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MIDDLE SCHOOL MATHEMATICS STUDENTS’
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WHOLE-CLASS DISCUSSIONS: THEIR BELIEFS,
GOALS, AND INVOLVEMENT

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MIDDLE SCHOOL MATHEMATICS STUDENTS’ MOTIVATIONS
FOR PARTICIPATING 1N WHOLE-CLASS DISCUSSIONS:
THEIR BELIEFS, GOALS, AND INVOLVEMENT
By

Amanda Jansen Hoffmann

A DISSERTATION
Submitted to
Michigan State University

in partial fulfilhnent of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Counseling, Educational Psychology, and Special Education

2004

ABSTRACT
MIDDLE SCHOOL MATHEMATICS STUDENTS’ MOTIVATIONS
FOR PARTICIPATING IN WHOLE-CLASS DISCUSSIONS:
THEIR BELIEFS, GOALS, AND INVOLVEMENT
By

Amanda Jansen Hoffrnann

Whole-class discussions in mathematics classrooms are considered to foster
active sense-making and intellectual autonomy among students. Through participating in
these discussions, students have the opportunity to develop skills of mathematical
communication, reasoning, and justification. However, middle school students may resist
participating in whole-class discussions if they perceive social consequences resulting
from this activity.

Research on mathematics classroom discourse typically focuses on the role of the
teacher in discourse, examining student variables as outcomes to measure the
effectiveness of the teachers’ strategies. Alternatively, in this study, students’ beliefs and
goals are examined for how they influence students’ participation in classroom discourse
rather than as outcomes.

I assessed beliefs and goals of 15 target students from two seventh grade
mathematics classrooms through one-on-one interviews and a Likert-scale survey
instrument. Students’ talk in interviews was analyzed through the use of a framework that
included imperative verbs to capture idealized states, repetition to capture emphasis, and

connections to affect to capture relative importance to the student. This framework

allowed for a more rigorous analysis of students’ beliefs in contrast to reporting any and
all of their responses to interview questions.

Students’ involvement in classroom discourse was described based on an analyses
of videotaped classroom discussions about four investigation problems from the
Connected Mathematics Project Standards-based mathematics curriculum.

Results from this study indicate that students’ involvement in classroom
discussions is influenced by their social goals and epistemological beliefs.

Students who believed they learned mathematics through a process of negotiation and
associated a low level of risk with participating in discussion were more likely to extend
their participation during an interaction, critique the thinking of their classmates, and talk
about mathematics at a high level of explicit meaning. There were also differences in
students’ involvement between the target students based on their classrooms.

This study illustrates how adolescence intersects with the mathematics reform
movement by taking into account students’ perspectives. Future research investigating
how beliefs and goals relate to students’ involvement in discussions may explain how a
classroom of students together supports the development of effective classroom

discussions.

COPyright by
Amanda Jansen Hoffmann
2004

ACKNOWLEDGEMENTS

I would like to express my appreciation for those who played a role in supporting
the work of my dissertation.

I would first like to express my gratitude to my dissertation chair and academic
advisor, Dr. Jack Smith. Since the beginning of my academic career at Michigan State,
he has been helpful, concerned, and thoughtful. I am particularly grateful for his careful
reading of multiple editions of each of these chapters, even while away on sabbatical
overseas. I consider myself fortunate to have Jack as my mentor.

Additionally, my committee members, Dr. Ralph Putnam, Dr. Sandra Crespo, and
Dr. Ellen Altennatt each played an important role in my work. Ralph and I had many
engaging discussions about analyzing the classroom discourse, in no small part supported
by his knowledge of studying classroom processes. Sandra consistently helped me feel
like a professional researcher when I was feeling small, and she saw possibilities in my
ideas even when they were first developing. Ellen’s analytic perspective on issues such
as units of analysis helped me see this study in new ways. They all asked challenging
and helpful questions throughout my journey. For each of them, I am very thankful.

Two funding sources — the MSU / Spencer Research Training Grant and the
Graduate School’s Dissertation Completion Fellowship — provided me with the
opportunity to develop and execute a complex study.

I am honored to have shared in the classroom experiences of the teachers and
students of these two classrooms in the 2002-03 school year. They have taught me much

more than what I was able to capture in these pages.

I have been blessed with a supportive group of friends and family. My mom, dad,
and sister have encouraged me to pursue my dreams and have been my biggest fans.
Thank you for being so proud of me. Many of my friends during graduate school have
helped me maintain my sanity and thoughtfulness during the past two years, and for that I
am grateful. I am especially appreciative of Jane Pizzolato’s continual encouragement
and for her critical eyes as my best first reader.

My husband, Eric, has been unconditionally loving and supportive throughout
graduate school. He gave me space when I needed it, reminded me to stop working when
I needed balance, made me laugh when I took myself too seriously, and believed in me
when I did not believe in myself. Words cannot thank him enough for being such a

wonderful partner.

Vi

TABLE OF CONTENTS

LIST OF TABLES ........................................................................................................... xi
LIST OF FIGURES ......................................................................................................... xii
CHAPTER 1
INTRODUCTION ........................................................................................................... 1
Theoretical Perspective ........................................................................................ 4
Students’ Motivations: Beliefs and Goals ............................................................ 6
Defining Beliefs ....................................................................................... 8
Range of Students’ Goals ......................................................................... 10
Students’ Involvement in Whole-Class Discussion ............................................. 11
Relations Between Beliefs, Goals, and Involvement in Whole-Class Discussion
.............................................................................................................................. 12
Author’s Brief History Toward This Dissertation ............................................... 14
Students’ Beliefs and Goals ..................................................................... 14
Students’ Involvement in Whole-Class Discussions ............................... 16
Goals of This Study ............................................................................................. 18
CHAPTER 2
LITERATURE REVIEW ................................................................................................ 20
Students’ Beliefs and Goals in Mathematics Classrooms .................................... 23
Students’ Beliefs about Learning Mathematics ....................................... 23
Beliefs Moderate the Process of Learning Mathematics ............. 24
Assessing Classroom Interventions: Developing Productive Beliefs
...................................................................................................... 27
Development of Students’ Beliefs About Learning Mathematics
...................................................................................................... 31
Categories of Mathematics-Related Beliefs ................................. 34
Students’ Goals in Mathematics Classrooms ........................................... 38
Learning Goals ............................................................................. 39
School-Related Goals: Beyond Learning Goals .......................... 40
Social Goals ................................................................................. 42
Patterns Among Beliefs, Goals, and Academic Risk ............................... 42
Discourse in Mathematics Classrooms ................................................................ 45
Describing Mathematics Classroom Discourse ....................................... 48
Form and Function ....................................................................... 49
Content ......................................................................................... 51
Best Practices: Mathematics Classroom Discourse ................................. 52
Quality of Mathematical Talk as Outcome .................................. 52
High Involvement as Outcome .................................................... 53
Students’ Motivation as Outcome ................................................ 54
Student Achievement as Outcome ............................................... 55
Student Characteristics and Involvement in Mathematics Classroom
Discourse .................................................................................................. 56

vii

Students’ Personality Traits and Dispositions ............................. 56

Socio-Economic Status and Race ................................................ 58
Gender. ......................................................................................... 59
CHAPTER 3
METHODS ...................................................................................................................... 61
Research Questions .............................................................................................. 62
Setting: School, Curriculum, and Classrooms ..................................................... 63
Standards-Based Mathematics Curriculum .............................................. 64
Mrs. Evans’ Fifth Hour Class .................................................................. 66
Ms. Carson’s Third Hour Class ............................................................... 67
Fidelity to CMP ........................................................................................ 68
Participants: Target Students ............................................................................... 68
Data Collection: Students’ Beliefs and Goals ...................................................... 71
Interviews ................................................................................................. 71
Survey Instrument .................................................................................... 73
Data Analysis: Students’ Beliefs and Goals ........................................................ 75
Analytic Framework: Analysis of Students’ Interviews .......................... 75
Survey Analyses ....................................................................................... 80
Data Collection: Students’ Involvement in Whole-Class Discussion .................. 81
Data Analysis: Students’ Involvement in Whole-Class Discussion .................... 84
Data Analysis: Relations Between Students’ Beliefs and Goals and Their
Involvement ......................................................................................................... 86
My Role in the Research Process ......................................................................... 86
Results .................................................................................................................. 90
CHAPTER 4
RESULTS: STUDENTS’ BELIEFS AND GOALS IN DISCUSSION-ORIENTED
MATHEMATICS CLASSROOMS ................................................................................. 91
Students’ Talk During Interviews: Beliefs and Goals ......................................... 92
Perceptions of Social Risk ....................................................................... 96
Epistemological Beliefs ........................................................................... 102
Relation Between Epistemological Beliefs and Level of Social Risk ..... 108
Academic and Social Goals ..................................................................... 110
Four Clusters of Beliefs and Goals: Variations Between and Within ..... 119
Diversity Within Clusters: Third Quarter Performance ........................... 127
Survey Analyses ................................................................................................... 128
Relations Between Beliefs ....................................................................... 129
Differences Between Classrooms ............................................................ 131
Target Students Relative to Their Classmates ......................................... 132
Summary .............................................................................................................. 134
CHAPTER 5
RESULTS: THE NATURE OF WHOLE-CLASS DISCUSSIONS AND TARGET
STUDENTS’ PATTERNS OF INVOLVEMENT .......................................................... 137
The Nature of Whole-Class Discussions in Two Mathematics Classrooms ........ 138

viii

Similarities ............................................................................................... 138

Commitment to Textbook Series ................................................. 138
Instructional Model ...................................................................... 139
Pursuit of Multiple Solution Methods .......................................... 141
Differences ............................................................................................... 144
Amount of Time Spent on Typical Activities .............................. 146
Contrasting the Nature of Classroom Talk .................................. 152
Target Students’ Patterns of Involvement ............................................................ 163
Level of Involvement ............................................................................... 165
Proportion of Interaction Segments ............................................. 167
Extended Interaction Segments .................................................... 168
Off-Topic Talk ............................................................................. 171
Nature of Involvement ............................................................................. 173
Hesitancy and Assertiveness ........................................................ 172
Positioning ................................................................................... 179
Students’ Mathematical Reasoning .............................................. 182
Statistical Analyses ...................................................................... 187
Summary: Nature of Whole-Class Discussion & Target Students’
Involvement ................................................................................. 188
CHAPTER 6
RESULTS: RELATIONS BETWEEN STUDENTS’ BELIEFS, GOALS, AND THEIR
INVOLVEMENT IN WHOLE-CLASS DISCUSSIONS ............................................... 192
Epistemological Beliefs, Social Risk, and Involvement ...................................... 194
Academic and Social Goals and Involvement ..................................................... 201
Complete Task ......................................................................................... 201
Appear Competent ................................................................................... 201
Gain Status ............................................................................................... 201
Behave ...................................................................................................... 202
Help Classmates ....................................................................................... 202
Clusters and Involvement .................................................................................... 203
Summary .............................................................................................................. 205
CHAPTER 7
DISCUSSION .................................................................................................................. 207
Review and Interpretations of Results ................................................................. 208
Range of Students’ Beliefs and Goals ..................................................... 209
Students’ Beliefs and Goals Influencing Their Involvement ................... 218
Teachers’ Discourse Practices Influencing Students’ Involvement ......... 222
Contributions From this Study ............................................................................. 224
Adolescent Development Intersecting with Mathematics Reform .......... 225
Revisiting Mathematics Anxiety .............................................................. 228
Revisiting Learning Goals ....................................................................... 229
Revisiting Risk in Academic Work ......................................................... 231
Methods: Analytical Framework for Analysis of Students’ Beliefs ........ 232
Limitations ........................................................................................................... 233

ix

Implications .......................................................................................................... 234
Teachers’ Knowledge of Students’ Beliefs and Goals and Their

Development ........................................................................................... 235

Classroom Interventions .......................................................................... 237

Future Research ................................................................................................... 239

Open Questions for the Field ................................................................... 239

Furthering My Own Research Agenda .................................................... 241

Summary .............................................................................................................. 245
Closing ................................................................................................................. 245
APPENDICES ................................................................................................................. 248
REFERENCES ................................................................................................................ 253

LIST OF TABLES

Table 3.1: Participant Information ................................................................................... 69
Table 3.2: Data Sources in relation to Research Questions ............................................. 71
Table 3.3: Survey Scales .................................................................................................. 74
Table 3.4: Analyzed Discussions ..................................................................................... 82
Table 4.1: Target Students in Each Belief Cluster ........................................................... 96
Table 4.2: Definitions for Perceptions of Social Risk ..................................................... 97
Table 4.3: Perceived Social Risk by Classroom .............................................................. 99
Table 4.4: Definitions of Epistemological Beliefs ........................................................... 103
Table 4.5: Epistemological Beliefs by Classroom ........................................................... 106
Table 4.6: Students’ Perceived Levels of Risk by Epistemological Beliefs .................... 109
Table 4.7: Definitions of Academic and Social Goals ..................................................... 110
Table 4.8: Academic and Social Goals of Target Students by Classroom ...................... 112
Table 4.9: Target Students’ Performance (Letter Grade in Mathematics) By Beliefs
(Epistemological Beliefs and Perception of Risk) ........................................................... 128
Table 4.10: Survey Results Across the Two Classrooms (N = 42), Scale Means, Standard
Deviations, and Reliabilities ............................................................................................ 129
Table 4.11: Intercorrelations Among Survey Scales ....................................................... 130
Table 4.12: Survey Results: Comparisons Between Classrooms .................................... 131
Table 4.13: Epistemological Belief by Classroom .......................................................... 132
Table 4.14: Survey results: Target Students in Relation to Their Respective Populationls33
Table 4.15: Survey Results: Target Students in Relation to their Class Scale Mean ...... 134
Table 5.1: Average Time Spent on Typical Activities during a Class Period ................. 148
Table 5.2: Average Time Spent on Typical Activities during a Lesson .......................... 150

xi

Table 5.3: Number of Interaction Segments per Day ...................................................... 150

Table 5.4: Time Spent (in minutes) on Whole-Class Discussion per Day ...................... 151
Table 5.5: Percentage of Off-Topic Interaction Segments .............................................. 153
Table 5.6: Mrs. Evans’s Class’s Student Initiated Interaction Segments ........................ 157
Table 5.7: Criteria for Students’ Levels of Involvement ................................................. 166

Table 5.8: Target Students’ Percentages of Participating Interaction Segments, Class Day

by Target Student ............................................................................................................. 167
Table 5.9: Number of Extended Interaction Segments for each Target Student ............. 171
Table 5.10: Classes by Target Students’ Levels of Involvement ..................................... 173
Table 5.11: Percentages of Hesitancy and Assertiveness by Target Student .................. 178
Table 5.12: Percentages of Interaction Segments Involving Positioning ........................ 182
Table 5.13: Definitions of Students’ Mathematical Reasoning ....................................... 183
Table 5.14: The Level of Mathematical Reasoning for Target Students ......................... 185
Table 5.15: Comparisons Between the Two Classrooms ................................................ 187

Table 6.1: Comparisons Between Belief Groups: Target Students’ Patterns of
Involvement, Version A ................................................................................................... 196

Table 6.2: Comparisons Between Belief Groups: Target Students’ Patterns of
Involvement, Version B ................................................................................................... 197

Table 6.3: Diversity Within Belief Groups, Patterns of Involvement by Target Students,
as Grouped by Beliefs ...................................................................................................... 199

Table 6.4: Students’ Belief Clusters, Individual Students by Belief Cluster and Classroom
.......................................................................................................................................... 203

Table 6.5: Differences in Involvement by Belief/Goal Clusters, Mean Percentages of
Involvement by Forms of Involvement and Cluster ........................................................ 204

Table 7.1: Epistemological Beliefs by Perception of Social Risk ................................... 211

xii

LIST OF FIGURES

Figure 4.1: Seventh Grade Students’ Motivations ........................................................... 94
Figure 5.1: Seventh Grade Students’ Involvement in Whole Class Discussions about
Mathematics ..................................................................................................................... 164
Figure 7.1: Interactions in a Mathematics Classroom ..................................................... 209

xiii

CHAPTER ONE

INTRODUCTION

Familiar images of middle school mathematics classrooms have not commonly
included whole-class discussions. A more typical image includes a teacher describing
procedures for students to learn and practice. The teacher delivers a body of knowledge
that students take in. Current reforms in mathematics education challenge this image,
suggesting alternative images of teaching and learning mathematics that promote, among
other things, Shifts in the roles of students and teachers (NCTM, 1989, 1991 , 2000).
These reforms suggest teachers should involve students in the process of socially (re-)
constructing mathematical knowledge. In this construction process, whole-class
discussions of problems and potential solutions play a prominent role.

Creating whole-class discussions that involve students in the local construction of
knowledge may be difficult in mathematics because of the need for students to acquire
knowledge of algorithms, understandings, and relationships established by the
mathematical community, and because of the expectations students may have about their
role as the receiver of knowledge in the classroom. Although these seem like logical
assumptions, there is more to learn about whether and how students experience these
difficulties. Insights from students in mathematics classrooms where whole-class
discussions consistently take place may illuminate opportunities that teachers have to
increase depth and breadth of student engagement in mathematics classroom discussions.

By whole-class discussions, I mean large group interactions during mathematics

class in which students present their solutions to a problem and alternative solutions are

contrasted to one another. In these discussions, students are invited to voice arguments
for or against the solutions. The National Council of Teachers of Mathematics [NCTM]
advocates discussion in their Standards documents (NCTM, 1989, 1991, 2000). They
encourage mathematics teachers to create‘environments for students to communicate with
one another, listen carefully, and critique each others’ ideas, in order to promote
autonomous thinking among students and to provide students with opportunities to learn
how to justify their thinking mathematically. However, the Principles and Standards of
School Mathematics (NCTM, 2000) also mentions challenges involved with holding such
discussions in the middle school setting:

For some students, participation in class discussions is a challenge. For example,

some students in the middle grades are often reluctant to stand out in any way

during group interactions. (NCTM, 2000, p. 61)
Students’ reluctance may be due to their perceived social repercussions of participation.
In other words, if a student participates and is incorrect, he may fear that their classmates
will pass negative judgment upon him. If a student participates and is either correct or
thoughtfirl about mathematics, she may be concerned that their peers will think she is
altogether too dedicated to school. During adolescence, the development of
metacogrrition leads to a greater awareness of what others might think, and a heightened
sensitivity to the possibility of being judged. Discussion-rich environments may pose
challenges not only for students who are reluctant to participate, but also for those who
lack the linguistic and social competencies to engage firlly in the classroom discourse.
Although the risks associated with publicly sharing one’s thinking may be a hindrance for

early adolescents’ participation in classroom discussions, it is possible that sharing

strategies for solving problems in mathematics class may benefit the middle school
student and meet some of their needs, overriding the associated risks.

As teachers engage students in talking about mathematics, they may assume that
their students are familiar with the ways classrooms function, yet lack awareness of the
complexity of students’ roles in the construction of the classroom micro-culture and the
larger school culture (Sarason, 1982). Teachers may not know how the history of their
students’ relationships with one another over preceding years, or even from the cafeteria
at lunchtime that day, influences their social interactions about the subject matter during
class. Research characterizing students’ perspectives on the social dimension of the
classroom, among other issues, can provide insight into the development of classroom
micro-cultures. Lubienski (2000a, 2000b), for example, has shown that working class
middle school students prefer more direction from the teacher than their middle class
peers in discussion-intensive mathematics classrooms.

In this study, I examined two seventh-grade classrooms where whole-class
discussion took place on a regular basis. In these classrooms, I examined and analyzed
the range of students’ perspectives, in terms of their beliefs and goals, and whether and
how these beliefs and goals related to their involvement in whole-class discussions.
Students’ involvement in whole-class discussion may relate to their beliefs and goals in
ways that help teachers and mathematics educators gain an understanding of why some
students get involved in discussion and others do not, or why some students adopt
productive ways of talking during class discussion and others do not. For example,
students who have beliefs about the nature of mathematics such that they view the

domain as flexible, pursuing multiple solution paths to a problem, may be more likely to

participate in an extended discussion around alternative methods for solving a problem.
Students without this belief may resist getting involved in such a discussion. The primary
goal of this study is to capture how junior high students experience Standards-based
mathematics settings, in terms of their involvement in whole-class discussions, their
beliefs and goals in these settings, and how their actions and beliefs interrelate.

In this chapter, I first describe the theoretical perspective orienting this study.
Then I define the constructs in this study — students’ beliefs, goals, and their involvement
in whole-class discussions — and describe my orientation to studying relations between
them. Next, I share some of my personal history to illustrate my own journey toward this
study. Finally, I describe the goals of this study and the structure of this dissertation.
Theoretical Perspective

This project is designed to situate research on students’ motivation in the activity
of whole-class discussion and the subject matter of mathematics. Mathematics educators
have called for analyses of students’ motivation that take into account subject matter and
issues related to the mathematics reform movement (e. g., Middleton & Spanias, 1999).
Analyzing beliefs and goals in light of a specific activity and subject matter allows for the
potential of developing new frameworks of beliefs and goals to study with respect to
students’ experiences as learners. When the study of motivation is situated in a context,
the beliefs and goals students express may be more specific to the activity and subject
matter, and less general in comparison to constructs such as learning goals (Ames, 1992;
Dweck, 1986).

Following symbolic interactionism (Blumer, 1969, as cited in Cobb, Yackel, &

Wood, 1993), in this study, I assume that beliefs and goals have a reflexive relationship

with classroom discourse practices. On the one hand, beliefs shape students’ involvement
and participation. On the other hand, beliefs change as discourse practices change. For
students to participate in a particular way, such as critique the solution of a classmate,
their beliefs and goals would need to support this behavior. They may hold the belief that
a student has the authority to evaluate, or the goal to help a classmate. If students are
encouraged to change how they participate, their beliefs and goals may shift after trying
on new behavior. However, students’ beliefs and goals could relate to their discourse
practices in a range of ways.

We therefore conjectured that students develop specifically mathematical

beliefs and values that enable them to act as increasingly autonomous

members of the classroom mathematical community as they participate in

the negotiation of sociomathematical norms. .. Once again, this conjecture

is open to empirical investigation. (Cobb, Stephan, McClain, &

Gravemeijer, 2001, p. 124)

Empirical investigation of how students’ beliefs and goals are related to their
involvement in classroom discourse practices would allow for verifying, enriching, or
problematizing current theories in mathematics education about relations between
students’ beliefs, goals, and discourse practices.

Mathematics education researchers have begun to examine links between
students’ beliefs, goals, and their classroom discourse practices, but coordinating
analyses of individual students’ perspectives with analyses of the social setting of the
classroom is a complex endeavor. While some of this work has been taken up (e. g,

Bowers & Nickerson, 2001; Cobb et al., 2001; Lo, Wheatley, & Smith, 1994; Stephan,

Cobb, & Gravemeijer, 2003), these studies have placed more emphasis on the analysis of
the classroom practices, complementing these analyses with case studies of small
numbers of students (between two and five). In this study, I coordinated analyses of
individual students’ perspectives with the analysis of social settings by shifting the
emphasis onto individual students and including more cases (N = 15) of students’
involvement, beliefs, and goals, with complementary analyses of classroom practices.

My emphasis on individual cases of students focuses more on a cognitive analysis
of a plurality of individuals than on the activity of a collective (Stephan et al., 2003). A
primary focus on individual students allows for mapping out the diversity of students’
perspectives in their social context. An examination of a wide range of students’ views is
limited with an analysis of only two to five students. Another difference between this
study and others that have attempted to coordinate analyses of the individual and social
setting is that other studies have focused on ways of thinking and talking about
mathematics specifically, rather than my focus on beliefs and goals related to students’
roles as learners. The practices of each local classroom community are also described
briefly in this study, in contrast to studies that emphasize descriptions of classroom
practices over individual perspectives.
Students’ Motivations: Beliefs and Goals

Research on students’ experiences in Standards-based mathematics settings have
tended to focus on student achievement (e. g., Huntley, Rasmussen, Villarubi, Sangtong,
& Fey, 2000; Reys, Reys, Lapan, & Holliday, 2003; Riordan & Noyce, 2001; Senk &
Thompson, 2002). Notable exceptions include studies of students’ attitudes, beliefs, and

achievement and how they change in the context of new curricula or pedagogical

approaches (e. g., Wood & Sellers, 1997). There exists a variety of methods for exploring
students’ experience of Standards-based curricula beyond achievement, including
capturing their voices and telling their stories (e. g., Boaler, 1998; Holt et al., 2001) or
interpreting their dispositions, beliefs, attitudes, or motivational goals from self-reported
survey data or interview data. The field of mathematics education has more to understand
about the students’ experiences with Standards-based curricula, since these curricula are
relatively new. There may be more diversity existing among students’ perspectives than
is currently known, as each student may interpret the same activity in a classroom from
their own unique viewpoints.
...each individual person in the classroom creates his own unique construction of
the rest of the participants, of their goals, of the interactions between herself and
the others and of all the events, tasks, mathematical contents which occur in the
classroom. (Bishop, 1985, p. 26)
Examining how classrooms that use Standards-based curricula are seen and felt by
students is important work, as students’ experiences have not received sufficient attention
from educators (Erickson & Shultz, 1992).
One goal of the NCTM Standards and the National Research Council (Kilpatrick,
Swafford, & F indell, 2001) is to promote “productive” dispositions among students.
...the tendency to see sense in mathematics, to perceive it as both useful and
worthwhile, to believe that steady effort in learning mathematics pays off, and to
see oneself as an effective learner and doer of mathematics.(Kilpatrick et al.,

2001,p.13l)

Helping students develop particular dispositions would support students’ motivations for
engaging in learning or doing mathematics. We do not know, however, which particular
dispositions support students’ involvement in class discussions during mathematics class.

From an educational psychologist’s perspective, people’s motivation to engage in
particular behaviors may be explained through expectancy-value models (Eccles &
Wigfield, 2002). Expectancies include beliefs about one’s capabilities, addressing the
question, “Can I do this?” Additionally, expectancies may include epistemological beliefs
about the nature of knowledge or subject matter. Since implementing whole-class
discussion in mathematics classrooms can promote shifts in expected roles of teacher and
students, this study focuses on students’ epistemological beliefs with respect to the
process of learning mathematics: the process of learning and students’ and teachers’
roles. Values may include reasons for doing the activity, addressing the question, “Why
do I want to do this?” Studying students’ beliefs and goals in mathematics classrooms
implementing whole-class discussion could provide insight into their dispositions and
motivations for involvement in Standards-based settings.

Defining beliefs. Beliefs are considered to be an ill-defined construct
(Schoenfeld, 1992; Thompson, 1992). One struggle with this construct involves parsing
the definition of beliefs as exclusive fi'om other phenomenon, such as knowledge,
attitudes, or values. While it might be tempting to assume that beliefs is a term for which
there exists a shared understanding, Furinghetti and Pehkonen (2001, 2002) have shown
that researchers’ definitions are not consistent, one to another.

For the purpose of this study, beliefs are defined as students’ personal knowledge

(Polanyi, 1958) about the process of learning mathematics that serve as the “assumptions

from which individuals make decisions about the actions they will undertake”
(Kloosterman, Raymond, & Emenaker, 1996, p. 39). This personal knowledge is
generally not subject to objective scrutiny and/or justification, so there is little sense in
asking if or evaluating whether beliefs are “right” or Wrong.” Beliefs are an important
element in students’ learning, as they are a part of their schema through which they make
sense out of the world.

...the beliefs students construct, the overall goals they establish, and the contexts

in which they do mathematics are their attempts to find a Viable way of operating

in the classroom. (Cobb, 1986, p. 8)

Students’ attempts to make sense out of the learning process are cognitive interpretations,
dependent upon what an individual knows (knowledge) or assumes to be true (beliefs),
and this sense-making activity is provoked in the social setting of the classroom. Beliefs
may also have an emotional response tied to them (Mandler, 1989).

Some researchers refer to beliefs as “generally stable” (McLeod, 1992), while
others (cf., diSessa, Elby, & Hammer, 2002; Furinghetti & Pehkonen, 2002; Pehkonen &
Furinghetti, 2001) describe beliefs as more contextually dependent. This study examines
students’ beliefs during a relatively short period of time, and I suggest that there is
substantial stability of these students’ beliefs during the time period of the study, due to
the short time period and the level of commitment the students appear to express.1 Beliefs
may be held at varying degrees of commitment. However, for the purpose of this study, I
examined beliefs for which students express at a strong level of commitment, as these

beliefs may also be those that they act upon.

 

' Whether the same beliefs are stable over much longer periods of time is not addressed in the study and
therefore an open question.

Range of students’ goals. I operationally define students’ goals as the specific
reasons they cite for their actions, which are more specific than the broader psychological
assumptions through which students interpret their world (beliefs). Goals are the desired
end-states that orient students’ actions—what actions are intended to achieve. Students
may have multiple goals for participating in whole-class discussions during mathematics
class, not all of which may appear to be obviously related to their beliefs about learning
mathematics, but can directly affect how students choose to participate when talking
about mathematics. Some potential goals students might act on include: (1) learning
goals, such as whether students are motivated by a need to perform well or develop a
deep understanding (Ames, 1992; Dweck, 1986); (2) social goals, e. g., whether students
are motivated by gaining the attention of a peer or looking good in front of others and
presenting one’s self in a particular way (Goffrnan, 1959); (3) efficiency goals, such as
finishing the problems as soon as possible so as not to have homework; and (4) academic
but non-mathematical goals, as students may have a goal of focusing on their
mathematics, but a stronger goal toward something else, such as memorizing their
vocabulary for another class. Students holding these goals may also hold goals more
closely related to developing a deeper understanding of mathematical content.

Students’ beliefs and goals may exist in clusters, supporting one another to create
motivational paths (Pintrich, 2003) for students’ involvement in whole-class discussions.
These motivational paths may be the set of beliefs that provide the psychological context
for decision-making and the goals providing an internal stimulus for action. Additionally,
multiple beliefs and goals held in isolation may set up competing priorities for the

student.

10

Students’ Involvement in Whole-Class Discussion

Students may act on their beliefs and goals during their involvement in whole-
class discussion. Reforms in mathematics education, both national and international, call
for improving students’ proficiencies in mathematical communication. In the United
States, the National Council of Teachers of Mathematics has addressed the significance
of discourse in mathematics classrooms (NCTM, 1991):

Discourse refers to the ways of representing, thinking, talking, agreeing, and

disagreeing that teachers and students use to engage... The discourse embeds

fundamental values about knowledge and authority. Its nature is reflected in what
makes an answer right and what counts as legitimate mathematical activity,

argument, and thinking. (p. 20)

Through their involvement in mathematics classroom discourse, students attempt to
communicate about mathematics, which makes their thought process more public,
allowing teachers to assess their students informally and allowing students to practice
reasoning about mathematics. Making one’s thinking public is a practice that some
mathematics teachers are attempting to engage in with their students (e.g., Larnpert,
2001). Since discussions are a fairly new practice in mathematics classrooms, recent
research has examined them.

In the mathematics education research literature, researchers have attempted to
document what these whole-class discussions should or could look like in classrooms
(e. g, Chazan, 2000; Heaton, 2000; Larnpert, 2001). Julianne Turner and colleagues
(Turner et al., 1998) determined that upper-elementary mathematics classroom teachers

who fostered hi gh-involvement classrooms scaffolded instruction through techniques

11

such as transferring responsibility. Knuth and Peressini ( 2001) contrasted the differences
between univocal and dialogic discourse (following Wertsch, 1991) - an attempt to
convey a specific message versus more two-way, give-and-take communication,
respectively —— in their writing for mathematics classroom teachers.

In this study, rather than focusing on how teachers can foster productive whole-
class discussions about mathematics, I examine students’ contributions during these
discussions, along with an analysis of their beliefs and goals. This analysis of students’
contributions during whole-class discussion allows for an investigation of what students
do in Standards-based mathematics classrooms, providing another look at their
experiences with the implementation of standards-based mathematics curricula. My
analysis of the range of students’ contributions during whole-class discussion included an
examination of their levels of reasoning about mathematics as they communicate in this
curricular setting (of, Stein & Lane, 1996). Additionally, I analyzed students’
contributions in terms of indications that they are taking on an active, autonomous role in
their classroom, such as a willingness to assert themselves or to critique their classmates’
solutions (Walen, 1994).

Relations Between Beliefs, Goals, and Involvement in Whole-Class Discussion

Certainly, there is no reason to expect any simple relationship (or any set of
relationships) between students’ beliefs or goals and their participation in whole-class
discussions. On the one hand, classroom practices may moderate relations between
beliefs and actions, as classroom norms set up expectations for what constitutes
appropriate behavior, and beliefs are aligned due to the process of negotiating the norms

(Cobb et al., 1993). On the other hand, individual variation among students, such as

12

socio-economic status (Lubienski, 2000a, 2000b) or previous history as a student and
home influences (Ridlon, 2001), may moderate the relation between students’ beliefs and
their participation in mathematics classroom discussion. My students’ histories prior to
their seventh grade mathematics classroom may result in wide variation among their
beliefs and goals. Additionally, students who express a particular belief may involve
themselves in discussion in a variety of ways. For example, students who express the
belief that mathematics problems have more than one solution may participate by
offering an alternative solution, or they may listen carefully to the discussion to make
sure they do not miss hearing one of the several solution methods being discussed.
Attempts to study links between students’ beliefs and their actions (or their
teachers’) are common among researchers who adopt a cognitive perspective on learning.
In mathematics education, students’ beliefs have been studied in relation to their problem
solving behaviors (Cobb, 1985; Schoenfeld, 1985, 1988) and group work (Kloosterman et
al., 1996). Schoenfeld (1985, 1988) found that high school students who held a belief that
they should be able to solve mathematics problems quickly, in 12 minutes or less, also
exhibited a lack of persistence when working on challenging problems. Cobb (1985)
demonstrated that a first-grade student who had an ego-involvement learning goal
orientation also had an instrumental view of mathematical knowledge, while another
student with a task-involvement learning goal orientation had a more relational
understanding of mathematics. Kloosterrnan and colleagues (1996) determined that fourth
through sixth grade students developed preferences over time for group work in
classrooms where the teacher implemented group work, suggesting that students’ beliefs

about group work were related to teachers’ practices. This dissertation study focused on

13

examining relations between students’ beliefs and goals and their involvement in
classroom discussions rather than engagement in problem solving activity or group work.
Author’s Brief History Toward This Dissertation

My intellectual evolution toward this study began even before I started my
graduate studies. Some of the curiosities leading to this study began from my time as a
classroom teacher, and even my time as a mathematics student. Below, I briefly discuss
some of my personal history as it relates to the origin of this dissertation. My initial
motivation, which came out of my teaching experience, helps to explain how and why I
see the results of this sort of research as relating to classroom practice.

Students’ beliefs and goals. The origin of this dissertation lies in my curiosity
about students’ relationships with the school subject of mathematics. These curiosities
stemmed from my experiences as both a junior high public school teacher and a
mathematics education researcher.

I taught junior high mathematics for three years. I sometimes heard my students
express extreme reactions to mathematics and my class. They had reactions such as
looking forward to coming to class, expressing that it was their favorite class, as well as
resisting the activity by not doing homework or participating. Some students started off
disinterested in learning and left with a better attitude. Others challenged me for giving
them too much homework when they would have rather been playing sports afier school,
perhaps growing disinterested in mathematics. I became particularly interested in how a
teacher could help students overcome mathematics anxiety, as I seemed to experience a
few successes with some of my students, and perhaps helped to foster new cases of

mathematics aversion among other students. As a teacher, it was tempting to assume my

14

students would all develop the beliefs and goals I wanted them to hold, but every student
appeared to interpret their experience in my classroom differently. I would teach
similarly in different class periods, but each class period had a slightly different
atmosphere. Depending on the make-up of students in the classroom and the time of day,
some class periods felt alive and engaged, others felt easily distracted and scattered, and
others felt quieter and perhaps even dull at times. Individual variation between students’
experiences in mathematics classrooms has fascinated me fiom my time as a classroom
teacher.

As a researcher, early in graduate school I interviewed high school students about
their experiences moving out of Standards-based mathematics curricula and into a more
traditional mathematics setting (Smith & Berk, 2001; Smith, Herbel-Eisenmann, Jansen,
& Star, 2000; Smith et al., 2001). I did not expect consistent responses fi'om all of the
students, but I was taken aback at the wide range of reactions students had to the
curricular shifi. Some students were relieved they were in a setting where the teacher was
more prescriptive, since their performance on high school mathematics was recorded on
their transcripts, and direct instruction from the teacher helped these students understand
their teacher’s expectations. Other students missed the opportunity to voice their thinking
during class, and seemed to be annoyed the teacher was telling them so much rather than
letting the students try to do the mathematics, so they felt as if some intellectual power
was taken from them. Again, there was a great deal of diversity among the students’
perspectives, but as my role with students shifted from teacher to researcher, I was given

privileged access to more of these students’ thinking.

15

Central to both of these experiences was my interest in taking students’
perspectives seriously. I believe students have valuable insights into their learning
experiences. While I do not believe teachers should always act to satisfy students’ felt
needs or address their emotional reactions, gaining a sense of what students think about,
and mapping the range of students’ perspectives, could inform important instructional
thinking and decisions.

Students’ involvement in whole-class discussions. I have not had extensive
exposure to whole-class discussions during mathematics class prior to doing this study.
As a mathematics student, I did not experience learning through discussion. The
mathematics I studied in high school was structured by the Saxon textbook series (Saxon,
2003). In these classrooms, the teacher demonstrated a new skill each day, and in the
homework we practiced a few exercises of the new skills and reviewed all previously
taught skills. I was successful in mathematics in high school. As a mathematics major in
college, I chose to retake calculus in order to provide myself with a solid foundation in
the subject, even though I had already taken two years of calculus in high school and
earned excellent grades. At the university, I experienced reform-oriented mathematic
courses. I took Harvard Consortium calculus (Hughes-Hallett, Gleason, & McCallum,
1994) through multivariable calculus, and had a similar textbook series for differential
equations, but whole-class discussion did not appear to me as an important focus in these
courses. Instead, as a student, I noticed an increased emphasis on story problems, graphs,
and the use of graphing calculators, and during class I noticed that it seemed to take a
longer time and many chapters to get to what I thought were the “real” ideas in calculus,

such as the chain rule, as I recalled from my first pass through calculus in high school.

16

As a teacher of general mathematics and Algebra I at the junior high level, I was
introduced to teaching through the use of whole-class discussion through the district’s
mathematics specialist, who would come visit my classroom and model demonstration
lessons with my students. She would teach two of my three seventh grade class periods
about once a month. I would participate in the first one as a student, observe the second
time, and I would teach the lesson and she would observe me during the third class
period. When I did not have these model lessons to follow, I struggled to implement
whole-class discussions in my mathematics classroom. I had a hard time coming up with
good questions to pose to the students. I found it challenging to select mathematical tasks
that were rich enough to provoke thoughtful discussions. At times, I believed it was a
good idea to engage students in talking about mathematics, but, as a novice teacher, I
sought to maintain control of my classroom, and I viewed opening up the floor to their
ideas as a somewhat chaotic experience that I would rather avoid, especially with the
pressure I was feeling to raise my students’ test scores. It was the general sense among
my colleagues that it was more efficient to raise test scores through teaching mathematics
by direct instruction.

As a researcher, I have come to believe there is a great deal to understand about
students’ experiences in whole-class discussions during mathematics class. On the one
hand, I read reform documents and research articles suggesting how discussions could
help students become autonomous learners of mathematics. On the other hand, I recall
from teaching that what is helpful for one student can be threatening for another. I now
believe more empirical evidence is required for making sense out of the range of

mathematics students’ experiences with whole-class discussion. University teacher

17

educators and reform documents encourage teachers to foster discussions in their
mathematics classrooms, but not all whole-class discussions are created equal, nor do all
students experience these discussions about mathematics in the same way. I have read
many research reports in which students’ beliefs are “read off” of their discourse
practices in mathematics clasSrooms. I designed this study to examine what more we
might have to learn about students’ beliefs in discussion-oriented mathematics
classrooms fi‘om talking with students about their experiences as well as observing and
documenting their participation in them.
Goals of This Study

From these prior experiences and relevant research, I designed this study to
examine the following questions: How do students in early adolescence experience
classrooms that use Standards-based mathematics curricula? In particular, what motivates
these students to participate in whole-class discussion? My analysis has three primary
outcomes: (a) A framework for students’ beliefs and goals that describes the diversity of
students’ beliefs and goals emerging from work with Standards-based mathematics
curricula; (b) a description of the range of ways that these students involved themselves
in whole-class discussions, including their mathematical reasoning and communication
strategies that may reveal students’ sense of autonomy with respect to learning and doing
mathematics; and (c) an examination of the relations between students’ beliefs and goals
and their involvement in whole-class discussion.

This study also has a series of secondary goals. I explored whether students’
involvement was related to their classroom settings and local classroom practices. I

wanted to develop an analytical framework for assessing students’ beliefs based on their

18

talk in interviews. I wanted to design the study as a case of a coordinated analysis
between individual and social levels of learning and doing mathematics in school, with
the awareness that analyses at each level, individual and social, would present challenges.
I wanted the results of this study to allow me to revisit the conceptualizations of
motivational and socio-emotional constructs, such as epistemological beliefs and
mathematics anxiety, by studying them and similar constructs in Standards-based
mathematics settings.

The body of the dissertation is made up of six additional chapters. Chapter Two,
that follows, relates this study to previous research and unanswered questions in the fields
of mathematics education and educational psychology. In Chapter Three, 1 describe my
methods of data collection and analysis. While collecting the data was relatively
straightforward, scoring and analyzing the results, particularly those that concern how
beliefs and goals relate to involvement in whole-class discussion, was a more involved
process. Chapters Four through Six present the results. I describe the results of the
analyses of students’ beliefs and goals in Chapter Four - primarily from interview data
and secondarily from survey data. In Chapter Five, I present the results of the analyses of
students’ involvement in whole class discussion, preCeded by an introduction to the
discourse practices in each classroom, and ending with analyses of whether target
students participated differently depending on their classrooms. The results of the
analyses of relations between students’ beliefs and goals and their patterns of
involvement are described in Chapter Six. Finally, I discuss my interpretations of the

results, suggest future research, and present implications of the results in Chapter Seven.

19

CHAPTER TWO

LITERATURE REVIEW

This study was designed to examine seventh grade students’ beliefs and goals
about learning mathematics as related to their involvement in whole-class discussions.
The goals of the study lie at the intersection of at least five significant areas of
educational research: (a) the mathematics reform movement in the US, including the
development of NCTM Standards-based curricula and teaching practices; (b) adolescent
development; (c) beliefs about learning mathematics; ((1) students’ motivations, such as
their academic and social goals; and (e) discourse in mathematics classrooms.

To bound the task of reviewing the research literature relevant to this study, I
chose to discuss studies at the intersection of at least two of the five areas listed. Each of
the five tOpics listed above would merit its own literature review, so examining work at
the intersection of these topics approximates the boundaries of this study. In a few
instances I also included foundational pieces that were relevant to only one of the five
themes. An example of a foundational piece would be a frequently cited reference (e.g.,
Mehan’s (1979) book in which he describes the Initiate-Respond-Evaluate interaction
sequence in classroom discourse). Generally, I placed an emphasis on peer reviewed
empirical research articles. However, I additionally included some chapters from edited
books.

There are bodies of research that I could discuss in this chapter, but I have chosen
not to, at least in great detail. I could include a discussion of research on teachers’ beliefs

(e.g., Aguirre & Speer, 2000; Good, Grouws, & Mason, 1990; Stipek, Givvin, Salmon, &

20

MacGyvers, 2001; Thompson, 1984; Thompson, Phillip, Thompson, & Boyd, 1994),
because they could influence how teachers structure whole-class discussions, but I will
not, since this line of work has been pursued to some detail already.2 Likewise, I could
discuss the growing body of research assessing the impact of the NCTM Standards, such
as whether students studying mathematics using Standards-based curricula perform better
than, or at least equal to, students studying mathematics in traditional curricular settings
(e. g., Huntley, Rasmussen, Villarubi, Sangtong, & Fey, 2000; Reys, Reys, Lapan, &
Holliday, 2003; Riordan & Noyce, 2001; Senk & Thompson, 2002; Wood & Sellers,
1997), but I will not, since this study is not designed to be an evaluation of Standards-
based mathematics curricula. Additionally, I could discuss research that examines
students’ beliefs in terms of their diversity of conceptions about a particular concept in
mathematics (e. g., Szydlik, 2000; Thompson, 1994), but this study is designed to capture
the socio-emotional context of learning mathematics in a discussion-oriented setting
rather than students’ understandings of a mathematical concept.

This study focuses on individual variation in adolescent students’ beliefs and
goals related to learning and doing mathematics, and how these beliefs and goals relate to
their involvement in discussion-oriented mathematics classrooms. Rather than examine
what is normative, or most common, among the students in the two mathematics
classrooms that I visited, I wanted to map the diversity of students’ perspectives. This is
in contrast to researchers who choose a perspective on students such that they a set of
beliefs that correspond with patterns in classroom talk. The danger of focusing only on
the most common patterns across students is the possibility of minimizing some students

perspectives in the discussion of improving mathematics classrooms — when those

 

2 For literature reviews, see Pajares (1992) or Thompson (1992).

21

students’ experiences out against the grain of what is normative. If teachers and
researchers hope to promote mathematics learning among all students, it seems
appropriate to examine the range of students’ perspectives rather than only those that are
the most common.

One of the challenges with studying students’ perspectives is interpreting their
beliefs and goals through a deficit model, and discussing some beliefs and goals as more
productive or beneficial than others. So, another way of talking about normative, rather
than what is most common in a classroom, is relative to an external view of what is good
and productive. I strive to break away from this sense of normative as well and honor the
students’ perspectives as a worthy starting point rather than a potentially unproductive
one. Rather than critique the students’ perspectives, I would like to rest the critique on the
goals of the mathematics reform, allowing the students’ experiences in Standards-based
curricular settings provide a lens for revisiting NCTM’S recommendations for increased
public classroom discourse.

My purpose in this chapter is to review literature at the intersection of research on
Standards-based mathematics classrooms, adolescents’ learning in school settings,
students’ beliefs about leaming mathematics, students’ goals in mathematics classrooms,
and discourse in mathematics classrooms. In order to discuss this wide range of literature,
I will first review research literature relevant to students ' beliefs and goals, as it intersects
with either Standards-based mathematics settings or adolescents’ learning. Then, I will
review research literature relevant to the study of classroom discourse, as it intersects
with teaching and learning in Standards-based mathematics settings, adolescence, or

students’ beliefs and goals.

22

Students’ Beliefs and Goals in Mathematics Classrooms

A psychological perspective on student learning involves an examination of
mental structures, such as beliefs and goals. Beliefs are students’ expectations and
assumptions about the process of learning and doing mathematics that provide a
psychological context for decision making and behaving in classrooms. Goals are what
students want to accomplish in their actions —- the states of affairs that actions are
designed to achieve.

Studying students’ beliefs involves an analysis of the assumptions and
expectations students hold and act upon related to the process of learning and doing
mathematics. Studying students’ goals differs fi'om studying students’ beliefs, as goals
are more specific objectives students’ hold in relation to particular activities rather than
learning and doing mathematics more generally. Beliefs provide a psychological
backdrop for learning, while goals orient actions more directly. Beliefs may be held at a
more unconscious level than goals, as students may not think much about what they
believe about mathematics, and as a result are not very aware of their own beliefs (Lester,
2002). They may not be cognizant that others could believe differently. Regardless of
their level of awareness, students hold multiple beliefs and goals at varying levels of
commitment, acting on different goals depending on the situation. I will first discuss
research on students’ beliefs about learning mathematics, and then I will discuss research
on students’ goals for learning in mathematics classrooms.

Students’ Beliefs about Learning Mathematics
Beliefs have been studied extensively in mathematics education; a literature

review discussing the range of research on students’ beliefs about mathematics was

23

published ahnost 20 years ago (Underhill, 1988), while a book discussing the role of
beliefs in mathematics education was published recently (Leder, Pehkonen, & Tomer,
2002). Students’ beliefs have been studied in mathematics education for two primary
purposes: (a) to understand how student thinking moderates the process of learning
mathematics and (b) to assess teaching interventions, such as Standards-based curricula
and teaching practices.

Beliefs moderate the process of learning mathematics. The study of students’
beliefs as an important phenomenon in the learning process began as an effort to move
away from process-product orientations in educational research. A large body of research
accumulated in the 1970’s and early 1980’s, the results of which demonstrated that
certain teaching practices led to improved student outcomes, such as achievement or time
on task (e. g., Clark et al., 1979). Process-product research was complexified as
researchers began to study how students’ thinking played a role in the learning process
(e.g., Peterson & Swing, 1982; Winne & Marx, 1982).

Researchers have examined mathematics students’ beliefs in terms of how they
moderate the process of learning and motivation to learn. For example, with respect to
motivation, Stodolsky, Salk, and Glaessner (1991) suggested students’ beliefs about the
nature of the school subject are related to their learning goals. Cobb (1985) demonstrated
this relationship through two case studies of first grade students, illustrating that a student
with an ego-involvement learning goal, such as a focus on performance, expressed that
mathematics problems and procedures were unrelated to one another. Another student

with a task-involvement learning goal, or a persistence in learning the material, expressed

24

beliefs that mathematics procedures were related. Beliefs about the nature of the school
subject co-occurred with particular learning goals for these students.

Based on the assumption that students’ Operational definitions of the school
subject would be a hidden factor in students’ confidence and perceived usefirlness toward
mathematics, Kouba and McDonald (1991) studied over 1,200 students across elementary
grades (K-6) to determine whether they saw different situations as involving mathematics
(or not). They also conducted classroom interviews to identify the characteristicsthat
distinguished the different ways in which students defined mathematics. Their results
indicated that K-6 students believed mathematics is a narrow domain, primarily about
numbers and operations, an exclusive domain, such that it only occurs in school and is
isolated from other school subjects, and is an upwardly shifting domain, such that once
the students consider the situation to be automatic and easily known, the students no
longer perceive the situation as mathematical. One concern these researchers expressed
was that if students believed that mathematics is always difficult, they may discount a
portion of their understandings about mathematics, and then their confidence would
decrease.

Relations have been found between students’ beliefs and their problem solving
behaviors. Schoenfeld (1985, 1988, 1989) found that high school students who held a
belief they should be able to solve mathematics problems in 12 minutes or less also
exhibited a lack of persistence when working on challenging problems. This belief has
been referred to as “quick learning.” It has also been found to be a strong predictor of
high school GPA; the less students believed in quick learning, the higher GPA they

earned (Schommer, Calvert, Gariglietti, & Bajaj, 1997). With regards to undergraduates’

25

problem solving behaviors, Pajares and Miller (1994) found that self-efficacy was more
predictive of success in problem solving than math self-concept, perceived usefulness of
mathematics, prior experience with mathematics, or gender. In another study of
undergraduates, Lerch (2004) found that lack of confidence and previous lack of success
in mathematics led to a lack of persistence with solving a problem. If they did persist,
students continued with unsuccessful strategies when working on unfamiliar problems,
suggesting a dependency upon particular solution strategies for specific problem types.
Thus, beliefs have been shown to provide a psychological foundation for problem solving
that leads to persistence, or lack thereof, and higher performance in school.

Most of the studies of whether and how students’ thinking during instruction,
including their beliefs, influence mathematics learning have been conducted prior to the
current U. S. mathematics education reform movement. A recent exception is the work of
the QUASAR proj ect [Quantitative Understanding: Amplifying Student Achievement
and Reasonsing] (Henningsen & Stein, 1997; Stein & Lane, 1996). Although QUASAR
did not explicitly focus on the analysis of students’ beliefs, their research acknowledges
that students’ drinking can mediate their experiences with mathematical tasks. They
proposed that a mathematical task exists at three levels: as represented in the cunicular
materials, as set up by the teacher in the classroom, and as implemented by students in
the classroom. Further, students’ dispositions toward learning mathematics (which
include their beliefs) are conjectured in their framework to be a factor that influences
students’ implementation of a task. Additional work is needed that explores whether and
how students’ beliefs influence students’ learning in reform-oriented mathematics

classrooms, particularly with respect to specific activities such as participation in whole-

26

class discussion, in order to build upon the research previously conducted on individual
students’ problem solving.

Different kinds of beliefs can affect the learning of mathematics. Beliefs about the
nature of mathematics may provide a psychological foundation for holding additional
beliefs, such as confidence or interconnectedness, breadth, isolation, and difficulty of the
school subject, in clusters with learning goals. Beliefs about the process of learning
mathematics, such as whether mathematics can be learned quickly, and beliefs about the
self as a learner of mathematics may affect students’ performance and problem solving
behaviors. Epistemological beliefs, such as those about the nature of knowledge and the
process of learning, may moderate students’ learning processes.

Assessing classroom interventions: Developing productive beliefs. Since the
mathematics reform movement in the US strives to promote certain productive beliefs
among students, mathematics education researchers have also studied students’ beliefs as
an outcome for evaluating the effectiveness of a teaching intervention. If some students’
beliefs are considered to be problematic, e. g., that mathematics problems can always be
solved quickly or that the school subject is narrow and made up of isolated facts and
procedures, then changing the culture of mathematics classrooms in schools may promote
more productive beliefs. Researchers have analyzed whether implementing Standards-
based teaching or curricula has resulted in changes in students’ beliefs. While some of
this research has addressed the development of pre-service teachers’ beliefs (Ambrose,
Clement, Philipp, & Chauvot, 2004; Cooney, Shealy, & Arvold, 1998), this section will
address research on students’ beliefs in the settings of interventions related to

mathematics reform. Belief change may be considered among one of the outcomes of

27

teaching interventions related to mathematics reform, acrOss age groups: elementary
school (Kloosterman, Raymond, & Emenaker, 1996; Nicholls, Cobb, Wood, Yackel, &
Patashinck, 1990), high school (Boaler, 1997; 1998), and post-secondary school (Hofer,
1999)

Nicholls et al. (1990) conducted a study of Six second grade classrooms, one
taught according to constructivist views of mathematics and five that were taught more
traditionally, to examine the dimensions of second grade students’ theories about success
in school mathematics. Students in the non-traditional classroom expressed a higher task-
orientation, where the goal is to gain understanding rather than be superior to others (ego-
orientation), in comparison to the students in the traditional classrooms. Also, students in
the non-traditional classroom were more likely to express the belief that success depends
on effort, attempts to understand, and cooperation with peers, in comparison to students
in the more traditional classrooms.

Kloosterrnan et a1. (1996) studied elementary students’ beliefs for three years,
starting when they were in first through fourth grades, and found that beliefs about
learning and doing mathematics were relatively stable over time. These students were at a
school that taught mathematics through problem solving. Results of this study showed
students’ beliefs were strongly tied to their classroom environments. Students approved
of cooperative learning, for example, only when their teachers were using it. These
students also expressed a narrow belief about the usefirlness of mathematics and tended
to like mathematics more as it became more difficult, appreciating the challenge of

mathematics.

28

Boaler (1997; 1998) compared high school students’ experiences over three years
from two schools with similar demographic profiles, but with different curricular settings:
one school with open-ended mathematics activities and the other with a more traditional
textbook approach. She found that students in the open-ended setting were more likely to
express enjoyment in doing mathematics and to appreciate thinking for themselves over
memorizing. Students in the traditional setting were more passive about their learning,
were more likely to have a set view of mathematics as a disconnected collection of
exercises, rules, and equations, and viewed mathematics as relating less to the world than
other school subjects.

Hofer (1999) also contrasted students in two different curricular settings, but she
compared college undergraduates at one university who experienced different forms of
Calculus: one that emphasized active and collaborative learning both in and out of class
and primarily focused on word problems, and the other a more traditional approach of
lecture and demonstration. The students in the non-traditional calculus course were found
to have more sophisticated beliefs about mathematics; they were particularly less likely to
believe that doing mathematics involves getting a right answer quickly. Achievement was
positively correlated with sophistication in mathematical beliefs (as in Schommer et a1.
(1997), and students with sophisticated beliefs about mathematics were more likely to
have mastery orientations toward learning mathematics.

Some studies reporting students’ reactions to teaching interventions have assessed
their perspectives more broadly than in terms of their beliefs alone. Bay, Beem, Reys,

Papick, and Barnes (1999) studied sixth and seventh grade students’ reactions to

29

Standards-based mathematics curricula.3 In letters students wrote about their experiences
with these curricular series, students reported positive experiences with hands-on
activities, group work, and the new content, an increase in application problems, and
improvement in attitudes toward mathematics. However, the diversity across classrooms,
in terms of students’ perspectives on the difficulty of the curricula, the use of technology,
and their assessments of their own progress, suggests implementation varied, and success
of the curricula with students is dependent upon the teacher. Reactions from older
students were reported in their own voices (Holt et al., 2001), as high school students
commented on their experiences with the Interactive Mathematics Project [IMP] — one of
the NSF -funded Standards-based high school curricula. Five female students said that
they appreciated IMP’S emphasis on problem solving, group work, and communication
skills, and expressed a need for more practice doing algebraic manipulation. They
generally believed the program did prepare them for college. In both cases, students were
able to identify salient features of new mathematics curricula and mentioned ways in
which they benefited from their experiences learning from it.

Since these six studies focus on assessing curricular or teaching interventions
designed to improve students’ learning of mathematics, the results emphasized the
commonalities in students’ experience, whether the intervention was the classroom or
school. In each case, the interventions had some impact on students’ beliefs that was
assumed to be positive. It appears that students develop beliefs somewhat consistent with
their teachers’ implementation of a curriculum or other reform-oriented teaching practice.
Additionally, these studies all took place in settings where non-traditional mathematics

teaching and curricula were novel, as these approaches are relatively new. One question

 

3 Connected Mathematics Project and Sixth Through Eighth Mathematics, now titled MATH Thematics.

30

they raise is what shape students’ experiences and beliefs will take in non-traditional
mathematics settings after they have been in them for extended periods of time, as their
perspectives may shift afier novelty wears off.

Development of students’ beliefs about learning mathematics. Researchers
who study students’ beliefs about mathematics have been curious about the factors that
influence belief development. The goals of the mathematics reform are based on the
assumption that teaching and cunicular interventions can make a difference in students’
beliefs and learning. Beginning with the scholarship of Doyle (1983, 1988) and moving
into the work of the QUASAR project (Henningsen & Stein, 1997; Silver & Stein, 1996;
Stein, Grover, & Henningsen, 1996; Stein & Lane, 1996), one premise is as follows:

...the work students do, which is defined in large measure by the tasks teachers

assign, determines how they think about a curriculum domain and come to

understand its meaning. (Doyle, 1988, p. 167)

Thus, continued exposure to certain types of mathematics tasks, perhaps those at a higher
cognitive demand, those that afford flexible solutions, or those that are more related to
the world outside of school, may promote productive beliefs among mathematics
students. Also, teachers’ implementation of mathematics tasks may influence students’
beliefs. However, there are a number of issues related to individual diversity among
students that also support the development of beliefs, in addition to classroom influences,
such as socio-economic class, race, the culture of students’ home environments, and
gender differences.

Franke and Carey (1997) found that demographics (particularly race and socio-

economic status [SES]) may moderate students’ belief development, as they studied

31

elementary school students’ perspectives on what it means to learn mathematics with a
curriculum that has a problem solving emphasis (Cognitively Guided Instruction [CGI]
(Carpenter, Fennema, & Franke, 1996)). Contrasting students who were predominantly
White and middle class with students who were predominantly African American and
low SES, they found that the White, middle class students were more likely to say that
doing mathematics involved spending a long time on a problem and that mathematical
success was determined by the strategy one used. In contrast, African American, lower
SES students said that mathematical success was determined through obtaining correct
answers and by speed and accuracy. These findings imply a need for further study on the
development of students’ beliefs in Standards-based settings, as curricula alone does not
appear to predict the development of particular beliefs.

Ridlon's (2001) case study of a seventh grade student revealed the influence of
family on students’ beliefs and resistance to participate in learning mathematics from a
problem centered mathematics program. The case focused on a student from a rural
farming community. His parents were third generation soybean farmers. His mother was
involved in school, calling the teacher four times in a nine week period and writing two
notes, on her own initiative. Mark did not volunteer in class initially, and became more
withdrawn throughout the nine-week period. He refirsed to write in a math journal, and
resisted working in groups. The researcher determined that “the idea of speaking your
mind and multiple correct answers was confusing for him” (p. 59), and that Mark’s “well-
established habit of bowing to authority and accepted procedures without making sense

of them” was strengthened by his southern, rural upbringing. In this case, the curriculum

32

and teaching style were unsettling to the students’ beliefs, which the researcher attributed
to his cultural background.

Lubienski (2000a; 2000b) taught a socio-economically diverse group of 18
seventh grade students in one class using the Connected Mathematics Project curricular
series, and determined that their views about whole-class discussions differed based upon
their socio-economic backgrounds. Hi gher-SES students considered the discussions to be
a helpfirl forum for exchanging ideas and were conceptually oriented during discussions.
Lower-SES students preferred more teacher direction and were more often focused on
giving correct answers to specific problems. Her research suggests that students’ socio-
economic status moderates students’ beliefs about learning in discussion-oriented
mathematics classrooms.

There is conflicting evidence on gender differences in students’ beliefs about
learning mathematics. Studies from the 1970’s and 1980’s (e. g., Fennema & Peterson,
1985; F ennema & Sherman, 1976) indicate differences between males and females in
terms of confidence in mathematics and perceived usefulness. However, gender
differences in beliefs may be disappearing, as more recent studies investigating gender
differences in beliefs do not show gender differences. For example, Vanayan, White,
Yuen, and Teper (1997) only found gender differences in terms of more boys saying that
they were good at mathematics, but both boys and girls said that they liked mathematics,
and no other gender differences in beliefs were found, such as relevance of mathematics.
A study of a girls-only middle school mathematics class revealed that the students’

experience enhanced their ability to learn math and their view of themselves as

33

mathematicians (Streitmatter, 1997), suggesting that girls’ beliefs may improve in single
gender classrooms.

This growing body of research has begun to explore the ways in which students’
demographic backgrounds may shape the development of students’ beliefs about learning
mathematics. But it is also possible that the age and developmental challenges of students
may influence the character of their beliefs. The age of the student can affect what they
are cognitively capable of understanding, as well as the students’ heightened sense of
how their classmates perceive them (Elkind, 1978). Edwards and Ruthven's (2003) study
is an example of how the age of the student can influence their beliefs. They replicated an
approach taken by Kouba and McDonald (1991), asking students whether certain
everyday situations involved mathematics, with junior high and high school students
rather than elementary school students. These older students’ beliefs about the domain of
mathematics were broader than the elementary students fi'om the previous study. The
developmental trajectory of the student is another factor that contributes to the diversity
of students’ beliefs.

Categories of mathematics-related beliefs. A wide range of beliefs related to
learning mathematics can be studied. DeCorte, Op'tEynde, and Verschaffel (2002) and
McLeod (1992) proposed two somewhat similar, but slightly different, frameworks for
studying students’ beliefs in mathematics classrooms. DeCorte et al. (2002) distinguished
between three categories of mathematics-related beliefs: beliefs about mathematics
education, beliefs about the self in relation to mathematics, and beliefs about the social
context. Beliefs about mathematics education include beliefs about the nature of the

school subject, about the process of learning and problem solving, and beliefs about

34

mathematics teaching. Beliefs about the self in relation to mathematics learning include
motivational beliefs, such as self-efficacy beliefs, control beliefs, task value beliefs, and
goal orientation beliefs. Beliefs about the social context include students’ beliefs about
their classroom norms, their expectations for their teacher’s role, their role, and their
classmates’ roles in their current classroom, and beliefs about aspects of their classroom
culture that are specific to mathematical activity, such as what counts as a different
solution. Alternatively, McLeod (1992) proposed categorizing beliefs in terms of those
that are about the nature of mathematics, about the self, about mathematics teaching, and
about the social context, separating those that are about the nature of mathematics from
those about the process of teaching.

Epistemological beliefs are assumptions about the nature of knowledge and the
process of knowing. Hofer and Pintrich's (1997) review of research of epistemological
beliefs research, synthesized epistemological beliefs into four categories, two that refer to
the nature of knowledge and two that address the nature of knowing. Beliefs referring to
the nature of knowledge include certainty of knowledge (whether knowledge is fixed or
fluid) and simplicity of knowledge (whether knowledge is an accumulation of isolated
facts or highly interrelated concepts). Beliefs addressing the nature of knowing include
source of knowledge (whether knowledge resides in an external authority or within the
student’s ability to construct knowledge) and justification for [mowing (how individuals
evaluate knowledge claims). These beliefs could be stated in domain-specific terms,
using mathematical as a modifier for knowledge, such as beliefs of the simplicity of
mathematical knowledge — whether mathematics is a coherent system or a collection of

isolated pieces.

35

Whether beliefs are domain-specific (localized to the school subject of
mathematics) or general (pertaining to a range of subject matters) is currently an active
and open question. In support of the latter view, Schommer and Walker (1995) found that
students held a range of conceptions at consistently sophisticated levels across the
domains of both mathematics and social studies, including conceptions of knowledge as
less certain or simple, learning as a not quick process, and one’s ability not being fixed.
However, Buehl, Alexander, and Murphy (2002) present challenges to this work. They
found evidence supporting the domain-specificity of certain beliefs (e.g., knowledge
utility or value — mathematics was more related to other areas than history) when survey
items were worded in reference to disciplines.

Another way of describing epistemological beliefs about the process of learning
would be in terms of women’s ways of knowing (Belenky, Clinchy, Goldberger, &
Tarule, 1986). Three of these ways of knowing may be particularly relevant to students
who participate in whole-class discussions: received, procedural, and constructed.
Received knowers conceive of themselves as capable of receiving, even reproducing,
knowledge from the all-knowing external authorities but not capable of creating
knowledge on their own. Procedural knowers are invested in learning and applying
objective procedures for obtaining and communicating knowledge. Constructed knowers
View all knowledge as contextual, experience themselves as creators of knowledge, and
value both subjective and objectives strategies for knowing. Mathematics students may
be less likely to express evidence of subjective knowing, as the nature of mathematics is
rarely perceived as entirely subjective. Additionally, if they participate in whole-class

discussion, they are not exhibiting silence, or experiencing themselves as mindless and

36

voiceless and subject to the whims of external authority. While these may be considered
women ’s ways of knowing, they may have characteristics that apply to both young men
and young women. Epistemological beliefs have typically been studied in college
students and adults, but if younger students have moved beyond relying on receiving
knowledge from an authority, this may be significant enough to document. Researchers
have documented students’ epistemological beliefs as early as fourth grade (Johnston,
Woodside-Jiron, & Day, 2001).

Beliefs about the self as a learner of mathematics have already been explored in
great detail. One example is the breadth of research on self-efficacy. Self-efficacy in
mathematics is whether or not students believe they can control or regulate their own
learning and master academic tasks in mathematics (Bandura, 1993, 1997), and these
self-beliefs of efficacy play a role in students’ motivation. Students with a strong sense of
self-efficacy with respect to a particular domain or task may be more likely to attribute
their success to their own efforts. Additionally, students may be more likely to put effort
into activities that they believe they can succeed in. Efficacy beliefs may be unrealistic,
over-estimates or under-estimates of one’s capabilities. They have been found to
determine goal setting, activity choice, willingness to expend effort, and persistence
(Bandura, 1997).

Even though self-efficacy in mathematics is challenging to assess (Paj ares &
Miller, 1995), self-efficacy is considered to moderate students’ mathematics problem
solving more effectively than other variables, such as self-concept, usefulness of
mathematics, or gender (Pajares & Miller, 1994). This relationship may be

developmental, as Kloosterman et al. (1996) found a lack of relation between self-

37

confidence and achievement for first grade students, but a strong relation between the
variables for third grade students. Additionally, Pietsch, Walker, and Chapman (2003)
found self-efficacy in mathematics to be more highly related to mathematics performance
than self-concept among secondary school students. As self-efficacy has been found to be
a strong predictor of engagement in problem solving and achievement in mathematics, it
may also relate to students’ participation in whole-class discussion during mathematics
class.

Overall, while a wealth of research on students’ beliefs has been taken up in
mathematics education, new questions still remain. While research has suggested that
students’ epistemological beliefs play a role in mathematics learning, such as relations to
performance goals and problem solving behaviors, it is not clear how epistemological
beliefs would be related to students’ experiences of learning in Standards-based
mathematics classrooms. While there is some evidence that students develop more
sophisticated epistemological beliefs in Standards-based settings, how do these beliefs
then influence students’ future experiences with learning mathematics?

Students’ Goals in Mathematics Classrooms

Recall that goals, in contrast to beliefs, are what students want to achieve through
their actions, rather than broader assumptions students use for interpreting a situation
(beliefs). Students may be more cognizant of, and perhaps more able to articulate, their
goals than their beliefs. The study of whether and how goals influence students’
behaviors and achievement in school has been taken up more by researchers in the field
of educational psychology than those in mathematics education. Many educational

psychologists have either situated their studies in mathematics classrooms, or suggested

38

that their studies of adolescent motivation could be relevant to mathematics learning.
They have primarily studied students’ goals in relation to achievement. As with the
research on beliefs, these researchers have assumed that some goals are more productive
than others and that classrooms can be structured in order to help students develop more
productive goals. Research on students’ goals in school has been dominated by a focus on
learning goals, but recently has begun to expand the focus onto students’ social goals.

Learning goals. Two primary distinctions have been made between students’
goals for learning academic content: the difference between (a) seeking positive
evaluations of one’s competence and avoiding negative evaluations of one’s competence
or (b) a focus on mastering tasks and increasing one’s understanding of the content.
These goals have fallen under a variety of labels, including ego-involved and task-
involved goals (Nicholls et al., 1990), performance and learning goals (Dweck, 1986),
and performance and mastery goals (Ames, 1992). For the sake of clarity, I will refer to
the goals related to evaluation of performance as performance goals and the goals related
to understanding content as mastery goals.

An additional distinction has been made between two types of performance goals:
performance-approach and performance-avoidance goals. Students who hold
performance-approach goals want to be recognized positively for their competence, and
students who hold performance-avoidance goals try to avoid looking incompetent
(Midgley, Kaplan, & Middleton, 2001).

The development of learning goals has been studied among middle school
students (e. g., Middleton, Kaplan, & Midgley, 2003). Mastery and performance goals

appear to be fairly stable over time during adolescence. However, students who expressed

39

high self-efficacy and performance-approach early in middle school shifted toward
performance avoidance goals later in middle school. These students may have a need to
protect their self-efficacy, and if it is threatened, they may avoid engaging in increased
competition or unfamiliar material.

Meyer and Turner (1997) found that fifth and sixth grade students in problem
solving environments fell primarily along two dimensions: challenge-seekers and
challenge-avoiders. Challenge-seekers expressed mastery goals, higher self-efficacy than
challenge-avoiders, and a tolerance for failure, while challenge-avoiders expressed a high
negative affect after failure, performance goals, lower self-efficacy, and used surface-
level strategies when solving mathematics problems. The researchers suggested following
up these studies of students’ motivations in mathematics classrooms with additional
research in order to examine whether the patterns hold in other settings, as their studies
focused on a small number of classrooms.

In another study, Summers, Schallert, and Ritter (2003) determined that middle
school mathematics students expressing a low level of mastery goals were more
influenced by comparisons to close fiiends than to other students in the class, suggesting
issues of relatedness are integral to students who do not have high mastery goals.

School-related goals: Beyond learning goals. Adolescents may also strive to
attain a range of school-related goals, not only learning goals. These include task-related
goals, cognitive goals, or social relationship goals (Wentzel, 1999). Task related goals are
focused on learning content. Cognitive goals are related to seeking challenges. Social
relationship goals are pursued in order to feel connected to others. Sharing of

mathematical strategies in discussion could address task-related goals of mastering

40

subject matter or meeting a standard of achievement. Coming up with an alternative
solution to share may address cognitive goals of engaging in creative thinking or
satisfying an intellectual-challenge. Sharing one’s perspective during mathematics class
may also support social relationship goals of gaining approval from others, establishing
personal relationships with teachers or peers, or cooperating with classmates. Desiring
the opportunity to succeed in the task of discussing mathematics problem strategies may
be related to more than one school related goal, and goals other than mastery or
performance goals.

Additionally, students may orient their classroom behavior to meet personal
needs. In self-determination theory (Deci & Ryan, 1985), these include the need for a
sense of relatedness to others, for autonomy, and for competence. Of course, pursuit of
these needs is only possible a range of suitable classroom environments. Discussing
strategies during mathematics class can foster a sense of relatedness, as students may
offer their thinking in support of the thinking of another student in order to help each
other. The opportunity to share strategies can foster a sense of autonomy, as students are
encouraged to share their own thinking in these mathematics classroom discussions.
Mathematics classroom discussions are also an opportunity for students to demonstrate
competence, as students can Show what they know through these discussions.
Researchers have begun to study students’ autonomy in learning mathematics (e. g.,
Kamii, 1985; Yackel & Cobb, 1996), and their self-efficacy and confidence (e. g., Pajares
& Miller, 1994; Pietsch et al., 2003). However, mathematics education research could do

more to take up issues of relatedness, particularly in light of NCTM’S recommendations

41

for increased interactions in classroom discourse, since increase in discussion may lead to
increased opportunities for public performances and evaluations.

Social goals. Although it may make intuitive sense that students’ social goals and
sense of relatedness are significant elements of students’ experiences in school settings,
researchers have only recently begun to address their role in the learning process. For
example, in students from 3rd — 6th grade, their sense of relatedness to parents, teachers,
and peers was found to contribute to classroom engagement (Furrer & Skinner, 2003).
This suggests that the quality of relationships experienced by students in the classroom
has implications for their academic success.

Additionally, students’ goals are often assessed Via surveys, and social goals have
not usually been included on survey instruments. When studying middle school students’
motivations inductively through listening to students’ talk in interviews, Dowson and
McInemey (2003) discuss five social goals upon which students operate in social
contexts: social affiliation, social approval, social responsibility, social status, and social
concern. The authors suggest students’ motivation be conceptualized as a process of
managing multiple goals, both academic and social goals, as has Wentzel (1999).
Listening to how students talk about their motivations may reveal a broader range of
goals beyond the traditionally studied learning goals, and students may discuss multiple
goals specific to the activity they are participating in, such as whole-class discussions
about mathematics.

Patterns Among Beliefs, Goals, and Academic Risk
Doyle’s (1983) review of the role of academic work on students’ learning states

that since academic tasks are embedded in an evaluation system, students work on these

42

tasks under conditions of ambiguity and risk. It is possible that risk is heightened not only
when the task is designed to assess understanding and is less opinion oriented (Doyle,
1983), but also when it takes place in a public setting. Whether students experience a
classroom to be a risky environment may be related to their beliefs and goals as well as
their opportunities to participate during class.

If students perceive a task to have a high level of risk, they may act in such a way
that reduces the risk of the task. Risk-reducing actions may include as performing only
the minimum requested problems, restricting the amount of input given to the teacher to
minimize the risk of exposing errors or in order to elicit assistance, getting a teacher or
another student to answer on their behalf (Doyle, 1983). Perception of risk involved in
publicly discussing a task may be related to whether and how students choose to
participate in whole-class discussions.

This sense of risk can be considered in relation to students’ beliefs and behaviors
in school settings. For example, students’ perceptions of risk may be related to their
experiences with teachers in the classroom, as Ryan, Gheen, and Midgley (1998) found
that Sixth grade mathematics students with lower self-efficacy who may otherwise avoid
seeking help were more likely to do so in a classroom with a teacher who attended to
students’ socio-emotional needs. This implies that teachers who attend to socio-emotional
aspects of the classroom environment may support students’ learning.

In order to assess students’ experiences in discussion, it seems important to also
consider students’ social goals, in addition to their learning goals. Students’ learning
goals seem particularly important given both the activity of discussion and the

developmental time frame of adolescence. During adolescence, students may have a

43

heightened sense of how others perceive them, or a sense of an imaginary audience
(Elkind, 1978). As a participant in dialogue with others, people are naturally attuned to
issues of presentation of self.
Let us turn now from the others to the point of View of the individual who
presents himself before them. He may wish them to think highly of him, or to
think that he thinks highly of them, or to perceive how in fact he feels toward
them, or to obtain no clear-cut impression; he may wish to ensure harmony so that
the interaction can be sustained, or to defraud, get rid of, confuse, mislead,
antagonize, or insult them (Goffinan, 1959, p. 3).
Individuals engaging in discussion with others, then, are not only thinking about the
content they are discussing, but also about how they are viewed by the other participants
in the dialogue and their relationships with them. Considering students’ learning goals
alone does not effectively capture the range of possibilities of students’ social goals
during whole-class discussion about mathematics during middle school, such as a sense
of harmony among the group.
But it can be challenging to hypothesize which patterns of student involvement
may relate to students’ beliefs and goals. This is partially because what may seem like a
significant connection between beliefs, goals, and involvement to a researcher may not be
a Significant connection to the student. For example, recent research on 11-13 year-olds’
prosocial behaviors indicated that students perceive a wider range of behaviors than are
normally studied to be part of their prosocial activity, such as standing up for others,
encouraging others, helping others develop skills, including others who are left out, or

being humorous (Bergin, Talley, & Hamer, 2003). Traditional research on prosocial

44

development in youth has focused on behaviors such as sharing, helping, volunteering.
Since researchers may not make the same connections between beliefs, goals, and
involvement that students make, including students’ perspectives on their involvement in
classrooms may allow for a more thorough description of these potential relations than
would be possible through observation alone.
Discourse in Mathematics Classrooms

As I mentioned in Chapter One, the National Council of Teachers of Mathematics
has made recommendations for mathematics teaching in order to improve student
learning. The image of mathematics classrooms promoted by NCTM looks quite different
from asking students to imitate and mimic procedures as presented by the teacher. In an
often quoted passage, the authors of the Professional Standards for Teaching
Mathematics state:

We need to shift toward classrooms as mathematical communities — away from

classrooms as simple collections of individuals; toward logic and mathematical

evidence as verification — away from the teacher as the sole authority for right

answers; toward mathematical reasoning — away from merely memorizing

procedures; toward conj ecturing, inventing, and problem solving — away from an

emphasis on mechanistic answer-finding; toward connecting mathematics, its

ideas and its applications — away from treating mathematics as a body of isolated

concepts and procedures... (NCTM, 1991, p. 3)
NCT M’s image sees the classroom as a community of learners who reason and

communicate about mathematics together to verify their ideas. Recommendations for

45

creating such mathematics classrooms include orchestrating discourse between the
teacher and students.

The teacher of mathematics should promote classroom discourse in which
students: listen to, respond to, and question the teacher and one another; use a
variety of tools to reason, make connections, solve problems, and communicate;
initiate problems and question; make conjectures and present solutions; explore
examples and counterexamples to investigate a conjecture; try to convince
themselves and one another of the validity of particular representations, solutions,
conjectures, and answers; rely on mathematical evidence and argument to
determine validity. (NCTM, 1991, p. 45)

So, in these classrooms imagined by the NCTM, students are expected to talk about
mathematics in particular ways, including initiating problems, presenting solutions, and
questioning the thinking of others and trying to verify the ideas of their classmates and
teacher with evidence and reasoning. Classrooms fostering these sorts of interactions are
considered to be an advance over classrooms that emphasize imitating procedures,
because students may develop autonomous thinking such that they are able to
communicate, reason, and justify their ideas about mathematics to themselves and others.

Classroom discussion is considered to be a valuable practice in many subject

matters, not mathematics alone. John Bruer (as cited in Cazden, 2001) suggests five
hypotheses as to why promoting classroom discourse would be an effective teaching
strategy across the curriculum: (a) thinking is made public, and skilled thinkers can
model their thinking to others; (b) the task of thinking can be distributed among the

classroom community; (0) dialogue involves both language comprehension and language

46

production, and production is considered to be more demanding, so participating in
discussions may result in greater depth of processing; ((1) classroom discourse sends the
message that thinking and intelligence are valued; (e) discourse does not simply make
thought visible, but thinking is internalized discourse. These hypotheses were developed
from information processing and social psychology.

If discourse is thought about in terms of students’ opportunities to try on new
ideas about mathematics, talking about mathematics is an opportunity to learn through
the process of internalization, as described by Vygotsky (1978). Appropriation, a term
introduced by Leontev (1981), is also sometimes used to describe this process of the
transformation of knowledge, and at times this term is preferred because it acknowledges
the two directions — students can appropriate the ideas of their teachers and teachers can
appropriate the ideas of their students. However, discussing learning in terms of
appropriation or internalization does not effectively explain why or how individuals
construct knowledge differently through discourse. Studies of classroom discourse could
benefit from addressing the cognitive and socio-emotional diversity among learners who
participate in classroom discourse.

Research on classroom discourse in mathematics classrooms has begun to
acknowledge and describe the role that individual students have in shaping mutually
constituted classroom norms (e. g., Stephan, Cobb, & Gravemeijer, 2003), but it is still
common among researchers who study mathematics classroom discourse to focus on
describing classroom discourse in normative terms or the role of the teacher in

orchestrating effective classroom discourse.

47

Describing Mathematics Classroom Discourse

As teachers and researchers attempt to foster classroom productive and powerfirl
discourse about mathematics, careful descriptions have emerged in the literature. Lampert
(1990), Yackel and Cobb (1996), and Kazemi and Stipek (2001) each describe
characteristics of discourse in mathematics classrooms designed to alter the roles of the
teacher and the student and promote autonomous thinking about mathematics concepts
rather than procedures. Lampert (1990) illustrated how she and her fifth grade students
talked about exponents, including the need to talk about talking as well as mathematics
content, the challenge of negotiating the meaning of exponents, and determining that
finding the answer is not the signal to stop thinking. Yackel and Cobb (1996) described
classroom norms that influenced first grade students’ learning of place value.’ These
norms were termed “sociomathematical,” since they were at the intersection between
expectations for social interactions and appropriate ways of doing mathematics.

Sociomathematical norms described by Yackel and Cobb (1996) include using
mathematics to justify explanations rather than social status and authority, conceptual
rather than procedural explanations, and reflecting on the explanations of students.
Kazemi and Stipek (2001) extended this work by describing sociomathematical norms in
fourth and fifth grade classrooms where teachers pressed for conceptual learning by: (a)
moving beyond procedural descriptions to mathematical arguments, (b) seeking to
understand relations among multiple strategies, (c) discussing errors in order to
reconceptualize a problem, explore contradictions in solutions, and pursue alternative
strategies; and (d) collaborative work also involves individual accountability and seeking

consensus through mathematical argumentation. In these studies, the researchers agree on

 

’ “Norm” refers to regular patterns of interaction during a discussion.

48

the importance of talking about talking with students, such as continuing the conversation
once the answer has been found and talking about more than the procedures for solving
the problem. Additionally, discussing students’ strategies themselves, and looking for
relations between them, is considered fruitful for developing mathematical
understandings.

Describing classroom discourse, however, has been a part of the practice of
educational researchers outside of mathematics and before the NCTM reforms. There has
been an interest in understanding the nature of talk in classrooms to helping students
become a part of the culture of their classroom, as practices of talking in school may be
different than those at home (Shultz, F lorio, & Erickson, 1982), and educators may be
concerned with helping students adopt the discourse of the discipline (Lemke, 1990). One
challenge of studying talk in classrooms has consistently been finding the balance
between studying the structure of the talk and accounting for the subject matter in the
discourse. The tension lies between an analysis of the form and function and the content
in the talk.

Form and function. Talk in classrooms may be examined in terms of its form
and function. One common way of describing the form of classroom talk is in terms of
whether or not the interactions follow a traditional form of teacher-initiated questions,
student responses, and teacher evaluations (I-R-E) (Mehan, 1979). Talk that follows this
form is considered to function as a message of the teacher as authority, in terms of who
controls what is discussed and who determines the accuracy and validity of students’
responses. While I will not describe an exhaustive review of the ways talk in classrooms

can be studied in terms of form and function, I will present some of the forms and

49

functions of talk that have been studied with adolescents, or in mathematics classrooms,
or both.

The form of talk can be studied in terms of the settings in which talk takes place,
who is speaking to whom, and the ways talk is structured in particular interactions or
utterances. Classroom discourse has been studied in different settings, such as public talk
in front of the class or private talk between the teacher and the student (e.g., Hart, 1989)
as well as either large group (e. g., Hamm & Perry, 2002; Nathan & Knuth, 2003) or small
group discussions (e. g., Nussbaum, 2002). The form of talk is sometimes described in
terms of whether the interactions are monologic, between a teacher and one student, or
dialogic, involving multiple students such that they address one another directly
(N ystrand, 1997). Specific utterances within interactions have also been examined for
form, such as whether and how teachers revoice, or repeat, students’ utterances in order
to call attention to them or appropriate them in discussion (O'Connor, 1998). The form, or
structure, of talk can be analyzed at many levels down to the level of word choice.

Analyzing the function of talk is more interpretive than analyzing the form of talk,
and analyzing the form of talk is often the way researchers operationalize analysis of the
function of talk. One example was mentioned above, in the case of the I-R-E form
indicating the function of communicating teacher’s authority. Monologic and dialogic
forms of talk have been utilized by researchers studying the function of talk in
mathematics classrooms in terms of whether autonomous thinking is promoted among
students. Hamm and Perry (2002) analyzed whether the function of talk in six first grade
mathematics classrooms was at a high level of “mathematical discourse” -— talk not

entirely scripted by the teacher where ideas were exchanged — by looking at the degree to

50

which the dialogue was reciprocal. Nathan and Knuth (2003) examined one sixth grade
teacher’s classroom for monologic and dialogic talk over two years to document changes
over time. In both studies, teachers were challenged to move beyond monologic talk.

Content. As an alternative to describing the form and function of classroom talk,
researchers have also described the content of the talk in classrooms, which allows
researchers to capture more of the subject matter in the discourse. Hamm and Perry
(2002) also analyzed the discourse in the six first grade classrooms in terms of the degree
of mathematical analysis, or the level at which students engaged in higher order thinking
about mathematics, such as searching for mathematical patterns, making mathematical
conjectures, evaluating, arguing, and inventing original procedures. Lubienski (2000a;
2000b) studied her own teaching of a seventh grade mathematics classroom using the
Connected Mathematics Project textbook series, and analyzed students’ involvement in
whole-class discussions in terms of the content of their talk, including whether the
students discussed answers to problems, in terms of how to get an answer, with a focus
on answer over method, or whether the student talked about patterns, such as pointing out
the existence of a pattern or explaining a pattern. Describing the content of classroom talk
captures subject matter, but sometimes at the expense of describing how the opportunities
to talk about the content may arise in the interactions, which may be afforded by an
analysis of form and function.

Whether the analysis is on the form and function of classroom discourse or the
content of the talk in classroom discourse, there are challenges with studying classroom
discourse in mathematics classrooms. Analyzing and describing the form and frmction of

talk may compete with a focus on accounting for the subject matter in the talk.

51

Additionally, since teachers play a prominent role in classroom discussions, even those
who attempt to abide by the recommendations of the mathematics reform movement,
analyses of classroom discourse often illuminate more about the teacher’s talk than
students’. Examining the nature of talk in classrooms does not necessarily provide insight
for why and how some students are more involved than others, as it does not account for
the agency of individual students.

Best Practices: Mathematics Classroom Discourse

Research on mathematics classroom discourse has also moved beyond to
describing the nature of discussions to assess the effectiveness of discussions in
mathematics classrooms. Effectiveness has been operationalized in terms of a range of
outcomes: (a) the quality of the mathematical talk, such as whether the talk maintained
cognitive demand or mathematical precision, (b) whether students were highly involved,
(c) the reported motivation of the students, and ((1) students’ achievement. If NCTM
recommends implementing discussion in classrooms, the assumption that discussions are
beneficial should be evaluated according to a variety of outcomes.

Quality of mathematical talk as outcome. The effectiveness of a discussion may
be assessed with respect to the quality or level of mathematical talk in the discussion
itself. Henningsen and Stein (1997) examined four middle school mathematics
classrooms and determined factors that assisted students’ engagement at high levels of
cognitive demand, such as scaffolding and consistently pressing students to provide
meaningful explanations or make meaningful connections, and factors that led to the
decline of cognitive demand, such as the removal of challenging aspects of the problem,

lack of time, or inappropriateness of the task. Removal of challenging aspects of the

52

problem and scaffolding or pressing students occurred as part of classroom discussion.
Nathan and Knuth’s (2003) study of a sixth grade teacher’s attempt to include more
dialogic talk in her classroom resulted in a decrease in the mathematical precision of the
talk among the class. Certain features of classroom discussions may support students’
attempts to talk about mathematics more than others, such as scaffolding, that may be
challenging to incorporate when the teacher is initially trying to shift her role in
classroom discussions.

High involvement as outcome. Discussions in mathematics classrooms have also
been evaluated with respect to whether or not students are highly involved. Turner et al.
(1998) examined at least four classroom discussions from each of seven fifth and sixth
grade mathematics classrooms, and analyzed the self-reports of students’ involvement
from six target students in each classroom. She determined that three of the seven
classrooms had higher involvement from the target students. Two of the high-
involvement teachers demonstrated scaffolding in terms of negotiating understanding,
adjusting instruction in response to students, and transferring responsibility. They also
held students accountable for their own understanding provided intrinsic supports for
students’ motivation, such as evoking students’ curiosity, providing encouragement, and
advocating risk-taking. The third high involvement teacher did not focus on negotiating
understanding, but did focus on transferring responsibility and intrinsic supports. The
students in the high involvement classrooms expressed that their skills were suited for
working at the level of challenge in their classrooms. The students in the low
involvement classrooms reported their skills as exceeding the classroom challenges, and

these classrooms were characterized as adhering to I-R-E sequences, emphasizing

53

procedures and extrinsic supports for motivation. These results from the field of
educational psychology are somewhat consistent with the results from mathematics
education: effective classroom discussions move away fi'om IRE and procedural talk
toward a focus on students’ ideas, but Turner’s work contributes that the role of affect
and motivational support plays a significant role in classroom discourse when assessing
students’ involvement.

Students’ motivation as outcome. Students’ self-reports of their motivation are
another outcome that may be used to assess the effectiveness of discussion. Turner and
colleagues (Turner, Meyer, Midgley, & Patrick, 2003; Turner et al., 2002) examined sixth
grade mathematics classrooms and found that, while instructional discourse similarly
focused on students’ understanding of mathematics, the classrooms differed in terms of
the support for students’ autonomy and intrinsic motivation. The students in the
classroom with more supportive motivational discourse reported less negative affect and
self-handicapping, or avoidance behaviors, than did students in the classroom with less
supportive motivational discourse. Students reported using avoidance strategies less often
in classrooms that they perceived as emphasizing learning, understanding, effort, and
enjoyment. These findings suggest that environments that support mastery over
performance combine cognitive and affective components of teaching and learning.

Turner and colleagues have made an effort to integrate an individual viewpoint
with a social perspective on studying involvement, with their use of self-reports of
students’ motivation and involvement as outcomes. However, their perspective on
students’ motivation does not go beyond students’ learning goals, and students’

motivation may be broader than these goals.

54

Student achievement as outcome. Classroom discussions about mathematics
may also be assessed in terms of their impact on student learning or achievement. Hiebert
and Wearne (1993) assessed the relationships between teaching and learning of
mathematics in six second-grade classrooms in one school. Two of these classrooms
focused on constructing relationships over practicing procedures. These two classrooms
had higher gains on the achievement measures of place value understanding and both
routine and novel computation. The researchers determined similarities in the
instructional features of these two classrooms that appear to have supported the students’
learning in comparison to the four more traditional classrooms. The students in the
“learning” classrooms received fewer problems, spent more time with each problem,
were asked more questions asking them to describe and explain alternative strategies, and
both talked more often and used longer responses. The researchers concluded that both
instructional tasks and classroom discourse support students’ learning, and they were not
surprised that particular instructional tasks occurred together with forms of discourse.

Through examining a range of outcomes, such as student achievement,
motivation, involvement, and the quality of talk about mathematics during classroom
discussion, researchers have determined characteristics of effective classroom
discussions, such as explaining alternative strategies and looking for relationships
between them, scaffolding students’ efforts to explain their thinking about mathematics,
and scaffolding intrinsic motivation. Additionally, the nature of the task supports the
quality of discourse. Characteristics of ineffective discussions about mathematics include
removing challenging aspects of the task, not spending enough time or too much time on

a task, emphasizing procedures, following an I-R-E sequence, and scaffolding extrinsic

55

motivation. However, these characteristics of classroom discussions emphasize what the
teacher could do, or could avoid doing, but say little about whether and how students’
characteristics relate to their involvement in classroom discussions about mathematics.
Student Characteristics and Involvement in Mathematics Classroom Discourse

Teachers may not be aware of the challenges students face with discussions
during mathematics class. Lampert (1990) reported the range of ways her fifth grade
students were challenged by learning in a discussion-oriented setting. Some students
preferred to look to either the teacher or a more knowledgeable classmate as an authority.
Some did not realize that using a rule is different from explaining why it works. Students
kept silent for a variety of reasons, such as not having the words to express their thinking
out loud, lacking the courage to share their thinking, or because they copied another
student’s work. Social power and deciding by majority was an effective means of
justifying mathematical ideas, for some students. Finally, some students appeared to
believe that if something was wrong with their reasoning, then something was wrong
with them as an individual. Some of these issues decreased over the year in her
classroom, but the challenges that her students experienced provide insight for why some
mathematics students may hesitate to get involved in whole-class discussion, or not
benefit from their involvement. Not all students experience whole-class discussions about
mathematics in the same way. A range of student characteristics may contribute to how
students experience and participate in these discussions, such as personality traits,
dispositions, race, socio-economic status, or gender.

Students’ personality traits and dispositions. To date, mathematics educators

have not explicitly analyzed whether and how students’ personality traits or dispositions,

56

including their beliefs or attitudes, shape students’ involvement in whole-class
discussions. However, educational psychologists have conducted studies of whether and
how students’ personality traits or dispositions shape students’ involvement in discussion
more generally (Nussbaum, 2002; Nussbaum & Bendixen, 2003).

Nussbaum (2002) examined whether extroverted students in small group
discussions with other extroverts participated differently than a small group of introverted
students. Four extroverted sixth grade students and four introverted sixth grade students
discussed issues about urban planning. These students were sorted into these two groups
based on their responses to a personality questionnaire. The extroverted students’
involvement had significantly more contradictions and counterexamples and a greater
tendency to use conflictual discourse during the small group discussions. In contrast, the
introverted students worked with one another more collaboratively. The study was
replicated with 16 undergraduate pre-service teachers taking an educational psychology
course, and the results were similar: introverted students were more likely to participate
in collaborative discourse while extroverted students were more likely to challenge one
another.

Nussbaum and Bendixen (2003) used self-reported data (rather than observations
of students’ involvement in discussion) to analyze whether undergraduates’
epistemological beliefs, desire for warmth in relationships, and need for cognitive
challenge predicted whether they were more likely to avoid or approach arguments. Their
results demonstrated that epistemological beliefs, such as certain and simple knowledge,

and desire for warmth were paired with avoiding arguments, while the need for cognitive

57

challenge occurred with approaching arguments. Additionally, assertiveness predicted
both approaching and avoiding arguments.

These studies are two of the few examining whether and how students’
personality traits or dispositions relate to their involvement in discussions. However,
neither of these studies took into account the subject matter involved in the discussions.
Although the results of these studies suggest introverted students are less likely to
challenge one another during discussion, as they prefer collaborative talk, and that
epistemological beliefs, need for warmth, and cognitive challenge affect students’
willingness to participate or avoid arguments, it is not clear which aspects of students’
disposition may relate to what forms of talk during students’ involvement in mathematics
class.

Socio-economic status and race. Classroom discussions are only effective when
they benefit students fi‘om all racial and socio-economic backgrounds. White (2003)
documented how two third grade teachers used classroom discourse to promote the
mathematical learning of their diverse students, including over a third African-American
and Hispanic students. Over the school year, the teachers came to value students’ ideas in
the discussions, explored students’ answers, incorporated students’ background
knowledge, and encouraged student-to-student communication. According to White,
“these teachers’ practices help dispel the myth that African American and Hispanic
students must be told how to think about and solve mathematics problems” (p. 51), and
this study serves as an existence proof of the possibility of effective classroom

discussions with diverse groups of students.

58

However, Lubienski (2000a; 2000b) studied students from diverse socio-
economic backgrounds, looking at individual students’ involvement rather than the
collective discussions about mathematics, and determined that students fiom lower socio-
economic backgrounds participated differently from the higher SES students. Lower SES
students required more external direction from their teacher, and the higher SES students
participated in the discussions as a forum for exchanging ideas. Lubienski’s work
suggests that discussions in mathematics classrooms may be more aligned with middle
class cultures. There is more to understand about how students’ backgrounds relate to
how they interpret and experience discussions in mathematics classrooms.

Gender. Some researchers have found that male and females may participate
differently in mathematics classroom discussions. For example, Hart (1989) studied
seventh graders and determined that more boys than girls participated in public
interactions with the teacher, as opposed to private interactions; were more likely to
volunteer during discussions; were more likely to call out; and were more likely to share
their thinking when it was incorrect. Her results suggested that if certain forms of
participation are more valued and privileged in mathematics classrooms, males may
benefit more from the experience of discussions. However, more recent literature on
differences between how different genders participate in discussions more generally
suggests that the differences are often small (Goldsmith & F ulfs, 1999). Although this is
not an extensive representation of the research on gender differences in involvement in
whole-class discussion, I mean to suggest that there is more to understand about how the

young men and women involve themselves in discussion during mathematics class.

59

In summary, there is a need to coordinate studies of individual students’ beliefs
and goals with their involvement during mathematics class discussions. Such analyses
can illuminate students’ role in creating classroom norms of interaction during
mathematics class. Research attempting to coordinate the analyses of the individual and
social aspects of experiencing class discussion have focused only on a few cases, such as
Lo, Wheatley, and Smith's (1994) case study of one third-grade student. This student
developed increasingly sophisticated understandings of arithmetic over time while he
actively assisted classmates with sense-making during discussion by indicating when he
' did not understand or when he disagreed with a classmate. With respect to beliefs about
participating in discussions, he expressed that sharing ineffectively was a form of
dishonesty. He was comfortable with the chaos that went along with disagreement in
discussion, instead finding it to be intellectually stimulating. Looking beyond a case of
one student can allow for describing a broader range of the beliefs relevant to learning
mathematics in discussion-oriented settings.

A range of student characteristics may influence how students involve themselves
in discussion during mathematics class. Mathematics educators have begun to study
whether and how differences in race, socio-economic backgrounds, and gender influence
students’ involvement, while the influence of students’ personality traits and dispositions
on their involvement in mathematics classroom discussions remains largely unexplored.
Additionally, accounting for subject matter in the analysis of students’ involvement
continues to be a challenge. Research on students’ motivation in learning mathematics
needs to be reconceived in light of reform-oriented mathematics classrooms (Middleton

& Spanias, 1999).

60

CHAPTER THREE

METHODS

This study is a snapshot in time of 15 target students’ beliefs and goals and their
involvement in whole-class discussions in two different mathematics classrooms during
the Spring of 2003. I studied students’ beliefs and goals through my interpretation of their
self-reported data (one-on-one interviews and Likert-scale surveys), and I studied
students’ involvement and participation in whole-class discussion through videotaped
records of students’ behaviors during mathematics class. These data were analyzed both
at the individual student level and at the aggregate levels, grouped by both classroom and
groups based on shared beliefs and goals, in order to assess relations between students’
beliefs and goals and their involvement in whole-class discussion. As stated in Chapter
One, I was most interested in examining students’ motivations for participating in
discussions during mathematics class.

I did not have specific conjectures as to the nature of the relations between
students’ beliefs and goals and their involvement in whole-class discussion. Thus, the
data collection focused on open-ended approaches to gathering self-reported data for
studying beliefs and goals: interviews rather than forced-choice survey instruments. This
follows the suggestion of Aikenhead, Fleming, and Ryan (1987) who critique the use of
Likert-scale instruments in studying students’ perspectives. Considering students’
perspectives is important because teachers could benefit from an increased awareness of
the complexity of students’ roles in the classroom micro-culture and the larger school

culture (Sarason, 1982), and the social-emotional consequences of engaging students in

61

discussion in mathematics class remain largely unexplored (Chazan & Ball, 1995).
Research suggests that teachers could improve upon their assessment of their students’
motivations (Givvin, Stipek, Salmon, & MacGyvers, 2001).
Research Questions
In order to study adolescents’ motivations for participating in Standards-based
mathematics classrooms, I designed my methods of inquiry to address the following
primary research questions:
1. What are students’ beliefs and goals?
2. What are students’ patterns of involvement in their whole-class discussions
about mathematics?
3. What are the relations between students’ beliefs, goals, and their patterns of
involvement?
Each of the primary research questions was supported by a series of secondary questions,
which were as follows:
1. What are students’ beliefs and goals with respect to learning mathematics in
discussion-oriented classrooms?
a. What are the relations between students’ beliefs and goals?
b. To what extent are the target students’ beliefs representative of the
populations of their classrooms?
2. What are students’ patterns of involvement during whole-class discussions about
mathematics?
a. What are the similarities and differences between the discussions in each

of two mathematics classrooms?

62

b. How do students’ patterns of involvement differ by classroom?
3. What are the relations between students’ patterns of involvement and their beliefs
and goals?
a. Do students who express particular beliefs involve themselves differently
in whole-class discussions?
b. Do students who express particular goals involve themselves differently in
whole class discussions?
0. Do students who express particular clusters of beliefs and goals involve
themselves differently in whole-class discussions?
The purpose of this chapter is to discuss the context and approach to my inquiry.
First off, I will describe the setting of the study, including the two classrooms and the
target students. Then, I will discuss methods of data collection and analysis as related to
each of the three primary research questions.
Setting: School, Curriculum, and Classrooms
This study took place at Two Rivers5 Middle School (grades 6—8), the single
middle school in a school district serving a rural community in Mid-Michigan. I spent
approximately 100 total hours in two mathematics classrooms at Two Rivers Middle
School during the 2002-03 school year as a participant-observer for the purpose of this
study.
The school is located in a small town, according to the National Center for
Education Statistics. The occupations of the parents and adult community members of
this town are a mix between farmers and commuters to the more urban settings, which are

located approximately 30 miles from their town. The school enrolls approximately 440

 

’ All names are pseudonyms.

63

students each year, approximately 150 students per grade. In 2001 , 98.4% of the student
body was White, 0.7% Native American, 0.7% Hispanic, and 0.2% Black. 12.4% of the
student body received free or reduced-price lunch under the National School Lunch
Program as a result of low family income.6

Standards-based mathematics curriculum. I selected this school because the
students’ experience with Standards-based curricula and teaching was not novel. The
mathematics cunicular experience for students in this school district starts with the
Investigations textbook series (TERC, 1998) in elementary school and moves into the
Connected Mathematics Project [CMP] textbook (Lappan, Fey, Fitzgerald, Friel, &
Phillips, 1997) in middle school. These two curricular series are complementary in that
their development was funded by the National Science Foundation in response to the
calls of the NCTM Standards (1989, 2000). Most studies of students’ experiences with
Standards-based texts are conducted during early stages of implementation. In contrast,
these students only know Standards-based mathematics curricula.

As there is a low attrition rate in this school,7 most of the students at Two Rivers
Middle School have studied mathematics with a Standards-based text from early in
elementary school until the present. The implications of this include students
experiencing problems in real-world contexts and experiencing teaching practices that
included, to some degree, small group work and whole-class discussions for most of their
time in school.

Another reason for selecting this school was the teachers’ experiences with this

textbook series. The school was one of the initial sites to implement CMP, and has used

 

6 Statistics from: http://www.ses.standardandpoors.com/ (04/05/03)
7 Participants reported attending school with their classmates since kindergarten, generally.

64

the textbook series since it was written. The mathematics teachers at this school across
the grade levels have participated in professional development with the authors of CMP,
worked with the authors as consultants, and have presented about their teaching with
CMP to other teachers across the country. The administration and the teachers
demonstrated overarching support for this mathematics curriculum.

In 2002-03, the school was piloting the revision of CMP texts. As a result, the
students had less colorful texts and the teachers did not have teachers’ editions. The
teachers mentioned not having teachers’ editions affecting them in terms of not always
being told the larger mathematical concept of the investigation and not having specific
guiding questions to use in whole-class discussion. Since they would normally have this
information in a teachers’ edition, sometimes they would email or call the textbook
authors for insights.

Two of the three seventh grade teachers at this school participated in this study
during the 2002-03 school year. I selected two teachers because more than one classroom
would afford a broader range of students’ beliefs and goals, as well as patterns of
involvement in whole-class discussions, that would not be solely shaped by the teacher. I
did not choose to work with the third teacher in order to keep the sample size manageable
for an exploratory study, and because, based on communication with these two teachers,
other teachers at this school, and members of the textbook authors who were familiar
with this site, the third teacher was relatively less faithful to the curriculum and did not
attempt to coordinate with the other two teachers. Below I describe the two teachers and
their classrooms that participated in this study, Mrs. Evans’s fifth-hour class and Ms.

Carson’s third-hour class. These two teachers’ class periods were selected because they

65

were both seventh grade sections and they were relatively close to one another in time of
day, and also near each teacher’s planning period or lunch hour, in case I needed time to
interact with the teachers.

Mrs. Evans’ fifth-hour class. Mrs. Evans participated in this study during her
16th year of her teaching career. She was certified to teach elementary school and was in
her 9th year of teaching middle school mathematics at Two Rivers Middle School and of
teaching with the CMP texts, primarily seventh and eighth grade. Mrs. Evans was one of
the original teachers invited to pilot the revised 7th grade CMP curriculum. She worked
with the curriculum developers during the school year and summertime to design and
carry out professional development for CMP teachers as well as evaluate new problems
and construct assessment items. In her experience at this middle school, she had also
taught some sections of science. I informally observed two of her classes, a seventh grade
and eighth grade section, on an average of one day a week during the 2001-02 school
year prior to this study. The purpose of this pilot work was to characterize features of
whole-class discussion about mathematics. During 2002-03, all of her classes were
mathematics, three eighth grade sections and two seventh grade sections.

Her fifth-hour seventh grade CMP students participated in this study. The class
periods were 58 minutes long, beginning at 12:21 pm, and it was the second class period
after the seventh graders’ lunch and the second to last period before the end of the day. In
the fall, this class consisted of 24 students, 10 males and 14 females. In the spring, 27
students were in the class, 15 females and 12 males. As students’ schedules changed
between fall and spring semesters, there were 10 new students in the class and 7 students

were no longer in the class. Mrs. Evans mentioned to me on more than one occasion that

66

her 5th hour was her bi ggest challenge to manage in terms of classroom behavior after the
students’ schedules changed for spring semester, potentially due to the combination of
the time of day and the mix of personalities in the class.

Ms. Carson’s third-hour class. Ms. Carson participated in this study during her
4th year of teaching, and she taught with the CMP text for all of these years. She is
certified to teach at the secondary level. 2002-03 was her second year of teaching at Two
Rivers Middle School, as she changed schools after her first two years of teaching. She
learned how to teach with the CMP curriculum through coaching from another
experienced CMP teacher during her first two years of teaching at her previous school,
and through collaborations with her experienced colleague at Two Rivers Middle School,
Mrs. Evans. They talked almost daily at lunch, before school, after school, or between
classes about how their students were performing on similar units and about their
approaches to teaching different units. They appeared to collaborate with each other more
than with the other seventh and eighth grade mathematics teacher. Mrs. Carson has also
participated in multiple professional development programs Sponsored by the CMP
textbook developers. During 2002-03, Ms. Carson also taught three sections of eighth
grade math and two sections of seventh grade.

Her third-hour seventh grade class participated in this study. This class period was
also 58 minutes long, beginning at 9:41 am. In the fall, the class consisted of 21 students,
11 females and 11 males. In the spring, 18 students were in the class, 10 males and 7
females. As students’ schedules changed between fall and spring semeSters, there were 2
new students in the class and 5 students were no longer in the class. Mrs. Carson

mentioned to me on more than one occasion that upon the schedule change for the spring

67

semester, her 3rd hour happened to be her lowest performing class academically. This
class was also her smallest enrolhnent section.

Fidelity to CMP. Both teachers were committed to following the CMP texts,
utilizing the range of materials and teaching suggestions made by the authors, such as the
text’s reflection questions, and having their students keep a notebook that included their
responses to these reflection questions and also their work on investigation problems,
homework, and vocabulary words. The teachers did not supplement the textbook with
worksheets of practice problems. The teachers did implement partner quizzes, as
suggested by the textbook authors, about once a chapter.

Participants: Target Students

The analyses in this study involve data fi'om 15 target students during the Spring
of 2003. I selected seven students (out of 21) in Ms. Carson’s classroom and eight (out of
24) students in Mrs. Evans’s classroom to be target students. I initially selected 20 target
students in the Fall of 2002, 10 from each classroom through purposeful sampling (Miles
& Huberman, 1994; Patton, 1990) along the dimensions of gender and fi‘equency of
participation in classroom discussion, as judged from my fall observations. I shared the
selection of participants with the teachers to inquire as to whether I also would have
diversity along the dimension of achievement. The teachers both agreed that I had a
broad representation of the range of students in the classrooms, based on achievement
(performance on the first quiz) and participation. Table 3.1 presents a list of participants

by teacher, gender, and level of observed participation in the fall.

68

Table 3.1: Participant Information

 

 

 

Psuedonym Teacher Gender Fall Involvement
Abby“ Carson F Low
Allen Carson M Moderate
Allison Carson F Moderate
Colleen Carson F Low
Hannah Carson F Low
Laura“ Carson F High
Max Carson M Moderate
Pete Carson M Low
Shawn"I Carson M High
Tim Carson M High
Alex Evans M High
Alyssa Evans F Low
Becky Evans F Moderate
Bill Evans M High
J on" Evans M Low
Marissa Evans F Low
Molly Evans F High
Robert“ Evans M Moderate
Steve Evans M Moderate
Tricia Evans F Moderate

*F all participant only.

There were approximately three students at each level of involvement from both
classrooms in the fall, with one additional low involvement student from Ms. Carson’s
class and one additional moderately involved student fiom Mrs. Evans’s class. These
students were representative of the demographics of their school in terms of race: all of
the target students were White, as were most of the students at this school. I did not
obtain data as to whether any of the participating target students received free or reduced
lunch. Due to the changes in classroom populations at the beginning of the Spring

semester, as described above, three of the fall target students from Ms. Carson’s class and

69

two of the fall target students from Mrs. Evans’s class were not included in the analyses
of this study from the spring.

This study focused on the data during the Spring semester primarily due to the
surprisingly large (to me) change in classroom population. Initially, I intended to study
change over time in students’ beliefs and goals and their involvement in whole-class
discussion. The high turn over at semester was an additional variable I had not accounted
for in the design of this study, and changes in target students’ participation may have
been influenced by the change in classroom population. In this study, then, I did not
examine the change over time in students’ beliefs and goals or their involvement in
whole-class discussion. Rather, this is a descriptive study of how students’ express beliefs
and goals related to learning mathematics at one point in time. Results describe whether
and how these beliefs and goals relate to students’ involvement in whole-class
discussions during this point in time: late February through March, 2003, while the
students were studying an algebra CMP unit: Moving Straight Ahead.

Recall that I designed the study in order to address three primary research
questions. The following table illustrates how the data sources correspond with my three

research questions:

70

Table 3.2: Data Sources in relation to Research Questions

 

 

Research Question Data Sources

1. What are students’ beliefs and goals 1. a. Interviews with 15 target
with respect to learning students.
mathematics in discussion-oriented 1. b. Likert-scale survey instruments
classrooms? given to population of each

classroom.

2. What are students’ patterns of 2. Videotaped observations of whole-
involvement during whole-class class discussions from each
discussions about mathematics? classroom.

3. What are the relations between 3. Analyses of above data sources

students’ patterns of involvement

and their beliefs and goals?

 

Data Collection: Students’ Beliefs and Goals
I had two data sources for data on students’ beliefs and goals: interviews and
surveys. I will first discuss the interviews, and then I will discuss the surveys.
Interviews. Interviews were conducted in late March or early April with the 15
target students, and they were one-on—one, between the target student and myself. Each
interview lasted between 30 and 45 minutes, and was conducted during the student’s
math class in a small room either in the school’s library or the front office, depending on

availability.

71

The interview was designed to elicit data on students’ beliefs about learning
mathematics. I began the interview with open-ended questions to assess students’ beliefs
about learning mathematics, such as describe a successful mathematics student or a good
mathematics teacher. One of the open-ended questions, about describing a successful
mathematics student, was inspired by Spangler (1992). As the interview continued, I
asked the students to elaborate on some of their responses to items from a survey
instrument (5=True, l = Not At All True) that students completed in early March,
following the technique of Aikenhead et a1. (1987).

Some interview questions were more conducive to eliciting data on students’
beliefs and goals about learning mathematics in discussion-oriented settings. Examples of
interview questions that consistently elicited data relating to students’ motivations to
participate in whole-class discussion about mathematics included:

0 Let’s say that your school needed to hire a new 7th grade math teacher, and
they wanted to get some students’ ideas about what would make a good
7th grade math teacher. What would you tell them?

0 Let’s say a student moved into your classroom, and they wanted to know
what they had to do to be successful in math, what would you tell them?

0 Elaborate on your response to this survey item: When my teacher asks a
question in math class, it is important that I explain how I did the problem,
not give my answer.

0 Elaborate on your response to this survey item: To work on math
problems, I have to be taught the rules and steps, or else I can’t solve

them.

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0 Do you think about yourself as somebody that likes to talk about their
ideas in class or somebody that would rather listen to other people’s ideas
or in between?

A complete list of interview questions can be found in the appendix.

Survey instrument. The survey instrument, Students ’ Mathematical Views
[SMV], was a forced choice Likert-scale instrument with 63 items (see Appendix B for
items and scales). The SMV survey was administered after a quiz during the students’
mathematics class. I selected these scales based upon themes in previous research on
students’ beliefs and goals associated with learning and doing mathematics, such as
research on students’ confidence in mathematics (Kloosterman, 1991), students’ task and
ability orientations (Ames, 1992; Dweck, 1986), students’ autonomy in learning and
doing mathematics (Y ackel & Cobb, 1996), the usefulness of mathematics (Boaler,
1997), the interconnectedness of mathematics (Gfeller, 1999), whether students View
mathematics as conceptual or calculational (Thompson, Phillip, Thompson, & Boyd,
1994) or procedural or relational (Skemp, 1978), and whether students considered the use
of multiple methods when solving a problem to be an important part of doing
mathematics (Weinstein, 2000).

The scales on the survey drew from previously published instruments. I designed
the survey from scales used in a previously constructed survey or based upon single items
fi'om instruments used in previous studies. The nine scales were confidence (F ennema &
Sherman, 1976), task orientation (Midgley et al., 1996), ability orientation (Midgley et al,
1996), process versus product (Schoenfeld, 1989), autonomy and authority (Grouws,

1994; Schoenfeld, 1989), usefulness (Fennema & Sherman, 1976; Grouws, 1994),

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structure (Grouws, 1994), conceptual versus procedural (Grouws, 1994; Schoenfeld,

1989), and multiple methods (Grouws, 1994; Schoenfeld, 1989). The confidence, task

orientation, ability orientation, structure, and usefulness scales were imported intact fi'om

previously established instruments, and the process versus product, autonomy and

authority, conceptual versus procedural, and multiple methods scales were written by

myself based upon items used in previous research. Table 3.3 presents the name of each

scale (with its eventual reliability scores), its definition, and a sample item.

 

 

 

 

Table 3.3: Survey Scales
Title of scale Definition Sample Item
(0!)
Confidence I am good in math and 27. I am sure that I can learn
(0.91) successful in math class vs. I am math.
not able to learn or be successful
10 Items in math class.
Task Orientation I value developing a deeper 2. I like doing problems in
(0.83) understanding of mathematics math class that I'll learn
and working hard. from even if I make a lot of
6 Items mistakes.
Ability Orientation I value getting good grades and 3. I want to do better than
(0.87) being recognized for my other students in my math
6 Items successes when learning. class.

 

Process vs. product
(0.39)

The process (or the product) is
most important to focus on when
learning mathematics.

16. When my teacher asks a
question in math class, it is
important that I explain how
I did the problem, not just

 

 

 

6 Items give try answer.

Autonomy & authority Students are capable of 40. To work on math

(0.43) discovering mathematical problems, I have to be
patterns vs. the teacher should taught the rules & steps, or
provide the students for clear else I can’t solve them.
rules for doing the math

7 Items problems.

Usefulness The extent the mathematics we 22. Math is a worthwhile

(0.67) do in school is related to life subject for me.

8 Items outside of school

 

 

 

74

 

Table 3.3, continued

 

 

 

Structure Mathematics is an interrelated 21. Finding answers to one
(0.56) set of ideas VS. made up of type of math problem
isolated ideas cannot help you solve other
8 Items types of problems.
Conceptual vs. Mathematics is primarily about 8. The math that I learn in
procedural ideas vs. mathematics is school is mostly a set of
(0.33) primarily about procedures rules to memorize
lltems
Multiple methods Mathematics problems have one 35. It is possible to approach
(0.37) right solution vs. multiple the same math problem in
solutions are possible for solving more than one way
iiltems the same problem.

 

 

 

 

 

Reliability scores were high for the confidence and ability orientation scales (above 0.80),
moderate for task orientation, usefulness, and multiple methods (above 0.60, but below
0.80), and low for the rest of the scales. The entire survey instrument can be found in the
appendix.
Data Analysis: Students’ Beliefs and Goals
In order to analyze students’ beliefs and goals, I analyzed both the interviews with
students and their survey responses. Interviews were the leading data source, since I was
not certain I was aware of the range of students’ beliefs and goals prior to this study in
Such a way that would allow me to design a survey instrument. Therefore, the survey was
generally insufficient for addressing Research Question #1. I transcribed the audiotapes
0f the interviews myself. Rather than analyzing the interview data solely for content in
Students’ talk, I applied an additional level of rigor through the development of an
analytic framework for examining cues in talk that suggests the existence of beliefs.
Analytic framework: Analysis of students’ interviews. In complement to

studies of students’ beliefs and goals through the use of forced choice instruments,

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students’ talk in their interview statements was analyzed through an examination of three
types of language cues. Through a close analysis of students’ talk, the following language
cues appeared to provide evidence of beliefs and goals: (1) verbs expressed in terms of
idealized states or preferences, (2) affective statements, and (3) repetition.

The subjective knowledge of beliefs is considered to exist in the form of idealized
states. Abelson (1979) suggests that beliefs do not exist only at the level of personal
truths (Polanyi, 1958), but additionally manifest in the form of idealized states.

Belief systems often include representations of ‘alternative worlds,’ typically the

world as it is and the world as it should be. The world must be changed in

order to achieve an idealized state... (Abelson, 1979, p. 357)

Beliefs held in the form of idealized states focus on how the world should operate rather
than how it does operate. These idealized states may be recognized through a
participant’s use of stating the imperative Ge, “1 need to...” “I have to...” “I should. . .”)
to express their perception that their world Should operate in particular ways. The
conceptualization of beliefs as idealized states may lead to a sufficient (but not necessary)
criterion of imperative statements in the participant’s responses for indication of a belief.
Bills (1999) utilized similar linguistic cues, modal auxiliaries (may, might, can, could,
shall, should, will, would, ought, need), in order to study students’ attitudes when talking
about math.

An example of the use of imperative statements to express a belief can be seen in
this interview with a 7th grade student, Molly,8 who was asked to discuss her responses on
a beliefs survey instrument. She used a series of imperatives to describe the importance

of understanding in mathematics.

 

8 psuedonyrn

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AJH: OK. The math I learn in school is mostly a set of rules to memorize. You

put 1, not at all true, what do you think?

MH: It’s not just to memorize, you have to understand it.

AJH: OK.

MH: They’re not just, there are rules, but you have to know, like, why it’s a rule

and why it’s so important, so I don’t think it’s true that you just have to memorize

it. You also have to know its meaning and understand it.

AJH: Yeah, so memorizing’s not enough.

MH: No. Not in math. (10/10/02, emphasis added)

Her repeated use of“. . .you have to...” suggests a belief in understanding as an important
element in learning mathematics beyond memorizing procedures. The relationship
between the idealized states of beliefs and imperative statements may assist in analyzing
verbal data, such as interviews and questionnaire responses, for evidence of beliefs.
Similarly, goals may be expressed in terms of desires or preferences, such as “I want
to...” or “I hope to...” An examination of verb usage could reveal some evidence of a
belief or goal.

Carefully attending to participants’ expression of affect can also help the
researcher to establish whether the evidence lends itself to claims about beliefs, values,
goals, or attitudes. Hannula (2002) focused on students’ affective statements as evidence
of students’ attitudes, and beliefs are closely related to affect (McLeod, 1992). If students
discuss an emotional reaction, the psychological backdrop to their reaction is evidence of

their beliefs.

77

Another excerpt from Molly’s interview illustrates how a student may
demonstrate attitudes and beliefs in conjunction through her affective statements.

MH: I like the way she teaches in that class.

AJH: What do you like about it?

MH: I like that she put, she makes it frmny, and she gets the whole class involved,

so, I don’t know, it makes it a little bit more frm to learn to do. (10/10/02)
Molly’s attitude toward learning mathematics was a positive reaction, as she enjoys her
teacher’s approach and her experience of learning. This enjoyment appears to be
connected to her belief that the teacher should express her sense of humor and get the
students involved in the class. Again, these statements alone are not sufficient evidence
for a belief, but are selected to illustrate how looking at affective statements can reveal
relations between beliefs and attitudes. Similarly, affective statements could provide
firrther evidence for a goal, as students may emphasize reasons for their behavior with a
sense of affect. Stronger evidence for a belief or goal occurs when evidence of affect
converge with additional evidence of verbs indicating idealized states or preferences.

Tannen (1989) suggests that one of the functions of repetition in discourse is to
aid comprehension through providing redundancy. Repetition can also be thought of as a
form of emphasis as seen in songs, poetry, or oratorical discourse. Instances of repetition
in the students’ interview statements, especially when triangulated with affective
statements and statements of idealized or preference verbs, may also suggest a belief or
goal, as the repetition may indicate a form of emphasis. For example, we see above in

Molly’s affective statement the repetition of “she makes it funny,” and “makes it a little

78

bit more frm.” Also, in the example of her idealized statement, she repeats that
memorizing is not important, echoing the survey statement.

In order to identify themes in the content of students’ beliefs and goals, data
analysis was conducted through constant comparative analysis of interview transcripts
(Glasser & Strauss, 1967), with the use of the analytic framework of idealized states,
affect, and repetition to guide the process. Students’ talk in interviews had to meet two of
the three criteria in the analytic framework in order to be considered as expressing a
particular belief or goal in their interview talk. This process allowed for building codes
and allowing themes to emerge from the data, as an alternative to forced choice survey
instruments. I used the qualitative software NVivo to help me track my process and
organize my data. Codes were verified through the use of peer debriefing. Cobb &
Whitenack (1996) suggest two types of peer debriefers, those who are familiar with the
participants and the study and those who are not. I tested the credibility and resonance of
my analysis by sharing data with colleagues who were more and less familiar with this
study for feedback on the development of codes and relations between them.

The analyses of student’s interviews went through multiple cycles of analysis.
Initial code development was based on interviews conducted in the fall. I developed
general codes motivated by research on students’ motivation, such as self-efficacy,
learning goals, and codes for students’ beliefs about their own role and their teacher’s
role in learning mathematics. The spring interview questions focused more directly on
students’ participation in classroom discussions, which resulted in the revision of the
initial set of codes to include students’ social goals and perceptions of risk associated

with participating in whole-class discussions. Students’ beliefs about the teacher and

79

students’ roles in learning were collapsed into codes for students’ epistemological beliefs
about learning mathematics. Students’ perception of risk seemed to capture students’
motivations for participating more so than self-efficacy, based on the students’ talk in
interviews, so the self-efficacy code was eliminated. Specific codes used in this study,
and the results from these analyses, are discussed in Chapter Four (Results).

Survey analyses. I collected survey responses from all students in both
classrooms (N = 42 in the Spring), and I imported the numerical data into SPSS for
statistical analyses. I wanted to address two sub-questions with the analysis of survey
data. How did the target students’ beliefs compare to the populations of their classrooms?
Do students in different groups (classrooms, groups by shared beliefs or goals) express
different beliefs? I used unpaired t-tests to compare groups of students for each of the
scales presented earlier. I used a significance level of p < 0.05 to determine differences
between groups.

In order to compare target students to the populations of their classrooms, I
compared the target student’s mean on each scale to their class mean on that scale in
terms of standard deviations. If the student’s mean was within his or her class’s standard
deviation, I considered the target student’s mean to be similar to the classroom
population. If the target student’s mean was more than one standard deviation above or
below the mean, I considered the target student to be different fiom the class population.
This identification of target students into three categories relative to the views of their
entire class (above, within, or below the class mean) was useful to see the extent to which

target students expressed similar beliefs to their classmates.

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Data Collection: Students’ Involvement in Whole-Class Discussion

I visited each classroom and videotaped observations of all whole-class
discussions between February 10 and March 7, 2003. I videotaped whole-class
discussions during the CMP unit Moving Straight Ahead. These days encompassed work
on the series of problems fi'om the first two Investigations from the unit. An Investigation
is a set of two, three, or four exploratory problems along a theme, both a thematic context
and a mathematical theme. I did not videotape after March 7, because the teachers did not
want to be videotaped during the third and fourth investigations, because these
investigations were being pilot tested as part of the revisions of the CMP texts.

From this corpus of video records, I analyzed four consecutive instances of
whole-class discussion fi'om each classroom. In addition to choosing these four days
based on the consecutive coverage of material, I asked each target student to watch three
to five minutes of video from their classroom’s discussion of Problem 1.3, and I asked
whether this was a typical whole-class discussion for their class. They all agreed that the
video captured a typical class discussion. So, based on their reactions, I claim that these
days were representative of the whole-class discussions in these two classrooms. I did not
analyze the first few days of videotaping in order to give the students time to get used to
the camera being around, as these days involved some initial attention-seeking moves
from the students that faded somewhat over time. Table 3.4 describes which days of

whole-class discussions that I analyzed.

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Table 3.4: Analyzed Discussions

 

Ms. Carson Mrs. Evans

 

2/20/03 (Problem 1.3, 42 min discussion) 2/20/03 (Problem 1.3, 45 min)

2/21/03 (ACE day, 7 min) 2/21/03 (Problem 1.3 cont, 33 min)
2/25/03 (Problem 2.1 & 2.2, 15 min) 2/25/03 (Problem 2.1 & 2.2, 30 min)
2/26/03 (Problem 2.3, 20 min) 3/03/03 (Problem 2.3, 20 min)

 

There are evident lapses in time between videotaped discussions of problems. February
20 and 21 were a Thursday and Friday, respectively. On Monday, February 24, both
classes used the time for seatwork and other class business. The days between Tuesday,
February 25, and Monday, March 3, for Mrs. Evans’ class included an “ACE. day’” on
February 26, a teacher in-service day on February 27 and a day off from school on
February 28 for the end of the marking period.

The problems discussed over these four days included problems 1.3, 2.1 & 2.2 (as
a joint problem), and 2.3. The mathematical content of these problems involved linear
relationships. This content was significant to the research, because students’ transitioning
into algebra during middle school may provide new challenges for participating in whole-
class discussion. I will describe the problems in more detail below.

Problem 1.3 was about pledge plans for participating in a walkathon. In the
problem, the students are encouraged to compare three different pledge plans.

Leanne says that each sponsor should pay $10 regardless of how far a person

walks. Gilberto says that $2 per kilometer would be better because it would bring

in more money. Alana points out that if they ask for too much money, not as

 

9 An A.C.E. day is a class period spent working on problems similar to the investigation problems as
seatwork.

82

many people will want to be sponsors. She suggests that they ask each sponsor for

$5 plus 50 cents per kilometer.

In the text, students are asked to create a table, graph, and an equation to represent how
much money would be taken in by each of these pledge plans. Additionally, the text
asked students to locate particular values, such as how much a sponsor would owe for 8
kilometers, and the students are requested to write new equations, such as a pledge plan
with a larger or smaller rate than those Shown, and to describe a pledge plan that is not
linear. In both teachers’ implementations of this problem, students were asked to describe
how they knew the pledge plans were linear based on the table, graph, or equation, but
this question was not a part of the problem in the text.

Problems 2.1 and 2.2 were about two brothers, Henri and Emile, who were going
to race each other. The older brother, Emile, learned at school that his walking rate was
2.5 meters per second, and he timed his little brother Henri and determined his walking
rate to be 1 meter per second. Henri challenged Emile to a race, and Emile wanted to
design a race that Henri would win, but would still be close. In problem 2.1, students
were asked to come up with a suggested distance for the race such that Henry would still
win, but in which Emile would come close to winning. In problem 2.2, students were
asked to justify their answer with a table and a graph. They were also asked to create
equations and relate the y-intercepts in the equations to the y-intercepts in the graphs.

In problem 2.3, students were asked to compare price plans for either a
membership to a movie theater (a $49 membership fee and $1 per movie) or to pay $4.50
for each movie. Students were asked to compare the plans for a certain number of movies

(20), a certain amount of money ($120), to determine the number of movies in which the

83

cost of both plans would be equal, and to explain why the relationships in the plans are
linear.

Across these problems, a range of concepts and skills were discussed, including
graphing a line, constructing an equation for a linear relationship, creating a table for a
situation, comparing relationships to determine which one was smaller or lower for a
particular portion of the domain, determining whether a relationship was linear, and
moving between representations.

I Videotaped whole-class discussions over these problems with one digital camera
on a tripod. I placed the camera in one Spot in the classroom, either in the back of the
classroom or the fi'ont comer of the classroom, moving the camera to catch the image of
the speaker who had the floor. I zoomed in on a speaker if necessary for identification,
such as if it wasn’t clear who was speaking because the student’s back was to the camera.
I also drew a floor plan of the classroom to identify the speakers when I would watch the
videotape. The microphone on the camera was sufficient for picking up the dialogue
during whole class discussion.

Data Analysis: Students’ Involvement in Whole-Class Discussion

In order to prepare the classroom data for analysis, I transferred the digital
videotapes to .MPEG video files using Dazzle hardware and Movie Star software. I
downloaded these .MPEG files to compact disks for portability. I imported these files
into T ransana software for transcribing and qualitatively analyzing videotape data. This
software allowed me to use the computer’s keyboard to stop and start the video while

transcribing, and allowed me to timestamp the transcript to match up with the video. I

84

transcribed the clips of whole-class discussion myself, making notes along the way about
target students.

My unit of analysis for coding classroom discourse was an interaction segment. I
define an interaction segment as either a student or teacher initiated series of turns in an
interaction around a consistent topic. Interaction segments were a useful unit for the
analysis of the content of students’ talk during discussion because they encompassed both
questions and replies. I segmented all transcripts into interaction segments. Then I coded
all of the interaction segments that involved target students’ vocal participation. Target
students participated in 72% of the overall interaction segments from Mrs. Evans’s class
and 65% of the overall interaction segments from Ms. Carson’s class. More than one
student may have participated in an interaction segment.

I was interested in developing codes for capturing students’ autonomy, initiative,
or confidence in their talk, as well as instances of mathematical justification. I selected
these themes before analyzing the data because the goals of conducting discussions
during mathematics class are often to promote active sense making and intellectual
autonomy among students as well as promoting mathematical reasoning and
communication. To examine these themes, I developed codes through a constant
comparative process (Glasser & Strauss, 1967). Specific codes, and the results fiom these
analyses will be presented in Chapter Five (Results), as they were indeed results of

intensive work with the video data.

85

Data Analysis: Relations Between Students’ Beliefs and Goals and Their
Involvement

My primary goal in this study is to analyze whether students who express similar
beliefs and goals also participate in similar ways during whole-class discussion. In order
to pursue this question, I created groups of students based on their shared beliefs or
shared goals. For each belief or goal, there were two groups of students. Usually these
groups were constructed by putting the students who did express a particular belief or
goal in one group and those who did not express the belief or goal into another group.
Then I counted the number of times the student participated in a particular type of
interaction segment, calculated the percentage of participation according to that type of
interaction segment for each target student, and then calculated the mean and standard
deviations of percentages of participation in each type of interaction segment for the
belief or goal groupings of students. To determine whether the “belief groups” of students
participated differently, I calculated unpaired t-tests. Results from these analyses are
presented in Chapter Six (Results).

I conducted similar analyses, grouping students by teacher rather than beliefs or
goals, in Order to explore whether or not target students participated differently by
classroom rather than belief group. These results are presented in Chapter Five.

My Role in the Research Process

Since I acted as both a participant-observer in these classrooms and an interviewer

of these students, I would like to be explicit about my role in these classrooms and my

relationship with the teachers and their students.

86

Initially, when I arrived in the fall, students in both classrooms labeled me as the
“camera lady.” Their exposure to me was as a person who stood behind a video camera
for two to three weeks each semester during their math class. I had the opportunity to
introduce myself to each classroom in the fall and invite the students to participate in the
study, but other than that, I had no interaction with the class as a large group. Mrs. Evans
referred to me by first name to the students, while Ms. Carson referred to me as Mrs.
Hoffrnann to the students. Also, the school librarian made me a badge with my name on
it, similar to those worn by the staff at the school, but the students who didn’t work with
me directly still only referred to me as the “camera lady.”

My videotaped observations were focused on whole-class discussions, so I moved
out from behind the camera during other parts of the class period. When students were
working independently or in small groups during seatwork, I would wander from table to
table, as both classrooms had tables with three or four students at a table. Sometimes I
took on the role of observer of these groups, saying nothing, but more often I would
interact with the students. I let the teachers know that they could consider me a resource
during seatwork time, because I am a former junior high mathematics teacher, I have a
bachelor’s degree in mathematics, and I am familiar with the textbook series. When I was
interacting with the students, I would ask questions such as, “How do you know you’re
right?” whether or not the student was correct. I would also ask them how they solved the
problems. Sometimes students would thank me for “helping” them, which I found
interesting, because I never gave the students explicit instructions for how to solve the
problems. I interacted with the students by asking questions rather than telling them how

to work out problems.

87

Students appeared to value the attention they perceived to receive from interacting
with me. Students in the class periods I visited would declare themselves as “special” to
the students in the preceding and following class periods during the passing period as I
was brining my equipment in and out of the classroom. Students in class periods that I
did not visit would ask me when I was going to videotape them. I do not see this
“specialness” as interfering with the videotaped discussions, as I did not observe
differences between the discussions on the days I videotaped (after the initial day or two
of attention-seeking behaviors) and the days I was observing without a camera before or
after conducting an interview. When completing survey instruments, students worked
quietly and intently, sometimes putting comments in the margins about their opinions
about the items. When I was interviewing students, they would ask me who was getting
interviewed next. I would tell them that students who asked to be interviewed would not
be getting interviewed. However, I had already selected target students, and I sometimes
interviewed students who had asked to be interviewed. It seemed as though my extended
time spent in the classroom helped the target students open up with me during the
interviews, since I was not a stranger and had invested in getting to know them and their
classmates.

I spent time talking with the teachers before and after class. Ms. Carson had a
preparation period immediately after my observation of her class, during fourth hour,
which intersected with Mrs. Evans’s lunch period. I would often debrief with Ms. Carson
about the day as I took down my camera equipment, and she would share with me what
she thought did and did not go well, in her opinion, during that class period. I responded

to these comments in an effort to empathize with her concerns, but I did not offer her

88

advice. Then, I would set up my camera in Mrs. Evans’s classroom during her lunch
hour. Sometimes these two teachers would chat with each other during this time period,
and other times I would talk with Mrs. Evans. Our conversations were mostly about what
was going on in Mrs. Evans’s life outside of the class period I was observing, such as
with her children or with the graduate level course she was taking that semester. These
conversations also focused on procedural aspects of the study, such as whether the next
day would be appropriate for videotaping and for scheduling times to administer the
survey or scheduling times to interview target students.

Some of my conversations with the teachers included me reassuring them that this
study was about their students rather than an evaluation of their teaching. These sorts of
reassuring conversations would occur usually after administrating a survey, and the
teachers would ask whether their students did well on the survey. This question surprised
me at first, because I did not look at the students’ responses on the survey items as an
evaluation of the teachers. However, it appeared as though they wanted their students to
respond in particular ways to the survey, and they would have considered some responses
to be better than others, while I did not have the same value associated with these
responses, nor did I have the same stakes associated with the students’ responses on the
survey that these teachers may have. Similar conversations also occurred after
interviewing some of the students. I responded to their questions by assuring them that no
one particular response was “better” than another, based on my orientation to the data. I
did not share individual students’ responses with the teacher, nor did they ask to see
them. The teachers may have been concerned with how my dissertation results would be

perceived by those who were not supportive of the CMP textbook series.

89

Results

The following three chapters present the results from these analyses. In Chapter
Four, I present the results of interviews with 15 target students and their expressions of
their beliefs and goals related to learning mathematics and participating in whole-class
discussion. Additionally, survey analyses are presented in Chapter Four. In Chapter Five,
I present the results of the analyses of four days of whole-class discussion in each of the
two classrooms. Common patterns of involvement among target students are presented,
as well as results as to whether or not target students participate similarly or differently in
each classroom. In Chapter Six, I present the results of the analyses of students’
involvement in discussion, and whether students involved themselves differently

depending on the beliefs or goals they expressed.

90

CHAPTER FOUR
RESULTS: STUDENTS’ BELIEFS AND GOALS IN

DISCUSSION-ORIENTED MATHEMATICS CLASSROOMS

Students’ motivations in general, and their beliefs and goals in particular, are not
commonly studied in a situated manner. If the study takes place in a particular subject
matter, it is usually to isolate a variable rather than to study the learning of that particular
content. This study focused on mathematics learning because students may associate
particular epistemological beliefs with this content that differ from their beliefs about
other subject matters, such as social studies or English. This study is situated in
discussions-oriented mathematics classroom because of NCTM’s recommendations for
promoting classroom discourse. What were these seventh grade students’ beliefs and
goals for participating in mathematics classroom discussions?

I structured this study of students’ motivations to include their beliefs about
learning mathematics, as these beliefs create the psychological context of what is possible
for students in their classroom activity, and their academic and social goals related to
their involvement in whole-class discussion during mathematics class. Although I
collected both interview and survey data from my participants, the majority of the data
presented in this chapter are from the interviews with the 15 target students. In this
chapter, I will present the target students’ beliefs and goals first fiom my analysis of their
talk in interviews, followed by analyses of students’ responses on survey instruments.
Findings from students’ interview data will be depicted by (a) describing students’ beliefs

and goals (b) presenting how beliefs and goals co-occurred, or clustered, for individual

91

students, and (c) briefly addressing potential relations between beliefs and students’
performance in their mathematics classes. Then, I will present the following survey
results from the population of students in these two classrooms: (a) relations between
beliefs, (b) differences between classrooms, and (c) whether the target students were
representative of their class.

Students’ Talk During Interviews: Beliefs and Goals

The principal goal for this study was to pursue an understanding of the nature of
students’ motivations to participate in discussions about mathematics. I wanted to focus
my analysis on a bottom-up examination of students’ talk in interviews, because whether
and how students’ motivations may support and constrain students’ involvement in
discussions during mathematics class remains largely unexplored. However, I had some
of my own conjectures that these students might express based on prior research and my
own experiences working with middle school students.

My selection of survey scales reveals my conjectures for the beliefs and goals that
these students would express. For example, I conjectured that students would be
motivated to participate through their learning goals, as assessed by the task orientation
and the ability orientation scales on the survey instrument. I also expected that students
who were mastery-oriented might participate in order to learn material, while those who
had more performance-oriented goals may be participating for recognition or to avoid
appearing unintelligent. Additionally, I conjectured that students would be motivated to
participate through their belief that mathematics was a flexible discipline, as assessed by
the multiple methods scale on the survey instrument. I conjectured that students who

believed that mathematics was a flexible discipline would participate in order to share

92

alternative solutions. While I had these conjectures, I did not assume that I knew enough
about students’ beliefs and goals in these settings to test hypotheses, so I relied more on
open-ended interview data to assess students’ beliefs and goals. The interviews analyzed
in this chapter were conducted in mid-March through early-April 2003.

I hypothesized that students’ beliefs and goals would vary as much within
classrooms as they did between them, as students have histories as learners and doers of
mathematics prior to their experiences in these classrooms, but I wanted to investigate
this assumption empirically in both survey and interview analyses. The surveys analyzed
in this chapter were collected in mid-March 2003.

The 15 target students expressed four clusters of beliefs and goals in their talk
during interviews. Each cluster consists of (a) the students’ perception of the level of risk
associated with participating in whole class discussion (social risk), (b) students’ beliefs
about learning mathematics (epistemological beliefs), and (c) students’ academic and
social goals, such as their task goals (their investment in completing the task efficiently),
their goal to appear competent publicly, goal to gain status among their classmates, their
goal to behave appropriately, and their goal to help their classmates. These four clusters
were far fi'om evident at the conclusion of the interviews. They only emerged after many
cycles of analysis and complementary reading of relevant research literature.

I describe these four clusters in terms of thematic titles: learn the material in a
public forum, gain attention, do the right thing, and help others while overcoming social
risk. Figure 4.1 represents the clusters of students’ perceptions of social risk,

epistemological beliefs, and academic and social goals.

93

Figure 4.1: Seventh Grade Students’ Motivations

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

P t' f E . t l . a1 Academic&
ercep Ion o prs emo ogrc S 'al G a1
Social Risk Belief °"’ ° 3
f N F fl Ila—- Complete
Low Social Negotiated <: Task
Risk knowing i "~\..\ r i
R J k J ‘J Appear
\3.,,‘.-°‘° L Competent J
r \ F N ......... r 1
fl...” ..uuuuuuonuzi G ’ Status
High Social Received ”3.1...." L am
Risk knowing ......... J
K 1 L ..... Behave
_J
N
Help
Classmates
L

Clusters of Beliefs and Goals

- """""""" : Learn the Material in a Public Forum
............ : Gain Attention

..................... : Do the Right Thing

 

: Help Others While Overcoming Social Risk

 

Initially, two clusters of students were formed based on relations between students’

perception of social risk and epistemological beliefs. Learn the material in a public forum

and gain attention, are clusters in which students perceived a low level of risk associated

with sharing their thinking publicly during a whole-class discussion about mathematics.

These students also did not believe learning mathematics was a process of receiving

94

knowledge — that is, of taking in the knowledge presented by an authority, such as the
teacher or a knowledgeable peer. Rather, they believed that that learning mathematics
was a process of negotiating ideas. Differences between these two clusters lay in
students’ academic and social goals, as the students in learn the material in a public
forum expressed goals of completing the task, while the students in gain attention were
less task oriented and more interested in appearing competent and gaining status for that
competence. Students in the other two clusters, do the right thing and help others while
overcoming social risk, are clusters in which students expressed a higher level of risk
associated with involvement mathematics class discussions. They also expressed the
epistemological belief that learning mathematics was a process of receiving knowledge
from an external source. But there were differences between these two clusters in the
students’ social goals. The students in help others while overcoming social risk shared a
social goal of helping their classmates, while students in do the right thing shared social
goals of behaving appropriately, appearing competent, and gaining status among their
classmates. The latter group did not take on the goal to help others; they focused on their
own learning.

These four cluster categories were not distributed equally across the 15 target
students. Overall, as Table 4.1 shows, there were fewer gain attention and do the right

thing students across the two classes.

95

Table 4.1:

Target Students in Each Belief Cluster

 

Ms. Carson’s students

Mrs. Evans’s students

 

 

 

 

 

Learn the material in a Tim, Allison Alex, Steve, Becky
public forum

Gain attention Molly, Bill

Do the right thing Max, Pete Marissa

Help others while Allen, Colleen, Hannah Alyssa, Tricia

overcoming social risk

 

 

 

The more common clusters among these target students were learn the material in a

public forum (N = 5) and help others while overcoming social risk (N = 5). Only one

cluster, gain attention, was limited to target students in one of the classrooms — though

given the small sample size, little can be concluded from this empty cell.

Before discussing the four clusters in greater detail, I will discuss the nature of the

students’ talk during interviews with respect to each of the beliefs and goals that make up

these clusters: perception of social risk, epistemological beliefs, and their academic and

social goals. I will also address the analytic process for claiming the student held a belief

or a goal in order to provide some justification, beyond the general methods pursued, for

the clusters.

Perceptions of Social Risk

During my interviews with target students, I asked them questions about their

experiences during classroom discussions, such as whether they would rather listen to

others talk or participate in the whole-class discussion. When students described their

96

 

preferences for involvement in whole-class discussion, a number of them (N = 6)

mentioned that participating in discussion felt threatening, because they could be

incorrect in front of others. I concluded that these students perceived participating in

whole-class discussion to be a high-risk activity. Other students did not View

participation as a threatening experience. Instead, they perceived participating in

classroom discussion to be a moderate to low risk activity. Table 4.2 summarizes the

definition for these beliefs about social risk and presents a quote to illustrate the nature of

students’ talk for each belief.

Table 4.2: Definitions for Perceptions of Social Risk

 

 

 

 

 

 

 

 

 

Perception of Definition Illustrative Quote
Social Risk

High Sharing thinking “...when I’m put on the spot, I kind of
publicly is go off track. I don’t know how. Every
threatening, because time I’m out on the spot in fiont of an
of the chance of audience, I just pm and I can’t really
being incorrect in think straight.” (Allen, 3/19/04, #70,
front of classmates. Ms. Carson)

Moderate Students were Talk from interview demonstrated
somewhat concerned some threat, but also some
about being incorrect, indifference.
but not consistently.

Low A lack of concern for “...if I get something wrong, then I can
being incorrect in see what I did wrong, and they’ll, like,
front of their they’ll help me and s_how me how to
classmates. glgjt.” (Molly, 3/13/03, #29, Mrs.

Evans)

 

As discussed in Chapter Three (Methods), one utterance or one turn in an

interview was not enough evidence for determining whether a student held a belief. I

used an analytic framework consisting of three criteria: repetition within the interview,

imperative verbs, and relation to affect. Allen and Molly’s quotes above illustrate some

of these frames in the underlined phrases (emphasis added).

97

 

Allen expressed the idea of being “put on the spot” repeatedly and a sense of
“panic” associated with this experience. He was speaking about his experiences
participating in class discussions. The high sense of affect, expressed through use of
“panic," indicates a high sense of risk associated with sharing his thinking during class.
He also repeated this feeling of being “put on the spot” in other turns during the
interview, with comments such as, “. . .going up in front of everybody, that just makes me
nervous” (3/19/04, #90). This increased affect around sharing one’s thinking publicly was
a common indicator of a high sense of risk in students’ talk.

Molly talked about being “wrong” in front of classmates by emphasizing the
benefits rather than a sense of negative affect. She talked about getting help and being
shown how to do an incorrect problem. Additionally, when asked directly about whether
she minded being incorrect in fiont of her classmates, she spoke as follows:

I’d like to see if I, because I could be right, and even if I’m not, I realll don’t feel

bad, because other people aren’t always right, either, so it, just, it doesn’t reflv

bother me. (3/13/03, #232, Mrs. Evans).
She repeated this sense of not feeling “bad” and that it “doesn’t really bother” her to be
incorrect. Since she mentioned benefits with being publicly corrected, mentioned not
associating negative affect with being incorrect in front of classmates, and spoke
consistently about this issue more than once in the interview, she appears to have a low
sense of risk associated with sharing her thinking in front of her classmates.

Students from both classrooms expressed varying levels of perceived risk. Table

4.3 illustrates the levelof risk reported by the target students in each classroom.

98

Table 4.3: Perceived Social Risk by Classroom

 

 

 

 

Ms. Carson’s class Mrs. Evans’s class
High Perceived Risk Allen, Colleen, Hannah, Alyssa, Marissa, Tricia
Moderate Perceived Risk Max, Pete
Low Perceived Risk Allison, Tim Alex, Becky, Bill, Molly,
Steve

 

 

 

 

 

The six students who spoke about perceiving a high level of risk associated with
sharing their thinking during class discussion all talked Similarly to Allen. They
expressed feeling threatened when they were not sure that they were correct. For
example, Colleen talked about only sharing her thinking when she was certain she was
correct.

Colleen: It depends on the problem. Usually I just like to sit and listen.

AJH: OK.

Colleen: If I know. _I_know the correct answer, Tm how I got it, then I would

kind of say it. (3/26/03, #61-63, Mrs. Carson)

The repetition of “I know,” suggests a level of confidence she must have before joining
the large group conversation. Otherwise, she expressed a preference for observing and
taking it all in, as she said “usually I just like to sit and listen.” Another target student,
Tricia, spoke about how it was important for successful mathematics students to
volunteer their thinking during class. When I asked her whether she volunteered her
thinking during class, she spoke as follows:

Tricia: Sometimes. I’m kind of really shy, so I’m like super-conscious about when

it comes to answering in front of people. I get, like, all nervous and stuff.

 

AJH: So, what do you mean, can you say a little more about that?

Tricia: Well, like, in math, I used to be, like, if a teacher called on me or

99

something, and I had my hand raised, my fgce would turn all red, and I’d get
really nervous and I’d start sweating, until I got the answer out, and, like, it, I was,

like, always nervous th_at it would be wrtmg. And, I’m not so much nervous any

 

more, but I’m still kind of worried about if the answer’s wrog, like, I’ll get, like,
messed up or something. (3/18/03, #52-54)
Tricia talked about her experience of being incorrect in fi'ont of the class with many

9, ‘6

affective terms (“really shy, super-conscious,” “worried”) with repetition (“nervous”),
suggesting that these experiences of sharing her thinking have an impact on her and are
threatening. Also, the repetition of “wrong” in her talk helps illustrate her fear: she does
not want to be incorrect in fi'ont of the class.

The seven students who spoke about perceiving a low level of risk associated with
sharing their thinking during class discussion spoke about the benefits associated with
sharing their thinking and did not mention threat associated with sharing their thinking
during class discussion. Becky described a successful mathematics student in the
following manner:

And when you have to do problems, don’t just sit there. You W get into the

conversation in order to actually get it yourself and me you understan_d it, don’t

just understand it like how other kids do it. (3/27/03, #42, Mrs. Evans)
For Becky, to understand (or to “actually get it”) means having her own ideas, as
indicated by the repetition of “understand.” Putting her ideas on the floor during the large
group discussion is a way for her to realize these ideas, as indicated by the use of

imperative verbs such as “have to” and “make you.” Another target student, Steve, spoke

about the importance not just paying attention, but also participating.

100

Both, really, because, like, you c_an_3 just sit back the whole time and just let
everybody else teach you. You’ll know how to do it, but you wouldn’t hgve tried
it, you wouldn’t l_cnow if it would work or not. So, if you’re out there, Mg to
throw out your ideas, you could actually find for all, a new way of doing a math
problem. (3/26/03, #92, Mrs. Evans)
Steve’s repeated use of “try” suggested that even some effort on the students’ part to
participate was necessary, because otherwise, “you wouldn’t know if it would work or
not.” Also, the use of the strong verb of “can’t” in reference to the process of receiving
knowledge showed the importance he placed on participation.

The two students whose talk suggested a moderate level of perceived risk
expressed both feeling threatened and seeing the benefits of sharing their thinking
publicly. For example, when I asked Max directly whether he was comfortable sharing
his thinking if he was not certain that he was correct, he spoke as follows:

Uh, no. I’m not right, I’m not going to raise my hand, if I don’t know if I’m right

or something. If I get it wrong and then she says it’s wrong, I don’t care but if I

 

think I got it right, then I’ll raise my hand, but if I don’t l_mow. then I probably
won’t. (3/26/03, #259, Ms. Carson)
While he didn’t have a strong sense of affect associated with being publicly incorrect (“I
don’t care. . .”), Max did consider whether or not he might be correct, by his repetition of
whether or not he is “right” or whether he “knows.” Sometimes he thought he would be
correct when he was not, and this did not bother him. Max did not choose to participate if
he did not think his answer was correct enough to share.

The difference between students with a moderate sense of risk and a high sense of

101

risk is that, while both groups hesitate to participate unless they are correct, those with a
moderate sense of risk do not have a sense of negative affect or threat connected with
being wrong in front of their classmates or teacher. Another target student, Pete,
demonstrated this when I asked him what it was like to volunteer to share his thinking.

Pete: Kind of, like, weird. You’re not sure whose is right. Theirs is right, or yours

is right, or you’re going to get wrong. That’s what.

AJH: OK. And do you mind that?

Pete: Not really. (3/25/03, #82-84, Ms. Carson)

While he recognized that he could be publicly incorrect, this did not appear to bother
him, other than it being “kind of, like, weird.”

The difference between students with a low and moderate sense of risk was that
the students with a low sense of risk explicitly mentioned benefits to sharing their
thinking during class discussion. Students with a moderate sense of risk did not talk about
these benefits in their interviews. Students with a high sense of risk mentioned the
perceived threat associated with involvement in whole-class discussion repeatedly
throughout the interview and often in response to an open ended question rather than a
question specifically about participation, while those with a moderate sense of risk did
not mention feeling threatened repeatedly. I also found that these differing levels of social
risk correlated with different views on what it takes to learn mathematics.
Epistemological Beliefs

As discussed in Chapter Three (Methods), my interview questions were also
designed to reveal evidence of students’ beliefs about learning mathematics. I asked the

students several open—ended questions to assess these beliefs, including what a successful

102

mathematics student should do and what a good teacher does. As a result, students’ talk
was rich with data about the process of learning mathematics.

Students talked about the process of learning mathematics either as receiving
knowledge from an authority (received knowing) or as negotiating between their thinking
and the ideas of others (negotiated knowing). Received knowing is similar to the
epistemological belief in Belenky, Clinchy, Goldberger, & Tarule (1986). Students with
this belief expressed that the work or wisdom of others was central to the process of
learning mathematics, and they did not talk about themselves as capable of creating
lmowledge. Negotiated knowing is a blend between Belenky et al.’s procedural and
constructed knowledge. Any departure from received knowing was considered an
advance, and the nature of mathematics as a school subject may not lead as naturally to
constructed knowing. Students with this belief had less reliance on authority. Instead,
they looked for feedback on their own thinking. Table 4.4 summarizes the definition for
these beliefs and presents a quote to illustrate the nature of students’ talk for each belief.

Table 4.4: Definitions of Epistemological Beliefs

 

 

Epistemological Definition Illustrative Quote
Belief
Received Knowing involves “...you gotta pay attention so you
knowing: obtaining knowledge know what’s going on, and if you
External source of from an outside don’t, then you’re pretty much lost,
knowledge authority. and you won’t be able to really gatih

up real fast, it might take you a while
to catch up, so you gotta reallxpay
attention and you ggtg listen a lot.”
(Allen, 3/19/03, #46, Ms. Carson)10

 

 

 

 

 

 

 

'0 (Psuedonym of student, date of interview, line number of turn, teacher’s pseudonym)

103

Table 4.4, continued
Negotiated source Knowing involves “. . .if you talk through it, and you,

 

 

of knowledge getting feedback on like, talk about it you might realize
one’s own ideas something you did wrong, if you

and comparing the talk about it, you might say, oops, I
ideas of others with timesed when I was supposed to
your own. divide or something.” (Molly,
3/13/03, #58, Mrs. Evans)

 

 

 

 

 

Allen’s quote contains some repetition and imperative verbs. His use of “pay
attention” and “catch up” repeatedly suggests not only the importance of staying focused,
but also a sense of difficulty associated with not staying focused. He said that if he did
not pay attention, he would be lost, which suggests that he needed to hear what is going
on around him to receive the knowledge. Additionally, the importance placed on paying
attention is indicated by his use of “gotta” (or “have to”). From this quote, we can
conjecture that Allen holds a belief of knowledge coming from an external source. At
least one other turn during the interview that met two of the three criteria was necessary
for claiming that a student held a belief.

Additional evidence for Allen’s belief in knowledge residing in an external source
was in how he talked about the teacher’s role in learning mathematics: “. . .you’ve [the
teacher] go_tta keep them [the students] under control, and you’ve gghta, like, kind of,
well, you’ve m kind of get their attention.” (3/19/03, #114) His repeated use of “gotta”
in reference to how the teacher needs to help the students stay focused was another
example of an imperative verb, emphasizing the importance of helping the students pay
attention. Though he focused on the teacher’s role, the implications of that role were
consistent with a student’s role of primarily paying attention to the teacher. This quote
demonstrates repetition within in the interview of the importance of students paying

attention. This idea repeated in other instances throughout his interview as well. This

104

sense of a need to pay attention over other activities during mathematics class suggested a
reliance on an external source of knowledge.

In contrast, Molly’s talk indicated that knowledge did not come exclusively from
an external source, as seen through the repetition and affect in her quote in the table
above. She repeated “talk through it” and “talk about it [the problem]” in the quote,
indicating a commitment to share her own thinking about the mathematics problem. She
also repeated “might,” as in “you might realize” and “you might say, oops.” While this
verb is more hesitant than an imperative verb, the content of her talk suggests that talking
about the problem can have a benefit, potentially learning more about the problem. Her
use of “might” could be because sometimes she was correct, and did not need to realize
what she did wrong. Her use of “oops” suggests affect associated with being incorrect,
and that she may not have had the opportunity to get her thinking corrected if she had not
shared it. Her knowledge of mathematics appears to come both from her own thinking
and from others, as she shares her thinking for feedback.

Additional evidence for Molly’s belief in knowing mathematics through
negotiation rather than an external source came from quotes in her interview, such as
when she was asked about how often students should be asked to share their thinking in
mathematics class.

In, like, I think every situation, because mitt people, like, learn djfifm ways

and mpg problems might have (him troubles. Like, I might find one

problem really easy or another problem really hard, so you need to explaifin it,

even if you do get it, even if it is hard. (3/13/04, #97, Mrs. Evans)

 

105

While she acknowledged that explaining is not easy (“even if it is hard”) and that
sometimes she did not need to be corrected when she explained (“even if you do get it”),
she also recognized that others could benefit from her explanations. Her repetition of
“different” indicates a sense of the range of learners’ understandings, and how some
students will understand one explanation over another. Thus, she said with an imperative
verb, “you need to explain it.” If students explain, others can learn if the problem was
hard, so students should try to figure it out for themselves and share, both for their own
benefit, and the benefit of others. For Molly, knowing was a process of negotiation of
ideas between the students and the teacher.

Students from both classrooms expressed evidence of both epistemological beliefs
in about equal numbers (Table 4.5).

Table 4.5: Epistemological Beliefs by Classroom

 

 

 

 

Ms. Carson’s class Mrs. Evans’s class
Received knowing Allen, Colleen, Hannah, Alyssa, Marissa, Tricia
Max, Pete
Negotiated knowing Allison, Tim Alex, Becky, Bill, Molly,
Steve

 

 

 

 

Although relatively more students in Ms. Carson’s class expressed knowing to be a
received process and more students in Mrs. Evans’s class expressed knowing to be a
negotiated process, these samples are small and represent less than half of each class. In
addition, my sampling within classes was not necessarily random. It is not reasonable to
claim that students fiom Ms. Carson’s class were more likely to believe in received
knowing.

The eight students who spoke about knowing as a process of receiving knowledge

gave similar accounts to Allen’s. For example, Marissa also spoke about the importance

106

of paying attention. When asked whether she would choose to participate in the
discussion or listen during class, she responded as follows:

I think paying attention. I 1&2, just, watching people more than I like jumping

right in, arguing, getting in the fight, all that good stuff. I think I’d rather just 199k

and pay attention and just, yeah. (3/31/03, #80)
Marissa expressed affect and a preference for watching, looking, and paying attention
repeatedly. Additionally, when asked what a good student needed to do, her first response
was, “I think, um, they have to pay attention.” Then she talked about not being
intimidated by the teacher’s jokes and the need to take notes, but then she revisited the
importance of paying attention. “I think you just have to pay attention. Keep yourself on
the ball, keep watching everybody.” (3/31/03, #40). She repeated the imperative verb of
“have to” as she repeated her statements about paying attention. Attention was an
important issue for about half of the target students when they talked about the process of
learning. These students did not talk about the importance of having their own ideas and
sharing them with others, often devaluing their own ideas in their talk. I interpret talk
about paying attention to the teacher as indicating beliefs of received knowing. Such talk
indicated that students were listening rather than speaking and focusing on the authority
in the classroom as the teacher rather than the students.

The seven students for whom learning was a process of negotiation likewise
spoke similarly to Molly. They did not emphasize paying attention in their talk about
learning mathematics. Rather, they talked about getting feedback on their own thinking.

Tim described how successful students participate in class.

107

I’d say, listen carefirlly to the teacher and other people when they share ideas and
participate by, just participate, because sometimes when you do answer
something, it just clicks in your head and then you know what you’re talking
about after you think about it a little bit, so just listen more in class, and then
you’ll be able to come up with more irkg about how easy to be able to do
problems. (3/13/03, #35, Ms. Carson)
He repeated the verb “participate,” and spoke in terms of commanding students to
participate in his verb tense. Also, he mentioned that in the activity of participating, the
student could come up with his or her own ideas, as they could “just click in your head.”
Another instance when Tim mentioned a preference for sharing his thinking in class was
in the following quote:
Well, a lot of times I want to know if my ideas are wrong or right, and if they’re

wrong, I want to know so I can correct the next time that I do it, so I like to be

 

picked on, just to, like, raise my hand, then she picks on me. Il_ike to be picked on
just as much as everybody else does. Maybe more. (3/13/03, #49)
He spoke of wanting to know and preferring to be called on by the teacher. This repeated
preference suggests a commitment to the practice of sharing his thinking for the sake of
being corrected. The preference suggests affect supporting his belief in knowing as a
negotiated process. He even claims that he wants his ideas to be critiqued in order to
determine if his thinking is correct or not. This was representative of many of the
negotiated knowers.
Relation Between Epistemological Beliefs and Level of Social Risk

Students’ epistemological beliefs about learning mathematics were coherently

108

related to their beliefs about the risk associated with sharing their thinking publicly
during large group discussion in mathematics class. Students who expressed that the
potential of being wrong in front of their classmates was somewhat threatening to them
also believed that learning mathematics was a process of receiving knowledge. Students
who expressed that involvement in class discussions was a low risk activity also believed
that learning mathematics was a process of negotiating knowledge. Table 4.6 displays the
target students by perceived levels of risk and epistemological beliefs.

Table 4.6: Students’ Perceived Levels of Risk by Epistemological Beliefs

 

 

 

 

High level of risk Moderate level of Low level of risk
risk
Received knowing allen, colleen, max, pete
hannah, ALYSSA,
MARISSA, TRICIA
Negotiated knowing tim, allison, ALEX,
BECKY, BILL,
MOLLY, STEVE

 

 

 

 

*Lower case: Ms. Carson’s class. Upper case: Mrs. Evans’s class

This relation between students’ epistemological beliefs and perceived level of risk
associated with classroom participation held true for students in both classrooms. This
consistency of a shared level of risk within the groups of students who expressed a
similar epistemological belief grouped the target students into two categories: (a) those
who expressed a belief in received knowing and expressed a high or moderate level of
risk associated with classroom participation and (b) those who expressed a belief in
negotiated knowing and expressed a low level of risk associated with classroom
participation. The relation between students’ perceived level of social risk and

epistemological beliefs establishes an initial set of two co-occurring beliefs, for two

109

 

clusters of beliefs.

Academic and Social Goals

Target students varied within these two clusters of beliefs in terms of their

academic and social goals. During the interviews, students expressed five different goals:

(a) an academic goal of completing the task efficiently (b) a social goal of appearing

competent in fi'ont of their classmates, (c) a social goal of gaining status for

improvements in their competence or ((1) their behavior, and (e) a social goal of behaving

appropriately, and helping their classmates. Table 4.7 summarizes the definitions for

these goals and presents a quote to illustrate the nature of students’ talk for each goal.

Table 4.7: Definitions of Academic and Social Goals

 

 

 

 

 

 

 

 

Goal Definition Illustrative quote
Complete Focus on own learning over Becky: I hk_e having to d_o
task classmates’. Emphasized a high them [math problems], like

priority of wanting to get own work writing them down, reading
finished efficiently rather than them, instead of, like, having to,
supporting their classmates’ like, Mrs. Evans says it, like,
understandings. she reads it when you do it as a
class. I guess I like that, like,
sometimes, but not all the time.
I’d rather sit down and dpit
myself. (3/27/03, #124, Mrs.
Evans)
Appear Show others that you understand or Harmah: Like, if I understand
competent can do the mathematics. Not something, and meme else in
necessarily about demonstrating my cla_r_ss doesn’t. or nobody
improvement in one’s competence. e_lse in my clas_s doesni, I’m
confident that I can do better
and maybe I can get more done.
(3/25/03, #90, Ms. Carson)
Gain status Focus on receiving recognition. Gain Max: I want to prove it that I

 

attention for having important
contributions about mathematics, earn
increased status from demonstrating
improvement in class, or gain
recognition for appropriate behavior.

 

know it better than I was.
l_asitime I didn’t really know a
l_qt_and this time I’m doing
pretty good. (3/26/03, #90-92,
Ms. Carson)

 

110

 

Table 4.7, continued

 

Behave

Follow teacher’s expectations for
good behavior. Not necessarily about
gaining recognition for this choice to
behave appropriately.

Marissa: Yeah, another thing
with the class is that people
disrupt, and that kind of throws
you off, about train of thought,
and during the problem, Mrs.
Evans has to stop, and it’s like
ah, you just want to yell out
something. Yeah, that’s another
thing, is people, I think that’s
kind of, not really intimidating,
but it’s just disrupting and ru_de.
_R_ud_e. (3/31/03, #62, Mrs.
Evans)

 

 

 

Help
classmates

 

Focus on classmates’ learning. Teach
classmates if you know more than
they might.

 

Tricia: Um, you should try to
hplp, like, explain it to people,
and give out your ideas, and,
like, it’s OK if they’re not right,
because people in the class will
Mp you understand them.
They’ll explain it to you, they’ll
h_elp you, like, figure it out and
stuff. So I think you should try
to volunteer as much as you
can. (3/18/03, #50, Mrs. Evans)

 

Among these five goals, one was more academic than the others: complete the

task. This goal was explicitly about their mathematical work. The other goals addressed

concern for themselves in the social context of the classroom (appear competent, gain

status) and concern for others (behave, help classmates). Some social goals were more

prominent in target students fi'om one classroom over another, while the academic goal of

complete the task was not found among target students in Ms. Carson’s class. Table 4.8

displays which target students expressed particular academic and social goals.

111

 

Table 4.8: Academic and Social Goals of Target Students by Classroom

 

 

 

 

Complete Appear Gain Status Behave Help
Task Competent Classmates
Ms. Carson’s class Allen Max Colleen Allen
Hannah Pete Hannah Allison
Max Max Colleen
Pete Pete Hannah
Tim Max
Pete
Tim
Mrs. Evans’s class Alex Alex Bill Alyssa Alyssa
Alyssa Becky Marissa Becky Bill
Becky Bill Molly Marissa Tricia
Molly Marissa Steve
Steve Molly
Tricia

 

 

 

 

 

 

 

In Ms. Carson’s class, all of the students expressed a desire to help classmates, and
ahnost all expressed a goal of demonstrating their competence. None of them
demonstrated a goal for completing the task in their talk. In Mrs. Evans’s class, more than
half of the students demonstrated this goal in their talk. Many of her students also
expressed the goal of appearing competent as well. The other social goals were
demonstrated by at least some of the students in Mrs. Evans’ class. It is not necessarily
the case that these are the social goals that are the most normative among Mrs. Evans or
MS. Carson’s classrooms. Rather these are the prominent social goals among the target
students. The extent to which the target students are representative of the population of
their respective classrooms will be addressed later in the chapter.

In the proceeding paragraphs, I will discuss the evidence necessary for attributing
each of the academic and social goals: complete the task, appear competent, gain status,

behave, and help classmates.

112

Complete the task. Students’ goal of completing the task stood out in contrast to
their social goals, not only because it was the one academic goal students expressed, but
because of the students’ focus on their own individual learning rather than the collective
learning of the class. Students with a goal of completing the task usually did not mention
wanting to help classmates during their interview. They emphasized a high priority of
wanting to get own work finished rather than supporting their classmates’
understandings. These students appeared to be interested in completing their work, and
did not mention attaining a deep level of understanding.

Becky’s quote in Table 4.7 above expressed a preference that she “liked” and
“would rather” spend time doing problems than talking about them and reading about
them as a class. She also repeated this idea within her interview. Both the repetition and
connection to affect reveal a level of commitment to wanting to spend time working on
the problem on her own.

Additionally, Alex talked about wanting to get his work done in class because he
did not want to take homework home. This meant that he would prioritize his own work
over helping classmates.

Alex: Um, I like to get my homework done before, so then I won’t have anything

to do at, like, home or something, but, like, I usually try and get thpt done. and

then I’ll help ‘em. So, like, if somebody asks me a question, I’ll be, like, uh, hold
on, just a second. I gotta finish this. (04/03/04, #50, Mrs. Evans)
He demonstrated his commitment to this idea through repetition of getting the work
“done” or “finished,” and through his use of an imperative verb, “gotta.” Alex wanted to

finish his work to have time for outside activities, while Becky would rather be given the

113

time and space to work alone. She believed working alone allowed her to complete her
work.

Students with this goal talked more about completing the tasks than gaining a
deeper understanding of the content. This talk had evidence of an individual focus toward
the task than the perspectives of other students.

Appear competent. Students’ whose talk suggested a goal of demonstrating
competence wanted to let others know that they understood or could do the mathematics.
This goal was inherently social because it was centrally concerned with communication
with others, as opposed to the focus above on mathematical tasks.

In her illustrative quote from Table 4.7, Hannah expressed feeling more confident
when she understood a math problem and other students did not. She spoke of other
students not understanding with repetition. Her talk about confidence was evidence of
affect. Having the opportunity to look better in comparison to the class was important to
her. Her interest in comparing herself to classmates approaches a concern for
appearances.

Similarly, Marissa spoke about doing better than her classmates and feeling more
confident as a result.

Marissa: I think when you feel like you’ve done better th_an the cla_sa, I Like—it

when teachers have on their progress report the class average, and then you’re

like, y_ay, I did better than the whole gasp did from the class average. Because
then it can just boost your confidence a little bit and feel, glam. Go me.

(3/31/03, #133, Mrs. Evans)

114

In both classrooms, the teachers would periodically publicly post students’ grades, with
student-selected nicknames rather than their own names, and students could examine how
they were doing in relation to one another. Marissa’s repetition of “doing better than the
class” and her use of affective phrases both indicate the importance of this goal for her.
Her concern for appearance is revealed in the public reporting of these grades, as all
students received these grade printouts, and their grades were posted regularly on a
bulletin board under student-selected nicknames.

Students with the goal of demonstrating competence spoke about their own
performance in relation to others, and the opportunity to appear as if they are doing better
than others, but not necessary demonstrating that they have improved in mathematics.

Gain status. Students whose talk revealed a concern for gaining status were
focused on recognition from others. This recognition took several forms, including
attention for contributing important ideas about mathematics during the class discussing,
earning increased status from demonstrating improvement in class, or gaining recognition
for appropriate behavior. They want to do mathematical work, but they also want to be
recognized for it.

Max’s quote in Table 4.7 demonstrated his commitment to gaining status by
contrasting how he “knows it better” now with how “last time I didn’t really know a lot,”
repeating this idea in additional turns in the interview. He also expressed a desire to
“want to prove that I know it.” This desire is a preference, indicating a sense of affect
supporting the goal.

Molly explained that she liked sharing her thinking in class because she enjoyed

getting attention for her thinking about mathematics.

115

Molly: . . . I likajp t_alk in front of people, so people, like, pay attention to me and
everything... I really l_ikitp talk to people and tell them what I think of it.
(3/13/03, #52, Mrs. Evans)
She repeated how much he “likes to talk in float of people,” because of the attention she
receives when she has the opportunity to express herself. Both this repetition and
connection to affect demonstrate a level of commitment to the goal of gaining status.

Students’ gaining status talk was focused on how others might View them, in
terms of their improvement in their mathematical competence, behavior, or the
opportunity to gain their attention. The difference between this social goal and that of
demonstrating competence or appropriate behavior more generally is how students’
talked about wanting to be noticed for improvement in their behavior or competence in
mathematics class if they had the goal of gaining status.

Behave. Some students’ talk particularly emphasized the goal of satisfying their
teacher’s expectations for good behavior. These students may have differed from those
who were interested in gaining status for this behavior in that they were not concerned
with recognition for their own choice to behave appropriately, but more concerned that
their peers to choose to behave more appropriately.

For example, in Marissa’s quote in Table 4.7, she repeats the words “disrupt” and
“rude.” Additionally, she expressed affect in how someone might want to yell out, in
frustration with those who are rude. It was not so much her own behavior she was
concerned with, but of the ideal student who should be behaving appropriately, herself

and others included.

116

When asked about being a successful mathematics student, another student, Pete
emphasized good behavior, not because it would help him understand, but because it
would help him raise the participation portion of his grade (in Ms. Carson’s class).

Pete: Uh, participate a lot. Pay attention. And don’t talk while the teacher’s

mpg. .. because if you don’t participate, then she’ll just call on you, and if you

weren’t payp'ng attention, then you might not get it right or might have no clue
what she’s talking about. (3/25/03, #26-32, Ms. Carson)
He repeated “pay attention,” emphasizing this as an activity of what a good student
should do, but accompanied with “don’t talk while the teacher’s talking.” This suggests a
sense of taking one’s turn to talk when it is most appropriate. He did not always talk
about paying attention in light of assisting his learning.

Help classmates. Finally, other target students were concerned not just for their
own learning, but that of their classmates. They spoke of assisting others who did not
understand. It was as if they did not view the teacher as the only one in the class who
could instruct others about mathematics, but that students had a role in this process as
well.

In Table 4.7, for example, Tricia spoke of how students ideally “should try to
help.” This helping behavior was not one-way. She intended to give help in recognition
of others’ help to her, past and future. Most of the other students with this goal also
expressed this symmetric social relationship. Not only did she use an imperative verb, but
she repeated these words, suggesting a commitment to this goal. She spoke about this

idea in other turns during the interview as well.

117

Another student, Colleen, also spoke similarly about how She tries to help her
classmates when they do not understand.

Colleen: Well, like, if people are struggling, then I like to gaise my han_d, like,

give them a boost.

AJH: What do you mean?

Colleen: Like, if they’re, like, struggling, like, they know the answer, but they

can’t say how they got it or how they got it, but they don’t know the answer, I’ll

Q'se my hand to, like, kind of jump in a little bit. (3/26/03, #71-73, Ms. Carson)
She repeated that students might be “struggling” and that her response would be to “raise
her hand” and “jump in” or “give them a boost.” She also expressed this with affect,
saying that she “liked to” do this. Students expressing the goal of helping classmates
spoke of whether others understood and if they could have a role in helping them
understand.

Students’ goals: Summary. Students’ talk in interviews demonstrated evidence
for five goals, one academic and four social goals: complete the task, appear competent,
gain status, behave, and help classmates. When I examined relations between these goals,
students’ epistemological beliefs, and perceptions of social risk, I determined four
clusters of beliefs and goals for these target students, as illustrated in Figure 4.1. Clusters
were first determined through differences in combinations of epistemological beliefs and
social risk. Then, students were grouped again based on shared goals. Below, I describe

in further detail the variations between and within these clusters.

118

Four Clusters of Beliefs and Goals: Variations Between and Within

As described initially at the beginning of this chapter, these 15 students held four
different clusters of beliefs and goals: learn material in public forum, gain attention, do
the right thing, and help others while overcoming social risk. Each of these clusters is a
set of co-occurring beliefs and goals as expressed by two or more target students.
Students within each cluster also diverged, not sharing other goals, creating diversity
within clusters as well as between them.

Learn material in public forum. Students in this cluster shared the perception
that involvement in whole-class discussion was a low risk activity and the belief that
learning mathematics involved negotiating knowledge. Additionally, all of the students
who expressed the academic goal of completing the task are in this cluster. These beliefs
and goals appear to focus on a concern for learning the content, suggesting that the risk of
sharing one’s thinking is beneficial. Alex, a member of Mrs. Evans’ class, demonstrates
this perspective.

AJH: How important is to you to be asked to explain what you think? If you were

never asked to explain what you think, would that...

Alex: Uh, you’d just, I don’t think you’d learn as much. Other kids wouldn’t learn

as much, you wouldn’t learn as much.

AJH: How wouldn’t you learn as much by not being asked to explain?

Alex: Because you can learn more once you have other people’s opinions

sometimes. And, like, because if you’re not getting something, and then if you say

something, it might just spark something in your mind. (4/03/03, #225-228, Mrs.

Evans)

119

His use of “you,” including “1 don’t think you’d learn as much,” and “if you’re not
getting something,” suggests that he’s focused on whether or not he understands the
material. Additionally, his talk about both learning from “other people’s opinions” and
his own ideas, such as a “spar ” in his mind, suggests a belief in a negotiated source of
knowledge that rests in both internal understandings and those of others. Other students
who shared this perspective talked about getting their questions addressed by their
teacher, such as Steve, who mentioned little to no concern about putting his question in
front of his classmates.

AJH: If a new student moved into your class and they had to know what they

needed to do to be successful, what would you say?

Steve: Pay attention. Do all your work. And, uh, like, don’t be afraid to ask

questions. Don’t be afraid to, you know, jump right in to a question that she

asked, you know. Always try to be in the group, you know? In the big group

discussion.

AJH: Oh, yeah?

Steve: Yeah. Because then you’ll understand it more and you’ll interact with the

question more. (3/26/03, #69-72, Mrs. Evans)
He recognized that some students might “be afraid” to jump in to the discussion, but he
encourages them to think otherwise, as he has found benefits to his own learning from
doing so.

Target students in this particular cluster appeared to value input from their teacher

over their classmates. Their lower sense of risk when sharing thinking publicly in class

120

appeared to support a willingness to trust their own thinking, but still value the input their
teacher in developing their thinking about mathematics.

The salient beliefs among students in this cluster is the low level of social risk
combined with a belief in the process of knowing as negotiation, as the students
expressed that it was worth participating in order to learn. The students in this cluster
from Mrs. Evans’ class also shared a task orientation goal, or a focus on their own
understanding over the need to help others. All of the students in this cluster from Ms.
Carson’s class valued helping others. These students did not end up in the cluster of help
others while overcoming social risk because they expressed a lower level of risk.
Additional variation among students within this cluster included the goal of helping
others and demonstrating competence. With respect to competence, Allison and Steve
specifically said they were not concerned with demonstrating competence or appearing as
one of the smartest students in the class.

Gain attention. In addition to believing in knowing as a process of negotiation
and associating a lower social risk with participation, some students also expressed a
social goal of gaining status. This cluster is similar to the previous cluster, learn material
in a public forum, but the students expressed an additional goal for receiving attention.
This goal was expressed with a level of commitment so strong that it distinguished these
two students, Molly and Bill, as appearing as slightly different than the others in
previously described cluster. These students were also consistently concerned with
demonstrating their competence.

Molly wanted to get attention for her ideas about mathematics, as she said, “. . .I

like to talk in front of people, so people, like, pay attention to me and everything”

121

(3/13/03, #52, Mrs. Evans), while Bill, another student in Mrs. Evans’s class, seemed to
want to demonstrate that he was more than the class clown. At the end of the interview,
in order to follow up on some of his comments, I asked him directly about whether he
participated in order to receive attention. He spoke as follows:

Bill: Uh, sometimes I do, and, like, [mentions students’ names], and there’s a

couple people, three or four people who do it because they want attention. But,

uh, sometimes I do it because, like, I want people to know that I know how to do
the problem, and I’m just not, like, fooling around and stuff all the time, like,
some people think. (3/25/03, #129, Mrs. Evans)
He wanted not only to get people’s attention because of his sense of humor, but also
specifically so they would think of him as more than someone who was “fooling around
and stuff all the time.”

Molly and Bill expressed consistent epistemological beliefs of negotiating
knowledge, perceptions of low social risk, and social goals of gaining status and
demonstrating competence. They differed with respect to the goal of helping classmates,
as Molly did not express this goal to the extent that Bill did. No target students from Ms.
Carson’s class were in this cluster.

Do the right thing. The second two clusters both include target students with the
epistemological belief of knowing as a process of receiving knowledge and a moderate to
high perception of risk associated with Sharing their thinking during class discussion. In
this cluster, do the right thing, students also consistently held the social goal of gaining
status, demonstrating competence, and demonstrating appropriate behavior. These

students especially expressed a concern with behaving appropriately in their mathematics

122

class in order to seek a level of social status. Further evidence from Marissa included
talking about how others might view her.

I don’t want to be, to put down or to make people uncomfortable. Because I

wouldn’t, I don’t know, I try to be fiiendly, I don’t know if I am all the time, but,

um, it’s not nice. The world doesn’t respect you if you put people down, it doesn’t

help your social career or your business career, either. (3/31/03, 60, Mrs. Evans)
Her consideration for whether her classmates are uncomfortable was directly related to
how her behavior would reflect upon herself, whether it would help her “social career,”
which indicates a concern for status.

Target students in this cluster were from the group of students who believed in
received knowing, perceived a higher sense of social risk, and consistently held social
goals for gaining status, demonstrating competence, and demonstrating appropriate
behavior. Among the students within this cluster, there was diversity in whether they
would help others, paralleling the diversity in the gain attention cluster. Those from Ms.
Carson’s class expressed the goal of helping others, while the student in this cluster from
Mrs. Evans’s class did not.

Help others while overcoming social risk. Most of the students who expressed a
belief in knowing as a process of receiving knowledge, or obtaining knowledge fi'om an
external source, were in this cluster. All of these students expressed a high level of social
risk. Additionally, students in this cluster consistently expressed a social goal of wanting
to help their classmates, while students in the previous cluster did not, and they did not

consistently express a goal of wanting to gain status or behave appropriately. They talked

123

about competence more in terms of avoiding appearing incompetent than appearing
competent.

Students’ sense of social risk and their goal of helping one another appeared to
interact. It appeared as though these target students identified with what it is like not to
understand, and were interested in helping others who feel this way. This excerpt from
Tricia’s interview combined two illustrative quotes presented earlier in this chapter. They
occurred together as part of a longer exchange in her interview, demonstrating how
intertwined the goal of helping was for students with a high perception of social risk.

Tricia: Um, you should try to help, like, explain it to people, and give out your

ideas, and, like, it’s OK if they’re not right, because people in the class will help

you understand them. They’ll explain it to you, they’ll help you, like, figure it out
and stuff. So I think you should try to volunteer as much as you can.

AJH: OK. Um, do you find that you ever volunteer?

Tricia: Sometimes. I’m kind of really Shy, so I’m like super-conscious about when

it comes to answering in front of people. I get, like, all nervous and stuff.

AJH: So, what do you mean, can you say a little more about that?

Tricia: Well, like, in math, I used to be, like, if a teacher called on me or

something, and I had my hand raised, my face would turn all red, and I’d get

really nervous and I’d start sweating, until I got the answer out, and, like, it, I was,
like, always nervous that it would be wrong. And, I’m not so much nervous any
more, but I’m still kind of worried about if the answer’s wrong, like, I’ll get, like,
messed up or something. (3/18/03, #50—54, Mrs. Evans)

She spoke of understanding that it was valuable to give help, that she appreciated when

124

others helped her, and that she is shy of being incorrect in front of her peers, although
less shy now than she used to be.

Students in this cluster also had beliefs in received knowing. When asked whether
she was someone who participated in the discussions or not, she said, “I’m more towards
the listening” (3/18/03, #80). She said that, in order to be successful in mathematics,
students should pay attention.

Pay attention, because sometimes some of the problems are really harp, and yap

don’t get, like, parts of it, like yesterday in the worksheet we were working on,

um, Mrs. Dole said the Meat part was on the tables, to figure out the equation.

And, like, if you pay attention in class, like I got the equation right away...

(3/18/03, #44)

This focus on taking in knowledge, such as how to figure out the hardest equations,
indicated evidence of received knowing.

Other students spoke similarly to Tricia, such as Hannah from Ms. Carson’s class.
Hannah said that helping others was important, when she said, “if people don’t
understand it, then you should give how you do it.” (3/25/03, #102, Ms. Carson)
However, she also revealed a high level of social risk in her talk with statements such as,
‘Sometimes I’ll share, but sometimes I just like to listen to what other people think, I’ll
think what I did was wrong” (3/25/03, #58, Ms. Carson). Her lack of faith in her own
thinking combined with the phrase “sometimes I’ll share” suggested a hesitancy to
participate due to a heightened sense of risk associated with the activity.

Hannah was also a received knower. When I asked her how she would face a

brand new math problem, she said that she could not work on it without being told how to

125

do it.

Well, if you just give me the problem, and I don’t know how to do it. like, if you

gave me this book and told me to do it, I wouldn’t bgarble to do it, because I

wouldn’t underatand how to. But if you told, if you taught me how to do it. and

how to get througfl, then I’d be able to nrohaliy do it. (3/25/03, #142)

She emphasized that she would not know how, would not be able to, and would not
understand how to do a new problem through her repetition of these similar phrases. She
said that if she was “told” then “taught” how to do the problem, then she could probably
do it. Her use of probably indicates some lack of faith in her own skills even after being
taught. She appears to be more reliant on an authority to learn mathematics than her own
ideas.

Students in this belief cluster appeared to be concerned primarily with avoiding
the appearance of being incompetent, with their shared perception of a high social risk
and a social goal of helping their classmates. They expressed that they were generally not
interested in participating because of the potential threat of being incorrect in front of
peers and their belief that learning occurred in the process of receiving knowledge rather
than negotiating. However, these target students consistently said that if they were to
participate, it would be to help their classmates, perhaps because they identified with not
being able to understand. These students also were received knowers. Target students in
this cluster did not consistently share social goals of demonstrating competence or

appropriate behavior.

126

Diversity within Clusters: Third Quarter Performance

While this study was not designed to explore whether students’ beliefs and goals
related to their achievement in mathematics class, it is reasonable to examine briefly
whether students’ beliefs and goals related to their achievement. I collected students’
third quarter grades from each classroom as evidence of their achievement, as the rest of
my data was also gathered during this academic quarter.11 It may be reasonable to assume
that achievement moderates students’ beliefs and goals, thus explaining their
participation, if there is a relation between students’ beliefs and goals and their
participation. However, within each cluster of beliefs and goals, there was diversity
among students’ third quarter performance.

Relations between students’ grades and their beliefs are presented below. Are the
students who express higher levels of perceived social risk and epistemological beliefs of
received knowng also the students who are less successful in mathematics? Are the
students who are the most successful in mathematics also the students who believed in
negotiated knowing and lower levels of perceived social risk?

Since the relation between students’ epistemological beliefs and perception of
social risk primarily determined the clusters, Table 4.9 illustrates students’ third quarter

performance in relation to these beliefs.

 

" I acknowledge that there are potential problems with using class grades as achievement data, as they
include indices of effort, such as participation grades, as well as performance on mathematical tasks.
However, since some students look at course grades as feedback on their performance, they may provide a
useful distinction between participants.

127

Table 4.9: Target Students’ Performance (Letter Grade in Mathematics)
By Beliefs (Epistemological Beliefs and Perception of Risk)

 

 

 

 

 

 

 

 

 

 

A B C D E

Negotiated tim, ALEX, BILL allison,
& Low Risk BECKY, STEVE

MOLLY
Received & max pete
Moderate
Risk
Received & MARISSA, allen, colleen,
High Risk TRICIA ALYSSA hannah

 

*lower case: Ms. Carson’s class. UPPER CASE: Mrs. Evans’s class
Within belief clusters, there was diversity among students’ performance. Generally,
higher performing students were in the clusters of those who believed in knowing as
negotiation and a lower perception of risk, but students with these beliefs also earned
grades as low as C’s. Two students who earned B’s during third quarter expressed a high
perception of risk and a belief in knowing as a received process, but otherwise the
students with these beliefs were the lower performing students.
Survey Analyses

In addition to interviewing 15 target students, I surveyed all students in both
classrooms (N = 42) using a Likert-scale instrument that assessed a set of beliefs about
and orientations toward learning mathematics. Through my survey analyses, I addressed
the following secondary questions: (a) Are there relations between students’ beliefs at the
group level? (b) Do students beliefs differ by classroom? (0) To what extent are the target
students representative of their classrooms?

Recall from Methods (Chapter Three), that I used scales from Fennema-Sherman
(1976) in order to assess their confidence in mathematics and whether they believe

mathematics is useful; I used scales from the Conceptions of Mathematics Inventory in

128

 

order to assess the structure and interconnectedness of mathematics; scales item the
Patterns of Adaptive Learning Survey (Midgley et al., 1996) were used to assess
students’ task and ability orientations; and I designed four scales based on single items
from Schoenfeld (1989) to assess whether students focused on the process or product,
whether they focused on concepts while learning mathematics, whether they believed
they could learn autonomously or had to rely on an authority, and whether they believed
mathematics problems could be solved using multiple methods or solution paths. The
reliability of three of these nine scales was _>_ 0.70 (Cronbach’s a): confidence, task
orientation, and ability orientation. However, I will report survey results for all scales.
Means, standard deviations, and reliabilities (Cronbach’s a) are shown in Table 4.10.

Table 4.10: Survey Results Across the Two Classrooms (N = 42)
Scale Means, Standard Deviations, and Reliabilities

 

 

 

 

 

 

 

 

 

 

 

 

Scale Mean SD. Cronbach’s o.
(1) Confidence 3.16 0.34 0.91
(2) Task Orientation 3.53 0.79 0.83
(3) Ability Orientation 3.03 1.07 0.87
(4) Process over Product 3.15 0.36 0.39
(5) Autonomy 3.48 0.54 0.43
(6) Usefulness 3.00 0.33 0.67
(7) Structure 3.09 0.49 0.56
(8) Conceptual 3.33 0.63 0.33
(9)Multiple Methods 3.16 0.43 0.37

 

 

 

 

Relations Between Beliefs.

I analyzed whether there were relations between beliefs across both classrooms in
order to determine whether any of the beliefs assessed through survey scales would
cluster together. Intercorrelations could reveal alternative belief clusters not assessed
through the analysis of students’ talk during interviews. Table 4.11 illustrates the

intercorrelations for the nine survey scales.

129

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130

While a number of inter-scale correlations were significant, if scales with reliabilities
below 0.50 are excluded, only one of the correlations was significant. There was a
significant positive correlation (0.31) between confidence and structure. In other words,
the students with a higher sense of self-confidence in mathematics were more likely to
believe that concepts in mathematics were interconnected, and students with lower levels
of self-confidence were more likely to believe that ideas in mathematics are isolated from
one another. It is unclear if the beliefs as measured on the survey are generally unrelated,
or if low reliability scores led to lack of relations.
Differences Between Classrooms

In order to determine whether the students in the two classrooms expressed
similar or different beliefs in their survey responses, I calculated means and standard
deviations for each classroom, and calculated two-tailed t-test comparisons between the
class means, as seen in Table 4.12.

Table 4.12: Survey Results: Comparisons Between Classrooms

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Means (Standard Deviations)
(1)* (2) (3) (4) (5) (6) (7) (8) (9)
Carson 3.21 3.44 2.92 3.07 3.53 3.05 2.99 3.19 3.07
N=16 (0.44) (0.57) (1.19) (0.36) (0.55) (0.42) (0.51) (0.57) (0.39)
Evans 3.13 3.59 3.11 3.21 3.46 2.97 3.15 3.41 3.21
N=26 (0.27) (0.90) (1.01) (0.36) (0.54) (0.27) (0.48) (0.66) (0.46)
*p<0.05 .

The classrooms were significantly different on one scale: confidence. Otherwise the

 

classrooms were not significantly different on any of the other scales. However, I see this
significant difference as occurring more due to the standard deviations than a large

difference between scale means. Differences between the confidence scale means were

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less than 0.1, which does not appear to be a practically significant difference. AS a group,
the students in the two classrooms did not appear to hold different beliefs about learning
mathematics, other than their confidence in mathematics.

Additionally, I examined differences between classrooms in terms of students’
beliefs as expressed in interviews. I assessed whether the target students from each
classroom were distributed between the two epistemological belief categories differently
than expected by chance. Due to the low numbers, expected values less than ten (2 x 2
matrix), I used Fisher’s Exact Test.

Table 4.13: Epistemological Belief by Classroom

 

 

 

 

Received Knowing Negotiated Knowing
Ms. Carson’s Class Allen, Colleen, Hannah, Tim, Allison (2)
Max, Pete (5)
Mrs. Evans’s Class Alyssa, Marissa, Tricia (3) Alex, Becky, Bill, Molly,
Steve (5)

 

 

 

The two-sided Fisher’s exact test did not confirm that the beliefs in these two classrooms
occurred more frequently than we would expect due to chance (p = 0.315). It may be
equally likely that students with epistemological beliefs of knowing as a process of
receiving knowledge would be in Ms. Carson’s classroom as in Mrs. Evans’s classroom.
Target Students Relative to their Classmates

In order to determine whether the target students were similar to their classmates
in terms of their responses on the survey, I compared each target student’s mean to their
class means for each survey scale. Recall that Ms. Carson’s class had seven target
students, while Mrs. Evans’s class had eight target students. In Table 4.14 below,
students’ means were categorized as “within” if they were within one standard deviation

above or below the class mean; otherwise, the student was counted as either above or

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below the mean. Table 4.14 illustrates the number of students in each classroom who
were within the class mean, above the mean, and below the mean.
Table 4.14: Survey Results: Target Students in Relation to Their Respective
Populations

Frequency Within, Above, and Below their Class Mean

 

 

 

 

 

 

 

 

 

 

 

 

Ms. Carson’s class Mrs. Evans’s class
Within Above Below Within Above Below

(1) Confidence 4 3 0 5 1 2
(2) Task Orientation 6 0 1 6 1 l
(3) Ability Orientation 3 l 3 5 2 l
(4) Process over Product 4 2 1 6 2 0
(5) Autonomy 5 l l 5 1 2
(6) Usefulness 4 2 1 7 1 0
(7) Structure 6 0 l 7 0 1
(8)Conceptual 6 1 0 7 1 0
(9) Multiple Methods 3 2 2 6 l 1

 

 

 

 

 

 

 

 

In Mrs. Evans’s class, for every survey scale, five or more target students were within
one standard deviation of their class mean. In Ms. Carson’s class, five or more target
students were within one standard deviation of their class mean for four out of nine
scales: task orientation, autonomy, structure, and conceptual. Four or three of the target
students were within the mean on the other five survey scales. More of the target students
in Mrs. Evans’s class were within the mean survey responses than those in Ms. Carson’s
class. In each class, most target students were within the mean. The 15 target students
appear to be reasonably representative of each classroom of students. Table 4.15 shows

which target students responded above and below their class mean.

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Table 4.15: Survey Results: Target Students in Relation to their Class Scale Mean

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Survey Scale by Students
L0) 1 1211(3) l (4) l (5) l (6) l (7) l (8) l (9)

Ms. Carson’s class
Allen B B
Allison B B B
Colleen B B A A
Hannah A
Max A B A A A
Pete A B B
Tim A A A A

Mrs. Evans’s class
Alex B A‘ A B
Alyssa A B B A B A
Becky A
Bill
Marissa A A
Molly B B
Steve B A
Tricia A

 

 

 

 

 

 

 

 

 

 

 

 

A=Above, B=Below

In Ms. Carson’s class, three target students were either above or below the class mean on
four or more of the survey scales: Colleen, Max, and Pete. In Mrs. Evans’s class, two
target students were either above or below the class mean on four or more survey scales:
Alex and Alyssa. The rest of the target students (four in Ms. Carson’s classroom, six in
Mrs. Evans’s classroom) were within the class mean on four or more survey scales.
Generally, target students were within their class’s mean scores on the survey.
Summary

Analyses of students’ talk during their interviews revealed that their
epistemological beliefs interacted with their perceptions of social risk, and that social
goals factored strongly into their motivations to participate during whole-class
discussion. Students expressed that participating in whole-class discussion has benefits,

such as opportunities to learn content through being corrected or hearing new ways of

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solving problems, opportunities to demonstrate competencies in both their mathematical
understandings and behavior, as well as opportunities to gain recognition for these
competencies, and opportunities to be supportive and helpful to classmates. However,
some students found whole-class discussions to be threatening due to the potential of
being publicly incorrect in front of their classmates and teacher.

The 15 target students could be separated into four clusters of beliefs and goals
based on their perceptions of social risk, epistemological beliefs, and academic and social
goals. I refer to these clusters as: learn content in a public forum, gain attention, do the
right thing, and help others while overcoming social risk. Two of these clusters, learn the
material in a public forum and gain attention, appear to be more self-focused while the
other two appear to be more community-focused. The students in the self-focused
clusters emphasized goals that benefit the student, such as learning content, gaining
status, or appearing competent. The goals of the students in the community focused-
clusters included behaving and helping classmates. On the one hand, the students with
beliefs of negotiated knowledge and low social risk may have productive beliefs that
allow them to take advantage of opportunities to learn in discussion-oriented mathematics
classrooms. On the other hand, students with social goals of helping and behaving may
have productive goals related to fostering a cohesive classroom community.

Students’ beliefs did not appear to be bound by classrooms, though there were
some differences between classrooms among students’ goals. Both classrooms included
target students who expressed three out of the four clusters of beliefs and goals. Fisher’s
Exact Test did not reveal significant differences between the classrooms based upon

epistemological beliefs. Eight of the nine survey scales were not Significantly different.

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All of the target students in Ms. Carson’s classroom expressed a goal for helping
classmates, while this was not the case among Mrs. Evans’s target students. None of Ms.
Carson’s target students expressed a goal for completing the task, while this goal was
mentioned by more than half of the target students in Mrs. Evans’s class.

Third quarter grades did not appear to completely explain differences in students’
beliefs. While the higher performing students expressed beliefs about negotiated knowing
and low social risk, there were also some students who earned C’s in this category.
Students with beliefs of received knowing and high social risk earned grades ranging
from B’s to D’s, with more of the lower performing students holding these beliefs.
Students who earned B’s and C’s expressed both types of beliefs, which suggests that
students with moderate grades could hold either type of belief.

In the proceeding chapter, I will situate these students’ participation in context by
describing the nature of discussions in each classroom, and then I will present the results

that demonstrate how the target students participated in these whole-class discussions.

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CHAPTER FIVE
RESULTS: THE NATURE OF WHOLE-CLASS DISCUSSIONS AND

TARGET STUDENTS’ PATTERNS OF INVOLVEMENT

In the previous chapter, data suggested that target students’ beliefs and goals were
not completely shaped by their current classroom experiences. Students’ beliefs and goals
varied within each classroom, and some students with similar beliefs and goals were in
different classrooms. As the overarching goal of this study is to examine whether and
how students’ beliefs and goals relate to their participation in whole-class discussions, the
next logical step is to examine the patterns of participation that took place in each
classroom. The results presented in this chapter address the following question: How do
the target students participate in whole-class discussions? I chose to examine students’
involvement in whole-class discussion rather than engagement in other activities, such as
small group discussions and individual problem solving, because of the particular push to
encourage large group discussion in the NCTM reform movement and the potential threat
adolescents might experience when expected to speak about their thinking in front of
their peers.

In order to examine students’ involvement in whole-class discussions, I first
compare and contrast the nature of discussions in the two classrooms in the beginning
sections of this chapter. Similarities and differences in classroom discussions, including
the amount of structure or flexibility in each classroom, reveal students’ opportunities to

participate in each setting. Then, I compare and contrast the target students’ participation

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in the latter sections of the chapter in terms of their level of involvement and the nature of
their involvement.
The Nature of Whole-Class Discussions in Two Mathematics Classrooms

Before presenting the target students’ patterns of involvement, 1 illustrate the
nature of whole-class discussions in these two mathematics classrooms. Since these two
teachers Shared a commitment to faithfully implementing their Standards-based
mathematics curricula, they may have conducted their whole-class discussions somewhat
similarly. These similarities may have provided a common experience for structuring
students’ opportunities to engage in whole-class discussion. Each teacher may vary in
how they implemented these discussions as well, and these variations may have created
differential opportunities for students to involve themselves in particular forms of
classroom talk.

Similarities

Whole-class discussions in these two classrooms had some similarities that
potentially shaped students’ involvement. These similarities included the teachers’
commitment to whole-class discussions, the teachers’ commitment to following the
textbook series, using the CMP instructional model, and pursuing multiple solution
methods for solving or interpreting a mathematics problem.
Commitment to Textbook Series

The teachers expressed similar levels of commitment to implementing discussions
and involving students in the construction of meaning during mathematics class (personal
communication, 08/02). One way they both put this level of commitment intro practice

was by dedicating a portion of time to whole-class discussion for every lesson.

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Essentially, whole-class discussions were occurring in both classrooms. At the minimum,
if students wanted to talk about mathematics in whole-class discussion, they had the
opportunity in both settings. However, I will present later in this chapter that the teachers
dedicated different amounts of time to whole-class discussion. These two classrooms
were both settings in which students were invited to talk about mathematics.

Also, the teachers demonstrated commitment to the CMP textbook series. They
communicated with each other in order to stay on a similar schedule, and each used the
investigation problems as the foundation for their lessons. Classroom discussions, within
a day or two of each other, were about similar mathematics problems, since students were
working on the same mathematics tasks in small groups or individually before the
discussions. Additionally, both teachers utilized features of the textbook series including
reflection questions at the end of each investigation, assigned A.C.E. problems for
homework, and did not supplement the textbook series with additional practice problems.
Lessons in both classrooms were based on the same mathematical tasks, those directly
from the textbook series.

Instructional Model

The teachers both implemented CMP’S recommended Launch, Explore, and
Summarize instructional model (Lappan, Fey, Fitzgerald, Friel, & Phillips, 1996), which
is another example of their commitment to the textbook series. According to this model,
Launch involves introducing an investigation problem to students, Explore involves class
discussion of students’ ideas from working on the investigation in small groups or
individually, and Summarize involves wrapping up the discussion, helping students take

away the main ideas. This Launch, Explore, Summary model takes the place of the

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teacher’s presentation of content in a more traditional mathematics setting, in which the
teacher models steps for solving a problems and provides the students with opportunities
for guided practice. Instead, during the Launch, students are introduced to a problem to
work on. During Explore, students spend time working on these problems either
individually or in small groups, and they also talk about their thinking during whole-class
discussion. This is different from guided practice time during a traditional lesson, since
students have not been given explicit instruction for how to solve the problems, but
instead are invited to try the problems on their own first. During whole-class discussion,
students’ ideas about how to solve the problems serve as a replacement for the teacher’s
prescriptions of the steps for solving the problems. Then, during Summarize, with the
teacher’s guidance, the class determines which new concepts and approaches to problem
solving came out of working on that investigation problem, similar to a closure of a
traditional lesson, but perhaps with more student involvement.

Along with the shared Launch, Explore, Summarize instructional model, both
teachers structured the mathematics class periods with similar activities each day. Typical
activities in each of these classrooms included: warm-up problems at the beginning of the
class period, grading homework, introducing a new problem for investigation (Launch),
seatwork over new problems (Explore) or homework problems, and large group
discussion about the seatwork (Explore and Summarize). Although this structure was
adhered to in both classrooms, later in this chapter I will demonstrate differences in the
implementation in this structure, including the time spent on each activity in the class

period and the nature of the talk during whole-class discussion.

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Pursuit of Multiple Solution Methods

One Similarity in the nature of classroom discussions was that both teachers
solicited multiple methods for solving or interpreting the mathematics problems during
whole-class discussion. In Problem 2.3 from the second investigation in Moving Straight
Ahead, students examined whether or not it was a better deal to get a membership to a
movie theater or to pay per movie. Both classrooms discussed multiple solution methods,
but there were differences in what counted as an alternative solution in each classroom,
such as whether the focus was on calculation paths or various representations. For
example, Ms. Carson solicited additional responses from students in the following
manner:

T: OK, did anybody else get 69 dollars for a member to see 20 movies?

{Ss raise their hands}

T: Did anybody do it differently than what Tyler did? Holly, what did you do?

(interaction segment #2, 2/26/03)
Ms. Carson consistently solicited an alternative perspective from students by asking,
“Did anybody do it differently from [student’s name]?” Ms. Carson did not request
alternative perspectives through any other types of questions. Her consistent use of the
verb “do” in these questions when she solicited multiple solution methods indicates a
focus on different calculation paths over other conceptions of multiple solution methods.

In contrast, Mrs. Evans’s requests for alternative perspectives on the problems
involved requesting more than alternative calculation methods, and also involved

reflecting back students’ perspectives for feedback. In the following example from the

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whole-class discussion of Problem 2.3, Mrs. Evans’s request for an alternative method
involved a request for a different perspective other than an additional calculation path.

T: So I could have used that calculator and table. It happened at 14, they were

both 63 dollars. Did anybody find that 14 in a different way than table?

Jim: Um, yeah, I did. I did it on a graph. (Interaction segment #20, 3/03/03,

Evans)
Mrs. Evans’ use of “a different way than table,” and her listing of what they could have
used (calculator and table) indicates that alternative forms of representations also counted
as multiple solution methods. These representations included the use of the calculator for
calculating values or using an equation, finding the values in a table, using the graph, as
mentioned by Jim. Bringing in alternative representations in addition to multiple
calculation paths indicates that Mrs. Evans’s whole-class discussions encompassed a
broader perspective for what sounds as an alternative solution. Using a table for solving a
problem rather than a graph is a way of taking a new lens on the relationship, perhaps
seeing something about the relationship differently as a result of the alternative
representation, while an alternative calculation path may not shed new light on the
relationship, but rather allows students to become more proficient in their calculation
skills.

Another way Mrs. Evans requested multiple solution methods was in her requests
for students to evaluate the solutions of others. In the following example, Mrs. Evans
asked Molly to discuss her critique of another student’s solution and invited others to

share their viewpoints as well.

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T: Molly is saying she doesn't like this one, say it one more time, 'cause I'm not

sure it's all been heard.

Molly: because, 10, it's saying that for every kilometer you go, you're adding 10

more, so if you go 2, you're gonna have 20, and it's really only 10.

T: so you would disagree with that one.

Molly: Yeah.

T: How about other people? (Interaction segment #32, 2/21/03, Evans)

Mrs. Evans’s use of “like,” in reference to whether Molly “liked” the problem she was
critiquing was one way that she invited students to share alternative perspectives on the
problem. Then, once one student put her ideas on the floor, rather than evaluating her
response, Mrs. Evans asked what others thought. While this is not the same as asking
whether anyone did the problem differently, it is a request for alternative perspectives on
the problem. In the act of critiquing one another, students may suggest an alternative
solution method to contrast with the method they are critiquing.

Both classrooms explored multiple methods for examining linear relationships,
with differences in what counted as an alternative solution. Ms. Carson explicitly
solicited alternative calculation paths, while Mrs. Evans additionally requested alternative
representations, as well as asking students to evaluate one another’s solutions.

The similarities between these two classrooms may provide common experiences
for these seventh graders around participating in whole-class discussion about
mathematics. Both classrooms had discussions about mathematics over the same
mathematical tasks. Both teachers implemented a similar instructional model, as

advocated by the textbook series, and a similar series of typical activities during a class

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period. Additionally, multiple solution methods were pursued in both classrooms. It is
unclear whether the similarities between the two classrooms, relative to more traditional
settings, or the differences between the whole-class discussions play a stronger role in
considering the development of students’ beliefs and goals and their participation
practices. Compared to the population of mathematics teachers across the nation, these
two teachers may be relatively similar, but there were also some differences between
them, beyond what counted as an alternative solution in each classroom.

Differences

AS mentioned above, while these two teachers implemented the Launch, Explore,
Summarize instructional model, the teachers varied in their implementation of the
instructional model. For example, during the Launch in Ms. Carson’s class, the teacher
read the problem to the students from the textbook, while Mrs. Evans’s class spent
slightly more time on this event in the model, and this time was spent in terms of the
teacher retelling the problem in an elaborated form, connecting back to previous
investigation problems, with the students interj ecting jokes or comments. The Explore
and Summarize lesson elements included the whole—class discussions, and based on
variations in the discussions, Ms. Carson’s class appeared to be more structured, while
Mrs. Evans’s class appeared to be more flexible.

In the Explore and Summarize portions of Ms. Carson’s lessons, the whole-class
discussions appeared to be able to be predicted by the pages of the textbook, as the
content of the interaction segments consistently followed the order of each of the sub-
problems in the investigation problem (part a, b, c, etc.). Exploratory work around the

investigations appeared to happen less during whole-class discussion and more during

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seatwork time, when the teacher would go around from table to table (with three to four
students at a table, and students were invited to work together, but they also had the
option to work alone) and consult with students. More time was spent on seatwork during
Explore in Ms. Carson’s class than in Mrs. Evans’s class. The purpose of the whole-class
discussion appeared to be having students check their answers from their work on the
investigations, to get feedback as to whether they were right or wrong. The teacher
evaluated whether the solutions were correct rather than other students, and this was seen
in the adherence to the Initiate-Respond-Evaluate (Mehan, 1979) model of discourse
during whole-class discussion in Ms. Carson’s class. During these discussions, students
rarely contributed off-topic comments, and the teacher was the initiator of new topics of
discussion rather than students. Based on how the content of the talk followed the
structure of the sub-problems of the investigation problems, the adherence to the I-R—E-
model, the role of the teacher as the primary evaluator, and the lack of student initiated or
off-topic talk, Ms. Carson’s whole-class discussions appeared to be highly structured.

In Mrs. Evans’s class, the whole-class discussions also covered all of the sub-
problem in the investigation, but the events in the discussion were less predictable. At
times, students initiated new topics of conversation, such as asking for clarification about
a particular idea, going back to a previous problem, or revising their vocabulary. The
teacher took up these suggestions to varying degrees rather than moving to the next sub-
problem. It appeared that the teacher would follow up on the student-initiated topics if
these aligned with her mathematical agenda for the day, or if they seemed particularly
mathematically poignant. Students had not completed or clarified their thinking by the

time the classroom discussion took place; these discussions appeared more exploratory in

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nature than those in Ms. Carson’s class, as students sometimes revised their own
contributions during discussion. Sometimes the class did not finish discussing an
investigation in one class period, and they would continue talking about it in the next
class period. This did not generally happen in Ms. Carson’s classroom. Mrs. Evans’s
class spent more time on whole-class discussion than Ms. Carson’s class, and relatively
less time on seatwork. Additionally, students in Mrs. Evans’s class would not only
initiate mathematical topics during the discussion, but would also make off—topic
comments. Since the discussion did not always follow the order of the sub-problems in
the investigation, students initiated discussion topics, both mathematical and off-topic,
and the whole-class discussions sometimes extended beyond the class period, if
necessary, the discussions in Mrs. Evans’s class appeared to be more flexible.

In order to systematically examine the apparent differences between the two
classrooms, I pursued two lines of analysis: (a) I identified segments of activity in the two
classrooms and compared and contrasted the amount of time spent on each activity, and
(b) I identified elements of classroom discourse that would allow me to assess the
structure or flexibility of the whole-class discussions. The class periods analyzed below
included the days spanning five consecutive investigation problems from Moving Straight
Ahead.

Amount of Time Spent on Typical Activities

As mentioned above, Mrs. Evans’s class spent more time on classroom
discussion, while Ms. Carson’s class spent more time on seatwork. In order to examine
the amount of time spent on discussion or seatwork in these two classrooms in more

detail, I measured the time spent on each of the typical activities: warm-up problems at

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the beginning of the class period, grading homework, introducing a new problem for
investigation (Launch), seatwork over new problems (Explore) or homework problems,
and large group discussion about the seatwork (Explore and Summarize). I compared and
contrasted the time spent at the level of the class period and at the level of each lesson,
since the Launch, Explore, Summarize sequence sometimes occurred over multiple days.
Time spent over the average class period. In order to examine the time spent on
each activity at the level of a particular day, I averaged the time spent on each activity for
ten observed days spanning five consecutive investigation problems. The only activity
that happened with regularity in each classroom every day was the warm-up problems on
the overhead at the beginning of each classroom. Otherwise, homework problems may or
may not have been graded in class. Some days, students turned in the homework
problems and the teacher would grade them. Also, on some days the entire class period
was taken up by seatwork with little to no classroom discussion, while other days the
class period mostly consisted of whole-class discussion, with little to no seatwork. Given
this variability day to day, I pursued the contrast between classrooms by examining the
average time spent on each activity across ten days of observation. The following table
represents a typical day in each classroom, as calculated by taking the average time spent

on an activity across the observed days. ’2

 

‘2 Days included in this analysis include 2/11, 2/13, 2/18-2/21, 2/24-2/26, & 3/03.

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Table 5.1: Average Minutes Spent on Typical Activities during a Class Period

 

 

 

 

 

 

 

Ms. Carson’s Mrs. Evans’s

Class Class
Warm-up problems / 10.6 15.4
class business
Grade homework 3.3 5.4
Introduce new problem 3.1 3.3
(Launch)
Seatwork (Explore) 24.5 8.9
Whole-class discussion 15.4 21.4
(Explore & Summary)
TOTAL“ ~57 ~54

 

 

 

 

 

*Totals do not add up to 58 minutes due to transition times between activities.

Similarities in the ways the two classrooms used time included the warm-up
problems (or “openers”), grading homework, and introducing the new problems (launch).
Each of the classrooms opened the period by working on a set of 1-5 problems that as
students were getting settled. The teacher would then go over these problems with the
class and talk about any class business. During the four weeks of classroom observation
in the spring semester, Mrs. Evans’ class spent about five minutes more on the warm-up
problems and class business than Ms. Carson’s class. Neither class spent much of the
period grading homework or discussing answers to homework problems. Instead, the
teachers would more often grade students’ homework themselves and hand them back to
the students with feedback.

The principal differences between the two classrooms were in the time spent on

seatwork and whole—class discussion activities. Ms. Carson’s class spent about three

148

times as much time on seatwork. During seatwork in Ms. Carson’s class, students often
worked alone, but they also had the option to ask questions to the other two to three
students sitting at the table or of the teacher. Mrs. Evans’ class did not spend as much
time on seatwork. When students worked on investigation problems during seatwork time
in Mrs. Evans’s class, they usually worked with a partner or in small groups, in contrast
to the students in Ms. Carson’s class who worked more independently. Mrs. Evans spent
extended periods of time sitting down and talking about the investigation problem with
one group or another during this time, spending time with one or two of the five or six
small groups, in contrast to Ms. Carson who walked around the room taking questions
from students with raised hands from almost every small group. Whole-class discussion
time also appeared to be different in each classroom, with Mrs. Evans’ class typically
spending more time on discussion than Ms. Carson’s class — approximately 50% more
time.

Time spent over the average lesson. As mentioned earlier, each day varied, so
characterizing each classroom in terms of a typical day could be misleading, as
sometimes no time was spent on an activity, such as seatwork or discussion, on a
particular day. The unit of a “lesson” encompassed the Launch, Explore, Summarize
sequence for an investigation problem, and this sequence usually took place over the span
of two or thee days. The following table illustrates the average time each class spent on
introducing new problems, doing seatwork, and in discussion over problems 1.2 to 2.3 in

the Moving Straight Ahead textbook, a total of five problems.

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Table 5.2: Average Time Spent on Typical Activities during a Lesson

 

 

 

 

 

 

 

 

Ms. Carson’s Mrs. Evans’s

Class Class
Introduce new problem 6.5 9
(Launch)
Seatwork 33.25 18
(Explore)
Whole-class discussion 30.75 45.25
(Explore & Summary)

 

At the level of a lesson, similar patterns follow as those seen at the level of a class period:
each class spent a relatively small amount of time introducing the problem, with Ms.
Carson’s class spending less time on this activity. Ms. Carson’s class spent more time on
seatwork (almost twice as much as Mrs. Evans’ class) and Mrs. Evans class spent more
time on large group discussions (approximately 1.5 times as many minutes).

Interaction segments and time spent on discussion per day. The additional
time Mrs. Evans’s class spent on discussion provided students in her class with more
opportunities to talk about mathematics in front of the whole class. Another way to
demonstrate that the students had increased opportunities to participate in Mrs. Evans’s
class would be to compare the number of interaction segments in the whole class
discussions. Table 5.3 illustrates the number of interaction segments in each classroom
across four days of classroom discussion.

Table 5.3: Number of Interaction Segments per Day

 

2/20/03 2/21/03 2/25/03 2/26/03 3/03/03 Average

 

Ms. Carson’s class 81 16 41 24 41

 

Mrs. Evans’s class 68 83 53 28 58

 

 

 

 

 

 

 

 

 

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Mrs. Evans’s class, on average, had the opportunity to participate in almost 50% more
interaction segments, as Mrs. Evans’s class had an average of 58 interaction segments
over these four days and Ms. Carson’s class had 41. Three of the four days in Mrs.
Evans’s classroom had 50 or more interaction segments, while only one of the days in
Ms. Carson’s classroom had 50 or more interaction segments. Looking across the four
days, there was also variability in the amount of interaction segments for both
classrooms. Ms. Carson’s class had greater variability, with a range between 16 and 81
interaction segments. This variability in interaction segments was generally paralleled in
the amount of time spent on discussion during these four days.

Table 5.4: Time Spent (in minutes) on Whole-Class Discussion per Day

 

2/20/03 2/21/03 2/25/03 2/26/03 3/03/03

 

Ms. Carson’s class 42 7 30 13

 

Mrs. Evans’s class 45 33 36 20

 

 

 

 

 

 

 

 

Generally, more interaction segments occurred during a longer duration of class
discussion. Exceptions, when increased interaction segments were not accompanied by
increased in time spent in whole-class discussion, were related to the pace of the
discussion, as more interaction segments could occur over a similar time period with
more overlapping talk and less time between turns. The highest number of interaction
segments, 83, occurred during a 33 minutes discussion in Mrs. Evans’s class rather than
the longest times spent on classroom discussion 42 or 45 minutes. Again, there was more
variability in the amount of time spent on class discussion in Ms. Carson’s class, with a
range between 13 and 42 rrrinutes. Three of Mrs. Evans’s whole-class discussions were at

or above 30 minutes on these four days, while two of Ms. Carson’s fell into this category.

151

An examination of the number of interaction segments per day and the time spent on
whole-class discussion per day also illustrates that Mrs. Evans’s students had more
opportunities to participate in class discussion.

The increased time spent on whole-class discussion and the decreased time spent
on seatwork in Mrs. Evans’s class appears to be related to the flexibility of the nature of
the discussions. A teacher who is willing to dedicate more time to discussion may also be
more willing to follow up on students’ initiation of discussion topics, either mathematical
or off-topic. Such follow up on student-initiated topics did not occur in Ms. Carson’s
class, as students did not initiate topics for discussion. It is not clear whether more time
was spent on discussion because Mrs. Evans was willing to follow up on students’
contributions or if Mrs. Evans followed up on students’ contributions because she was
willing to spend more time on whole-class discussion. Additionally, if Ms. Carson’s class
spent more time on seatwork, it is possible that the students had the opportunity for more
flexible discussions with the teacher during this activity, so the purpose of the whole-
class discussions in Ms. Carson’s class may have instead focused on making sure the
students were correcting their answers for the investigation problem. The amount of time
spent on discussion appears to be related to whether the whole-class discussion was more
structured or flexible.

Contrasting the Nature of Classroom Talk

In addition to the differences in the allocation of time, I selected three elements of
classroom talk to analyze in order to assess whether Mrs. Evans’s classroom was more
flexible and MS. Carson’s was more structured. These characteristics were created from a

bottom-up comparison of my observations of the two classrooms, as well as an analysis

152

of the literature on classroom discourse: (a) the opportunities for off-topic discussion, (b)
how students would obtain the floor, and (c) the extent the discussions maintained or
deviated from the IRE model (Mehan, 1979).

Variability in amount of off-topic discussion. In order for an interaction
segment to be coded as “off topic talk,” the entire interaction had to consist of content
separate fiom the mathematics under discussion; it was not coded as off topic if only part
of the segment addressed alternative content.

In MS. Carson’s class, the topic under discussion in the large group was always
the mathematics content, during the four days analyzed, as well as during my informal
observations throughout the study. Only one-percent of the exchanges on 2/20/03 were
off the mathematical topic in Ms. Carson’s class from the four days of analyzed whole-
class discussion, and it was two students mentioning that they were missing the
vocabulary sheet.

In Mrs. Evans’ class, by contrast, while much of the large group discussion .
focused on mathematics, some of the interaction segments also consisted of off-topic talk,
such as whether the students were required to take notes, whether they would have
homework, or generally teasing one another.

Table 5.5 illustrates the percentage of off-topic interaction segments for both
classrooms across four days of whole-class discussion.

Table 5.5: Percentage of Off-Topic Interaction Segments

 

2/20/03 2/21/03 2/25/03 2/26/03 3/03/03 Average

 

Ms. Carson’s class 1% 0% 0% 0% 0.25%
Mrs. Evans’s class 7% 17% 13% 7% 1 1%

 

 

 

 

 

 

 

 

 

 

153

Generally, in Ms. Carson’s class, students do not engage in off-topic talk, while in
Mrs. Evans’s classroom, students engage in off-topic talk in approximately 10% of the
interaction segments. While this may seem like a significant chunk of interactions to
spend during the classroom discussion on off-topic talk, it is possible that the Mrs.
Evans’s willingness to allow off-topic talk creates a comfortable space for student’s to
initiate talk about mathematics as well.

The following examples illustrate that students primarily initiated the off-topic
talk in Mrs. Evans’s class, but the teacher also would engage in the off-topic talk as well.
Some of these interaction segments addressed classroom management, such as the
example below.

T: Seth.

Steve: Can you tell Alyssa to stop clicking her pens and everything?

T: Please stop clicking your pens.

(Interaction segment #6, 2/20/03)

In other cases, students were attempting to negotiate the amount of work they

were expected to do.

Alyssa: Are we going to have homework tonight?

T: Yes.

Alyssa: No!

Gary: Yes! (Interaction segment #10, 2/21/03, Evans)

Others were about interpreting the text, such as a series of interaction segments
discussing how to pronounce a character’s name from a CMP investigation, Gilberto,

which was an unusual name for White students in a rural setting to encounter.

154

T: Yes?

Sally: Um, I told my dad, the Gilberto thing.

T: Uh, huh.

Sally: And, um, he says that it’s Gilberto {emphasis on the G sound}.

T: We can pronounce it that way if you want. (Interaction segment #8, 2/21/03,

Evans)
Some of the off topic interactions were initiated by the teacher as well, some related to
classroom management, and others were joking around with students. The example
below is a case of the students and teacher making jokes together.

T: I lost my opener. {in a funny voice}

Bill: I lost my brain a long time ago.

Alyssa: You never had a brain.

Albert: We lost our can opener.

{Ss laugh.) (Interaction segment #65, 2/21/03, Evans)

Approximately one-third of the off topic interaction segments were initiated by
Mrs. Evans and the rest were initiated by students, suggesting that the teacher may have
accepted, and perhaps implicitly encouraged, slipping back and forth between off-topic
and on-topic talk. This focus on both mathematical and non-mathematical talk suggests a
more informal tone in Mrs. Evans’ whole-class discussions than in Ms. Carson’s whole-
class discussions. It is possible that the informal tone led to students also being more
likely to initiate talk about mathematics as well as talk about off-topic issues.

Variability in amount of students’ attempts to obtain the floor. Students

initiated some interaction segments; to qualify as “student initiated” segments, students

155

 

Ed‘s :Ii amid .

either had to call out to get the floor, or brought up a topic for discussion, when called
upon by the teacher, that diverged fiom the question asked by the teacher. Additionally,
the segment did not qualify as student-initiated if the solicitation was not picked up on by
the teacher or other students; instead, it was considered to be an attempt at student-
initiation, also coded as assertive talk in terms of individual students’ patterns of
involvement. Some of the segments that were coded as “student initiated” were about
mathematics and others were off-topic.

In Ms. Carson’s class, students raised their hand to be called on, and generally did
not speak until recognized formally by the teacher. Out of the four analyzed whole class
discussions, 4% of the interaction segments were student initiated. Since there were so
few student initiated interaction segments, there was not a typical segment. In the
following example, Tim initiated an interaction segment correcting a calculation.

Tim: There’s negative 3 up there.

Max: It should be negative 1 point 5.

T: Because we’re trying to get a negative 3 up here?

Tim: Yeah.

T: So I should have a negative 1 point 5?

Ss: Yeah.

T: Then, plus a negative three, I get... (Interaction segment #13, 2/21/03, Carson)
In this case, the students put the idea on the floor, and Ms. Carson took the idea up by
asking follow-up questions to clarify what the students said. Then she took the floor back
from the students, redirecting the discussion, as she asked, “and if I look at my table,

when X is 2, what does it say for Y?”

156

In Mrs. Evans’ class, student initiated interaction segments were substantially
more frequent. Table 5.6 illustrates the percentage of interaction segments that were
initiated by students across the four analyzed days of classroom discussion.

Table 5.6: Mrs. Evans’s Class’s Student Initiated Interaction Segments

 

2/20/03 2/21/03 2/25/03 3/03/03 Average

 

Student-initiated segments 49% 36% 17% 14% 29%

 

 

 

 

 

 

 

 

21% of the total amount of student initiated interaction segments was also off-tOpic,
while the rest of the segments were about mathematics. Since examples of off-topic
segments were presented above, the following are examples of student initiated
interaction segments about mathematics. In the first example, Becky initiates an
interaction segment by asking a question about the format of a proposed equation, asking
about the order of the symbols, specifically whether the equal sign could go closer to the
end rather than the beginning of the equation.

Becky: Why don’t you just do 2 times K equals M?

T: Could I do 2 times K equals M?

Jim: No.

T: Can I write it in the other direction?

Ss: No. Yes.

T: What tells me that that’s OK?

Alex: Uh, because it’s Y equals and that equals Y. (Interaction segment #22,

2/21/03, Evans)

157

At times, student initiated segments were similar to Becky’s segment above: A student
posed a question to the teacher, and then the teacher reflected the question back to the
rest of the class. Other instances of student-initiated segments were students putting their
ideas on the floor, when the teacher was collecting a range of students’ perspectives. In
the example below, the teacher initially asks the class, “What about this one?” The
teacher calls the class’s attention to a particular students’ suggested equation, y = 10, for
expressing a relationship such that a participant in a walkathon would collect a flat rate of
$10 from those who are interested in donating rather than a fee per kilometer. Mrs. Evans
is asking the class to evaluate this equation. Alex presents his opinion, then Jamie also
offers her opinion, without being solicited explicitly.

T: What about this one?

[Omittedz joke from Bill.]

Andrew: I don't get it. I don't get how, like, how would you write that. Like, it's
not mathematical. It doesn't seem, like, mathematical.

T: It doesn't seem mathematical to say money equals 10.

Andrew: Yeah.

Josh: Yeah, it does.

T: Yes.

Jamie: I don’t think it shows enough information about the kilometers, like if
someone had that equation, they could just, they would be, like, what? Huh? It
wouldn’t show enough information.

T: So you like this one because it shows what Bill was saying, that she ain’t
getting anything for those kilometers?

Jamie: It shows, like, the table. Like, you could make a table out of it.

158

(Interaction segment #36-38, 2/21/03, Evans)

Since Jamie presented her thinking about the problem right after Alex’s, not in response
to an explicit solicitation for alternative methods or perspectives, Jamie’s contribution is
more student initiated than teacher initiated. Mrs. Evans at times took on the role of a
moderator, collecting a range of students’ perspectives, such as Alex’s and Jamie’s above
and reflecting them back to the class. Mrs. Evans reflected back Jamie’s response by
saying, “So you like this one because it shows what Bill was saying, that she ain’t getting
anything for those kilometers?” These questions called attention to the student initiated
responses and also provided the student the opportunity to elaborate on his or her
response. Mrs. Evans’s role was less that of an explicit evaluator, and more of an implicit
evaluator, as she would call attention to particular student contributions through her
questions. More student initiated interaction segments occurred in Mrs. Evans’s class,
perhaps in part due to the openings for off topic talk, and also in part due to the teacher’s
acceptance for collecting several student responses to one question.

Variability in adherence to I-R-E discourse structure. A common interaction
pattern in classrooms is the Initiate-Respond-Evaluate structure for an interaction
segment, where the teacher initiates a question, the student responds, and the teacher
evaluates (Mehan, 1979). Mrs. Carson’s class used the I-R-E structure with only slight
deviations, while Mrs. Evans’s class rarely used the I-R-E structure, and deviated more
dramatically from this model. I will present examples of the ways in which each teacher
deviated from this model to demonstrate that Mrs. Evans was firrther away fiom the LR-

E structure.

159

Interaction segments in Ms. Carson’s classroom deviated slightly fiom the I-R-E
structure, in formats such as extended I-R-E’s, or initiate-respond-initiate-respond-
evaluate. In the example below, Ms. Carson asks how to find a dependent variable if they
know a particular value for an independent variable.

T: Um, 8 kilometers, and we want to know how much each person would make if

they walked 8 kilometers. For Leanne, how much? I

Allen: 10 bucks.

T: How do you know?

Allen: Because she’s, because, like, she doesn’t even have to walk. She, like, a

sponsor has to pay 10 bucks no matter how far she walks.

T: OK. (Interaction segment #46, 2/20/03, Carson)

This example is very close to the I-R-E structure. The interaction segment begins with the
teacher asking a question. A student responds. The teacher than asks a follow-up
question. The student responds with an elaborated answer, and the teacher evaluates
(“OK,” meaning “correct”). These interactions were typical in Ms. Carson’s class. After
responding to a question with an answer, the teacher replied with a follow up question
about the process (e. g., “How do you know?”), and the student would then explain their
process for obtaining the answer. This I-R-I-R-E structure is not exactly like the I-R-E
structure, but is similar, and was the primary deviation from the I-R-E structure in Ms.
Carson’s classroom.

Interaction segments in Mrs. Evans’s classroom deviated more dramatically fiom
the I-R-E structure in that the teacher did not evaluate students’ responses as directly, and

more than one student would respond. A typical deviation from the I-R-E structure in

160

Mrs. Evans’s class involved the teacher initiating a question, then collecting a range of
students’ responses. The teacher then reflected the responses back to the students, asking
them to evaluate. The teachers’ role was one of revoicing (O'Connor, 1998) and asking
for clarification. In the following example, Mrs. Evans asks the students for the equation
for a relationship, the relationship mentioned above about the participant in the
walkathon who wants to collect pledges at a flat rate of $10.

T: Let's talk about Leanne. With Leanne, we have the same variables of money

and kilometers. How do you get the money for Leanne?

Jim: Oh! I know!

Becky: You get 10 dollars the whole time.

T: I get 10 dollars all the time. How do I write that as an equation?

Becky: 10.

Jim: You go money equals 10 dot K, or also known as time.

T: Becky says 10. Jim says 10 dot K.

Alex: Um, OK, I wrote Y equals zero times X plus 10.

Bill: That's what I put.

Molly: Um, the 10, Jim’s... wouldn't work, because that's saying for every that

kilometer that you go, you add 10 more. That's not going to work.

T: So, you don't like this one? I'm not sure that everyone at that back table is

listening.
As this example illustrates, Mrs. Evans would ask a question (e. g., “How do you get the
money for Leanne?”), and a number of students would reply, as if it was understood that

questions were posed to the group rather than one student at a time. Additionally, the

161

teacher would intervene to reflect back some of the collected responses (“Becky says...
Jim says. . .”) and focus the class’s attention on another student’s critique (“S0, you don’t
like this one?”). This participation structure, collecting students’ responses and critiques,
deviated highly from the I-R-E structure, and did not occur in Ms. Carson’s classroom.

Since students would interject and initiate interaction segments, joking would
occur periodically, more than one student would respond to a teacher initiated question,
and the teacher would not directly evaluate student solutions, Mrs. Evans’s class’s
interaction segments rarely aligned with the I-R-E structure. It was challenging to label
any of the interaction segments in Mrs. Evans’s class as following the I-R-E structure,
particularly due to the lack of explicit evaluation by the teacher. In the following
example, Mrs. Evans inquires as to how to set up a table.

T: ...So where did you put the 45 head start on here?

S: At zero.

T: At the zero, because it wasn't at zero, zero, like the other ones. (Interaction

segment #17, 2/25/03)
Mrs. Evans asked a question, a student answered, and she revoiced, elaborating slightly.
This was the way Mrs. Evans presented her evaluation of students’ answers. These
interaction segments, teacher initiates a question, student responds, teacher revoices, was
the closest approximation to the I-R-E structure in Mrs. Evans’s classroom, although the
specific evaluation portion is implicit in the last turn. This form of interaction segment
occurred less than 5% of the time in each class period, as more than one student usually

participated in an interaction segment.

162

Summary: Differences between whole-class discussions. Mrs. Evans and MS.
Carson’s classrooms differed in the amount of off-topic talk, student-initiated talk, and
deviation fi'om the I-R-E model, suggesting that Mrs. Evans’s class was more flexible
than Ms. Carson’s, as perceived in my classroom observations. Varying degrees of
student initiation and teacher control in the classroom talk may affect whether students
have the opportunity to act on their beliefs and goals. For example, in classrooms where
it is more common for students to initiate an interaction segment, it may be less risky to
ask a question during whole-class discussion, while in a classroom in which students
rarely initiate interaction segments, it may be more of a challenge to ask a question. A
student may have beliefs or goals that support the importance of asking questions during
whole-class discussion, but certain settings may make this act more possible than others.
However, students could experience each of these settings differently; a classroom that
may be inviting and comfortable to one student may be threatening to another, which
suggests no immediately obvious mapping between students’ beliefs and goals and each
of these classroom settings. The relations between the target students’ motivations and
their involvement in classroom discourse will be discussed in more detail in Chapter Six.

Target Students’ Patterns of Involvement

Although there are similarities and differences between the duration and nature of
classroom discussions, in order to examine whether and how beliefs and goals influenced
classroom talk, I also analyzed whether and how the target students each involved
themselves in the whole-class discussions. I selected and analyzed four patterns of
student involvement that were potentially comparable among the target students across

both classrooms. These were: (3) their level of involvement; (b) their hesitancy and

163

assertiveness in their talk; (c) their positioning in terms of how they critiqued others’
ideas; and (d) their mathematical communication in terms of their explanations and
justifications. These patterns of involvement became the focus of this analysis because
they each occurred in varying degrees across both classrooms, and because they bring the
analysis closer to the central issue of how beliefs and goals correlate with patterns of
participation, in these 15 students. Figure 5.1 summarizes the ways I assessed students’
involvement in whole-class discussion.

Figure 5.1: Seventh Grade Students’ Involvement in
Whole Class Discussions about Mathematics

 

Proportion of segments

Persistence: number of extended
involvement 0 Relevance: whether the segments were
on the mathematical topic

 

 

 

 

 

 

 

Hesitancy or assertiveness
Positioning: whether or not students

Nature Of participate in critiquing classmates’
Involvement solutions

 

 

 

0 Mathematical reasoning: level of
meaning

 

 

 

In order to assess the degree to which students participated at all, I assessed their level of
involvement in terms of three factors: (a) the percentage of interaction segments they
participated in, (b) whether they persisted in their talk to be involved in more extended
interaction segments, and (c) whether their talk was about a relevant mathematical topic.

I assessed the nature of students’ involvement in terms of their hesitancy and

164

assertiveness since these particular kinds of involvement could be evidence of students’
confidence while talking during class. I also assessed the nature of their involvement in
terms of positioning, or whether or not the students participated in critiquing a solution or
mathematical idea on the floor. Positioning and critiquing is a way of interacting in
mathematics classrooms that may promote autonomous sense-making and reasoning.
Finally, I assessed the nature of students’ involvement in terms of their mathematical
reasoning, characterizing their explanations and justifications in terms of whether and
how students focused their talk on procedures and calculations or concepts and meaning.
Results for these four patterns of involvement — level of involvement and three factors in
the nature of students’ involvement -— across the four days of whole-class discussion are
presented below for each target student in both classrooms.

Level of Involvement

As discussed in Chapter Three (Methods), one dimension I considered in selecting
the target students was their frequency of participation. So, it was not surprising when
these same students were involved at varying degrees when I observed in the spring.
However, I selected the target students in September, so their level of participation in the
spring may have changed since the fall.

In order to analyze students’ level of involvement in the spring, I focused on three
factors: (a) the proportion of interaction segments in which they participated, (b) whether
their participation in an interaction was extended in those segments (amounting to taking
more than two turns in an interaction), and (c) whether their talk was about mathematics

or off-topic in those segments. Table 5.7 below summarizes how determinations of low,

165

moderate, and high were made for assessing students’ level of involvement along these

 

 

 

 

 

 

 

 

 

criteria.
Table 5.7: Criteria for Students’ Levels of Involvement
Percentage of Extended Non-Mathematical
Segments Participation Talk
2 2 turns
High 2 10% No Low
5 - 10% Yes Low
Moderate 5 - 10% No Low
10% + No High
5 5% Yes Low
Limited _<_ 5% No Low
5 — 10% No High

 

 

 

 

In order to assess the level of target students’ participation as limited, moderate, or high, I
first determined the percentage of segments they participated in, then determined whether
they participated in any extended segments, and finally determined whether they
participated in a high or low amount of mathematical talk. Target students were
considered to participate at a high level involvement amounted to participating in either a
high percentage (above 10%) of segments or a moderate percentage (5-10%) of segments
with extended participation. Limited involvement amounted to participating in either a
low percentage of segments (less than 5%) or a moderate percentage of segments (5-
10%) with a high amount of non-mathematical talk. Target students who participated at a
moderate involvement either participated for a moderate percentage of interaction

segments, for a high percentage of segments but with a high amount of non-mathematical

166

 

talk, or for a low percentage of segments with extended participation. Below I will
discuss target students’ participation along each of the criteria for level of involvement:
percentage of segments, extended participation, and non-mathematical talk.
Proportion of Interaction Segments

In order to determine the percentage of interaction segments that the target
students participated in, I counted the number of interaction segments in which a target
student participated for each of the four analyzed whole-class discussions and divided
that number by the total interaction segments for that discussion. There may have been
more than one target student participating in each interaction segment. Then, I found the
average percentage of interaction segments that the target student participated in across
the four days. Table 5.8 illustrates the percentage of interaction segments that the target
students participated in across the four analyzed whole-class discussions.

Table 5.8: Target Students’ Percentages of Participating Interaction Segments

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Class Day by Target Student

2/20/03 2/21/03 2/25/03 2/26/03 3/03/03 Average
Ms. Carson’s class
Allen 8.6% 12.5% 9.8% 0% 7.3%
Allison 9.9% 6.3% 9.8% 0% 6.5%
Colleen 3.7% 0% 2.4% 4.2% 3.4%
Hannah 7.4% 6.3% 4.9% 4.2% 5.7%
Max 21.0% 31.3% 14.6% 8.3% 18.8%
Pete 9.9% 12.5% 0% 8.3% 7.7%
Tim 11.1% 25% 7.3% 12.5% 14%
Mrs. Evans’s class
Alex 4.4% 12% 1.9% 35.7% 13.5%
Alyssa 4.4% 9.6% 1.9% 10.7% 6.7%
Becky 2.9% 9.6% 7.5% 3.6% 5.2%
Bill 22.1% 25.3% 34% 14.3% 23.9%
Marissa 0% 0% 0% 0% 0%
Molly 7.4% 4.8% 7.5% 10.7% 7.6%
Steve 20.1% 21.7% 11.3% 7.1% 15.1%
Tricia 0% 0% 1.9% 3.6% 1 .4%

 

 

 

 

 

 

 

167

 

There were students in both classrooms that participated in every percentage range. Five
students participated in a high percentage (210%) of interaction segments (Max and Tim,
in Ms. Carson’s class, and Alex, Bill, and Steve, in Mrs. Evans’ class). Seven students
participated in a moderate percentage (5-10%) of interaction segments (Allen, Allison,
Hannah, and Pete, in Ms. Carson’s class, and Alyssa, Becky, and Molly, in Mrs. Evans’
class). Three students participated in a limited proportion (55%) of interaction segments
(Colleen in Ms. Carson’s class and Marissa and Tricia in Mrs. Evans’ class). On this
basis, the classes, as represented in the focal students, look approximately equivalent.
However, fi'equency of participation was not considered sufficient for calculating
students’ level of involvement, particularly because frequency of participation may reveal
only how often the teacher called on the student and does not take into account whether
the talk was on task. I also considered whether and how students participated within the
interaction segments, including whether they participated for an extended time or whether
the talk was about mathematics.
Extended Interaction Segments

Some of these students participated in more extended segments than others.
Students who participated in an interaction segment by taking more than one turn, and
did so over more than one large group discussion (at least two days out of four), were
considered students with more extended participation in their level of involvement. The
following example, from Mrs. Evans’ class on 2/21/03, demonstrates a case of Becky’s
extended participation.

T: I owe them 2 dollars. So if I show up at the walkathon, and I don't walk at all,

what happens?

168

Becky: You get 5 dollars.

T: Where does it say I get 5 dollars if I don't walk at all?

Becky: It's starting at 5.

T: It's starting out at 5 dollars. What happens if I walk a kilometer?

Jim: You lose two of them!

Becky: You have 3.

T: I walk another kilometer?

Ss shout answer. (Interaction segment #75, 2/21/03, Evans)
In this interaction segment, we see Becky continuing her participation, even when not
called on directly, when other students are also taking up the floor, such as Jim. This
extended participation indicates a persistence on Becky’s part to continue to be a part of
the conversation about this mathematical topic. This persistence is indicative of
engagement on the student’s part. If a student is willing to persist in talking about the
topic, they are likely also engaged in thinking about it, although there may be students
who are engaged in thinking about the topic who are not talking.

Additionally, the following example, form 2/21/03, demonstrates extended
participation in an interaction segment from Ms. Carson’s class.

T: OK. Doesn't count by anything. Good. So, which tables did you decide would

represent linear relationships? Max?

Max: Uh, 1 and 4.

T: Why 1?

Max: Because, um, on the y-axis it's always 3, so it's always at a constant rate.

T: What is the constant rate?

169

Max: Uh, 3.

T: It's always at 3? But the constant isn't three, what is the rate?

Max: Oh.

T: Meaning what is it changing by, Max?

Max: Nothing.

T: So the number is?

Max: Zero.

T: The rate, it does have a constant rate of zero, it's always three, so that one's

linear. What's my equation for that one? Table one. Albert? (Interaction segment

#6, 2/21/03, Carson)
In both classrooms, the teacher’s follow-up questions appeared to be posed in such a way
that either the student who was most recently speaking was invited to continue, or any
other student was welcome to participate. On the one hand, answering a teacher’s follow-
up question may not appear like persistence, but more like responding directly to a
question, students did not always respond to these follow-up questions, sometimes
waiting for other students to respond instead.

Students who participated in two or more extended interaction segments were
those who persisted in their talk, according to this analysis. Table 5.9 demonstrates how
many total extended interaction segments that the target students participated in across all

four analyzed whole-class discussions.

170

Table 5.9: Number of Extended Interaction Segments for each Target Student

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Percentage Extended
Extended Interaction
Interaction Segments
Segments

Ms. Carson’s class
Allen 6.7% 1
Allison 7.7% 1
Colleen 0% 0
Hannah 7.7% 1
Max 16.7% 5
Pete 9.1% 1
Tim 27.8% 5

Mrs. Evans’s class
Alex 26.9% 7
Alyssa 6.3% 1
Becky 23.1% 3
Bill 24.1% 14
Marissa 0%
Molly 6.3% 1
Steve 30.3% 10
Tricia 0% 0

 

 

 

 

In Ms. Carson’s class, two students persisted in their participation: Max and Tim. Both of
these students participated at a high percentage. In Mrs. Evans’s class, four students
persisted in their participation: Alex, Becky, Bill, and Steve. Alex, Bill, and Steve
participated at a high percentage, but Becky participated for a moderate percentage of
interaction segments, and persistence in her participation changed her level of
participation fi'om moderate to high. No other students’ level of participation changed as
a result of considering persistence, or participating in extended interaction segments.
Off-Topic Talk

In addition to percentage of interaction segments and extended participation, or
persistence, students’ talk was assessed in terms of relevance, or participation in

mathematical or non-mathematical talk. Target students’ interaction segments were

171

examined in terms of whether they participated in non-mathematical, or off-topic, talk
during the whole class discussion.13 Students with a high amount of non-mathematical
talk participated in off-topic interaction segments for more than half of their total
interaction segments. Students with a low amount of mathematical talk participated in
off-topic interaction segments for less than half of their total interaction segments.

Most students did not participate at a high level of non-mathematical talk. No
target students in Mrs. Carson’s class participated in non-mathematical talk. Only Max
participated in any non-mathematical talk across the four analyzed days of whole-class
discussion in Mrs. Carson’s class, and he only participated in one non-mathematical
interaction segment. Max also participated in the most interaction segments out of all of
the target students in Mrs. Carson’s class. In Mrs. Evans’s class, two students participated
in a high amount of non-mathematical talk: Bill and Alyssa. Bill also participated in the
most interaction segments out of all of the target students in his class. Since Bill and
Alyssa both participated in non-mathematical talk for half or more of their interaction
segments, their level of participation changed from the percentage range to the next
lowest level. While Bill participated in a high percentage of interaction segments, his
high level of non-mathematical talk shifted his level of participation down fi‘om high to
moderate. Alyssa participated in a moderate percentage of interaction segments, but her
high level of non-mathematical talk shifted her level of participation down from moderate
to low.

The following table presents the results for the target students’ levels of

involvement by classroom. The three factors were taken into consideration: proportion of

 

’3 Off-topic interaction segments were discussed previously in the chapter, when the classrooms were
contrasted.

172

interaction segments, persistence (extended interaction segments), and relevance (lack of
non-mathematical talk).

Table 5.10: Classes by Target Students’ Levels of Involvement

 

Ms. Carson’s class Mrs. Evans’s class

 

High Max, Tim Alex, Steve
Involvement

 

Moderate Allen, Allison, Hannah, Pete Becky, Bill, Molly
Involvement

 

 

Limited Colleen Alyssa, Marissa, Tricia
Involvement

 

 

 

Not surprisingly, the distribution of students across levels of participation is somewhat of
a bell-curve distribution, with more students participating at a moderate level of
participation, and less at high and limited levels. In addition to the fact that this variation
is to some degree naturally occurring in most classrooms, I selected target students in part
to achieve this sort of distribution.
Nature of Involvement

Looking beyond the level of students’ involvement, or the frequency of their
participation, I also examined the nature of students’ involvement in terms of three
factors: hesitancy or assertiveness, positioning, and mathematical reasoning. Each of
these factors, and evidence for students’ participation according to these factors, will be
presented below.
Hesitancy and Assertiveness

One of the ways I assessed the nature of students’ talk during whole-class

discussions was by examining whether they were hesitant or assertive during interaction

173

 

segments. I conjectured that students’ assertiveness could indicate a high degree of
confidence or self-efficacy in mathematics, while students’ hesitancy could indicate a
lower degree of confidence or self-efficacy in mathematics. Alternatively, assertive or
hesitant interaction segments may be fostered more in one classroom than another.

Students’ participation in an interaction segment was coded as hesitant if they
exhibited some hedging or backing down. For example, Allison’s hesitancy in the
interaction segment below from Ms. Carson’s class is an example of a student backing
down fi'om an interaction:

T: OK, any other estimates? Allison?

Allison: Um, I think it's 3, never mind.

T: Nick? (interaction segment #38, 2/20/03)
Allison backed down from the interaction by starting to provide an answer, but then
stopped, ending her turn with “never mind.” Backing down from an interaction was
considered a form of hesitancy because the student did not stand by her initial answer.

Students also demonstrated hesitancy by hedging in their talk when expressing
solutions. The example below is a case of hedging from Ms. Carson’s class.

T: Let's do this. Look at Alana's first. Where do we see the y-intercept in her

equation? Allen?

Allen: You see it at, uh, like, in between like, um, 3 point, uh, I don't know how to

explain it. Hold on. It's like, we see it around, like, 5x and 10y. You see it.

T: OK, what I'm talking about, Allen, is this point right here. OK? This point right

here. Where do I see that in her equation?

Allen: OH! OK. Um.

174

T: Or maybe I can't? Max? (Interaction segment #25, 2/20/03, Carson)

In this example, Allen’s expression of “I don’t know how to explain it. Hold on,”
indicates hedging, or a sense of buying time to figure out what he thinks. Hedging was
considered evidence of a lack of certainty on the student’s part about their answer, since
the student did not respond as if he immediately knew the answer.

Participation in an interaction exchange was coded as assertive differently in each
classroom, as the differences in structure and teacher control in the classrooms created
different possibilities for exhibiting assertive behavior. In Ms. Carson’s classroom,
assertive talk looked like soliciting the floor or speaking without being called upon. In the
example below, Allen comments aloud in response to the teacher’s statement and is not
called upon.

T: one thing that I did not see anybody in this class do, but I did see in 2nd hour,

was the graph.

Allen: I know, I saw that up there.

T: If I were to make a graph, what would go up here? Mara? (Interaction segment

#1, 2/25/03, Carson)

When students commented aloud without being called upon in Ms. Carson’s class, the
teacher rarely pursued their comments with a follow-up question. Since students’
comments were not pursued by the teacher, students may have been less likely to initiate
interaction segments.

Behaviors such as soliciting the floor or speaking without being called upon were
more normative for all students in Mrs. Evans’ class, so assertive talk instead looked like

initiating a new topic of conversation. Some of these instances of assertive talk were also

175

coded as off-topic talk. In the example below, Alyssa initiates an off-topic interaction
segment.

Alyssa: Do you want us to write this down?

Ss: Yes, she just said that. She just said yes.

Gary: Oh, my God. Tells how much our group pays attention.

Bill: It's not our group, it's just.

Gary: Those girls. (Interaction segment #16, 2/25/03, Evans)
Alyssa’s initiation of a question about whether they had to take notes was a shift fi'om the
mathematical topic on the floor, as prior to Alyssa’s question, the class was discussing
whether it would help to represent a particular relationship from a story problem in
tabular form. In this example, the students followed up on Alyssa’s question rather than
the teacher. The teacher or classmates may not have followed up additional instances of
assertive talk, such as jokes. In the example below, Bill makes a joke.

T: So, I'm feeling like yesterday's discussion, even though it got a little weird, it

might have been helpful for more people besides Steve.

Bill: I want a bronto burger. (Interaction segment #5, 2/21/03, Evans)
This comment, and other similar jokes, was spoken loud enough to make it to the floor,
so it was considered to be an attempt at a student initiated interaction segment. Attempted
student interaction segments were coded as assertive talk because the student was taking
the initiative to begin an interaction. The turn did not have to be followed up on in order
to be coded as assertive talk.

Other instances of assertive talk were more about mathematics, in order to pursue

a question or clear up one’s own confusion. In the example below, Mrs. Evans asks the

176

students for their perspective on an equation, one student evaluated the equation , and the
teacher revoiced his response. Alex then initiates a question about the form of the
equation itself.

T: What about this one?

Bill: That's a loser one, too.

T: That's a loser one, too.

Ss: No...

Alex: I don't get it. I don't get how, like, how would you write that. Like, it's not

mathematical? It doesn't seem, like, mathematical.

T: It doesn't seem mathematical to say money equals 10.

Alex: Yeah.

Jim: Yeah, it does. (Interaction segments #36-37, 2/21/03, Evans)
In this example, the teacher followed up on Alex’s statement with revoicing. The next
series of interaction segments pursued Alex’s concern. In Mrs. Evans’s class, students
sometimes refocused the talk to pursue their own questions about the mathematics under
discussion.

The following table presents the percentages of the target students’ interaction
segments that were coded as hesitant or assertive out of their individual totals of

interaction segments.

177

Table 5.11: Percentages of Hesitancy and Assertiveness by Target Student

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I Hesitancy I Assertiveness

Ms. Carson’s class
Allen 27% 33%
Allison 3 l % 0%
Colleen 20% 0%
Hannah 31% 0%
Max 3% 27%
Pete 0% 9%
Tim 6% 22%

Mrs. Evans’s class
Alex 0% 46%
Alyssa 0% 38%
Becky 0% l 5%
Bill 0% 62%
Marissa 0% 0%
Molly 0% 13%
Steve 0% 39%
Tricia 0% 50%”

 

 

 

 

 

Only target students in Ms. Carson’s classroom expressed hesitancy, and all but one of
the target students in Ms. Carson’s class exhibited some expression of hesitancy in their
talk. All of the male target students in Ms. Carson’s class exhibited assertive talk. All but
one of the target students in Mrs. Evans’s class exhibited assertive talk, and Mrs. Evans’s
target students expressed higher percentages of assertive talk than those in Ms. Carson’s
class. Some of the students who exhibited hesistancy also exhibited assertiveness. For
example, Allen sometimes spoke up without being called upon. These striking
differences between the two classrooms in terms of assertive and hesitant patterns of
involvement suggests that classroom practices may also play a strong role in shaping this
form of involvement, but the next chapter will also examine whether beliefs and goals

also shape this form of involvement.

 

'4 1 out of 2 contributions

178

Positioning

The nature of students’ involvement was also examined in terms of their
positioning. In these classrooms, students were invited to varying degrees to critique their
classmates’ solutions. I refer to this form of talk as positioning, as through the act of
critique, they agree and disagree with the thinking of their classmates, aligning
themselves for and against each other. In the following example from Ms. Carson’s class,
Tim positions himself against Albert’s solution.

T: The rate, it does have a constant rate of zero, it's always three, so that one's

linear. What's my equation for that one? Table one. Albert?

Albert: Wouldn't it be y equals zero x?

T: You guys agree?

Ss: mm, hmm

Tim: No, Y equals 3.

T: Allen?

Allen: That's what I put, I put y equals 3.

Pete: Yeah, I had that, too.

T: Any other thoughts?

Tim: If you had y equals zero x, it'd be all zero.

T: It's be all zero. And it's not all zero, is it?

Tim: No.

T: It's always 3. Y is always 3, there's your equation for table 1. Max, you also

said 4, right? (Interaction segments #7-8, 2/21/03, Carson)

179

 

t n. NH.“ ..l ‘i.!lljne .

 

In this segment, Tim’s involvement was coded in terms of “disagree” for his form of
positioning, because he was the first one to disagree, and he offered a justification for his
response. Ms. Carson’s question, “You guys agree?” may have invited students to
evaluate the solution on the floor. Such questions were not commonly posed in Ms.
Carson’s class, as interaction segments in this class were more often in the form of the I-
R—E structure, with the teacher evaluating directly.

Positioning appeared to be more normative in Mrs. Evans’ class, as students
offered critique without being solicited by the teacher, and the teacher explicitly
encouraged critique with which comments she highlighted in the talk and the nature of
her questions. For example, students were discussing whether or not a particular
relationship was linear, and the equations for representing these relationships. The
problem was about pledge plans for a walkathon. One person, Leanne, did not collect
money per kilometer, but instead asked people to donate a flat fee of $10. Three forms of
equations were proposed: y=10. y= 0x + 10. y=10x.

Molly: Um, the 10, Jim’s... wouldn't work, because that's saying for every that

kilometer that you go, you add 10 more. That's not going to work.

T: So, you don't like this one? I'm not sure that everyone at that back table is

listening.

... [classroom management talk omitted]

T: Molly is saying she doesn't like this one, say it one more time, 'cause I'm not

sure it's all been heard.

Molly: because, 10, it's saying that for every kilometer you go, you're adding 10

more, so if you go 2, you're gonna have 20, and it's really only 10.

180

T: so you would disagree with that one.

Molly: Yeah.

T: How about other people? (Interaction segments #30, 32, 2/21/03, Evans)

In this case, Molly is providing her critique of the y = 10x solution. She began
positioning herself against that solution, and the teacher noticed that the class was not
attending to her talk, so through classroom management talk, the teacher attempted to
focus the class on her disagreement.

Students also agreed with the solutions of their classmates. For example, in the
same discussion over Leanne’s pledge plan, Bill positioned himself to be for, rather than
against, a solution presented by Alex (y = 0x + 10):

T: Bill.

Bill: I agree, with Alex, because, uh, he ain't getting paid per kilometer.

...[jokes and clarification question by student omitted]

T: And Bill was saying he liked that one, but I missed why. Why did you like that

one?

Bill: Because, uh, she ain't gonna run, uh, she ain't getting money whenever she

walks, but she is getting 10 dollars.

T: So you like saying it's zero times the number of kilometers, and then she just

gets 10 bucks.

Bill: Yeah, she gets 10 bucks for the fun of it. (Interaction segments #33, 36,

2/21/03, Evans)

Again, Mrs. Evans focused the talk back to a student’s move to critique, encouraging a

justification for the position that was not initially presented.

181

The following table illustrates students’ positioning in the two classrooms, with a
percentage of times they positioned themselves out of their total interaction segments.

Table 5.12: Percentages of Interaction Segments Involving Positioning

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I Positioning

Ms. Carson’s class
Allen 20%
Allison 0%
Colleen 0%
Hannah 0%
Max 0%
Pete 9%
Tim 1 1%

Mrs. Evans’s class
Alex 19%
Alyssa 6%
Becky 8%
Bill 5%
Marissa 0%
Molly 38%
Steve 1 2%
Tricia 0%

 

 

 

 

Positioning occurred among more target students in Mrs. Evans’s class than those in Ms.
Carson’s class. Only three students, all boys, participated in positioning in Ms. Carson’s
class. The only students in Mrs. Evans’s class who did not participate in positioning
rarely participated at all over the four analyzed whole-class discussions. Marissa and
Tricia did not participate in positioning, and each participated in only two interaction
segments over the four days.
Students’ Mathematical Reasoning

The content of students’ contributions in their talk also varied in how they
reasoned about mathematics. For this phase of analysis, interaction segments were only
coded if they contained mathematical reasoning. Students’ mathematical reasoning in an

interaction segment was coded in terms of three levels: low, moderate, and high levels of

182

explicit meaning. These levels were inspired by QUASAR’S (Henningsen & Stein, 1997 ;
Silver & Stein, 1996; Stein, Grover, & Henningsen, 1996; Stein & Lane, 1996) analyses

of problem solving in Standards-based mathematics classrooms, but modified given the

nature of the content in these discussions. The conceptual talk in these mathematics

lessons was around the idea of how one could know whether a relationship was linear.

Table 5.13 illustrates the coding of students’ mathematical reasoning:

Table 5.13: Definitions of Students’ Mathematical Reasoning

 

 

 

 

 

 

 

 

Level of Code Definition Illustrative Example
Mathematical
Reasoning
Low Explicit Facts Student answers T: Say it loud. The y-intercept.
Meaning a question such What definition do we have for
as naming a the y-intercept? Hannah?
feature on a Hannah: Where the line
graph or crosses the y-axis.
definition (Interaction segment #31,
previously 2/25/03, Carson)
discussed.
Procedures Student describes T: What about Gilberto? For
without how he or she him to make 10 dollars?
meaning solved a problem Allison?
by talking about Allison: OK, what I did is that
calculations. I did 10 divided by 2, and I got
5.
T: Super...
(Interaction segment #54,
2/20/03, Carson)
Moderate Procedures Student describes T: what about Alana?
Explicit with meaning how he or she Colleen?
Meaning solved a problem Colleen: 5 plus point 50 K
by talking about equals C.
why the T: What does that mean?
calculations were Colleen: 5 plus 50 cents every
appropriate or kilometer, you get 5 dollars,
why and how times every kilometer, you get
they work. 50 cents.
T: Good. . ..
(Interaction segment #16,
2/20/03, Carson)

 

183

 

Table 5.13, continued

 

 

 

 

 

 

why a
relationship is
linear or not.

 

Interpreting Student describes T: ...where do I, if I have my
representations how to read a table of time, here’s my
representation distance, for Emile and Henri,
such as a graph how do I see this point of
or table. intersection on the table,
(Reading an because some of you did make
equation is coded the table, where do I see that?
as “procedures Tim?
with meaning”) Tim: When it’s on the graph,
or, the table, when it says 75
and 75 on the distance.
T: Yeah, so they’re both the
same. ..
(Interaction segment #29,
2/25/03, Carson)
High Explicit Concepts Student talks T: Steve.
Meaning about a solution Steve: OK, thank you. See,
in terms of didn’t she, didn’t someone
concepts such as give her a pledge for 5 dollars?
constant rate or T: That’s what she’s saying,
in light of the yeah.
problem context Steve: If someone gave her a
in a way that is pledge for 5 dollars, that’s just
not as closely like the zero, you know?
tied to the (Interaction segment #20,
procedures. 2/20/03, Evans)
J ustifying Student presents T: OK, Molly, other thoughts.
linearity an argument for Molly: For Alana, the reason

why I thought it was, um,
linear, I don’t know if it is,
because a unit rate is when it’s
at 1, and you really shouldn’t
count the zero, because it’s not
really at that, you know what I
mean, because, like, one
kilometer is her unit rate, that’s
what she gets per kilometer, so
that’s why.

(Interaction segment #9,
2/20/03, Evans)

 

Although there were two codes for each level, results are presented for each student in

each classroom according to each of the three levels. Percentages of student talk at each

184

 

level were calculated for number of coded segments out of his or her total segments that
were coded as mathematical talk. Table 5.14 presents the results for levels of
mathematical talk for each student and the average percentages for each class. Students
were assigned an overall level of explicit meaning based upon which category they had a
higher percentage in. If the student was within 10% between two categories, he or she
was assigned both categories overall.

Table 5.14: The Level of Mathematical Reasoning for Target Students

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Low Explicit Moderate Explicit High Explicit Overall
Meaning Meaning Meaning

Ms. Carson’s
class
Allen 58% 34% 8% Low
Allison 54% 46% 0% Low / Moderate
Colleen 20% 60% 20% Moderate
Hannah 73% 27% 0% Low
Max 50% 50% 0% Low / Moderate
Pete 64% 27% 9% Low
Tim 41% 59% 0% Moderate
Class average 51% 43% 6% Low / Moderate
Mrs. Evans’s
class
Alex 24% 47% 30% Moderate
Alyssa 40% 40% 20% Low / Moderate
Becky 36% 45% 1 8% Low / Moderate
Bill 42% 29% 20% Low
Marissa 0% 0% 0% None
Molly 31% 15% 54% High
Steve 19% 50% 32% Moderate
Tricia 0% 0% 0% None
Class average 24% 58% 23% Moderate

 

In Ms. Carson’s class, moderate was the highest level of explicit meaning among
the target students’ mathematical reasoning. Three of the students reasoned at a low level
of explicit meaning, while two others reasoned between low and moderate. No students

in Ms. Carson’s class reasoned at a high level of explicit meaning. Only three of Ms.

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Carson’s seven students participated in any high explicit meaning interaction segments,
and two of them participated high explicit meaning interaction segments for less than
10% of their total interaction segments that included mathematical reasoning. These two
students, Allen and Pete, were also the only two students who positioned themselves in
Ms. Carson’s class. The third student, Colleen, also reasoned at a higher explicit level of
meaning than her classmates, but did not position herself. Procedures without meaning
was the most frequent form of mathematical reasoning among the target students in Ms.
Carson’s class.

More target students in Mrs. Evans’s classroom reasoned at a higher level of
meaning than those in Ms. Carson’s class. Justijying linearity, reasoning at the highest
level of explicit meaning, occurred among the target students in Mrs. Evans’s class.
Procedures with meaning was the most frequent form of mathematical reasoning for the
target students in Mrs. Evans’s class, which was higher than the most frequent form of
reasoning among the target students in Ms. Carson’s class, procedures without meaning.
One of the students in Mrs. Evans’s class reasoned at a high level of explicit meaning,
while no students in MS. Carson’s class reasoned overall at a high level of explicit
meaning. Only one of the students reasoned at a low level of explicit meaning, in contrast
to three in Ms. Carson’s class. Two of the students reasoned between low and moderate
and two also reasoned at a moderate level, similarly to the students in Ms. Carson’s class.
Molly, the student who reasoned at the highest level, also positioned most often. The two
students who reasoned at a moderate level also positioned at levels above 10%, which

were the next two most frequent participators in positioning after Molly.

186

Statistical Analyses

In order to test these analyses of whether target students’ patterns of involvement
differed statistically between the two classrooms, I conducted unpaired t-tests to compare
the target students’ mean percentages in each classroom on the different patterns of
involvement. Results fi'om these analyses can be seen in Table 5.15.

Table 5.15: Comparisons Between the Two Classrooms:
Target Students’ Patterns of Involvement

 

 

M % t(13) p d
(SD)
Carson Evans
Level of Involvement
Frequency 9.26 8.94 9.13 0.93 0.05
(4.77) (7.93)
Extended Interaction Segments 10.94 14.61 0.64 0.53 0.36
(8.86) (12.69)
Nature of Involvement
Assertive 13.00 32.88 2.09 0.06 1.16
(14.14) (21.29)
Hesitant 16.73 0.00 3.53 0.01 ** 1.96
(13.47) (0.00)
Positioning 5.71 11.00 0.96 0.36 0.53
(7.89) (12.57)
Explicit Mathematical Meaning
Low 51.43 24.00 3.13 0.01M 1.73
(17.18) (16.69)
Moderate 43.29 28.38 1.61 0.13 0.89
(14.12) (20.67)
High 5.29 21.75 2.28 0.04* 1.27
(7.63) (17.65)

 

Note: Percentages out of each target students’ totals.

*p<.05. **p<.01.

187

These analyses provide another lens for determining whether target students participated
differently in each classroom. Target students’ involvement in terms of frequency,
extended interaction segments, and positioning did not appear to differ between the two
classrooms. Participation in assertive interaction segments and segments involving
moderate level of explicit mathematical meaning did not differ much, but perhaps
slightly, between classrooms. Interaction segments involving hesitancy and high and low
levels of explicit mathematical meaning were significantly different between the groups
of target students in each classroom.

Summary: Nature of Whole-Class Discussion & Target Students’ Involvement

Before discussing whether and how target students participation practices related
to their beliefs and goals in Chapter Six, I will revisit and summarize the main findings of
this chapter. I analyzed both similarities and differences between the whole-class
discussions in these two classrooms as well as patterns of involvement for the target
students.

These two classrooms shared a model of instruction, Launch, Explore,
Summarize, used the same mathematical tasks from the CMP textbooks, and each
devoted time to whole-class discussion. However, Ms. Carson’s class spent more time on
seatwork, while Mrs. Evans’s class spent more time in whole-class discussion. Both
classrooms pursued multiple solution methods, but MS. Carson’s class emphasized
multiple ways of calculating an answer, while Mrs. Evans’s class also discussed
alternative representations. What counted as an alternative or different solution varied

between the two classrooms.

188

Ms. Carson’s whole class discussions were more structured than Mrs. Evans’s.
The discussions in Ms. Carson’s classroom followed the I-R-E structure closely, while
Mrs. Evans’s classroom deviated more dramatically fiom the I-R-E structure,
alternatively collecting a range of student responses, less evaluation directly from the
teacher, and some student initiated questions. In Mrs. Evans’s classroom, there were
student initiated interaction segments and off-topic talk, which rarely happened, if ever,
in Ms. Carson’s classroom.

The target students’ level of involvement was similar in both classrooms, with
most participating at a moderate level, and a few in each classroom participating at a high
and limited level.

The nature of the target students’ involvement was assessed in terms of their
hesitancy and assertiveness, positioning, and mathematical reasoning. In terms of their
hesitancy'and assertiveness, only the target students in Ms. Carson’s class expressed
hesitancy, while all but one of Mrs. Evans’s target students expressed assertiveness. Four
out of seven target students, all boys, in Ms. Carson’s class expressed assertiveness. The
differences in hesitancy were supported by statistical analyses.

With respect to positioning, three out of seven target students engaged in
positioning in Ms. Carson’s class, while six out of eight target students engaged in
positioning in Mrs. Evans’s class. The three target students who were involved in
positioning in Ms Carson’s class were either at moderate or high levels of involvement,
but one of the students at a high level of involvement did not participate in positioning.
All three of these students engaged in assertive talk. The two target students in Mrs.

Evans’s class who did not engage in positioning were at low levels of involvement. One

189

the one hand, positioning appears to relate to level of involvement, but on the other,
positioning may be moderated by teacher, as Mrs. Evans appeared to encourage more
positioning than Ms. Carson. Statistical tests did not support differences in positioning
between the groups of target students in each classroom.

The target students’ mathematical reasoning differed between the two classrooms,
with more target students engaging in higher explicit levels of meaning in Mrs. Evans’s
classroom. Mathematical reasoning generally appeared to be related to positioning, as the
students with higher levels of mathematical reasoning also participated in higher
percentages of positioning. One exception is Colleen, from Ms. Carson’s class, who
reasoned at a moderate level of explicit meaning, but did not position herself. However,
she also participated at a limited level of involvement, the only student in Ms. Carson’s
class to do so. On the one hand, target students’ levels of mathematical reasoning appear
to relate to their engagement in positioning. On the other, mathematical reasoning could
also be shaped by the teacher’s practices, since the target students in Mrs. Evans’s class
reasoning at higher levels. This difference could also be due to sampling. The differences
in students’ levels of mathematical reasoning were supported by statistical analyses.

The differences in students’ patterns of involvement between these two
classrooms suggests that classroom practices may play a role in shaping these target
students’ patterns of involvement, particularly in terms of hesitant talk, and low and high
explicit mathematical reasoning. However, there were other patterns of involvement that
did not appear to be different between classrooms. Students’ individual agency may lead

to some patterns of involvement to be moderated by factors such as their beliefs and

190

goals. In the next chapter, I will explore whether students’ beliefs and goals relate to

these participation practices.

191

CHAPTER SIX
RESULTS: RELATIONS BETWEEN STUDENTS’ BELIEFS, GOALS,

AND THEIR INVOLVEMENT IN WHOLE-CLASS DISCUSSIONS

The overarching goal of this study was to examine whether and how students’
beliefs and goals influence students’ participation in whole-class discussions about
mathematics. In this chapter, I will examine whether groups of students who shared
similar beliefs participated similarly or differently in class discussions. It may be
reasonable to expect that students’ participation in whole-class discussions about
mathematics would be shaped by classroom discourse practices or the ways of
participating promoted by the teacher rather than by the students’ beliefs about learning
mathematics. From this perspective, it is more the participation in these different
classrooms that shaped students’ beliefs rather than the reverse. In Chapter Five, I
explored students’ patterns of participation from this perspective. In this chapter, I
explore relations between students’ beliefs, goals, and their involvement. Such analyses
could provide insights about students’ roles in the development of classroom discourse
practices.

The analyses in Chapter Four showed that the students in the two classrooms did
not differ dramatically in their beliefs and goals. The 15 target students express four
different clusters of beliefs and goals in their interviews. The first distinction between
these four clusters is that two of them, learn the material in a public forum and gain
attention, contained negotiated knowers and low social risk students, while the other two

clusters, do the right thing and help others while overcoming social risk, contained

192

received knowers and high social risk students. Students’ academic and social goals then
distinguished between each pair of clusters. Learn the material in a public forum
included a completing the task goal, and gain attention had a gain status goal, while they
both shared an appear competent goal. In addition to goals to appear competent and gain
status, students in do the right thing had a goal to behave. The students in help others
while overcoming social risk held a social goal of helping others. Out of the four clusters,
learn the material in a public forum is focused on mastering mathematics, while the other
three are predominantly focused on social concerns. Gain attention is more self-focused,
while do the right thing and help others while overcoming social risk are more
community-focused.

Students’ beliefs and goals were not distributed differently than we would expect
due to chance between classrooms. In their survey responses, the population of students
from both classrooms differed on only one of the nine scales (confidence). These data
suggested either that students could hold beliefs and goals independent fi‘om their
classroom experiences, or that both classrooms supported similar beliefs and goals among
the students.

In Chapter Five, I examined the extent to which the target students’ patterns of
involvement varied between the two classrooms. Students’ involvement varied in terms
of the nature of their involvement, particularly in terms of the hesitancy expressed in their
discourse, and low and high levels of explicit meaning expressed when talking about
mathematics during whole-class discussion. In MS. Carson’s class, more target students
expressed hesitancy in their talk and a low level of explicit meaning. In Mrs. Evans’s

classroom, more target students talked about mathematics at high level of explicit

193

meaning. Many patterns of involvement were more similar than different between the
two classrooms, including indicators of the level of students’ involvement, such as
frequency and participation in either extended or off-topic interaction segments.
Additionally, the nature of students’ involvement did not vary between classrooms in
terms of their assertive talk, positioning, and moderate level of explicit meaning
expressed when talking about mathematics.

In this chapter, I examine whether target students participated differently in terms
of the beliefs and goals that they expressed in interviews — that is, I directly explore the
issue of whether individual students’ expressed beliefs and goals influence their
participation in discussion. Rather than grouping the students by classroom, in this
chapter, I group the target students by the beliefs and goals they expressed in the
interviews. I first examine whether patterns of involvement differed with respect to
epistemological beliefs and the perceived level of social risk associated with participating
in whole-class discussion.” Then, I examine whether students who express particular
academic and social goals participate differently than those who do not. Finally, I
examine whether involvement in whole-class discussion about mathematics appears to
vary based on the four clusters of beliefs and goals discussed in Chapter Four. ’
Epistemological Beliefs, Social Risk, and Involvement

In Chapter Four, the target students’ interview responses led to two belief clusters
depending on their expression of (a) epistemological beliefs and (b) perception of risk
associated with participating in whole class discussion, when moderate and high risk

students were collapsed into one cluster. Eight target students talked about learning as a

 

'5 As social risk and epistemological beliefs co-varied in this sample almost perfectly, there was little to
gain from looking how each related to students’ participation.

194

process of receiving knowledge and associated a high level of risk with participating,
while seven target students talked about learning as a process of negotiation and
associated a lower level of risk with participating.

In order to analyze whether these two groups of students participated differently, I
conducted unpaired t-tests to assess for differences in the mean percentages for each
pattern of involvement. I set the level of significance to be a two-tailed p value < .05. I
calculated each student’s percentages for frequency differently than I calculated his or her
percentages for the various patterns of involvement. Frequency percentages were
calculated by dividing the target student’s total interaction segments by their classroom’s
overall total of interaction segments across the four analyzed discussions. Means and
standard deviations of these frequency percentages were calculated for the two belief
groups. Percentages for the other patterns of involvement were calculated by dividing the
number of interaction segments in which the target student exhibited a particular pattern
of involvement (e. g., hesitant talk, positioning) by the target student’s total interaction
segments. Using each student’s individual total for these percentages approximates what
would be normative for that student. Means and standard deviations were calculated for
the two belief groups for each pattern of involvement. Table 6.1 displays the mean
percentages of the two groups for the nine patterns of involvement fi'om Chapter Five, the

t-value of the difference between the means, p-value, and effect size of the difference.

195

Table 6.1: Comparisons Between Belief Groups: Target Students’ Patterns of

Involvement, Version A

 

 

M % t(13) p d
(SD)
Received Negotiated
Knowing & Knowing &
High Risk Low Risk
Level of Involvement
Frequency 6.79 l 1.72 1.56 0.14 0.86
(5.72) (6.55)
Extended 5.80 20.88 3.69 0.01" 2.05
(5.79) (9.80)
Off Topic 14.48 18.09 0.31 0.76 0.17
(26.02) (18.03)
Nature of Involvement
Assertive 19.57 28.24 0.81 0.43 0.45
(19.81) (21.74)
Hesitant 10.10 5.19 0.76 0.46 0.42
(13.38) (11.47)
Positioning 4.42 13.62 1.73 0.11 0.96
(7.23) (12.26)
Explicit Meaning
Low 38.09 36.58 0.13 0.90 0.07
(28.37) (13.34)
Moderate 29.73 41.68 1.24 0.24 0.69
(21.47) (14.62)
High 7.18 21.72 1.96 0.07 1.09
(8.78) (18.90)

 

Note: Percentages out of each target students’ totals.

p < .05. "p < .01.

Table 6.1 demonstrates that negotiated knowers and low social risk students were more

likely to participate in extended interaction segments. The p-values for positioning and

talking about mathematics at a high level of explicit meaning suggest that there may be

some differences between the belief groups in these patterns of involvement, as the p-

values are near the significance cutoff.

196

To follow up on these potential differences between belief groups, I calculated the
target students’ percentages for patterns of involvement using different totals. Instead of
examining the expected percentage for the individual student, by dividing out of the
individuals’ totals, I divided out of their classroom total interactions. Calculating the
percentages out of the classroom total interactions could approximate how often the
target student would participate in that pattern of involvement during a class period.
Table 6.2 displays the new mean percentages for the nine patterns of involvement
(frequency remains unchanged), the t-value of the difference between these means, p-
value, and effect size of the difference.

Table 6.2: Comparisons Between Belief Groups: Target Students’ Patterns of
Involvement, Version B

 

M % t(13) p d
(SD)
Received Negotiated
Knowing & Knowing &
High Risk Low Risk

Level of Involvement

Frequency 6.79 11.72 1.56 0.14 0.86
(5.72) (6.55)

Extended 0.67 2.69 2.45 0.03* 1.36
(1.02) (2.06)

Off Topic 0.67 2.95 1.36 0.20 0.75
(1.49) (4.51)

Nature of Involvement

Assertive 1.46 4.35 1.43 0.18 0.80
(1.86) (5.38)

Hesitant 0.77 0.44 0.63 0.54 0.35
(1.08) (0.92)

Positioning 0.36 1.35 2.43 0.03* 1.35
(0.65) (0.91)

197

Table 6.2, continued

Explicit Meaning

Low 2.88 3.14 0.19 0.85 0.11
(2.94) (2.10)

Moderate 2.11 3.38 1.13 0.28 0.62
(2.56) (1.63)

High 0.29 1.54 2.77 0.02* 1.54
(0.31) (1.24)

 

Note. Percentages out of total interaction segments across the four days in each class.
(Carson: 162. Evans = 232.)
*p < .05. **p< .01.

The results in Table 6.2 Show that negotiated knowers with a low perception of
risk were more likely to extend their interaction segments, engage in positioning, and use
a high level of explicit meaning in their talk. The differences between the means for these
patterns of involvement in Table 6.2 may not be impressive when viewed in terms of the
sizes of the percentages, but if we examine how the proportional relationships between
the percentages, we see more dramatic differences. The negotiated knowers and low risk
target students participated in extended and positioning segments approximately four
times as often as the received knowers and high risk target students, and participated in
talking about mathematics at high levels of explicit meaning over fives times as often.

The analyses presented in Tables 6.1 and 6.2 demonstrate that participation in
extended interaction segments occurred more often among negotiated knowers and low
social risk students. Involvement in positioning and talking about mathematics at high
levels of explicit mathematical meaning was also somewhat more likely to occur among

these students. These three patterns of involvement are more related to target students’

beliefs rather than their experiences in their classrooms.

198

There was some diversity in these patterns within belief groups. Table 6.3
presents the individual students’ percentages in each belief group for extended,
positioning, and high explicit meaning.

Table 6.3: Diversity Within Belief Groups
Patterns of Involvement by Target Students, as Grouped by Beliefs

 

 

 

% Out of Individual Totals
Extended Positioning High Explicit Meaning
Received Knowing & High Risk
allen 6.67 20.00 8.33
ALYSSA 6.25 6.25 20.00
colleen 0.00 0.00 20.00
hannah 7.69 0.00 0.00
MARISSA 0.00 0.00 0.00
max 16.67 0.00 0.00
pete 9.09 9.09 9.09
TRICIA 0.00 0.00 0.00
Mean 5.80 4.42 7.18
(SD) (5.79) (7 .23) (8.78)
Negotiated Knowing & Low Risk
ALEX 26.92 19.23 29.41
allison 7.69 0.00 0.00
BECKY 23.08 7.69 18.18
BILL 24.14 5.17 19.35
MOLLY 6.25 37.50 53.85
STEVE 30.30 12.12 31.25
tim 27.78 11.11 0.00
Mean 20.88 13.26 21 .72
(SD) (9.80) (12.26) (18.90)

 

*Lower case: Ms. Carson’s class. Upper case: Mrs. Evans’s class

For example, Max extended his interaction segments at a more similar level of frequency

to the negotiated knowers and low social risk students. Molly and Allison did not extend

their interaction segments as often as the other students in their belief group. Allen

199

participated in positioning more similarly to negotiated knowers and low social risk
students in contrast to the students in his own belief group. Allison was the only student
among the negotiated knowers and low social risk students who did not participate in
positioning at all. Allison and Tim, both from Ms. Carson’s class, did not engage in
talking about mathematics at as high levels of explicit mathematical meaning as the other
students in their belief group, the rest of whom were in Mrs. Evans’s class. Additionally,
Alyssa and Colleen talked about mathematics at higher levels of explicit mathematical
meaning. Alyssa was from Mrs. Evans’s class, while Colleen was in Ms. Carson’s class.
Colleen was the only student in Ms. Carson’s class who talked about mathematics at a
high level of explicit mathematical meaning. While the differences between belief groups
are statistically significant, individual anomalies are worth noting. In particular, the
differences between belief groups for high levels of eXplicit mathematical meaning
appears to be explained somewhat more by classrooms than beliefs, when diversity
within the groups are examined.

Looking back at Tables 6.1 and 6.2, it is noteworthy that the belief groups did not
differ in other patterns of involvement. Target students’ frequency of participation, off-
topic talk, assertive talk, hesitant talk, and talk about mathematics at low and moderate
levels of explicit meaning did not differ according to belief group. The patterns of
involvement that did differ by classroom — students’ assertive talk, hesitant talk, and talk
about mathematics at a low level of explicit meaning — did not differ according to the
belief groups. However, students’ talk about mathematics at a high level of explicit

mathematical meaning differed by both belief groups and classrooms.

200

Academic and Social Goals and Involvement

Target students also were divided into groups based on their academic or social
goals. In the interviews, target students expressed academic goals about completing their
tasks social goals about appearing competent, gaining status, appropriate behavior, and
helping their classmates. In this section of the chapter, I examine whether target students’
involvement differed in terms of whether or not they expressed any of these five
academic or social goals. I conducted unpaired t-tests between the group of students who
expressed each goal and the group that did not for all nine involvement categories.

Complete task. Five students of the 15 students were focused on completing their
tasks as a goal during whole-class discussions. All were in Mrs. Evans’s class: Alex,
Alyssa, Becky, Molly, and Steve. Students who expressed this goal participated
significantly differently from those who did not with respect to positioning and high
levels of explicit meaning. (p < 0.05). These students positioned more often (16.56% in
comparision to 4.54%) and used higher levels of explicit meaning more frequently
(30.54% compared to 5.68%) when talking about mathematics than those who did not
express a goal of completing the task.

Appear competent. Eleven students discussed the goal of appearing competent
during whole-class discussions. From Ms. Carson’s class, Allen, Hannah, Max, Pete, and
Tim expressed this goal. From Mrs. Evans’s class, Alex, Becky, Bill, Marissa, Molly, and
Tricia expressed this goal. Students with this goal did not participate significantly
different from those who did not express this goal on any of the forms of involvement.

Gain Status. Five students were concerned with gaining some form of status,

either in terms of their mathematical ideas, their improved behavior, or their improved

201

academic performance. From Ms. Evans’s class, Max and Pete were concerned with
status, while Marissa, Molly, and Bill were concerned with status in Mrs. Evans’s class.
Students who expressed this goal did not participate significantly differently fi'om those
who did not express this goal on any of the forms of involvement.

Behave. Eight target students were concerned with behaving appropriately during
whole-class discussion. Alyssa, Becky, Marissa, and Tricia expressed this goal in Mrs.
Evans’s class. From Ms. Carson’s class, Colleen, Hannah, Max, and Pete, fi'om Ms.
Carson’s class, expressed this goal. Students with this goal participated significantly
differently (p < .05) with respect to extended segments (7.85% to 18.54%) and
positioning (2.88% to 15.02%). The students who did not express the goal of behaving
appropriately involved themselves more often in these forms of involvement. This is
consistent with the received knowers and high social risk students. Becky was the only
student expressing this goal outside of this belief group.

Help classmates. Ten target students wanted to participate in whole-class
discussion in order to help their classmates. From Mrs. Evans’s class, Alyssa, Bill, and
Tricia expressed this goal, and all of the students in Ms. Carson’s class expressed this
goal. Students with this goal participated significantly differently (p < .05) from those
who did not with respect to high levels of explicit meaning. Those expressed the goal of
helping classmates involved themselves less often in talking about mathematics with high
levels of explicit meaning (7.68% as compared to 26.54%).

This is the opposite of the results from the group of students with the goal of
completing the task. The students who did not express the goal to help others were Alex,

Becky, Marissa, Molly, and Steve, from Mrs. Evans’s class. This is almost the same

202

group as the students with the goal of completing the task. Marissa is in the group with
the helping goal, but Alyssa was in the group with the goal of completing the task. It may
be the case that students who were more self-focused and academic focused were more
likely to talk about mathematics at high levels of explicit mathematical meaning.

Out of these five social goals, three appeared to be related to students’
involvement in whole-class discussion about mathematics: Complete the task, behave,
and help classmates. The goals of appearing competent and gaining status did not appear
to be related to students’ patterns of involvement. Students with the goal of completing
the task participated more often in critiquing or positioning during discussion and talked
about mathematics at a high level of explicit mathematical meaning. Alternatively,
students positioned less and engaged in less extended segments if they had the goal of
behaving, and they were less likely to talk about mathematics at a high level of
mathematical meaning if they had the goal of helping their classmates.

Clusters and Involvement

Students expressed four clusters of beliefs and goals in their interviews: Learn the
material in a public forum, gain attention, do the right thing, and help others while
overcoming social risk. Table 6.4 displays which students fell into which cluster.

Table 6.4: Students’ Belief Clusters
Individual Students by Belief Cluster and Classroom

 

 

 

 

 

 

Ms. Carson’s students Mrs. Evans’s students
(I) Learn the material in a Tim, Allison Alex, Steve, Becky
publicj'orum
(2) Gain attention ’ Molly, Bill
(.9 Do the right thing Max, Pete Marissa
(4) Help others while Allen, Colleen, Alyssa, Tricia
overcoming social risk Hannah

 

 

 

203

 

To examine whether expressing a particular cluster of beliefs and goals was related to
how students’ involved themselves in whole-class discussion about mathematics, I
explored the percentages of average involvement for each cluster. Table 6.5 presents
these average percentages across students in each cluster for the nine forms of
involvement.

Table 6.5: Differences in Involvement by Belief/Goal Clusters
Mean Percentages of Involvement by Forms of Involvement and Cluster

 

 

(1) (2) (3) (4)
Level of Involvement
Frequency 10.03 15.55 8.72 5.46
Extended 23.15 15.20 8.59 4.12
Off Topic 12.82 31.25 0.00 22.5
Nature of Involvement
Assertive 24.63 37.29 1 1.92 24.17
Hesitant 7.27 0.00 1.1 1 15.49
Positioning 10.03 21.34 3.03 5.25
Explicit Meaning
Low 34.73 41.19 37.88 38.21
Moderate 49.50 22.21 25.76 32.12
High 15.77 36.60 3.03 9.67

 

Ideally, I would run an AN OVA to determine differences within and between groups.

However, one of the clusters, gain attention (2), has only two members, which does not

allow me to run such an analysis.

204

Based on percentages alone, these students apparently involved themselves
differently depending on which cluster they expressed. Students in cluster (2), gain
attention, differed the most dramatically from the other clusters, participating more
fiequently, in more off-topic interactions, more assertive interactions, utilized positioning
more often, as well as both low and high levels of explicit meaning in their mathematical
talk. Findings related to cluster (2) should be interpreted with hesitancy, because only
two students are included in this group. Cluster (1), learn the material in a public forum,
included a group of students who were more likely to participate in extended interactions
and talk about mathematics at moderate levels of explicit mathematical meaning.
Students who participated in hesitant talk were more likely to be in cluster (4), help
others while overcoming social risk. Students in cluster (3), do the right thing, never
participated in off-topic talk.

Summary

Through comparing a series of groups of students, involvement in discourse
appears to be related to students’ beliefs with respect to the following forms of
involvement: extended interaction segments, positioning, and the use of high levels of
explicit meaning. Diversity within belief groups suggests that students’ talk about
mathematics at high levels of explicit levels of mathematical meaning may be more
different by classroom.

When students were grouped by their goals, differences between the groups were
along those same three patterns of involvement: extended interaction segments,
positioning, and high levels of explicit mathematical meaning. The students with a more

self-focused goal of completing the task were more likely to position themselves during

205

discussion and talk about mathematics at a high level of explicit mathematical meaning.
The patterns of involvement of students with other self-focused goals, appearing
competent and gaining status, did not differ. The involvement of students with
community focused goals, behaving and helping, did differ. Students with a goal of
behaving were not as likely to position or participate in extended segments. Students with
a goal of helping were less likely to talk about mathematics at a high level of explicit
mathematical meaning.

There were also differences in patterns of involvement when students were
grouped by clusters. The involvement of the students in cluster (2), gain attention,
differed most markedly, but there were only two students in this cluster. Cluster (1), learn
the material in a public forum, contained students who were more likely to extend their
interactions. Students in cluster (3), do the right thing, never participated in off-tOpic talk.
More hesitant talk took place by the students in cluster (4), help others while overcoming

social risk.

206

CHAPTER SEVEN

DISCUSSION

This study was designed to explore whether and how middle school students’
beliefs and goals influenced their involvement in whole-class discussions, in contrast to
research that focuses on the collective aspects of whole-class discussion (e. g., Stephan,
Cobb, & Gravemeij er, 2003) and the teacher’s efforts to structure the classroom discourse
(e. g., Nathan & Knuth, 2003). This study’s primary contribution is a set of relationships
between students’ beliefs and goals and their patterns of involvement. In particular,
students who believed in negotiated knowing and low social risk participated in more
extended interaction segments, increased positioning, and at a higher level of explicit
mathematical meaning. Students who believed in received knowing and high risk were
more likely to participate in these shorter interaction segments and to talk about
mathematics at lower levels of explicit meaning. Such results provide an explanation of
how students assist in establishing or maintaining classroom norms. Students have
developed beliefs and goals, and associated expectations for behavior, as a result of
previous educational experiences. Then, they may enter a new classroom and expect to
act on these beliefs and expectations for behavior. Teachers play a prominent role in
structuring the whole-class discussions, but students also shifi the nature of whole-class
discussion by initiating their own questions or evaluating a classmates’ solution even if
not explicitly requested by the teacher.

Some of the additional contributions of this study include a framework of middle

school students’ beliefs and goals in discussion-oriented mathematics classrooms;

207

revisiting constructs such as mathematics anxiety in light of mathematics reforms; and an
analytic framework for the analysis of students’ beliefs. Based on students’ interview
data, a framework of students’ beliefs and goals in discussion-oriented classrooms should
include their social goals as well as their beliefs about learning mathematics. Previous
studies of mathematics anxiety have not taken place in reform-oriented mathematics
classrooms, and these results suggest that discussion-oriented classrooms may foster a
performance anxiety in mathematics classrooms for some students. When analyzing
students’ beliefs fi'om interview data, perspectives from discourse analysis, such as the
use of repetition (Tannen, 1989), can lead to stronger claims.

In this chapter, I will first review and interpret the results fi'om this dissertation.
Then I will discuss the contributions of this study, the limitations of the study, the
implications of these results for mathematics classroom practice, and future research
suggested by these results.

Review and Interpretations of Results

Figure 7.1 illustrates the relations between the results that I will discuss in this
chapter: (a) the range of students’ beliefs and goals that they bring into their mathematics
classroom, (b) whether and how students’ beliefs and goals influenced their involvement
in whole-class discussions, and (c) whether and how the classroom discourse practices or
the teachers’ discourse practices influenced students’ involvement in whole-class

discussions.

208

Figure 7.1: Interactions in a Mathematics Classroom

Teacher

(C)

 

Mathematics
Classroom
Discourse

 

 

 

(bl

(a) Student 1 Student 2 Student 3 Student n

Range of Students’ Beliefs and Goals

One of the goals of teaching mathematics is to foster productive dispositions for
learning and doing mathematics. Dispositions include students’ beliefs, their attitudes,
and their emotions. Beliefs provide the psychological backdrop for students’ goals. Many
studies of students’ motivation and beliefs in mathematics classrooms were conducted
before the NCTM reforms. Research on students’ motivation in mathematics classrooms
has not taken into account students’ beliefs and goals in relation to participating in
classroom discussions—a key element of classroom practice proposed in these reforms.
The range of students’ beliefs and goals in discussion-intensive mathematics classrooms
should be mapped and described in order to understand the motivations of all students.

In this study, an examination of students’ beliefs and goals about learning
mathematics in discussion-intensive mathematics classrooms, the following patterns were

observed among the 15 target students’ beliefs and goals: (a) epistemological beliefs

209

correlated with their perception of risk associated with involvement in classroom
discussion, and (b) target students expressed both academic and social goals that
supported their involvement in classroom discussion.

Epistemological beliefs and perception of risk. The 15 target students in this
study expressed beliefs and goals in four different clusters. Two of the clusters, learn the
material in a public forum and gain attention, included the epistemological belief of
learning as a process of negotiating knowledge as well as lower perceived level of risk
associated with participating in whole-class discussions about mathematics. Students in
the other two clusters, do the right thing and help others while overcoming social risk,
expressed the belief that learning mathematics involved receiving knowledge from an
authority and that participating in discussion involved a higher level of risk.

So epistemological beliefs and perception of risk correlated for the target students
in sensible ways. Students who saw benefits to sharing one’s ideas during mathematics
class discussion, the negotiated knowers, were not threatened by the idea of participating.
For these students, the benefits of involvement outweighed the risk of being incorrect in
front of their peers. Alternatively, some students did feel somewhat threatened by the
idea of participating in class discussions. These students were the received knowers. They
may have found safety in the belief that the process of learning mathematics involves
receiving knowledge from an authority. This belief may protect them from sharing ideas
in front of classmates and allow them to avoid being publicly incorrect. Ifthe student did
not view himself or herself as one of the authorities in the classroom about the current

mathematics problem under discussion, they may not interact during class in particular

210

ways. Future research could examine the development of students’ epistemological
beliefs in relation to their perceptions of social risk.

Table 7.1: Epistemological Beliefs by Perception of Social Risk

 

 

 

 

 

 

Perception of Social Risk
Epistemological High Low
Beliefs
Received Avoid Involvement Devalue Involvement
(N = 7) (N = 0)
Negotiated F ace-Saving Involvement Approach Involvement
(N = 0) (N = 3)

 

 

 

Table 7.1 indicates that the target students’ expression of beliefs in this study did
not fall along all possible relations between epistemological beliefs and perception of
social risk. While the participants’ interview data from this study demonstrated relations
between epistemological beliefs and perceptions of social risk, additional relations may
be revealed among future participants. For example, students who believed that learning
mathematics involved negotiation could have expressed a higher sense of social risk,
even though none of the negotiated knowers in this study expressed perceptions of higher
social risk. Negotiated knowers could believe that knowing mathematics requires the
sharing and interchange of ideas, but might still see this practice as somewhat
threatening. I conjecture that the involvement of students who believe in negotiated
knowing and experience high social risk may be observed as face-saving behaviors, such
as not always participating when incorrect or backing out of an interaction. Also, future
participants may believe that learning mathematics involves receiving knowledge and
also express a low level of social risk. These students might not be intimidated by the

idea of participating in discussion, but do not see the purpose of doing so, given their

211

epistemological beliefs. I conjecture that these students may devalue involvement. While
none of the participants in this study expressed the other two possible combinations
between epistemological beliefs and perceptions of social risk, other middle school
mathematics students may hold them.

Future research could examine students’ epistemological beliefs and perceptions
of social risk in a variety of mathematics classrooms. Expanding the analysis into a wider
range of classrooms from this study would provide the opportunity to expand this
dissertation’s framework of students’ beliefs and goals.

No previous study has examined relations between students’ epistemological
beliefs and perception of risk. One previous study examined relations between two
second grade students’ beliefs about the nature of mathematics, a domain specific
epistemological belief, and their learning goals (Cobb, 1985). If learning goals are related
to students’ epistemological beliefs, they may also be related to students’ perception of
risk. However, Cobb’s study is different fiom this dissertation, as the students were
younger, there were only two cases analyzed in the article, and the constructs were
different. Researchers could examine relations between students’ perception of risk and
their learning goals, as well as relations between students’ beliefs about the process of
learning mathematics and their beliefs about the nature of mathematics.

Students’ goals: Academic and social. The four clusters differed due to
students’ goals, both academic goals and learning goals. Students in learn the material in
a public forum (N = 5) expressed the academic goal of wanting to participate in
discussion in order to complete the task as well as a goal of appearing competent in front

of classmates. Students in the other three clusters did not cite goals related to completing

212

their mathematical tasks when they shared their reasons for participating in whole-class
discussion. In the cluster gain attention, students (N = 2) expressed goals of appearing
competent and gaining status for their thinking. These two clusters of students both
expressed competence goals, but one cluster was more focused on completing content,
while the other was attending to gaining status. The students in the group learn the
material in a public forum were the only ones whose principal focus was learning the
content; the others were focused on social goals.

The two other clusters, do the right thing and help others while overcoming social
risk, also were made up of students with similar epistemological beliefs and perceptions
of social risk, and these clusters differed in terms of the goals that students expressed.
Students in the cluster of help others while overcoming social risk (N = 5) were focused
on the goal of helping their classmates, while students in the cluster of do the right thing
(N = 3) were concerned with issues of appropriateness and appearance, such as appearing
competent, gaining status, and behaving. The focus in the former group was clearly
social, where the latter group was more focused on themselves as individuals.

Students’ may particularly attend to social issues in discussion-intensive
mathematics classrooms during middle school. Both the activity of whole-class
discussion and the developmental challenges of adolescence could alert students to attend
to social goals. Students’ social goals reflected some element of performance or
awareness of how others perceive them, whether it was in terms of appearing competent
intellectually, getting recognized by others for the content of their thinking, appearing

socially appropriate, or helping their classmates learn.

213

For students who held it, the academic goal of completing the task seemed to
relate to their view of learning as a negotiated process. Ifstudents wanted to figure out
what they needed to know in order to get their work done, they could focus their
participation in order to get the information they need to help them accomplish this goal.
The goal of completing the task may have appealed to some students’ sense of efficiency.
They may have valued efficiently finishing the task over any risks involved in
participation.

Target students’ goals could also be considered either self- focused or community-
focused. The goals of learning content, appearing competent, and gaining status were
more self-focused. Students’ efforts to achieve these goals are not out of concern for their
classmates, except if they indirectly ask questions or share mathematical thinking that
could potentially provide insights. These actions seem secondary to the goal of appearing
competent, perhaps more important for gaining status, and most likely for learning the
content. Though it is somewhat counter-intuitive, the class may benefit indirectly from
students who act on their self-focused goals. For example, if a student asks a question in
order to help himself or herself complete their assignment, another student may have had
that same question and therefore benefit from the response, from the teacher or from
peers. In contrast, behaving and helping classmates were community-focused goals.
Good behavior allows more students to focus on paying attention to the discussion rather
than other distractions. Helping their classmates is a supportive action of one’s
classmates, as it is an effort to help them learn mathematics.

It is noteworthy that the students who expressed what might be considered the

more “productive” epistemological beliefs and perceptions of social risk—at least as

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viewed fi'om the reforrner’s perspective—(negotiated knowing and low risk) appear to
hold more self-focused goals, such as completing the task, demonstrating competence,
and gaining status. Compare this focus on oneself to the call by NCTM, “We need to shift
toward classrooms as mathematical communities” (NCTM, 1991, p. 3). The social nature
of a community involves considering the needs of others and interacting out of
consideration of these needs. The students who may have expressed the more productive
beliefs toward learning mathematics were less focused on considering the needs of the
others in their classroom community.

In contrast, the target students in this study who spoke of learning mathematics as
a process of receiving knowledge and of participating in discussion as a high-risk activity
may not have expressed the more “productive” beliefs about learning mathematics, but
they may have expressed more productive social goals for fostering a sense of
community in their classrooms. These students were more likely to express social goals
of helping classmates and behaving. These goals suggest a focus on the needs of others:
helping others learn and establishing an atmosphere conducive to learning though their
good behavior. Even though these students do not express a belief in the negotiation of
knowledge, they may still participate in the activity of classroom discussion due to an
interest in supporting their community.

A range of beliefs and goals support the emergence of norms in mathematics
classrooms, from epistemological beliefs about negotiating mathematical knowledge to
social goals about helping classmates and behaving. Social goals are important factors in
the development of a mathematical community. Attending to social processes in the

classroom can allow for more harmonious discussions. If no students were focused on

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behaving, then the discussions may flow less effectively. However, there are elements of
risk taking associated with sharing one’s thinking in a public space, so getting any initial
ideas on the floor from students during discussion requires at least some of the students to
be less concerned with the social consequences associated with taking these risks.
“Productive” beliefs and goals in a mathematical community may include beliefs about
learning mathematics, such as negotiated knowing and low social risk, as well as less
self-focused goals, such as helping classmates and behaving.

Teachers could consider how to honor productive goals of helping others and
behaving while considering the learning beliefs of these students (received knowing and
high social risk). For example, teachers who make an effort to balance out the time spent
in whole-class discussion with time spent in small group discussion may be considering
the level of risk involved in whole-class discussion. Students can meet their goal of
helping their classmates by working together in smaller groups, and they may find small
group work to be less risky.

I did not explicitly design this study to assess social goals. Data on social goals
arose from questions such as those inquiring why students chose to get involved in class
discussions. If I asked more general questions about whether and why the student wanted
to learn mathematics, I could have learned about a wider range of social goals, such as
social afliliation, social approval, social responsibility, social status, and social concern
(Dowson & McInemey, 2003). Target students’ goal to help others could be considered a
social concern goal, a concern for assistng others in their academic development. Their
goal to gain status is similar to the goal of social status, or to achieve academically to

attain social position. Additionally, students’ goal to appear competent may be aligned

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with either performance goals or social approval goals, depending on how the student
talked about his or her goal to appear competent. Finally, target students’ goal to behave
is similar to the social goal of acting out of social responsibility, to meet social role
obligations or to follow social and moral rules. Target students did not extensively
mention social goals of social afliliation, to enhance a sense of belonging, or social
approval, to gain the approval of peers, teachers, or parents.

However, one student, Marissa, mentioned defining herself against a group of
students, which would be the antithesis of a social afliliation goal. She mentioned to me
that she did not participate in class often because she didn’t want to be like one of those
“preppy people.” I asked her what she meant by this:

...to me preppy people are people who can brag about what they wear, brag about

their grades, brag just brag. I don’t know. I don’t like that, to brag about things.

Maybe I do, I don’t know, maybe I don’t catch it. I don’t think I do (3/31/03,

#66)

She did not want to come across as a certain type of person. Sometimes participating in
mathematics class felt like bragging to her, because she would have been showing off
about what she knew. She made the effort to not affiliate herself with students that she
saw doing this. Marissa was the only student whose interview talk hinted at social
affiliation goals, or anti-affiliation, in this case. Adolescents may be particularly focused
on issues of affiliation and anti-affiliation, as they wrestle with the question, “who am I in
relation to others?” However, I interpreted this data as indicating a goal of behaving, or,
in Dowson & McInemey’s terms, social responsibility, since not bragging may also be

refraining from an action that may be offensive to others.

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In the Principles and Standards for School Mathematics, NCTM suggests
“. . .some students in the middle grades are often reluctant to stand out in any way during
group interactions,” (N CTM, 2000, p. 61) This study sheds light on some adolescents’
goals that could explain this potential reluctance. They may hold self-focused goals, such
as those related to gaining status and appearing competent, due to their concern with an
imaginary audience during adolescence (Elkind, 197 8). Additionally, students may define
appropriate behavior in ways that would cause them to hesitate to participate, as the
students from working class backgrounds in Lubienski’s research (2000a, 2000b) or
students from rural farming communities, as the student in Ridlon’s case study (2001).
More research is needed to explore whether and why middle school students may be
reluctant to participate in mathematics classroom discussions, and to determine
characteristics of classrooms in which middle school students become less reluctant to
participate.
Students’ Beliefs and Goals Influencing their Involvement

Students’ involvement in whole-class discussion appeared to be influenced by
their beliefs and goals, particularly their participation in extended interaction segments,
positioning, and high levels of explicit mathematical meaning. Students who expressed
epistemological beliefs of negotiating knowledge and a perception of lower social risk
were more likely to participate in these forms of involvement. Target students who
expressed social goals of completing the task, gaining status, and behaving appropriately
were more likely to participate in interactions with high levels of explicit mathematical
meaning. Positioning occurred more frequently among target students who expressed the

goal of completing the task and less frequently among target students who expressed the

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air-if" “H J

goal of behaving appropriately. Extended interaction segments occurred less frequently
among these target students as well. These three patterns of involvement — extended
segments, positioning, and high levels of explicit mathematical meaning — appear to be
influenced by students’ beliefs and goals.

Extended segments and positioning. Students who participated in these
interaction segments held the floor for multiple turns and critiqued the solutions of others.
These ways of talking could suggest that these students have developed mathematical
autonomy.

...autonomy is defined with respect to students’ participation in the practices of

the classroom community. In particular, students who are intellectually

autonomous in mathematics are aware of, and draw on, their own intellectual
capabilities when making mathematical decisions and judgments as they
participate in these practices (Kamii, 1985). These students can be contrasted with
those who are intellectually heteronomous and who rely on the pronouncements

of an authority to know how to act appropriately (Yackel & Cobb, 1996, p. 43).
If autonomy can be read off of students’ involvement in whole-class discussions, those
who participating in extended interaction segments and positioning may appear
autonomous, since they expressed a stronger commitment to the ideas on the floor. But I
would argue that an examination of behavior alone is not sufficient for determining
whether students are mathematically autonomous. A range of beliefs or goals may
support the same action. However, out of the target students in this study, more students
who participated in extended and positioning interaction segments also expressed beliefs

and goals that could suggest intellectual autonomy.

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Students with epistemological beliefs about negotiated knowing, perception of
low social risk, and a goal of completing the task were more likely to be involved in
extended interaction segments and positioning. These beliefs and goal may indicate
autonomy, particularly in light of students’ epistemological beliefs. Negotiated knowing
was partially determined by whether students suggested that their ideas were important to
examine and compare to others’ ideas in the process of learning mathematics. In the act
of examining their own ideas, they drew on their own intellectual capabilities.

Students who hold beliefs and goals that lead to positioning and extended
interaction segments (negotiated knowing, low risk, complete the task) may not be
threatened by the idea of others critiquing them. Instead, they see this activity as a part of
the learning process. They may not mind when they are critiqued, and they are less
hesitant to critique the thinking of others.

Those less likely to position or participate in extended interaction segments were
students with beliefs about received knowing, perceptions of high social risk, and those
with the goal of behaving. They may be treating their classmates as they would prefer to
be treated. They may prefer not to be critiqued themselves, so they do not participate in
the activity of critiquing others. Additionally, their belief in received knowing suggests
that they depend on an authority for learning, such as their teacher. While they may
acknowledge that they can learn from, and even help, their classmates, they may not act
as a knowledgeable authority through positioning.

High levels of explicit mathematical meaning. Students were more likely to
involve themselves in mathematical talk with higher levels of explicit meaning if they

expressed beliefs about negotiated knowing and low social risk and goals of completing

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the task, gaining status, and behaving appropriately. Students with beliefs about
negotiated knowing and low social risk may attain a depth of mathematical meaning in
their talk through the process of relating their solutions to others. Students with a goal of
completing the task may be invested in attaining a deeper understanding than students
without this goal. A goal of gaining status may lead students to want to prove that they
know mathematics well, resulting in higher explicit meaning. A goal of appropriate
behavior may lead students to talk about mathematics with high levels of explicit
meaning if they perceive that such talk is expected in their classroom.

Attending to whether and how students’ beliefs and goals are related to their
involvement in whole-class discussions provides insight into why some students may
pick up on the discourse practices encouraged by particular teachers. Their beliefs and
goals may fit with those being promoted by the teacher, and this fit can result in case of
involvement. Additionally, some students may shape the discourse practices of the
classroom by initiating some patterns of involvement not explicitly encouraged by the
teacher. For example, Allen and Tim initiated some efforts to position and critique in Ms.
Carson’s classroom. Ms. Carson did not invite this type of participation as purposefully
as Mrs. Evans did. Instead, Ms. Carson tolerated the student’s contribution, and then took
back the roles of initiating and evaluating. Mrs. Evans used questions directed to students
such as, “What do we think?” and these questions may be likely to evoke more critique
and positioning. The teacher may not encourage positioning or extended segments, but
students may assert their involvement in such a manner due to their expectations they
developed from previous experiences in mathematics classrooms where teachers

requested critique.

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Teachers’ Discourse Practices Influencing Students’ Involvement

Students’ involvement may also be influenced by the classroom discourse
practices, as the teacher may take on a particular role in structuring these practices.
Target students’ involvement in each of these two classrooms differed in terms of their
hesitancy and low and high levels of explicit mathematical meaning. The teachers may
have influenced students’ participation by structuring the classroom talk.

Levels of explicit mathematical meaning. Mrs. Evans and Ms. Carson each may
have influenced the level of explicit mathematical meaning in their target students’
interactions. Target students from Mrs. Evans’s class were more likely to participate at
high levels of explicit meaning, while target students from Ms. Carson’s class were more
likely to participate at low levels of explicit meaning. These teachers differed in terms of
their use of the I-R-E participation structure as well as the nature of the questions they
posed to their students.

Interaction segments in Ms. Carson’s classroom were more aligned with the I-R-E
structure than those in Mrs. Evans’s class. Ms. Carson would take on the tasks of posing
questions to the students and the role of evaluating students’ responses. Rather than
evaluating students’ responses, Ms. Evans would ask the students, “What do we think?”
Encouraging students to evaluate one another’s thinking may have helped students’ talk
move to a higher level of explicit meaning.

Others questions that the teachers posed to students could have influenced the
level of explicit mathematical meaning in students’ talk. Ms. Carson would ask her
students question such as “How did you solve this problem?” or “How do you know if

you’re correct?” Students would often respond by describing procedures, or lower levels

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of explicit meaning. In contrast, Mrs. Evans would ask her students questions such as,
“How do we know this is a linear relationship?” or “If Molly solved this with a graph,
how would someone else solve this with a table?” These questions may be more likely to
promote conceptual or relational responses fi'om students, which would be at a higher
level of explicit meaning.

Hesitant talk. While some of the teacher’s discourse may have influenced the
level of explicit mathematical meaning in the students’ talk, other features of their talk
may have influenced the less mathematical features of students’ talk, such as their
hesitancy. Students who expressed hesitancy were those who either hedged in their talk
or backed out of an interaction. Teachers’ discourse practices may have encouraged
students to not be hesitant, such as affinning their participation through revoicing
(O’Connor & Michaels, 1998) and following up on students’ attempts to direct the
classroom talk. Alternatively, teachers’ discourse practices may have been implicitly
accepting of students’ hesitancy, through maintaining control of the talk through I-R-E
interaction segments.

Mrs. Evans was more likely to revoice students’ contributions and follow up on
the students’ questions and comments during large group discussion. These discourse
practices appeared to be well established by the spring. If students’ ideas are implicitly
made valuable through the teacher’s appropriation of them, students may feel as if their
thinking is affirmed. If students’ thinking is affirmed, they may not have a reason to
express hesitancy in their interactions.

As mentioned previously, Ms. Carson was more likely to use the I-R-E format in

her interaction segments. This teacher took on a more authoritative role during whole-

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class discussions. Students in this classroom may have felt less empowered to participate
in this classroom, and backed out of interactions or hedged during interaction when they
were unsure.

These results suggest that the differences in how teachers structure classroom talk
could influence their students’ involvement in whole-class discussion. Teachers may not
be cognizant of the ways in which their talk could have supported higher levels of
explicit mathematical meaning in students’ talk. Supporting the development of students’
mathematical communication and reasoning would involve assisting students to
communicate with high levels of explicit mathematical meaning. Teachers who ask
questions that promote high levels of explicit meaning and structure discussions that
support conceptual depth are not always easy to find when studying mathematics
classrooms (e.g., Hamm & Perry, 2002).

Contributions from this Study

The results of this study contribute to and extend the research literature in
mathematics education and educational psychology. For mathematics educators, these
results provide new content for (a) discussing how issues of adolescent development
intersect with the mathematics reform movement, as well as (b) revisiting the construct of
mathematics anxiety. For educational psychologists, these results suggest the importance
of revisiting the nature of constructs, such as (c) students’ learning goals and ((1)
academic risk, within the context of subject matter and specific activities. Additionally,
researchers interested in discourse analysis or analyses of students’ thinking may engage
in dialogue around (e) the analytic framework I presented for the analyses of students’

beliefs. I will discuss each of these five contributions below.

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Adolescent Development Intersecting with Mathematics Reform

Through the analyses of students’ beliefs and goals, it appears as though students
operate on academic and social goals simultaneously, such as students who are interested
in completing their task and appearing competent to others. The goal of appearing
competent may be heightened during adolescence due to students’ development of
metacognition and increased awareness that others may be thinking about them and
potentially judging them. Social goals may have increased importance in the middle
school years over the elementary school years. During adolescence, a significant task for
young people is to develop an identity that fits them. As they attempt to work on their
senses of self, they define themselves as alike and apart from others, and this activity is
largely at work among students in middle school classrooms. Additional research could
explore the change over time in students’ social goals in mathematics classrooms.

Students’ social goals may provide the opportunity to move beyond their beliefs
about learning mathematics. Seven of the eight students who expressed beliefs that
knowing mathematics involves receiving knowledge also expressed the social goal of
helping their classmates. It may be that students could begin to move away from received
knowing as they make attempts to help their classmates. If their efforts to help others
through sharing their ideas in whole-class discussion, they could receive feedback on
their thinking through this involvement and begin to relate their ideas to those of others.
Then students may begin to experience, appreciate, and come to believe in the process of
negotiating knowledge in mathematics through attempts to help. Future research could

explore the development of adolescents’ epistemological beliefs in mathematics

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classrooms, including whether students’ social goals act as a mechanism to support the
development of their epistemological beliefs.

It is common for mathematics educators to briefly acknowledge that socio-
emotional factors influence students’ learning of mathematics. However, socio-emotional
aspects of adolescence are more than a footnote in seventh graders’ experiences of
learning mathematics. Rather, they are central to the process of learning. If students’
perception of risk in the mathematics classroom is aligned with their beliefs about the
process of learning, and epistemological beliefs may be also related to their achievement
(Schommer et al., 1997) or their learning goals (Cobb, 1985), then attending to socio-
emotional factors in the mathematics classroom could promote more productive learning
goals or perhaps promote increased achievement.

The Principles and Standards of School Mathematics (NCTM, 2000) discusses
both content and process standards for teaching and learning mathematics. The content
standards (e.g., algebra, geometry, measurement) include detailed recommendations for
what teachers should address with their students across grade bands. In contrast, the
process standards are less detailed in their recommendations across grade bands. For
example, the appendix of the PSSM does not break down process standards by grade
bands, but it does discuss content standards by grade bands.

In the larger body of the text, the process standard of communication for the 6-8
grade band begins to address some of the developmental concerns of adolescence.

During adolescence, students are often reluctant to do anything that causes them

to stand out from the group, and many middle-grades students are self-conscious

and hesitant to expose their thinking to others. Peer pressure is powerful, and a

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desire to fit in is paramount. Teachers should build a sense of community in

middle-grades classrooms so students feel flee to express their ideas honestly and

openly, without fear of ridicule (NCTM, 2000, p. 267)

There is a need for more detailed recommendations for teachers to consider when they
are working to encourage students to “feel free to express their ideas honestly and openly,
without fear of ridicule.” One of these recommendations could include considering
whether to refer to students’ solutions by attributing them to a specific student (e.g.,
“Becky said...” versus “one student said. . .”), as some students may appreciate being
singled out, while others might not. Also, sometimes teachers are encouraged to develop
student-to-student dialogue in their classrooms (Nathan & Knuth, 2003), but teachers
should also reflect on potential social implications of such a practice, as students may call
on their friends or call on people in order to single them out unfavorably. The
mathematics education community could benefit from addressing social implications of
whole-class discussion.

Educational psychologists have begun to explore whether and how upper
elementary school teachers’ motivational discourse supports students’ motivation (Turner
et al, 1998, 2003). Supportive motivational discourse in the classrooms from these studies
involved evoking students’ curiosity, providing encouragement, commenting on progress,
responding positively to errors, advocating risk taking, and talking about challenge as
desirable. Classrooms with this supportive motivational discourse were more likely to
have increased involvement from students, students were more likely to perceive tasks as
challenging rather than threatening, and students were less likely to report self-

handicapping behaviors. Attending to relations between teachers’ motivational discourse

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in the analysis of classroom talk is one way that researchers have acknowledged the role
of affect in leaming. Future research in mathematics education should do more to account
for socio-emotional factors in the process of students’ learning, particularly the socio-
emotional factors most relevant to students in their respective stages of development,
such as issues of competence in adolescence.
Revisiting Mathematics Anxiety

Mathematics anxiety as a psychological construct received much attention in
mathematics education in the 1970’s and 1980’s. Sheila Tobias wrote an article in Ms. in
the late 1970’s (Tobias, 1976) that called public attention to mathematics anxiety. Her
article stated that the use of the term “anxiety” cast women’s feelings about mathematics
as pathological, since more women suffered from mathematics anxiety. After this article
it appeared, educators considered it advisable to focus on math anxiety as a psychological
state rather than as a defect (Tobias and Weissbrod, 1980). Mathematics anxiety has since
been studied in relation to achievement and performance (Hembree, 1990; Ma, 1999).
However, the construct may be under-conceptualized, as mathematics anxiety has
similiarities with mathematics avoidance (Dew, 1984), low confidence in mathematics,
(Fennema & Sherman, 1976) or low self-efficacy in mathematics (Cooper, 1991).

Mathematics anxiety appears to be related to students’ test anxiety (Hendel,
1980), such that the two are difficult to separate (McLoed, 1992). The anxiety may be
rooted in performance, or a fear that one may not perform as well as he or she would like
on tests in general and mathematics in particular. Opportunities to perform would be

more threatening for a student who is experiencing mathematics anxiety.

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Given the performance component of mathematics anxiety, the results of this
study suggest that whole-class discussions about mathematics may foster another form of
mathematics anxiety among students, social mathematics anxiety. Social mathematics
anxiety would then be the fear of being incorrect in front of peers. It may occur among
students who perceive an extremely high sense of social risk in discussion-intensive
mathematics settings. It is possible that teachers’ supportive motivational discourse can
serve to reduce the development of this anxiety.

Both the performance dimension of mathematics anxiety and the increased
opportrmities for student performance in Standards-based mathematics settings suggests
future research on the nature and development of mathematics anxiety. Anxiety about
mathematics may be socially constructed through interactions in the mathematics
classroom. Students who express anxiety may be reacting to a classroom that emphasizes
performance or competition, either explicitly or implicitly. If students experience
mathematics anxiety as a perceived personal deficit (e.g., I am afraid I won’t be able to
do this, because I don’t think I am good at it.), then that deficit may be most worrisome
and painfirl in the adolescent years when students are actively comparing themselves to
others.

Revisiting Learning Goals

The results from this study may appear to parallel previous research in
educational psychology on students’ learning goals. The target students revealed
competence goals of avoiding the appearance of incompetence (performance-avoidance
goals) as well as being gaining status for their ideas and appearing competent

(perforrnance—approach goals) in their interview talk. The students did not express goals

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related to understanding the material (mastery goals) as clearly. Rather, they talked about
completing the task before them, which may be more of an efficiency goal.

While performance and mastery goals are common constructs in the study of
student learning, these constructs may be too generally conceived to be explanatory.
These goals may have specific components that, once broken down, may provide insights
that are helpful for considering learning process in specific subject matters, such as
mathematics, and in relation to specific activities, such as involvement in whole-class
discussions. Dowson & McInemey (2003) determined that students’ goals were multi-
dimensional constructs, including affective and behavioral components as well as
cognitive components, after interviewing 86 middle school students, ages 12-15. For
example, students may demonstrate performance goals16 through behaviors that relate to
measuring performance relative to classmates, such as working harder than necessary to
secure a higher grade. Additionally, affective reactions could indicate performance goals,
particularly those that may be maladaptive for engaging in academic work, such as
debilitative anxiety about schoolwork. Finally, Dowson and McInemey described
cognitive components of performance goals, indicated by a variety of relatively shallower
cognitive processes, such as copying notes from the blackboard without thinking about
them. Future research could examine components of performance, mastery goals, and
social goals in various subject matters, as Dowson and McInemey did not account for
subject matter in their study.

Components of performance goals that target students expressed were along the

affective dimension, manifested in students’ expressions of their perceptions of social

 

'6 Dowson & McInemey (2003) did not distinguish between performance-approach and performance-
avoidance goals.

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risk and overlapping with social goals. Target students also expressed a goal of wanting
to appear competent. This goal may be related to performance goals, and it goal also has
features of a social goal, of being interested in gaining approval for one’s ability, based
on how students talked about wanting to appear competent. Examining students’ goals
through their talk may reveal more of the components of the goals that relate to specific
activities or subj ect matters.

Revisiting Risk in Academic Work

Students’ perceived level of risk associated with participating in whole-class
discussions about mathematics could be related to the risk inherently involved in
academic work (Doyle, 1983). Academic risk is has been discussed in light of the
evaluative nature of classrooms. Teachers evaluate student performance constantly, and
their parents and classmates may be made aware of these evaluations.

Doyle (1983) writes about academic tasks being associated with dimensions of
ambiguity and risk. Less ambiguous tasks are those for which a precise answer may be
pre-defined, or for which a method of generating an answer is readily available. The level
of ambiguity is considered to be inherent in the task. Risk is a feature that Doyle
attributes to the teacher, with respect to the stringency of the evaluative criteria and the
possibility of meeting these criteria.

Results from this study suggest that students may not only see risk in the
academic work that is graded, but also in the academic work that is evaluated more
informally, such as their interactions about mathematics in the public forum of whole-
class discussions. Students may not only be concerned with informal evaluations from the

teacher, but also those from their classmates. Future analyses of the risk involved with

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academic work should address both graded and ungraded academic work, and evaluations
from both teachers and classmates.
Methods: Analytical Framework for Analysis of Students’ Beliefs

It is a challenge to claim with confidence that a student holds a “belief.” Whether
or not a student is aware of his or her beliefs, students’ level of commitment toward the
beliefs, the stability of the beliefs, and the inter-relations and overlapping of beliefs all
complicate the process of analyzing students’ beliefs. Through designing this study, I
developed an analytic framework for analyzing students’ talk in interviews in order to
claim with greater confidence that students were expressing a belief, inspired by previous
analyses of students’ talk in order to infer their attitudes (Bills, 1999). According to the
framework developed for this dissertation, repetition (Tannen, 1989) was indicative of
commitment. Use of imperative verbs, such as “need to” or “have to,” suggested the
idealized state that beliefs may hold (Abelson, 1979). Affective statements suggested
beliefs, due to the strong relations between beliefs and affect (Hannula, 2002). Recall that
before claiming that a student held a belief, I analyzed the student’s interview talk in
order to determine whether their talk met any two of the three criteria explicated above.

This framework for inferring students’ beliefs from their talk may not be the only
relevant framework for its purpose. However, being explicit about one’s methods for
analyzing and inferring beliefs from students’ talk adds a level of rigor in support of
claims. This framework afforded a distinction between talk students used when
answering my questions for the sake of being polite, and the talk that revealed stronger
evidence of their beliefs. Additionally, being explicit about methods supports the

opportunity for replication of studies. Future research on students’ beliefs should

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continue to be explicit about methods for inferring beliefs from students’ (or teachers’)
talk.
Limitations

While this study contributes to the research literature, there are limitations to the
generalizability of these results. Future research is needed to determine whether these
results about middle school students’ involvement in whole-class discussions about
mathematics hold only for White seventh graders in rural communities in a Midwestern
state, or only among students in schools or classrooms that use Standards-based
mathematics curricula.

Certain background data may have been reasonable to collect, but was not
collected, in this study. Lubienski’s (2000a, 2000b) research on students’ participation in
a seventh grade Connected Mathematics Project classroom explicitly examined how
social class appeared to influence students’ involvement. SES data was not collected for
the participants in this dissertation study, which eliminated the possibility of analyzing
for patterns among students from particular social classes. While it is not realistic to
expect to analyze for every background dimension in one study, the potential parallels
between Lubienski’s study and this dissertation did not become clear to me until I was
nearing the end of my data analysis. Future research could extend these two studies in
order to determine whether social class appeared to shape students’ epistemological
beliefs about the process of learning as well as perception of social risk.

Additionally, the analyses of students’ beliefs and goals relied on my
interpretations of students’ interview data. While I have argued earlier in this manuscript

that this is a strength, due to the openness to students’ perspectives that interviews can

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provide, as well as the analytic framework I can employ to analyze students’ talk, I want
to acknowledge the complexities associated with interviewing students. For one, these
seventh grade students had the opportunity to observe me talking with their teachers
informally before and after class. While I told students that whatever they shared with me
would be kept confidential, it is possible that they would remain skeptical about this, and
censor themselves in light of their awareness of my relationship with their teacher. Also,
they had a relationship with me outside of the interview setting, since I was in their
classrooms regularly in both the spring and previous fall. To a certain extent, their
interview responses could have been about their attempts to develop a positive
relationship with me as the researcher and classroom visitor. For example, if students are
concerned with good behavior, they may have images of what good behavior during an
interview or appropriate interview responses should be. While I believe that my analytic
fi'amework assists in sorting some of this “good student” talk out of the analysis, if they
repeated the talk enough during the interview, their responses mentioned out of
appropriateness concerns may be included in these results. In spite of some of the
drawbacks of interview analyses, they can also afford access to students’ drinking that
Likert-scale surveys do not, as can open-ended questionnaires (Aikenhead, 1987).
Implications

While I intend to share the results of this study with the teachers I worked with, at
the time of drafting this manuscript, I have not yet done so. My current thinking about
how the results of this study inform mathematics classroom practice are based upon my
previous interactions with these two classroom teachers during data collection. I recall

two questions from them that influence how I view the implications of this study. I will

234

 

share these questions, and explain how this dissertation helped me think about these
questions, below. Implications fiom this dissertation address knowledge for teachers
about students’ beliefs as well as classroom interventions for potentially influencing
students’ beliefs and goals.
Teachers’ Knowledge of Students’ Beliefs and Goals and Their Development
Frequently after I administered surveys, and sometimes after I interviewed
students, the teachers would ask me, “How did they do?” I was initially surprised by this
question. I did not have strong preferences for how I would prefer the students to respond
on either the survey or the interviews. These teachers had a different sort of investment in
the students’ responses. They teach a middle school Standards-based mathematics
curriculum that is under scrutiny by parents and high school teachers. They students’
responses in surveys and interviews could be another outcome that to be used by
someone, perhaps even me, as ammunition either for or against CMP, or for or against
their teaching strategies. I see their question as reasonable, given the nature of the data I
am collecting. In order to respond to their question, I intend to share my results with these
teachers, and ask them what they think. How do they think the students did? What do
they make out of my interpretations of the data? Would they interpret it differently?
Research has demonstrated that teachers and students make different judgments of
students’ motivations (Givvin et al., 2001), so it would not be a surprise if these teachers
had not identified the same beliefs and goals for these particular students. If these beliefs
and goals are important to the target students, it would be interesting to learn whether
they are important to their teachers. If they are not, perhaps teachers could benefit fi’om

knowledge of students’ beliefs and goals, in terms of which beliefs and goals students

235

 

may act upon when involving themselves in whole-class discussion. With this
knowledge, teachers may interpret their students’ involvement differently than they did
before, and perhaps structure their classroom conversations differently as a result.

Teachers could also benefit from knowledge about the development of students’
beliefs and goals. While this study did not explicitly address development of students’
beliefs and goals, such knowledge could assist teachers by releasing a burden of feeling
responsible for completely shaping students’ dispositions. Students’ beliefs and goals
develop over long periods of time as a result of their cultural experiences and home lives
outside of school in addition to their classroom experiences. During interviews, many
students cited experiences from previous years when justifying their perspectives. For
example, when students talked about the importance of being able to solve problems with
more than one method, they shared experiences from elementary school, particularly
multi-digit multiplication or addition. Students’ histories in classrooms over time
influence their beliefs. Current teachers may be planting seeds of influence that may take
time to manifest. Teachers have a great deal to be responsible for, so results from
research on the development of students’ beliefs could help them see both the
possibilities for influencing students’ beliefs as well as the limited influence they may
have. Such insights, while making them feel less powerful and effectual, may also release
a bit of the burden of wanting to influence dispositions in addition to teaching content.

I would not interpret the results of this dissertation as an evaluation of the
effectiveness of CMP or of these teachers’ practices. Several of students’ patterns of

involvement appear to be influenced by their beliefs and goals, and these beliefs and

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we. _..., ‘ VI

 

goals could have developed outside of school as well as in school, in previous classrooms
as well as their current classroom.
Classroom Interventions

There may be ways in which teachers can structure classroom discussions that
influence students’ involvement in them, which may, in turn, influence students’ beliefs
and goals related to learning mathematics. I remember an instance when one of the
teachers asked me about what I thought she could do to help her students get involved in
classroom discussion. The teachers’ second question, then, was “What can I do?” Rather
than answering her, I asked her what she thought she should do, as I didn’t have my
thoughts formulated in that moment, and I was concerned about interfering with the study
itself.

The results of this study suggest classroom interventions for supporting teachers’
efforts to structure their classroom discourse in order to influence students’ talk about
mathematics. Recall that target students’ involvement differed by classroom in terms of
hesitancy and level of explicit mathematical meaning. Hesitancy may be considered a
social element of talk, while level of explicit meaning is about the mathematics in the
talk.

In order to encourage and foster students’ autonomous thinking about
mathematics, it may be worthwhile to attempt to reduce their hesitancy in their
involvement. There was less hesitancy in Mrs. Evans’ classroom. In Mrs. Evans’
classroom, there was off-topic or non-mathematical talk, some humor, and also students
took the opportunity to shift the discussion topics. It may have been possible that the off-

topic and humor allowed the students to feel comfortable to initiate discussion topics.

237

Additionally, fostering supportive motivational discourse (Turner, 1998; 2003), fostering
courage and humility among the classroom community (Lampert, 1990), and talking
about expectations for talking (Y ackel & Cobb, 1996) may assist in creating a classroom
community in which students feel safe to contribute their own ideas and speak with less
hesitancy.

Many researchers share the goal of wanting to understand how to promote class
discussions in which students participate at a high level of explicit meaning (e.g.,
QUASAR, 1996; Hamm & Perry, 2002). Results from this study suggest that teachers
should ask students questions that are relational and conceptual rather than procedural
and to minimize the focus on correct answers. Additionally, teachers could shift their role
such that students are asked to evaluate the solutions of their classmates. However,
previous research has shown that initial attempts to shift the authority of the teacher may
decrease the level of explicit meaning in the talk (Nathan & Knuth, 2003). Teachers may
need to learn skills for scaffolding students’ critique and positioning in order to maintain
higher levels of explicit mathematical meaning in the talk.

A challenge in structuring classroom discussions may be the need for decisions of
which to focus on first: creating a welcoming community that may reduce hesitancy or
foster high explicit meaning in students’ mathematical talk. With adolescent students’
focus on social goals, it appears to be important to teach students how to be in
community with one another as well as teach them academic content. There is more to
learn about the delicate balance between teaching students how to participate in a
mathematics learning community and teaching students mathematics, as the two are

interrelated and inseparable in discussion-intensive classrooms. Given the middle school

238

 

student population, it may be fi'uitful to invest heavily in the development of classroom
community earlier in the school year, which could provide a foundation that then opens
up the opportunity to take risks related to sharing their ideas about mathematics.

It is not clear which comes first, involvement or changes (even minor
adjustments) in beliefs. Students may develop new patterns of involvement and trying on
ways of thinking about mathematics through talking about mathematics, and then their
beliefs may shift in line with these ways of talking and thinking about mathematics.
Alternatively, students may develop new beliefs and goals given life outside of the
classroom, or even new events within the classroom, and shift their involvement
accordingly.

Future Research

While this dissertation provides new insights on students’ beliefs, goals, and
involvement in mathematics classroom discussions, more questions remain to be
explored. There are many open questions for the fields of mathematics education and
educational psychology in general—some of which are implicit in the previous sections
of this chapter. Additionally, there are some specific next steps I intend to undertake in
my own research program in order to extend these results. Below I will first present the
range of open questions, and then I will discuss my own next steps as a researcher of
these issues.

Open Questions for the Field

While this study may contribute to the research literature, the results also suggest

unanswered questions for the field of mathematics education and educational psychology.

Mathematics educators could continue to take up questions related to socio-emotional

239

factors in the learning process. While this study was a descriptive study of relations
between students’ beliefs and goals and their involvement in whole-class discussions,
future research is needed to take up questions related to the development of these beliefs
and goals and the classroom processes that influence students’ involvement.
Importance of socio-emotional factors in mathematics learning. This study
described some of the socio-emotional factors among seventh graders as they learn
mathematics in discussion-intensive settings. Socio-emotional issues have not been taken
up as frequently as research on student cognition in mathematics education (Lubienski &
Bowen, 2000). What are the additional socio-emotional factors that are a part of students’
learning experiences in Standards-based mathematics settings? How might these socio-
emotional processes facilitate or inhibit learning? In particular, do Standards-based
classrooms foster or curb mathematics anxiety or academic risk? If so, how? If social
goals are an important factor in students’ experiences in Standards-based mathematics
classrooms, what are the behavior, cognitive, and affective components of these goals?
How do the answers to these questions change as students develop and age?
Development of students’ beliefs and goals. The relations among students’
beliefs and goals as described in this study suggest that even though these beliefs and
goals may co-occur at one point in time, they may not have all developed at once. Which
came first, and how did the development of an earlier belief or goal support the
development of firture beliefs and goals? Do students’ epistemological beliefs predict the
development of social risk, or vice versa? If so, which epistemological beliefs, those
about the nature of the subject matter or the process of learning the content, predict the

development of social risk? Do students’ social goals act as a mechanism to promote the

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IWJ Y-

 

 

development of epistemological beliefs? As students act on the social goal of helping
their classmates, do they start to value the process of negotiating knowledge?

Examining students’ beliefs and goals in a variety of mathematics classrooms —
those with both Standards-based and non-Standards-based curricula, those in a range of
geographic locations — would allow for contrasts. These contrasts would allow for an
examination of which classroom processes play a stronger role in belief and goal
development in mathematics classrooms. Which classroom processes foster particular
beliefs and goals?

Age contrasts across elementary, middle, and high school would allow for an
examination of whether adolescence is a time when students are reluctant to participate
and why. Do students’ social goals indeed become heightened in adolescence? If so, are
any social goals heightened more than others? Do some of these social goals become less
important as students age?

Furthering My Own Research Agenda

As I advance my own research agenda, I will build upon this study. My
immediate next steps will extend the results from my dissertation by examining students’
beliefs, goals, and involvement in additional middle school classrooms in different
settings and in multiple subject matters. Not only will I diversify my analyses, but I will
also incorporate contrasts that allow me to compare students’ beliefs, goals, and
involvement in different academic domains. Further steps include the following: (a)
extending these analyses to focus on pre-service teachers’ beliefs, goals, and involvement
in their mathematics education courses, (b) studying the range of adolescent young

women’s ways of leaming mathematics, (0) studying teachers’ assessments of students’

241

(if—{‘3 flfl"‘-

 

beliefs and goals compared to their own assessments, and (d) the development of
students’ beliefs and goals.

Extending these results: Generality & domain specificity. As mentioned
previously, this study was conducted with two classrooms in the same school, with 15
target students, all of whom were White and from a small Midwestern town. These
results are not necessarily generalizable to all populations. A small-scale descriptive
study allows for developing models and new hypotheses to test in larger samples. It is
possible that the results from this dissertation will hold in other settings, but examinations
of other classrooms and communities would allow me to expand the framework of beliefs
and goals in light of new data. Additionally, new relations between students’ beliefs and
goals and their involvement may manifest in with more students in more classrooms.

I would also like to extend this study by conducting similar analyses in other
academic domains, such as English, social studies, or science, to contrast with students’
experiences in mathematics classrooms. This would allow for analyses of the domain
specificity of students’ beliefs and goals and their relation to students’ involvement. The
issue of domain specificity of epistemological beliefs is currently under debate in the
field of educational psychology. Also, the examination of other academic domains could
shed light on the specific nature of mathematics anxiety.

Pre-service mathematics teachers’ beliefs, goals, and involvement. Taking the
analyses into other populations could also extend these results. The field of mathematics
education is interested in improving teacher education. Research on whether and how
pre-service teachers’ beliefs and goals influence their learning, particularly their

involvement in discussions, could contribute to understanding pre-service teacher

242

 

 

learning. Would results differ for pre-service elementary teachers in contrast to pre—
service secondary teachers?

Adolescent young women’s ways of learning mathematics. An issue left
unexamined thus far in this dissertation is whether and how sex differences factor into
relations between students’ beliefs and goals and their involvement. Target students’
involvement appeared to be influenced by their beliefs and goals, but were these beliefs
and goals explained by sex differences? Certainly there were more young women who
expressed beliefs about received knowing and high social risk, but there were also young
men in this category. While there were more young men who expressed beliefs about
negotiated knowing and low social risk, there were some young women who also
expressed these beliefs.

Rather than generating claims that describe what the majority of adolescent young
women appear to believe, it may be beneficial to examine the diversity within young
women’s beliefs. Future research could develop a framework of adolescent young
women’s ways of leaming mathematics. Such a framework would be useful for breaking
down gender stereotypes of young women in mathematics classrooms.

Teachers’ assessments of students’ beliefs and goals. Students’ beliefs and
goals are influenced by what teachers expect from them—though teachers’ influence may
change as students grow older. Teachers may implicitly make assumptions about the
beliefs and goals that their students hold and treat them accordingly. I am interested
examining teachers’ assessments of students’ beliefs and goals and comparing their
assessments with students’ assessments of themselves, similar to Givvin et al. (2001).

Such a study could also serve as an intervention for teachers, helping them develop

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awn—“M ‘-

 

knowledge of students’ beliefs and goals in mathematics classrooms. Teachers could
benefit fi'om being more aware of their students’ beliefs and goals, or at least how their
assessments may not be similar to students’ assessments of themselves, so that teachers’
implicit expectations of students do not serve as self-fulfilling prophecies in unproductive
ways.

Development of students’ beliefs and goals. Why do some students develop
beliefs about negotiated knowing and others develop beliefs about received lmowing
while simultaneously experiencing mathematics learning in reform settings? There may
be important variations between teachers’ practices in reform settings that promote the
development of certain beliefs and goals over others. But other developmental factors
may be situated more in the lives of students than in the beliefs and practices of their
teachers. For example, how do factors in students’ lives outside of school contribute to
the development of their beliefs and goals, such as socio-economic class (Lubienski,
2000a, 2000b)? And how might dimensions of personality, e.g., introversion vs.
extroversion, interact with the developmental challenges of adolescence in the very
public setting of the classroom—especially those where whole classroom discussion is an
important activity?

This dissertation study serves as a foundation for a breadth of future research in
my research program. While I have learned ways in which seventh grade students’ beliefs
and goals influence their involvement, I am left with many new questions as a result of

conducting this study.

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l'VT‘“""“"‘V r-

 

Summary

The perspectives of the target students in this study expressed beliefs and goals
related to seeking and avoiding opportunities to involve themselves in whole-class
discussion. Students’ motivation in mathematics classrooms has been typically studied in
terms of performance and mastery goals, self-efficacy, or self-handicapping. These
students spoke about their experiences in ways that suggested other constructs. Their talk
in interviews illustrated that epistemological beliefs and social goals were pursued
simultaneously in their attempts to get involved with or resist participation in
mathematics classroom discussions. Mathematics educators could benefit from attending
to students’ social goals, as creating an effective classroom learning community involves
attention to social harmony as well as conceptual depth in the subj ect-matter. Students’
motivations for avoiding or approaching involvement were related to whether they were
involved in positioning, or critiquing their classmates’ solutions, extended interaction
segments, and talk at a high level of explicit mathematical meaning. While their teachers’
practices also supported a high level of explicit mathematical meaning, students’ beliefs
may predict who is more likely to involve themselves in these interactions. Continued
efforts to examine relations between students’ beliefs and goals and their involvement in
discussions may explain how a distribution of diverse students can work together to
support the development of classroom norms.

Closing

The perspectives of the target students in this study expressed beliefs and goals

related to seeking and avoiding opportunities to involve themselves in whole-class

discussion. Students’ motivation in mathematics classrooms has been typically studied in

245

Aw If..."

 

ffimfl-‘“.‘ ..-'—-‘ -

terms of performance and mastery goals, self-efficacy, or self-handicapping. These
students spoke about their experiences in light of other constructs. Their talk in
interviews illustrated that epistemological beliefs and social goals were pursued
simultaneously in their attempts to get involved with or resist participation in
mathematics classroom discussions. Mathematics educators could benefit from attending
to students’ social goals, as creating a classroom community is both about social harmony
as well as conceptual depth. Students’ motivations for avoiding or approaching
involvement related to whether they were involved in positioning, or critiquing their
classmates’ solutions, extended interaction segments, and talk at a high level of explicit
mathematical meaning. Although their teachers’ practices also supported a high level of
explicit mathematical meaning, students’ beliefs may predict who is more likely to
involve themselves in these interactions. Continuing to examine relations between
students’ beliefs and goals and their involvement may explain the how a distribution of
diverse students can work together to support the development of classroom norms.
Students’ experiences in mathematics classrooms should not be reduced to
achievement measures alone. The socio-emotional factors that support and constrain
students’ involvement are inextricably related to the learning of mathematics content.
Students do not just learn methods and processes in mathematics classrooms, they
learn to be mathematics learners and their learning of content knowledge cannot
be separated from their interactional engagement in the classroom, as the two
mutually constitute one another at the time of learning. (Boaler, 1999, p. 380,

italics in original)

 

246

After the school year ends, how students felt about themselves during their mathematics

class may remain long after the formulas fade.

247

211-i ' 2 _ii

 

APPENDIX A

Interview Protocol, Spring 2003

Introduction:

I have been spending time in your classroom recently and back in September and October
because I am trying to better understand how kids learn math in order to improve math
textbooks and help teachers. So, I am trying to understand very carefully, as a researcher,
how YOU think. Let’s pretend I’m trying to create a robot who thinks just like you, who
thinks and learns just like you learn. When you answer my questions today, try to give
me as many details as you can in order to help me make that thinking machine. OK?

How do you feel math has been going for you since the last time I saw you?

- What has been going well? What do you think has been helpful about the class
lately? What has helped you learn?

- Can anything about your class be improved to make your experience in class
better? If you wanted to make the situation a better learning experience, what
could your teacher do? What could your peers do? What could YOU do?

Say a new student moved into your school and they wanted to know what they had to do
to be successful in your math class. What would you say?

- What exactly are you paying attention to? Remember that we want to make a
robot that thinks like you!

- People talk about paying attention to a lot of different parts of the class. Some
people say that they pay attention to the teacher’s explanations of the
problem’s directions, some say that they listen carefully to a classmate’s
explanation of a problem, others say that they listen carefully to the teacher
explaining the steps for how to calculate the answer to a problem, and there
still more things to pay attention to. Do you pay attention to any of those
things? Are any of them more important than the others? Can you try to
specifically describe what you are paying attention to?

- Do you do anything else in class besides PAY ATTENTION? Do you ever
talk or share your thinking about a problem? How often?

- What is more important for your learning — to talk through your thinking with
someone or to listen to the thinking of another? Why would you say so?

Say that your school needed to hire a new 7th grade math teacher and they were asking
students for advice on what would make a good 7th grade math teacher. What would you
say?

- Does your current teacher do any of these firings? Do you wish she would?

- Which one of your teachers did something like this, if anyone?

Would you rather listen to the ideas of others or share your own ideas? Why would you
say so?

Elaboration on responses to survey:

3. I want to do better than other students in my math class.
- Why? (or why not?)

248

 

- Do you compare yourself to other people in the class? Does this affect how
you learn?
- How important is doing better than other students to you?
- How important is it to you to make sure no one in the class is left behind when
moving on to a new problem? How about moving on to a new book?
16. When my teacher asks a question in math class, it is important that I explain how I
did the problem, not just give my answer.
- (If it is important to them to explain...) How important is it to explain your
thinking? Does it help you learn or not?
- When do you think you should be asked to explain and share your thinking?
In what ways?
- When are you asked by your teacher to explain your thinking? Does this help
you learn?
35. It is possible to approach the same math problem in more than one way.
- Is it important to you to be able to solve a math problem in more than one
way? Why or why not?
- Does being able to solve a math problem in more than one way help you learn
anything or does it really matter?
40. To work on math problems, I have to be taught the rules & steps, or else I can’t solve
them.
- Can you try a new problem without help and be able to solve it?
- If you need some help to get started, what kind of help do you need? Please be
specific as you can. Again, remember that we need to try to create a robot to
think just like you think!

249

 

APPENDIX B

Students’ Mathematical Views Instrument: 9 Scales, 63 items

 

Not at all True Not Very True Somewhat True Mostly True True

 

 

Confidence scale:
1. I feel secure about attempting math problems.
27. I am sure that I can learn math.
36. I think I could handle more difficult math.
46. I can get good grades in math.
63. I have a lot of self-confidence when it comes to math.
41. I’m no good in math.
14. I’m not the type to do well in math.
28. Even though I study, math is unusually hard for me.
55. Most subjects I can handle OK, but I have a knack for messing up math.
61. Math is my worst subject.

Task Orientation scale:
2. I like doing problems in math class that I'll learn from even if I make a lot of
mistakes.
15. An important reason why I do my work for math class is because I like to
learn new things.
26. I like work in math class best when it really makes me think.
37. An important reason why I do my work for math class is because I want to
understand it.
47. An important reason I do my work for math class is because I enjoy it.
56. I do my work for math class because I'm interested in it.

Ability Orientation scale:
3. I want to do better than other students in my math class.
10. I would feel successful if I did better than most of the other students in my
math class.
25. I'd like to show my math teacher that I'm smarter than the other kids in this
class.
38. Doing better than other students in this math class is important to me.
48. I would feel really good if I were the only one who could answer the teacher's
questions in (math) class.
57. It's important to me that the other students in this math class think that I am
good at my work.

 

250

Process vs. Product scale:
4. When my teacher asks a question in math class, I have to say the right answer
to answer it correctly.
16. When my teacher asks a question in math class, it is important that I explain
how I did the problem, not just give my answer.
24. Just because I got the wrong answer doesn’t mean I don’t know how to do the
problem.
39. If I get the wrong answer, I do not know how to do the problem.
49. I want to hear how other students did the math problem, not just what they got
for an answer.
58. I do not think it is important for students to share how they did a math
problem, only the answers they got.

Autonomy & Authority scale:
5. In math, it is possible to discover things by myself.
11. Math problems can be solved by thinking carefully, not just by the math rules
I learn in school.
23. I cannot discover things about numbers in math class without being taught.
40. To work on math problems, I have to be taught the rules & steps, or else I
can’t solve them.
42. In math, the teacher has the answer and it is the student's job to figure it out.
50. When a classmate and I don't agree on an answer, we can usually think
through the problem together until we have a reason for what is correct.
29. When a classmate and I don't agree on an answer in math, we need to ask the
teacher or check the book to see who is correct.

Usefulness scale:
6. I'll need math for my future work.
17. Math has very little to do with my life.
22. Math is a worthwhile subject for me.
30. Taking math classes is a waste of time.
32. Knowing math will help me earn a living.
43. Math will not be important to me in my life's work.
51. I will use math in many ways as an adult.
59. I expect to have little use for math when I get out of school

Structure scale:
7. Tables and graphs have little to do with other things in math like formulas and
equations.
12. Often a single mathematical idea will explain many equations or steps for
doing a problem.
21. Finding answers to one type of math problem cannot help you solve other
types of problems.
31. Math is mostly thinking about relationships among things (for example:
ntunbers, points, and lines).

251

33. There are few connections between the different mathematical topics I have
studied (for example: measurement and fractions).

44. Ideas learned in one math class can help you understand material in the next
math class.

52. Math consists of many unrelated topics.

60. Most mathematical ideas are related to one another.

Conceptual vs. Procedural scale:
8. The math that I learn in school is mostly a set of rules to memorize
18. The math that I learn in school is a way to think about the world around me
20. The math that I learn in school is mostly about computation, like addition,
subtraction, multiplying, or dividing.
34. The math that I learn in school is mostly a set of steps to remember
53. The math that I learn in school is go_t just a set of rules.
45. The math that I learn in school is about seeing new relationships in data
64. The math that I learn in school is mostly about ideas

Multiple Methods scale:
9. A math problem can only be done correctly in one way
19. A math problem can be solved correctly in more than one way
35. It is possible to approach the same math problem in more than one way
54. It is not possible to approach the same math problem in more than one way.
13. Once I get an answer I know is correct, looking at a second way to do the
problem is pointless
62. When your method of solving a problem is different from your teacher's
method, your method can be as correct as your teacher's.

252

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