider discrete mathematics as an important change in the new document.
Over the next year Sandy encountered a number of different opportunities
to learn about discrete mathematics. She would draw from her learning that year
to teach other teachers ideas and practices for teaching discrete mathematics in
the context of several state-sponsored workshops the following year. Inside
these workshops, Sandy communicated her evolving ideas about how teachers
could introduce discrete mathematics into their own teaching. Below, I describe
how Sandy developed her ideas about discrete mathematics, what she tried to

learn, what directed her attention there, and how her learning progressed.

Learning About Discrete Mathematics

Shaping Sandy’s direction for learning about discrete mathematics was
her organizational strategy for her workshops. Sandy simply found the content
strands in framework to be a helpful way to cut across the multidimensional
nature of the state’s reform agenda. Inside each content strand, Sandy could
address a wide range of issues, ideas, and practices she believed represented the
reform agenda. In the interest of finding suitable curriculum materials for
teachers to work with at the workshops, Sandy began a process of learning about
discrete mathematics. She set out to locate investigative-type activities she
thought could represent discrete mathematics in the elementary grades.

Sandy had difficulty finding any materials she felt certain represented
discrete mathematics. Two problems surfaced. First, Sandy had few ideas to
draw from to conduct a search for curriculum materials. She really didn’t know
what mathematical ideas she was looking for. Second, none of the materials

Sandy encountered were labeled as discrete mathematics.

170

Obviously, Sandy had very little to go on. And she found out very
quickly that she was not alone in her struggles. Sandy decided to consult a
number of other teachers, a district math specialist, and a secondary mathematics
teacher to help her locate materials. Yet, she found very little help in identifying
curriculum materials that focused on discrete mathematics. No one that she
encountered seemed to have good ideas about materials she could use to
accomplish her goals (Interview, 1 /93).

Sandy eventually consulted a state-level mathematics educator, someone
involved as a developer of the new framework. He also was unable to offer
Sandy suggestions about elementary curriculum materials (Interview, 1 /93).
Early on Sandy found herself in a position of trying to make decisions about
good curriculum materials by herself even though she had very little knowledge
of the subject matter to go on.

Sandy decided to return to the framework and to the Curriculnm
Standards (1989) for guidance. She searched each document trying to dig out
everything she could find about discrete mathematics. Most of the information
Sandy found in the documents was geared toward secondary teachers. Reading
the information was very difficult for her though she began to formulate some
ideas about what discrete mathematics involved. She read and re-read the
material over and over, searching for clues about the content. Below I

characterize what Sandy learned as reported in several interviews.

The Referm Decuments

Sandy’s reading of the reform documents represented her first dip into the
mathematical ideas of discrete mathematics. She seemed to be using the
paragraphs much like a textbook, trying to teach herself something about the

mathematics. What Sandy found was that the documents were of little help to

171

her. She complained that they were very confusing. She found brief descriptions
of the term discrete and several rationales for introducing discrete mathematics
into the school curriculum. She found a listing of some of the central ideas, most
of which were unfamiliar to her. Sandy complained, "there just isn't much detail
in there and much of it I just don't understand" (Interview, 1 /93).

Sandy also found a few examples offered as illustrations of the types of
counting problems one encounters in discrete mathematics. Unfortunately,
Sandy had great difficulty making sense of the examples. One problem read as

follows:

For example, a complex network of one-way streets can be
represented geometrically by a directed graph, which in turn
can be interpreted algebraically as a matrix. An i-j entry in the
matrix is 1 if and only if corresponding vertices are adjacent (i.e.

connected by an edge); otherwise, the entry is 0.

By representing the graph as a matrix S and then multiplying S
by itself, students can use S squared to determine the number
of two-stage routes connecting the various paths of points.
Students can generalize this procedure to graphs of any size and
computer software can be used to compute powers to the
corresponding route matrices, which can then be analyzed to
determine numbers of multiple-stage routes as well as other

characteristics of networks (NCTM, 1989, p. 177).

This paragraph represented to Sandy an opportunity to learn what discrete

mathematics involved. This particular problem focused on directed graphs. The

172

question asks how many two-stage paths there are connecting one point on a
diagram to another. The authors suggest that matrices can be used to model the
connecting paths and the mathematics of matrices can be used to solve the
problem. An illustration of this process is included.

Sandy’s work with this problem led to frustration. She knew little about
matrices. She decided she would try to learn about matrices in order to
understand this example. She consulted other teachers and on several occasions
even asked me for help (Interview, 1 /93). The process was ongoing and took a
great deal of time.

As Sandy focused extensively on the mathematics of matrices, how to set
them up, add and multiply them, she began to learn mathematics she had not
encountered prior. She became familiar with setting up a matrix to represent a
problem situation. She became quite comfortable with multiplying two matrices.
It seemed that Sandy’s efforts advanced her understanding of matrices. Yet, in
an interview Sandy could not explain how the matrix solution related back to the
original problem in the text. At that point in the process, her learning did not
function to promote an understanding of the combinatorial counting ideas
underlying the problem.

Because Sandy was entirely unfamiliar with the mathematics of matrices,
she centered much of her attention there. The mathematics of combinatorial
counting never really came to the surface, at least it never became focal. Sandy’s
learning did function to promote the idea that matrices were connected to
discrete mathematics. She teetered on the idea that matrices is an example of
discrete mathematics. And, she decided that discrete mathematics was very
difficult to learn (Interview, 1 /93).

Earlier I suggested that Sandy was very resourceful as a learner. She

creatively sought out learning experiences that she believed would help her put a

173

picture of discrete mathematics together. At least two ideas described in the
documents attracted Sandy’s attention. One involved contrasts drawn between
continuous and discontinuous mathematics. The other was about whether the
integers was an example of a system of numbers used in discrete mathematics.

Sandy’s puzzling about these ideas resulted in a number of different
occasions to talk to others about discrete mathematics (Interview, 1 /93). Sandy
arranged to talk to other mathematics educators and in these contexts she would
grapple with a variety of mathematical ideas. She also encountered a rationale
for including discrete mathematics that was not discussed in the documents.

Below, I describe what Sandy encountered in these settings.

Cenversatiens With gfiher Educaters

Sandy decided to consult the ”experts” to help her understand the
mathematical issues she encountered in reading the reform documents. Because
she viewed individuals with direct responsibility for conceptualizing, writing,
and implementation phases of the framework as experts with the ideas, she
contacted individuals in these roles to help her learn more about discrete
mathematics (Interview, 2/93).

After meeting with one person and asking specifically about the
characterization of continuous and discontinuous mathematics, it was mutually
decided that he could not be of much help to her. He directed her to someone he
thought could help, a leading mathematics educator involved in conceptualizing
the new framework.

Sandy set up a meeting. My description of what took place is organized in

relationship to the mathematics Sandy encountered on these occasions.

*Contrasts between continuous and discontinuous mathematics

174

*Number systems

*Relational theories

Continuous and discentinuous mathematics. Sandy read that discrete

mathematics was unlike other, more familiar mathematics because of its
discontinuous qualities. In an interview Sandy explained her confusion with this
notion, at least in part, involved what the terms continuous and discontinuous
described. Sandy had already searched the policies for any information she
could find. She read in the new framework, "Discrete mathematics, the study of
systems with separate (discrete) entities, is contrasted with systems involving
continuous quantities" (p.149). And she read in the Cnrricnlnm Standards on p.
176, "Whereas the physical or material world is most often modeled by
continuous mathematics, that is, the calculus and prerequisite ideas from
algebra, geometry, and trigonometry, the non-material world of information
processing requires the use of discrete (discontinuous) mathematics."

Sandy’s confusion prompted her to ask a question. She explained, "so I
asked him [mathematics educator at the state-department] about continuous and
discontinuous mathematics and the connection to computers." And, I asked,
”whether the terms continuous and discontinuous described the entire system of
numbers underlying each kind of mathematics or if the terms described the
elements of the underlying system” (Interview, 8/93).

Sandy described the discussion as difficult for her. She explained,

"he communicates at a secondary level. I had to be really honest
and say, wait, wait, I am an elementary teacher." She confided, ”we
both really had to struggle to talk. He had to lower, well not lower,

but unpack his vocabulary so that we could communicate and that

175

was challenging for both of us. When I needed it, he tried to clarify
things for me and I would try to pull from my background but

really I couldn't" (Interview, 1 /93).

Even though Sandy was frustrated and suggested the meeting was less than
ideal for her learning about discrete mathematics, she seemed to take away
several ideas from this meeting. She explained, "he went into ideas where he
talked about the integers from zero to infinity. I know now it [discrete
mathematics] can involve continuous data but the solutions to problems are
different, they are finite unlike in algebraic functions where the solutions often
go on and on.” Sandy clarified further, "there is something about algebra and
discrete math, they are different, one is based on continuous numbers, like with
functions, and the other, I am not sure, this is where I get stuck," (Interview,
2/92)

Sandy’s comments reflected some circling around the ideas she had pulled
from the documents. She was in the process of questioning what the differences
might be in the sets of numbers or data underlying discrete mathematics and the
more familiar mathematics of algebra that she had some ideas about. She
clarified for herself that the terms continuous and discontinuous not only
referred to the systems of numbers underlying the different mathematics but also
to the kinds of solutions algebra and discrete mathematics offer. Despite these
advances, Sandy remained stuck on the question of what the systems underlying
discrete mathematics might be like.

Underlying number systems. Sandy read in the Framewerk (1992), and
the Curriculnm Standards (1989), that discrete mathematics is, ”the study of
systems with separate (discrete) entities, [and] is contrasted with systems

involving continuous quantities" (p. 149 Curriculum Standards). She also read,

176

"Discrete in this context means focusing on discrete and separate entities rather
than on measures of continuous quantities" (p. 84 Framework). From reading
these statements Sandy came to a question. She wondered whether the integers
represented an example of the kind of number system reformers described as
underlying discrete mathematics. Sandy knew a great deal about the integers
and could relate well to what they were like. At first she conjectured, "the
integers are not part of discrete math because they are continuous, they go on
and on.” Yet, she puzzled, "but each integer is distinct and separate from the
next integer” (Interview, 8 / 93).

Sandy pulled out the following statements from the framework pointing
to them in an interview, ”it does not mean that everything not continuous is to be
considered discrete mathematics. Arithmetic with integers, for example is
treated under number, not under discrete mathematics" (p. 84 Framework).
Sandy concluded, ”if arithmetic with integers does not fall under discrete
mathematics then the integers must not be an underlying system in discrete
mathematics” (Interview, 8/93). The documents suggested to Sandy that the
integers do not underlie the mathematics of discrete mathematics.33

Once again, Sandy turned to the leading mathematics educator at the state
department. He seemed to contradict this conclusion. Sandy concluded after her
conversations, ”the integers are an example of a number system underlying
discrete mathematics in the same way the real numbers underlie most of

algebra” (Interview, 8/93). Sandy seemed more settled with this understanding.

 

33Interestingly, recall in chapter two, in an interview with a mathematician specializing in
discrete mathematics, he noted that the counting numbers and/ or integers are perhaps the best
starting point for beginning the work of combinatorial counting, a central idea of discrete
mathematics, mostly because people have familiarity with the ideas of the system.

177

Relatienal theeries.34 Sandy explained, ”I wanted to follow up on some of
the ideas in my other meeting [with the leading mathematics educator] and I still
needed to find activities I could use [at the workshop],” (Interview, 8 / 93).
During a meeting with the district math specialist Sandy asked about relational
theories. She complained to the district math specialist about her previous
conversations with the leading mathematics educator that "he rattled on about
relational theories. I basically thought this is impossible" (Interview, 8 / 93). The
district math specialist decided she would help Sandy by trying to explain what
recurrence relations involved. Sandy recalled ”she pulled a secondary textbook
out and identified several relational theories and began explaining them step-by-
step," (Interview, 8 / 93).

In an interview, Sandy was unable to describe what recurrence relations
involved and she could not explain how they fit with discrete mathematics. Yet,
she indicated that most of the time she spent with the district math specialist
involved discussion of these ideas. Sandy was convinced that she just could not
understand recurrence relations and thought, "this stuff is not translatable to the
elementary level." She described her meeting with the district math specialist as
difficult and complained, ”I just couldn't get it” (Interview, 8/93).

Sandy felt the conversations she had with other mathematics educators
were overwhelming. She felt frustrated in her efforts to locate avenues for
understanding discrete mathematics. She mostly thought that what had
happened was not very useful.

Yet, Sandy managed to connect with an interesting piece of information

she had gathered from these conversations. This information, in itself seemed to

34Recurrence relations involve the construction of formulas for expressing the relationship
between terms in a sequence or series as a function of one or more of the previous terms.
Examples include compound interest formulas, home mortgage formulas or a formula for
expressing the more famous Fibonacci sequence. Sandy termed these ideas as relational theories.

178

have a dramatic impact on her learning. Sandy encountered a different rationale
for introducing discrete mathematics into school mathematics that she had not
read in the documents. Instead of fostering a stronger desire to learn more about
discrete mathematics, it convinced Sandy that there was less importance for her

as an elementary teacher to learn about discrete mathematics. Sandy recalled,

"he [leading mathematics educator] explained the strand was really
added for the secondary level. He explained that the algorithms for
the inclusion- exclusion principle, pigeon-hole principle were not
falling under the strand of number strength very well and they
were being overlooked in the secondary curriculum. So, they
decided to enlarge the scope of what ought to be taught and added
discrete mathematics. At least that is what I recall" (Interview,

1/93).

Sandy placed a lot of stock in this information. She admittedly knew nothing
about the mathematical principles she mentioned here, but she became
convinced that discrete mathematics was introduced mostly for secondary
grades. At that point Sandy seemed to shift her efforts to learn about discrete
mathematics in low gear. She no longer was preoccupied with setting up
learning opportunities for herself. She had been convinced reformers’
introduced the new strand to enlarge and re-emphasize what was not getting
adequate attention in the secondary curriculum.

Interestingly, Sandy did not abandon the idea'of introducing discrete
mathematics at the elementary grades altogether. She explained, the Framewerk
(1992), ”suggests the introduction of discrete math at all levels because of the

belief in California that some work can be done at all levels (Interview, 5/93).

179

She rationalized, ”of course, there is always some work that can be done at the
elementary level, but it doesn’t have to be very extensive.” Sandy commented
further, ”no one had any suggestions for what those activities might be, at the
elementary level, so I had to do this myself" (Interview, 5/93).

It seemed Sandy had come full circle. She was convinced mostly of two
things. She felt that consulting others had not been very promising or useful for
learning about discrete mathematics. She also believed that discrete mathematics
was very difficult for her to learn. Consequently, Sandy decided not to involve
others and instead would return to search for elementary curriculum materials. I
turn now to focus on Sandy’s experiences in relationship to two pieces of
curriculum materials she located for learning about and representing discrete

mathematics for the elementary grades.

Curricnlum Materials

Sandy decided to look again at the Framewerk (1992). She explained, ”So,
I re—read the Framewerk (1992), again and I said okay I’m gonna pick just one
idea in here, first, permutations and combinations. Then I will work on the next
idea, maybe unions and intersections after that” (Interview, 2 / 93). Sandy
selected combinations and permutations as a starting point. She remembered her
experience with the ideas in high school algebra. She began searching for any
activities she could find that focused specifically on combinations and
permutations including an old high school textbook. Her goal was to locate an
activity she could use as an example of teaching discrete mathematics. The
activity also had to be an investigation, learning by doing experience that

teachers could use with their own students.

180

Sandy found in her own collection of materials several mirneographed
sheets of paper, a Marilyn Burns activity she referred to as the ”ice-cream
problem/’35

She explained,

"early on I grabbed at all kinds of stuff based on my gut instinct, I
thought this idea of combination and permutation was a good one. 4
But then, it wasn't labeled anything, it was just this kind of
extracurricular piece to the math content of the text. Now, I guess

we can call it discrete mathematics" (Interview, 2/93).

Sandy decided she would use the ice—cream problem as an example of a discrete
mathematics problem with teachers in her workshop. To do so, she knew she
would need to explore the mathematics of the problem for herself. The materials
she had consisted of two worksheets labeled combinations and permutations at

the top. Each sheet posed a problem.

Kids love ice cream cones. How many different two scoop cones
can you make with chocolate and strawberry ice cream”? Draw a

picture of each cone that is different.

The second sheet had the following problem:
How many different two scoop cones can you make with three

flavors? The flavors are chocolate, strawberry and vanilla.

 

35The activity is currently situated inside a series of activities titled Brown Bag Series and is
designed to focus on the ideas of pattern and function. The activity is not identified in these
materials as discrete mathematics but in fact, is an example of a combinatorial counting problem.
Say how these are different.

181

The materials Sandy had did not specify how to solve the problems.

There were cones drawn on the paper simulating ice-cream cones. Students were
to construct various two-scoop ice-cream cones making decisions about whether
to count cones that had the same flavor for both scoops or reverse-ordered
flavors as different ice-cream cones. For example, in the first problem, a decision
for whether to count chocolate on top and strawberry on bottom as different
from a cone where the flavors are reversed is important. Organizing outcomes
using some type of system for keeping track of the different cones would make it
easier to determine when all cones have been accounted for.

Sandy’s experiences working on this problem by herself and later with
other teachers at the workshop fostered for her several new ideas. Sandy
decided that the ice-cream problem was more about learning how to count things
than getting the right answer. She contrasted this idea with her prior experience

in her high school algebra class. She recalled,

"Traditionally, in an algebra class, the teacher gave us a formula for
combinations, a formula for permutations and you were supposed
to be able to use them, not understand them. What I like about this
now is that it is more about learning how to think about things,
learning how to count and organize things, a way to count things,

and if it is taught in this way it is more powerful" (Interview, 8 / 93).
Sandy seemed to realize that the mathematics of combinations and permutations

would involve more than memorizing a definition or a formula for calculating

the right answers.

182

Sandy also focused on the idea of order and whether order makes a
difference in determining what to count. She explained, ”first, you need to
decide if a chocolate on top and vanilla on bottom is the same cone as vanilla on
top and chocolate on bottom.” And, "if you decide that order does matter then
what and how you count is different than if you decide it doesn't" (Interview,
8/ 93).

Sandy also realized the importance of being systematic in counting. She

reflected on an incident in her teaching,

"did you notice how Mary Jean came up and she instantly had a
strategy for organizing the cones? And yet, Mike couldn't organize,
he was all over the place, he couldn't set up any pattern using the
three flavors. So, we had to think about what would be a way to
organize the process and later I asked him now is there a better
way to organize and communicate the data? And two or three
others came up and finally we had organized the scoops in a way
that we all could determine that we had covered the possibilities"

(Interview, 2/ 93).

Sandy’s learning revolved around the notions that order and patterns in
counting combinations and permutations matter. She argued that systematic
counting provided certainty that all possibilities were accounted for. Sandy
illustrated in a diagram the number of different ice-cream cones using a pattern
of combining all possibilities with vanilla first, then strawberry, then chocolate.
Sandy showed there were nine possibilities if two-scoops of the same flavor were

counted, and 6 if not. She illustrated there were only 3 possible outcomes if for

183

example chocolate on top and vanilla on bottom was considered the same cone
when the order was reversed (Fieldnotes 8 / 93).

Sandy also defined combination and permutation. She explained that
permutation meant, "order is important" and combination meant, "order is not
important." She explained there were 6 permutations and 3 combinations of ice—
cream cones for the problem involving two scoops and three flavors of ice cream.

Sandy’ understanding of the algorithms for calculating the number of
different combinations or permutations of a set of objects remained unclear.
Recall that she had memorized the algorithms in high school. At this point she
could not recall the algorithms but explained, "I know the algorithm is just a
shorthand version of this process.” And, "I know I should have the talent to say,
now here's the algorithm and here's how I can prove it works, but that is really
hard for me to figure out" (Interview, 8 / 93).

Although Sandy was unable to remember the algorithm or provide an
explanation of how it connected to the counting process she demonstrated, she
did imagine for the first time that there were connections. Sandy realized that
the formula she had memorized years ago was rooted in the process she now
used for formulating the outcomes. The formulas somehow seemed less of a
mystery to Sandy. She commented that she no longer thought of the algorithms
as if they had been, ”made up out of the air” (Interview, 8 / 93).

Sandy’s View of the importance of the algorithms also had changed. She
explained, ”you know the power of all this is really more about how you can
decide how to count, you decide if vanilla over chocolate is different from

chocolate over vanilla, and if so, then you decide how to count those, and so that

184

ll

 

 

is what is empowering to me, not the algorithm. Then maybe one day we can
think about proving the algorithm" (Interview, 8/93). 36

Sandy’s work with the ice-cream cone problem functioned to help her
create an experience with discrete mathematics for other teachers. Sandy used
what she learned from the ice-cream problem to facilitate other teachers' learning
about the ideas of combination and permutation. I turn now to focus on the
workshop experience Sandy offered other teachers. I use this context both as a
site to illuminate what Sandy learned about combinatorial mathematics and as
an example of the learning opportunity other teachers were offered for learning
about these ideas. I begin with a brief discussion of what the workshop was like
for participants more generally. I then focus on the particular segment of

instruction Sandy offered teachers for learning about discrete mathematics.

Teacher Workshops

Sandy’s enthusiasm at the workshop was indescribable. Her style
promoted a renewed sense of energy and excitement. Bells, whistles, flashing
lights and even magic were part of the show. Sandy’s enthusiasm seemed to rub
off on participants as well. Teachers sitting slumped in their chairs awaiting a
lecture type workshop seemed to pop-up to see what was about to happen next.
Heads turned, smiles appeared, everyone was trying to see what Sandy was up

to. There was laughter and enjoyment.

 

36Interestingly, the power Sandy seems to be referring to, involving decision making for what
and how to proceed with counting, runs counter to the notion of power as I described it in my
sketch of discrete mathematics in chapter three. I suggested that power from a mathematician’s
point of view may be rooted in an understanding of the generalizations that can be drawn across
similar kinds of counting situations, leading to the formulating of algorithms that do the work of
counting for you. This is not to suggest mathematicians do not also consider understanding the
tedious nature of the counting process valuable, but more to suggest that the power of this
mathematics is related to the efficiency and effectiveness of the theories, formulas and algorithms
and assuming the more tedious nature of the work Sandy is highlighting.

185

Much like her third-grade practice, Sandy organized the workshop in
short segments focusing on each of the eight content strands. During each
twenty minute segment Sandy asked teachers to engage in one or two short
versions of student activities that she believed represented the eight content
strands. This approach Sandy inherited from her work with Project AIMS. The
underlying philosophy of AIMS rested on the belief that teachers could learn
about instructing their students by actively engaging in the activities for
themselves. In the process teachers would learn the mathematics and a
pedagogy for teaching those ideas. It was from this perspective that Sandy tried
to engage teachers in investigation-type activities. She asked them to make
believe that they were elementary students participating in real classrooms.

Sandy’s goals were multidimensional. She wanted to provide teachers
with activities they could take back and use in their own classrooms. She also
hoped teachers would learn the mathematics underlying each activity. Sandy
counted on the investigations to prompt teachers to ask questions and promote
discussions about the mathematical ideas. She knew that conversations would
arise concerning appropriate changes in activities for various grade-levels. She
also expected participants to ask questions concerning pedagogical issues. Sandy
often tried to offer teachers ideas about where students might have difficulties,
pointing out likely trouble spots and what she would do to help students in
those instances (Fieldnotes, 8/93).

Sandy also wanted teachers to think about the reform agenda. She
explained to teachers that the investigation-type activities she offered are
representative of the kinds of student learning experiences characterized in the
new mathematics framework. Sandy argued that the idea of getting kids to work
collaboratively in problem solving experiences is part of the new pedagogy

proposed in the new framework (Fieldnotes, 8/93).

186

Sandy made selections about activities using two criteria. First, the
activity had to correspond to one of the eight content strands of the framework.
Sandy wanted teachers to have experiences that cut across the content areas of
the framework. As a result, participants at the workshop often found themselves
doing mathematics they currently were not teaching. For example, several
teachers expressed that they did not teach mean, median and mode to their
students. Sandy promoted these ideas as part of the probability and statistics
strand. All teachers at this workshop agreed that they were not currently doing
anything in their own teaching with discrete mathematics (Fieldnotes, 8 / 93).

The second criteria Sandy used was whether the activity embraced a
learning-by-doing phiIOSOphy. Sandy embraced this philosophy for students
and for teachers' learning. She personally viewed the philosophy of learning by
doing as representative of reformers’ vision of teaching and learning
mathematics. And she believed that activities like those designed by Project
AIMS, Marilyn Burns and a few others would engage teachers in learning a new
pedagogy that they could use in their own classrooms (Interview, 8/93).

In the segment on discrete mathematics, Sandy focused on two activities,
both counting problems. The first was the ice-cream cone problem described
earlier. Teachers worked with paper cut-out ice-cream cones and flavors. They
were asked to construct as many two-scoop cones as possible using the three
flavors provided. Sandy suggested they keep track of their work in whatever
fashion they thought necessary. After a few minutes, Sandy asked groups of
teachers to share what they had done with the larger group. Before looking
closely at what happened, I describe the second activity.

The second activity, by Project Aims, was titled Teddy Bear Dresses the
Sasens. A handout posed the question how many different outfits can Teddy

make from 2 hats and 3 sweaters? Teachers were asked to dress bear cut-outs in

187

as many different outfits as possible using cut-out clothing of three sweaters and
two hats. After a few minutes, Sandy suggested teachers try variations of the
problem and provided additional cut-out clothing for that purpose. The
variations participants may have tried were not discussed in the larger group.

As a pair, the two activities introduced teachers to the discrete
mathematics content strand. -The time that teachers engaged in these activities
fell within the 20-30 minute framework Sandy devoted to each strand. She
indicated to teachers that the time she allowed for teachers to work with the
activities was abbreviated in comparison to what she thought appropriate for
students. Discussions were brief, mostly questions not about the mathematics.
Teachers’ interactions with each other or with Sandy were usually cut-short by
Sandy in the interest of getting to everything.

I turn now to focus more closely on participants’ experiences with the two
activities. Sandy's introduction to the ice-cream cone problem immediately
grabbed teachers' attention. She entered the room with a huge bucket of ice-
cream asking participants whether they liked ice-cream. She began scooping out
ice-cream cones for participants making sure each one had two scoops on their
cone. As Sandy dished out ice-cream, she directed participants attention to the
hand-out in front of them.

Group leaders had passed out paper cut-outs of ice cream cones and cut-
outs of three flavors of ice cream: chocolate, vanilla, and strawberry. Posed on
the overhead projector screen was the following question, ”How many different
two-scoop cones can you make using three different flavors of ice-cream:
chocolate, strawberry, and vanilla?” Participants began immediately

constructing the cones with their cut-outs, and immediately questions arose.

188

Participant:

Sandy:

Participant:

Sandy:

Participant:

Participant:

Participant:

What about a vanilla on top and bottom cone?

I don't know, do you think that is a possibility?

We don't have the cut-outs for that.

But we could trade and provide that.

We could define that the cones have to have different
flavors

I think we should count them because you can buy a
double vanilla cone.

And, what about vanilla on top and chocolate on the
bottom, is that the same as chocolate on top and vanilla

on the bottom.

In an interview, Sandy commented that she was pleased with all the

questions teachers asked. She admitted that she really liked this activity because,

”the decisions students would have to make and how the rules for the problem

can naturally evolve out of the group rather than the teacher making up all the

rules.” She explained, "I like the fact that in this activity the students can decide

what goes and what doesn't" (Interview, 8/93). Sandy let the questioning

continue before she made a suggestion.

Sandy:

Well, let's decide a few things. Raise your hands if you think
chocolate on top and vanilla on the bottom should be counted
the same as vanilla on top and chocolate on the bottom.
[teachers raise their hands]

What about different. [other teachers raise their hands]

After counting Sandy announced:

Okay, I guess we are saying they are different so that means we

189

have to count each one. And, I guess we will say that the cones

have to have different flavors.

Sandy based her determination of whether reverse-ordered cones were
different cones on the basis of teachers’ hands. She then instructed participants
to return to their work. Even though Sandy proposed the process of voting for
deciding the parameters of the problem, she explained in an interview that she
viewed this interaction to be a nice opportunity for students, ”to actually design
the circumstances of the mathematics in the problem” (Interview, 8 / 93). In
particular, Sandy thought these circumstances helped students to focus on the

issue of order in counting. In an interview, she explained,

In one way, as it is decided that two cones with opposite
order are each to be counted, the role of order becomes a
significant point. In fact, the question of order determines
what is to be counted and what is not. In this sense the role
of order becomes significant for students defining what the
counting problem involves. Had this decision been made
differently, the results would be different. For instance, if
opposite-ordered cones were considered to be the same
cone, the number of possible arrangements would be three.
As the problem stands there are six possible arrangements.
So order figures as a significant factor in the problem

(Interview, 8/93).

After a few minutes, Sandy asked if someone would share their results.

One participant came up to the board announcing that he had six cones.

190

Sandy: So, what's your pattern?

Participant: Well, I put vanilla on the bottom and then paired it with
each of the other two flavors. Then, I put chocolate on the
bottom and paired it the same way. I did the same
thing for strawberry.

Sandy: Good, in other words your pattern helped you keep track

of the possibilities. Did anyone do it differently?

Another participant comes to the front and draws a different picture.

Sandy: So what is your pattern?

Participant: It's pretty much the same but I used a different drawing.

Sandy: Good. Once again it is the pattern that allowed you to
keep track.

Sandy emphasized the idea of order and the role of looking for a pattern
to keep track of arrangements. She worried little about participants getting a
correct answer. Sandy believed that what took place contrasted sharply with
traditional views of teaching and learning. Instead of the teacher telling
participants how to proceed, Sandy perceived the teacher as standing back,
interjecting only at critical points, and participants taking charge of their own

learning. She commented,

We often give tasks where we prescribe the rules for them.
For example, if your goal is to have them count by twos, you tell

them twos are multiples of two or count every other number or

191

whatever rule you want to prescribe. When we pose the question
of here's an ice-cream cone with two scoops and three different
flavors to use, how many different ones can you make, there are no
rules. Instead, the rules evolve based on a consensus of the group.
And that is empowering students to a different level of

understanding (Interview, 8 / 93).

After participants presented results, Sandy quickly explained the second
activity. It was clear she was rushed for time. She quickly passed out materials
and explained that this activity is similar to the previous one. The materials
consisted of a handout with the posed problem, bear shape cut-outs, cut-out
sweaters and hats. Sandy noticed the time and said there would not be time to
actually dress the bears but that students in participants’ classrooms should be
given time to do so. She then commented that order is not an issue in this
problem because there is no question of where the hat or the sweater should go.
Sandy then displayed very quickly a process of six different ways to dress the
bear using three sweaters and two hats. Sandy explained that the first sweater
would be matched with each of the hats, then the second sweater with each of
the two hats and finally the third sweater with each hat. Sandy reminded
participants they could increase the difficulty of the problem by adding more
hats, sweaters or including trousers. She quickly wrapped up the segment

making the following points.
Sandy: With permutations the order is important.

With combinations it is not.

So, would this be permutation or combination?

192

[Sandy points to the six arrangements of outfits for the

bear]
Participant: combination [in chorus - many do not answer ]
Sandy: yes, the order was not important. With the ice-cream

cone problem was it permutation or combination?
Participant: permutation [once again in chorus only this time a few
participants responded combination but more loudly
permutation was heard]
Sandy: Yes, the order was critical. But we also could do
combinations and say that order doesn't matter.
Kids love this stuff. I think it is really hard for us but I

find it is much easier for them.

After these comments, Sandy moved on to the next activity relating to a
different content strand. Participants’ opportunity to learn about discrete
mathematics and how to introduce it into their teaching consisted of the
interactions described here, at least for this workshop. Participants were offered
two examples of fairly non-traditional counting problems, some ideas for how to
work on those problems, some ideas about the importance of patterns and some
sense of whether the solution would be considered a combination or a
permutation of the set.

Sandy, in an interview, said she felt satisfied with what she offered
participants for a first experience in thinking about discrete mathematics. She
claimed her ideas offered participants a starting point for thinking about
combinations and permutations. And she indicated that she hoped participants
would find other opportunities to continue developing ideas about counting and

more broadly discrete mathematics (Interview, 8/93).

193

Teaching Combinatorial Counting For Understanding

The case study of Sandy's efforts to introduce discrete mathematics into
elementary school teaching illuminates what it is like for a teacher to learn an
unfamiliar topic in mathematics. It reflects mostly a difficult and bleak set of
circumstances. The opportunities Sandy encountered for learning about discrete
mathematics were difficult at best. Yet despite the difficulties, Sandy assembled
some ideas about counting that would serve as a basis for introducing discrete
mathematics into the elementary mathematics curriculum. And she offered what
she learned to other teachers as well.

Was the experience Sandy offered at the state-sponsored workshop
consistent with what the state had in mind for introducing discrete mathematics?
Did Sandy provide an example of what it might be like to teach discrete
mathematics for understanding? In the remainder of this chapter, I address these
questions from the perspective of the theoretic framework underlying this
dissertation. In other words, I consider whether Sandy’s learning experiences
offered what she would need to teach discrete mathematics for understanding. I
begin by considering the mathematical ideas Sandy grappled with. I compare
and contrast the central ideas of her learning experiences with my analysis of
what may be central for knowing and doing combinatorial mathematics. My
purpose is to check out whether Sandy’s learning experiences centered on or
overlapped with the central tenets of combinatorial mathematics. My aim is to
see whether Sandy acquired the subject-matter knowledge she would need to

teach combinatorial counting for understanding.

194

The Mathematics This Teacher Fecused m

Recall that Sandy’s mathematics learning in the context of formal
coursework was quite limited. She described her high school experiences as
mostly procedural in nature. In college, she had only one course focusing on
mathematics, an elementary math methods course. More recently Sandy learned
mathematics in the context of her training with Project AIMS.

The experiences Sandy had previously were ineffective in terms of
preparing her to learn about continuous and discontinuous mathematics. She
also had great difficulty learning about recurrence relations. Recall that the
district math specialist considered relational theories to be essential for
understanding discrete mathematics. Sandy could make little sense of the ideas
she encountered as central to discrete mathematics. It would require that she go
far beyond the more procedural ideas she had previously encountered. Sandy
would have to draw on a wider range of knowledge about mathematics that she
currently did not have. She would need to understand finite and infinite sets,
continuity and infinity, as well as the idea of functions.

Given that Sandy knew of no formal courses of study aimed at learning
about discrete mathematics, she arranged for a variety of different experiences
she thought would be useful. She proceeded to learn about a curious mix of
mathematical ideas mostly in the contexts of conversations with other educators.
She encountered continuous and discontinuous mathematics, the integers,
combinations and permutations, order and patterns, and combinatorial counting
ideas. Most of the experiences she encountered were frustrating for her.

Sandy's learning experiences ricocheted in multiple directions, stopped
and started, shifting in focus over the course of time. Eventually she isolated
herself from other educators and began focusing on a small set of curriculum

materials designed to offer students experiences with non-traditional counting

195

problems. She cut off the opportunity to learn from others and relied solely on
what she could gather for herself. She eventually progressed into mathematical
ideas she had no previous experience with. She learned what she could teach
herself about the underlying ideas of order and patterns for systematically
keeping track of outcomes (Interview, 8/ 93). What she learned contrasted
sharply with her earlier experiences of memorizing formulas and getting right
answers (Interview, 1 / 92).

Sandy relied on her own insights for making connections between her
learning and what the documents proposed. She came to define discrete
mathematics as, ”learning to organize and count outcomes, it is how one reasons
through counting problems,” (Interview, 8/ 93). Her definition, although narrow
from the perspective of the larger field of discrete mathematics, at the same time
captures a very central piece of what discrete mathematics is. This definition
contrasts sharply with her initial hunches, ”I think it is math in the real world”
and ”maybe it is connected to the elements chart in science” (Interview, 1 /92).

Given the progress Sandy made, a question remains. Did Sandy learn
what she would need to teach combinatorial counting for understanding? To
address this question, one must first consider another. What would it look like to
teach combinatorial mathematics for understanding in the elementary grades?
Unfortunately we have few examples at the elementary grades to consider.
Consequently, there is much uncertainty around the question.

What is possible is to speculate about what it would mean to understand
combinatorial counting questions, in particular the questions Sandy explored
with other teachers at the workshop. Jerome Bruner argues that all students can
be taught all subject matters in "intellectually honest" ways (Bruner, 1977, 1990).
Bruner’s argument rests on the idea that the integrity of both the subject matter

and the learner must be respected. My purpose in this discussion would be to

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uncover what it might mean to respect the integrity of the mathematics of
combinatorial counting to see if Sandy's learning and teaching centered on those
ideas. My analyses would be limited to the mathematics embodied in the two
counting problems focusing Sandy's work.

In formulating the analysis, I draw from my work in chapter 3, NCTM’s
Yearbook, Discrete Mathematics Acress the Curriculum, K-12, (1991a), several
discrete mathematics textbooks, an interview with a mathematician, and my own

learning about discrete mathematics-”’7

Combinaterial Mathematics

Combinatorics, a sub-category of discrete mathematics, is the study of a
particular kind of counting. It involves looking for relationships and patterns as
well as uncovering techniques in counting various arrangements, configurations,
or combinations of mostly discrete objects. Combinatorics is valued as a science
in itself and for the contributions it offers real-world contexts and other sciences.

Sandy’s teaching focused on two combinatorial counting problems. To
understand the mathematics that underlies her teaching, I begin by exploring the
mathematics of combinatorial counting questions. Such questions ask how many
different combinations or arrangements of objects can be generated from an
original set. More simply, how many different ways are there to combine the
elements of a set. An example might be how many different phone numbers can
be created using 9 digits and a specified organization of those digits. Another

example might be how to arrange children in a classroom so that each child can

 

37My earlier analysis of proposals to introduce discrete mathematics is situated inside chapter
three, part two of this work. My own learning about combinatorial counting has been informed
by my own teaching, discrete mathematics textbooks, and reading NCTM’s 1991 yearbook on
discrete mathematics. In addition, I learned a great deal from Bruce Sagan, a mathematician at
Michigan State University, who took the time to explain his views and perceptions of what
discrete mathematics involves.

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be paired with a different partner each week, insuring that everyone has a chance
to work with everyone else. The counting process in each of these examples is
rather complex and quite tedious in nature, requiring much insight and strategy.
It requires a kind of reasoning that allows generalizations to be made so that
results can be determined without wading through the tedious counting process.
This kind of reasoning, combinatorial reasoning, would be at the heart of
combinatorial counting. It is embodied in the formulas and algorithms
mathematicians derived to formulate the counting process, the same formulas
highlighted in mathematics textbooks.

The idea that efficient and accurate results can be accomplished by a
means simpler than actually counting every possibility is an appealing goal.
Mathematicians make their living working out such puzzles. Mathematicians
doing research in the field of combinatorial analysis are interested in
conjecturing, formulating, and justifying theories about counting. Theories about
counting are mostly valued on the basis that they can provide good information
on complicated counts while avoiding the tedious nature of the counting process.
From this standpoint, combinatorics involves formulating good hypotheses for
making counts, justification for what the techniques accomplish, and proving
that the results they offer are accurate. Combinatorics as a science has become
increasingly useful for managing current problems and questions within our
culture. It has become one of the fastest growing areas of mathematical study.

Reasoning in ways that make tedious and complex counting problems
more manageable would be an important characteristic of what it means to
understand combinatorial analysis. Recall that Sandy's high school experiences
involved memorizing formulas and solving all the problems and applications in
the text. Yet, she never really understood the formulas in terms of the

mathematics or why they worked. Like most teachers Sandy would not bring

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this knowledge to her opportunities to learn about discrete mathematics.
Instead, she would have to be offered opportunities to understand what
underlies the formulas she memorized. Her knowledge at that time would
naturally be bounded by procedures, the typical preparation of most teachers.

Sandy's learning experiences pressed her to consider the ideas of
combinations and permutations in ways devoid of formulas. Her teaching
focused on the idea of order and the importance of identifying patterns for
making accurate counts. In contrast with her high school experiences, Sandy
made much progress. She experienced a significant departure from earlier ideas
she had about non-traditional counting. Yet, did she develop ideas about what it
would mean to reason combinatorially? I examine this construct much more
closely in the paragraphs that follow.

Below is a speculative list of elements that are characteristic of
combinatorial reasoning. My findings are based on my reading of the NCTM
yearbook on discrete mathematics, my reading of several discrete mathematics
textbooks, and my interviews with Bruce Sagan a mathematician specializing in
discrete mathematics. The list is not intended to be exhaustive but instead offers

a View of what may be central to the combinatorial counting process.

Essential Elements of Combinatorial Reasoning
* knowing when counting arrangements is the question, the nature of
particular kinds of counting problems
" knowing what to count, different problems involve counting
different things and in different ways
" understanding the tedious nature of counting
"’ formulating systematic ways to keep track of the counting process

* identifying patterns in counting

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* Knowing when you have counted all possibilities
" considering whether generalizations are appropriate
" formulating generalizations

" considering the boundaries to those generalizations

Each element involves reasoning about a particular aspect of the counting
process. I have italicized one particular element for the purpose of drawing
attention to it. Reasoning about when you have all possible outcomes is a critical
aspect toward generalizing results. And, as I argued above, generalizing
counting ideas is central to combinatorial mathematics. It moves mathematical
knowing from concrete examples to abstract theories. It formalizes knowing as a
process of reasoning.

Although there is much left untouched and unexamined in what I have
offered, the elements of combinatorial reasoning can be useful for comparing and
contrasting the central ideas Sandy encountered in learning about combinatorial

counting as well as her teaching of combinatorial mathematics to other teachers.

An Appraisal of A Teacher Leader's
Teaching the Mathematics Of Counting

Sandy’s teaching touched on at least three elements on the list. She
offered participants an experience with the tedious nature of the counting
process in the context of two non-traditional counting problems. She centered
learning on the idea of order although I will argue later that Sandy's sense of
when and how order mattered is problematic. And, she focused on identifying
and using patterns for the purpose of counting systematically.

Unexamined elements in Sandy's teaching included when counting
arrangements is the question, whether generalizations seemed appropriate, the

business of generalizing, or considering the boundaries of generalizations. Even

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more problematic was that Sandy’s teaching left the issue of reasoning about
when all possibilities were accounted for unattended. This aspect, because it was
left unattended, did not set the learner up to generalize information about the
counting process. In effect, teachers lacked the opportunity to conjecture about
the process, generalize their understanding, or uncover any limits to claims.

An important question is whether Sandy thought of her efforts as a step in
a larger learning process, one that would lead to an overall view of the
mathematics of combinatorial counting, perhaps across a series of connected
episodes of teaching? A simple answer would be yes, of course. Yet, Sandy’s
workshop was a one-shot opportunity.

Would the experience Sandy offered set the learner up to connect to the
elements that were lacking? I address this question by first revisiting Sandy's
teaching at the workshop. Each of the counting problems Sandy offered
involved a fairly non-traditional kind of counting question, especially when
compared to traditional school mathematics curricula. Sandy’s practice
emphasized hands-on, learning by doing activities. Manipulatives, small group
discussion, and emphasis on the ideas of order and systematic counting were
central features of the learning experience. For many participants, Sandy’s
teaching might represent an entirely new approach to teaching mathematics
especially when compared to their own teaching.

Yet, below the surface of Sandy’s teaching were critical problems with the
mathematics she offered. NCTM’s yearbook on discrete mathematics concurs
with Sandy’s idea that counting should be emphasized in the early grades K-3
(NCTM, 1991a, p. 18). The yearbook (p. 19), departs from Sandy’s ideas in that
emphasis should be placed on understanding the nature of different kinds of
counting questions. Sandy’s teaching gave little attention to the nature of the

differences underlying the two counting questions she offered. Mathematically,

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each question involved entirely different counting structures, resulting in very
different kinds of generalizations.

Consider the ice-cream cone problem. Sandy asks participants at the
workshop to find the number of different two scoop cones (two element sets)
that can be created from three flavors of ice-cream (a three element set). The
question involves taking the elements from the set {v, c, 5}, corresponding to
vanilla, chocolate and strawberry, and arranging the elements in as many ways
as possible using two element. A single set of three elements serves as the basis
from which all possible two-element sets can be created. The possibilities
include {v, c}, {v, s}, {c, s} if no repetition of any flavor is permitted. Sets such as
{v, v} repeat the same element and {c, v} reverses the order of a set already
created. These sets may or may not be included in the count depending on the
parameters of the problem. If they are included, the number of possible
solutions changes. Mathematically, generalizations result from taking three
elements, two at a time.

The second problem is a different kind of counting question. There are no
decision points about repeating the same element or reverse-order. A bear is to
wear only one sweater and one pair of pants, worn in only one way. In this
problem, order in a set is not relevant. There are two distinct sets that serve as
the basis from which all arrangements can be formed. One set contains three
sweaters, each of a different color. The second set contains two pairs of pants
also of different colors. For example, {white, red, green} and {black, brown}. The
sets are disjoint. Arrangements involve taking one element from each set to form
a two-element set. There are six and only six possible arrangements pairing one
sweater with one pair of pants. It would not make sense to consider a {white,
black} arrangement as different from a {black, white} arrangement. The outcome

represents the same arrangement.

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Sandy's teaching did not address the differences in the underlying
structures of the two problems. She treated the questions generally, almost as if
they were the same type of counting problem. The issue becomes more obvious
and relevant when the goal is to generalize the reasoning process. The second
problem involves counting arrangements that can be formed by joining one
element from each of two disjoint sets in a particular way. Mathematicians
generalize these findings in what is known as the product rule. The product rule
states that the total number of arrangements can be found by multiplying the
number of elements from each set, in this case 2 x 3 or 6 possibilities.

The nature of these differences is significant when generalizing becomes a
central goal. Differences can be revealed by examining underlying structures
and the influence of those structures on the counting process. Participants at
Sandy’s workshop were not offered an opportunity to consider the nature of
these differences. Nor were they offered any opportunity to form generalizations
of each case.

Sandy’s practice has the potential to set the learner up for misconceptions
about what combinatorial mathematics is. By de-emphasizing the nature of the
differences in the problems offered Sandy ignored a critical element required for
generalizing the reasoning process. Teachers did not encounter the opportunity
to reason about when all possibilities are accounted for. By ignoring this element
of the counting process, the learner is not set up to generalize results, even if
offered in subsequent learning contexts. Recall from my sketch in chapter three
that the central purpose of combinatorial mathematics is to find ways to make
difficult counts without the tedious step—by-step nature of counting.

My analysis is not intended to suggest that what Sandy did do is incorrect
or unimportant. I also am not suggesting that teaching should move quickly and

rapidly to formalizing findings. What I am suggesting is that teachers would

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need the opportunity to explore the underlying structures of different kinds of
counting questions and work to explain how they know when all arrangements
are accounted for if they are to understand what underlies combinatorial
mathematics. Without this experience, generalizations will more than likely
remain as formulas to be memorized and not understood. Sandy’s teaching does
not set the learner toward this direction. Her teaching about the ideas of
combination and permutation became unknowingly corrupt in practice. If Sandy
understood the central tenets of combinatorial reasoning, she would be better
prepared to examine those ideas with other teachers.

My analysis suggests that Sandy's teaching overlaps only slightly with
what it might mean to teach combinatorial counting ideas for understanding.
And, Sandy’s learning experiences did not provide the substantive knowledge
she would have needed to build her capacity to teach combinatorial mathematics
in ways that emphasize understanding. The opportunities Sandy encountered
offered bits and pieces of knowledge with little attention to how any of the ideas
fit together toward an overall view of. combinatorial mathematics. Sandy’s story
uncovers what may be a deep rooted contradiction. On the one hand,
policymakers argue that teachers must have an explicit, deep knowledge of the
subject matter if they are to teach for understanding. Yet, proposals in the
reform documents suggest that discrete mathematics can be introduced into the
school curriculum in a relatively easy fashion, by way of what is already there.
Recall the rhetoric, ”teachers at all levels are already teaching a variety of

mathematical ideas considered important to a study of discrete mathematics,”

(NCTM, 1989).

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CHAPTER SIX

RE-FORMING MATHEMATICS TEACHING
AND THE TEACHER'S OPPORTUNITIES TO LEARN

Introduction

In the two previous chapters I focused separately on Sandy’s responses to
two different strands of reform ideas. I suggested that Sandy's learning
experiences in each case did not effectively close the gap between her practice
and the teaching envisioned in the reform documents.

My purpose in this chapter is to understand why. The state developed a
multitude of levers and supports to press teachers toward desired practices.
Sandy was motivated and committed to the state's goals. Yet, my appraisal of
her teaching in both case studies revealed little common ground between the
two. To understand why, I examine Sandy's professional development activities
across the case-studies. I compare and contrast the opportunities to learn that
Sandy encountered in light of policy goals. I explore what the occasions were
like, what they offered Sandy substantively and structurally, and to what extent
they offered ideas and practices envisioned in the reform documents. My aim is
to develop a portrait of the circumstances and conditions that mediated Sandy's

learning about the proposals she responded to.

A Pedagogical Framework
In chapter 1 I discussed the assumptions about policy-practice relations
underlying this study. I suggested that the "policy as pedagogy" framework was

not the common view. Historically policy and its implementation has been

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described as putting or placing recommended changes into practice by methods
of transmission, implantation, or mutual adaptation (Fullan, 1992; McLaughlin,
1976). In contrast, I assumed that learning was the central activity of policy
implementation (Cohen 8: Barnes, 1993a; Cohen 8: Ball, 1990). Thus, I viewed
Sandy’s efforts to reform her mathematics teaching from a cognitive frame. I
assumed that implementation rested for the most part on what and how she
learned. My analysis in this chapter rests on these assumptions as well.

In this chapter, my goal is to examine the relationship among three central
elements of the framework: the subject matter of reform, the teacher as the
learner of reform, and the teacher's opportunities to learn. I begin by sketching
ideas about the curriculum of reform. I compare and contrast the cases to see
what the proposals involve, including recommendations for how changes might
be accomplished as well as the implications for teacher learning. I then focus on
the learner of reform. I develop a sketch of Sandy’s personal history with each
strand of reform ideas. I look to see what the similarities and differences are
across the mathematics topics and what the implications are for Sandy as a
learner. Finally, I focus on Sandy’s opportunities to learn. My analysis aims to
uncover whether and how Sandy’s opportunities to learn challenge and extend
her prior knowledge and teaching experiences in ways that foster understanding

and enactment of the curriculum of reform.

A Cnrriculum ef Referm I

There were three central kinds of recommendations made across the
proposals for altering the content and pedagogical practices for teaching

computational skills and discrete mathematics.

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*Introducing new content
*Teaching familiar mathematics in new ways
*Introducing problem solving contexts and

conceptual orientations to teach all content

I explore each of these recommendations below.

Innoducing New Centent

Introducing new content involves the proposal to teach new mathematics
content, ideas not already part of the school mathematics curriculum. This
includes ideas such as students’ strategies for computing, non-traditional
algorithms for computing, conceptual basis of algorithms, and combinatorial
counting principles such as the inclusion-exclusion principle or the pigeon-hole
principle as examples. Introducing new content would require teachers to learn
new mathematics. For example, introducing the conceptual basis of traditional
algorithms would require that teachers learn the conceptual underpinnings of
addition, subtraction, multiplication and division. Teachers would also need to
learn how to develop those ideas for students across a series of lessons that aim
to support students' understanding and proficiency with computation.

An example of introducing new content to support the goal to teach
discrete mathematics would involve combinatorial counting ideas. Most
elementary teachers would not already understand combinatorial mathematics.
Not only would the teacher need to learn the mathematics of counting, she
would also need to learn how the ideas work together to support an overall view
of combinatorial mathematics. Consequently, introducing new content into the
existent mathematics curriculum implies that teachers would need to learn much

new subject-matter including how mathematical ideas connect.

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Further complicating matters is the treatment of content as discrete
entities in the reform documents. The mathematics across the cases illuminates
this problem. The mathematics of combinatorial counting and computation is
actually more connected and alike than one might first imagine. Although each
involves uncommon ground, they share ideas as well. The overlap can be
imagined by comparing a traditional counting question and a combinatorial
counting question. For example, traditional counting questions ask for a total
when 4 things are added to 2 things or 4 things are multiplied by 2 things.
Combinatorial counting questions extend counting into a new realm asking, for
example, how many different arrangements are possible when there are 4 things
and 2 things to choose from. Combinatorial questions typically ask for possible.
configurations, arrangements or combinations. Computation typically involves
finding sums, differences, products, quotients, and additionally percents and
fractions. Conceptually, the ideas of grouping and combining connect the two.
Fundamentally, the mathematics of combinatorial counting is an extension of
traditional counting ideas. Most elementary teachers would not bring these
insights to reading the documents. At the same time, the documents make no
mention of any connections across content even as they aim to alter the

fragmented approach to teaching mathematics in schooling.

Teaching Familiar Mathematics in New Ways

This recommendation requires teachers to reconsider the familiar
mathematics they already teach. Historically, traditional computing algorithms
have defined the computational curriculum in public schooling. Policymakers
argue that there are a variety of new computational skills that must be taught,
estimation is one example. Estimation should be taught as a computational

technique in itself, not as a checking strategy. Estimation is a familiar idea to

208

most teachers. Usually, estimation is taught as a checking strategy. To transform
this aspect of their teaching, teachers would have to transform their ideas about
estimation and develop new ideas for teaching estimation to students.

In reconsidering familiar mathematical ideas, teachers are pressed to
explore the underlying structures of familiar ideas they already teach. For
example, understanding what underlies computational algorithms implies an
investigation of the structures of routine procedures and pedagogy used to
support learning of routine and taken for granted procedures. Teachers would
have to explore such questions as what are the conceptual ideas underlying
routine computation? Other issues would involve whether traditional
algorithms should be taught and if so, to what level of complexity and mastery?
Should drill and practice be used to foster fluency with those skills and is speed
and accuracy still valued?

There are also examples of reconsidering familiar mathematics in discrete
mathematics. The reform documents suggest that discrete mathematics can be
introduced into the mathematics curriculum by way of familiar ideas already
taught. Venn diagrams is listed as an example and are probably familiar to most
teachers. Yet, they are typically used to illustrate relationships among sets. The
ideas of union and intersection are often focal. Venn diagrams are also useful for
illustrating basic counting principles in combinatorial mathematics. For
example, the inclusion-exclusion principle asserts that the number of elements in
two finite sets A and B can be found by combining the elements of set A with the
elements of set B and subtracting what is in common. Another way to state this
principle is to find the total number of elements in both sets one must include all
elements of both sets and exclude those elements in common. To illustrate,

consider the Venn diagram below:

209

 

 

 

Set A Set B

 

Figure 7.
Venn Diagram

Set A has 7 elements, set B has 10. A and B, when added, have a total of 17
elements. Yet, four of those elements are common to both sets. Thus, 17 - 4
yields 13 elements total in the two sets. The Venn diagram is useful for showing
the relationship between A and B and the total number of elements. Although
Venn diagrams still represents relationships between sets, the counting principle,
for many, will be an entirely new mathematical idea. It would be impossible to
introduce combinatorial counting ideas by way of Venn diagrams without
knowledge of the underlying counting principles. Venn diagrams do not
represent the mathematics of discrete mathematics. They are useful to display
relationships and counting principles. But they work in coordination with other
knowledge to introduce the ideas of discrete mathematics. Much learning of
new subject matter would be required if discrete mathematics is to be introduced
by way of what is familiar.

Teaching familiar mathematics in new ways requires more than
reexamining familiar content. It also requires reconsidering the pedagogical
practices for teaching familiar mathematics. For example, traditional forms of
practice often result in learning that is rule-bound. Such practices contribute to
other problematic issues such as students' view that mathematical knowing

resides in the teacher or text and not in students' sense making. Many teachers

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teach unaware of these issues. Such issues point to the third aspect of the
curriculum of reform: the recommendation to introduce new pedagogical

practices.

Intreducing New Pedagegical Practices

The proposed changes across the strands intertwine content and

 

pedagogy so that what teachers would have to do and learn is not so neatly
either content or pedagogy. To this point, my discussion has focused mostly on
changes in content. Yet, changing the content of the elementary school
mathematics program would not be sufficient to create the kind of teaching
imagined. Interconnected to changes in content are changes in pedagogy. In
other words, teaching students non-routine algorithms, estimation as a
computing strategy, or the use of new technologies for computational work in
traditional, didactic, lecture-type formats wOuld not accomplish the teaching
imagined. New content must be introduced in relationship to proposed changes
in pedagogy. Essentially, computational skills teaching would be done in
problem-solving settings. Teachers would need to learn to facilitate
computational goals simultaneously with problem solving goals. To illustrate

further, I draw from the Teaching Standards 1991, p. 26.

Teachers must assess the extent to which skills play a role in the
context of particular mathematical topics. A goal is to create
contexts that foster skills developments even as students engage in
problem solving and reasoning. For example, elementary school
students should develop rapid facility with addition and
multiplication combinations. Rolling pairs of'dice as part of an

investigation of probability can simultaneously provide students

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with practice with addition. Trying to figure out how many ways
36 desks can be arranged in equal-sized groups and whether there
are more or few possible groupings with 36, 37, 38, 39, or 40 desks -
presses students to produce each numbers’ factors quickly. As they
work on this problem, students have concurrent opportunities to
practice multiplication facts and to develop a sense of what factors

are.

The vision of students working to determine possible desk arrangements
requires students to work on multiplication facts. This practice would be
dramatically different from modal teaching where students memorize
multiplication facts using worksheets or other drill and practice activities.
Underlying the change is the argument that what students learn is
fundamentally connected with how they learn it.

What a teacher might need to learn to make sense of this paragraph is not
straightforward. In addition to learning constructivist ideas about learning,
teachers would also need to learn new discourse patterns requiring substantial
shifts in teachers’ and students’ roles. Teaching as telling would no longer
dominate classroom discourse. Instead, students would work on mathematical
problems in small and large groups, talking to each other and to the teacher.
Teachers would have to learn when to tell students something as opposed to
facilitating students’ construction of an idea. These sorts of changes require
substantial new insights into mathematics and pedagogy.

Interestingly, the text above intertwines the mathematics content of the
two strands focusing this study. Trying to figure out how many ways 36 desks
can be arranged in particular sized groups is a combinatorial counting question.

Yet, unless and until the teacher has ideas about what combinatorial counting

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involves, this paragraph most likely would get overlooked as an example of
introducing discrete mathematics. In most cases, teachers would miss this
occasion as an opportunity to learn about discrete mathematics.

The curriculum of reform is immense. The analyses above only begins to
unpack what teachers would need to learn to enact the proposals. The examples
sketch only some of the problems that teachers would encounter if they were to
give serious attention to the recommendations. Yet, a very important insight
arises from this work. What there is for teachers to learn, the vastness of what is
there, contributes greatly to the disjuncture between envisioned ideas and
practices and the teacher's capacity to reform.

I turn now to focus on Sandy as a learner. I look across the cases to

compare and contrast what she brings as a learner to reforming her teaching.

The Teacher as Learner

Research on teacher learning suggests that Sandy’s interpretations of the
proposed changes are naturally influenced by what she brings as a learner. To
understand Sandy’s interpretations of the proposed changes, I explore her
personal history surrounding each strand of reform ideas and how her history

may have functioned in making sense of the proposed changes.

What Dees The Teacher Bring?
The columns in the table list the ideas that Sandy brought to her learning

experiences for each strand. The lists are organized for the purpose of
comparing the ideas across the cases. For example, content knowledge in one

strand is organized across content knowledge in the other strand.

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Table 1. What Sandy Brings

 

Computational Skill

Discrete Mathematics

 

 

Sandy brings ideas about what computational
skill is - mostly proficiency with traditional
algorithms, mental computation and using

those skills in problem solving contexts

Sandy brings ideas about techniques for
computing , i.e. paper pencil techniques,
borrowing, carrying, etc. She brings ideas
about getting the right answers, speed
accuracy, and efficiency. Sandy brings ideas
about why computational skill is included in
the elementary school curriculum; because it is

a useful skill in daily life

Sandy brings a very detailed picture of what
the computational curriculum involves and
how it might be situated within the school

curriculum

no ideas about what discrete mathematics is
no ideas about what the central questions of
discrete mathematics involve or why the

questions are important to ask

no ideas about the techniques or strategies
used for examining those questions
no ideas about what discrete mathematics

explores

no ideas for how topics, ideas might be

included in the school curriculum

 

 

214

 

Table 1 (continued)

 

 

Ideas about the conceptual underpinnings of
the algorithms / and other conceptual ideas
related to the four operations. Sandy also
brings ideas about the important elements of
computational skill such as getting the right
answers, knowing facts and algorithms, speed

and accuracy.

Sandy brings ideas about the underlying
systems of numbers. She understands what is
involved when operating within different
systems of numbers. She has ideas about place
value, different bases, operations with

decimals, fractions, etc.

Sandy brings an entire curriculum including a
set of firmly entrenched instructional practices

she believes develops computational skill

 

some familiarity with only a few of the central
ideas identified in the Framework as
important to discrete mathematics. For
example Sandy has some familiarity with the
ideas of Venn diagrams, deductive logic, and
combination and permutation. Sandy's
familiarity is not inside the context of discrete

mathematics.

no ideas about the system of numbers that
underlie discrete mathematics. Although
Sandy has ideas about the integers as a
number system and arithmetic as a study on
that system she does not know whether or
how the integers fit with discrete mathematics.
She has some familiarity with continuous

mathematics at least algebra

some ideas for what instructional practices
might best represent discrete mathematics
including problem solving and investigation

type teaching methods

 

The contrast between the strands is striking. Like many teachers, Sandy

already had many ideas and understandings related to the content and

instructional practices of basic computational skills. Sandy brought little, if any,

215

 

specific ideas about discrete mathematics. What she did bring, she had little
ideas for how it might help her to learn about discrete mathematics.

Sandy brought firmly grounded ideas about what computation involves
and how children become proficient at it. For example, Sandy brought to her
learning about the proposed policy changes with the View that computational
skill involves proficiency with traditional paper-pencil algorithms and some
limited mental calculation as well. Sandy also brought to her learning the idea
that memorization using drill and repeated practice exercises are best for
learning those skills. Recall in the case study how Sandy encouraged students’
use of traditional paper-pencil algorithms even in settings where computations
could have been made more easily using alternative strategies. She also stressed
the importance of speed and accuracy with traditional computational algorithms
using students’ participation in a pizza party as a motivational factor.

Sandy also brought a very detailed picture of what the third grade
computational curriculum should include. For example, she had specific ideas
about which algorithms should be taught in the third grade and to what level of
complexity and mastery. Recall that Sandy thought mastery of the division
algorithm did not belong in the third grade arguing that conceptual grounding
should dominate third grade instruction of division (Interview, 1 /93). Yet, she
believed that mastery of the traditional multiplication algorithm was an essential
goal in third grade, not only to four digits but also with complicated carrying
procedures involving numbers with many zeros.

In addition, Sandy brought very specific ideas about the instructional
practices necessary for developing computational skills. They included a
number of gimmicks to help students remember difficult procedures and deal
with common errors. Recall the phrase ”slash, burn it off,” for remembering to

cross out a zero and replace it with a nine. Sandy often relied on catchy phrases

216

to help students memorize computational techniques as well as help students
avoid common errors in the procedures.

Sandy had specific ideas about what it would mean to understand
computational algorithms conceptually. Consider her ideas about multiplication
as ”adding fast” and division as ”fair-sharing.” Sandy’s ongoing work with
Project AIMS contributed greatly to how she thought about conceptual
understanding and conceptually orienting the school mathematics curriculum.
Project AIMS assumes a hands-on, learning by doing philosophy. Sandy’s ideas
about problem solving and contextualizing mathematics evolved in relation to
her work with Project AIMS. Yet, my analysis in the case study suggests Sandy’s
ideas and practices overlapped only slightly with policy ideas relating to
conceptual understanding and contextualizing mathematics.

In contrast, Sandy had very few ideas to work from for learning about
discrete mathematics. In fact, she claimed she knew so little that she had trouble
making any sense of what was written in the reform documents. She concluded
the documents confused her and were not very helpful for developing a view of
discrete mathematics or how to introduce it into the school curriculum. Even
though the documents suggested that much of what discrete mathematics
involves is already largely part of the school curriculum, Sandy had no ideas for
what this would be. She admitted she was unable to make sense of the proposals
and had no ideas for how the mathematics she was already teaching connected

to discrete mathematics.

Prier Knewledge as Prerequisite Knewledge

Sandy’s prior knowledge and teaching experiences function as
prerequisite knowledge for making sense of new forms of practice. As a

resource, prerequisite knowledge significantly affects a teacher's capacity to

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reform. Sandy brought very different ideas and experiences across the topics of
computation and discrete mathematics to draw from. Her ideas about
computation ran deep and were mostly contrary to the proposals. As
prerequisite knowledge, much would need to be challenged, uprooted and
discarded if Sandy were to embrace ideas and practices compatible with policy.
Sandy’s prior knowledge for learning about discrete mathematics was highly
underdeveloped, predictably so. She had great difficulty making sense of the
examples and illustrations offered in the reform documents.

In contrast, the cases represent opposite situations from the standpoint of
what the learner brings. On the one hand, Sandy’s prior knowledge and
experience surrounding computational goals was enormous and very specific.
Yet, most of her ideas and practices ran contrary to the proposed changes. In the
case of discrete mathematics, Sandy’s knowledge was so underdeveloped that
even reading the reform documents was puzzling and strange for her.

These findings have very different implications for Sandy as a learner.
Sandy’s prior knowledge and beliefs must function as a basis on which to build
her understanding of the proposed changes. Yet, as prerequisite knowledge,
what Sandy brought was often mismatched with what she would need to make
sense of the ideas offered across the opportunities to learn that she encountered.
Recall the example of Sandy trying to make sense of the illustration in the
Curn’culum Stendards (1989), involving a directed graph for counting the
number of different paths between two points. The example was offered as an
illustration for introducing discrete mathematics. Prerequisite knowledge for
making sense of this illustration would include fairly well-worked out ideas
about matrices. Yet, like most elementary teachers, Sandy did not bring ideas

about matrices to her opportunity to learn from this illustration. Her learning, at

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least in this context, remained limited to what she could muster up for
understanding matrices.

In some instances Sandy’s prior knowledge may have functioned to
promote a mismatch in her understanding of the policy proposals. For instance,
the policy suggests that the computational curriculum be conceptually oriented.
This proposal implies that teachers would either need to bring a conceptual
understanding of what underlies computation or they would need to develop
those ideas. In Sandy’s case, she brought well-worked out ideas about the
conceptual underpinnings of the four operations. However, I described in the
case study that her ideas and understanding did not coincide with what the
policies described as conceptual understanding. In this example, Sandy’s
prerequisite knowledge functioned to promote a different view of conceptually

orienting the computational curriculum than what the policy described.

Prier Knewledge and Framing a Ceurse ef Actien

Across each strand, there is a striking difference in how Sandy framed
what she would need to do to respond to the proposals. Although I argued that
the recommendations across the strands implied much new learning for the
teacher, Sandy framed her work with discrete mathematics to involve learning
new content and her work with computation to involve a shift in instructional
time. In each case, Sandy’s framework for attending to the proposals focused on
only bits and pieces of the range of different recommendations. This
separateness, uniqueness, and attention to only aspects of the proposed changes
created a kind of multiplicity effect on her learning. Because Sandy had few
ideas about how discrete mathematics and computational skills connected
mathematically, she perceived each strand to involve a unique course of action,

with little, if any, overlap.

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Recall that Sandy had already created a conceptually-oriented and
problem-solving approach to teaching mathematics. To that end, she had
minimized traditional practices aimed at the rote-learning of computational
algorithms. Sandy’s reading of the new framework interacted with what she had
already established. She interpreted the proposals to suggest that she place more
emphasis on conceptual and problem solving goals and that she should nearly
extinguish all traditional practices surrounding computation. For Sandy, the
recommendations implied a shift in emphasis and time, not a reconstruction of
her ideas and practices.

Once problems arose in her practice, Sandy changed her ideas about what
she would need to do. As she began to search for opportunities to reconsider her
practice, she became increasingly convinced that her initial interpretations of the
policy were wrong. After much learning, Sandy interpreted the proposals to
suggest she should return to practices she once had abandoned. However, what
Sandy learned did not challenge her prior knowledge and beliefs about
computation. She put aside her more traditional ideas about computation as she
focused on the conceptual underpinnings of the four operations. She assumed
that emphasizing conceptual understanding and problem solving would also
promote students’ fluency with computation. Sandy's more traditional ideas
were never really challenged, only shelved temporarily. There were other
aspects of the proposals that went unexamined as well. For example, Sandy
hardly responded to the recommendation that a variety of new computational
skills be introduced. Estimation, for example, went mostly unexamined as a
computational strategy. Sandy also did not emphasize technology mostly
because students were not permitted to use calculators on state exams nor did

they have consistent access to calculators. The changes Sandy made only

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touched the edges of the proposals and in part, because of what she brought to
her learning about the proposals.

Sandy framed her initial work with the proposals to introduce discrete
mathematics to involve a very different course of action. Here, she framed the
proposed changes as a matter involving learning new mathematics for herself.
For over a year, Sandy would learn what she could about discrete mathematics.
Because Sandy believed she had already developed a pedagogy that emphasized
a problem solving and conceptual orientation to learning mathematics, she did
not perceive the recommended changes surrounding discrete mathematics to

include, at least for her, any pedagogical changes.

A Teacher's Qppertunities Te Leam
(Links Between Policy And Teaching Practice)

My analysis of the curriculum and the learner of reform suggests there is a
natural disjuncture between the ideas proposed in policy and the teacher’s
capacity to understand the recommended changes. I have suggested at least
three factors contributed to this disjuncture. The curriculum of reform is vast, a
wide territory of new ideas and practices, most strikingly different than modal
teaching. This sets the stage for an enormous agenda for teacher learning, most
of which is not well-defined. What Sandy brings as a learner contributes further
to this gap. I argued there exists a mismatch between Sandy’s prior knowledge
and beliefs and the prerequisite knowledge needed to understand proposed
ideas. This factor alone makes it highly unlikely for any teacher to understand
the proposed ideas without particular kinds of learning. Sandy's prior
knowledge and teaching experiences also interacted with how she framed what

she would need to do to enact the proposed changes in her teaching. Because

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Sandy perceived no common ground across the proposals she consequently
framed each strand to involve distinct agendas for change.

From a pedagogical perspective, what Sandy learns should function to
close this gap. In other words, the opportunities to learn that Sandy encountered
should function to promote and sustain the kinds of support Sandy would need
to create the kind of teaching envisioned in policy. In other words, what Sandy
learns should bring her up to speed for making sense of the illustrations in the
reform documents. Her learning should function to meet her at the point of her
prior knowledge and beliefs. They should foster a way to sensibly frame the
reform agenda to insure that she learns the central tenets in ways that are
manageable and profitable. In essence, Sandy’s opportunities to learn should
function to extend and sustain a knowledge base sufficient to build and support
her capacity to teach in ways imagined in the policy.

The previous two chapters illuminate that Sandy's opportunities to learn
did not function to close the gap. Instead, the learning experiences she had may
have created an even greater distance between the teaching imagined and
Sandy’s practice. My purpose in this section is to carefully examine Sandy's
opportunities to learn, to better understand why they were mostly unproductive
at supporting Sandy's efforts to invent and sustain the kind of teaching
imagined.

I described in chapters four and five the variety of different situations that
Sandy created or encountered to help her understand and enact the proposed
changes focusing this study. I described, for example, how Sandy arranged for
conversations with other educators, how she made use of curriculum materials
and participated in state-sponsored inservice activities. The case studies teach us
that the teacher is likely to make use of a wide variety of learning experiences,

each contributing to the teacher's thinking and the changes made in teaching. In

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this chapter, I consider the occasions for Sandy's learning as a whole. Across the
case studies I examine the qualities and conditions of the set of experiences
Sandy had for learning what she could about the state's efforts to reform
mathematics education. My purpose is to understand what these experiences
offered for building and supporting Sandy’s capacity to reform her teaching.

Whether the occasions were intentional, incidental, planned, unplanned,
formal or less formal, each experience that I uncovered as altering Sandy’s beliefs
or knowledge about computation or discrete mathematics, I considered as an

”opportunity” for her learning. To clarify, I offer the following explanation.

We construe "learning opportunities" to be those experiences, kinds
of work, and interactions that create images and insight, that
generate disequilibrium and curiosity, that offer the possibility of
change and growth. Their quality is shaped by many factors, the
time allotted to them, the engagement they engender, the
possibilities for collaborative work or thought they offer, and the
worth that participants believe they have. Examples may include
courses, workshops, conversations, reading, using a new

curriculum, and teaching experiences. (Ball 8: Cohen, 1995a)

I identified six different kinds of occasions across the case studies that, for
Sandy, fostered new images or insights, created curiosity or disequilibrium, and
resulted in changes in her thinking or teaching. Some of these occasions
involved hours across a number of years, while others lasted only minutes.
Several began prior to our work together and continued during data collection.
Whether Sandy created the occasions herself or she participated in an experience

offered, the categories I list below are representative of the multiple kinds of

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learning experiences Sandy had. Each category represents one or more occasions
that contributed significantly to Sandy’s thinking and altered her ideas and
practices regarding her teaching of basic computational skills or introducing
discrete mathematics into her teaching. In effect, Sandy’s ideas and practices
changed in relationship to her interaction on these occasions.

Drawing on interview and observation data, the occasions most central to

Sandy’s learning across the strands were :

"Sandy’s conversations with leading mathematics educators,
*reading reform documents,

*doctoral studies,

"curriculum materials,

*teaching experiences,

*formal professional development activities

I include Sandy’s ongoing work with Project AIMS and the state-sponsored
workshops she attended as specific examples of the formal professional
development activities she participated in.

Prior to examining the categories I have listed here, I explain briefly
several of the issues that arose in doing the analyses. Characterizing the
substantive focus of Sandy’s opportunities to learn was treated differently across
the categories above. When there were differences in what an opportunity
offered substantively from my view as compared with Sandy’s view I had to
decide what to do. For example, my analysis of what the policy suggested for
reforming basic computational skills instruction was quite different than what
Sandy made of it. Independently, I searched the reform documents for all that I

could find relating to computational skills instruction. My analysis suggested

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there was much more offered substantively than what Sandy reported. When
this occurred, I included both of reports as the basis on which to examine what
Sandy’s opportunities to learn offered substantively. Sandy’s teaching
experiences, reading the reform documents, the curriculum materials she used,
and the professional development experiences she participated in represented
similar kinds of data. In each of these categories, I collected data independent of
Sandy’s self-report and included that data in my analysis. In effect, my
determinations of what the opportunities offered substantively were influenced
by what I could learn about them independent of Sandy's report and yet were
inclusive of Sandy's report.

There were a few occasions that I had little or no independent data on.
For instance, I had no opportunity to observe or interview anyone participating
in Sandy’s doctoral studies. Sandy’s self report and my analysis of the course
syllabus and assignments was the basis of my determinations of the substantive
focus of these opportunities. Sandy’s conversations with other educators is an
example where no independent data was possible. In these instances, it would
have been difficult to collect data on what the opportunity offered because they
were either unplanned or not appropriate to contact people to ask about the
substantive nature of the conversations. I used solely Sandy's self-report to make
determinations about the substantive focus in these instances.

I point to these issues prior to proceeding with my analysis to discuss how
they are reflected in the analyses. It would be problematic to suggest that all of
what was available to Sandy substantively or otherwise is reflected in the
analyses. Because not all of teachers' opportunities to learn are planned or
visible, it is not always possible to collect independent data on what is offered.
Thus, in some cases, a teacher's self-report has to serve as the basis of the data.

The data also does not reflect what else may have been available for teachers'

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learning but not taken advantage of. In this study, an available opportunity
implies that Sandy created or encountered the opportunity to learn. I decided
not to conduct any wider investigation of the professional development activities
offered in other contexts that Sandy did not participate in because my aim was
to understand what mediated a teacher's learning and change and how that
happened. I was less interested in what could have happened had Sandy heard
about or decided to participate in other learning experiences. In making this
choice, the analyses does not reflect information or insights into opportunities to
learn that Sandy elected not to participate in or had no knowledge of.

This last point leads to another. Making decisions about what counts as
an opportunity to learn is not straightforward. It was not as simple as just
noticing that Sandy encountered and participated in a learning opportunity.
Sandy, for example, constructed ideas about computational skills and discrete
mathematics in relationship to experiences that were not designed for that goal.
Recall that she learned about automaticity in a doctoral course. Although the
course was developed to promote learning about curriculum and instructional
design, what Sandy learned from the course had a significant impact on her ideas
about reforming her teaching of basic computational skills. This example begins
to blur the boundaries of what typically is thought of as an opportunity for
teachers to learn about reform ideas. Not all learning opportunities can be
discerned as opportunities to learn ahead of time. It must be shown that what
the teacher learned had a significant impact on the teacher's thinking and
practice in relationship to specific reform ideas. Thus, reflected in the analyses is
both what a teacher learns from more formal professional development activities
as well as opportunities the teacher thought relevant and useful for her learning.

Despite the complexities I discuss here, the data is very robust and strong

enough to bear up to the questions underlying this study. Sandy’s self-reports

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cuts across all the categories I identified as affecting her learning and is a strong
indicator of what each learning experience offered. My independent analysis of
four of the categories provides further insight into what the occasions offered for
learning about the ideas focusing this study. In only two of the categories I relied
on Sandy’s report and any artifacts of the experience that I could collect after the
experience had occurred. On these occasions, my independent analyses was not
possible due to the spontaneous nature of the experience.

To establish the nature of the six categories of learning opportunities that

mediated Sandy’s learning, I introduce three analytic categories:

-the big ideas
-consideration of what Sandy brings

—Sandy’s views of what the opportunity offered

These categories correspond to three critical factors that research on teacher
learning has suggested shape what and how teachers learn: the content or
substance of the learning experience, the interaction of the learners’ prior
knowledge and beliefs, and the interaction of the learners’ perception of what the
experience offers. I begin by examining the substantive focus predominant

across Sandy’s opportunities to learn.

The Big Ideas

In the table below, I contrast the big ideas that were predominant in
Sandy’s opportunities to learn about reforming her teaching of computational
skills with the central tenets offered in the reform documents. The central tenets
of the proposals to reform teaching of computational skills is drawn from my

analyses of the proposed changes in chapter three. My views of the predominate

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ideas embodied in Sandy’s opportunities to learn are drawn from my analysis of
the big ideas across her opportunities to learn (see Appendix A). My purpose is

to see if and where there is overlap.

Table 2. Central Reform Ideas And The Central Ideas of Sandy’s Opportunities
to Learn About Computational Skill Instruction

 

 

 

Teaching Computational Skills Central Ideas of Sandy’ 3
Central Ideas of the Reform Learning Opportunities
Computational Curriculum Computational Curriculum

extensive mental computation

estimation Emphasis on conceptually
non-traditional computing strategies understanding ideas
calculator and computer technology students’ algorithms
traditional algorithms some mental computation

basic number facts

 

 

Pedagogical Approaches: Pedagogical Approaches:
contextualize use conceptual orientation to
emphasize understanding operations more important
link conceptual and procedural balance conceptual aims with
thinking strategies for learning facts more traditional practices
practice under certain conditions Contextualize skills

 

There is a marked difference between the central ideas of reform and the

central ideas of Sandy’s opportunities to learn. Sandy’s opportunities to learn

228

emphasize the importance of understanding the mathematical ideas underlying
the four operations. Estimation as a computing strategy, technology, and
alternative algorithms were not focal points in Sandy’s opportunities to learn.
Further, there was no evidence that Sandy was offered any ideas about
alternative algorithms such as adding back, doubling, or grouping tens. Even
though these strategies were mentioned in the reform documents, overall Sandy
had no opportunity to think through the ideas or the implications of the ideas for
her teaching. In contrast, estimation, technology, alternative algorithms as well
as a variety of other computing skills are central to the proposals for change. The
only overlap occurred around ideas involving students’ algorithms and mental
computation. For example, Sandy’s use of Marilyn Burns and other curriculum
materials encouraged her to focus on students’ construction of algorithms and
mental computation. At the same time, little attention was given to any
pedagogical issues surrounding teaching students’ strategies, such as what to do
when students’ strategies produced incorrect results or how to encourage
discourse among students around new strategies.

Although there was overlap around the idea of conceptual understanding,
there also was a significant departure around the construct of conceptual
understanding. For example, Sandy’s opportunities emphasized the importance
of understanding addition, subtraction, multiplication and division. In
particular, the curriculum materials she used focused on conceptual orientations
to learning each operation. Sandy’s reading of the reform documents
emphasized understanding as opposed to memorizing steps to algorithms.
Recall that Sandy’s teaching in relationship to division focused on the idea of
equal distribution. She was not challenged to explore, for example, the
underpinnings of the division algorithm. Nor was she challenged to think about

linking conceptual orientations and procedural orientations of the four

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operations. She also did not encounter ideas about how conceptual
understanding can be used in place of traditional algorithms. None of Sandy’s
learning opportunities attended explicitly to these ideas.

Sandy’s conversations with other educators focused on the notion of
balance. Sandy’s understanding of balance ran completely contradictory to
proposed ideas. She interpreted balance to imply the separate and isolated
treatment of procedural and conceptual knowledge, a balance in terms of
instructional time. None of the opportunities that I had independent data on
contradicted Sandy’s interpretation, even though the reform documents stated
explicitly that the isolated treatment of computation was not compatible with the
views expressed in the policy. Sandy did not encounter the idea of
simultaneously linking the two in practice, a different interpretation of balance.

There also was overlap around the idea of contextualizing mathematics.
Yet, once again, there was great variation in what Sandy encountered and what
the proposals suggested. My analysis of the reform documents suggests that
contextualizing computational skills is at the core of recommendations for
reforming instruction of basic computational skills. Contextual use involved the
idea of situating learning basic computational skills in relation to contexts where
those skills arise. The purpose is to understand that contexts play an important
role in how mathematics gets done. For instance, if the context requires an
estimate, then estimation would be an appropriate computational technique. My
analysis of Sandy’s work with Project AIMS provides an example of the variation
in meaning that Sandy encountered. Sandy drew primarily from her training
with AIMS and her use of AIMS materials to develop her ideas about the role of
context in learning mathematics. The primary goal of AIMS materials is to

integrate science and mathematics through explorations of real-world

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phenomena. From this standpoint, mathematics learning is not the central goal.38
Further, underlying AIMS materials is a View of mathematics that is very
different from what is described in the reform documents. Mathematics is
viewed more like a tool-box where ideas and skills are developed and then used
in the service of these investigations. From this View, proficiency with
computation is learned elsewhere and then brought to bear on investigations.

In practice the distinction is striking. In both views, contexts are defined
as real-world or mathematically-oriented contexts. Students might begin by
formulating conjectures for how to solve problems. When computational skills
are necessary to solve a problem, AIMS materials assume students have
developed those skills and bring them to the ongoing problem-solving
opportunity. The relationship between context and computational skill is not
primary. If students' skills are weak, work outside the context is assumed
necessary. In reform-oriented teaching, the context impacts the computational
work students will need to do. Here, students rely on the context to help them
decide how to compute most efficiently and decide whether accuracy is an issue.
Students have concurrent opportunities to develop problem solving and
computational goals. The mismatch is striking. Yet, of the four opportunities I
collected independent data on, none focused on any distinctions.

One idea that Sandy did not encounter in relationship to her learning
opportunities was the role of practice for developing computational skills.
Practice, although discussed in the reform documents, was not central or
explained explicitly. Sandy’s opportunities to learn reflected no attention to

whether or what the role of practice in learning basic computational skills should

 

38I base these claims on my observations and analysis of Project AIMS curriculum materials
provided teachers at a Project AIMS workshop and my analysis of a written statement by Dr.
Arthur Wiebe, founder of Project AIMS , explaining a model of mathematics underlying Project
Aims materials.

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be. There was no evidence that suggested Sandy even considered the notion of

practice in any other light than what modal teaching portrayed.

At this point, I shift my focus and turn to explore the big ideas of Sandy’s

opportunities to learn about introducing discrete mathematics. As before, the

table below contrasts the big ideas of Sandy’s opportunities to learn about

introducing discrete mathematics into the school mathematics program with the

proposed ideas characterized in the policies.

Table 3. Central Reform Ideas And The Central Ideas Of Sandy’s Opportunities
to Learn About Introducing Discrete Mathematics

 

Introducing Discrete Mathematics

Central Ideas of the Reform

Central Ideas of Sandy's

Opportunities to Learn

 

 

Rationale:
computer technology connection
more real-world connections
modernizing school mathematics

strategic site for other reform ideas

What is Discrete Math?

"discrete implies separate or distinct

a'suggests contrast continuous vs.

discontinuous mathematics

Rationale

secondary issue

What is Discrete Math?

discrete as not continuous

some attention to continuous

vs. discontinuous mathematics

 

 

232

 

Table 3. (Continued)

 

*identifies many central ideas central ideas include Venn diagrams,
but mostly does not specify how ideas combination and permutations , but does not
fit or represent discrete math provide ideas for how they fit in

the larger structure of discrete
"suggests discrete math involves math, relational theories
alternative ideas about counting and

gives illustrations of typical problems

Situating Discrete Math Inside Curriculum Situating Discrete Math:
introduction done informally elementary included
mixed messages for elementary insert topics across - no ideas
not a separate course provided for how this might be
insert topics, central ideas done

‘no attention to how ideas might be situated so that
discrete mathematics might be represented as a

coherent field of study.

 

 

Pedagogical Approaches: Pedagogical Approaches:
problem solving approach problem solving approach
emphasis on ideas emphasis on ideas

 

 

Rationales for why discrete mathematics should be introduced into the
school curriculum were different. The idea that discrete mathematics should be
introduced into the secondary curriculum was encountered in the context of

Sandy's conversations with other educators. Sandy’s efforts to introduce discrete

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mathematics into her practice subsided and became less important, less valued
aspects of reforming her mathematics teaching.

Across the category what. is discrete mathematics, there is good overlap.
For example, the ideas of continuous and discontinuous mathematics overlap.
The overlap was mostly because Sandy arranged for opportunities specifically to
explore these ideas. For example, she arranged for conversations with other
educators to explore discontinuous and continuous mathematics and she
selected curriculum materials she believed focused on combinations and
permutations. Yet, the opportunities Sandy encountered to explore these ideas
did not delve deeply into any idea. In part, Sandy’s underdeveloped knowledge
of discrete mathematics made it difficult to draw on what she would need to
make sense of the mathematics offered. Also problematic was the lack of
continuity across learning experiences. Overall, Sandy's experiences lack any
long term view of what she would need to learn about discrete mathematics to
introduce it into her teaching. She described her learning experiences as
frustrating and confusing. Eventually Sandy centered her efforts on learning
about two combinatorial counting problems leaving much of the subject matter
unexplored.

In terms of situating discrete mathematics in the school curriculum,
Sandy’s opportunities offered little. None of Sandy’s opportunities to learn
provided her with detail for how this might be done. The reform documents
suggested that discrete mathematics could be introduced by way of some of the
familiar ideas already getting attention in the elementary curriculum. Venn
diagrams is an example. The curriculum materials suggested that counting
problems could be used as a kind of stand alone activity with little concern for
connecting ideas across teaching. There was little concern for how these

activities connected to other mathematics as well.

234

In terms of pedagogy, the overlap was congruent. The big message was
that discrete mathematics should not be introduced as formulas and procedures
to be memorized for getting right answers. Instead, conceptual and problem
solving orientations should be emphasized. Sandy’s learning in relationship to
Marilyn Burns and Project AIMS materials emphasized compatible views. They
offered a hands-on, learning by doing, investigations approach to learning
mathematics. At the same time, there was a great deal of mismatch in meaning
around the notion of contextual use and conceptual understanding.

The nice overlap occurred because Sandy used the reform documents
much like a textbook, ideas to guide her efforts to learn what she would need. At
the same time, Sandy’s opportunities to learn did not delve deeply into any of
the mathematics identified in the documents. In addition, the documents did not
portray combinatorial reasoning or generalizations as central to the mathematics
of combinatorial counting.

There are essentially two critical findings revealed in the contrast between
the big ideas of Sandy’s opportunities to learn and the central ideas described in
the reform documents. My analysis suggests that much of the reform agenda
was untouched in the contexts of Sandy’s opportunities to learn. In addition,
there is a mismatch across several big reform ideas, specifically conceptual
understanding and contextual use. My analysis of Sandy’s opportunities to learn
about introducing discrete mathematics revealed similar problems. Even though
there seemed more overlap across the big ideas, the central tenets of the
mathematics of counting went untouched across Sandy’s opportunities to learn.

I turn now to consider the relationship between Sandy’s prior knowledge
and teaching experiences and her opportunities to learn. In particular, I examine
whether and how Sandy’s opportunities to learn considered her personal history

with each strand.

235

Connections to A Teacher's Prior Knowledge and Boliefs

I suggested earlier that Sandy’s prior knowledge and beliefs represent
Opposite situations from the standpoint of teacher learning. Sandy brought a
great many ideas and practices related to computational skill teaching, most of
which was firmly entrenched and well-grounded in her teaching. Sandy’s prior
ideas and practices run in stark contrast with the proposed changes. It was clear
that much would need to be challenged and uprooted if new ideas and practices
were to replace them. In the case study, I described how Sandy tried a new
approach to teaching computation. When problems surfaced, she returned to the
more traditional ideas and practices she had abandoned. From a pedagogical
standpoint, Sandy’s learning should have functioned to challenge her prior ideas
and practices creating much disequilibrium around traditional forms of teaching.
At best, Sandy put aside her ideas only temporarily, leaving much unchallenged.

The reform documents reflect the important role of prior knowledge and

experience on new learning.

”Instead, in many situations individuals approach a new task with
prior knowledge, assimilate new information and construct their
own meanings. They will accept new ideas only when their old
ideas do not work or are inefficient. This constructivist active View
of the learning process must be reflected in the way much of
mathematics is taught.” (National Council of Teachers of

Mathematics, 1989, p. 10).

My analysis assumes the same insights that surround students’ learning would

also be true about practitioners’ learning about reforming teaching. From this

236

standpoint, the analyses considers whether and how Sandy's opportunities to
learn about reforming her teaching of computational skills challenged her to
rethink her existent ideas and practices.

The situation surrounding discrete mathematics is somewhat different.
Sandy brought very little specific content knowledge that she could draw on for
making sense of the proposals to introduce discrete mathematics. For this topic,
there was very little to uproot or challenge. Instead, Sandy brought more global
ideas such as the notion that mathematics should be taught for understanding
and that conceptual grasp was important to the process. From a pedagogical
perspective, Sandy’s opportunities to learn should function to increase her
capacity to teach discrete mathematics for understanding. To do so, Sandy's
subject-matter knowledge would need to be extended sufficiently to include
topics in discrete mathematics. This would involve identifying fruitful points of
connection, such as the integers or the mathematics of traditional counting,
specifically for the purpose of building and sustaining new subject-matter
knowledge and teaching practices.

Assuming that I have characterized Sandy’s personal history with each
topic fairly well, I examine the categories of opportunities to explore whether
and how Sandy's prior knowledge and teaching experiences are considered.

Reforming eomputational skills instruetion. The table below summarizes
the central ideas of Sandy’s opportunities to learn in contrast with the ideas
Sandy brought to her learning about reforming her teaching of basic

computational skills.

237

Table 4. Central Ideas of Sandy’s Opportunities to Learn About Reforming
Teaching of Computational Skills And What Sandy Brings

 

Sandy's Opportunities to Learn

What Sandy Brings

 

 

Computational Curriculum

emphasis on understanding operations

 

conceptually

some attention to students’ algorithms and mental

computation

 

Computational Curriculum

Sandy brings ideas about what computational skill
is - mostly proficiency with traditional algorithms,
some mental computation and use of those skills to

solve problems.

why it is important to learn computational skills

Sandy brings a very detailed picture of what the
computational curriculum involves and how it is
situated in the school curriculum. For example,
Sandy has a well-worked out vision for the desired

level of complexity and mastery for each algorithm

ideas about the conceptual underpinnings of the
algorithms / and other conceptual ideas related to
the four operations

Sandy also brings ideas about what the important
elements of computational skill include - such as
getting the right answers, knowing facts and

algorithms, speed and accuracy with those

Sandy brings ideas about differences in underlying
systems of numbers and how computational skill
changes in relation to each system. She brings ideas
about place value, different bases, operations with

decimals, fractions, etc.

ideas about estimation, how to estimate and when

 

238

 

Table 4 (continued)

 

 

Pedagogical Approaches

conceptual orientation to four operations as more
important than procedural orientation - emphasis
on how to conceptually orient the curriculum

the notion of balancing conceptual aims with more
traditional practices

contextualize skills

 

Pedagogical Approaches

Sandy brings instructional practices she believes
develops computational skill - this includes isolated
drill and practice on basic facts, routine algorithms,

and mental computing.

Sandy brings knowledge of common student errors
and misconceptions with traditional computing
algorithms

She brings ideas about the practices she believes

helps students’ overcome those

 

The predominate idea across Sandy’s opportunities to learn emphasized

teaching addition, subtraction, multiplication and division conceptually and in

problem solving contexts. At the same time, the opportunities she encountered

gave little attention to why or how traditional ideas and practices are

problematic. There was no evidence that Sandy explored the underpinnings of

traditional algorithms, whether they remained valued, and if so, to what level of

complexity or mastery. Sandy’s views about proficiency with routine algorithms

went untouched as she explored the four operations conceptually.

Sandy’s experiences learning new computational skills, such as students’

algorithms or more extensive mental computation, also did not challenge her to

rethink any of her existent ideas and practices. Although Sandy thought that

students’ algorithms and mental computation were to replace the preoccupation

with traditional algorithms, she was not challenged to focus on any problems

that might arise as a result of these changes.

There was nothing in Sandy’s opportunities that challenged her ideas

about speed and accuracy with computing. Sandy continued to believe that

239

 

traditional algorithms provided the most efficient and accurate method for
computing. This idea runs contrary to views discussed in the documents.
Sandy’s very detailed view of the desired level of complexity and proficiency for
each algorithm also went unexamined.

Sandy’s ideas about estimation also went unchallenged. She brought very
traditional ideas about estimation, how to estimate and for what purpose.
Across the opportunities she encountered, only one, the documents, suggested
that estimation be viewed as a computational skill itself. Even though Sandy
read the proposals surrounding estimation, she missed the idea that estimation
should be treated differently. She continued to emphasize estimation as a
checking strategy for routinely computed answers asking students to use
estimation to see if their exact answers were ”in the ball park.”

Sandy brought very traditional pedagogical ideas for developing students’
computational skills. For example, she brought the idea that isolated drill and
practice on basic facts and algorithms help develop proficiency. She banked on
the idea that, ”as students come to conceptually understand procedures and
learn why certain steps make sense, fluency will develop as a result,” (p. 56
Framework). Sandy interpreted this statement to suggest that a conceptual
orientation to the four operations would also result in computational proficiency
for students. She developed a practice she thought represented this idea. Yet,
when problems arose she abandoned this approach and returned to traditional
practices. Sandy’s opportunities to learn did not focus on the problems that
arose in practice. None of the opportunities she encountered emphasized
analyzing teaching and connecting findings to reform ideas. Sandy’s
interpretations of the changes she made in her teaching went unchallenged.

Overall, Sandy's prior knowledge and teaching experiences were not

challenged. The central focus of her learning experiences involved figuring out

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the correct balance between more traditional, skill-oriented teaching and reform-
based teaching. It would have been interesting to see what would have
happened had Sandy become convinced that traditional practices were extremely
problematic and had to be abandoned. Would she have returned so easily? By
not uprooting Sandy’s existent ideas and practices, they remained a legitimate
choice, especially when supported in other learning contexts.

Introducing diserete mathematics into the eurrieulum. I turn now to

examine Sandy’s opportunities to learn about discrete mathematics.

 

Table 5. What Sandy Brings And The Central Ideas of Sandy's Opportunities to
Learn About Introducing Discrete Mathematics

 

 

 

 

 

What Sandy Brings Sandy’s Opportunities to Learn
What is Discrete Mathematics? What is Discrete Mathematics?
no ideas about what discrete mathematics involves meaning of discrete
no ideas about what the central questions of discrete some attention to continuous as opposed to
mathematics involve or why the questions are valued discontinuous mathematics

no ideas about the techniques or strategies used for
examining those questions

no ideas about what discrete mathematics explores central ideas including Venn diagrams, deductive

 

logic and counting combinations and permutations

 

241

 

Table 5 (continued)

 

 

 

some familiarity with algebra

some familiarity with a few of the central ideas
identified in the Framework and Standards,. For
example Sandy had some familiarity with the ideas of
Venn diagrams, deductive logic, and combination and
permutation. Her familiarity is not inside the context of

discrete mathematics.

no ideas about number systems underlying discrete
mathematics. Although Sandy has ideas about the

integers it is not in relation to discrete mathematics.

Situating Discrete Mathematics

no ideas for how topics, ideas might be introduced into

the school curriculum

Situating Discrete Mathematics

 

elementary included

insert topics across - no details provided

 

242

 

 

Table 5 (continued)

 

 

 

 

Pedagogical Approaches Pedagogical Approaches
no ideas for what instructional practices might best problem solving approach
represent discrete mathematics conceptual understanding emphasized

Sandy brings many ideas for what it might mean to
teach using a problem solving approach where ideas are
central (as opposed to a procedural approach where

rules and algorithms are seen as central)

no ideas about common student errors or
misconceptions with any of the ideas or techniques

central to discrete mathematics

 

 

Sandy brought to her learning experiences some familiarity with algebra,
some familiarity with several of the mathematical ideas listed in the documents
as central to discrete mathematics, extensive knowledge of traditional counting
procedures, familiarity with the counting numbers and the integers. Sandy's
opportunities to learn focused on a variety of mathematical ideas Sandy knew
little about. For example, Sandy’s conversations with leading mathematics
educators focused on continuous and discontinuous mathematics. Although
Sandy found the ideas difficult to comprehend, she developed some sense of the
contrast drawing on what she already understood about algebra. The connection
represented an important insight into discrete mathematics especially in light of
Sandy's initial conjecture that discrete mathematics might have something to do

with the elements chart in science.

243

 

Sandy’s familiarity with the integers was another site of connecting to her
prior knowledge and teaching experience. Sandy’s teaching of the integers was
rooted in the mathematics of arithmetic and set theory. Initially, Sandy was
confused about whether the integers were connected to discrete mathematics.
She arranged for conversations with other educators to explore this issue.
Consequently Sandy learned that the integers represented a system underlying
the mathematics of discrete mathematics.

Sandy’s experiences with Project AIMS provided her multiple ideas about
teaching pedagogy. The reform documents warned that discrete mathematics
should not be included if taught in an algorithmic, procedurally-oriented way.
AIMS activities rarely gave attention to algorithms or formulas. Instead,
activities were designed to support investigative, hands -on learning experiences.
And although the materials offered little pedagogical guidance, Sandy’s
preparation as an AIMS consultant supported her use of these materials.

There were a number of learning experiences that ignored Sandy’s prior
knowledge and teaching experiences. The curriculum materials Sandy used is an
example. Sandy was completely unfamiliar with non-traditional counting ideas.
Her experiences with combinations and permutations in high school involved
memorizing algorithms and getting the right answer. Yet the materials Sandy
used, although they offered new insights into the idea of order and patterns for
formulating counts, at the same time did not connect Sandy's prior knowledge
about the algorithms to any reasoning process or the notion of generalizing
reasoning processes. None of Sandy's opportunities to learn pressed her to
consider any differences across counting problems that would lead to different
generalizations explaining the different algorithms she had once memorized.

As Sandy began to clarify and extend some of her ideas about discrete

mathematics, she encountered mathematics she had no prior experience with.

244

For example, several of the conversations she had with other educators focused
on recurrence relations. Sandy explained that she listened, to be polite, but really
had no ideas for how recurrence relations fit with discrete mathematics. She
recalled that even though several relational theories had been explained to her,
step by step, she had no way to understand the meaning. This illustrates that
even though there was overlap around the idea of recurrence relations, Sandy's
prior knowledge was not tapped in a way that offered the opportunity to
formulate connections that would advance her ideas about discrete mathematics.

Sandy’s opportunities to learn, the content and structure offered,
portrayed discrete mathematics as lists of ideas and topics, mostly unfamiliar to
Sandy. Venn diagrams is an example. Yet, none of Sandy's learning experiences
suggested how Venn diagrams might be used to introduce discrete mathematics.
Venn diagrams remained unexamined across Sandy's learning experiences,
except that Sandy organized Venn diagrams under discrete mathematics.
Although Venn diagrams may have been a useful site to extend Sandy's ideas
about counting, she had no opportunity to capitalize on her prior understanding.

Essentially, there was very little continuity across Sandy’s learning
experiences. None of the opportunities Sandy encountered were organized in
ways to build on previous learning. This happened on occasion by chance.
However in most circumstances, the learning opportunity lacked the potential
for Sandy to form connections to new knowledge by way of what she already
understood. In effect, Sandy was not offered the possibility to learn about
discrete mathematics as a coherent field of study, involving specific kinds of
questions, systematic methods for analyzing those questions, or standards by
which to judge the products of those questions.

I turn now to the third element of the analyses.

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The Teacher's Views of What the Opportunities To Learn Offered

The data suggests, through a number and variety of statements in
interviews, particular views that Sandy held in relationship to the opportunities
to learn she encountered. My aim in this section is to establish some sense of
what those views were and how they may have interacted with what Sandy was
learning. Specific findings focus only on those opportunities that had sufficient
data to support any claims.

Sandy's characterized her learning in the context of her doctoral studies as
”research-based” and ”global understandings” (Interview, 1/93). Sandy viewed
her doctoral work as a particular kind of resource, one that provided expert
knowledge and global understanding of teaching and learning. Recall that
Sandy considered the idea of automaticity to be ”research-based.” In
combination with the emphasis on computation embodied in the state’s
mandated testing program, Sandy concluded proficiency with traditional
computational skills remained a high priority for elementary teaching.

Sandy viewed Project AIMS to offer a ”teaching methodology.” She
argued that AIMS materials framed mathematics teaching and learning very
differently than traditional practices. Sandy Viewed AIMS methodology to be
compatible with the visions of teaching and learning portrayed in the reform
documents. The documents described a kind of teaching that emphasized
exploring mathematical ideas and students’ constructing knowledge in
cooperative learning groups. Sandy viewed Project AIMS materials as putting
the ideas into practice (Interview, 5/92). Sandy's interpretations of the reform
agenda were colored by her views of what Project AIMS offered.

Sandy’s views of the framework shaped what and how she would learn
about reforming her teaching. Sandy viewed the framework as a ”critical piece

of work, designed specifically for teachers,” and ”visionary, hopeful” (Interview,

246

2/93). Unlike many teachers, Sandy’s copies of the Framework (1992), and the
Currieulum Standards (1989), did not lay on a shelf collecting dust. Instead, she
read constantly, scribbling notes inside the text. Sandy read and re-read sections
hoping to dig out something she didn't see before. Yet, even as Sandy thought of
the documents as visionary, she also believed they offered a specified program
for changing her teaching. She searched for teacher directives, prescriptions for
practice. Any vagueness or ambiguity was interpreted as her own personal
inability to make sense of the proposals.

This search for certainty colored Sandy's interactions across the six
categories of learning experiences. Sandy's perceptions reflected the framework
as teacher directives and correct practices. And she viewed the opportunities she
had to learn as occasions to uncover specified plans of action for changing her
teaching. Recall that Sandy’s decision to return to traditional teaching practices

represented an instance where she had, ”finally gotten it right,” (Interview,

8/93).

The Qualities And Conditions of A Teacher's
Opportunities To Learn About Reform-Based Teaching

To conclude this chapter I consider findings across the three analytic
categories to appraise Sandy's educational experience as a whole, characterizing
the nature of the support she encountered for responding to the state's efforts to

reform mathematics education.

Mismateh in Content

Most of Sandy’s learning experiences were grounded in ideas and
assumptions about mathematics teaching and learning that were fundamentally
different from those underlying the reform documents. The analyses above

reflects that some of the differences were epistemological, whereas others were

247

more conceptual and philosophical. An example would be the idea that
computational skills teaching should be approached in contexts that support
understanding why specific skills are useful. Another example is the idea of
reasoning combinatorially. There were many other ideas and assumptions that
also were overlooked or under examined. Essentially, there is a clear mismatch
in ideas and practices central to the reform agenda and Sandy's opportunities to
learn. Overall, Sandy’s professional development experiences lacked strong
connections to many of the central ideas and practices described in the reform
documents. This mismatch contributes greatly to the teacher's capacity to

understand and enact reform ideas in teaching.

Few nnctintthTahr'Priranl an Tah' Ex in

A teacher's prior knowledge and teaching experiences profoundly
influence what a teacher learns. This study underscores the significance of this
aspect of teacher learning in the context of policy proposals. The cross-case
analyses illuminates that Sandy's prior knowledge and teaching experience was
different across different topics in mathematics. For example, Sandy's prior
knowledge and teaching experiences in relationship to computational goals were
deeply entrenched and functioned as a barrier to long-lasting change. Yet, the
above analyses reveals that Sandy's opportunities to learn did not function to
challenge or explicitly explore her prior knowledge and teaching experiences
surrounding computation. This was also the case for discrete mathematics. For
the most part, the opportunities to learn that Sandy encountered failed to
support Sandy's efforts to form strong connections between the knowledge,
skills, and dispositions that she brought as a learner to the ideas and practices
envisioned in the reform documents. As a resource, Sandy's prior knowledge

and teaching experiences were mostly ignored.

248

Laek of Continuity Aeross Learning Experienees

Sandy’s decisions to participate in specific professional development
activities were based on what she alone determined to be important. They were
not chosen based on any previously developed design or specific intervention.
Sandy located resources as she became aware of specific learning needs. This
awareness occurred spontaneously, often in contexts not intended as support for
learning about reform issues. Little overall attention was giVen to whether or
how Sandy's experiences served to create an overarching View of the proposals
to reform computational skill teaching or introduce discrete mathematics. These
findings suggest that Sandy's experiences were characterized by a lack of
continuity in learning experience. Other than what she could provide herself,
specific learning needs were not addressed in any systematic way.39

Sandy’s conversations with other educators, in part, is an exception. In
these instances Sandy focused the occasion directly on her needs as a learner. At
the same time, lacking in the experience was any larger view of what would be
required for Sandy as a learner. Recall that many of her conversations with other
educators about discrete mathematics resulted in experiences with Sandy trying
to understand recurrence relations or matrices. Sandy recalled these experiences
as, ”very frustrating” and ”offering little for understanding discrete
mathematics.” Even in these instances, continuity across learning experiences

was decidedly lacking.

 

39John Dewey develops the idea of continuity of learning experience in his writings. See
WM. Macmillan Publishing Co. 1963. Dewey argues in principle that
continuity (united with interaction) measures the significance of educative experiences.
Continuity is concern for how future experiences might be shaped by what is learned at each and
every stage of the learning process. Thus, as a discriminating factor, continuity helps to judge
what is educative and what may be mis-educative on the basis of what the experience moves one
toward and into.

249

Few rtuniti s to nn ct L arnin t Teachin Practi

Sandy’s own teaching served as the only site for exploring
recommendations for change in the context of teaching. Even as she may have
had multiple opportunities to imagine the proposed changes in practice -- she
had little opportunity to try out new patterns of instruction and explore what
happened as a consequence. Overall, Sandy’s opportunities to learn provided
very little guidance with what the proposals implied for her teaching. Sandy
determined what the proposals suggested as changes in her teaching. She made
sense of what happened as a result of those changes. And although this work
was certainly influenced by what she was learning in other contexts, she alone
had to make sense of the new instructional patterns she introduced into her
teaching and the impact of the changes on students' opportimity to learn.

These conditions violate an important principle about learning that
policymakers hope Sandy understand. Context plays an important role in the
learning experience. Applying this principle to reforming teaching suggests that
the teaching context will be an important factor in what the teacher learns.
Connecting teacher learning opportunities more directly to the teaching context,
whether by observation and discussion of videotaped teaching or one's own
teaching, offers teachers a richer context for learning the practical implications of

the proposed changes (Ball 8: Cohen, 1995a).

Absenee of Putposeful Teaehing

Sandy made decisions about what and how she would need to learn
almost exclusively on her own. Recall that Sandy’s decision to focus on discrete

mathematics was motivated by her responsibilities to educate other elementary

250

teachers about the new framework. Sandy’s efforts to learn more about teaching
computation was motivated by the difficulties she encountered in her practice.

Of the six categories of opportunities I investigated, none were designed
purposefully to explore the ideas and practices Sandy decided were important.
In effect, Sandy’s learning about these proposals went on subsequent to other
learning agendas. As a result, Sandy's learning experiences lacked purposeful
teaching about the ideas and practices to reform her teaching of computation and
introduce discrete mathematics into her teaching. In effect, the experiences
offered little specific guidance for accomplishing the goals Sandy identified as
central. And although Sandy may have mustered an abundance of resources to
support her efforts, the experiences she developed encouraged isolation in the
learning process (see Lortie, 1975, p.70). Within these experiences Sandy had to
foster for herself a wider range of wisdom and expertise to support her
understanding and enactment of the proposals she focused.

As a learner, Sandy confronts a paradox. She must make judgments about
what is useful to learn and either create the circumstances for learning it or
recognize the opportunity when it arises in contexts not purposefully designed to
support such learning. To effectively select and judge opportunities to learn,
Sandy would have to bring well-worked out ideas about the proposals in order
to make good decisions in the contexts of her learning opportunities. Yet, as a
learner in the context of reform, Sandy brings knowledge, skills, and dispositions

that run counter to the ideas and practices she must learn.

Litt mm L arnin mmuni
Across several of the categories of opportunities to learn, there were many
other educators involved in Sandy's learning experience. This included other

teachers, other doctoral students, state officials and mathematics educators. The

251

potential for Sandy to work collaboratively with others on the ideas of reform
seems readily available. Yet, most of these environments provided at best a
social setting for Sandy to work alone on the ideas of reform. Because much of
what Sandy learned about teaching computational skills and introducing discrete
mathematics was situated in contexts not designed for exploring those proposals,
her learning process coexisted, along side other learning agendas, progressing
mostly in isolation of any others participating in the same environment.

Sandy’s doctoral studies and the professional development activities she
participated are good examples of environments that involved other learners but
at the same time aimed at agendas other than the proposals focusing this study.
Sandy’s efforts to understand the reform documents, her own teaching practice,
and the curriculum materials she used went on without the benefits of
community. Sandy's conversations with other educators may be an exception.
She reported that these conversations offered the opportunity to work with
others on the specific proposals she tried to understand and enact.

Some of the isolation in Sandy's learning experience was created by Sandy
herself. For example, Sandy decided to focus exclusively on curriculum
materials to help her learn about discrete mathematics after she encountered
much confusion and frustration in learning with others through the
conversations she arranged. Although the curriculum materials offered Sandy
ideas, at the same time, she made sense of those ideas in isolation of others

purposefully cutting herself off from any wider range of wisdom or expertise.

A Seareh For Certainty

Sandy’s ideas about computational skill teaching and introducing discrete
mathematics evolved over long periods of time and in relationship to a variety of

resources that supported her learning. I have described the experience for Sandy

252

as a kind of wandering, a selection process bounded by what she could
understand herself and what she encountered in the contexts of the resources she
put together. At the same time that Sandy's experience is characterized by much
uncertainty, the analyses also revealed that Sandy perceived the work to involve
certainty, a search for right answers for her teaching. An important aspect of this
process involved the unpleasantness she experienced, an emotional response to
the difficult circumstances she encountered. Sandy perceived the discomfort as
her inability to ”get it right.”

*****$****************$***

These findings explain how the gap between instructional policy and the
individual teacher's sense making and teaching can remain wide. Even though
the teacher makes use of ample resources and opportunities, there is a lack of
coherence and coordination in the teacher's educational experience as a whole.
The opportunities to learn that Sandy encountered offered a less than whole
view of the recommendations for change. They barely skimmed the surface of
the proposals and failed to challenge and extend the knowledge, skills, and
dispositions she brought to the experience. Teacher's subject-matter knowledge
was mostly assumed. There were few opportunities for the teacher to connect
learning to the practice of teaching. For the most part, Sandy had to figure out
for herself what the proposals implied for her teaching and whether the changes
she made improved students' opportunity to learn.

The analyses also suggests that the gap may have widened further. Not
only did Sandy's opportunities to learn fail to foster what she needed to imagine
and create the teaching portrayed in the documents, on several occasions the
opportunities Sandy encountered fostered learning that supported entrenched,
traditional patterns of instruction. Overall, the occasions that Sandy had for

learning about the state's goals did not challenge or explore any differences or

253

discrepancies in one form or another form of teaching. Instead, Sandy resolved
differences and discrepancies on her own. Often, Sandy's analysis resulted in a
reinterpretation of policy. In effect, the opportunities to learn that Sandy
encountered fostered, and at times supported, a veering off in directions

incompatible with the state's ideas of reform-based teaching.

254

CHAPTER 7

LINKING INSTRUCTIONAL POLICIES TO
PROFESSIONAL DEVELOPMENT OPPORTUNITIES FOR TEACHERS

Introduction

Beginning in the mid-1980s, the State Department of Education in
California produced a massive effort to further statewide reform in mathematics
education. Policymakers specified, in a new framework, essential characteristics
of empowering mathematics programs and described how changes in
curriculum, teaching, and students' learning could be accomplished. State
officials aligned multiple levers to press teachers toward desired practices. The
state sponsored and supported the development of new curriculum materials, for
example. These materials, known as replacement units, were tied closely to the
goals of the framework and were intended to take the place of larger segments of
the curriculum. The state argued that if real change were to take place, teachers
had to have available large chunks of curriculum that interrelated recommended
changes and could be inserted into the sequence of topics normally taught. In
addition, textbooks, for the first time ever, had to be tied directly to the goals of
the framework. Other units-such as those written by educational entrepreneur
Marilyn Burns—were also recommended.

The state developed a new assessment program as well. Test developers
experimented with alternative forms of assessment. The goal was to align
assessment and the framework in an effort to apply greater pressure on districts
to conform to the state's goals. Teacher certification and professional

development were also tied in. The state required teachers to be recertified every

255

five years by obtaining professional development credits and passing the
California Basic Essential Skills Test.

Professional development was also targeted. The state took an assertive
role in educating teachers about the framework. The legislature offered funding
to support school-based teacher improvement projects. Funding was also
pursued for larger scale projects such as the California Mathematics Project and
the Middle Grades Mathematics Renaissance. These projects, among others,
created an infrastructure of professionals to provide teachers with leadership and
professional development. Four-week summer institutes, one—shot workshops,
and multiple other opportunities were offered to support teachers' efforts toward
reform-based change.

California had mobilized a remarkable array of levers toward reforming
mathematics teaching in the state. Still, such change must be effected in
classrooms, in the thoughts and practice of the teacher. Policy analyst Milbrey
McLaughlin reminds us, "Change ultimately is a problem of the smallest unit"
(McLaughlin, 1987). In other words, the policy that's delivered depends finally
on the individual at the end of the line. This dissertation offers policymakers and
educators a window for viewing what happened as a policy was transformed by
the individual teacher's interpretations and responses. We can see that this
teacher, Sandy Wise, became deeply involved in the state's goals. We see that
she was motivated and willing to embrace policy objectives. She understood that
the state was attempting to align policy and practice in ways never before
imagined. She spent countless hours reading the framework, analyzing the new
CAP tests and curriculum guides, trying out new curriculum units, and changing
her practice. The case studies in Chapters 4 and 5 provide examples of the deep
effects on Sandy's thinking and practice. In many respects Sandy was successful

at reform. She brought the conviction to learn what was needed to understand

256

and enact the state's recommendations. She attended several state-supported
professional development activities. She used Marilyn Burns' unit on
multiplication. She even supported other elementary teachers by offering
inservice opportunities to learn about the new framework.

Yet, this dissertation reminds us that despite great strides on the part of
committed state officials, policymakers, educators, and a teacher, the disjuncture
between policy and practice can still remain wide. In this work, we have the
opportunity to view how change happens at the level of the individual teacher.
By focusing on the individual, we can see the interplay between a teacher's
thoughts and actions and what she encounters across a range of opportunities to
learn, including many state-sponsored activities. The analysis in Chapter 6
uncovers aspects of practice that make it particularly difficult for the teacher
responding to policy. We saw, for example, that the teacher can remain isolated
despite great effort to interact with others. As Sandy began locating people and
opportunities to help her understand and enact the framework, she encountered
multiple interpretations of the proposed changes and different paths for learning
what the state recommended. Many paths were contradictory. Essentially, the
teacher had to figure out for herself what to participate in, how to follow up, and
whether and how the ideas she encountered connected to the state's goals.

My analysis and conclusions focus on teachers like Sandy--those working
as entrepreneurs, seeking opportunities for their own learning as they make
sense of the reforms swirling around them. I suggest that this is what the

situation was like for many teachers in California.40 And although we already

 

40Even though much of the current literature and efforts at school improvement focus on groups
of teachers, collegiality, the importance of professional community etc., this study reminds us
that many teachers will approach reform essentially alone. Even as teachers interact with many
colleagues, there remains a dimension of aloneness in constructing and developing the ideas of
reform-based teaching. Understanding what the individual teacher encounters remains an
important aspect of understanding the overall picture of reform-based change. This work helps

257

understand that isolation and individuality in teaching are problematic, this
work suggests that even though much may be done to create collegiality and
press systemic efforts to reform, what the individual teacher encounters is
somewhat different. I found that even though multiple resources and levers
were aimed at pressing and supporting teachers toward desired practices, much
of what the individual teacher would need to learn was overlooked or not
addressed adequately. In particular, the subject-matter knowledge a teacher
would need to learn to teach computational skills differently or introduce
combinatorial counting into practice was not addressed adequately. The teacher
had to locate and develop opportunities to learn the mathematics for herself.
What she created or encountered was often situated in contexts not specifically
designed to teach the content knowledge needed. The teacher therefore had to
draw out and connect mathematical ideas she believed would be useful. More
problematic still was that most of the occasions relied upon embodied
assumptions about mathematics and teaching that ran counter to policy goals.
These circumstances place teachers in a difficult spot. In order to locate
and develop opportunities to learn that are grounded in ideas and practices
argued for in policy, a teacher would have to bring well-worked-out ideas about
reform-based teaching. Yet, we already know from research on teacher learning
that teachers are more likely to bring forth knowledge, skills and dispositions
that run counter to the ideas and practices envisioned in the policies of the
eighties and early nineties. Because of the vastly different nature of practice
most teachers are accustomed to, they have had little opportunity to understand
and experience the ideas and practices proposed. These circumstances create a

contradictory learning environment for the individual teacher. The conditions

 

us to see how the individual teacher organizes and structures much of their own educational
opportunities in the context of broad—based efforts to reform.

258

can limit rather than foster the teacher's capacity to develop deeper insights into
policy and create the practice imagined.

Suppose for a moment that US. students encountered similar
circumstances for learning high-school algebra. As learners, students would not
understand the central ideas of algebra or see how the ideas fit together as a field
of study. Yet, imagine that students had to design and direct an investigation
into the field. Fortunately, there are some resources available. Students might
dip into textbooks, talk to a variety of educators, and work with each other to
develop some ideas. Interest, curiosity, and connections would guide progress.
Students’ ideas would evolve. Yet, what would guarantee that students actually
developed an appreciation of the mathematics of algebra? Even in cases where
very resourceful students made use of an abundance of resources, the
opportunity to learn would be limited by what individuals could locate or
develop for themselves. Missing would be any wider range of wisdom or
guidance about what to learn or how that learning might be accomplished.
These circumstances would seem ridiculous for US. students. We expect that
students are supported by a knowledgeable teacher and a well-designed
curriculum. Would teachers require similar conditions?

This chapter focuses on this question. Suppose that educators and
policymakers were to continue in the direction the state of California was
heading -- that is, that they might continue to take teachers seriously as learners.
What does this study suggest about the teacher's needs as learner? My aim is to
draw together findings in this study, with other research on teacher learning and
research on policy and practice, to suggest how the ties between efforts to reform
instruction, teachers' learning, and teachers' practice might be strengthened. I
propose that more attention be given to what links policy to the individual

teacher's sense making and enactment of proposed changes. I begin by arguing

259

that the nature of the relationship between policy, teacher learning, teaching
practice is dynamic. I argue that within the dynamic, there exists an interplay of
factors that affect what a teacher learns and therefore can make of policy. I
provide four examples of factors that had a significant impact on Sandy's
learning. The lack of attention to these factors, coupled with the nonsystematic
nature of the professional development available, reveals the lack of coordination
and coherence in efforts to support teachers' responses to policy. To conclude
the chapter, I sketch several ideas for forming more solid links between policy,
teacher learning, and teaching practice, and I suggest directions for future

research.

Responding to Reform Initiatives:
Focusing at the Level of the Individual Teacher

Policymakers and educators are not sufficiently aware of the long-term
and complex nature of the relationship among policy, teacher learning, and
teaching. Many underestimate what the implementation process involves at the
level of the individual teacher. And even less is known about what affects a
teacher's responses to policy.

This study has several important contributions to our understanding of
these issues. Below, I suggest that learning is the most central activity of policy
implementation for the individual teacher. I argue that the relationship between
policy, teacher learning, and teaching is dynamic as opposed to linear. Not only
does policy affect practice, but what a teacher learns from practice greatly affects
what the teacher thinks a policy recommends. I then argue that a teacher's
learning is also complicated by a set of factors that, knowingly or unknowingly,
will affect a teacher's responses to new frameworks, assessments, curriculum
guides, and other levers designed to promote the state's goals. The four

examples sketched in this chapter stood out in the case studies both because of

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the significant impact they had on this teacher's learning and because of how,
together, they illuminate the effects of factors not typically coordinated within
efforts to promote policy. They include: the teacher's analysis of the impact of
changes on students' learning; learning in contexts not intended to press policy
objectives; the teacher's personal stance toward learning; and the topic-specific

nature of the teacher's prior knowledge and teaching experience.

The Dmamic Nature of the Learning Process

This study underscores the importance of recognizing and understanding
what happens once the teacher tries out new instructional strategies. Sandy
generated ideas about what to change and then monitored what happened as a
consequence of the changes, focusing particularly on the impact on students'
learning. Recall that Sandy was completely willing to suspend her deeply held
knowledge and beliefs about teaching computation in order to try other
instructional strategies. As she began working with the new strategies, she
assessed that students' proficiency with computing greatly declined. This
analysis had a very powerful influence on subsequent changes she made in her
teaching and was pivotal to her interpretation of what the state proposed for
reforming computational skills instruction.

These findings illuminate that there is both an interactive and iterative
quality to the relationship between instructional policy and teaching practice,
one critically influenced by a teacher's continual learning. The idea that learning
impacts a teacher's actions in the classroom, and that a teacher's analysis of the
changes made affects the teacher's interpretation of policy, represents a clear shift
in more conventional ideas about policy implementation. Straightforward
transmission and adoption, for example, do not conceptualize teacher learning as

the core activity affecting how policy is implemented. These models either

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assume away or ignore the dynamic nature of the relationship between policy
and the teacher. They far undervalue, for example, the activity of refining
instructional strategies over time, as new learning is factored in. We saw in
Sandy's case that her teaching of the conceptual underpinnings of computation
went through several rounds of revision, in part, because of what she learned
from students' interaction with the new strategies. Later, she reinstated
traditional teaching practices once she factored in her analysis of students'
proficiency with computation. What she learned from her practice ultimately
fostered a new interpretation of policy as well. These events might be
understood or interpreted as teacher disinterest, or worse yet incompetence,
when the central frame of analysis is not teacher learning.

The work of McLaughlin and others did describe implementation to
involve a dynamic relationship between policy and the teacher (McLaughlin,
1976). The model of mutual adaptation, for example, suggested that the
relationship between policy and the teacher was two-way and that each shaped
the other. Yet, within that frame, the core activity was not conceptualized as
teacher learning. At the same time, this work set the direction for future research
to examine more closely the relationship between policy and the teacher. The
EPPS research recognized the central role that teacher learning played in the
policy implementation process and began identifying and studying various
aspects of the relations among policy, teacher learning, and practice (see for
example, Ball et al., 1994; Cohen & Ball, 1990a; Peterson, 1990; Wilson et al. 1996).

This work is a continuation of earlier EPPS research. Like earlier studies,
this work suggests that learning is the core activity of policy implementation for
the teacher. Findings illuminate that policy is not a static set of new ideas for
teaching. Instead, the ideas of policy grow and change as the teacher interacts

with them. Chapters 4 and 5 serve as examples of the changing nature of policy

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and practice as a teacher's prior knowledge and beliefs interact with learning
about new proposals for teaching. The case studies portray implementation as
an evolutionary process, one where ideas and practices evolve slowly and
unevenly in a teacher's thought and action in the classroom. More subtle and
dramatic changes that the teacher made in her teaching were not the end-product
of policy. Instead, changes in teaching represented only one aspect of a more
broad-based, ongoing process of learning. Other activities such as unpacking the
proposals, generating teaching strategies, and analyzing the implications of
changes represented other aspects of the implementation process. Yet, at the
core of each of these activities was the teacher's continual learning and
investigation of changes in teaching.

What I describe suggests a model of implementation that is substantially
different from more common views. Policy does not arrive in teaching practice
but instead is evidenced by continual growth and change in the teacher's ideas
about teaching and about policy. This shift requires that we abandon
assumptions underlying straightforward transmission or unadulterated
adoption. Instead, the policy and practice relationship is conceptualized as one
of teaching and learning (Ball et al., 1994; Cohen & Barnes, 1993a). This implies
an intermingled process, a dynamic of back and forth, with each aspect of the
relationship impacting the other. In this model, it would become natural to
think, for example, that changes in teaching signal the onset of new learning and
reinterpretation of policy.

If we conceptualize the policy and practice relationship as one of teaching
and learning, then what we mean by "professional development" would also
change, in that opportunities for teacher learning would include many things

besides traditional professional development and inservice events. Before

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turning to these issues, I continue to focus on the complexity involved for the

teacher in learning from external efforts to reform.

Factors Affeeting Teaeher Learning
Recall from Chapter 1 that policymakers in California initially

underestimated the significance of teacher learning in relationship policy.
Policymakers tended to rely on changes in standards, guidelines, testing, and
textbooks to support the desired goals. They assumed that mere exposure and
awareness of ideas coupled with other levers of standards-based reform would
accomplish the state's goals. These levers were relied upon as much or more
than teachers’ professional development.

This study reminds us that even if professional development had been
targeted at the onset, there would have been much more for policymakers and
educators to understand to adequately address teachers' needs as learners. This
study reveals there are multiple factors that interact and affect what a teacher
learns. Some of the influences are intentional levers, developed specifically to
promote the policy (i.e., mathematics frameworks, curriculum guides,
professional development activities). These factors played a significant role in .
Sandy's ongoing development of ideas and practices. Yet, the analysis in
Chapter 6 also reveals that there were multiple other influences impacting
Sandy's thoughts and actions as much or more than intentional levers. Some of
these involved resources Sandy used because of connections she made between
policy ideas and ideas offered in the context of the learning occasion (i.e.,
doctoral studies, conversations with other educators). Others were rooted within
the teacher as an individual learner and others in the school context (i.e., Sandy's
prior knowledge and teaching experiences, Sandy's and the principal's views of

good mathematics teaching). Although each of the factors identified in Chapter 6

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had a significant impact on Sandy's thought and practice, the following four
stand out because they represent the powerful effect of factors not typically

expected, or coordinated within efforts, to press desired goals and practices.

Analysis of the Impaot of Changes on Student Learning

I mentioned earlier, in my discussion of the dynamic nature of the
learning environment, the important role that Sandy's analysis of the impact of
changes on students' learning played in shaping her understanding and
enactment of the policy. I reiterate here the significance of this analysis on
Sandy's practice. Sandy observed and analyzed what happened in her teaching
as a result of the changes she made. She formed judgments about whether and
to what extent students' learning opportunities were improved. She also made
judgments about whether her findings fit with the learning goals envisioned in
the reform documents. This analysis proved to be the most pivotal information
Sandy would rely upon for reforming her teaching of computation. Her return

to traditional ideas and practices rested almost exclusively on the analysis.

Learnin nt xt Not Int nd t Pr 5 P lic b' tiv

We also saw in the case studies that Sandy's learning in situations not
intended to press policy objectives had a significant impact on what she made of
the state's recommendations. For example, Sandy learned in unplanned, private
conversations with other educators that discrete mathematics was intended only
for secondary education. My analysis in Chapter 6 suggests that although
teachers respond to the multiple levers designed specifically to press desired
practices (i.e., curriculum guides, state testing program, reform documents), they
are influenced equally or more by resources not designed or intended to support

the state's goals. From this standpoint, teachers often learn about reform-based

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teaching through experiences that are grounded in differing assumptions, aims,
and purposes than those underlying policy.

Sandy's work with Project AIMS is also a good example. Sandy used
Project AIMS curriculum materials to support changes she wanted to make in
her teaching of multiplication and teaching of discrete mathematics. I found that
the philosophical and epistemological views of mathematics and science learning
underlying AIMS materials were different from those described in the new
framework. Sandy negotiated the meaning of any differences on her own, or

they acted as a subconscious influence on her understanding of the state's goals.

The Teaeher's Personal Stanee Toward Learning

A somewhat less obvious influence on Sandy's interpretation of policy
was her stance toward her own personal learning. Although Sandy argued that
learning involved a constructivist kind of process, her stance for learning herself
was more a search for certainty (McDonald, 1992). Sandy operated as if the
policy embodied the right answers for her teaching, and her job was to figure out
what those answers were. This view contrasts sharply with other views.
Shulrnan, for example, argues that policy ideas are more moral and political
imperatives requiring degrees of teacher autonomy so that professional
judgment can function comfortably (Shulrnan, 1983).

Sandy's stance contributed to an emotional response to learning that was
unproductive for her. She reported that she often felt bad because she lacked
important knowledge and that these feelings contributed to her sense that she
was not a good teacher. She reported a sense of remorse and regret over errors
she felt she had made in her teaching. Lord (1993), Schifter (1993), and others
report on an emotional response to teacher learning that fosters and underlies

crucial learning about changing teaching. Lord describes a productive

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disequilibrium for teachers and argues that these experiences are necessary so
that teachers can confront their histories in teaching with an eye toward the
policy vision. He claimed that the experience is often uncomfortable or difficult
for learners, as firmly held, deeply entrenched beliefs and knowledge are
challenged or discarded. For Sandy, the uneasiness she experienced failed to
provoke questioning or crucial learning about deeply held beliefs and
knowledge. Instead, it created a sense of self-doubt and frustration. These
feelings perpetuated her stance toward certainty and an isolatiOn in her learning.
Recall in Chapter 5 that Sandy responded to the frustration she experienced by
turning away from other educators and toward curriculum materials developed

by Marilyn Burns.

Topie-Specifie Differences in Teaehers' Prior Knowledge and Teaching
Experience

Research on teacher learning has already shed much light on how
teachers' prior knowledge and teaching experiences interact with teachers'
learning to teach, acting both as a contributor and an obstacle (Lortie, 1975; Ball,
1988). Findings from this study suggest that teachers' prior knowledge and
teaching experiences are topic-specific and interact differently on teachers' sense
making of different policy proposals and in teachers' learning experiences. I
argued in Chapter 6 that differences in Sandy's prior knowledge and teaching
experiences were mostly ignored as most content knowledge was assumed. Yet,
one of the most striking contrasts in the cross-case analyses was the topic-specific
differences in Sandy's prior knowledge and teaching experiences and the impact
of these differences on Sandy's responses to the proposals focusing on

computational skills and combinatorial mathematics.

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*fi*$*¥***$*$***$$*$$

To this point I have focused at the level of the individual teacher's sense
making in relation to policy. I have identified several aspects of the learning
environment that make it particularly difficult for the individual teacher to learn
about reform-based change. Below, I continue to identify aspects that
complexify matters for the teacher as learner, only now, I turn to focus at the

level of teachers' opportunities to learn.

The Non-Systematie Nature
of Teachers' Opportunities to Leam

In Chapter 6 I described the learning environment for teachers responding
to policy as lacking both coherence and coordination. I argued, for example, that
Sandy encountered a mismatch between the ideas and practices she encountered
in the contexts of her learning opportunities and what she needed as a learner to
understand and enact the ideas envisioned in the framework. In response, Sandy
tried to locate or develop opportunities to learn that she thought would attend to
needed learning by herself.

These circumstances place teachers in a paradoxical position. The
situation requires that teachers bring, rather than build, the capacity to select and
develop opportunities to learn that will focus on the central tenets of policy.
Furthermore, it requires that teachers bring the capacity to monitor and
understand factors affecting their learning, in order to manage the effects toward
the goal of producing reform-based teaching. Given the wide gap between
modal teaching and envisioned practice, it would be highly unlikely that any
teacher would bring such capacity. For one thing, most teachers would not bring
deep insights into what reform-based teaching looks like, how to create it, or

what is necessary to learn. Dewey and others made the argument long ago that

268

learners, by themselves, are not in a position to design their own learning
experiences because they do not bring a knowledge of the means and ends of an
experience that is truly educative (Dewey, 1938, Schwab, 1978). These scholars
recognized that the teacher, as a learner, is not likely to provide what is needed
to create an educational experience that insures that the learner will encounter
and develop the central tenets of reform-based change.

The professional development experiences Sandy encountered also gave
little attention to a principle about learning that policymakers and educators
hope teachers will learn and embody in their own teaching. The reform
documents promoted the idea that learning occurs best when situated inside
contexts where learning naturally arises. The implication of this principle for
learning to teach differently is that teachers would need to learn in contexts that
situate learning in the practice of teaching, where such learning is used. One
problem that Sandy encountered was that while teaching, there were other
competing goals that made it difficult to focus on required learning. Researchers
suggest that whether by observation and discussion of one's own practice,
another's practice, or video tape of practice, such contexts can offer richer
opportunities to link up reform ideas with the practical implications of those
ideas (Cohen, 1989; Ball 8: Cohen, 1995a). Opportunities such as these were
lacking in Sandy's educational experience.

As educators, we already understand that the conditions described are not
optimal for teachers. If teachers such as Sandy were to remain designers of their
own educational experience, they would have to become skilled at selecting and
judging among learning experiences in order to link up opportunities that attend
more directly to the ideas envisioned in policy and connect learning to the
practice of teaching. This would not be an easy or straightforward process. In

Chapter 3 I highlight the complexity and variation on the ideas and practices

269

envisioned in the reform documents. The case studies in Chapters 4 and 5
further illuminate that the teacher is likely to encounter multiple interpretations
of proposals and various directions for responding to the state's goals. Chapter 6
reveals aspects of teachers' opportunities to learn that make it difficult to learn
what may be needed. The paradox that teachers encounter in guiding their own
educational experience, as well as the need to connect learning to the practice of
teaching, are two examples of the complexity involved.

More problematic still is that many teacher educators and policymakers
do not recognize the complexity, and think the problem is a much simpler one.
Many suggest that because there already exists such a wide array of professional
development for teachers, teachers would only need to participate to develop
greater capacity to improve teaching. Overcoming such beliefs is yet another
obstacle when thinking about what may be needed to change these conditions.
This research suggests that more participation would not equate to improved
capacity. Although Sandy participated in and developed a wide variety of
professional development activities, there was little that offered systematic,

coherent learning about reform-based teaching.

*as»***»s*s*

What I have identified might be thought of as a set of links between
instructional policy and teaching practice. These links are examples of what's
involved as the individual teacher's sense making is connected to reform-based
ideas. How a teacher manages the interactive and iterative nature of the learning
required, as well as the interplay of factors that affect teacher learning, will
certainly impact the teacher's capacity to learn what is needed. The paradox that

teachers encounter as they select and judge learning experiences, as well as the

270

necessity of connecting learning to teaching practice, will also impact the
teacher's capacity to learn what is needed.

Findings in Chapter 6 suggest that Sandy's opportunities to learn, for the
most part, did not recognize or attend to the links I describe. Little, if any,
attention was given to the dynamic nature of the learning required. Sandy
worked mostly alone in analyzing the impact of the changes she made in her
teaching. None of Sandy's professional development opportunities focused on
making sense of the impact of changes she made in her teaching. Sandy
analyzed the impact on students' learning experience. Most of the professional
development activities Sandy encountered assumed knowledge of mathematics.
As a consequence, Sandy developed the opportunities she had for learning
combinatorial counting and the conceptual underpinnings of computation. And
even though learning in contexts not intended to press policy goals had a
powerful effect on Sandy's understanding of the state's goals, what she
encountered in those contexts often ran contrary to policy objectives.

The inattention to the dynamic nature as well as the factors affecting
teacher learning, create a complexity in the learning environment that teachers
like Sandy will work through on their own. The paradox that the teacher
encounters and the lack of opportunity to connect learning to teaching practice
further complicate matters for teachers. The inattention to these links reveal the
lack of coherence and coordination in a teacher's educational experience in the
context of efforts to reform.

These findings suggest that even if policymakers and state officials make
great strides in producing more coherent visions of change and multiple levers to
press teachers toward desired practices, it would not be enough to affect practice.
Attention would have to be given to the teacher's learning environment: asking

how to create conditions that will foster learning that is not so detached from

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policy objectives. This implies attending to the sources of weakness inherent in
teachers' opportunities to learn. The remaining pages of this chapter draw
together findings in this study with related research to offer several ideas for
attending to the specific links identified in this chapter. My overall aim is to
conceptually develop directions that would likely lead to more systematic and
coherent learning about reform-based teaching for teachers. Thus, I speculate
about what would need to be different in the learning environment if the links I
described were not so weak. I sketch proposals that recognize and attend to the
dynamic nature and multiple factors that affect teacher learning. I discuss
possible ways to address the paradox inherent in professional development
experiences as well as the lack of connection between learning and teaching.

And I suggest promising directions for future research.

Creating Stronger Learning Links Between
Instructional Policies and Teachers' Opportunities to Learn

Some might suggest that as a starting point, teachers would need more
coherent and simplified visions of change. In other words, some would start by
trying to improve policy itself. An important first step may be to let go of the
idea that policy can create the teaching hoped for. A general finding from
revisiting the RAND Change Agent study, 10 years later, is that "It is exceedingly
difficult for policy to change practice, especially across levels of government"
(McLaughlin, 1990). This study concurs with this finding and offers a particular
perspective of why it remains true. In Chapter 3, I described the difficulties in
formulating a view of envisioned practice. The analysis revealed that a teacher's
prior knowledge and teaching experience affects greatly what the teacher can
learn from new frameworks, texts, or tests. What the teacher brings to ”reading”
policy ultimately determines whether any of the proposals can be recognized or

understood. More problematic still, is that reform documents, in particular, are

272

under-specified, but for good reason-~they are not intended as programs for
practice. My work in Chapter 3 provides specific examples of how a single
proposal might be interpreted in multiple ways, each with its own merits to
judge. Perhaps more important, this study reveals the lack of coherence and
coordination that teachers encounter across professional development
experiences. More specifically, it illuminates a set of weak links in the teacher's
learning environment in the context of external efforts to reform.

Thus, as instruments of change, policy can inspire, encourage, and paint
grander visions of teaching. At the same time, policy may not be the most
promising avenue for addressing teachers’ needs as learners. It may be more
productive for policymakers and educators to ask whether and how policy can
facilitate effective opportunities for teachers to learn about reform-based change.

Educational researchers are beginning to focus on what it might take to
improve teachers' opportunities to learn. Many have recommended massive
reform of professional development opportunities for teachers (Sarason, 1993).
Some argue that reform of teachers' professional development forms the most
serious unsolved problem for policy and practice in American education today
(Sykes, 1996). Wondering about how teachers' opportunities to learn can be
improved implies consideration of a complex set of issues involving reasoning
among the subject matter of reform, theories for learning it, the teacher as
learner, and the contexts in which teachers work (Schwab, 1978; Shulrnan, 1987).

Although the challenges are significant, much research and promising
projects are already underway. Cohen and Barnes, for example, argue that the
key to change is that reform itself must be framed as a set of educational
opportunities designed to embody the sorts of teaching and learning that
reformers wish to promote (Cohen & Barnes, 1993). They suggest it would be far

more likely to produce desired patterns of teaching and learning in classrooms if

273

teachers could experience those patterns for themselves first-hand. In other
words, teachers' opportunities to learn would have to be fundamentally different
than what Sandy had. Yet, how would they need to be different? This question
points to what this study suggests should be a central focus of future research --
how can the environments of instruction for teachers and policy work together to
create new contexts that sufficiently meet teachers' learning needs in the context
of external efforts to reform? To the extent that we can identify and understand
what is involved, we can begin to imagine what it would take to create better
opportunities to learn for teachers.

This study offers several important contributions to what may be needed.
I begin by focusing on the non-systematic nature of teachers' opportunities to
learn. I argue that a combination of freedom and control over teachers' learning
experiences may be needed to attend to the paradox and lack of connection
between learning and practice. I suggest that guidance can function as a form of
control that can assist teachers in making selections of what to participate in, and
forming judgments about teaching that does and does not fit well with reform-
based ideas. I offer two examples of guidance. I propose that clearly stated
learning goals can function as a form of guidance, especially in relationship to
contexts not intended as support of policy objectives. Recall that this factor had a
powerful effect on Sandy's understanding and enactment of the framework. At
the same time, there was little in place to guide her selections and judgments of
experiences not intended to press policy goals. As another example, I suggest
that teacher leaders, if well-prepared, can function effectively as a form of
guidance. I discuss what this research suggests is important as preparation for
such as role.

I then turn to focus on the dynamic nature of the learning process

Specifying how the interactive and iterative nature of the learning required might

274

be attended. I suggest that teachers need more opportunities to learn that focus
on what happens as changes are made in teaching. The purpose would be to
strengthen the connection between the ideas of policy and teaching practice. In
addition, I propose a set of four learning objectives, recommending that these
objectives be taken up within the contexts of teachers' opportunities to learn. The
goal would be to foster learning that connects what a teacher is likely to bring as
a learner to what teachers are offered in the contexts of opportunities to learn. I
identified three additional factors affecting teacher learning in this work: the
teacher's analysis of the impact of changes on students' learning; a teacher's
personal stance toward learning; and topic-specific differences in a teacher's
prior knowledge and teaching experience. To attend to these links, I propose
that teachers' opportunities to learn focus directly on analyzing the impact of
changes on students' learning experiences, including attention to teachers' beliefs
about how students' opportunities to learn will be improved. I then suggest that
teachers' opportunities to learn attend to the individual nature of the
disequilibrium teachers experience. And finally, I argue that teachers'
opportunities to learn must attend to topic-specific differences in teachers'

subject-matter knowledge.

Attn in t th Nn tma' Natur
of Teaehers' Professional Development Opportunities
In my discussion of the nonsystematic nature of teachers' professional
development opportunities, I argued that teachers encounter a paradoxical
situation. Teachers must select and judge opportunities to learn, with few ideas
and little first-hand experience with the ideas and practices underlying policy.
These circumstances, in addition to the lack of opportunities that connect

learning more directly to teaching practice, create difficulties for the teacher to

275

work out on her own. Below, I offer two directions for strengthening a teacher's
capacity to select and judge opportunities to learn and form stronger connections

between learning and teaching practice.

A Combination of Freedom and Control

' Many policymakers, educators and researchers argue that because
changing teaching is highly specific and context dependent, teachers would
necessarily need to craft for themselves what they would need to learn and do to
reform their teaching. The analyses across this work suggest the opposite. When
teachers are left to develop for themselves a course of study to learn about
reform-based change, the educational opportunity becomes restricted. The case
studies help us to see how this happens. Even as teachers are highly sensitive to
the contexts in which they work, they also bring knowledge, skills and
dispositions that can act as powerful barriers to change. Teachers must confront
and extend what they already know and believe about mathematics teaching if
they ever hope to understand and produce the teaching envisioned in policy. A
teacher's opportunities to learn most certainly will play an important role. Yet,
these circumstances raise an important question regarding the degree of freedom
teachers have to select, develop and judge opportunities to learn.

The degree of freedom a teacher has to choose and make judgments about
their own professional development experiences can limit the teacher's
educational opportunity. This claim is likely a controversial one, given that
teachers are professionals and are obligated to do what is needed to do their job
well. Yet, Dewey warns there can be no greater mistake than to treat freedom in
learning as an end in itself. He argues that freedom is only powerful when it
functions to frame purposes, to judge wisely, and to evaluate desires by the

consequences that result from the action (Dewey, 1963, p.64). This study

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illuminates what happens when a teacher has the freedom to select and order
experience to carry desired ends into operation. Sandy experienced freedom to
the extent that much of what she learned went uninterrupted and unchallenged
by policymakers' ideas. This was both a function of her individual choices as
well as what she encountered as professional development opportunities. In her
educational experience, freedom and control over learning functioned
disproportionately and unproductively. Yet, what would it mean to provide
each element? And in what form could control be offered?

At the same time, we have learned from past efforts at reform that policy
can't mandate what matters at the local level (McLaughlin, 1990). This finding
suggests that control would probably not work well in the form of a mandate.
Instead, past studies suggest that local capacity, expertise, organizational
routines, and resources available to support planned change efforts generate
fundamental differences in the ability of practitioners to plan, execute or sustain
an innovation (McLaughlin, 1990). These findings suggest that control might
best be offered in the form of a resource, such as guidance or expertise.

One form of guidance that would likely have helped teachers such as
Sandy would be clearer articulations of learning goals for teachers. I argued that
it is difficult for the teacher to recognize and comprehend reform ideas in the
contexts of opportunities to learn, because they have so few first-hand
experiences with the principles and practices underlying policies that call for
substantial departure from traditional teaching. Clear articulations of learning
goals potentially could offer teachers guidance for planning and making
decisions about what is important to learn and what to participate in.

Educational researchers have begun to outline the vastness and

enormity of the learning required for teachers to respond to the flood of

277

policies in the late eighties and early nineties (Ball, 1997; Wilson et. al,

1996). The paragraph below highlights what may be needed.

The teaching that reformers seem to envision thus would require
vast changes in what most teachers know and believe. Teachers
would have to revise their conception of learning, to treat it as an
active process of constructing ideas rather than a passive process of
absorbing information. They would have to rediscover knowledge
as something that is constructed and contested rather than handed
down by authorities. They would have to see that learning
sometimes flourishes better in groups than alone at one's desk with
a worksheet. And in order to learn, teachers would have to unlearn
much deeply held knowledge and many fond beliefs. Such
learning and unlearning would require a revolution in thought,
and scholars in several fields have shown that such revolutions are
very difficult to foment. Moreover, once teachers' academic
knowledge and conceptions of learning changed, they would then
have to learn how to teach differently. (Cohen & Barnes, 1993a, p.
246)

Given the variety and vastness of what is believed important for teachers'
learning, it may be worthwhile to focus, at least initially, on a smaller subset of
learning goals for teachers. Sandy's responses to the different strands of
proposals focusing this study suggest that a teacher will narrow reform, to a
smaller subset of the agenda, in order to define a starting place and make the
workload more manageable. It may be worthwhile for policymakers and

educators to plan in advance for this scaling down by asking what may be the

278

most salient ideas for teachers to begin efforts to reform. The case studies in this
work underscore the importance of this point. Sandy scaled down the agenda
for herself and gave attention predominantly to only a few select ideas and
proposals. Whether her decisions focused on the central tenets or most
promising aspects of the reform agenda is an open question.

Sandy's emphasis on discrete mathematics, for instance, may or may not
be a priority at the elementary level. If it is, then clearer ideas and articulations
of what is important for elementary students to learn about discrete mathematics
would be an important step toward identifying clearer learning goals and
guidance for teachers' learning. Sandy's work with computation reminds us of
the stranglehold that traditional computational skills instruction has on teachers’
ideas about good mathematics teaching (Putnam & Geist, 1994; Wilson, 1997).
One could argue that initial efforts should focus directly on the goal to teach
computational skills differently. In California, policymakers and educators took
a less than direct approach to underscoring the importance of this topic-specific
area of mathematics. Yet, teachers’ histories with teaching computation tell a
very different story about the role that computation will play in teachers' efforts
to change their practice. I argued that Sandy's ideas about computation were not
uprooted, hardly even challenged, despite great effort to reform computational
skills instruction. The idea of problem solving as a context for developing
computational skills, in particular, was barely examined. Had Sandy had a clear
set of learning goals within this topic-specific area, it may have directed her
efforts toward the more central tenets of reforming computational skills teaching.

The goal then would not just be about making better learning experiences
for teachers, although that would be paramount, but it would also be about
offering teachers a form of guidance, a lens to help teachers, teacher leaders,

principals and others to select and judge opportunities to learn. If clear

279

articulations of learning goals were available, they could assist educators in
recognizing what is essential to making sense of reform ideas. Had Sandy had
clearer ideas about what and how she would need to learn, this could have
helped to focus her efforts on central aspects of the reform agenda.

Policy levers do not offer such guidance. In Chapters 4 and 5 I argued
that even opportunities specifically designed to press teachers toward desired
practices offered conflicting messages of reform-based teaching. The state’s
testing program, for example, painted a very different View of the mathematics
important for students to know and do than, for example, replacement units.
The need for such a lens is underscored further when we take into account that
teachers learn a great deal from occasions that are not intended as support
toward the state's goals. In these settings teachers could use the lens to examine,
select and then judge opportunities to learn on the basis of how they connect to
learning goals for teachers. Teachers could sort through ideas and opportunities,
forming priorities, and seeking out learning experiences that were well-matched
with the principles important to understand the teaching and learning

underlying instructional policy.

The Important Role and Preparation of Teaeher Leaders

Guidance might also be offered in the form of well-prepared teacher
leaders. In Chapter 5 I described the opportunities that Sandy offered other
teachers for learning about the new framework in the contexts of state-sponsored
workshops. An important question that surfaces from the study of her efforts to
teach other teachers is whether Sandy was sufficiently prepared to teach others
about reform-based teaching. After all, there is no formal, organized preparation
for such a role. Sandy acquired a reputation for being an excellent mathematics

teacher. Many would argue that she is a likely candidate for teacher leader. Yet,

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I have argued that the teacher, as a learner, is not well-positioned to guide even
her own learning. For one thing, the policy proposals sketch dramatic
departures from modal teaching, making it very difficult for any teacher to have
deep knowledge of what is involved. Furthermore, most teachers would not be
knowledgeable of the factors that affect teachers' learning. Sandy, for example,
did not acknowledge or address teachers' prior knowledge and dispositions
toward teaching computation at the workshops she offered.

Yet, what if conditions were optimal and formal preparation for teacher
leaders did exist? What would such programs involve? Few projects to date
have explored this issue. Even less is known about what teachers would need to
know to facilitate other teachers' learning. At the same time, this study
illuminates that even though Sandy is an exemplary elementary teacher, she
could have benefited others in her role as teacher leader had she had much
greater understanding of what underlies reform-based teaching and well-
grounded ideas about how teachers might learn the ideas of reform.

There are several projects underway that consider the preparation of
teacher leaders an important aspect of any change effort. One project describes
four objectives as central goals of teacher leaders' knowledge: theoretical
knowledge in mathematics education, methods of integrating new technologies,
enhancing mathematical knowledge and didactic repertoire and developing
leadership skills (Even, 1994). The objectives are aimed at developing
accomplished professionals who understand teacher learning as well as goals
and objectives for improving mathematics education.

Cohen & Barnes point out that such professionals would also have to
develop relationships with teachers that combine trust and critical reflection

(Cohen & Barnes, 1996, p.247). They stress that teacher leaders would have to

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understand the difficulties involved as teachers unlearn and encounter the
uncertainties of changing teaching practice.

Another project underway, DlVII (Developing Mathematical Ideas), has
given much attention recently to preparing teacher leaders (Schifter, 1998). The
project is looking carefully at what may be involved for teachers learning to
facilitate other teachers' learning in relationship to DMI curriculum materials.
DMI developers have created multiple supports for teacher leaders including a
mentoring program and two-week institutes that explore DMI curriculum and
teacher learning issues. Researchers studying this effort identified the
importance of teacher leaders learning to facilitate "openings" in the curriculum,
places in the discourse where participants' prior knowledge and beliefs
potentially become central targets of exploration (Remillard & Geist, 1998).

This study adds further to these insights. In particular, teacher leaders
would need to understand the important role that a teacher's analysis of the
impact of changes on students' learning will play in sustaining any change effort.
They would have to understand and be prepared to support teachers in
formulating this analysis and sketching further directions for refining
instructional strategies. In addition, teacher leaders would have to be prepared
to support teachers who are likely to bring a stance of certainty to their
opportunities to learn. More specifically, they would have to understand the
individual nature of the disequilibrium a teacher is likely to experience. Another
important aspect of the teacher leader's knowledge would be understanding the
topic-specific nature of teachers' prior knowledge and teaching experiences, as
well as the impact of differences on learning about reform-based teaching. I
expand on each of these aspects of teacher leaders' knowledge as I discuss

proposals that focus at the level of the individual teacher's sense-making.

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Proposals to Improve Teaehers' Opportunities to Learn

I turn now to focus on the individual contexts of teachers' opportunities to
learn. Earlier I argued that inattention to the dynamic nature and the factors
affecting the teacher's learning experience creates complexities that the teacher
has to figure out alone. Here I speculate about what would be needed if these
links were attended to and woven into the contexts of teachers' opportunities to
learn. I sketch four objectives that potentially could strengthen a teacher's

capacity to understand and enact reform-based teaching.

Learning to Analyze the Impact of Change

We saw in the case studies that a teacher's responses to policy do not stop
once the teacher changes her teaching. Instead, a back and forth process of
learning and change continues. Sandy formed and reformed her practice in
relationship to much new learning. Yet, there seemed a critical point in her
learning process that later became pivotal to Sandy's understanding and
enactment of the policy. This involved Sandy's analysis of the impact of change
on students' learning. Although much was done speculatively, the actual
analysis of the consequences of change, based on the actual interactions with
students in real classrooms, had a much greater impact on Sandy's interpretation
of reform-based teaching. More specifically, Sandy's analysis reflected her belief
that students' proficiency with computation remained a high priority, equal to
that of conceptual understanding. It would have been interesting to see whether
Sandy's analysis would have been different had students' learning goals been
reassessed and factored into her analysis of the impact of the changes she made

in her teaching.

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It may be useful for teachers to have multiple opportunities to learn that
focus directly on making sense of what happens once changes are made. This
aspect of the teacher learning process deserves much more attention, especially
in future educational research. We need to know much more about how teachers
analyze the impact of changes, what they look for, and how they assess student
learning goals in the context of change, including what might be used as
evidence for improving learning opportunities. Whether teachers observe and
analyze other teachers trying out similar strategies, examine video tape of one's
own teaching or another's teaching, or become involved in ongoing discussions
about the impact of change, these occasions could offer teachers a richer context
for learning to analyze the impact of change.

One stumbling block to creating such opportunities would be getting past
the belief that once teachers learn they can easily introduce proposals into
teaching. Policymakers, educators, and teacher leaders would first have to come
to understand that changes in teaching are not the end product of policy but
instead mark the onset of critical new learning about reform-based teaching.
Accepting this as reality for the individual teacher would not be an insignificant
thing. This step alone could begin to fundamentally change ideas about what
professional development for teachers would need to be. For example, one
change might involve overcoming the idea that a teacher, single-handedly and
simultaneously, could attend to the competing goals of implementing new
instructional strategies, working with the constraints and uncertainties of
classroom teaching, assessing students' learning experiences, and attending to
their own learning agenda. An appropriate shift might be that teachers need
multiple opportunities to unpack each goal and attend to each aspect, drawing

information together over time to further refine teaching strategies.

284

Another change might be that other teachers or teacher leaders would
need to become involved in the individual teacher's practice. Perhaps by
observing teachers try out new instructional strategies and helping them make
sense of what happens, especially in analyzing students' learning experiences.
Other teachers or teacher leaders might also become involved by teaching with
the teacher or supporting the teacher through ongoing conversations and
decision making, perhaps around video taped teaching. The overall idea would
be to support teachers' efforts to systematically inquire into teaching as a group
as well as individually. From this standpoint, teachers would learn to work the
ideas of reform into teaching collectively and help each other analyze the effects
of new strategies. The impact of such learning could strengthen a teacher's
capacity to connect reform-based teaching ideas to the practical realities of
teaching. Involvement in an ongoing study group, for example, or interaction
with a teacher leader could have been quite useful in helping Sandy analyze the
impact of the changes she made, especially in making sense of the problems that
arose in her practice. Study groups can offer teachers an opportunity to explore
reform ideas in the context of their own or another's teaching experiences over

time (Featherstone et al., 1993a, 1993b).

F trin Blif A utIm rvin tud nt' rtuniti Larn

A significant aspect of Sandy's analysis, of the impact of changes she made
in her teaching, had to do with her beliefs about whether students' opportunities
to learn were improved. Recall that Sandy, in the second year, delivered a mix of
innovative and traditional teaching strategies to address learning objectives for
students. Her decision to reinstate traditional forms of teaching would signal to
many a failure to implement policymakers' ideas. Yet, I argued that the mix of

old and new strategies was Sandy's interpretation of policymakers' ideas. And I

285

argued that her interpretation rested largely on her analysis that students'
opportunities to learn were not necessarily improved by increasing emphasis on
conceptual and contextual goals. She reinstated drill and practice on number
facts and algorithms to round out students' overall experience, claiming that
these changes better represented what policymakers had in mind.

This aspect of Sandy's learning experience points to the importance of
attending more explicitly to teachers' beliefs about how students' opportunities
to learn will be improved. It may be necessary to focus directly on teachers'
beliefs about how reform-oriented teaching improves students' learning
experiences in the contexts of teachers' opportunities to learn. I argued that
Sandy relied on herself for making sense of students' learning experiences. She
noticed that the changes she made assumed mastery, proficiency, accuracy and
speed with computing. Because she valued these goals, she evaluated students'
experience , at least in part, on the basis of whether these goals were attended.
Yet, I argued that the reform documents indicated these goals were less
immediate and conceptual and contextual learning were seen as more central.
Sandy's beliefs about whether students' opportunities to learn were improved
rested on misguided ideas about how students' opportunities to learn would be
improved. These issues were taken for granted on any wider scale, perhaps the
were assumed as part of buying into the reform package.

Teachers' beliefs about improving learning experiences for students may
need to be a central focus of teachers' opportunities to learn. Inattention to these
beliefs, for Sandy, fostered the idea to balance the old and new forms of teaching.
In a worse case scenario, inattention might suggest that reform is optional. If
teachers had opportunities that attended explicitly to their ideas of whether and
how students' opportunities to learn were improved, it could also encourage

teachers to work through problems in practice rather than return to traditional

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strategies. Fostering beliefs about how traditional forms of teaching and reform-
based teaching do and do not "improve" students' opportunities to learn can
function as a pivotal point in teachers' decision making. Overall, teachers may be
far less likely to discard or compromise new teaching strategies because they
understand the implications of both reform-based teaching and traditional

teaching on students' learning opportunities.

Fostering A Stance of Critique And Inquiry

Ball and others argue that because policy goals involve work that is
uncertain and underdetermined, a stance of critique and inquiry, one of asking,
debating, formulating and exploring conjectures and deliberation would be far
more productive for teachers (Ball, 1996; Lord, 1995). Fostering a stance of
critique and inquiry would therefore be a critical element of the learning
environment as teachers work to interpret policy and progress into various
phases of learning, change, and refinement of reform-based teaching ideas.

In this study, I argued that this teacher approached reform with a stance
of certainty. And although Sandy participated in several collegial learning
environments, few offered an opportunity to experience a stance of critique and
inquiry first-hand. In addition, when Sandy did participate in environments that
were less certain and embraced more complex views of learning, Sandy often
withdrew and developed learning experiences that were more isolated, claiming
the more collegial environments were less productive because they produced
much frustration and confusion.

Sandy's experience raises many questions about how a teacher's stance
toward learning might be altered. To the extent to which this is an individual or
a matter of professional community is an important question. We already have

evidence to suggest that hearing how others interpret and manage issues

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regarding change can support teachers as they work to change their own
personal stance toward learning (Lord, 1994; Lieberman, 1995; Little, 1993).
Teachers’ questioning and exploration when situated in common learning
communities can be a central source of support for teachers as they struggle with
relevant issues of changing teaching. At the same time, opportunities to learn
that value a stance of critique and inquiry can represent difficult situations for
teachers that bring a stance of certainty. Sandy withdrew into materials such as
curriculum units and the framework when she encountered difficulties in
learning with others. Interviews revealed that she did not recognize collegial
environments or differences in stance a crucial aspect of her own learning.

It may be that teachers need a combination of pressure to participate and
support within environments that offer an opportunity to experience a stance of
critique and inquiry. The only way teachers can have a personal experience
interacting in collaborative environments is if they are required to participate in
collegial opportunities to learn that embody a stance of critique and inquiry. If
Sandy were to experience first-hand how interactions with others increase the
potential of her own learning, she must participate in environments that offer the
opportunity to recognize differences in stance. At the same time, adequate
support must also be offered, so that effective collaborative work can become
part of the teacher's personal learning experience. Recall that the disequilibrium
Sandy experienced created self-doubt and frustration rather than questioning
and debate. If Sandy's experience were to be more productive, she would have
to experience disequilibrium differently, recognizing it as an essential element of
the process, learning to question and debate the issues creating the uneasiness.
This means she would have to continue to participate and work in these

environments rather than turn away from them. And the environment must be

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supportive in ways that press Sandy gently toward a shift away from certainty
and toward a stance of critique and inquiry.

Several non-traditional projects underway count the construction of these
environments where issues can be discussed and debated as essential elements of
an educative atmosphere (Ball, 1996, Brown, 1994; Featherstone et al., 1993a,
1993b). If supportive environments such as these were developed in both
traditional and non-traditional structures and teachers were required to
participate in them, it could be an effective way to strengthen teachers' capacity
to approach reform-based change with a stance of critique and inquiry. Wide-
scale participation would build teachers' capacity to recognize and understand
environments that do and do not value such a stance (see Lortie, 1975, p.70).
Developing such capacity can function to guide teachers in their selections and

judgments of opportunities to learn.

Subjeet-Matter Foeused, Topie-Speeifie Oppormoities to Leam

An important contribution of this study is recognizing the significance of
topic-specific differences in teachers' prior knowledge and teaching experiences
and the impact of those differences on teachers' responses to reform-based
change. Findings suggest that just as changing teaching across subject areas is
not a general proposition, changing teaching within subjects is also not a general
proposition. Although others have pointed to the importance of considering
subject-matter specific aspects of learning to teach (Shuhnan, 1986; Stodolsky,
1988), this study suggests we must go one step further than previous studies
have argued important. Teachers' topic-specific, subject matter knowledge
affects what they can learn in a climate of educational reform. I found that
within the same subject-matter there are significant differences in both what

teachers bring and what teachers would need to learn to teach different topics in

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mathematics for understanding. Recall that Sandy often found herself trying to
understand how to introduce discrete mathematics into her teaching lacking the
necessary prerequisite knowledge of mathematics needed to understand the
examples and illustrations offered. Sandy's opportunities to learn overall did not
attend to her prior knowledge, skills, and dispositions and whether they set her
up productively to learn in the circumstances offered.

One implication of these findings is that subject-matter focused, topic-
specific learning opportunities may be required for teachers to learn to change
their teaching of any specific mathematics topic targeted by policy. To date,
topic-specific aspects of changing teaching is a relatively untouched and
unexamined area in educational research. Much more research would be needed
to characterize and understand the nature of these differences and the impact on
teachers learning to change their teaching. However, there is much in the
learning theory literature that does bear on these findings. Situated learning
theory supports the principle that learning is dependent on elements of the
context and content (Brown et. al, 1989, Lave 8: Wenger, 1991). Furthermore, if
differences in subject-matter represent a critical variable in the learning
environment, it would follow to reason that topic-specific differences are crucial
aspects to attend to as well.

If we were to attend to the topic-specific nature of teacher learning, we
would need to know much more about how differences impact teachers'
responses to policy. Although policies often nominate specific-topics in
mathematics considered important for student learning, what teachers bring and
would need to learn to teach those topics is not addressed. Tracking down the
nitty-gritty of what teachers would need to learn to change their teaching of any
topic in mathematics would not be a straightforward task. My analysis in

chapter 3 contrasts extant ideas and practices with policy proposals for two

290

different topics. This process may be useful for drawing out both what teachers
would likely bring in relationship to specific topics, as well as ideas about what
policy proposes teachers should change. Figuring out where and how the two
might connect can be useful toward developing and articulating learning goals
for teachers within topics.

Yet, much more research would be needed to understand the role of
differences across topics and the impact of differences on the learning agenda for
teachers. In the case of discrete mathematics, for example, Sandy had few ideas
to draw from for making sense of the proposed changes. She knew little about
discrete mathematics and had no experience teaching it. Essentially, Sandy was
unprepared to select and configure a course of study for introducing discrete
mathematics into elementary mathematics teaching. Yet, as a teacher leader, she
had to design an overall plan in order to teach other teachers about the new
framework. Recall in chapter 5, I argued that there were more fruitful sites than
relational theories or discontinuous mathematics that could have focused
Sandy's learning experiences. Sandy's response to withdraw from these
experiences and to focus on the ice-cream cone problem seemed a more
promising direction because the ideas connected more closely to ideas she
brought to the learning experience. At the same time, Sandy worked alone
making sense of the mathematics underlying the problem. My examination of
the opportunities to learn that Sandy offered others revealed that combinatorial
reasoning was completely side-stepped as central aspect of teaching
combinatorial counting for understanding.

In the case study of Sandy changing her teaching of computational skills,
there was much to be challenged, up-rooted and transformed in Sandy's thought
and practice. Researchers have already pointed to the importance of un-learning

in changing teaching (Ball, 1988). Yet, we have few ideas of what un-learning

291

would involve for specific topics in mathematics. We saw that this teacher
became stuck around issues of mastery, speed, accuracy, proficiency in
computing, as well as memorization of number facts and algorithms. And even
though Sandy went on to learn new ideas and practices for teaching
computational skills, she easily returned to traditional ideas when problems
arose in her practice. This easy return suggests that Sandy had little evidence of
the impact of traditional forms of teaching on students' learning experiences.
Although Sandy heard the arguments for why traditional forms of practice, such
as drill and practice and the importance of speed and accuracy in computing,
limit students' flexibility with computational skills, she had no first-hand
experiences that produced evidence of the impact of traditional forms of practice.
Sandy's experience lacked an important connection that would be required to
sustain and refine new forms of practice. Had she been convinced of the impact
of traditional forms of teaching, it may have been more likely that Sandy would
continue to examine alternatives instead of returning to traditional patterns. For
Sandy, traditional forms of practice remained a legitimate option.

Findings such as what I describe here can be very useful for developing
topic specific learning goals for teachers. Educators and teacher leaders would
then have to work to understand the relationship between what teachers bring
across targeted topics and what teachers would need to know and do to teach
those topics differently. Ultimately, the aim would be to produce opportunities

to learn that attend to differences and the impact on a teacher's responses to

policy.

Change At the Level of The Smallest Unit
In this study, the conversation about the policy and practice relationship is

situated within the realm of the teacher's learning and teachers' professional

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development. I argued that even if the state could find ways to align its vision
more effectively across policy instruments, it would not have helped this teacher
respond to the policy any better. Although systemic alignment, regulation,
incentive, restructuring, as well as many other factors, may, in fact, be part of the
overall plan, this study illuminates the importance of attending to the
educational issues that affect a teacher's understanding and enactment of policy.
Only then are we likely to improve teaching for US. children (Ball & Cohen,
1995a, 1995b; Little, 1993; Sykes, 1996; Smylie, 1997, Wilson et al., 1996).

Since the onset of this work, much has changed in California. It was just a
short time after the 1992 framework that reform began to disintegrate. Wilson, in

a recent paper describing the events in California, writes:

” groups such as HOLD (Honest Open and Logical Debate), a
parent and community organization concerned with what they

called the "'discovery-based constructivist math characterized by
the framework began to appear at public hearings and on the
World Wide Web. Their concerns were many -- that the "new

mat " ways of teaching were untested and unproved, that the large
scale empirical work done on mathematics instruction
demonstrated that direct instruction was more effective, that the
frameworks emphasized mathematical appreciation, not

mathematical content knowledge, that the new tests being

advocated were "subjective" . . .

Wilson explains that time and again two issues arose in the worries of
groups like this: the lack of attention or de-emphasis on basic and computational

skills and the sacrifice of conversations about content in the name of

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conversations about pedagogy (Wilson, 1997). By 1995, the new superintendent
of schools ordered a task force to examine the state of affairs in mathematics
education. The task force issued reports calling for ”balance” and' arguing for
more ”basic skills” and traditional mathematics instruction along with an
increased emphasis on problem solving. Ultimately, a new committee began
work on a new mathematics framework- a more balanced approach - for
mathematics education. The New York Times reported in November of 1997,
that the California Board of Education had endorsed a new set of standards for
elementary and middle school students that emphasized the more traditional
drill and practice approach over a new method that was characterized by
complex word problems.

This dissertation offers policymakers and state officials a portrait of one
teacher's efforts to attend to many of the issues dominating conversations and
current debate about these reforms and the more recent return to traditional
practices. The portrait illuminates the complexity involved for the individual
teacher in the context of strong systemic efforts to reform. It illustrates that there
are a variety of different components to a teacher's practice and that there is
much for the teacher to think about and do to revise, develop, and compose in
the course of efforts to improve instruction.

Further, it shows that the mathematics framework, and the other levers
aligned by the state, are not the only influences on a teacher's practice, and that
many of the influences are not always recognized or coordinated. Findings also
reflect that what the teacher encounters as professional development experiences
can be insufficient for meeting the individual learning needs of the teacher. And
to make up for the lack, the teacher will likely develop and coordinate learning
experiences for herself to attend to what she believes is needed. However, the

analyses across this dissertation suggest that the teacher, as a learner, may not be

294

well-positioned to do this alone. This claim represents a fundamental statement
about who the teacher is in the context of external efforts to reform. The teacher,
as the learner, will likely bring knowledge, skills, and dispositions toward
teaching that run counter to reform goals. Thus, it would be very difficult for the
teacher to provide the guidance needed to insure that learning experiences were
coordinated and more coherent to the teacher.

And yet, what the teacher brings and musters as resources and learning,
will have a great deal to do with whether reform efforts continue to press
forward at the level of the teacher's practice. In Sandy's case, she brought, for
example, the conviction that learning was necessary, a stance of certainty toward
learning, a view of reform-oriented teaching, and a particular view of
mathematics grounded in the philosophy of Project AIMS. What Sandy brought
influenced her responses to the framework in the form of shaping her course of
action, what she thought she would need to learn, and where she would turn for
guidance. I argued in the case studies that Sandy often relied upon resources
that resonated well with her prior knowledge and teaching experience, in part,
because she had direct access to these resources. Many of these occasions
embodied assumptions about mathematics teaching and learning that were
incompatible with the ideas and practices described in the reform documents.

As a result, Sandy's understanding of reform-oriented teaching was hardly
challenged as most assumptions were left unacknowledged and unexamined
across learning experiences.

Thus, the question arises as to how teachers' educational experience might
need to change, so that it offers a more coordinated and coherent approach
toward the goal of reform-oriented teaching. I have sketched several ideas in this
chapter as a closing to this work. In essence, I have suggested that any efforts to

effect and sustain change at the level of the teacher's practice would be based on

295

understanding better the relationship between the teacher's needs as a learner
and the environments of instruction for teachers, and working to create more

meaningful links between the two.

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Appendices

297

Appendix A

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Appendix A

The table below summarizes the big ideas predominate across the opportunities
to learn that Sandy encountered for learning about discrete mathematics and
basic computational skills. I prioritize the ideas by what the opportunity

emphasized most, substantively.

Table X: Predominant Ideas Emphasized Across Sandy's Opportunities to Learn

 

Basic Computational Skill Discrete Mathematics

 

 

. . relational theories and algorithms
Conversations With

Balance basic skill instruction with
Other educators

conceptually oriented curriculums what discrete means

Emphasis on technology in the discrete mathematics is included for
classroom although not directly the secondary curriculum to attend
associated to computational skill to ideas not currently getting taught

 

 

 

 

299

 

 

Reading Reform

Documents

 

Estimation, technology , alternative

algorithms as computing strategies

Conceptually orienting

computational curriculum

Contextualize skill instruction

 

Introduced as a new strand of
content important to include at all

levels of the curriculum

ideas are mostly in the current
curriculum but would need different

emphasis and organization

Include discrete math because it is
the math of our time - more current
than other mathematics already

being taught

Limited ideas about what discrete

math is

general points about pedagogical
approaches i. e. should not be
algorithmic or involve the
memorization of formulas, and
instead should encourage a

conceptual understanding

mentioned a number of ideas central

to discrete mathematics

 

300

 

 

Doctoral Studies

Bloom’s automaticity ideas

Emphasis on constructivist ideas

about learning

Emphasis on cognition,
understanding and problem solving
as curriculum goals as opposed to

memorization of routine procedures

none

 

Curriculum

Materials

No direct attention to proficiency

with computation
Emphasis on contextualizing and
conceptually understanding of four

operations

Alternative computing strategies

the ideas of combination and

permutation

looking for patterns in counting

what discrete mathematics involves

alternative pedagogy for teaching

ideas (moves away from traditional

explain and tell types of

instructional patterns)

 

 

Teaching experiences

 

Difficulties and uncertainties of
conceptually focused instruction of
four operations to promote

proficiency with computation

More direct attention to

computational goals

 

ideas of combination and

permutation

Strategies for counting outcomes

how some decisions in mathematical
problems can be turned over to the
learner shaping the mathematical
situation. Sandy interprets this to

suggest her teaching is less didactic

 

301

 

 

 

, no direct attention to computational Examples of different kinds of
Professronal

skill but offer ideas for how to counting problems
development Contextualize mathematical content
inside problem solving settings Alternative pedagogy for the
teaching of mathematics , mostly
Ideas about mathematics, its involving problem solving.
usefulness, and what conceptual

ideas are related to the four

operations

suggests alternative pedagogy for

the teaching of mathematics more

 

 

 

generally

 

The ideas in each category represent the substantive focus of each of the
six different categories of opportunities to learn. My determinations were based
on interview data and therefore factored in Sandy’s sense of what the big ideas
were as well. My own perceptions of what the opportunities offered also
contributed to my determinations. Data sources included my observations of
Sandy’s teaching, my analysis of the curriculum materials she used, my analysis
of artifacts from her doctoral coursework, my reading of the reform documents,
and my observations of the professional development activities she engaged in

including Project AIMS and the state-sponsored workshop.

The Substantive Focus of Sandy’s Learning Experienees For Changing Her
Teaehing Of Computational Skills

The predominate idea across the six different occasions for reforming
instruction of basic computational skills focused on the notion that students must
understand the mathematics of computation conceptually. Each of the six

categories involved some aspect of what it would mean to conceptually

302

understand and conceptually orient the elementary mathematics curriculum.
Sandy’s learning experiences centered on the idea of conceptually understanding
the four operations. For example, the reform documents suggested that
emphasis be shifted away from the preoccupation with calculating answers and
move toward understanding how operations work and are useful. The
curriculum materials Sandy used focused on the conceptual underpinnings of
the four operations. Project AIMS materials focused on using computational
skills in the service of solving real-world problems.

Sandy’s learning experiences also focused on the idea of contextualizing
mathematics. Sandy’s work with Project AIMS and her reading of the reform
documents focused more generally on learning mathematics in relation to
problem solving experiences. Some of the other materials Sandy used focused on
students’ strategies for computing in problem solving contexts. At the same
time, none of the experiences she encountered focused on whether or how the
traditional computational algorithms might be treated in relationship to problem
solving. My analysis in chapter three revealed that the reform documents did
not attend directly to the traditional computational curriculum, its problems or
what may still be reasonable, leaving the reader to dig out and figure out why
recommendations were proposed.

Most of Sandy's learning opportunities did not give attention to the
proposals to introduce a variety of new computational skills. Interestingly, the
only opportunity Sandy had that emphasized estimation as a new computing
technique was what she read in the reform documents. Yet from a policy
standpoint, estimation was a big idea. Estimation, for the most part, remained
untouched across Sandy’s learning experiences. Recall in the case study that she

returned to practices that used estimation mainly as a checking strategy.

303

Technology also did not get attention across Sandy’s opportunities to
learn. Although technology was discussed in the context of Sandy’s
conversations with other educators, the other categories of learning experiences
that Sandy had gave only rhetorical consideration to these proposals.

Sandy’s learning experiences did focus on students’ algorithms. The
curriculum materials she used promoted the idea that students naturally derive
alternative strategies for making computations. At the same time, Sandy’s work
with students’ computational strategies was the primary source of the difficulties
she encountered in her teaching. Recall that although Sandy encouraged
students to compute using their own strategies, students were often slow and
constructed strategies that did not produce accurate results. In response Sandy
demonstrated the traditional algorithm and suggested that students use it in
place of their own strategies. Sandy’s opportunities to learn did not focus any
attention on how she might manage the problems that arose in her practice.

Sandy encountered the idea of balance. In her conversations with other
educators, Sandy encountered that mathematics teaching should emphasize both
skill learning and conceptual understanding. For instance, in her doctoral
studies, Sandy encountered the idea of automaticity, suggesting to her that rote
memorization of particular mathematical skills is central to learning
mathematics. Across Sandy’s opportunities to learn the emphasis was on
conceptual and problem solving approaches to teaching. At the same time there
was emphasis on a balanced approach to the teaching and learning of
mathematics. In combination, the two ideas folded in comfortably, one along
side the other. Sandy drew from her learning experiences that both the
conceptual and the procedural knowledge in mathematics still mattered and that
pedagogically each must be attended. Because Sandy encountered little

opportunity to learn about how each could be attended within the same learning

304

experience, she concluded each must be attended and in ways already available
and understood. In her doctoral studies, Sandy encountered the idea of '
automaticity, suggesting that rote memorization of particular mathematical skills
is central to learning and doing mathematics. This knowledge reinforced her
sense that computational skill learning should be approached in rote, drill
oriented learning opportunities. As another example, Sandy encountered no
opportunities for learning how number fact acquisition might be nested in

problem solving contexts where learning specific facts made sense.

The Substantive Focus of Sandy's Opportunities to Learn About Teaehing

Diserete Mathematics

Sandy’s opportunities to learn about discrete mathematics focused on a
wide variety of mathematics. Yet most of the occasions focused on mathematical
ideas Sandy had little or no prior experience with. For example, Sandy’s
conversations with other educators focused on the ideas of continuous and
discontinuous mathematics, relational theories (recurrence relations), and
counting principles Sandy had no prior experience with. The curriculum
materials she used focused on combinatorial counting problems. At the same
time they did not explore any underlying ideas of combinatorial counting such
as combinatorial reasoning. Instead, Sandy's experience with the materials
focuses on the idea of order and patterns. Recall that Sandy’s reading of the
reform documents focused on matrices.

Sandy encountered contradictory information concerning whether
discrete mathematics should get attention in the elementary school curriculum.
Recall that she read in the framework that discrete mathematics should be
treated as any other strand. Yet, she learned from a leading mathematics

educator that discrete mathematics was introduced for the secondary

305

curriculum. The reform documents also seemed contradictory giving explicit
attention to discrete mathematics only in secondary focused sections.

Sandy encountered the idea that discrete mathematics should pot be
taught in ways that emphasized learning rules, memorization of formulas, or
symbol manipulation. Sandy’s learning experiences focused on the idea that
teaching discrete mathematics in ways that emphasized traditional patterns was

not appropriate.

306

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