PROSPECTIVE TEACHERS’ EMERGING PEDAGOGICAL CONTENT
KNOWLEDGE DURING THE PROFESSIONAL SEMESTER:
A VYGOTSKIAN PERSPECTIVE ON TEACHER DEVELOPMENT
by
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 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. JoAnn 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.......................................................................................13
The Zone of Proximal Development......................................................14
Implications of Vygotsky’s 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
TEACHER’S 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 Ann’s 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 Ann’s Students: More Knowing Others?....................................93
Discussion..........................................................................................................95
References..........................................................................................................98
Appendix..........................................................................................................102
PART IV: THE CYCLE OF MEDIATION: A TEACHER EDUCATOR’S
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
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 reformminded 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 SecondGrade 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.
CLASSROOM
OBSERVATION
FOCUSED
REFLECTION
TEACHING EPISODE
instructional conversation
CLASSROOM
OBSERVATION
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 prealgebra. 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, UnderwoodGregg 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
subtract eighty-one from both sides. O. K., so eighty-
you
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
to Marta.) Marta? (No response.) O. K., Allyson,
turns
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
card walks to the front.) All right. Very good. So what
did was we replaced s, our variable. We bumped her
“9”
we
(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 selfperceived 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 thirtyeight 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
mathematically. So you’ve got to show me
can back this up, saying it’s fourteen. If
if this was the unknown (she
to back it up
an equation that
I’m solving for a variable,
indicates the number of
diagonals of a heptagon).... O. K.,
paper and start
so get out a pencil and piece of
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., 120128). 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
Bartolini-Bussi, M. G. (1991). Social interaction and mathematical
knowledge. In F. Furinghetti (Ed.), Proceedings of the Fifteenth Annual
Meeting of the International Group for the Psychology of Mathematics
Education (pp. 1-16). Assisi, Italy.
Bauersfeld, H. (1988). Interaction, construction, and knowledge:
Alternative perspectives for mathematics education. In T. Cooney & D. Grouws
(Eds.), Effective mathematics teaching (pp. 27-46). Reston, VA: National Council
of Teachers of Mathematics and Lawrence Erlbaum.
Cobb, P. (1995). Mathematical learning and small-group interaction: Four
case studies. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical
meaning: Interaction in classroom cultures (pp. 25-127). Hillsdale, NJ: Lawrence
Erlbaum.
Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997). Reflective
discourse and collective reflection. Journal for Research in Mathematics
Education, 28(3), 258-277.
Cobb, P., Yackel, E., & Wood, T. (1991). Curriculum and teacher
development: Psychological and anthropological perspectives. In E. Fennema,
T. P. Carpenter, & S. J. Lamon (Eds.), Integrating research on teaching and
learning mathematics (pp. 83-119). Albany, NY: State University of New York.
Cobb, P., Yackel, E., & Wood, T. (1992). Interaction and learning in
mathematics classroom situations. Educational Studies in Mathematics, 23(1),
99-122.
Cole, M., & Wertsch, J. (1994). Beyond the individual-social antimony in
discussions of Piaget and Vygotsky [On-line]. Available Internet:
http://www.massey.ac.nz/~ALock/virtual/colevyg.htm.
Erickson, F. (1986). Qualitative methods in research on teaching. In M.
Wittrock (Ed.), Handbook of research on teaching (pp. 119-161). New York:
Macmillan.
Holzman, L. (1996). Pragmatism and dialectical materialism in language
development. In H. Daniels (Ed.), An introduction to Vygotsky (pp. 75-98).
London: Routledge.
Lo, J., Wheatley, G., & Smith, A. (1991). Learning to talk mathematics.
Paper presented at the Annual Meeting of the American Educational Research
Association, Chicago, IL.
Lotman, Y. (1988). Text within a text. Soviet Psychology, 26(3), 32-51.
Minick, N. (1996). The development of Vygotsky’s thought. In H.
Daniels, (Ed.), An introduction to Vygotsky (pp. 28-52). London: Routledge.
National Council of Teachers of Mathematics. (1989). Curriculum and
evaluation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1990). Professional
standards for teaching mathematics. Reston, VA: Author.
Peressini, D., & Knuth, E. (in press). Why are you talking when you
could be listening? The role of discourse and reflection in the professional
development of a secondary mathematics teacher. Teaching and Teacher
Education.
Potter, J., & Wetherell, M. (1987). Discourse and social psychology.
London: Sage.
Steffe, L., & Tzur, R. (1994). Interaction and children’s mathematics. In P.
Ernest (Ed.), Constructing mathematical knowledge: Epistemology and
mathematical education (pp. 8-32). London: Falmer.
Thompson, P. (1994). The development of the concept of speed and its
relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The
development of multiplicative reasoning in the learning of mathematics (pp.
179-234). Albany, NY: State University of New York.
Underwood-Gregg, D. U. (1995). Gender-related differences in
interaction patterns in elementary school Inquiry Mathematics classrooms.
Dissertation Abstracts International, 57, 07A. (University Microfilms No.
AAI9638247).
Vygotsky, L. (1986). Thought and language. (A. Kozulin, Trans.).
Cambridge, MA: Massachusetts Institute of Technology. (Original work
published in 1934.).
Wertsch, J. (1988). L. S. Vygotsky’s “new” theory of mind. The American
Scholar, 57, 81-89.
Wertsch, J., & Toma, C. (1995). Discourse and learning in the classroom:
A sociocultural approach. In L. Steffe & J. Gale (Eds.), Constructivism in
education (pp. 159-174). Hillsdale, NJ: Lawrence Erlbaum.
Wertsch, J., & Tulviste, P. (1996). L. S. Vygotsky and contemporary
developmental psychology. In H. Daniels (Ed.), An introduction to Vygotsky
(pp. 53-74). London: Routledge.
Wood, T. (1995). An emerging practice of teaching. In P. Cobb & H.
Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in
classroom cultures (pp. 203-227). Hillsdale, NJ: Lawrence Erlbaum.
Wood, T., Cobb, P., & Yackel, E. (1991). Change in teaching mathematics:
A case study. American Educational Research Journal, 28 (3), 587-616.
APPENDIX
COOPERATING TEACHER ASSESSMENT
OF 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
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.
same

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
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.
or
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, BenPeretz 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 FeimanNemser 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 threehour 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.
CLASSROOM
OBSERVATION
FOCUSED
REFLECTION
TEACHING EPISODE
instructional conversation
CLASSROOM
OBSERVATION
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 problemsolving 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 stepby-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, “Twentyfour. 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 problemsolving 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
0
7
11
16
24
17
14
14
28
0
5
0
8
5
8
0
Mathematical knowledge
Knowledge of student
understanding
Classroom management
Student-teacher
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
Ben-Peretz, M., & Rumney, S. (1991). Professional thinking in guided
practice. Teaching and Teacher Education, 7(5), 517-530.
Borko, H., & Mayfield, V. (1995). The roles of the cooperating teacher
and university supervisor in learning to teach. Teaching and Teacher
Education, 11(5), 501-518.
Bowman, N. (1979). College supervision of student teaching: A time to
reconsider. Journal of Teacher Education 30(3), 29-30.
Cobb, P., Yackel, E., & Wood, T. (1991). Curriculum and teacher
development: Psychological and anthropological perspectives. In E. Fennema,
T. P. Carpenter, & S. J. Lamon (Eds.), Integrating research on teaching and
learning mathematics (pp. 83-119). Albany, NY: State University of New York.
Feiman-Nemser, S. (1983). Learning to teach. In L. Shulman & G. Sykes
(Eds.), Handbook of teaching and policy (pp. 150-170). New York: Longman.
Feiman-Nemser, S., & Buchmann, M. (1987). When is student teaching
teacher education? Teaching and Teacher Education 3, 255-273.
Frykholm, J. (1996). Pre-service teachers in mathematics: Struggling with
the Standards. Teaching and Teacher Education 12(6), 665-681.
Gallimore, R., & Goldenberg, C. (1992). Tracking the developmental path
of teachers and learners: A Vygotskian perspective. In F. Oser, A. Dick, & J.
Patry (Eds.), Effective and responsible teaching: The new synthesis (pp. 203221). San Francisco: Jossey-Bass.
Hedegaard, M. (1996). The zone of proximal development as a basis for
instruction. In H. Daniels (Ed.), An introduction to Vygotsky (pp. 171-195).
London: Routledge.
Jones, G., Rua, M., & Carter, G. (1997, March). Science teachers’
conceptual growth within Vygotsky’s zone of proximal development. Paper
presented at the meeting of the American Educational Research Association,
Chicago, IL.
Richardson-Koehler, V. (1988). Barriers to the effective supervision of
student teaching: A field study. Journal of Teacher Education 39(2), 28-34.
Rogoff, B. (1990). Apprenticeship in thinking: Cognitive development in
social context. New York: Oxford University.
Steffe, L. (1983). The teaching experiment methodology in a
constructivist research program. In M. Zweng, et al. (Eds.), Proceedings of the
Fourth International Congress on Mathematical Education (pp. 469-471).
Boston: Birkhauser.
Steffe, L. (1991). The constructivist teaching experiment: Illustrations and
implications. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics
education (pp. 8-32). London: Falmer.
Vygotsky, L. (1978). Mind in society. (M. Cole, S. Scribner, V. JohnSteiner, & E. Souberman, Trans.). Cambridge, MA: Harvard University.
(Original work published in 1934.).
Zeichner, K. (1993). Designing educative practicum experiences for
prospective teachers. Paper presented at the International Conference on
Teacher Education: From Practice to Theory, Tel-Aviv, Israel.
Zimpher, N., deVoss, G., & Nott, D. (1980). A closer look at university
student teacher supervision. Journal of Teacher Education 31(4), 11-15.
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 postconference. 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 ASSESSMENT
OF 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.
LIST OF REFERENCES
Bakhurst, D. (1996). Social memory in Soviet thought. In H. Daniels (Ed.),
An introduction to Vygotsky (pp. 198-218). London: Routledge.
Balacheff, N. (1990). Beyond a psychological approach of the psychology
of mathematics education. For the Learning of Mathematics, 10(3), 2-8.
Ball, D. (1988). Research on teacher learning: Studying how teacher’s
knowledge changes. Action in Teacher Education, 13(10), 5-10.
Ball, D. (1990). Prospective elementary and secondary teachers’
understanding of division. Journal for Research in Mathematics Education,
21(2), 132-144.
Ball, D. (1992). Magical hopes: Manipulatives and the reform of math
education. American Educator, 46, 14-18.
Ball, D. (1993). With an eye on the mathematical horizon: Dilemmas of
teaching elementary school mathematics. Elementary School Journal, 93(4), 373397.
Ball, D., & Mosenthal, J. (1990). The construction of new forms of
teaching: Subject matter knowledge in inservice teacher education. (Report No.
90-8). East Lansing, MI: National Center for Research on Teacher Education.
(ERIC Document Reproduction Service No. ED323208).
Bartolini-Bussi, M. G. (1991). Social interaction and mathematical
knowledge. In F. Furinghetti (Ed.), Proceedings of the Fifteenth Annual
Meeting of the International Group for the Psychology of Mathematics
Education (pp. 1-16). Assisi, Italy.
Bartolini-Bussi, M. (1994). Theoretical and empirical approaches to
classroom interaction. In R. Biehler, R. Scholz, R. Strasser, & B. Winkelmann
(Eds.), The didactics of mathematics as a scientific discipline (pp. 121-132).
Dordrecht, The Netherlands: Kluwer.
Bauersfeld, H. (1988). Interaction, construction, and knowledge:
Alternative perspectives for mathematics education. In T. Cooney & D. Grouws
(Eds.), Effective mathematics teaching (pp. 27-46). Reston, VA: National Council
of Teachers of Mathematics and Lawrence Erlbaum.
Bauersfeld, H. (1994). Theoretical perspectives on interaction in the
mathematics classroom. In R. Biehler, R. W. Scholz, R. Strasser, & Winkelmann
(Eds.), The didactics of mathematics as a scientific discipline (pp. 133-146).
Dordrecht, The Netherlands: Kluwer.
Ben-Peretz, M., & Rumney, S. (1991). Professional thinking in guided
practice. Teaching and Teacher Education, 7(5), 517-530.
Berenson, S., Van der Valk, T., Oldham, E., Runesson, U., Moreira, C., &
Broekman, H. (1997). An international study to investigate prospective
teachers’ content knowledge of the area concept. The European Journal of
Teacher Education, 20(2), 137-150.
Bogdan, R. C., & Biklen, S. K. (1992). Qualitative research for education:
An introduction to theory and methods. Boston: Allyn & Bacon.
Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D., &
Agard, P. C. (1992). Learning to teach hard mathematics: Do novice teachers
and their instructors give up too easily? Journal for Research in Mathematics
Education, 24(1), 8-40.
Borko, H., & Mayfield, V. (1995). The roles of the cooperating teacher
and university supervisor in learning to teach. Teaching and Teacher
Education, 11(5), 501-518.
Bowman, N. (1979). College supervision of student teaching: A time to
reconsider. Journal of Teacher Education 30(3), 29-30.
Brown, C.A., & Borko, H. (1992). Becoming a mathematics teacher. In D.
Grouws (Ed.), Handbook of research on mathematics teaching and learning: A
project of the National Council of Teachers of Mathematics (pp. 209-239). New
York: Macmillan.
Carpenter, T. P., Fennema, E., Peterson, P. L., & Carey, D. A. (1988).
Teachers’ pedagogical content knowledge of students’ problem solving in
elementary arithmetic. Journal for Research in Mathematics Education, 19, 385401.
Cobb, P. (1995). Mathematical learning and small-group interaction: Four
case studies. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical
meaning: Interaction in classroom cultures (pp. 25-127). Hillsdale, NJ: Lawrence
Erlbaum.
Cobb, P., Boufi, A., McClain, K., & Whitenack, J. (1997) Reflective
discourse and collective reflection. Journal for Research in Mathematics
Education, 28(3), 258-277.
Cobb, P., & Bauersfeld, H. (Eds.). (1995). The emergence of mathematical
meaning: Interaction in classroom cultures. Hillsdale, NJ: Lawrence Erlbaum.
Cobb, P., & Steffe, L. (1983). The constructivist researcher as teacher and
model builder. Journal for Research in Mathematics Education, 14(2), 83-94.
Cobb, P., Wood, T., Yackel, E., & McNeal, B. (1992). Characteristics of
classroom mathematics traditions: An interactional analysis. American
Educational Research Journal, 29(3), 573-604.
Cobb, P., Yackel, E., & Wood, T. (1991). Curriculum and teacher
development: Psychological and anthropological perspectives. In E. Fennema,
T. P. Carpenter, & S. J. Lamon (Eds.), Integrating research on teaching and
learning mathematics (pp. 83-119). Albany, NY: State University of New York.
Cobb, P., Yackel, E., & Wood, T. (1992). Interaction and learning in
mathematics classroom situations. Educational Studies in Mathematics, 23(1),
99-122.
Cobb, P., Yackel, E., & Wood, T. (1993). Learning mathematics: Multiple
perspectives. In T. Wood, (Ed.), Rethinking elementary school mathematics:
Insights and issues. Journal for Research in Mathematics Education Monograph
(pp. 21-32). Reston, VA: National Council of Teachers of Mathematics.
Cole, M., & Wertsch, J. (1994). Beyond the individual-social antimony in
discussions of Piaget and Vygotsky [On-line]. Available Internet:
http://www.massey.ac.nz/~ALock/virtual/colevyg.htm.
Confrey, J. (1995). How compatible are radical constructivism,
sociocultural approaches, and social constructivism? In L. Steffe & J. Gale (Eds.),
Constructivism in education (pp. 185-225). Hillsdale, NJ: Lawrence Erlbaum.
Cooney, T. (1994). Research and teacher education: In search of common
ground. Journal for Research in Mathematics Education, 25(6), 608-636.
Daniels, H. (1996). Introduction: Psychology in a social world. In H.
Daniels (Ed.), An introduction to Vygotsky (pp. 1-27). London: Routledge.
Eisenhart, M. (1991). Conceptual frameworks circa 1991: Ideas from a
cultural anthropologist; Implications for mathematics education researchers. In
Proceedings of the Thirteenth Annual Meeting of the North American Chapter
of the International Group for the Psychology of Mathematics Education (pp.
202-219).
Eisenhart, M., & Borko, H. (1991). In search of an interdisciplinary
collaborative design for studying teacher education. Teaching and Teacher
Education, 7(2), 137-157.
Eisenhart, M., Borko, H., Underhill, R., Brown, C. A., Jones, D., & Agard,
P. C. (1993). Conceptual knowledge falls through the cracks: Complexities of
learning to teach mathematics for understanding. Journal for Research in
Mathematics Education, 24(1), 8-40.
Erickson, F. (1986). Qualitative methods in research on teaching. In M.
Wittrock (Ed.), Handbook of research on teaching (pp. 119-161). New York:
Macmillan.
Ernest, P. (1994). Social constructivism and the psychology of
mathematics education. In P. Ernest (Ed.), Constructing mathematical
knowledge: Epistemology and mathematical education (pp. 62-72). London:
Falmer.
Ernest, P. (1995). The one and the many. In L. Steffe & J. Gale (Eds.),
Constructivism in education (pp. 459-489). Hillsdale, NJ: Lawrence Erlbaum.
Even, R. (1993). Subject-matter knowledge and pedagogical content
knowledge: Prospective secondary teachers and the function concept. Journal
for Research in Mathematics Education, 24(2), 94-116.
Even, R., & Tirosh, D. (1995). Subject matter knowledge and knowledge
about students as sources of teacher presentations of the subject matter. Journal
of Mathematical Behavior, 29, 21-27.
Feiman-Nemser, S. (1983). Learning to teach. In L. Shulman & G. Sykes
(Eds.), Handbook on teaching and policy (pp. 150-170). New York: Longman.
Feiman-Nemser, S., & Buchmann, M. (1987). When is student teaching
teacher education? Teaching and Teacher Education 3, 255-273.
Frykholm, J. (1996). Pre-service teachers in mathematics: Struggling with
the Standards. Teaching and Teacher Education 12(6), 665-681.
Gallimore, R., & Goldenberg, C. (1992). Tracking the developmental path
of teachers and learners: A Vygotskian perspective. In F. Oser, A. Dick, & J.
Patry (Eds.), Effective and responsible teaching: The new synthesis (pp. 203221). San Francisco: Jossey-Bass.
Glaser, B., & Strauss, A. (1967). The discovery of grounded theory:
Strategies for qualitative research. Chicago: Aldine.
Hedegaard, M. (1996). The zone of proximal development as a basis for
instruction. In H. Daniels (Ed.), An introduction to Vygotsky (pp. 171-195).
London: Routledge.
Holzman, L. (1996). Pragmatism and dialectical materialism in language
development. In H. Daniels (Ed.), An introduction to Vygotsky (pp. 75-98).
London: Routledge.
Jones, G., Rua, M., & Carter, G. (1997, March). Science teachers’
conceptual growth within Vygotsky’s zone of proximal development. Paper
presented at the Annual Meeting of the American Educational Research
Association, Chicago, IL.
Lampert, M. (1990). When the problem is not the question and the
solution is not the answer. American Educational Research Journal, 27, 29-63.
Lampert, M. (1992). Practices and problems in teaching authentic
mathematics. In F. K. Oser, A. Dick, & J. Patry (Eds.), Effective and responsible
teaching: The new synthesis (pp. 295-314). San Francisco: Jossey-Bass.
Lo, J., Wheatley, G., & Smith, A. (1991). Learning to talk mathematics.
Paper presented at the Annual Meeting of the American Educational Research
Association, Chicago, IL.
Lotman, Y. (1988). Text within a text. Soviet Psychology, 26(3), 32-51.
McDiarmid, G., Ball, D., & Anderson, C. (1989). Why staying one chapter
ahead just won’t work: Subject-specific pedagogy. In M. C. Reynolds (Ed.),
Knowledge base for the beginning teacher (pp. 193-205). New York: Pergamon.
Merriam, S. (1988). Case study research in education: A qualitative
approach. San Francisco: Jossey-Bass.
Minick, N. (1996). The development of Vygotsky’s thought. In H. Daniels
(Ed.), An introduction to Vygotsky (pp. 28-52). London: Routledge.
National Center for Research on Teacher Education. (1988). Teacher
education and learning to teach: A research agenda. Journal of Teacher
Education, 39(6), 27-32.
National Council of Teachers of Mathematics. (1989). Curriculum and
evaluation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1990). Professional
standards for teaching mathematics. Reston, VA: Author.
Noddings, N. (1990). Constructivism in mathematics education. In R. B.
Davis, C. A. Maher, & N. Noddings (Eds.), Constructivist views on the teaching
and learning of mathematics. Journal for Research in Mathematics Education
Monograph (pp. 7-18). Reston, VA: National Council of Teachers of
Mathematics.
Oerter, R. (1992). The zone of proximal development for learning and
teaching. In F. Oser, A. Dick, & J. Patry (Eds.), Effective and responsible
teaching: The new synthesis (pp. 187-202). San Francisco: Jossey-Bass.
Patton, M. (1990). Qualitative evaluation and research methods.
Newbury Park, California: Sage.
Peressini, D., & Knuth, E. (in press). Why are you talking when you
could be listening? The role of discourse and reflection in the professional
development of a secondary mathematics teacher. Teaching and Teacher
Education.
Potter, J., & Wetherell, M. (1987). Discourse and social psychology.
London: Sage.
Richardson-Koehler, V. (1988). Barriers to the effective supervision of
student teaching: A field study. Journal of Teacher Education 39(2), 28-34.
Rogoff, B. (1990). Apprenticeship in thinking: Cognitive development in
social context. New York: Oxford University.
Romberg, T. A. (1992). Perspectives on scholarship and research
methods. In D. A. Grouws (Ed.), Handbook of research on mathematics
teaching and learning. New York: Macmillan.
Saxe, G. B. (1992). Studying children’s learning in context: Problems and
prospects. The Journal of the Learning Sciences, 2(2), 215-234.
Schaffer, H. R. (1996). Joint involvement episodes as context for
development. In H. Daniels (Ed.), An introduction to Vygotsky (pp. 251-280).
London: Routledge.
Schram, P., Wilcox, S. K., Lappan, G., & Lanier, P. (1989). Changing
prospective teachers’ beliefs about mathematics education. In C. Maher, G.
Goldin, & R. Davis (Eds.), Proceedings of the Eleventh Annual Meeting of the
North American Chapter of the International Group for the Psychology of
Mathematics Education (pp. 296-302). New Brunswick, NJ: Rutgers University.
Shotter, J. (1995). In dialogue: Social constructionism and radical
constructivism. In L. Steffe & J. Gale (Eds.), Constructivism in education (pp. 4156). Hillsdale, NJ: Lawrence Erlbaum.
Shulman, L. (1986). Those who understand: Knowledge growth in
teaching. Educational Researcher, 15, 4-14.
Shulman, L. (1987). Knowledge and teaching: Foundations of the new
reform. Harvard Educational Review, 57(1), 1-22.
Shulman, L. (1992). Research on teaching: A historical and personal
perspective. In F. K. Oser, A. Dick, & J. Patry (Eds.), Effective and responsible
teaching: The new synthesis (pp. 14-29). San Francisco: Jossey-Bass.
Simon, M. (1997). Developing new models of mathematics teaching: An
imperative for research on mathematics teacher development. In E. Fennema &
B. Nelson, (Eds.), Mathematics teachers in transition (pp. 55-86). Mahwah, NJ:
Lawrence Erlbaum.
Smith, M. (1987). Publishing qualitative research. American Educational
Research Journal, 24(2), 173-183.
Steffe, L. (1983). The teaching experiment methodology in a
constructivist research program. In M. Zweng, et al. (Eds.), Proceedings of the
Fourth International Congress on Mathematical Education (pp. 469-471).
Boston: Birkhauser.
Steffe, L. (1991). The constructivist teaching experiment: Illustrations and
implications. In E. von Glasersfeld (Ed.), Radical constructivism in mathematics
education (pp. 177-194). Dordrecht, The Netherlands: Kluwer.
Steffe L., & Tzur, R. (1994). Interaction and children’s mathematics. In P.
Ernest (Ed.), Constructing mathematical knowledge: Epistemology and
mathematical education (pp. 8-32). London: Falmer.
Taylor, L. (1993). Vygotskian influences in mathematics education, with
particular reference to attitude development. Focus on Learning Problems in
Mathematics, 15(2-3), 3-17.
Thompson, P. (1994). The development of the concept of speed and its
relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The
development of mathematical reasoning in the learning of mathematics (pp.
179-234). Albany, NY: State University of New York.
Thompson, P., & Thompson, A. (1994). Talking about rates conceptually,
part I: A teacher’s struggle. Journal for Research in Mathematics Education,
25(3), 273-303.
Underwood-Gregg, D. U. (1995). Gender-related differences in
interaction patterns in elementary school Inquiry Mathematics classrooms.
Dissertation Abstracts International, 57, 07A. (University Microfilms No.
AAI9638247).
Voigt, J. (1995). Thematic patterns of interaction and sociomathematical
norms. In P. Cobb & H. Bauersfeld (Eds.), The emergence of mathematical
meaning: Interaction in classroom cultures (pp. 163-201). Hillsdale, NJ:
Lawrence Erlbaum.
von Glasersfeld, E. (1987). Learning as a constructive activity. In C.
Janvier (Ed.), Problems of representation in the teaching and learning of
mathematics. Hillsdale, NJ: Lawrence Erlbaum.
Vygotsky, L. (1978). Mind in society. (M. Cole, S. Scribner, V. JohnSteiner, & E. Souberman, Trans.). Cambridge, MA: Harvard University.
(Original work published in 1934.).
Vygotsky, L. (1986). Thought and language. (A. Kozulin, Trans.).
Cambridge, MA: Massachusetts Institute of Technology. (Original work
published in 1934.).
Wertsch, J. (1988). L. S. Vygotsky’s “new” theory of mind. The American
Scholar, 57, 81-89.
Wertsch, J., & Toma, C. (1995). Discourse and learning in the classroom:
A sociocultural approach. In L. Steffe & J. Gale (Eds.), Constructivism in
education (pp. 159-174). Hillsdale, NJ: Lawrence Erlbaum.
Wertsch, J., & Toma, C. (1995). Discourse and learning in the classroom:
A sociocultural approach. In L. Steffe & J. Gale (Eds.), Constructivism in
education (pp. 159-174). Hillsdale, NJ: Lawrence Erlbaum.
Wertsch, J., & Tulviste, P. (1996). L. S. Vygotsky and contemporary
developmental psychology. In H. Daniels (Ed.), An introduction to Vygotsky
(pp. 53-74). London: Routledge.
Wilson, B., Teslow, J., & Taylor, L. (1993). Instructional design
perspectives on mathematics education with reference to Vygotsky’s theory of
social cognition. Focus on Learning Problems in Mathematics, 15(2-3), 65-86.
Wood, T. (1995). An emerging practice of teaching. In P. Cobb & H.
Bauersfeld (Eds.), The emergence of mathematical meaning: Interaction in
classroom cultures (pp. 203-227). Hillsdale, NJ: Lawrence Erlbaum.
Wood, T., Cobb, P., & Yackel, E. (1991). Change in teaching mathematics:
A case study. American Educational Research Journal, 28(3), 587-616.
Zeichner, K. (1993). Designing educative practicum experiences for
prospective teachers. Paper presented at the International Conference on
Teacher Education: From Practice to Theory, Tel-Aviv, Israel.
Zimpher, N., deVoss, G., & Nott, D. (1980). A closer look at university
student teacher supervision. Journal of Teacher Education 31(4), 11-15.
APPENDIX
COOPERATING TEACHER ASSESSMENT
OF 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 postconference. 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