ABSTRACT

PROSPECTIVE TEACHERS EMERGING PEDAGOGICAL CONTENT KNOWLEDGE DURING THE PROFESSIONAL SEMESTER:

A VYGOTSKIAN PERSPECTIVE ON TEACHER DEVELOPMENTby

MARIA 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 EDUCATIONRaleigh

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

Vygotskys Sociocultural Theory of Learning.............................................9

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

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

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

Social Interactions.......................................................................................13

The Zone of Proximal Development......................................................14

Implications of Vygotskys Sociocultural Theory

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

Teacher Education............................................................................................17

Teachers Beliefs and Knowledge.............................................................17

Learning How to Teach Mathematics.....................................................20

Teacher Development in Context............................................................21

Classroom Interactions....................................................................................23

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

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

In-Depth Interviewing................................................................................28

Participant Observation..............................................................................29

Teaching Experiments................................................................................30

PART II: METHODOLOGY.........................................................................................33

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

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

Data Collection..................................................................................................35

Data Analysis.....................................................................................................38

Role of the Researcher.....................................................................................40

PART III: MATHEMATICAL DISCOURSE IN A PROSPECTIVE

TEACHERS CLASSROOM: THE CASE OF A DEVELOPING

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

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

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

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

Process of Inquiry.............................................................................................49

The Research Setting..................................................................................49

Collecting the Data......................................................................................50

Analyzing Classroom Discourse...................................................................51

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

Process of Analysis......................................................................................54

Findings and Interpretations.........................................................................56

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

Early Pattern and Function in Resolving Students

Mathematical Dilemmas.......................................................57

Early Pattern and Function in Teaching a New Concept...............63

On Early Discourse and Mary Anns Practice....................................73

Indications of an Emerging Practice: Change in Pattern

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

The Problem-Solving Day.....................................................................75

Moving Forward in Classroom Discourse: Learning

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

Mary Anns Students: More Knowing Others?....................................93

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

References..........................................................................................................98

Appendix..........................................................................................................102

PART IV: THE CYCLE OF MEDIATION: A TEACHER EDUCATORS

EMERGING PEDAGOGY..........................................................................................107

Abstract.............................................................................................................108

Introduction....................................................................................................109

Rethinking the Role of Supervision: Education or Evaluation?........110

Collecting the Data: The Cycle of Mediation.......................................113

Pedagogy of the Teaching Episodes.......................................................116

Data Analysis...................................................................................................118

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

Instructional Conversation in Teaching Episodes

with Mary Ann.....................................................................119

Activating, 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 of

the Discussion.......................................................................124

Teacher Responsivity to Student Contributions...........................124

Connected 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

EDUCATORS 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: Vygotskys 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 EDUCATORS 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 teachers emerging pedagogical content knowledge during the professional semester. Vygotskys (1978) sociocultural perspective provides the theoretical framework for the study. Specifically, Vygotskys assertion that higher mental functioning is directly mediated through social interactions focused this study on the intermental context in which the prospective teachers practice develops during the professional semester, or student teaching practicum.

The nature of mathematical discourse embedded in social interactions in the prospective teachers classroom was analyzed as a window into the prospective teachers 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 teachers practice was considered. Analysis of pattern and function of classroom discourse substantiated an emerging practice, as the prospective teachers 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 teachers 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 teachers 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 teachers 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). Shulmans 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 ones 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 teachers 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, Vygotskys (1986) assertion that higher mental functions are directly mediated through social interactions suggests that the prospective teachers 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 ones (i. e., the prospective teachers) development within the zone of proximal development. The zone of proximal development is defined by Vygotsky (1978) as the distance between the [individuals] 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 teachers development.

This study is an investigation of the prospective middle school mathematics teachers 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 teachers 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 teachers construction of knowledge about teaching mathematics. Also, I examined the university supervisors role as a more knowing other in the prospective teachers emerging practice. Specifically, I considered what the pedagogy of supervision might resemble in order to open the prospective teachers 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 teachers mathematics classroom during the professional semester?

2. How does the university supervisor influence the prospective teachers 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 Vygotskys 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 ones theoretical assumptions concerning the nature of mind. In particular, the social constructivists 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.

Vygotskys Sociocultural Theory of Learning

According to Wertsch (1988), Vygotskys 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 protg 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 Vygotskys 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 Vygotskys 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 individuals 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: Vygotskys 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 ones 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 ones 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

Vygotskys 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 individuals 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 individuals 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 childs 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 childs 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 readers 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 Vygotskys 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 teachers 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 Vygotskys definition of the zone of proximal development requires an adult or more capable peer to foster ones development. However, Oerter (1992) distinguishes three contexts which can create ones 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 Vygotskys 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 ones construction of knowledge. According to Taylor (1993), Vygotsky also suggested that a students 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 Vygotskys 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, Vygotskys sociocultural theory guided my investigation of the prospective teachers emerging practice. As a formal theory, it provided an established language for communicating research, as well as an accepted format for investigation. More specifically, Vygotskys general genetic law of cultural development directed me to social interactions as a forum for the prospective teachers 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, Vygotskys construct of the zone of proximal development supports the use of intentional instruction during supervision to influence the prospective teachers 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, Simons 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

Shulmans 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 teachers 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 teachers 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 teachers 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 teachers ideas and practices concerning teaching for procedural and conceptual knowledge revealed a tension between the teachers 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 teachers support personnel. The researchers concluded that teaching for conceptual knowledge should enjoy consistent support from all of the professional participants in the student teachers 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 teachers underlying assumptions can guide the teachers 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 childrens constructions of mathematical knowledge, they suggest that the appropriate setting in which to ascertain teachers models is also the classroom. Their investigation of one teachers learning that occurred in the mathematics classroom indicated that the teachers beliefs about the nature of mathematics and learning were affected as she resolved conflicts between her existing teaching practices and the projects 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-Bussis (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 childrens constructions of mathematics, Steffe and Tzur (1994) analyzed social interactions attendant with childrens work on fractions using computer microworlds. They extended social interactions to mathematical interactions, with the latter including enactment or potential enactment of childrens 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 childrens 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 childrens constructions of mathematical knowledge (Cobb, 1995; Steffe & Tzur, 1994), it is theoretically feasible that interactions in the prospective teachers classroom should provide understanding of how knowledge about teaching mathematics is constructed. In this context, I interpret the prospective teachers 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 childrens 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 teachers 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 researchers 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 ones 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 researchers 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 subjects 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 researchers 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 students 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 Vygotskys 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 teachers role is to challenge the model, or explanation, of the students knowledge and examine how that model changes through purposeful intervention. This component is consistent with the Vygotskian (1986) notion of creating a students zone of proximal development and offering instructional assistance in order to effect the students 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.

Vygotskys (1986) studies of conceptual development in children indicate that teaching within the context of an investigation is not a new approach. His view that ones 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 Vygotskys inquiry. As such, the teaching experiment is particularly appropriate for studies that assume a Vygotskian theoretical framework for the purpose of understanding ones 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 teachers 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 teachers meetings with peers or the cooperating teacher. However, this study focused on one prospective teachers interactions with her students and the university supervisor.

It should be noted that, although the prospective teachers students would not typically be viewed as that teachers 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 Steffes (1991) use of a constructivist teaching experiment to elicit models of childrens 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 teachers pedagogical content knowledge.

Episodes of discourse in the prospective teachers mathematics classroom reflecting mediation of that teachers 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 teachers 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 teachers 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 teachers 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 teachers 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 Anns 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 Anns 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 teachers 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 TEACHERS 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 teachers 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 teachers 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 Anns 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 Anns 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 Anns 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 teachers 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 teachers 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 teachers classroom?

2. What does such discourse suggest about the prospective teachers pedagogical content knowledge?

3. How is the prospective teachers 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

Vygotskys (1986) sociocultural approach gives theoretical precedent to the place of discourse in an individuals development. According to Minick (1996), Vygotsky maintained that higher voluntary forms of human behavior have their roots in social interaction, in the individuals 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 individuals 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, Vygotskys 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 teachers 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 teachers 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 teachers 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 teachers practice. Furthermore, the nature of such discourse is a harbinger of the teachers internalized thinking about teaching mathematics. Under the umbrella of Vygotskys 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 students intramental functioning. This principle concerning the relationship between ones 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 teachers 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 teachers attempt to help a student resolve a mathematical dilemma. In the process of discourse, the teacher attempts to make sense of the students difficulty and decides on a course of action. As the instructional plan unfolds, the teacher tries to assess the students understanding and may subsequently modify the plan in order to influence that students thinking in a desired direction. In effect, the teachers behavior (as well as the students) is being mediated in the context of this interaction. What emerges for the teacher is a new awareness of how to address a students difficulty at some level of generality, an awareness that is reflected through variations in the teachers speech. The teachers 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 Anns 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 teachers 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 Anns 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 teachers classroom, the data corpus reflects broader issues in Mary Anns 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 Anns 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 teachers 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 teachers 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 someones 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 peoples talk. Pattern and function captured the nature of discourse in Mary Anns classroom and thereby revealed the essence of her developing knowledge about teaching mathematics. Furthermore, based on Woods (1995) process of documenting teacher learning in the classroom, I looked at shifts in pattern and function to establish Mary Anns construction of pedagogical content knowledge.

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