PROSPECTIVE TEACHERS’ EMERGING PEDAGOGICAL CONTENT KNOWLEDGE DURING THE PROFESSIONAL

SEMESTER: A VYGOTSKIAN PERSPECTIVE ON TEACHER DEVELOPMENT

byMARIA LYNN BLANTON

A dissertation submitted to the Graduate Faculty of North Carolina State University

in partial fulfillment of the requirements for the Degree of

Doctor of Philosophy

MATHEMATICS EDUCATION

Raleigh 1998

APPROVED BY:

Dr. Glenda S. Carter Dr. Jo-Ann D. Cohen

Dr. Lee V. Stiff Dr. Karen S. Norwood Co-Chair of Advisory

Committee

Dr. Sarah B. Berenson

Co-Chair of Advisory Committee

DEDICATION

To my family.

PERSONAL BIOGRAPHY

The author was born August 7, 1967, to Tommy and Patricia

Blanton. She was raised in Willard, NC. She received her Bachelor of

Arts degree in mathematics with secondary teacher certification and

Master of Arts degree in mathematics from the University of North

Carolina-Wilmington (UNCW).

After teaching at UNCW, she moved to Raleigh, NC, to attend

graduate school at North Carolina State University. Here, she

received her Ph. D. in Mathematics Education in 1998. While a

student, she worked as a teaching assistant in the Mathematics

Department and as a research assistant in the Center for Research

in Mathematics and Science Education.

ACKNOWLEDGMENTS

I would like to thank my family for their continued support

through all my years of school. I am especially grateful to have

parents that I can count on for anything and everything. They have

always provided a weekend haven from the rigors of graduate

school. Lisa and Joey have helped maintain my perspective through

laughter. My niece, Rachel, and nephew, Joseph, have reminded me

that the most important things in life are not always measured by

academic success.

I thank Dr. Wendy Coulombe for “paving the way” for me. She

has been a valued friend and mentor. I thank Dr. Draga Vidakovic

and Dr. Susan Westbrook for being unofficial committee members.

Their advice has always been insightful and challenging.

I would like to thank members of my committee, Dr. Lee V.

Stiff, Dr. Jo-Ann Cohen, and Dr. Glenda Carter, for being a part of

this process. I extend a special thanks to Dr. Carter for our

numerous impromptu discussions on Vygotsky. She was a

tremendous “more knowing other”.

I would like to thank my co-chair, Dr. Karen Norwood, for her

unique contribution. She motivates me to pursue my own practice

with unapologetic enthusiasm. To this end, she was always willing to

extend her expertise, as well as her classroom supplies.

Most importantly, I would like to thank my major advisor, Dr.

Sally Berenson. She introduced me to a national and international

research community in mathematics education through an extensive

apprenticeship in the Center for Research in Mathematics and

Science Education. It has been an invaluable experience. Most

especially, she placed an intellectual trust in me throughout the

dissertation process. I sincerely appreciate that trust, as well as the

guidance and encouragement that accompanied it.

TABLE OF CONTENTS

Page

LIST OF

TABLES.......................................................................................................

...ix

LIST OF

FIGURES....................................................................................................

.....x

INTRODUCTION.........................................................................................

.................1

PART I: LITERATURE

REVIEW...............................................................................8

Theoretical Framework....................................................................................8

Vygotsky’s Sociocultural Theory of Learning.............................................9

General Genetic Law of Cultural Development..................................10

Psychological Tools and Signs..................................................................11

The Role of Language.................................................................................12

Social Interactions...................................................................................

....13The Zone of Proximal

Development......................................................14 Implications of Vygotsky’s Sociocultural Theory

for this Study............................................................................16

Teacher Education..........................................................................................

..17Teachers’ Beliefs and

Knowledge.............................................................17Learning How to Teach

Mathematics.....................................................20Teacher Development in

Context............................................................21Classroom

Interactions....................................................................................23

Implications.......................................................................................................26

The Nature of Qualitative Inquiry................................................................27

In-Depth Interviewing................................................................................2

8Participant

Observation..............................................................................29Teaching

Experiments................................................................................30

PART II: METHODOLOGY.........................................................................................

33

Methodological Framework...........................................................................33

Participants.........................................................................................................35

Data Collection..........................................................................................

........35Data

Analysis.....................................................................................................38

Role of the Researcher.....................................................................................4

0

PART III: MATHEMATICAL DISCOURSE IN A PROSPECTIVE TEACHER’S CLASSROOM: THE CASE OF A DEVELOPING

PRACTICE......................................................................................................................42

Abstract...............................................................................................................43

Introduction......................................................................................................44

Teacher Learning Through Classroom Discourse....................................46

Process of Inquiry.............................................................................................4

9The Research

Setting..................................................................................49Collecting the

Data......................................................................................50Analyzing Classroom

Discourse...................................................................51Pattern and Function in Teacher-Student

Talk....................................51Process of

Analysis......................................................................................54Findings and

Interpretations.........................................................................56Early Pattern and Function in Classroom

Discourse...........................56Early Pattern and Function in Resolving Students’

Mathematical Dilemmas.......................................................57Early Pattern and Function in Teaching a New

Concept...............63On Early Discourse and Mary Ann’s

Practice....................................73Indications of an Emerging Practice: Change in Pattern

and Function............................................................................75

The Problem-Solving Day.....................................................................75Moving Forward in Classroom Discourse: Learning

to Listen.....................................................................................87

Mary Ann’s Students: More Knowing Others?....................................93

Discussion..........................................................................................................95

References..........................................................................................................98Appendix..........................................................................................................102

PART IV: THE CYCLE OF MEDIATION: A TEACHER EDUCATOR’SEMERGING PEDAGOGY..........................................................................................107

Abstract.............................................................................................

................108Introduction.......................................................................................

.............109Rethinking the Role of Supervision: Education or

Evaluation?........110Collecting the Data: The Cycle of

Mediation.......................................113Pedagogy of the Teaching

Episodes.......................................................116Data

Analysis...................................................................................................118

Findings and Interpretations.......................................................................119

Instructional Conversation in Teaching Episodes with Mary

Ann.....................................................................119Activating, Using, or Providing Background Knowledge

and Relevant Schemata......................................................120

Thematic Focus for the Discussion...................................................120

Direct Teaching, as Necessary.............................................................123

Minimizing Known-Answer Questions in the Course ofthe

Discussion.......................................................................124Teacher Responsivity to Student

Contributions...........................124Connected Discourse, with Multiple and Interactive

Turns on the Same Topic...................................................127

A Challenging but Nonthreatening Environment......................129

Instructional Conversation in Retrospect: More on the

Problem-Solving Day...........................................................130

Discussion.........................................................................................................131

References........................................................................................................134

Appendix..........................................................................................................137

LIST OF

REFERENCES.............................................................................................

.141

APPENDIX..................................................................................................

..................155

LIST OF TABLES

Page

PART IV: THE CYCLE OF MEDIATION: A TEACHER

EDUCATOR’S EMERGING PEDAGOGY

1. Conversational time used by participants in the teaching

episodes.............................................................................................

...124

2. Conversational time given to subject code during teaching

episodes.............................................................................................

...129

LIST OF FIGURES

Page

PART I: LITERATURE REVIEW

1. Higher mental functioning: Vygotsky’s general

genetic law of cultural

development..........................................................11

PART II: METHODOLOGY

2. The cycle of mediation in an emerging practice of

teaching.................38

PART IV: THE CYCLE OF MEDIATION: A TEACHER EDUCATOR’S

EMERGING PEDAGOGY

1. The cycle of mediation in an emerging practice of

teaching................116

ABSTRACT

BLANTON, MARIA LYNN. Prospective Teachers’ Emerging

Pedagogical Content Knowledge During the Professional Semester:

A Vygotskian Perspective on Teacher Development. (Under the

direction of Sarah B. Berenson and Karen S. Norwood.)

This investigation adopts an interpretive approach to study a

prospective middle school mathematics teacher’s emerging

pedagogical content knowledge during the professional semester.

Vygotsky’s (1978) sociocultural perspective provides the theoretical

framework for the study. Specifically, Vygotsky’s assertion that

higher mental functioning is directly mediated through social

interactions focused this study on the intermental context in which

the prospective teacher’s practice develops during the professional

semester, or student teaching practicum.

The nature of mathematical discourse embedded in social

interactions in the prospective teacher’s classroom was analyzed as

a window into the prospective teacher’s construction of knowledge

about teaching mathematics. The role of students as more knowing

others of the classroom norms for doing mathematics and how that

mediated the teacher’s practice was considered. Analysis of pattern

and function of classroom discourse substantiated an emerging

practice, as the prospective teacher’s obligations in the classroom

transitioned from funneling students to her interpretation of a

problem to arbitrating students’ ideas.

This study also explored the pedagogy of educative supervision

and the consequent role of the university supervisor in opening the

prospective teacher’s zone of proximal development. Classroom

observations by the supervisor, teaching episode interviews between

the supervisor and the prospective teacher, and focused journal

reflections by the prospective teacher, were coordinated in a process

of supervision postulated here as the cycle of mediation.

Understanding what interactions between the university

supervisor and prospective teacher might resemble in order to

promote the prospective teacher’s development within her zone was

central to this study. The resulting pedagogy of the teaching

episodes was consistent with instructional conversation (Gallimore &

Goldenberg, 1992). In this case, instructional conversation seemed

to open the prospective teacher’s zone so that her understanding of

teaching mathematics could be mediated with the assistance of a

more knowing other. This, together with the cycle of mediation,

suggests an alternative model for helping teachers develop their

craft in the context of practice.

INTRODUCTION

Historically, mathematics education has entertained diverse

views in an almost eclectic move toward a unified theory of learning.

Indeed, advances in cognitive psychology have prompted a shift from

stimulus-response models in which learning is defined by students’

perfunctory reactions to stimuli, to meaning-based models such as

constructivism, in which students are seen as actively and

individually creating their own knowledge (Noddings, 1990; von

Glasersfeld, 1987). Recently, as disciplines such as anthropology and

sociology have joined the quest for a comprehensive theory of

learning, emphasis on the more prevalent Western tradition of

individual knowledge construction has broadened to include the role

of culture and context in this process as well (e. g., Cobb &

Bauersfeld, 1995; Eisenhart & Borko, 1991; Ernest, 1994; Saxe,

1992; Shulman, 1992). The resultant theory, generally described as

social constructivism, has become a watchword for those who

espouse constructivist views that recognize contributions from social

processes and individual sense making in learning (Ernest, 1994).

For the most part, Vygotskian and Piagetian theories of mind have

dominated thinking in this area as scholars debate the primacy of

the social versus the individual in knowledge construction (e. g.,

Cole & Wertsch, 1994; Confrey, 1995; Ernest, 1995; Shotter, 1995).

In some cases, such debates have been discarded in favor of

theoretical perspectives that coordinate social and individual

domains in a complementary fashion (Cobb, Yackel, & Wood, 1993).

Mathematics education has led reform efforts in its attempts to

incorporate recent research in such disciplines as cognitive

psychology into an existing knowledge base to produce a codified

body of principles, or standards, for teaching and learning

mathematics. Most notably, the National Council of Teachers of

Mathematics (NCTM) Curriculum and Evaluation Standards for

School Mathematics (1989), which has theoretical roots in

constructivism, is grounded in two decades of research on students’

thinking about mathematics (Simon, 1997). According to Simon, a

strong research base on teacher development that parallels national

reform efforts in students’ mathematical development is currently

needed in the mathematics education community. It is not enough to

understand the process of learning mathematics; mathematics

educators must also understand the process of teaching mathematics

in reform-minded ways. Thus, the question becomes how can teacher

education programs integrate research in such disciplines as

cognitive psychology, sociology, and anthropology with that of

mathematics education to prepare a professional cadre of

mathematics teachers? More specifically, how can such programs

prepare inservice and prospective teachers to teach mathematics in

a manner consistent with the recommendations of the NCTM

Curriculum and Evaluation Standards? The NCTM Professional

Standards for Teaching Mathematics (1990) offers a timely response

to this question. Its stated purpose is to provide a set of standards

that

promotes a vision of mathematics teaching, evaluating

mathematics teaching, the professional development of

mathematics teachers, and responsibilities for professional

development and support, all of which would contribute to the

improvement of mathematics education as envisioned in the

Curriculum and Evaluation Standards (p. vii).

Furthermore, it advocates five major shifts in classroom perspectives

in order to promote students’ intellectual autonomy. In particular,

teachers’ thinking needs to shift

(a) toward classrooms as mathematical communities-away

from classrooms as simply a collection of individuals;

(b) toward logic and mathematical evidence as verification-

away from the teacher as the sole authority for right answers;

(c) toward mathematical reasoning-away from merely

memorizing procedures;

(d) toward conjecturing, inventing, and problem-solving-away

from an emphasis on mechanistic answer-finding;

(e) toward connecting mathematics, its ideas, and its

applications-away from treating mathematics as a body of

isolated concepts and procedures (p. 3).

Such recommendations reflect critical insights into teaching

mathematics and are consistent with the Curriculum and Evaluation

Standards.

Various long-term research agendas in mathematics education

directed towards prospective and inservice teachers are working to

address the need for a reform-driven research base in teacher

development (e. g., Ball, 1988; Berenson, Van der Valk, Oldham,

Runesson, Moreira, & Broekman, 1997; Carpenter, Fennema,

Peterson, & Carey, 1988; Cobb, Yackel, & Wood, 1991; Eisenhart,

Borko, Underhill, Brown, Jones, & Agard, 1993; Feiman-Nemser,

1983; National Center for Research on Teacher Education, 1988;

Schram, Wilcox, Lappan, & Lanier, 1989; Shulman, 1986; Simon,

1997). One such program has identified seven domains that

constitute teachers’ professional knowledge as content knowledge,

pedagogical content knowledge, general pedagogical knowledge,

knowledge of educational contexts, knowledge of curriculum,

knowledge of learners, and knowledge of educational aims

(Shulman, 1987). Shulman’s model continues to provide a conceptual

framework for other studies on teaching. Indeed, a number of

researchers in mathematics education (e. g., Ball, 1990; Berenson, et

al., 1997; Borko, Eisenhart, Brown, Underhill, Jones, & Agard, 1992;

Even & Tirosh, 1995; McDiarmid, Ball, & Anderson, 1989) recognize

that understanding of these knowledge domains, as well as the

consequent role of teacher education programs in teacher

preparation, is currently underdeveloped. They have accepted the

challenge this offers by studying various strands within each domain

as well as the connections that exist among them.

Of the seven components of this knowledge base for teaching,

pedagogical content knowledge was the focus of this study. Shulman

(1987) defines such knowledge as

that special amalgam of content and pedagogy that is uniquely

the province of teachers.... [It is] the blending of content and

pedagogy into an understanding of how particular topics,

problems, or issues are organized, represented, and adapted to

the diverse interests and abilities of learners, and presented

for instruction (p. 8).

Pedagogical content knowledge is recognized among mathematics

educators as playing a central role in one’s development from

learning mathematics to teaching mathematics (Ball, 1990; Borko, et

al., 1992; Even, 1993). Additionally, they acknowledge that our

understanding of this domain, as well as the integrated manner in

which it exists in the teaching process, is incomplete. Based on the

premise that the professional semester, or student teaching

practicum, is a pivotal context in which prospective teachers begin

to construct pedagogical content knowledge, this study considered

how the prospective mathematics teacher’s practice emerges during

this stage.

In order to understand the construction of pedagogical content

knowledge, I appealed to the theoretical lens of social

constructivism. Viewing mind metaphorically as social and

conversational, Ernest (1994) posits that people are “formed through

their interactions with each other (as well as by their internal

processes) in social contexts” (p. 69). This is no less true for

prospective teachers during the student teaching practicum. Indeed,

Vygotsky’s (1986) assertion that higher mental functions are directly

mediated through social interactions suggests that the prospective

teacher’s transition from mathematics student to mathematics

teacher does not occur apart from human interaction; rather, as a

result of it.

Such transitions can be characterized as a process of

acculturation resulting from one’s (i. e., the prospective teacher’s)

development within the zone of proximal development. The zone of

proximal development is defined by Vygotsky (1978) as “the distance

between the [individual’s] actual developmental level as determined

through independent problem solving and the level of potential

development as determined through problem solving under adult

guidance or in collaboration with more capable peers” (p. 86). This

suggests the importance of instructional assistance in the

prospective teacher’s development.

This study is an investigation of the prospective middle school

mathematics teacher’s emerging practice of teaching during the

professional semester. In particular, I first considered the nature of

mathematical discourse, or conversation, embedded in social

interactions in the prospective teacher’s mathematics classroom as

preliminary to the broader context of teacher development. The

nature of such discourse was expected to provide a window into the

prospective teacher’s construction of knowledge about teaching

mathematics. Also, I examined the university supervisor’s role as a

more knowing other in the prospective teacher’s emerging practice.

Specifically, I considered what the pedagogy of supervision might

resemble in order to open the prospective teacher’s zone of proximal

development and effect a change in practice. Thus, the following

questions were formulated to guide this research:

1. What is the nature of mathematical discourse in the

prospective teacher’s mathematics classroom during the

professional semester?

2. How does the university supervisor influence the

prospective teacher’s emerging practice of teaching

through the zone of proximal development?

LITERATURE REVIEW

This chapter begins with a discussion of the social construction

of knowledge as a theory of learning. It includes a detailed

examination of the sociocultural theory of Lev Vygotsky, which

provided the theoretical framework for this study. Attention is given

to the basic tenets of Vygotsky’s theory as well as various constructs

associated with it. Linkages between his theory and this study are

established. A review of current literature on the preparation and

development of teachers follows this. In connection with this, the

role of classroom interactions in the social construction of

knowledge is examined. Implications of this study in addressing the

limitations of existing research in teacher education are discussed.

Finally, the process of qualitative inquiry is described to support this

choice of research paradigm for the study.

Theoretical Framework

Shulman (1992) wrote that “knowledge is socially constructed

because it is always emerging anew from the dialogues and

disagreements of its inventors” (p. 27). This suggests an inherent

complexity of social constructivism. That is, social constructivism is

difficult to precisely define because it is subject to the varied

experiences and biases of its inventors. Ernest (1994) comments that

there is a “lack of consensus about what is meant by the term, and

what its underpinning theoretical bases are” (p. 63). He recognizes

that both social processes and individual sense making are central to

a social constructivist theory, and that the emphasis given to either

domain will vary depending on one’s theoretical assumptions

concerning the nature of mind. In particular, the social

constructivist’s view of mind will often have Piagetian or Vygotskian

roots, although one may rely on other perspectives more or less

compatible with these traditions. A Piagetian view prioritizes the

individual act of knowledge construction by interpreting social

processes as either secondary, or separate, but equal. Ernest

maintains that a Vygotskian theory of mind “views individual

subjects and the realm of the social as indissolubly interconnected”

(p. 69). He further explains that

mind is viewed as social and conversational because....first of

all, individual thinking of any complexity originates with, and is

formed by, internalized conversation; second, all subsequent

individual thinking is structured and natured by this origin;

and third, some mental functioning is collective (p. 69).

In this study, I have assumed a Vygotskian theory of mind. As such,

the remainder of this section will be used to outline the basic tenets

of such a theory and how it serves as the framework for this study.

Vygotsky’s Sociocultural Theory of Learning

According to Wertsch (1988), Vygotsky’s theory of mind

consists of three major themes. First, Vygotsky maintained that any

component of mental functioning is understood only by

understanding its origin and history. As Luria, a protégé of Vygotsky,

summarized,

in order to explain the highly complex forms of human

consciousness one must go beyond the human organism. One

must seek the origins of conscious activity....in the external

processes of social life, in the social and historical forms of

human existence (1981, as cited in Wertsch & Tulviste, 1996,

p. 54).

To this end, Vygotsky considered the life span development of the

individual (ontogenesis) and the development of species

(phylogenesis), as well as the associated sociocultural history. This

emphasis represented a shift from the traditional focus of his

contemporaries on the individuality of child development.

General Genetic Law of Cultural Development

Another major theme of Vygotsky’s theory is found in his

general genetic law of cultural development. This theorization of the

relationship between social and individual domains in higher mental

functioning emphasizes Vygotsky’s belief in the social formation of

mind: “Social relations or relations among people genetically

underlie all higher functions and their relationships” (Vygotsky,

1981b, as cited in Wertsch & Tulviste, 1996, p. 55). The general

genetic law of cultural development posits that an individual’s higher

(i. e., uniquely human) mental functioning originates in the social

realm, or between people, on an intermental plane. Internalization of

higher mental functions is then a process of genetic (i. e.,

developmental) transformation of lower mental functions to the

intramental plane, within the individual (Wertsch, 1988; Wertsch &

Toma, 1995). This process is illustrated in Figure 1. According to

Holzman (1996), the exact nature of this genetic transformation has

been a subject for much research. In particular, research in Soviet

psychology has produced a method of investigation known as the

microgenetic approach (from microgenesis). This approach involves

charting the transition from the intermental plane to the intramental

plane over the course of a brief social interaction in order to study

the process of change that occurs.

Figure 1. Higher mental functioning: Vygotsky’s general genetic law

of cultural development.

Psychological Tools and Signs

Finally, Vygotsky believed that higher mental functioning is

mediated by socioculturally-evolved tools and signs (Wertsch, 1988).

In particular, Vygotsky (1986) addressed human use of technical, or

physical, tools to illustrate the role of psychological tools in higher

mental functioning. He maintained that a physical tool acts as a

mediator between the human hand and the object on which it acts in

order to control natural, or environmental, processes. In an

analogous manner, psychological tools such as gestures, language

systems, mnemonic devices, and algebraic symbol systems, serve to

control human behavior and cognition by “transforming the natural

human abilities and skills into higher mental functions” (p. xxv).

According to Vygotsky, “humans master themselves from the outside

- through psychological tools” (p. xxvi).

Vygotsky studied signs as a special form of psychological tools

(Minick, 1996). Wertsch and Toma (1995) recognize this form as

well: “Of particular interest to [Vygotsky] were signs, which

constituted a broad category of mediational means used to organize

one’s own or others’ actions” (p. 163). These artificial stimuli include

such symbolic formations as social languages, mathematical systems,

and diagrams. Bakhurst (1996) describes tying a knot in a

handkerchief as a sign to invoke later rememberings. In this simple

illustration, the knot serves as a sign to control one’s behavior.

The Role of Language

Vygotsky (1986) viewed language as the most powerful

psychological tool for mediating higher mental functions. It is the

primary medium through which thought develops, making possible

the transition from the intermental plane to the intramental plane.

Furthermore, as a higher mental function, language is also subject to

the mediating effect of tools. Concerning this duality, Holzman

(1996) explains that the

dialectical role of speech is that it plays a part in defining the

task setting; this activity redefines the situation, and in turn,

speech is redefined. Language is viewed as both tool and result

of interpersonal [i. e., intermental] and intrapersonal [i. e.,

intramental] psychological functioning (p. 91).

In other words, language is unique in that it is both a mediating tool

and a mediated function.

Social Interactions

Vygotsky’s belief in the social origins of higher mental

functions and the mediating role of language in their development

underscores the importance of social interactions. Indeed, Vygotsky

argued that social interactions are the basis for an individual’s

development (Holzman, 1996). Minick (1996) explains that

Vygotsky turned to the primary function of speech as a means

of communication. [He] argued that the higher voluntary forms

of human behavior have their roots in social interaction, in the

individual’s participation in social behaviors that are mediated

by speech. It is in social interaction, in behavior that is being

carried out by more than one individual, that signs first

function as psychological tools in behavior. The individual

participates in social activity mediated by speech, by

psychological tools that others use to influence his behavior

and that he uses to influence the behavior of others (p. 33).

As an illustration, consider teaching a child to add fractions.

In the process of instruction, the teacher uses tools (e. g., language,

figural diagrams, and the real number system) to mediate the child’s

behavior or thinking. Once the child has appropriated this skill, he or

she then uses it in his or her own mathematical activity and

sometimes to influence the activity of peers. In this scenario, the

child’s development occurs within the context of social interactions.

While this illustration implies human-human interaction as a defining

characteristic of social interactions, participants in social

interactions are interpreted more broadly here to include

representations of ideas, such as those embodied in reading

materials. In this situation, the reader’s thinking is mediated through

written speech. Wilson, Teslow, and Taylor (1993) address this,

suggesting that the interactions between teacher and student can be

extended to “include interactions between learners and technology-

based tools and agents” (p. 81).

The Zone of Proximal Development

The zone of proximal development is one of the central

propositions of Vygotsky’s sociocultural theory. Daniels (1996)

describes this theoretical construct as the setting in which the social

and individual domains meet. Wertsch and Tulviste (1996) further

explain that the zone of proximal development has “powerful

implications for how one can change intermental, and hence

intramental, functioning” (p. 57). Change results from tool-mediated

activity such as instruction, that is, assistance by a more knowing

other offered through social interactions with the student. In turn,

instruction creates the zone of proximal development, which

stimulates inner developmental processes (Hedegaard, 1996). The

teacher’s task is to provide meaningful instructional experiences that

enable the student to bridge his or her zone of proximal

development. As such, the zone of proximal development is unique in

that it “connects a general psychological perspective

on...development with a pedagogical perspective on instruction” (p.

171).

A stringent interpretation of Vygotsky’s definition of the zone

of proximal development requires an adult or more capable peer to

foster one’s development. However, Oerter (1992) distinguishes

three contexts which can create one’s zone of proximal development:

intentional instruction (such as that given by a teacher or parent),

stimulating environments (such as books or materials for painting),

and play. He cites Vygotsky’s observations that children at play

create their own zones of proximal development: “In play the child

tries as if to accomplish a jump above the level of his ordinary

behavior” (Vygotsky, 1966, as cited in Oerter, 1992, p. 188). The

common thread is the presence of help in one’s construction of

knowledge. According to Taylor (1993), Vygotsky also suggested that

a student’s interactions with materials (e. g., manipulatives) can

enable that student to bridge the zone of proximal development for

deeper understanding. One can speculate that, had Vygotsky lived

long enough, his definition may have reflected this.

Implications of Vygotsky’s Sociocultural Theory for this Study

Eisenhart (1991) describes a theoretical framework as a

“structure that guides research by relying on a formal theory; that is,

the framework is constructed by using an established, coherent

explanation of certain phenomena and relationships” (p. 205). In this

sense, Vygotsky’s sociocultural theory guided my investigation of the

prospective teacher’s emerging practice. As a formal theory, it

provided an established language for communicating research, as

well as an accepted format for investigation. More specifically,

Vygotsky’s general genetic law of cultural development directed me

to social interactions as a forum for the prospective teacher’s

construction of pedagogical content knowledge. Furthermore, his

emphasis on the mediating affect of tools and signs, particularly

language, led me to investigate the role of language in that process.

Finally, Vygotsky’s construct of the zone of proximal development

supports the use of intentional instruction during supervision to

influence the prospective teacher’s development. According to

Manning and Payne (1993), “The mechanism for growth in the zone

is the actual verbal interaction with a more experienced member of

society. In the teacher education context, this more experienced

person is likely to be a supervising teacher, college supervisor,

teacher educator, or a peer who is at a more advanced level in the

teacher education program” (as cited in Jones, Rua, & Carter, 1997,

p. 6).

Teacher Education

As new theories of learning emerge, it becomes necessary to

rethink how we prepare prospective and inservice teachers. The

purpose of this section is to acquaint the reader with current studies

in teacher education with this objective. Cooney (1994) reports that

research in teacher education, more and more frequently situated in

interpretivist frameworks, emphasizes teachers’ cognitions and the

factors influencing those cognitions. He includes research on

teachers’ beliefs and conceptions, teachers’ knowledge of

mathematics, and learning how to teach, in this emphasis.

Additionally, Cooney credits the preeminence of constructivism as an

epistemological foundation of mathematics education for efforts to

reform teaching and teacher education. Regarding such reform,

Simon (1997) addresses the need for models of teaching consistent

with constructivist perspectives to serve as research frameworks for

mathematics teacher development. He postulates the Mathematics

Teaching Cycle, which characterizes the “relationships among

teachers’ knowledge, goals for students, anticipation of student

learning, planning, and interaction with students” (p. 76), as one

such framework. According to Cooney, Simon’s purpose is to

articulate explicit teaching principles based on constructivism “with

the intent that these principles will serve as organizing agents for

both research and development activities in teacher education” (p.

613).

Teachers’ Beliefs and Knowledge

Shulman’s knowledge base for teaching, developed through

research on how prospective teachers “learn to transform their own

understanding of subject matter into representations and forms of

presentation that make sense to students” (Shulman & Grossman,

1988, as cited in Brown & Borko, 1992, p. 217), has often provided a

framework for studying teacher development. Within this knowledge

base, content knowledge and pedagogical content knowledge have

received the most attention in educational research (Brown & Borko,

1992). In particular, Even (1993) has studied prospective secondary

mathematics teachers’ subject matter knowledge of the function

concept and its relationship to their pedagogical content knowledge.

A conclusion was that prospective teachers have a limited

understanding of functions, which is evidenced in their instructional

decisions. In addition, Even and Tirosh (1995) have investigated the

interconnections between secondary mathematics teachers’ subject

matter knowledge and knowledge about students and teachers’ ways

of presenting the subject matter. Their interviews with participants

revealed the need to raise the sensitivity of teachers to students’

thinking about mathematics. They further concluded that teacher

education programs should incorporate specific concepts from the

school curriculum to ensure that prospective teachers’ subject

matter knowledge is “sufficiently comprehensive and articulated for

teaching” (p. 18).

The National Center for Research on Teacher Education

(NCRTE) has implemented various research programs focusing on

elementary teacher preparation. Ball (1988) describes one project of

the NCRTE to investigate changes in prospective and inservice

teachers’ knowledge. This longitudinal study examined what

teachers are taught and what they learn, with an emphasis on

“whether and how their ideas or practices change and what factors

seem to play a role in any such changes” (p. 18). To do this, they

specified four domains of a knowledge base reflective of those

identified by Shulman: subject matter knowledge, knowledge of

learners, knowledge of teaching and learning, and knowledge of

context. Of these domains, Ball has focused on elementary and

secondary mathematics teachers’ subject matter knowledge,

identifying it as a central requisite for teacher preparation (Brown &

Borko, 1992). Observing such teachers’ representations of division

at the beginning of the teacher education program, she concluded

that their subject matter knowledge was often fragmented and rule-

dependent (Ball, 1990). Furthermore, Ball and Mosenthal (1990)

found that teacher educators often place less emphasis on this

knowledge domain, thus contributing to the dilemma.

Another program of the NCRTE addressed the nature of

elementary prospective teachers’ beliefs and knowledge about

mathematics, learning mathematics, and teaching mathematics, as

well as changes that resulted from their participation in a

coordinated sequence of innovative mathematics courses and

mathematics methods courses (Schram, et al., 1989). Analyses of

this longitudinal study showed that prospective teachers’ beliefs and

knowledge about mathematics, mathematics learning, and

mathematics teaching were positively affected by the course

sequence. However, the student teaching practicum revealed a

tension between their views as adult students of mathematics and

their instructional practices with children (Brown & Borko, 1992).

Learning How to Teach Mathematics

Feiman-Nemser (1983) has examined prospective elementary

teachers’ transition to pedagogical thinking. Such a transition is

characterized by a shift in the teacher’s thinking away from the

teacher and the content and toward students’ needs. Feiman-

Nemser and colleagues concluded that, alone, prospective teachers

can rarely see beyond what they want or need to do, or what

the setting requires. They cannot be expected to analyze the

knowledge and beliefs they draw upon in making instructional

decisions, or their reasons for these decisions, while trying to

cope with the demands of the classroom” (Brown & Borko,

1992, p. 217).

They maintained that the prospective teacher’s support personnel

should be actively guiding the prospective teacher and encouraging

him or her to analyze and discuss instructional decisions. This

conclusion has powerful implications for the role of the university

supervisor as the prospective teacher’s more knowing other during

the professional semester.

Elementary and secondary prospective teachers were the focus

of a program of study by Borko and colleagues that investigated

teachers’ thinking during the planning and instructional phases of

teaching (Brown & Borko, 1992). From this study, the researchers

identified several areas affecting success in learning to teach. In

particular, successful teachers exhibited careful planning that

anticipated students’ problems and provided strategies for

overcoming them, they demonstrated strong preparation in content,

and they held the view, supported by colleagues and administrators,

that the prospective teacher is responsible for classroom events.

In a related study, Eisenhart and colleagues (1993) studied

prospective teachers’ procedural and conceptual knowledge in the

process of learning to teach mathematics for understanding. Their

investigation of one student teacher’s ideas and practices

concerning teaching for procedural and conceptual knowledge

revealed a tension between the teacher’s stated commitment and the

reality of instruction, with instruction focusing on procedural

knowledge. Such a tension was echoed by the stated beliefs and

actions of the student teacher’s support personnel. The researchers

concluded that teaching for conceptual knowledge should enjoy

consistent support from all of the professional participants in the

student teacher’s experience in order to resolve these tensions.

Teacher Development in Context

Included in this review of research on teacher preparation and

development is a research program for inservice teachers known as

the Second-Grade Classroom Teaching Project (Cobb, et al., 1991).

This study is of particular interest because of its emphasis on

knowledge construction in the context of classroom interactions.

Additionally, the researchers’ use of a classroom teaching

experiment to effect changes in teaching practices supports the use

of such methodology in this study. Embedded within a theoretical

framework of constructivism that equally emphasizes the social

negotiation of classroom norms, the Second-Grade Classroom

Teaching Project addresses second-grade students’ construction of

mathematical knowledge, as well as the development of a

constructivist-based curriculum and the preparation of elementary

teachers to teach in a manner consistent with such a curriculum.

Concerning teacher development, Cobb and colleagues speculate

that

the phenomena of implicit routines and dilemmas suggest that

teachers should be helped to develop their pedagogical

knowledge and beliefs in the context of their classroom

practice. It is as teachers interact with their students in

concrete situations that they encounter problems that call for

reflection and deliberation. These are the occasions where

teachers can learn from experience. Discussions of these

concrete cases with an observer who suggests an alternative

way to frame the situation or simply calls into question some of

the teacher’s underlying assumptions can guide the teacher’s

learning (p. 90).

They also recognize that models of teachers’ constructions of

pedagogical content knowledge are needed. Furthermore, from

looking within the classroom to determine models of children’s

constructions of mathematical knowledge, they suggest that the

appropriate setting in which to ascertain teachers’ models is also the

classroom. Their investigation of one teacher’s learning that

occurred in the mathematics classroom indicated that the teacher’s

beliefs about the nature of mathematics and learning were affected

as she resolved conflicts between her existing teaching practices and

the project’s emphasis on teaching practices that promoted students’

constructions of mathematical knowledge.

Classroom Interactions

Given the recent attention to social constructivism as an

epistemological orientation, it follows that social interactions should

be represented in the research literature. In education, the idea of

social interactions in the classroom is intrinsically bound to such an

orientation. The purpose of this section is to inform the reader of

studies on classroom interactions, as well as discussions in the

literature concerning relevant theoretical perspectives.

Bartolini-Bussi’s (1994) theoretical predilections are more

Vygotskian than Piagetian; however, she argues for the acceptance

of complementarity as the basis for theoretical and empirical

research on classroom interaction in teaching and learning.

Complementarity separates the social and individual domains, yet

attaches equal importance to both. Bartolini-Bussi advocates the

freedom to “refer to approaches that are theoretically incompatible”

rather than yield allegiance to one system (p. 128). The latter can

potentially blind the researcher to “relevant aspects of reality...or

[introduce] into the system such complications as to make it no

longer manageable” (p. 130). Others echo this approach in their own

research (e. g., Cobb & Bauersfeld, 1995; Cobb, Wood, Yackel, &

McNeal, 1992).

In her theoretical discussion of research on classroom

interactions, Bartolini-Bussi (1994) cites studies on such interactions

in mathematics teaching and learning. This includes her own

research on the relationship between social interactions and

knowledge in the mathematics classroom, based on the

Mathematical Discussion in Primary School Project (see Bartolini-

Bussi, 1991). Also mentioned is work by Balacheff (1990) that

considers social interactions to understand how students treat

refutation in the problem of mathematical proof.

Elsewhere, using a teaching experiment to investigate

children’s constructions of mathematics, Steffe and Tzur (1994)

analyzed social interactions attendant with children’s work on

fractions using computer microworlds. They extended social

interactions to mathematical interactions, with the latter including

enactment or potential enactment of children’s operative

mathematical schemes. Furthermore, they examined both nonverbal

and verbal forms of communication as constituting mathematical

interactions. Consistent with their Piagetian roots, Steffe and Tzur

concluded that social interactions contribute to children’s

mathematical constructions, but are not their primary source.

Much of the research on classroom interactions using an

interactionist perspective comes from the individual and collective

efforts of Cobb, Bauersfeld, and their colleagues (e. g., Bauersfeld,

1994; Cobb, 1995; Cobb & Bauersfeld, 1995; Cobb, et al., 1992;

Voigt, 1995). Bauersfeld (1994) characterizes the interactionist

perspective as the link between the two extremes of individualism

and collectivism. The research traditions of symbolic interactionism

and ethnomethodology are prototypical of this perspective, which

establishes teachers and students as interactively constituting the

culture of the mathematics classroom. This perspective is

distinguished from the collectivist (e. g., Vygotskian) perspective, in

which learning is a process of enculturation into an existing culture,

and the individualistic (e. g., Piagetian) perspective, in which

learning is a process of individual change. Their work, like that of

many others discussed here, is positioned within elementary school

mathematics.

In an interactional analysis of classroom mathematics

traditions, Cobb and colleagues (1992) considered what it means to

teach and learn elementary school mathematics. Their approach

assumed that “qualitative differences in...classroom mathematics

traditions can be brought to the fore by analyzing teachers’ and

students’ mathematical explanations and justifications during

classroom discourse” (p. 574). I have made a similar assumption in

the present study. That is, classroom discourse is a catalyst for

elucidating qualitative differences in the emerging classroom

traditions of prospective mathematics teachers.

Research on interactions in the mathematics classroom

suggests an interesting analogy for research in the “teaching

mathematics” classroom (Cobb, et al., 1991). Just as research on

mathematics classroom interactions offers insights into children’s

constructions of mathematical knowledge (Cobb, 1995; Steffe &

Tzur, 1994), it is theoretically feasible that interactions in the

prospective teacher’s “classroom” should provide understanding of

how knowledge about teaching mathematics is constructed. In this

context, I interpret the prospective teacher’s classroom as the

various forums during the professional semester in which his or her

pedagogical content knowledge is mediated.

Implications

As the literature suggests, there is a growing research base

concerning the development of prospective teachers, as well as the

social construction of knowledge. However, more work integrating

these two areas is needed. In mathematics education, the balance of

research on prospective teacher development rests within the

elementary teacher population. Additionally, research on the social

construction of knowledge has been dominated by children’s

constructions of mathematical knowledge. As such, social

constructivism as an interpretive framework offers a rich basis for

research in mathematics teacher education. Specifically, we need to

consider how prospective teachers of all levels of mathematics

construct their knowledge of teaching. In addition, we need to find

new ways to “guide and support teachers as they learn in the setting

of their classroom” (Wood, Cobb, & Yackel, 1991, p. 611). By

adopting a Vygotskian perspective to investigate the prospective

middle school mathematics teacher’s emerging practice during the

professional semester and how that process can be encouraged

through external support, this study has addressed some of the

limitations of the existing research.

The Nature of Qualitative Inquiry

In addressing the possibility of alternative research models

with which to study teaching, Shulman (1992) looks beyond the

traditional focus of social science research in favor of a “move

toward a more local, case-based, narrative field of study” (p. 26).

This perspective reflects a growing genre of educational research for

which qualitative inquiry is appropriate. According to Cooney (1994),

the current emphasis in education on cognition and context has

produced a “rather dramatic shift away from the use of quantitative

methodologies based on a positivist framework to that of interpretive

research methodologies” (p. 613).

Qualitative research seeks to descriptively portray some

phenomenon under investigation through a “bottom up” approach in

which an explanation of the phenomenon emerges from the data.

Sometimes referred to as grounded theory, this approach is

succinctly illustrated by Bogdan and Biklen (1992) as the piecing

together of a puzzle whose picture is not known in advance, but

rather is constructed as the researcher gathers and analyzes the

parts. To accomplish this, the qualitative researcher is uniquely

positioned within the very process of the research, a role which

necessitates that any observations be filtered through the

researcher’s own interpretive lens. Understanding involves the

assumption that the world of inquiry is a complex system in which

every detail could further explain the reality under investigation.

Typically in qualitative research, an explanation for some type

of behavior is sought through an inductive process of spontaneous,

unstructured data collection (Bogdan & Biklen, 1992). A variety of

methods are available to the researcher for this purpose, any of

which may generate copious data that must be coded and analyzed

for presentation in a manageable form. The most prevalent of these

methods are in-depth interviewing and participant observation,

supplemented at times by artifact reviews. Although used less

frequently, teaching experiments offer a unique contribution to

qualitative research methodology as well.

In-Depth Interviewing

In-depth, open-ended interviewing is an essential tool of

qualitative research in which the researcher is “bent on

understanding, in considerable detail, how people such as teachers,

principals, and students think and how they came to develop the

perspectives they hold” (Bogdan & Biklen, 1992, p. 2). It is the

foremost medium through which the researcher gains access to

events in one’s mind that are not directly observable.

Patton (1990) has suggested three approaches to structuring

an interview for research purposes: the informal conversational

interview, the general interview guide, and the standardized open-

ended interview. The informal conversational interview has the

advantage of occurring as a natural extension of ongoing fieldwork

to the extent that the participant may not perceive the interaction as

an interview. The direction of the interview depends on events

occurring in a given setting and as such, predetermined questions

are not considered. The general interview guide offers a semi-

structured approach to interviewing through a checklist of relevant

topics to be discussed in some manner with each of the participants.

The most structured of the three approaches, the standardized open-

ended interview flows from a precisely worded set of questions

posed to each of the participants for the purpose of minimizing any

variations across interviews.

In common to all three approaches is the adherence to open-

endedness. It is essential that respondents be allowed to express

their perceptions in their own words, without consulting a

preconceived set of responses and without being guided by the

wording of an interview question.

Participant Observation

Bogdan and Biklen (1992) describe participant observation as

when the researcher “enters the world of the people he or she plans

to study, gets to know, be known, and trusted by them, and

systematically keeps a detailed written record of what is heard and

observed” (p. 2). The level of the researcher’s participation will vary

depending on the goals of the study, as well as any inherent

constraints of the research site. Concerning this participatory role,

Smith (1987) suggests that the “researcher must personally become

situated in the subject’s natural setting and study, firsthand and over

a prolonged time, the object of interest” (p. 175).

Observations made in the research setting are documented

through field notes, as well as audio recordings, audiovisual

recordings, or both. Although field notes can be broadly interpreted

to mean any data collected in the process of a particular study,

Bogdan and Biklen (1992) define it more narrowly as “the written

account of what the researcher hears, sees, experiences, and thinks

in the course of collecting and reflecting on the data in a qualitative

study” (p. 107). Typically, field notes taken during an observation are

hurried accounts of the events, people, objects, activities, and

conversations that are part of the setting. Ideally, this abbreviated

version is extended immediately after an observation into a full

description that includes the researcher’s reflections about emerging

patterns and strategies for further observations. This information is

often triangulated by the collection of documents or artifacts that

are relevant to the study. These items may be personal writings,

memos, portfolios, records, articles, or photographs. The review of

such artifacts is often regarded metaphorically as an interview.

Teaching Experiments

For some mathematics educators (e. g., Ball, 1993; Cobb &

Steffe, 1983; Lampert, 1992; Thompson & Thompson, 1994), a

particular phenomenon is best understood when the participatory

role of the observer enlarges to that of teacher, evoking a classroom-

based research model in which one studies mathematics learning by

becoming the mathematics teacher. Such action research describes

a “type of applied research in which the researcher is actively

involved in the cause for which the research is conducted” (Bogdan

& Biklen, 1992, p. 223). When the active involvement alludes to the

researcher as teacher, it generally refers to a teaching experiment.

In particular, Romberg (1992) defines the teaching experiment as a

method in which “hypotheses are first formed concerning the

learning process, a teaching strategy is developed that involves

systematic intervention and stimulation of the student’s learning,

and both the effectiveness of the teaching strategy and the reasons

for its effectiveness are determined” (p. 57).

Steffe (1991) describes the teaching experiment as directed

towards understanding the progress one makes over an extended

period of time. “The basic and unrelenting goal of a teaching

experiment is for the researcher to learn the mathematical

knowledge of the involved children and how they construct it” (p.

178). While his characterization refers specifically to children

constructing mathematical knowledge, it is appropriate to extend

this notion to include other learning situations, such as prospective

teachers constructing pedagogical content knowledge.

Steffe (1983) outlines three major components of the teaching

experiment as a methodology for constructivist research: modeling,

teaching episodes, and individual interviews. He uses models to

connote an explanation formulated by the researcher to describe

how students construct mental objects. His interpretation of

Vygotsky’s methodology prioritizes the development of such models

as a goal of teaching experiments. The teaching episodes involve a

teacher, student, and witness of the teacher-student interaction. The

teacher’s role is to challenge the model, or explanation, of the

student’s knowledge and examine how that model changes through

purposeful intervention. This component is consistent with the

Vygotskian (1986) notion of creating a student’s zone of proximal

development and offering instructional assistance in order to effect

the student’s conceptual change. Finally, Steffe suggests that

teaching episodes should be followed by individual interviews, which

differ from the former only in the absence of purposeful intervention

by the teacher with the student.

Vygotsky’s (1986) studies of conceptual development in

children indicate that teaching within the context of an investigation

is not a new approach. His view that one’s intellectual ability is more

accurately described as what can be accomplished with the help of a

more knowing other than what can be accomplished when working

alone shaped the nature of his investigations, often casting him in

the role of teacher. Although “the methodology of the teaching

experiment does not apply exclusively to a particular theory”

(Skemp, 1979, as cited in Steffe, 1983, p. 470), it describes the

nature of Vygotsky’s inquiry. As such, the teaching experiment is

particularly appropriate for studies that assume a Vygotskian

theoretical framework for the purpose of understanding one’s

development.

Finally, it should be emphasized that qualitative research

requires a philosophical perspective that is deeper than the methods

used. Methods are simply a vehicle in which the researcher can

travel from curiosity to theory. They alone do not define qualitative

research.

METHODOLOGY

Given the underlying tenet of this investigation that knowledge

is socially constructed through interactions with various mediating

agents, it was necessary to look within the various forums in which a

prospective teacher’s pedagogical content knowledge is mediated.

These include the mathematics classroom assigned to the

prospective teacher, meetings between the prospective teacher and

the university supervisor, as well as opportunities for reflection by

the prospective teacher. Other forums exist, such as the prospective

teacher’s meetings with peers or the cooperating teacher. However,

this study focused on one prospective teacher’s interactions with her

students and the university supervisor.

It should be noted that, although the prospective teacher’s

students would not typically be viewed as that teacher’s more

knowing others in terms of mathematical content, they are more

knowing others with respect to existing classroom norms. As such,

they will eventually generate contexts in which negotiation with the

teacher is required in order to achieve a taken-as-shared basis for

communicating mathematics in the classroom. The mediation of

pedagogical content knowledge occurring as a result of this was of

interest here.

Methodological Framework

A naturalistic mode of inquiry was adopted to address the

questions of this study. In particular, case studies incorporating

some of the design elements from the constant comparative method

(Glaser & Strauss, 1967) provided the methodological framework.

The constant comparative method can be described as a series of

steps that begins with collecting data and identifying key issues from

the data that become categories of focus. More data are collected to

explore the dimensions of such categories and to describe incidents

associated with them as an explanatory model emerges. The data

and emerging model are then analyzed to understand attendant

social processes and relationships. This is followed by a process of

coding and writing as the analysis focuses on core categories. The

entire process is repeated continuously throughout the data

collection as developing themes are refined (Bogdan & Biklen,

1992). The resulting explanation of the phenomenon under

investigation is often characterized as grounded theory in that it

emerges inductively from the data.

Here, the case studies of prospective middle school

mathematics teachers were treated as microethnographies. That is,

the studies were characterized by a sociocultural interpretation of

the data (Merriam, 1988), with the added assumption that each of

the prospective teachers’ classrooms would develop unique practices

for doing and talking about mathematics and mathematics teaching

(Underwood-Gregg, 1995). Additionally, the task of understanding

prospective teachers’ constructions of pedagogical content

knowledge during the professional semester called for a teaching

experiment. This was envisioned as an extension of Steffe’s (1991)

use of a constructivist teaching experiment to elicit models of

children’s mathematical constructions. In particular, the prospective

teacher, as student, was constructing pedagogical content

knowledge. The university supervisor, as teacher, assisted through

instruction.

Participants

Three prospective middle school mathematics teachers in their

final year of a four-year teacher education program at a large

southeastern university agreed to participate in this study. All three

had selected mathematics as an area of concentration; two had

opted for a dual concentration in mathematics and science. All were

members of a cohort of 47 students participating in an ongoing

investigation of the sociocultural mediators of learning during their

professional semester. The participants’ membership in this cohort

allowed the researcher increased accessibility to their mathematics

classrooms and, as such, was used as a selection criterion. The

participants, ranging in age from 21 to 24, included one European-

American female, one African-American male, and one European-

American male. They were selected to reflect diversity with respect

to race and gender. Additionally, all had average to above average

university academic experiences and were expected to successfully

complete their student teaching practicum.

Data Collection

The methodological framework of this study necessarily guided

the data collection. In particular, multiple methods appropriate

within a qualitative paradigm were used to collect data. Such

methods included participant observation, in-depth interviews, and

artifact reviews. In particular, the university supervisor observed

each of the three prospective teachers one day per week during two

different sections of a selected course for the twelve-week student

teaching practicum. During each visit, the prospective teacher

participated in a teaching episode interview. The observations were

planned by a telephone conference with the prospective teacher

prior to each visit. Field notes taken during the observations focused

on teacher-student interactions which indicated the prospective

teacher’s pedagogical content knowledge.

Episodes of discourse in the prospective teacher’s

mathematics classroom reflecting mediation of that teacher’s

pedagogical content knowledge became the focus of in-depth

interviews between the university supervisor and the prospective

teacher. In particular, the 45-minute interviews were used as

teaching episodes to further mediate the prospective teacher’s ideas

about teaching mathematics. New understanding resulting from the

episodes were used to generate alternative instructional strategies

for subsequent classes.

When teaching schedules permitted, the interview took place

between successive observations of same-subject instruction so as to

provide interventive mediation. Otherwise, it was scheduled after

the two classroom observations had occurred. Interview protocols

were modified as the study progressed to reflect the direction of the

data. All classroom observations and interviews were audiotaped and

videotaped. Finally, the participants were asked to write personal

reflections on mediation that occurred in classroom and interview

episodes of discourse.

The supervisory process of observation, teaching episode,

observation, and written reflection that the prospective teachers

experienced as part of this study is described here as the cycle of

mediation (see Figure 2). It is seen as cyclic in that new knowledge

about teaching mathematics should be reflected in future lessons as

the teacher’s practice emerges.

Other written artifacts including participants’ lesson plans and

related instructional materials, as well as teaching portfolios, were

included in the data corpus. Additionally, I audiotaped reflections

immediately following each visit in order to record my impressions

and ideas. Furthermore, each cooperating teacher was interviewed

twice during the practicum to obtain a more global picture of the

student teacher’s social context. Documents such as interview

protocols and consent forms necessary for the execution of this study

are included in the appendix.

Figure 2. The cycle of mediation in an emerging practice of teaching.

Data Analysis

The descriptive data corpus generated in this study was

analyzed inductively for themes emerging throughout the process of

data collection and as a result of working with the collected data.

Analysis in a qualitative research study is a systematic process of

sense-making that begins in the field (i. e., the place of data

collection). At this point, the purpose is to narrow the focus of the

study, to refine research questions, and plan sessions of data

collection in light of emerging themes. In this study, issues

concerning the prospective teacher’s pedagogical content knowledge

arising within episodes of discourse in the mathematics classroom

served to narrow the focus of inquiry during the data collection.

Given the dynamic process of becoming a teacher, it was expected

that the focus of research with each of the three participants would

be different. This, coupled with the extensive data corpus generated

by the study, required selecting one of the prospective teachers for

complete analysis after data collection. Hereafter, I will refer to that

participant as Mary Ann (pseudonym).

The analysis that occurred after the data had been collected

involved arranging the data into manageable pieces in order to

search for patterns, discover what was important, and decide what

to tell others (Bogdan & Biklen, 1992). This is often described by

qualitative researchers as finding the story in the data. To

accomplish this, transcripts from the audiovisual recordings of

observations and interviews with Mary Ann were reviewed for

episodes of meaningful interactions between Mary Ann and her

students or her university supervisor. Such episodes were noted and

further analyzed for the mediating role of conversation, or discourse,

in learning to teach mathematics. From this, appropriate segments

were selected for further analysis. Additionally, written artifacts (e.

g., journal reflections) supplementing these data were combed for

confirming or disconfirming evidence of assertions about Mary Ann’s

pedagogical content knowledge. Coding categories developed from

the analysis were refined through multiple sorts of the data. The

data were then analyzed longitudinally to determine how Mary Ann’s

ideas about teaching mathematics developed during the professional

semester as a result of social interactions. The process of analysis as

it relates to the specific questions of this study is outlined more

extensively in Part III and Part IV.

Role of the Researcher

A hermeneutical approach to research is subjective in that the

researcher, by choice, is situated within the context of the

investigation. As such, it is necessary here to discuss my role in this

investigation. In particular, I was both investigator of the study as

well as the university supervisor for the prospective teachers. While

this dual function of nonjudgmental observer and university

evaluator may seem incongruous, it served to minimize my intrusions

into the prospective teachers’ mathematics classrooms. This was

ultimately the greater priority, given the many challenges

prospective teachers already face during their practicum.

One of the advantages of this dual role is that it offered an

inside perspective from which to study the process of becoming a

mathematics teacher. Rather than doing research on prospective

teachers, I was involved in a collaborative effort with them to

improve their mathematics teaching. This view of teachers as

collaborators in research has become the norm as scholars recognize

the necessity of the teacher’s voice (Shulman, 1992). Others (e. g.,

Ball, 1993; Lampert, 1992) have used a similar approach in their

research by becoming teachers in the mathematics classroom.

In an analogous manner, I became the teacher for the

participants in a classroom where mathematics pedagogy was the

content. This allowed me to use instruction to create a zone of

proximal development for the prospective teachers during the cycle

of mediation. In this sense, I became the adult or more capable peer,

as conceived by Vygotsky (1986), for the prospective teachers.

MATHEMATICAL DISCOURSE IN A PROSPECTIVE TEACHER’S

CLASSROOM: THE CASE OF A DEVELOPING PRACTICE

Maria L. Blanton

North Carolina State University

Abstract

This investigation is a microethnographic study of a

prospective middle school mathematics teacher’s emerging practice

during the professional semester. In particular, a Vygotskian (1986)

sociocultural perspective on learning is assumed to examine the

nature of classroom discourse and its role in a teacher’s construction

of pedagogical content knowledge.

Classroom observations, teaching episode interviews, and

artifact reviews were used to document the practice of Mary Ann

(pseudonym) during the student teaching practicum. From the data

corpus, mathematical discourse embedded in classroom interactions

was analyzed with respect to pattern and function. Analysis of early

classroom interactions indicated that students’ awareness of

classroom norms for doing mathematics positioned them as Mary

Ann’s more knowing others, thereby contributing to a reciprocal

affirmation of the traditional roles of teacher and student. Moreover,

discourse seemed to play a dialectical role in Mary Ann’s

construction of pedagogical content knowledge, as her obligations in

the classroom transitioned from funneling students to her

interpretation of a problem to arbitrating students’ ideas.

The influence of Mary Ann’s interactions with her students on

her understanding of how to teach mathematics presents a challenge

to teacher educators to help teachers develop their craft in the

context of the classroom.

Introduction

In recent years, the preeminence of constructivism as an

epistemological orientation in mathematics education has directed

much attention toward understanding how students construct

mathematical knowledge (e. g., Bartolini-Bussi, 1991; Cobb 1995;

Cobb, Yackel, & Wood, 1992; Lo, Wheatley, & Smith, 1991; Steffe &

Tzur, 1994; Thompson, 1994). This focus has often led to interpretive

inquiries into classroom discourse as researchers seek to explicate

the nature of students’ mathematical thinking (e. g., Cobb, 1995;

Cobb, Boufi, McClain, & Whitenack, 1997). Since the National

Council of Teachers of Mathematics (NCTM) Curriculum and

Evaluation Standards for School Mathematics (1989) has prioritized

classroom communication as a facilitator of students’ mathematical

understanding, an ongoing research interest in discourse seems

assured. Indeed, a continued emphasis on classroom discourse is

pivotal to current reforms in mathematics education because it

informs not only our understanding of students’ thinking about

mathematics, but also teachers’ thinking about teaching

mathematics. Recent studies in the professional development of

mathematics teachers (e. g., Cobb, Yackel, & Wood, 1991; Peressini

& Knuth, in press; Wood, 1994; Wood, Cobb, & Yackel, 1991) have

broadened our vision of classroom discourse as a catalyst for teacher

learning. Cobb, Yackel, and Wood (1991) maintain that “it is as

teachers interact with their students in concrete situations that they

encounter problems that call for reflection and deliberation. These

are the occasions where teachers learn from experience (p. 90).”

However, the nature of classroom discourse and its concomitant role

in a teacher’s construction of pedagogical content knowledge is still

underdeveloped.

Wood (1995) addresses this deficit in the literature with an

interactional analysis of classroom discourse that situates the

teacher as the learner. In her study, classroom discourse is valued as

giving voice to the social complexities inherent in teaching in a

collective setting. By documenting patterns of interaction between

teacher and students as they negotiate their roles in the classroom,

discourse provides a verbal window into the teacher’s developing

practice. This genre of research on teacher development in situ

suggests an interesting parallel for the study of prospective teachers

during the professional semester, that is, the student teaching

practicum. Until this time, prospective teachers’ understanding of

how to teach mathematics is almost necessarily academic.

Prospective teachers may be primarily confined to university settings

which offer only decontextualized opportunities for developing their

craft. The professional semester offers the optimal context in which

knowledge of mathematics and mathematics teaching and learning

coalesce into an emerging practice for the neophyte teacher. Here,

my curiosity centers on the role discourse plays in this process.

Specifically, this study is guided by the following research questions:

1. What is the nature of mathematical discourse in a

prospective teacher’s classroom?

2. What does such discourse suggest about the prospective

teacher’s pedagogical content knowledge?

3. How is the prospective teacher’s pedagogical content

knowledge mediated through such discourse?

Since the notion of classroom discourse connotes a variety of

meanings, I specify it here to denote talk, or utterances, about

mathematics made by teacher and students in the classroom.

Teacher Learning Through Classroom Discourse

Vygotsky’s (1986) sociocultural approach gives theoretical

precedent to the place of discourse in an individual’s development.

According to Minick (1996), Vygotsky maintained that “higher

voluntary forms of human behavior have their roots in social

interaction, in the individual’s participation in social behaviors that

are mediated by speech [italics added]” (p. 33). Vygotsky extends

this idea in his general genetic law of cultural development, which

posits that an individual’s higher mental functioning appears first on

the intermental plane, between people, and is then genetically

transformed to the intramental plane within the individual. The

significance of this perspective is that it extinguishes traditional

boundaries between individual and social processes in order to forge

a view of mind constituted by both (Wertsch & Toma, 1995). Bateson

succinctly illustrates this notion of an extended mental system:

Suppose I am a blind man, and I use a stick. I go tap, tap, tap.

Where do I start? Is my mental system bounded at the hand of

the stick? Is it bounded by my skin? Does it start halfway up

the stick? Does it start at the tip of my stick? (Bateson, 1972,

as cited in Cole & Wertsch, 1994).

Therefore, Vygotsky’s belief in the social origins of higher mental

functioning embeds human consciousness in “the external processes

of social life, in the social and historical forms of human existence”

(Luria, 1981, as cited in Wertsch & Tulviste, 1996, p. 54). In the

external processes of the classroom setting, the teacher is also

subject to this social formation of mind. That is, the teacher’s

obligation to manage the intermental context of the classroom

generates opportunities for that teacher to learn as well. The activity

of teaching, of deciding what mathematical knowledge students need

and when meaning has been constructed, continually creates

dilemmas for the teacher to resolve in the process of classroom

instruction (Wood, 1995). Thus, understanding a teacher’s

construction of knowledge about teaching mathematics is inherently

linked to the social dynamics of the classroom.

Although Vygotsky theorized that higher mental functioning is

mediated by both physical and psychological socioculturally-evolved

tools (Wertsch, 1988), it was his belief in the primacy of language as

a mediating tool that drew my attention to classroom discourse.

Concerning language, Vygotsky further reasoned that, as a higher

mental function, language is itself subject to mediation. Holzman

(1996) explains this seeming conundrum:

The dialectical role of speech is that it plays a part in defining

the task setting; this activity redefines the situation, and in

turn, speech is redefined. Language is both tool and result of

interpersonal [i. e., intermental] and intrapersonal [i. e.,

intramental] psychological functioning (p. 91).

Such dualism lends further support to the centrality of discourse in a

teacher’s developing practice. That is to say, in the intermental

context of the classroom, it is primarily discourse, or the language

embedded therein, that mediates the teacher’s practice.

Furthermore, the nature of such discourse is a harbinger of the

teacher’s internalized thinking about teaching mathematics. Under

the umbrella of Vygotsky’s general genetic law of cultural

development, Wertsch and Toma (1995) maintain that the nature of

classroom discourse induces an active or passive stance on the part

of the student, which is subsequently echoed in that student’s

intramental functioning. This principle concerning the relationship

between one’s external and internal speech can be extended to the

teacher as well. In other words, the nature of classroom discourse

will be reflected in the teacher’s intramental thinking about teaching

mathematics. Finally, the effect of speech being redefined through

social interactions is then reflected in an emergent form of

languaging by the teacher. Therefore, language is central in a

cyclical process of development through which it mediates higher

mental functioning first intermentally, then intramentally. As

language voices that mediated higher mental functioning, the

process is renewed.

As an illustration, consider a teacher’s attempt to help a

student resolve a mathematical dilemma. In the process of discourse,

the teacher attempts to make sense of the student’s difficulty and

decides on a course of action. As the instructional plan unfolds, the

teacher tries to assess the student’s understanding and may

subsequently modify the plan in order to influence that student’s

thinking in a desired direction. In effect, the teacher’s behavior (as

well as the student’s) is being mediated in the context of this

interaction. What emerges for the teacher is a new awareness of

how to address a student’s difficulty at some level of generality, an

awareness that is reflected through variations in the teacher’s

speech. The teacher’s practice should increasingly reflect a depth of

experience born out of interactions with students.

Process Of Inquiry

I adopted an interpretive approach (Erickson, 1986) to

consider the developing practice of Mary Ann (pseudonym), a

prospective middle school science and mathematics teacher. Mary

Ann was in her final year of a four-year teacher education program

when asked to participate in this study. From our first meeting in

which I explained the purpose of my research, the professional

contribution that she could make, and my role as her university

supervisor, Mary Ann’s enthusiasm promised a partnership from

which we both could learn.

The Research Setting

I treated the case study of Mary Ann as a microethnography.

That is, viewing the classroom as a socially and culturally organized

setting, I was interested in the meanings that teacher and student

brought to discourse and how this shaped the teacher’s practice

(Erickson, 1986). Since such an approach presumes that classrooms

will develop as separate microcultures, I introduce the reader here

to the school community into which Mary Ann was acculturated as a

student teacher.

The county in which Mary Ann was assigned a student

teaching position is situated in a large urban area that supports 19

public middle schools, enrolling about 20,000 sixth-, seventh-, and

eighth-grade students. Mary Ann’s assigned school reflected a

relatively diverse student population of 1200. Progressive discipline,

site-based management, and the cooperation of parents and

community were hallmarks of its infrastructure. Outside of the

classroom, teachers worked in interdisciplinary teams to integrate

the various content areas. Within this system, Mary Ann was

assigned to a seventh-grade mathematics classroom in which she

taught general mathematics and pre-algebra. She was paired with a

cooperating teacher who provided a nurturing atmosphere for Mary

Ann.

Collecting the Data

Although my focus here is on discourse in the prospective

teacher’s classroom, the data corpus reflects broader issues in Mary

Ann’s developing practice. Specifically, participant observation, in-

depth interviews, and artifact reviews were selected as tools of

inquiry. Weekly visits with Mary Ann during the practicum were a

three-hour interval that consisted of a classroom observation,

followed immediately by a teaching episode interview, and finally, a

second classroom observation. Both observations were of Mary Ann

teaching general mathematics. Each visit was documented through

field notes and audio and audiovisual recordings.

Mary Ann was also asked to provide a copy of her lesson plan

along with any supporting materials, such as quizzes or activity

sheets, at each visit. Although these documents were viewed as

secondary data sources, I could not assume that key issues might not

later emerge from them. Additionally, Mary Ann was asked to keep a

personal journal in which she reflected on what she had learned

about her students, about mathematics, and about teaching

mathematics through the course of each visit. After each visit, I

audiotaped personal reflections about emerging pedagogical content

issues and how future visits could incorporate these themes as

learning opportunities for Mary Ann. In all, I had eight visits with

Mary Ann, followed by a separate exit interview. Finally, I conducted

two clinical interviews with the cooperating teacher to obtain a more

complete picture of Mary Ann’s classroom community (see

Appendix).

Analyzing Classroom Discourse

Pattern And Function In Teacher-Student Talk

I have outlined a process of data collection that is inclusive of

multiple influences in a teacher’s development. To examine the

questions posed in this study about classroom discourse, I focused

on classroom observations as the primary data source. Having

previously established the theoretical motivation for an analysis of

classroom discourse as a window into the student teacher’s

developing practice, I now turn to the specifics of such an analysis.

Discourse analysis rests upon the “details of passages of discourse,

however fragmented and contradictory, and with what is actually

said or written” (Potter & Wetherell, 1987, p. 168). The tendency to

read for gist, or to reconstruct the meaning in someone’s words so

that it makes sense to the reader or listener, should be resisted.

Because such an analysis is often tedious and unscripted, I have

attempted to concisely delineate that process here.

According to Potter and Wetherell (1987), there are essentially

two phases in discourse analysis: (1) identifying patterns of

variability and consistency in the data, and (2) establishing the

functions and effects of people’s talk. Pattern and function captured

the nature of discourse in Mary Ann’s classroom and thereby

revealed the essence of her developing knowledge about teaching

mathematics. Furthermore, based on Wood’s (1995) process of

documenting teacher learning in the classroom, I looked at shifts in

pattern and function to establish Mary Ann’s construction of

pedagogical content knowledge.

Current literature (e. g., Underwood-Gregg, 1995; Wood,

1995) provided insight into identifying patterns in classroom

discourse. Speaking from the traditions of ethnomethodology and

symbolic interactionism, Underwood-Gregg explains that obligations

felt by teacher and students in accordance with their perceived roles

in the classroom are enacted through various routines. Such

routines, most often embedded in language, comprise the patterns of

interaction in the classroom. For example , Mary Ann’s felt

obligation to clarify a student’s thinking was often enacted as a

routine in which she asked a series of instructional questions (i. e.,

those for which the teacher already knows the answer [Wertsch &

Toma, 1995]) designed to lead that student, step-by-step, to the

correct solution. Simultaneously, the student’s obligation to give the

teacher’s desired response sometimes led to a routine of guessing by

that student. Together, these routines comprised a pattern of

classroom interaction. Thus, identifying a pattern in the data

requires constructing its constituent parts, namely, the routines of

teacher and students that give rise to that pattern.

Identifying the function of discourse in the classroom leads to

a myriad of nuances in the teacher’s utterances which, in aggregate,

give voice to her mathematics pedagogy. Thus, drawing from the

work of Wertsch and Toma (1995), I appealed to Soviet semiotician

Yuri Lotman’s (1988) dichotomy of the function of text as univocal or

dialogic to provide a clarifying lens on this aspect of discourse.

Lotman broadly defines text as a “semiotic space in which languages

interfere, interact, and organize themselves hierarchically” (p. 37).

This includes written words, verbal utterances, and even art forms.

By univocal functioning, Lotman implies text that serves as a

“passive link in conveying some constant information between input

(sender) and output (receiver)” (p. 36).

As an illustration, consider teacher-student interactions in

which the teacher asks a series of instructional questions. In this

case, neither teacher nor student needs to actively participate.

Moreover, any discrepancy between what is transmitted and what is

received is attributed to a breakdown in communication. In contrast,

dialogic functioning refers to text that is taken as a “thinking

device”. That is, rather than being interpreted as an encoded

message to be accurately received, the speaker’s utterances serve to

generate new meaning for the respondent, who takes an active

stance toward the utterance by questioning, validating, or even

rejecting it (Wertsch & Toma, 1995). As such, it is likely that

students initiating and maintaining dialogic interactions may run

counter to typical (American) classroom norms, thereby making it

the responsibility of teachers and teacher educators to cultivate

dialogic functioning in the intermental context of the classroom.

Process of Analysis

Teasing out pattern and function from discourse data seemed

arduous at the outset. I began by transcribing audiovisual recordings

of classroom observations, inserting comments and questions as they

arose in transcription. In retrospect, these memorandums initiated

my sense-making of the data corpus. Using the conversational turn

as the basic unit of analysis, I combed the early transcripts to

identify a preliminary coding scheme that would describe the

purpose of Mary Ann’s utterances. For example, her questions

“What’s the common denominator between six and two?” and “How

did you figure out that six was the common denominator?” were

coded as “Request for Computation” [RFC] and “Request for

Procedure” [RFP], respectively. Such codes reflected Mary Ann’s

expectations of students as participants in mathematical discourse,

thereby providing insight into her thinking about teaching

mathematics. From this preliminary scheme, codes were refined or

discarded and new codes were added as subsequent data were

analyzed. (See Appendix for this coding scheme.)

To code the transcripts, each classroom observation was

divided into manageable sections based on naturally occurring

divisions in the sequence of classroom events. Such divisions were

signaled by a change in theme or direction, such as the conclusion of

class discussion on a particular problem. Sections were then coded

by conversational turn and the essence of interactions between Mary

Ann and her students was abstracted to get a sense of the routines

and patterns in the discourse. Additionally, sections were compared

in order to ascertain similarities and differences that suggested

changes in Mary Ann’s practice. The coding system represented my

first attempt at sorting the data and was eventually set aside as I

focused on the particulars of pattern and function in the discourse.

Once all of the transcripts had been coded, four classroom

observations representative of Mary Ann’s developing practice were

selected for further analysis. In deference to the cultural personality

intrinsic to individual classes, I chose all of these observations from

Mary Ann’s third period general mathematics class. Based on the

work of Underwood-Gregg (1995) and my own preliminary analysis, I

considered the routine actions that Mary Ann and her students

enacted subsequent to the following interdependent events:

a student posed a mathematical question

a student responded to a mathematical question

the teacher posed a mathematical question

the teacher responded to a mathematical question

From the four classroom observations, sections were selected as

representative of the routines and patterns manifested following

these events. These sections were then analyzed to characterize the

function of text as univocal or dialogic. Since function is identified by

the respondent’s passive or active interpretation of the speaker’s

utterance, it was necessary to look at each speaker’s utterance and

how it was subsequently interpreted (e. g., as a thinking device) by

the respondent. Additionally, I met periodically with my advisors and

other available faculty and graduate students to review the

audiovisual recordings and discuss the nature of discourse in Mary

Ann’s classroom, what it suggested about her pedagogical content

knowledge, and how it mediated that knowledge. Other data sources

(e. g., written artifacts) were perused for confirming or

disconfirming evidence concerning assertions generated through the

analysis.

Findings and Interpretations

Early Pattern and Function in Classroom Discourse

In this section, I discuss through transcription and analysis the

nature of early discourse in Mary Ann’s classroom and what such

discourse suggested about her pedagogical content knowledge while

in its infancy. Mary Ann’s early practice metaphorically identified

her as the captain of a ship, keenly obligated to navigate rough

waters for her students. Taking over the helm of the classroom when

all sailing seemed smooth only intensified her need to ensure

students’ cognitive calm. As Mary Ann anticipated mathematical

storms for her students, she often rushed to avert them by giving

information and explaining procedures, or changing the problem in

question altogether. As the captain, it was primarily her place to do

this. Indeed, she became the hero by skirting the hazards of

unknown waters. While this was a commendable role for Mary Ann,

it sometimes hindered students from steering themselves, as they

yielded the balance of responsibility to her.

Early pattern and function in resolving students’ mathematical

dilemmas. Throughout the practicum, Mary Ann’s usual custom was

to begin class with students’ questions from the previous night’s

homework, introduce new topics, and then close the lesson with

practice problems or a short quiz. The following excerpt from the

transcripts typifies the manner in which she addressed students’

mathematical questions during the early stages of her practice. In

this particular episode, a student (Allyson) has asked Mary Ann

about an exercise from homework. As was often the case when

working problems through whole-class discussion, Mary Ann copied

the exercise on the overhead projector [OP] and recorded

mathematical pieces of the ensuing discussion as students spoke.

(All names are pseudonyms.)

1 Teacher: O. K., what was the first step we want to do,

Allyson?

2 Allyson: Make it a zero?

3 Teacher: O. K., what’s the very first thing? What’s the very

first step yesterday? What did we want to do with that

variable?

4 Allyson: Isolate it.

5 Teacher: Isolate, and I want everybody to start using this

term, “isolate”. It’s a mathematical, algebra term

and I want you to learn how to use it. O. K., I know

you’re not used to seeing the variable on this side (right

side), so if you want to rewrite it, and just switch, you

can just switch it around like this (Mary Ann illustrates

on the OP.). That’s the same thing. O. K., so now we

want to isolate the variable, but what have we got to

do before we isolate the variable? (A student

indicates that they should evaluate the exponent.)

O. K., we want to get rid of that exponent. So what is

nine squared?

6 Students: Eighty-one. (One student says eighteen.)

7 Teacher: Who said eighteen? How did you get eighteen? I’d

like to know.

8 Student: I was thinking nine times two.

9 Teacher: O. K., remember that when you see nine squared,

that’s not nine times the exponent. That’s nine

times itself, and in this case you write nine down

twice. O. K., so then you’ve got one hundred and

twenty-one. O. K., so now how do we isolate the

variable? (Allyson’s response is inaudible to me.) O.

K., so subtract eighty-one, and I want you to start

using the term. When I ask, “How do you isolate the

variable?”, you say, “Subtract eighty-one from both

sides”. So you don’t have to say, “Subtract it from this

side, then subtract it over here”. Just tell me you

subtract eighty-one from both sides. O. K., so eighty- one

minus eighty-one?

10 Allyson: Zero.

11 Teacher: Zero. O. K., we have to line up the decimals, right?

So there’s an understood decimal behind eighty-one.

So five minus zero?

12 Allyson: Five.

13 Teacher: One minus one?

14 Allyson: Zero.

15 Teacher: Now we have to borrow, so that becomes zero

because we borrowed a whole. (Mary Ann pauses to

get the attention of several students who have

started to talk with each other.) Eight from twelve?

16 Allyson: Four.

17 Teacher: O. K., so you just have s equals...(her voice trails

off as she writes the final answer on the OP.)

The appearance of the correct solution signaled an end to the

episode and Mary Ann moved on to the next question.

Mary Ann began the dialogue outlined above by establishing

her approach for working the exercise, supplying Allyson with non-

mathematical, referent-laden hints that would prompt Allyson’s

recall of the procedure she needed to follow (1, 3). Allyson’s

unsuccessful attempt (2) to give the response that Mary Ann wanted

prompted Mary Ann to enact a “giving hints routine” (3). Allyson’s

obligation in this interaction was to guess the desired response,

upon which Mary Ann could move to the next phase. At this point,

Mary Ann initiated an “incremental questioning routine” in which

she asked a series of cognitively-small, closed, leading questions,

sometimes accompanied by her explanation, that funnelled Allyson

toward a final solution. To her credit, Mary Ann genuinely wanted

students to participate in the process of working the exercise.

However, at this point in her practice, she relied on questioning

strategies that required students primarily to compute simple

answers, recall information, or describe procedures previously

learned (e. g., What have we got to do before we isolate the

variable?, So what is nine squared?, [What is] eight from twelve?).

This type of question-and-answer interaction evoked a vertical

discourse between teacher and student that, given students’ willing

participation, quickly became a classroom norm for doing

mathematics.

The early pattern of interaction constituted by the routines of

Mary Ann and her students that unfolded when a student posed a

mathematical problem is summarized below:

Typical Early Pattern of Interaction

teacher writes the problem/exercise on the OP and sets the

direction for solving the problem by giving information and

asking leading questions

student guesses a response

teacher gives hints in order to get a particular response from

student(s)

student gives desired response

teacher repeats student’s response and asks a leading, follow-

up question.

With the exception of the first step, a variation of this pattern

typically repeated until a correct solution appeared.

This episode between Mary Ann and her students seemed to

indicate a predominantly univocal functioning of text. For example,

Allyson’s incorrect response (2) led Mary Ann to assume that her

original question (1) was either inaccurately transmitted or received.

This signaled Mary Ann to retransmit the message with more

accuracy, that is, give more suggestive hints (3). Allyson’s correct

response (4) then suggested that the message had been accurately

received and Mary Ann could continue (5). As Mary Ann

concentrated on demonstrating her thinking (e. g., 9, 11, 15), she

peppered her explanations with questions that served to check

accuracy in transmission (e. g., 13). Neither Mary Ann nor her

students seemed to treat a speaker’s utterance as something to be

questioned for the purpose of generating new thinking. In other

words, a respondent’s passive interpretation of a speaker’s

utterance designated the function of that utterance as univocal.

Although Mary Ann did question one student’s response (6) in a

seemingly dialogic fashion (7), her purpose was to dispel discrepant

thinking (9).

The obligation that Mary Ann felt to clarify Allyson’s thinking

positioned Mary Ann as the filter of discourse. That is, Mary Ann

initiated the exchange, decided what type of questions to ask, when

and to whom to ask these questions, and when an answer was

acceptable. The norm was for students to respond to the teacher’s

questions, not one another’s ideas. Orchestrating all of this is quite a

challenge, especially for the novice teacher. Although Mary Ann

seemed quite adept, the risk was in her controlling the discourse as

if somehow students were marionettes and she their puppeteer.

Rather than exploring students’ thinking, their ideas and strategies,

Mary Ann was intent on showing how she would have worked the

problem, fishing for student responses that would support her

interpretation. At this early stage in her practice, it seemed inherent

in her beliefs about teaching to be the center of information for her

students, weeding out responses that did not follow a teacher-

selected path for solving the problem at hand. As did all of her

efforts, this approach stemmed from an earnest desire to be a good

teacher.

Early pattern and function in teaching a new concept. On my

second visit with Mary Ann, I observed her teaching a lesson on

adding and subtracting algebraic expressions. I have included

lengthy transcripts from this lesson in order to preserve its integrity.

Mary Ann often tried to motivate new topics with a mathematical

activity that would pique students’ interest. Her opening activity for

this particular lesson reflected these efforts.

18 Teacher: (She hands an envelope to Laura.) You be Student

A, but don’t look at this. Hold it down. (She hands

an envelope to Debbie.) You be Student B. O. K., we

have two students, Laura is Student A, Debbie is

student B, and they’re working at a clothing store,

trying to make some extra money.... O. K., Student A

has an envelope that is one day’s pay. O. K.,

Students A and B are working at a clothing store

and they make the same amount of

money...for one day’s work. O. K., Student A has an

envelope that says “one day’s pay”. Student B has an

envelope that says “one day’s pay plus a three dollar

bonus”, so she got a little extra. Can you tell me

how much [Laura] has in her envelope without

looking in the envelope?

19 Laura: No.

20 Teacher: O. K., the amount is hidden, right? Because I won’t

let you open it. O. K., how much does Student B have?

(She looks around for a student who will respond.) O. K.,

what did you say Dianne? (Dianne’s reply is

inaudible to me.)

21 Teacher: O. K., she’s had one day’s pay with a three dollar

bonus. (Mary Ann’s intonation indicates Dianne’s

response was incorrect.) So she has Laura’s pay with a

three dollar bonus, right? (Various students begin

calling out responses.) O. K., so you know that she

has three dollars, so would she have three more

dollars than what Laura has?

22 Dianne: Yes.

23 Teacher: So we know that she has more than what Laura

has, right?

24 Dianne: Yes.

25 Teacher: O. K., Laura has one day’s pay and we know that

Debbie has one day’s pay plus three dollars. (She

begins to write information on the OP.) O. K., (to

Laura) I want you to open up your envelope and see

what you have.

26 Laura: Twenty dollars.

27 Teacher: Twenty dollars. Now Laura has twenty dollars, so

how much does Debbie have?

28 Students: Twenty-three.

29 Teacher: So y’all think she has twenty-three dollars?

30 Students: Yeah.

31 Teacher: So, we said Laura has twenty dollars. According to

what we’ve written here, she’s got twenty. If [Debbie’s]

got three dollars more, she should have twenty-three

dollars. (To Debbie) O. K., you can open your

envelope and see what you have.

32 Student: Yep [sic], she’s got twenty-three dollars.

One could justifiably argue that Mary Ann, not her students,

was the central player in this activity. A quick glance at teacher and

student routines confirms this. To her credit, Mary Ann seemed to

value the use of physical referents such as integer chips, geoboards,

graphing calculators, or her own creations, as a bridge to abstract

ideas. However, her exposition left little room for dialogic

interactions in the classroom.

From this activity, Mary Ann transitioned into the second

phase of her lesson, a review of the defining characteristics of

equations and expressions.

33 Teacher: O. K., what we’re doing today is talking about

expressions in addition and subtraction, and it’s

been a while since we talked about expressions

anyway, so I [wanted to] refresh your memory....

Ahhh, expression and equation, what is an

expression? Everybody just think about it for a

second. John?

34 John: An unfinished problem...

35 Teacher: O. K., we call it a phrase, an unfinished sentence

(she writes this on the OP). O. K., and what do we call

an equation? Does anyone know?

36 Sharon It was...

37 Teacher: Sharon, raise your hand if you want to answer.

(Turning to Kayla) Kayla? (Kayla’s response is

inaudible to me.) O. K., so it was a complete

number sentence. What is the one main difference

that I told you was between an equation and an

expression, John? (He does not know.) I told you one day

I was going to walk in here and I was going to look

like it. It’s the big difference between an equation

and an expression. (John’s response, inaudible to

me, is not what Mary Ann is looking for.) O. K.,

there’s one symbol that makes the difference. (After a

number of students raise their hands, some making

guttural sounds in order to be recognized, Mary Ann

turns to Marta.) Marta? (No response.) O. K.,

Allyson, can you help Marta out?

38 Allyson: Equal sign?

39 Teacher: An equal sign. Can you give me an example of an

equation, Sharon? (Sharon’s response is

inaudible to me.)

40 Teacher: (She writes Sharon’s response on the OP.) O. K.,

Sharon said that was an equation. Would y’all agree

with that? (Students offer mixed responses of “yes” and

“no”.) You wouldn’t agree with that? Why wouldn’t you

agree with that? (Mary Ann turns to one of the students

who disagreed with Sharon’s claim.) It’s a

complete number sentence, with an equal sign.

Maybe you’re thinking maybe if we wrote some things

like this (she writes on the OP). O. K., that’s an equation,

too. One’s numerical and one’s algebraic. Remember

we talked about that. (She turns her attention to the

whole class.) O. K., what would this be (she writes

another example on the OP)? An equation or

expression? Chris?

41 Chris: Uhm...expression?

42 Teacher: Expression. Why is it an expression?

43 Chris: Because it doesn’t have an equal sign.

This predominantly univocal exchange (to which reviewing

content easily lends itself) continued for several more minutes as

Mary Ann prodded students to recall information. It illustrates her

inclination to enact a routine of supplying non-mathematical

referents (e. g., I told you one day I was going to walk in here and I

was going to look like it) until students guessed her answer. Also,

students responded to Mary Ann, not their peers, thereby granting

her the mathematical authority. Mary Ann’s routine of repeating a

student’s correct response (the signal of affirmation), or meeting

incorrect responses with hints, explanations, or a request for peers

to assist, depicts a cultural norm of doing mathematics in her

classroom in which students looked to the teacher, not to

themselves, to explain, justify, or reject their ideas. Although

students’ acquiescence to this norm seemed to reinforce Mary Ann’s

practice, at one point she did begin to shift her obligation onto

students to argue Sharon’s claim and justify their own thinking (40).

When several students rejected Sharon’s claim, Mary Ann’s response

(You wouldn’t agree with that? Why wouldn’t you agree with that?)

seemed to indicate an attempt at dialogic interaction. At this point in

her practice, such an attempt was atypical. Furthermore, as Mary

Ann then tried to anticipate the student’s thinking (i. e., Maybe

you’re thinking if we wrote something like this.), the student was

unable to respond dialogically.

Following the review, Mary Ann led a whole-class discussion in

converting written expressions into symbolic form.

44 Teacher: O. K., it says (Mary Ann reads from the textbook),

“Jody is entering the pumpkin stacking contest at the

Pumpkin Festival. She’s hoping to balance three more

pumpkins in her stack this year than she did last year.”

So that’s kind of what we were just talking about.

Debbie had three more dollars than Laura did. O.

K.? So she wants more, three more, pumpkins this year.

(Mary Ann writes the problem on the OP.) O. K.,

we’re going to set this up in equation form using a

variable. Remember we talked about a variable?

We’re going to let n equal the number of pumpkins last

year. So we don’t know how many, so we’re just going

to give it a variable. We could have called that t or s

or a or b, whichever variable we want to call it. So she

wants three more. So when we think of more, do we

think of addition or subtraction?

45 Students: Addition.

46 Teacher: Addition. You’re going to add some things on. So

we know we’re going to add, and we know we want

three. It’s just like when we said we want one day’s

pay plus three dollars is what Debbie had. So this is

the number of pumpkins she had last year, plus the

three more she wants this year.

It is interesting to note that in this episode’s entirety, students

were asked only to determine if “more” implied addition or

subtraction. This underscores a recurring theme of univocal

discourse that positioned Mary Ann as the sender and students as

receivers of information. She continued this pattern of interaction

with a series of related tasks whereby, for each task, she read a

written or algebraic expression (e. g., a plus four vis-à-vis a + 4),

then asked instructional questions, sometimes offering explanations

and hints, in order to garner a particular response from students.

The following conversation highlights these interactions and further

supports the assertion that Mary Ann’s knowledge about teaching

mathematics prioritized teacher demonstration as a vehicle for

student learning.

47 Teacher: O. K., how would we say, using more than and

following our pattern, would we say a plus four?

Ron? We want to use our pattern that we have up here

(on the board). A plus four using more than? (Ron’s

response is inaudible to me.) See how I said three

more than n (referring to a previous problem)? I

rewrote that in words. I rewrote n + 3, the

expression n + 3, into words saying three more than n.

So how would I say this right here (i. e., a + 4) in

words using more than? Nunice, can you help

out?

A conversation later in the lesson illustrates what Mary Ann

had intended when she asked a student to solve a particular task.

48 Teacher: O. K., Tom I want you to do (i. e., convert to

symbolic form) the sum of a number z and five. O. K.,

let’s look for what symbol we’re going to use. We said

sum was what? Addition or subtraction?

49 Tom: Addition.

50 Teacher: O. K. Addition (Mary Ann writes a plus sign on the

OP.) O. K., where do you want me to put the z and five?

51 Tom: Z would go on that side (pointing to the left side).

52 Teacher: O. K., and the five would go over here (indicating

the right side)? (Tom nods agreement.)

In this episode, Mary Ann again enacts an incremental

questioning routine in order to funnel Tom to the correct solution. In

her request for Tom to “do the sum of a number z and five” (48),

Tom only had to associate the word “sum” with addition or

subtraction (for which, of course, he had a 50 percent chance of a

correct guess). In her eagerness, Mary Ann genuinely wanted Tom to

be successful. This characteristic of her teaching seems to partially

explain why she fractured the content into cognitively-small, leading

questions. It was as if her responsibility was to help students avoid

any of the struggles that, in reality, do (and should) accompany

mathematical inquiry.

After several more similar episodes, Mary Ann concluded the

lesson with a visual activity on evaluating expressions. She passed

out cards containing either a number, variable, or mathematical

symbol, to volunteers who had not participated on this particular

day. As she read a written expression (e. g., five more than s) aloud,

students with the corresponding parts (i. e., 5, +, and s) arranged

themselves at the front of the room. Occasionally prompted by Mary

Ann, they held their cards to indicate the expression s + 5.

53 Teacher: Now we’re going to work this out and find a value,

so I need whoever is going to make this sentence

complete and make it into an equation [to] come up

here. (The student with the “=“ card walks to the front.)

O. K., I want s to equal four. (The student with the

“4” card walks to the front.) I want to bump...s

and put four in. (Speaking to the student with the “s”

card) So you stand behind her (indicating the

student with the “4” card). I’m replacing the variable

s with the number four. Now we’ve got to find the

value, so whoever thinks they have the answer to

this, come on up. (The student with the “9” card walks

to the front.) All right. Very good. So what we did

was we replaced s, our variable. We bumped her

(indicating the student with the “s” card) and put in four.

We made a what? An expression or equation?

54 Students: Equation.

As with the opening activity of this lesson, Mary Ann again

purposed to situate an abstract idea in a concrete setting, this time

using students as visual referents to personify evaluating

expressions. Also as before, she assumed the responsibility of

explaining the process as well as the conclusions, leaving students

with only minimal input. Even so, this activity’s inclusion signaled

the importance Mary Ann attached to concrete experiences in

making mathematics meaningful for students.

On early discourse and Mary Ann’s practice. The early pattern

in classroom interactions that unfolded when Mary Ann taught new

concepts was equivalent in structure to the pattern exhibited when

she addressed students’ homework questions, outlined earlier in this

section. That is, whether Mary Ann or a student asked a question or

posed a task to be solved, Mary Ann typically established the

solution approach by giving information, asking leading questions, or

both (cf. 1, 44), then proceeded to direct students to the correct

solution through questions and hints (cf. 3-5, 37-39). Moreover, the

self-perceived roles of teacher and students in mathematical

discourse, manifested through their routine actions, led almost

exclusively to univocal classroom interactions.

What I observed in these early patterns of discourse is not

unlike those outlined elsewhere in the literature. In what Bauersfeld

(1988) describes as a funnel pattern, the teacher asks questions to

which he or she already has an answer. If a student gives an

incorrect response, the teacher then tells the correct response or

directs the student step-by-step to the correct answer. Underwood-

Gregg (1995) describes what Voigt has identified as an elicitation

pattern. In this, the teacher vaguely poses a question for which

students are obligated to offer a variety of answers. The teacher’s

need to direct how the question is to be answered creates the

obligation to follow students’ ideas that match those of the teacher,

or give hints in order to move students toward the teacher’s

thinking.

Such patterns of interaction in the classroom, as well as

discourse that is in essence univocal, have been documented in the

case of inservice mathematics teachers (e. g., Underwood-Gregg,

1995; Peressini & Knuth, in press; Wood, 1995). It is the occurrence

of such discourse from the outset of a prospective teacher’s practice

that is of note here. The student teacher undergoes a cultural

metamorphosis from learner of mathematics to teacher of

mathematics during the professional semester. If that student

teacher’s intramental thinking about mathematics is predominantly

say, univocal, then his or her initial teaching practice would reflect

this. That is, how one teaches mathematics is grounded in how one

thinks about mathematics. Mary Ann’s comments about the role of

problem solving in mathematics during an early teaching episode

identified a consistent link between her thinking about mathematics

and her early practice:

I know that math is one big word problem in itself, because

one thing builds on another. But I don’t look at it like that. I

look at math as just operations you go through, just like a

series of steps. You have to step on this step before you get to

the next one.

In this sense, Mary Ann’s early languaging in the classroom

seemed to be an external representation of her intramental thinking

about mathematics. This, coupled with the claim by Wertsch and

Toma (1995) that 80 percent of American classrooms bequeath

univocality to their students’ intramental thinking about

mathematics, made it likely that univocal discourse would dominate

Mary Ann’s early practice. Moreover, it seemed that the inertia

generated by univocal teacher-student interactions in Mary Ann’s

early practice held implications for her development. This intensified

the need to address her pedagogical content knowledge in its

infancy, in the context of her practice.

Indications of an Emerging Practice: Change in Pattern and Function

The “problem-solving day”. Although the pattern and function

that typified early languaging in Mary Ann’s classroom persisted

throughout the practicum, later discourse did substantiate emerging

patterns in her interactions with students, as well as a shift from

discourse grounded almost exclusively in univocal functioning. My

third visit with Mary Ann, later monikered the “problem-solving day”

because of the lesson’s focus, revealed such changes. I reiterate that

the purpose of the present study is not to address the role of

contexts external to the classroom on changes in Mary Ann’s

practice. Clearly, such contexts (e. g., interactions with the

university supervisor or cooperating teacher) shape the prospective

teacher’s thinking about teaching mathematics, as they did with

Mary Ann. Rather, the purpose here is to explore the nature of

interactions in Mary Ann’s classroom and how those interactions

mediated her pedagogical content knowledge.

The lesson on the problem-solving day dealt with the strategy

“working backwards” as a way to solve simple word problems. Mary

Ann had earlier insisted that she was uncomfortable with word

problems and did not want to teach this particular lesson, yet she

took considerable risks in an attempt to move from her previous

teaching paradigm. After addressing students’ questions from the

homework, she then asked students to work in dyads to solve the

following problem:

Problem 1: I’m thinking of a number that if you divide by three

and then add five, the result is eleven.

Removing herself as the mathematical authority, Mary Ann

seemed to want students to struggle with the problem through peer

interactions and to justify their thinking to one another before she

joined the process. Her attempt to renegotiate classroom norms in

resolving a mathematical question met with immediate resistance

from students as, almost imperceptibly, their role in doing

mathematics had shifted. The following conversation illustrates the

tension created by Mary Ann’s initial efforts to change her practice.

As it begins, a student has just asked Mary Ann if he should write

Problem 1 in his notes.

55 Teacher: If you feel like you need to write it down, write it

down. I just want you to solve it. I’m not going to

answer any questions, just solve it.

A student asked Mary Ann to check her solution. Mary Ann

responded by withholding closure:

56 Teacher: Well, that’s good. You need to write it down and

tell me how you solved it. You should be talking with

your partner. (To the class) Y’all love to talk. Now I’m

letting you talk.

The student again asked Mary Ann to check her solution.

57 Teacher: I’m not going to tell you if it’s right or wrong. I

want you to work it out. You can plug it back in and see

if it’s right.

Another student asked for help, yet Mary Ann continued to resist

intervening. Instead, she encouraged the student to work with her

partner.

58 Teacher: Did you consult with [your partner] and tell her

how you feel about it? (The student indicates she has.)

And she thinks that’s right? (The student again

indicates she has. More students raise their hands.)

No hands up. Just talk about it. (A student tells Mary

Ann she has the answer.) O. K. Good. Then y’all are

ready. (She turns her attention to a particular

dyad.) So have y’all talked about it? You got together?

(She moves to another pair.) Have you figured it out?

(They indicate they have.) And you both agree that this is

your number?

Mary Ann walked around the room several more minutes, stopping

periodically to promote students’ interactions. By the end of this

episode, the classroom resonated with a steady hum as students,

realizing Mary Ann’s intentions, began to communicate

mathematically with each other.

The whole-class discussion that followed reflected another

shift in Mary Ann’s practice, as she pointedly asked different groups

to share their solutions, and later their thinking, with the class.

Noting the first group’s correct response and immediately moving to

others for their solutions, Mary Ann appeared more interested in

understanding students’ thinking than in harvesting only correct

answers.

59 Teacher: O. K., our first group to finish was Debbie and

Susan, so they’re going to tell me the number they got.

(She writes their response “18” on the OP.) O. K.,

(turning to another group), what did you get?

60 Group: Six.

61 Teacher: O. K., what number did you get, Jack?

62 Jack: Eighteen.

63 Teacher: What number did you get (turning to another

group)?

64 Wendy: I got thirty-eight.

At this point, Mary Ann asked Debbie to explain her (correct)

solution of eighteen, to which Debbie responded with a procedural

account of her thinking (65). What seems noteworthy here is that,

by eliciting Debbie’s strategy, Mary Ann was relinquishing a role

which typically she felt obligated to fill.

65 Debbie: It says, ”If you divide by three and add five”, so you

do the opposite. You subtract five from eleven and

that’s six. Then you multiply six times three and

that’s eighteen.

Previously, a correct solution coupled with a correct procedure

would have signaled Mary Ann to repeat that procedure and then

move to the next task. However, in this instance, she turned back to

her students to try to further understand their thinking. After

making sense of the strategies used by those who had found the

correct solution, she asked several groups who had made

unsuccessful attempts to explain their thinking as well, reflecting a

departure from a practice in which she rarely countenanced

incorrect answers. The following episode depicts this.

66 Teacher: (Speaking to another dyad) How did you get

eighteen?

67 Student 1: Same way.

68 Teacher: You did this exact thing?

69: Student 1: No.

At the student’s hesitance to explain his group’s strategy, Mary Ann

turned to another pair frantically waving their hands in order to be

recognized.

70: Teacher: How did you get eighteen?

71: Student 2: We had a number. We said eighteen divided, three

will go into eighteen six times. Then we added five.

72 Teacher: O. K., so first of all you knew that the result had to

be eleven, so you said, “O. K., it’s eleven”. Then I told

you you had added five, so you had to think what added

to five will give you eleven. O. K.? I’m trying to help,

think like you were thinking. Is that what you did?

73 Student 2: Uh huh.

74 Teacher: (To another group) How did you get eighteen?

75 Student 3: Well, we got six.

76 Teacher: O. K., how did you get six?

77 Student 3: O. K., she (indicating her partner) got six because

she just added six to...(Student 3’s partner objects

but her response is inaudible to me). All she did

was added six to five.

78 Teacher: O. K., six to five, but what did you do with the

“three divided by”? (Student 3’s response is

inaudible.) See, it says “three divided by”, so if you

divide three into six, you’re going to get two and two

plus five is seven. O. K., who got the thirty-eight?

I’m curious to see who got thirty-eight. (Student 4

identifies himself.) Tell me how you got thirty-eight.

79 Student 4: I did s over 3.

80 Teacher: You did what?

81 Student 4: S over three.

82 Teacher: O. K., what now?

83 Student 4: Equals eleven.

84 Teacher: So what happened to the five?

85 Student 4: That’s what I said.

As the lesson continued, Mary Ann repeatedly positioned

Debbie as the mathematical authority, thereby allowing Debbie to

retain ownership of her ideas.

86 Teacher: The way Debbie chose to do the problem is what

we’re talking about today. She, well Debbie, you

tell me what you did. Is there any certain way you

can call maybe what you did, without using the book?

When you looked at this problem, where did you

start?

87 Debbie: Where did I start? I started at the answer.

88 Teacher: You started at the answer and then did what?

89 Debbie: And then I just went backwards.

90 Teacher: O. K., did everybody hear what she just said?

Debbie, repeat that one more time.

91 Debbie: I started at the answer and worked backwards and

did the opposite of, uhm, division and addition.

92 Teacher: So Debbie used a problem-solving strategy of

working backwards. That’s just one strategy. Some of

you used guess-and-check, and maybe you didn’t come

up with the right answer, but you were on the right

track. Some of you set up an equation.

Mary Ann’s comments in the interview prior to this class

revealed a different type of thinking about the use of multiple

approaches to solve a problem. Smiling sheepishly, she admitted,

I guess to me, like, I was always, give me a formula, or give me

a way to solve it, and I’ll solve it. And sometimes with word

problems there’s [sic] many different ways.... That puzzles kids

to think there might be more than one way. It always scared

me.... If I know that there...is more than one way, that scares

me. That’s weird, I know, but I feel like if there’s one way, I

can check it, and if I get it right, then I’m right, and I’m right,

and I’m right. That’s all there is to it.

Although she stated her receptiveness to students’ alternative

strategies, Mary Ann’s discomfort with exploring various routes to a

task’s solution was exhibited in early patterns of classroom

interactions where she, not the students, determined the solution

path (e. g., 48-52). That she was now willing to sacrifice the one

strategy she was comfortable with by the inclusion of other valid

processes seemed a significant shift for her.

After posing the following problem to students, Mary Ann once

again turned to Debbie.

Problem 2: The Blueberry Festival is held each Labor Day.

This year there are 89 entries. This is twice the number of last

year’s entries, plus seven. How many entries were in the

blueberry run last year?

93 Teacher: So what were some things when you were working

this other problem that you had to do? O. K., you told us

that you started at the end and you worked to the

beginning. So she started here and she went this way.

But what else did she have to do?

94 Debbie: First I had to write down what, like three divided

by...

95 Teacher: So you said three divided by (she writes this on the

OP). Then what did you say? (Debbie’s response is

inaudible to me.) So you add five, and the result was

eleven. Then what did you have to do? So now you

said you worked from the back end up. So what did you

have to do?

96 Debbie: Then I started with eleven and I subtracted five.

The discussion surrounding Problem 1 and Problem 2 reflects a

variation from typical early languaging in Mary Ann’s classroom. The

whole-class discussion about Problem 1 began with a univocal

sharing of students’ solutions (59-64). This transitioned into students

sharing their strategies (65-89) in a lengthy interaction with Mary

Ann. Furthermore, Mary Ann’s utterances (e. g., 72, 80, 82, 84)

indicated an emerging effort to focus on students’ ideas, not her

own. In particular, she seemed to dialogically question Student 4

(78-84) in order to achieve mutual clarity with him about the

problem’s solution. Previously, she would typically have interpreted

his incorrect solution of thirty-eight as the result of a transmission

error (i. e., univocally) which she was obligated to correct by

demonstrating her own strategy.

Debbie’s utterance (65) later prompted Mary Ann to attempt a

dialogic interaction with her (86-89). I describe this as an attempt

because, although Mary Ann’s questions (e. g., Where did you start?,

You...then did what?) solicited a procedural response from Debbie,

there seemed to be an underlying shift away from instructional

questions to questions that explored Debbie’s thinking.

In Mary Ann’s request, “She, well Debbie, you tell me what you

did”, Mary Ann started to appropriate Debbie’s ideas, then

reconsidered in order to externalize Debbie’s thinking, not

demonstrate her own (86, 88). Mary Ann later continued this

approach in an effort to situate the solution of Problem 2 within the

context of Debbie’s strategy [93-96]. While Debbie’s responses (87,

89, 91) suggested that she still interpreted Mary Ann’s utterances

univocally, I would emphasize that this interaction represented an

emerging form of languaging for both Mary Ann and her students. In

other words, dialogic interaction was not yet a classroom norm for

talking about mathematics for neither teacher nor student.

In concert with Peressini and Knuth (in press), I wish to clarify

my position that univocal discourse does have its place in the

classroom, albeit not at the expense of dialogic discourse. Their

conclusion that “all dialogic text must contain some univocal

functioning in order for clear communication to take place”

underscores the functional dualism of text argued by Lotman (1988).

However, as evidenced by Mary Ann’s early practice, there is a need

to cultivate balance in the function of text so that dialogic

interactions constitute a meaningful part of classroom discourse. The

discourse that characterized much of the problem-solving day

seemed to edge toward that preferred balance in which univocal and

dialogic discourse dualistically exist.

The routines enacted by Mary Ann and her students once

Problem 1 had been posed differed from those observed in her early

practice. Rather than giving hints and questioning incrementally to

lead students to a correct solution, Mary Ann enacted a “solicitation

routine”. In other words, she initiated the discussion by soliciting

solutions and procedures from students, focusing on their ideas and

strategies rather than her own. Furthermore, she seemed more

inclined to address students’ inappropriate responses by questioning

rather than telling (e. g., 78-85). In the absence of Mary Ann’s

routines such as giving hints, students were no longer obligated to

try to guess her thinking. Instead, they could share their solutions

and strategies as she requested. The pattern of interaction

constituted by these routines reflected a more interactive form of

languaging than that expressed in previous classroom discourse.

While this pattern was not fully adopted on the problem-solving day,

it did signal a shift in Mary Ann’s practice from the manner in which

student- or teacher-posed problems had typically been discussed.

This pattern is summarized below:

Emerging Pattern of Interaction

teacher writes the problem on the OP and asks students to

work in dyads for a solution

teacher asks various dyads for their solutions

student representative of each dyad responds

teacher selects dyad to explain their strategy

dyad representative responds

teacher comments, then selects another dyad to explain their

strategy

dyad representative responds

teacher comments, then selects another dyad to explain their

strategy

dyad representative responds

teacher questions dyad in order to understand their process

dyad representative responds

teacher selects another dyad to explain their strategy

dyad representative responds

teacher questions dyad in order to understand their process

dyad representative responds/clarifies thinking

teacher selects another dyad to explain their strategy

dyad representative responds

teacher questions dyad in order to understand their process

dyad representative responds/clarifies thinking

teacher addresses the validity of the various approaches.

During the remainder of the lesson, Mary Ann enacted the

familiar incremental questioning routine of asking leading, closed

questions (e. g., [What is] the inverse of divide?, Eleven minus five

is...?) to demonstrate similar problems to the whole class. Even so,

the experience of students being more actively engaged in discourse

seemed to open Mary Ann’s thinking to the value of dialogic

interactions. As she later reflected on the events that transpired

during this lesson, she wrote,

[At the beginning of the lesson], instead of throwing

information out, I let them figure the problem out in their own

style....To my surprise, one of the students performed the

problem exactly as the strategy suggested. Boy, was this a

memorable event. The pressure was lifted off of me.... Once

the students saw how one of their peers was able to solve the

problem, things were a lot more clear to all. I learned that

having the student come up with the solution means more to

the others than the teacher giving a long, drawn-out lecture.

This reflection supports the assertion that Mary Ann’s

pedagogical content knowledge was mediated toward a more

student-centered practice in the intermental context of the

classroom. In particular, where once she felt the obligation to give a

“long, drawn-out lecture” by “throwing information out”, she now

seemed to appreciate students thinking through a process with their

peers without a barrage of instructional questions from the teacher.

Moving forward in classroom discourse: Learning to listen.

Although Mary Ann’s emerging pedagogical content knowledge

exhibited a nonlinearity as she shifted between familiar and

unfamiliar routines, the events of the problem-solving day seemed to

anchor her flexibility for risk-taking in future discourse. An episode

several weeks later underscored this continuing growth in the

pattern and function of discourse in Mary Ann’s classroom. In an

investigation of the number of diagonals in a polygon, pairs of

students were given geoboards on which they were to form a

polygon (and all of its diagonals) by attaching rubber bands. As

students worked, Mary Ann recorded their findings on the board in

two columns, one showing the number of sides for a given polygon,

and the other its corresponding number of diagonals. After

determining the number of diagonals in a triangle, quadrilateral,

pentagon, and hexagon, students were asked to find a pattern that

would predict the number of diagonals in a heptagon without using

the geoboards. The following episode chronicles their ensuing

discussion.

97 Teacher: I want you to come up with a prediction, or a way

that we can figure out how many diagonals a heptagon

has without actually doing it on a geoboard. (Students

begin raising their hands.) I want everybody to

have a chance to think. Put your hands down.

Everybody talk with your neighbor. Think of a way....

I don’t want anybody forming a heptagon on the

board. I want you to do it thinking. Use your brain. (A

student asks a question that is inaudible to me.) No, just

talk it over with your partner. Y’all always want

to talk. I’m giving you a chance to talk.

The immediacy with which students appealed to Mary Ann for

some type of feedback and her consequent obligation to explain her

expectation that students negotiate with their partners suggests that

peer mediation did not yet constitute a shared understanding

between teacher and students for doing mathematics. Nevertheless,

Mary Ann persevered. After a reasonable amount of time had

passed, she congregated students’ attention for a discussion of their

thinking.

98 Teacher: So does everybody have a prediction, or has

formed a hypothesis, maybe?

99 Students: Yeah.

100 Teacher: Did you test it to see if it works?

101 Student 5: Yeah, it works.

102 Teacher: O. K., Susan what was your prediction? What do

you think about how many diagonals it’s going to have?

103 Susan: Fourteen.

104 Teacher: Fourteen. O. K., so Susan’s prediction is fourteen.

O. K., somebody else. Karen?

105 Karen: Fourteen.

106 Teacher: O. K., Karen thinks it’s going to be fourteen. O. K.,

Randy?

107 Randy: Fourteen.

108 Teacher: Fourteen. Christie?

109 Christie: Fourteen.

110 Teacher: Fourteen.

111 Student 6: People are raising their hand with the same

answer.

112 Teacher: Well, that’s fine. I want to hear what everybody

says.

113 John: I think twelve.

114 Teacher: You think twelve? Do you have something? Do you

have something to back up your prediction? [Some

way] how you want to test your hypothesis?

115 John: I did, but it’s wrong.

116 Teacher: Well, maybe not. Maybe if we test it. Lisa?

117 Lisa: I have twelve.

118 Teacher: O. K., so Lisa and John think it’s twelve. (Mary Ann

writes “12” on the board.) O. K., so John and Lisa,

since y’all got twelve, tell me how you got twelve.

119 John: Well, each one of those says increased by, uhm, a

number higher than each other, but...

120 Teacher: O. K., I didn’t understand that. So, you must have

said that not in my lingo. So, break it down.

121 John: Each one, when it’s increased by two, then

increased by another, each one is increased by one

and (at this point, John’s words become inaudible to

me) the number increased.

122 Teacher: Whoa! O. K., so you go....

123 John: You increase by two, then you add another one and

increase again by two. It’ll increase by three.

124 Teacher: O. K., so tell me. This (indicating one of the values

in the column containing the number of diagonals)

will increase by two. So we increase by two here.

Now tell me where I go from there.

125 John: Then you add one and increase by two again. Then

it increases by three.

126 Teacher: O. K., so you’re saying this increased by three. So

you add one to the former?

127 John: Yeah, and you keep doing that.

128 Teacher: Increase by four. So then what would your number

be here (indicating the unknown number of

diagonals)? If your prediction is you add one for every

time you add one here?

129 John: Fourteen. (His previous solution was twelve.)

130 Teacher: Fourteen. O. K., so somebody tell me, is there a

way, so you’re saying we’re going to have five here,

right? So how could we set this up in equation form to

get this number here (the unknown)? Is there a way

that you figured that out? (Susan raises here

hand.) Susan? (Her response is inaudible to me.) So is

this going to keep going in this order (indicating the

difference in consecutive numbers of

diagonals)? Two, three, four, five, six? Jack? (Jack

does not respond and Susan raises her hand again.)

Uhm, no, Jack is supposed to answer this. (After no

response from Jack, she turns to the class.) So we’re

saying that a heptagon is going to have fourteen

diagonals. O. K., if your hypothesis is correct, you’ve got

to back it up mathematically. So you’ve got to show me

an equation that can back this up, saying it’s

fourteen. If I’m solving for a variable, if this was

the unknown (she indicates the number of diagonals

of a heptagon).... O. K., so get out a pencil and

piece of paper and start computing. You’re

mathematicians and you’re scientists, and if

somebody asks you to test your hypothesis and

formulate your hypothesis, you’ve got to have some way

to back it up. I didn’t say this is right (i. e., fourteen).

I said this is what you’re making me buy into, or selling

to me. (After students start to work, she turns back to

John.) Do you have an answer?

131 John: Well, I have a hypothesis.

132 Teacher: O. K. (John’s response is inaudible to me.) O. K.,

John figured out that it was...fourteen.

133 John: Because it starts out at zero. Then you add one.

Then you add three on. Then you add four on.

134 Teacher: When you add nine and five, what do you get?

135 Student 7: Fourteen.

136 Teacher: Fourteen.

137 John: Oh, I thought it was twelve.

138 Teacher: (To Student 7) O. K., you say it’s fourteen. Prove it

to me.

For the remainder of the lesson, Mary Ann and her students

continued with this rich pattern of interaction. It stands in marked

contrast to the discourse that characterized her early practice. In

this episode, we see Mary Ann’s early tendency to ask leading

questions in order to demonstrate her thinking replaced with a

purpose to ask questions that make sense of students’ thinking. She

seemed to be learning to listen to her students dialogically. That is,

she seemed to be listening in order to generate new understanding,

not just determine if information had been correctly transmitted and

received (e. g., 120-128). This offers a compelling argument for Mary

Ann’s development as a teacher. Moreover, such discourse required

her to cede authority to her students, as she did with John. While she

risked vulnerability in doing this, her effort illustrates an ongoing

attempt to promote meaningful discourse in her practice.

As on the problem-solving day, Mary Ann again initiated a

routine of soliciting students’ solutions in the whole-class discussion

surrounding the problem of finding the pattern. Furthermore, where

earlier she may have solicited only correct solutions, it was the

introduction of an incorrect answer in this episode that finally got

her attention (102-114). This is not to say that correct thinking is not

a valued part of discourse. Indeed it is, and to suggest otherwise is

somewhat misleading. However, the activity of teaching must

extend beyond demonstrating correct procedures to include dialogic

interactions as well. Mary Ann’s later practice seemed to recognize

this need.

Mary Ann’s Students: More Knowing Others?

As a prospective teacher, Mary Ann was acculturated into a

mathematical community in which her students were already

members. Thus, students’ cognizance of that community’s existing

norms for doing mathematics positioned them as her more knowing

others. Clearly, Mary Ann’s students did not hold an overt agenda for

shaping her practice. Nonetheless, her sensitivity to students’

experiences while under her tutelage did yield a form of influence to

them. In particular, the early patterns of interaction observed in

classroom discourse led to a “reciprocal affirmation” of the

respective roles of teacher and student in the classroom. That is, the

cognitively simple questions that Mary Ann asked as she funneled

students toward a correct solution were often easily answered by

students (e. g., 11-17). As a result, students were affirmed in their

ability to do mathematics and their responses seemed to affirm Mary

Ann’s early practice.

The intermental context of the classroom thus served to direct

Mary Ann’s early languaging toward a more traditional paradigm of

giving information and inspecting the accuracy of transmission. In

effect, it mediated her intramental thinking about teaching

mathematics, that is, her pedagogical content knowledge. Indeed,

Vygotsky’s (1986) assertion that “higher voluntary forms of human

behavior have their roots...in the individual’s participation in social

behaviors that are mediated by speech” (p. 33) rang true for Mary

Ann’s early practice.

Interrupting the inertia that developed in univocal interactions

between Mary Ann and her students to make room for dialogic

discourse seemed crucial for her development. However, this

required Mary Ann to renegotiate classroom norms for doing

mathematics, to move away from rote question-and-answer

exchanges and toward interactions that probed students’ thinking.

Naturally, this met with initial resistance from students because they

were expected to assume an unfamiliar role in doing mathematics (e.

g., 55-58, 97). The resulting tension seemed to present a pivotal

juncture in Mary Ann’s development. It suggests a crucial point at

which other mediating agencies (e. g., university supervisor) can use

instructional assistance to support a prospective teacher’s efforts to

change what it means to do mathematics in his or her classroom.

Discussion

In its defense of new perspectives on teaching, the NCTM

Professional Standards for Teaching Mathematics (1990) outlined a

number of changes in teachers’ thinking needed to foster students’

intellectual autonomy. The shifts championed by this document

include a move toward verification through logic and mathematical

evidence and away from the teacher as the mathematical authority,

toward mathematical reasoning and away from memorization, and

toward hypothesizing and problem-solving and away from rote

answer-finding. Such recommendations must seem daunting to the

prospective teacher rooted intramentally in traditional norms of

doing mathematics. Indeed, change is not an easy process.

However, discourse in Mary Ann’s classroom did document an

emerging practice consistent with the views sanctioned by this

NCTM document. In particular, the univocal discourse that

characterized early languaging in her classroom was later tempered

with Mary Ann’s efforts to interact dialogically as she encouraged

students to hypothesize (e. g., 98-100) and justify their thinking with

mathematical evidence (e. g., 114-129) in order to solve non-routine

problems (e. g., 97). The patterns in classroom discourse expressed

this transition in Mary Ann’s pedagogical content knowledge as well.

Her image of the teacher as a mathematical authority, obligated to

funnel students exclusively to her own interpretation of a problem

through such routines as giving hints and incremental questioning,

gave way to a perception of the teacher as an arbiter of students’

ideas, obligated to solicit students’ thinking as a platform for

resolving mathematical dilemmas.

It is not my intention here to attribute such changes in Mary

Ann’s practice exclusively to classroom interactions. Rather, it is to

document the nature of such interactions and how they mediated her

pedagogical content knowlege. Contexts external to the classroom

shaped her practice as well. This raises an important issue that I

interject here and will pursue in Part IV. That is, how can teacher

educators provide the necessary scaffolding for the prospective

teacher so that mediation in the context of classroom discourse can

lead to a more effective practice?

The pattern and function of mathematical discourse in Mary

Ann’s classroom indicated her construction of pedagogical content

knowledge during the professional semester. In essence, Mary Ann’s

emergent languaging gave voice to the development of her

intramental thinking about teaching mathematics. Furthermore,

language in the intermental setting of the classroom mediated her

thinking about teaching mathematics because it exposed the nature

of students’ experiences in both affect and content. Therefore, the

dialectical role of language as articulated by Holzman (1996) was

evidenced in Mary Ann’s developing practice.

Implications of this study for teacher education center on

discourse. Specifically, we need to help prospective teachers

cultivate a practice that engages students in dialogical, as well as

univocal, classroom interactions. For the prospective teacher,

changing the nature of classroom interactions requires confronting

existing norms for doing mathematics. The resulting conflict places

students in a position to mediate the prospective teacher’s practice.

This is a critical juncture at which teacher educators can assist

prospective teachers in renegotiating the nature of classroom

discourse.

Furthermore, while the professional semester is an optimal

context to provide such assistance, the nature of discourse in a

prospective teacher’s classroom should be addressed in earlier

undergraduate settings as well. Indeed, the tool of language merits

the same attention in teacher education that physical tools (e. g.,

manipulatives) often enjoy. Ultimately, the mathematics teacher’s

ability to open a student’s zone of proximal development rests on the

nature of classroom discourse.

References

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APPENDIX

COOPERATING TEACHER ASSESSMENTOF THE

STUDENT TEACHER/COOPERATING TEACHER PARTNERSHIP

What were your goals and expectations when you entered this partnership?

How have these goals and expectations changed, if at all, during this practicum?

How did you perceive your role as cooperating teacher when you entered this partnership?

How has this perception changed, if at all, during this practicum?

Describe the nature of your partnership.

What do you think your student teacher learned from you?

Was there evidence that he or she successfully completed your perception of the practicum? If so, what?

What did you learn from your student teacher?

Describe your interactions with your student teacher. (E.g., Did you meet on a regular basis? Informally or formally? How did you negotiate your respective roles in the class?)

PRELIMINARY CODING SCHEME FOR DISCOURSE ANALYSIS

This scheme was developed based on the purpose of the

teacher’s utterance as ascertained during her conversational turns

in classroom discourse about mathematics. Multiple codes were

sometimes assigned to each utterance.

DQ: (Direct Question) Teacher asks a question to a particular

student.

RFI: (Request for Information) Teacher asks student(s) to

provide information that requires only rote recall (e. g.,

give definitions, acknowledge teacher’s solutions, respond to closed

questions).

RFPA (Request for Peer Assistance) Teacher asks other

student(s) to answer a question that a particular student

cannot answer.

RFC: (Request for Computation) Teacher asks student(s) to

perform a simple computation.

RFP: (Request for Procedure) Teacher asks student(s) to

explain procedure for obtaining a particular solution.

Not the same as RFJ (Request for Justification).

TCSR (Teacher Clarifying Student’s Response) Teacher poses a

question that paraphrases or repeats a student’s

response in order to verify her (teacher’s) understanding.

TP (Telling Procedure) Teacher tells/states a procedure or set of

fact(s) as a way of explanation, giving information, or

clarifying.

DRFPA(Denied Request for Peer Assistance) Teacher focuses on

one student’s participation when other peers are offering to

assist.

CFQ (Check for Questions) Teacher asks if student(s) has any

questions about a particular problem, or in general.

QSS (Questions Suggests Solution) Teacher asks leading

questions.

DP(Describing a Problem) Teacher is describing a problem for the

class to solve (e. g., reading a problem from the text) or

explaining directions for an activity.

CE(Communicating Expectations) Teacher is explaining what she

expects students to do in terms of homework, class

participation, and so forth.

HOA (Hands-on-Activity) Teacher uses a hands-on-activity.

This is to give additional information about a problem

students may be solving.

RFPS (Request for Problem Solving) Teacher asks student(s) to

solve a problem that requires higher order thinking

(beyond simple computation). May involve, e. g., modeling a

process with an equation, in order to solve.

RFJ (Request for Justification) Teacher asks student to justify

his or her thinking.

RTR (Request to Replicate) Teacher asks student(s) to

replicate a procedure with at most a minor alteration.

RTI (Request to Interpret) Teachers asks student(s) to

interpret information in order to answer a question.

THE CYCLE OF MEDIATION: A TEACHER EDUCATOR’S

EMERGING PEDAGOGY

Maria L. Blanton

North Carolina State University

Abstract

This investigation explores the pedagogy of educative

supervision in a case study of one prospective middle school

mathematics teacher during the professional semester. Educative

supervision as defined here uses the context of the prospective

teacher’s practice to challenge his or her existing models of

teaching. It rests on the Vygotskian (1978) tenet that the university

supervisor can guide the prospective teacher’s development to a

greater extent than the prospective teacher can when working alone.

Classroom observations by the university supervisor, teaching

episode interviews between the supervisor and prospective teacher,

and focused journal reflections by the prospective teacher were

coordinated in a process of supervision postulated here as the cycle

of mediation. The pedagogy of the teaching episodes, a central part

of this study, was closely aligned with instructional conversation

(Gallimore & Goldenberg, 1992).

The cycle of mediation suggests an avenue for effecting

prospective teachers’ development in the context of their practice. In

this study, perturbations experienced by the prospective teacher in

classroom discourse presented opportunities in supervision to

promote change in her practice. Moreover, instructional

conversation in the teaching episodes seemed to open the

prospective teacher’s zone of proximal development so that her

understanding of teaching mathematics could be mediated with the

assistance of a more knowing other.

Introduction

No one would seriously question the complexities of the

student teaching practicum. From a sociocultural perspective, the

practicum reflects the integration of often dissonant agendas of

teaching and learning that ultimately define a community into which

the student teacher is acculturated. It demands that the prospective

teacher negotiate tensions imposed by the juxtaposition of school

and university cultures in the context of a practice still in its infancy.

It is from the surfeit of pedagogical beliefs and practices constituting

this community that the student teacher’s practice emerges.

Despite these challenges, the practicum still promises the

optimal setting in which knowledge of content and pedagogy

coalesce in the making of a teacher. This possibility invites questions

about the ability of any agencies associated with the practicum to

effect teacher change. Of particular interest here is the role of

university supervision in that process. Specifically, is supervision an

effectual path to teacher development?

Furthermore, does supervision function as teacher education, or

does it instead reinforce pre-existing habits of teaching by focusing

on ancillary issues? Research on the supervision of student teachers

has produced a continuum of responses to these questions. While the

more skeptical suggest that we abandon supervision altogether

(Bowman, 1979), others argue that we must fundamentally alter the

way we supervise if we are to effect real change in the ways that

student teachers teach (Ben-Peretz & Rumney, 1991; Borko &

Mayfield, 1995; Feiman-Nemser & Buchmann, 1987; Frykholm,

1996; Richardson-Koehler, 1988; Zimpher, deVoss, & Nott, 1980).

Rethinking the Role of Supervision:

Education or Evaluation?

Historically, the role of supervision has likely tended toward

evaluative rather than educative interactions with student teachers.

That is, the traditions of supervision may be more closely described

by a perfunctory assessment of existing habits of teaching, buried

within an attention to classroom bureaucracy, rather than prolonged

interactions purposed to challenge those existing habits. Quite

possibly, this emphasis is a reflection of the chronological placement

of student teaching at the end of academic teacher preparation.

Furthermore, case loads that leave little time for one-on-one

interaction between the supervisor and student teacher often

relegate the supervisor to an evaluative role. However, Feiman-

Nemser and Buchmann (1987) challenge us to reconceptualize the

practicum (and hence supervision) as preparatory to future learning,

as educative rather than evaluative.

Research indicates that an educative stance is not currently

assumed in all supervisory relationships. In an investigation of

guided practice interactions between university faculty, cooperating

teachers, and student teachers, Ben-Peretz and Rumney (1991)

pinpointed the lack of professional reflection provided by support

personnel. They found instead that the authoritative demeanor

adopted by supervisors was met with passivity from student

teachers, resulting in little change in practice.

Elsewhere, Borko and Mayfield (1995) found that supervisors

focused on superficial aspects of teaching (e. g., paperwork, lesson

plans, behavioral objectives) and avoided in-depth discussions about

content and pedagogy, offering student teachers no specific

directives on how to change their practice. Concluding that

supervision seemed to exert little influence on student teachers’

development, they proposed instead that supervisors should actively

participate in student teaching and “challenge student teachers’

existing beliefs and practices and model pedagogical thinking and

actions” (p. 52). These recommendations might be seen to conflict

with the obvious physical parameters that constrain supervision.

However, an evaluative approach does not seem to engender

substantive change in teaching. In short, actively participating in

student teaching requires more than peripheral commitments by the

supervisor, but the result can be a practicum that functions as

teacher education rather than teacher evaluation.

Why should we consider an approach to supervision that

challenges student teachers’ models of teaching in the context of

their practice? First, it is within the demands of the classroom that a

student teacher’s internalized models of teaching are most readily

revealed (Feiman-Nemser, 1983). Such models, acquired through

years of classroom observations as a student, will persist throughout

the practicum if left unchallenged. Furthermore, any assumption

that desirable teaching habits necessarily derive from practice is

directly contradicted by existing research. For instance, Feiman-

Nemser cites studies in which successful student teaching was most

often equated with achieving utilitarian goals affiliated with

classroom management. This perspective on successful teaching

could likely impede any designs by teacher education programs to

infuse theory into practice. Feiman-Nemser and Feiman-Nemser and

Buchmann (1987) also report that student teachers tend to imitate

the persona of the school community into which they are

acculturated. Such behavior might reflect the specific habits of the

cooperating teacher or the more general attributes of the school

bureaucracy. Whether good or bad, this tendency could persist in the

absence of supervision that challenges student teachers’ models of

teaching.

Taken together, these findings point to supervision as pivotal

to teacher change. It is the supervisor who is most able to “provide

support and guidance for student teachers to integrate theoretical

and research-based ideas from their university courses into their

teaching” (Borko & Mayfield, 1995, p. 517). However, meaningful

supervision rests on reinterpreting the role of supervisor as teacher.

Sporadic visits by a supervisor whose primary function is to evaluate

peripheral characteristics of teaching seems to be an ineffective

route to changing practice (Borko & Mayfield, 1995; Frykholm,

1996; Zeichner, 1993).

From this position, I consider the place of educative

supervision in changing one student teacher’s practice. Educative

supervision is broadly defined here to mean supervision that

prioritizes the development of a student teacher’s pedagogical

content knowledge. Although student teaching is one of the most

widely studied components of formal teacher preparation, the

influence of supervision (particularly educative supervision) on

teacher learning is still unclear (Borko & Mayfield, 1995). Moreover,

understanding what educative supervision might resemble from a

sociocultural perspective remains virtually unexplored. As such, the

present study is guided by the following research questions:

1. What emerges as the pedagogy of educative supervision

during one prospective teacher’s professional semester?

2. What are the indications of the student teacher’s

development within the zone of proximal development?

Designing An Educative Approach to Supervision

As conceptualized in this study, educative supervision rests on

the Vygotskian (1978) tenet that the supervisor, as a more knowing

other, can guide the student teacher’s development to a greater

extent than the student teacher can when working alone. This

notion, theorized by Vygotsky as the zone of proximal development,

is unique in that it “connects a general psychological perspective on

[the individual’s]...development with a pedagogical perspective on

instruction” (Hedegaard, 1996, p. 171). As such, it lends theoretical

support to the use of intentional instruction during supervision.

Collecting the Data: The Cycle of Mediation

In order to study the influence of educative supervision on the

student teacher’s construction of pedagogical content knowledge, I

became the university supervisor of Mary Ann, a prospective middle

school science and mathematics teacher. Mary Ann was in her final

year of a four-year teacher education program for which ongoing

reforms in mathematics education are the unofficial mantra. She had

successfully completed her academic program and was eager to

begin the professional semester. Assigned to a seventh-grade

mathematics classroom in an urban middle school, Mary Ann was

paired with a veteran cooperating teacher who proved to be

extremely supportive. The cooperating teacher’s approach of sharing

her own wisdom of practice without stifling Mary Ann’s ideas led to a

positive, open relationship between them.

Mary Ann and I arranged weekly visits during the practicum

for what I conceptualized as an extension of Steffe’s (1991)

constructivist teaching experiment. That is, rather than eliciting

models of children’s constructions of mathematical knowledge, I was

interested in a teacher’s (i. e., Mary Ann’s) construction of

pedagogical content knowledge. Each visit consisted of a three-hour

sequence that began with an observation of Mary Ann teaching her

first period general mathematics class. Field notes taken during this

observation focused on episodes of discourse that reflected the

nature of her thinking about teaching mathematics. Immediately

following the observation, I collaborated with Mary Ann in a 45-

minute teaching episode to help make sense of these classroom

interactions. In particular, Mary Ann’s thinking about the

interactions, what they suggested about how students learn

mathematics, and consequently how subsequent lessons might be

modified to reflect this, were discussed. The visit concluded with a

second classroom observation of Mary Ann’s third period general

mathematics class. This provided the chance to document short-term

changes in Mary Ann’s practice as she taught the same subject to a

different class after a teaching episode. Additionally, Mary Ann was

asked to keep a personal journal in which she reflected on what she

had learned about her students, about mathematics, and about

teaching mathematics (see Appendix). Other written artifacts, such

as lesson plans, activity sheets, and quizzes, were collected as well.

At the conclusion of each visit, I audiotaped personal

reflections about emerging pedagogical content issues and how

future visits could incorporate these themes as learning

opportunities for Mary Ann. In all, I had eight visits followed by a

separate exit interview. Finally, I conducted two clinical interviews

with the cooperating teacher to obtain a more complete picture of

Mary Ann’s social context. These interviews were based on questions

concerning the cooperating teacher-student teacher partnership

that the cooperating teacher was asked to reflect on prior to the

meetings (see Appendix). Each visit, documented through field notes

and complete audio and audiovisual recordings, along with

supporting written artifacts and interviews with the cooperating

teacher, provided the data corpus for this investigation. The

supervisory process of observation, teaching episode, observation,

and written reflection that Mary Ann experienced as part of this

study is described here as the cycle of mediation (see Figure 1). It is

postulated in this study as a model for educative supervision.

Figure 1. The cycle of mediation in an emerging practice of teaching.

Pedagogy of the Teaching Episodes

The teaching episodes with Mary Ann were central to the

supervisory process outlined above. According to Steffe (1983), the

teacher’s role in an episode is to challenge the model of the

student’s knowledge and examine how that model changes through

purposeful intervention. This is consistent with the Vygotskian notion

of opening a student’s zone of proximal development and effecting

conceptual change through instructional assistance by a more

knowing other. Moreover, Manning and Payne (1993) suggest that

“in the teacher education context, this more experienced person is

likely to be a supervising teacher, college supervisor, teacher

educator, or a peer who is at a more advanced level in the teacher

education program” (as cited in Jones, Rua, & Carter, 1997, p. 6).

Intellectual honesty further mandated the pedagogy of the

teaching episodes. That is, since my purpose was to teach Mary Ann,

my own practice needed to be consonant with current reforms in

mathematics education. However, little is known about what it

means to supervise from this theoretical orientation. Moving away

from an authoritative voice, I turned to instructional conversation as

the underlying pedagogy. Instructional conversation stems from a

cultural ethos that emphasizes the use of narrative in an individual’s

development. Gallimore and Goldenberg (1992) and Rogoff (1990)

describe it as a primary means of assisted performance in preschool

discourse between parent and child. One’s way of life, embedded in

picture books and bedtime stories, is taught through conversation in

the context of familial relationships.

While formal schooling may seem far removed from this

setting, the essence of instructional conversation is a promising

technique in that context as well. Gallimore and Goldenberg (1992)

recognize that, traditionally, this form of teaching abates in school,

where teachers are more likely to dominate interactions and

students are less likely to converse with their teacher or peers. Part

of the difficulty of instructional conversation in the classroom is that

it involves the “paradox of planful intention and responsive

spontaneity” (Gallimore & Goldenberg, 1992, p. 209). Furthermore,

it requires teachers to shift from an evaluative role grounded in

“known-answer” questioning, to a facilitory role in which they elicit

students’ ideas and interpretations. Despite these challenges,

instructional conversation seemed an appropriate pedagogy with

which to engage Mary Ann in developing her craft.

Data Analysis

The teaching episodes with Mary Ann were selected as the

primary data source in this portion of the study. In order to

instantiate the pedagogy of these episodes as instructional

conversation, complete transcripts of four of the episodes were

coded by conversational subject using each speaker’s turn as the

basic unit of analysis. (See Appendix for a complete description of

the coding scheme.) The episodes were then quantified by a word

count to determine the emphasis given to each subject code and to

establish the amount of conversational time used by the university

supervisor (myself) and the student teacher. Additionally, transcripts

were examined for the use of known-answer questions and instances

of direct teaching by the supervisor.

Previously, I established evidence of long-term changes in

Mary Ann’s teaching during the practicum by documenting shifts in

the pattern and function of classroom discourse about mathematics.

Conclusions were based on the analysis of classroom observations

made during the practicum. In this portion of the study, the focus is

on the university supervisor as a mediating agent in the student

teacher’s practice. As such, transcripts from classroom observations

became a secondary source for corroborating short-term changes in

Mary Ann’s practice as a result of the teaching episodes. One visit,

selected as an exemplar of the cycle of mediation as a supervisory

model, was further analyzed to determine factors that promoted a

change in Mary Ann’s teaching. Specific excerpts from transcripts of

this visit (referred to later in the text as the “problem-solving day”)

are included to substantiate the results of the study.

Findings and Interpretations

Instructional Conversation in Teaching Episodes with Mary Ann

In an investigation of elementary students’ reading

comprehension, Gallimore and Goldenberg (1992) mutually

negotiated ten characteristics of instructional conversation: (a)

activating, using, or providing background knowledge and relevant

schemata; (b) thematic focus for the discussion; (c) direct teaching,

as necessary; (d) promoting more complex language and expression

by students; (e) promoting bases for statements or positions; (f)

minimizing known-answer questions in the course of the discussion;

(g) teacher responsivity to student contributions; (h) connected

discourse, with multiple and interactive turns on the same topic; (i) a

challenging but nonthreatening environment; and (j) general

participation, including self-elected turns. These characteristics

suggest what it might look like to supervise from a sociocultural

perspective. Those most representative of my instructional

conversations with Mary Ann motivate the following discussion on

how supervision from this perspective emerged during the

investigation. Excerpts from the problem-solving day are used to

situate these features within the context of the present study.

Activating, using, or providing background knowledge and

relevant schemata. Gallimore and Goldenberg (1992) maintain that

“students must be ‘drawn into’ conversations that create

opportunities for teachers to assist” (p. 209). An advantage of

educative supervision is that it can use the context of practice to

scaffold the student teacher’s emerging ideas about teaching. In

particular, episodes of classroom discourse became the nexus

between theory and practice in my instructional conversations with

Mary Ann. Using her classroom experiences as a referent seemed to

open her zone of proximal development and draw her into the

conversations. Indeed, Mary Ann became visibly passive when other

referents (e. g., my own experiences as a student teacher) were

introduced.

Thematic focus for the discussion. Gallimore and Goldenberg

(1992) also argue that “to open a zone of proximal development..., a

teacher has to intentionally plan and pursue an instructional as well

as a conversational purpose” (p. 209). By my third visit with Mary

Ann, I had identified a thematic focus on the nature of classroom

interactions that emerged after a mathematical task or question had

been posed. As discussed in Part III, observations prior to this visit

revealed predominantly univocal classroom interactions by which

Mary Ann funneled students toward her interpretation of the

problem at hand. The third visit presented an opportunity for

assisting Mary Ann in cultivating dialogic classroom interactions.

During the first period class that day, Mary Ann began a lesson

on “working backwards” as a problem-solving technique by giving

students a problem to work individually. “I’m thinking of a number”,

she said, “that if you divide by three and then add five, the result is

eleven.” After a short pause, Mary Ann began to dole out hints until

a correct solution appeared. After a student shared a procedure for

obtaining this solution, Mary Ann began a step-by-step account of

how to work backwards to find the answer. Analysis later showed

that she had interpreted students’ responses univocally, asking

cognitively-small questions (e. g., [What is] thirteen minus five?,

What is eight times three?) to align their thinking with her own.

Equating student feedback with understanding, Mary Ann’s

frustration surfaced later when the class attempted to solve a similar

problem.

1 Mary Ann: O. K., I’m thinking of a number [that], if you

divide by three and then add five, the result

is thirteen. So what would I first do just to get

an idea of what we’re talking about? Does

anybody know how we did the last one? (No one

responds to her questions.) O. K., what we need

to do first, step one, we need to write everything

down in the order in which we read it. So, we

start reading, “If you divide by three”, so we divide

by three. Then we’re going to add five. Then the

result is thirteen, and we want to work

backwards. So, what have we got to do when

we work backwards? (Again, students don’t

respond.) O. K., what was the word we used

when we talked about what we’ve got to do with all

of these [operations]?

2 Class: Inverse.

Univocal interactions between teacher and students continued

until a student produced a response of twenty-four. Mary Ann

concluded, “Twenty-four. So, that’s my answer. That is the answer. I

ask you what number did I start with, you’ll say what?” The students

were silent. She continued, “What number did I start with? The

problem says, ‘I’m thinking of a number’. What number am I thinking

of?” Hesitantly, students tried to guess the response, suggesting

various numbers that had occurred in the problem. Twenty-four

seemed to dominate, cueing Mary Ann to once again argue its

veracity. She repeated, “Twenty-four. That is your answer. You

worked backwards. You said thirteen minus five is eight and eight

times three is twenty-four.”

The perturbation that Mary Ann exhibited during this

interaction seemed to grow out of puzzlement that students did not

understand what she had carefully explained. This left her at a

pedagogical impasse. The challenge of the teaching episode that

followed (and future episodes) was to use such interactions to help

Mary Ann develop a sense of mathematics as a problem-solving

endeavor in which students struggled with unfamiliar problems and

justified their ideas through mathematical discourse with each other

and Mary Ann. In essence, the challenge was to help Mary Ann

create a classroom discourse in which dialogic and univocal

interactions dualistically existed. Using such interactions as a

thematic focus became an avenue for encouraging Mary Ann to

interact dialogically with her students. It provided an instructional

and conversational purpose that continued throughout the

practicum.

Direct teaching, as necessary. Given that students are more

likely to teach in ways they are taught (Borko & Mayfield, 1995;

Feiman-Nemser, 1983), I minimized instances of direct teaching in

the episodes with Mary Ann. Instead, I relied on “prompting,

modeling, explaining,...discussing ideas, [and] providing

encouragement” (Jones, et al., 1997, p. 4) to give structure to our

conversations. This emphasis is consistent with the pedagogy of

instructional conversation, which prioritizes students’ participation

in dialogue. Table 1 illustrates the amount of conversational time

used by Mary Ann during the teaching episodes. The results support

my intent to maintain a facilitory role that kept her at the center of

discourse. By this, it became Mary Ann’s responsibility to rethink her

teaching. What emerged was the opportunity for her to retain

ownership of her practice. Furthermore, this sense of ownership

seemed to heighten Mary Ann’s willingness to put new ideas into

practice.

Table 1

Conversational Time Used by Participants in the Teaching Episodes

Participant TE1 TE2 TE3 TE4

Student teacher 82 84 72 73

University supervisor 18 16 28 27

Note. Values represent percentage of time a given participant spoke

during a teaching episode. Percentages are based on word counts.

“TE” denotes a teaching episode.

Minimizing known-answer questions in the course of the

discussion. “When known-answer questions are asked, there is no

need to listen to a child or to discover what the child might be trying

to communicate” (Gallimore & Goldenberg, 1992, p. 209). An

imperative of the teaching episodes was to avoid the use of known-

answer questioning and instead, to interpret Mary Ann’s utterances

dialogically. As a result, the questions I posed to Mary Ann were

essentially open-ended. In the ensuing dialogue, Mary Ann was

expected to justify her thinking about teaching mathematics and her

consequent actions in the classroom, not passively respond to a

supervisor’s prompts.

Teacher responsivity to student contributions. The amount of

conversational time used by Mary Ann suggests that her

contributions were a priority in supervision (see Table 1). Moreover,

being responsive to her ideas required being sensitive to her zone of

proximal development as well. The following dialogue was

excerpted from an instance of intentional instruction with Mary Ann

during the problem-solving day. It illustrates the effort to maintain

sensitivity to her zone while guiding her thinking, to base

supervision on her understanding of teaching mathematics, not my

own.

3 Supervisor: Is this the kind of [math word] problem where you

could let two or three [students] work

together, and try to figure out how to do it, and

see what kind of method they come up with?

4 Mary Ann: That could be an idea. Maybe I could let them

work with the person beside them.

5 Supervisor:Do you think that is even feasible? If so, why? Or

why not?

6 Mary Ann: Two heads are always better than one, and the kid

next to you might be thinking of one way, but

might be stumped on how to do the next. But

you might be able to help him figure that out. The

only thing is that I don’t know if they (her

voice trails off). We’ll see, though. That might be a

way to try. I don’t know if they can handle that,

talking to each other.... They’re just talkers, all

the time. Maybe if I show them that they can

have some freedom like that (her voice trails off).

Mary Ann’s uncertainty toward this suggestion was manifested

as concern over classroom management. My role then became to

redirect the instructional conversation so that it was within her zone

of proximal development. This involved connecting her concerns

about students’ behavior with the alternative approach we were

negotiating (7).

7 Supervisor:Do you think they can handle working with a

problem that they can’t figure out, trying to solve a

problem in that sense?

8 Mary Ann: I think they would be more apt to keep their

attention on that problem if they’re working

with somebody rather than working by

themselves.

After probing Mary Ann’s understanding of the role of word

problems in mathematics and teaching mathematics, we revisited

the previous topic.

9 Supervisor:Would you be comfortable, for example, if you

came [in class],...[threw] out a problem, and [let]

students work it for a while, and try to figure

out how to come up with a solution?

10 Mary Ann: Yeah. That’s how I’m thinking about starting the

next class. We’ll have to go over homework

first because they’re having a quiz on that

tomorrow. And then just have that [math word]

problem up on the board, and then tell them

to solve it. Don’t introduce anything about working

backwards.

Transcripts strip the dimensionality of dialogue. Although it’s

not readily apparent, Mary Ann’s claim, “That’s how I’m thinking

about starting the next class” (10), was spoken with a sense of

reflection and ownership. It stood in sharp contrast with her initial

reticence (6). It should also be noted that this remark occurred over

halfway through the teaching episode, after much attention had been

given to Mary Ann’s thinking about problem solving and the nature

of interactions that surrounded a problem posed in class. While one

might argue that a didactical approach (in the American semantic) to

supervision would have been more efficient, I seriously question if it

would have led to Mary Ann’s commitment to try an alternative

strategy. However, instructional conversation seemed to open her

zone of proximal development cognitively and affectively, thereby

producing at least a short-term commitment to change.

Connected discourse, with multiple and interactive turns on

the same topic. Specific directives on how Mary Ann might alter her

instruction after a given mathematical task had been posed were

revisited several times within the teaching episode on the problem-

solving day. When I sensed that my directives were out of her zone

of proximal development, I steered to related subjects (e. g., her

perception of problem solving in mathematics), but eventually moved

back to this topic. Furthermore, this particular teaching episode

became a “hook” (Gallimore & Goldenberg, 1992), or referent, in

later conversations with Mary Ann.

Table 2 is provided as an overview of general topics addressed

in the teaching episodes. In particular, it summarizes the focus on

pedagogical content knowledge throughout Mary Ann’s practicum.

Specifically, instructional conversations with Mary Ann were

dominated throughout by discussions on topics coded as

“mathematics pedagogy”. This, coupled with discussions on other

subjects closely linked to pedagogical content knowledge (i. e.,

“mathematical knowledge” and “general pedagogy”), left little room

for the peripheral issues of school bureaucracy in our teaching

episodes.

Table 2

Conversational Time Given to Subject Code During Teaching

Episodes

Subject Code TE1 TE2 TE3 TE4

Mathematics pedagogy 20 51 53 47

General pedagogy 28 20 9

23

Mathematical knowledge 0 7 11

16

Knowledge of student 24 17 14 14

understanding

Classroom management 28 0 5

0

Student-teacher 8 5 8

0

relationship

Note. Values represent percentage of time the specified subject was

discussed in a teaching episode. Percentages are based on word

counts.

“TE” denotes a teaching episode.

A challenging but nonthreatening atmosphere. In action and

words, Mary Ann seemed at her ease during the observations and

teaching episodes. During a couple of observations, she even asked

for my input on a particular problem as she taught the lesson.

Moreover, the rapport we established early on seemed to contribute

to her responsiveness in the teaching episodes. Upon reflection, I

could have created a more challenging atmosphere for Mary Ann.

Indeed, the ongoing tension of supervision is understanding how to

strike an optimal balance that effectively challenges the student.

Instructional Conversation in Retrospect: More on the Problem-

Solving Day

The dissonance Mary Ann experienced in classroom

interactions presented opportunities in supervision to promote

change in her practice. However, she needed to own that change.

Instructional conversation in the teaching episodes became a conduit

to that ownership. On the problem-solving day, it seemed to extend

to Mary Ann a commitment to modify her practice. Mary Ann began

the lesson following the teaching episode as we had planned.

Departing from her previous strategy, she placed students in dyads

to solve the problem that had been assigned as individual seat work

in her earlier class. Removing herself as the sole authority, she

delayed closure so that students would begin to communicate

mathematically with each other. As one of the students began

explaining her group’s strategy for solving the problem, Mary Ann

looked at me in excited disbelief and mouthed, “Wow!” She

commended the student, “You just taught our lesson for today!”

Mary Ann’s expression told the story that her journal reflection later

confirmed.

Teaching this [to the first period class] was a real eye-opener

for me. I think I totally confused my students completely. I

tried to show them steps without letting them think about the

problem themselves.... [The next class] was different. After

[the university supervisor] and I talked about the lesson and

going over several suggestions, things seem [sic] to run much

smoother. Instead of throwing information out, I let them

figure the problem out in their own style.... To my surprise, one

of my students performed the problem exactly as the strategy

suggested. Boy, was this a memorable event. The pressure was

lifted off of me.... Once the students saw how one of their

peers was able to solve the problem, things were a lot more

clear to all. I learned that having a student come up with the

solution means more to the others than the teacher giving a

long, drawn-out lecture. Sometimes you need for things to flop,

so you can think up new ways to approach the situation.

From my observations, the problem-solving day was a first step

in Mary Ann’s attempts to interact dialogically with her students.

Furthermore, it seemed to anchor her willingness to take risks in her

practice based on ideas mediated through instructional conversation.

She continued to develop, albeit in a nonlinear fashion, toward a

practice which included dialogic as well as univocal interactions.

Discussion

This study investigates in part what it means to educate

student teachers from a sociocultural perspective during the

professional semester.

Cobb, Yackel, and Wood (1991) maintain that

teachers should be helped to develop their pedagogical

knowledge and beliefs in the context of their classroom

practice. It is as teachers interact with their students in

concrete situations that they encounter problems that call for

reflection and deliberation.... Discussions of these concrete

cases with an observer who suggests an alternative way to

frame the situation or simply calls into question some of the

teacher’s underlying assumptions can guide the teacher’s

learning (p. 90).

In this sense, the cycle of mediation became educative for

Mary Ann. Specifically, coordinating classroom interactions observed

during Mary Ann’s teaching with the instructional conversation of

the teaching episodes and Mary Ann’s reflections about her practice

converged to promote Mary Ann’s development within her zone.

Although such a process is arguably quixotic, it does suggest an

avenue for effecting prospective teachers’ development in the

context of their practice.

Furthermore, while it is difficult to establish a direct link

between instructional conversation and conceptual development

(Gallimore & Goldenberg, 1992), instructional conversation does

suggest an alternative pedagogy for educative supervision.

Specifically, it seemed to open Mary Ann’s zone of proximal

development so that her understanding of teaching mathematics

could be mediated with the assistance of a more knowing other.

Moreover, the notion that an individual’s intramental

functioning reflects the intermental context of the classroom

(Wertsch & Toma, 1995) suggests that instructional conversation

could mediate Mary Ann’s practice toward that type of pedagogy.

Simply put, students most likely teach in ways they are taught.

However, as a caveat, it should be noted that multiple influences

shape the prospective teacher’s emerging practice. This sometimes

limits, or even negates, the influence of the supervisor.

Understanding how all of these factors coalesce in the making of a

teacher is at best a delicate process. As such, this investigation is a

first attempt to understand that process from the supervisor’s lens.

References

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APPENDIX

EXPECTATION OF THE STUDENT TEACHER

Sept. 24

(Student Teacher)

As I have already mentioned to you, each of my observations will involve a (telephone) pre-conference, observing 2 classes weekly (when possible), and a post-conference. Except for the frequency, this should be typical for all student teachers. I would like for you to have a copy of your lesson plan to give to me on the day that I observe. Also, I would like a reflective journal entry for each of my visits. Below are some questions that I would like for you to address. Since this is only one entry per week (roughly), it should not be too demanding of your time. You do not have to do this separately from the journal requirements for Dr. S and/or Dr. N, but you may include my questions within their requirements (e.g., they may require more than one entry per week and you should fulfill that obligation, but I am only specifically looking for a single detailed entry corresponding to my visits that addresses the following questions. Where there is possible overlap, use it to your advantage.) If you have any questions, please let me know.

Questions to consider for your journal entries:

What student interaction(s) was/were the most memorable to you (during my observation)? (Please avoid interactions that deal with classroom management, etc. I am only interested in interactions as they relate to your teaching mathematics.) Why?

How (if at all) did this affect your instruction? How (if at all) did this affect your understanding of mathematics? What did you learn about (your) students as a result of this?

I have enclosed a consent form for you to sign. I will pick it up when I observe you.

Thank you!

Maria

COOPERATING TEACHER ASSESSMENTOF THE

STUDENT TEACHER/COOPERATING TEACHER PARTNERSHIP

What were your goals and expectations when you entered this partnership?

How have these goals and expectations changed, if at all, during this practicum?

How did you perceive your role as cooperating teacher when you entered this partnership?

How has this perception changed, if at all, during this practicum?

Describe the nature of your partnership.

What do you think your student teacher learned from you?

Was there evidence that he or she successfully completed your perception of the practicum? If so, what?

What did you learn from your student teacher?

Describe your interactions with your student teacher. (e.g., Did you meet on a regular basis? Informally or formally? How did you negotiate your respective roles in the class?)

CODING SCHEME FOR TEACHING EPISODES WITH MARY ANN

Mathematics pedagogy (MP). Conversation that addresses Mary

Ann’s learning about teaching mathematics as well as how she

teaches mathematics.

General pedagogy (GP). Conversation that addresses principles of

teaching that aren’t specific to mathematics (e. g., pacing

instruction, diversity in student learning).

Mathematical knowledge (MK). Conversation that addresses Mary

Ann’s knowledge about mathematics.

Knowledge of student understanding (KSU). Conversation that

addresses Mary Ann’s understanding of how students are or are not

understanding the content and how that directly affects her practice.

References to test performance are also designated KSU.

Classroom management (CM). Conversation that addresses non-

academic student needs (e. g., discipline, student health).

Student/teacher relationship (STR). Conversation that addresses

Mary Ann’s relationship with her students and how that influenced

instruction.

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APPENDIX

COOPERATING TEACHER ASSESSMENTOF THE

STUDENT TEACHER/COOPERATING TEACHER PARTNERSHIP

What were your goals and expectations when you entered this partnership?

How have these goals and expectations changed, if at all, during this practicum?

How did you perceive your role as cooperating teacher when you entered this partnership?

How has this perception changed, if at all, during this practicum?

Describe the nature of your partnership.

What do you think your student teacher learned from you?

Was there evidence that he or she successfully completed your perception of the practicum? If so, what?

What did you learn from your student teacher?

Describe your interactions with your student teacher. (E.g., Did you meet on a regular basis? Informally or formally? How did you negotiate your respective roles in the class?)

Consent to Release Information for Research Purposes

I, __________________________________________________, give permission for the contents of my journal, portfolio, surveys, videotapes, and any audiotapes to be used as a research resource for written professional reports. I also give my permission for information from interviews and conferences to be used. I understand that my name will never be used without my permission and that no information in written or verbal form can be used to punitively assess my student teaching performance. I also understand that all copies of audio tapes and video tapes will be destroyed two years from the end of the project. If any changes in this agreement are required, I must be contacted in writing.

Signature_____________________________________________Date_____________

Cooperating Teacher Agreement Form

I, ______________________________________________________, give permission for data collected from my classroom during this professional semester to be used as a resource for research of prospective education. I understand that the data collected will include videotaped classroom observations of the student teacher as well as interviews, surveys, and contents of the student teacher’s journal and portfolio. I also understand that my name will not be used in any way without my permission and that no identifying information in written or verbal form will be used. I also understand that copies of audiotapes and videotapes will be destroyed two years from the end of this project. If any changes in this agreement are required, I must be contacted in writing.

Signature___________________________________________Date________________

PRELIMINARY INTERVIEW PROTOCOL

Pick as many episodes from the classes I observed as I have time for. When possible, review these episodes on the video camera.

How did you feel about your lesson(s)?

What had you planned to do in this episode? Or had you planned anything?

What were you thinking during this episode?

Did you/would you change your instruction any as a result of this reflection? If so, how?

What did you learn about teaching in this episode?

What did you learn about mathematics?

What did you learn about teaching mathematics?

What did you learn about students? How will that affect your instruction?

EXPECTATION OF THE STUDENT TEACHER

Sept. 24

(Student Teacher)

As I have already mentioned to you, each of my observations will involve a (telephone) pre-conference, observing 2 classes weekly (when possible), and a post-conference. Except for the frequency, this should be typical for all student teachers. I would like for you to have a copy of your lesson plan to give to me on the day that I observe. Also, I would like a reflective journal entry for each of my visits. Below are some questions that I would like for you to address. Since this is only one entry per week (roughly), it should not be too demanding of your time. You do not have to do this separately from the journal requirements for Dr. S and/or Dr. N, but you may include my questions within their requirements (e.g., they may require more than one entry per week and you should fulfill that obligation, but I am only specifically looking for a single detailed entry corresponding to my visits that addresses the following questions. Where there is possible overlap, use it to your advantage.) If you have any questions, please let me know.

Questions to consider for your journal entries:

What student interaction(s) was/were the most memorable to you (during my observation)? (Please avoid interactions that deal with classroom management, etc. I am only interested in interactions as they relate to your teaching mathematics.) Why?

How (if at all) did this affect your instruction? How (if at all) did this affect your understanding of mathematics? What did you learn about (your) students as a result of this?

I have enclosed a consent form for you to sign. I will pick it up when I observe you.

Thank you!

Maria