THE PROOF IS IN THE PRACTICE? GRADUATE TEACHING ASSISTANTS AND FUTURE TEACHERS
By
Kimberly Cervello Rogers

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Mathematics Education
2012

ABSTRACT
THE PROOF IS IN THE PRACTICE? GRADUATE TEACHING ASSISTANTS AND FUTURE TEACHERS
By
Kimberly Cervello Rogers
In mathematics, engaging in reasoning-and-proving (RP) includes investigating
mathematical relationships, formulating conjectures, evaluating others' conjectures or
arguments, generating arguments, and communicating mathematical knowledge. These
mathematical processes are key habits of mind that help learners think critically within and
outside mathematics classrooms. Researchers and policymakers have reflected the centrality of
RP by emphasizing the importance of helping students in all mathematical content areas and in
all grade levels to engage in rich proving activities and become competent at generating and
evaluating mathematical arguments (NCTM, 2000, 2009; Yackel & Hanna, 2003). In contrast to
these recommendations, however, K-12 students’ initial experiences with formal proof
traditionally occur in high school geometry classes, where students often experience only one
slice of the range of RP processes with a focus on proving given statements using a formal twocolumn proof format (Harel & Sowder, 1998; Herbst, 2002). Prospective teachers of elementary
grades (PTEs) also typically have limited experiences developing and using proof, which could
hinder their future elementary students' engagement in these RP processes (Balacheff, 1988;
Martin & Harel, 1989). How are PTEs currently being prepared to provide rich opportunities for
their future students experience and understand RP?
Mathematics for elementary teachers courses are the primary site for supporting this
development of PTEs’ mathematical knowledge for teaching. Geometry and Measurement is a

common content slice for these courses, with explicit attention to the work of proof (Cannata &
McCrory, 2007; McCrory, Siedel, & Stylianides, 2008). This attention on RP is important for
developing PTEs’ ability to explain why mathematical relationships are true, as well as for
preparing them to help their students develop RP-abilities. By examining how graduate
teaching assistants (TAs) implemented RP mathematical tasks when teaching a Geometry and
Measurement course for PTEs, this study investigated the opportunities for PTEs to learn about
RP in such courses.
Specifically, six TAs, each assigned to teach their own sections of the course,
participated in the study. Through classroom observations and interviews, the way these TAs
engaged PTEs in RP and how TAs' conceptions of RP illuminate their instructional decisions
around RP tasks was examined. RP tasks were defined as tasks with potential to engage PTEs in
RP processes. Findings indicate that TAs engaged PTEs in a range of RP processes. For a plurality
of observed tasks, however, opportunities for PTEs to engage in RP were decreased. TAs often
decreased RP opportunities by stating a conjecture or proof before PTEs had a chance to
generate them. There are multiple factors (e.g., TAs’ conceptions of the purposes for teaching
about RP and how to facilitate class discussions) that influenced their instructional decisions.
This research has implications for professional development to support college mathematics
instructors’ teaching.

Copyright by
Kimberly Cervello Rogers
2012

I dedicate this work to
my gracious Lord and Savior, who inspires me;
my supportive husband, who encourages me;
and my loving family, who believes in me.
Thank You!

v

ACKNOWLEDGEMENTS
The printed pages of this dissertation represent far more than the culmination of years
of study. These pages also reflect the relationships with many generous and inspiring people I
have met since beginning my graduate work. The list is long, but I cherish each contribution to
my development as a scholar and educator: I am deeply indebted to my advisor and mentor,
Dr. Michael Steele, for his thoughtful, patient guidance throughout my doctoral program. Thank
you, Mike, for your generous words of wisdom, encouragement, and feedback that informed
this dissertation study and my research abilities. I am also grateful for my committee members,
Drs. Kristen Bieda, Ralph Putnam, and Clifford Weil, who brought expertise to my dissertation
and taught me how to conduct mathematics education research rigorously. Kristen, thank you
for your insights about collecting and analyzing the classroom observation data; your
collaboration in research and teaching has supported me in becoming a better researcher and
teacher educator. Ralph, your feedback about the interview protocols, the analysis of TAs’
conceptions, and earlier drafts of chapters was invaluable. Cliff, thank you for providing an
alternative perspective that informed my research design and writing.
This work would not have been possible without the TAs who agreed to participate in
this study. I owe my deepest gratitude to these six individuals who welcomed me into their
classrooms and took the time to answer my questions. It was a privilege and an honor to work
with and learn from all of you. To the mathematics department and course supervisor, thank
you for allowing me to study TAs under your care. I also acknowledge the funding support I
received from the College of Natural Science and Graduate School at Michigan State University.
I want to thank the faculty members, staff, colleagues, and friends from whom I have

vi

been blessed to learn and gather advice: Dr. Natasha Speer, thank you for your input in the
early stages of this research, pushing me to keep the scope and purpose of the study welldefined. Dr. Sharon Senk, as my course supervisor and instructor you provided rich learning
experiences and encouragement to pursue my research interests, thank you. Dr. Jack Smith, I
appreciated your detailed edits and comments on my course papers, helping me to learn about
and improve my writing. I also had the privilege of taking courses from a number of other
faculty members, including: Drs. Michael Shaughnessy, Robert Floden, Jennifer Kaplan, Susan
Melnick, and Steven Weiland, who refined my understanding of teaching, learning, and
research in mathematics education.
Many thanks to Lisa Keller, Margaret Iding, Jean Beland, and Becky Murthum for their
administrative support in the Program in Mathematics Education over these past five years. I
am thankful for all the graduate students I had the opportunity to work with at Michigan State,
as well. Special thanks to Sam Otten for his assistance with the development of coding schemes
and the coding of data. To my friends and colleagues, Drs. Jungeun Park and Sara Wyse:
Jungeun, I consider your friendship and willingness to dialogue about research and life
priceless. Sara, thank you for your consistent prayers, support, and encouragement—I am so
grateful we met during our first month in graduate school.
Last, but certainly not the least, I cherish the unconditional love and support I have
received from my family. To my parents, Douglas and Barbara Cervello, in-laws, Jack and Kathy
Rogers, and sisters, Angela Gonzalez and Michelle Cervello, thank you for encouraging me in my
aspirations. To my loving husband, Matthew Rogers, thank you for always being my ally and
helping me to remain in the Lord through this entire process.

vii

TABLE OF CONTENTS
LIST OF TABLES .......................................................................................................................... xii
LIST OF FIGURES ....................................................................................................................... xiii
CHAPTER 1
INTRODUCTION ...........................................................................................................................1
1.1 Focus of this Study .....................................................................................................3
1.2 Contributions to the Field ..........................................................................................6
1.3 Limitations .................................................................................................................6
1.4 Organization of the Dissertation ................................................................................7
CHAPTER 2
LITERATURE REVIEW ...................................................................................................................8
2.1 Examining Teachers’ Classroom Decisions..................................................................8
2.2 Examining Teachers’ Knowledge and Beliefs ............................................................10
2.2.1 Teachers’ Mathematical Knowledge for Teaching......................................11
2.2.2 Teaching about Reasoning-and-Proving in the Elementary Grades ............13
2.2.3 TAs Teaching Prospective Elementary Teachers about Reasoning-andProving ...................................................................................................15
2.2.3.1 Studying TAs’ Conceptions of Reasoning-and-Proving .................16
2.2.3.2 What is Known about TAs’ Potential Conceptions of Reasoningand-Proving ................................................................................19
2.3 Frameworks for Analyzing Teachers’ Classroom Practices ........................................20
2.3.1 Focusing on RP-related Classroom Tasks ...................................................21
2.3.2 Reasoning-and-Proving Framework ...........................................................24
2.4 Research Questions .................................................................................................30
CHAPTER 3
METHOD ...................................................................................................................................32
3.1 Context ....................................................................................................................32
3.1.1 The Design and Goals of the Course ..........................................................32
3.1.2 The TAs .....................................................................................................33
3.2 Instruments and Data Sources .................................................................................34
3.2.1 Data Sources for Studying TAs’ Conceptions of Reasoning-and-Proving .....34
3.2.2 Data Sources for Studying How TAs Enacted Reasoning-and-Proving Tasks
...............................................................................................................36
3.2.3 Data Sources for Studying How TAs’ Conceptions Illuminate Their
Instructional Decisions............................................................................38
3.3 Data Collection ........................................................................................................39
3.3.1 Textbook Analysis ......................................................................................39
3.3.1.1 Textbook Analysis: Unit of Analysis .............................................39
3.3.1.2 Textbook Analysis: Coding Procedure..........................................40

viii

3.3.2 Collecting Data about TAs’ Teaching Practices ...........................................42
3.3.2.1 Preobservation Survey Procedure ...............................................42
3.3.2.2 Classroom Observation Procedure ..............................................43
3.3.3 Collecting Interview Data ..........................................................................43
3.4 Data Analysis ...........................................................................................................44
3.4.1 Analyzing TAs’ Enactment of RP Tasks .......................................................45
3.4.1.1 Analysis of the Task-as-Written ...................................................46
3.4.1.2 Analysis of the Task-as-Set-up .....................................................46
3.4.1.3 Analysis of the Implementation of the Task ................................47
3.4.1.4 Analysis of the Enactment of the Task .........................................48
3.4.2 Analyzing TAs’ Conceptions of RP and How their Conceptions Illuminate
their Instructional Decisions ...................................................................50
3.4.2.1 Analysis of TAs’ Conceptions of RP in Mathematics, Teaching, and
Learning ......................................................................................50
3.4.2.2 Analysis of How TAs’ Conceptions Illuminate their Instructional
Decisions .....................................................................................53
CHAPTER 4
TEACHING ASSISTANTS’ CONCEPTIONS OF REASONING-AND-PROVING ....................................54
4.1 Analysis of All the TAs’ Conceptions of Reasoning-and-Proving ................................54
4.1.1 TAs’ Conceptions of the Purposes of Proof in Mathematics and in
Mathematics Teaching ...........................................................................55
4.1.2 TAs’ Conceptions of Why and What PTEs Should Learn about Reasoningand-Proving ............................................................................................60
4.1.3 TAs’ Conceptions of How to Teach about Reasoning-and-Proving .............64
4.2 Case Studies of TA’s Conceptions of Reasoning-and-Proving ....................................69
4.2.1 Preparing Prospective Teachers to Teach Others about RP Processes........70
4.2.2 Preparing Prospective Teachers to Produce Mathematical Explanations ...73
4.2.3 Helping MTH 2 Students Understand Formal Proofs ..................................75
4.2.4 Helping MTH 2 Students Learn RP Content Using Lectures ........................77
4.3 Conclusions about the Analysis of TAs’ Conceptions of RP .......................................80
CHAPTER 5
REASONING-AND-PROVING OPPORTUNITIES IN THE TEXTBOOK ...............................................82
5.1 Distribution of Opportunities for Reasoning-and-Proving Across Four Textbook
Chapters ............................................................................................................83
5.2 Types of RP opportunities Afforded in the RP Tasks .................................................85
5.3 Purposes of Explorations of Mathematical Relations, Conjectures, Evaluations, and
Proofs ................................................................................................................87
5.3.1 Purposes of Investigating, Conjecturing, and Evaluating ............................87
5.3.1.1 Considering the Analysis of the RPQs ..........................................87
5.3.1.2 Considering the Analysis of the RPTs ...........................................88
5.3.2 Purposes of Proofs ....................................................................................90

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5.4 Opportunities to Reflect on Characteristics of Proof and Proving Rather than to Do
Proof ..................................................................................................................91
5.5 Conclusions about the Textbook Analysis.................................................................92
CHAPTER 6
ENACTMENT OF REASONING-AND-PROVING TASKS ..................................................................95
6.1 Analytical Approach for Examining Enactment of Reasoning-and-Proving Tasks ......95
6.2 Overall Results from Analyzing the Enactment of Reasoning-and-Proving Tasks ......96
6.3 The Types of Reasoning-and-Proving Tasks Observed and How they were Enacted
........................................................................................................................ 101
6.4 Examining each TA’s Enactment of Reasoning-and-Proving Tasks .......................... 105
6.4.1 Individual TA’s Enactment of Reasoning-and-Proving Tasks across the
Semester .............................................................................................. 108
6.4.2 How Individual TAs Enacted Different Types of Reasoning-and-Proving
Tasks .................................................................................................... 109
6.5 Conclusions about the Enactment of Reasoning-and-Proving Tasks ....................... 112
CHAPTER 7
TEACHING ASSISTANTS’ INSTRUCTIONAL DECISIONS ............................................................... 115
7.1 TAs’ Decisions to Teach Reasoning-and-Proving Tasks ........................................... 116
7.1.1 How TAs Articulated their Selection of Reasoning-and-Proving Tasks ...... 116
7.1.2 Analysis of TAs’ Responses about Selecting Reasoning-and-Proving Tasks
............................................................................................................. 120
7.2 TAs’ In-the-Moment Instructional Decisions .......................................................... 124
7.2.1 Analysis of TAs’ Responses about In-the-Moment Instructional Decisions
............................................................................................................. 124
7.2.2 TAs’ Conceptions of RP and their In-the-Moment Classroom Decisions ... 126
7.2.2.1 How TAs who Aimed to Prepare PTEs to Teach others about RP
Processes Enacted RPTs ............................................................ 126
7.2.2.2 How a TA who Aimed to Prepare PTEs to Produce Explanations
Enacted RPTs ............................................................................ 128
7.2.2.3 How a TA who Aimed to Help MTH 2 Students Understand Formal
Proofs Enacted RPTs ................................................................. 129
7.2.2.4 How a TA who Aimed to Help MTH 2 Students Learn RP Content
Using Lectures Enacted RPTs ..................................................... 131
CHAPTER 8
CONCLUSIONS AND IMPLICATIONS ......................................................................................... 134
8.1 Implications and Recommendations for the Field .................................................. 135
8.1.1 Supporting TAs’ Enactment of Reasoning-and-Proving Tasks ................... 136
8.1.2 When is an Explanation a Justification? Developing a Reasoning-andProving Language ................................................................................. 138
8.1.3 Developing TAs’ Knowledge about PTEs and How RP Processes Prepare
PTEs to Teach ....................................................................................... 140

x

8.1.4 Developing Critically Reflective TAs ......................................................... 142
8.2 Future Research ..................................................................................................... 144
8.2.1 What about the Prospective Elementary Teachers’ Learning? ................. 145
8.2.2 Supporting TAs’ Teaching and Learning to Teach ..................................... 146
8.2.3 Collegiate Mathematics Teaching and Learning ....................................... 146
8.3 Conclusion ............................................................................................................. 149
APPENDIX A
PROTOCOLS AND SURVEY INSTRUMENTS................................................................................ 152
A.1 Observation Protocol ............................................................................................. 152
A.2 Preobservation Survey ........................................................................................... 154
A.3 Pre-Semester Interview Protocol ........................................................................... 155
A.4 Beginning-of-the-Semester Interview .................................................................... 156
A.5 End-of-Semester Interview Protocol ...................................................................... 157
A.6 Post-Observation Interview Protocol ..................................................................... 159
APPENDIX B
TEXTBOOK ANALYSIS AND CLASSROOM OBSERVATION CODING SCHEME ............................... 161
B.1 Coding the Task-as-Written ................................................................................... 161
B.2 Definitions for Components of Reasoning-and-Proving Framework Used in Task
Analysis ............................................................................................................ 162
B.3 Sections of Textbook Analyzed .............................................................................. 163
B.4 Coding the Task-as-Set-Up ..................................................................................... 164
B.5 Coding the Task as Implemented, Focusing on Individual or Group Work .............. 165
B.6 Coding the Task-as-Implemented, Focusing on Whole-class Discussions ................ 166
APPENDIX C
INTERVIEW CODING SCHEMES ................................................................................................ 167
C.1 Coding Interviews for Conceptions of RP in Mathematics ...................................... 167
C.2 Coding Interviews for Conceptions of RP in Teaching ............................................. 168
C.3 Coding Interviews for Conceptions of RP in Learning ............................................. 170
REFERENCES ............................................................................................................................ 172

xi

LIST OF TABLES
Table 1. Purposes of Proof in Mathematics and Teaching ..........................................................18
Table 2. Reasoning-and-Proving Framework (adapted from Stylianides, 2009) ..........................25
Table 3. Participants' Background Information ..........................................................................34
Table 4. Data Sources for Research Question 1: What are TAs' Conceptions of the Nature and
Purposes of RP in Mathematics, Teaching, and Learning? ...........................................35
Table 5. Classroom Observation Details for each TA ..................................................................42
Table 6. TAs’ Statements about Purposes of Proof in Mathematics and in Mathematics Teaching
....................................................................................................................................56
Table 7. Frequency of Purposes of Proof in Mathematics Articulated by TAs .............................57
Table 8. Frequency of Purposes of Proof in Mathematics Teaching Articulated by TAs ..............59
Table 9. Distribution of Reasoning-and-Proving-related Questions and Tasks............................83
Table 10. Categories for Types of Reasoning-and-Proving Opportunities in Tasks ......................89
Table 11. Opportunities for PTEs to Engage in RP processes within the 66 RP Tasks ..................89
Table 12. Types of Reasoning-and-Proving Tasks Observed: Opportunities to Prove in the Tasksas-Written ................................................................................................................. 102
Table 13. Enactment of Different Types of Reasoning-and-Proving Tasks ................................ 103
Table 14. Categories for How TAs Explained Why they Used Reasoning-and-Proving Tasks ..... 117

xii

LIST OF FIGURES
Figure 1. Model of major components of the study .....................................................................5
Figure 2. Framework for studying TAs' implementation of proof-related tasks ..........................23
Figure 3. Data source for Research Question 2: How do TAs enact RP tasks in a mathematics
content course for PTEs? .............................................................................................37
Figure 4. Data source for Research Question 3: In what ways, if at all, does understanding TAs’
conceptions of RP illuminate their instructional decisions regarding the enactment of
RP tasks? .....................................................................................................................38
Figure 5. Examples of how TAs talked about what and why they wanted students to learn about
reasoning-and-proving ................................................................................................61
Figure 6. TAs' conceptions of teaching and learning reasoning-and-proving in MTH 2 ...............63
Figure 7. How TAs talked about approaches for teaching reasoning-and-proving ......................66
Figure 8. Breakdown of RPTs and non-RPTs in the textbook ......................................................84
Figure 9. Number of tasks addressing RP processes in the textbook ..........................................86
Figure 10. Enactment of RP tasks, results from observing eighty-two tasks enacted .................97
Figure 11. Enactment of RP tasks including different types of decreases in RP demands ...........98
Figure 12. Individual TA's enactments of RPTs. Each circle represents one task observed during a
specific week of the semester: Ace, Isaac, and Laura ................................................. 106
Figure 13. Individual TA's enactments of RPTs. Each circle represents one task observed during a
specific week of the semester: Kelly, Evan, and Peter ................................................ 107
Figure 14. Major components of the study .............................................................................. 115
Figure 15. How TAs talked about their decisions to enact reasoning-and-proving tasks .......... 120

xiii

CHAPTER 1
INTRODUCTION
The process of reasoning-and-proving is fundamental to mathematics, both for
professional mathematicians and in mathematics classrooms. Descriptions of mathematicians’
practice and K-12 standards documents note that the proving process involves exploration of
patterns, which can lead to the generation of conjectures, and can then be tested and revised
or proven informally or formally (Lakatos, 1976; NCTM, 2009). The hyphenated term reasoningand-proving (Stylianides, 2008) denotes the range of activities including investigating patterns,
formulating conjectures, generating arguments, evaluating others’ arguments, and
communicating mathematical knowledge, which are “frequently involved in the process of
making sense of and establishing mathematical knowledge” (Stylianides, 2009, p. 259). These
reasoning-and-proving (RP) processes are key habits of mind that equip students to think
critically within and outside mathematics classrooms. Researchers and policymakers also
emphasize the centrality of RP through recommendations for students in all mathematical
content areas and in all grade levels, including elementary-grades, to engage in proving
activities and become competent at generating and evaluating mathematical arguments (Ball &
Bass, 2000; CCSSI, 2010; NCTM, 1989, 2000, 2009; Stylianides, 2007a; Yackel & Hanna, 2003).
The treatment of proof in K-12 classrooms, however, is often restricted to verifying
given statements using a two-column format (Ball, Hoyles, Jahnke, & Movshovitz-Hadar, 2002;
Harel & Sowder, 1998; Herbst, 2002). This emphasis represents only one aspect of the range of
RP processes, and contributes to the pervasive difficulties and limited views of proof held both
by K-12 students and their teachers (Alibert & Thomas, 1991; Balacheff, 1988; de Villiers, 1998;

1

Herbst, 2002; Martin & Harel, 1989). For example, teachers often favor empirical arguments
over deductive proofs, finding them more convincing or easier to follow, and are more likely to
favor form over explanatory power (Knuth, 2002a; Steele, 2006; Stylianides & Stylianides,
2009). These conceptions extend to pedagogical choices, as teachers are more likely to use an
empirical argument as compared to a proof if they find it more convincing (Knuth, 2002a;
Steele, 2006). The full range of RP processes of noticing patterns, making conjectures, and
forming arguments, however, can be accessible at the elementary (grades K-8) level, and as
such, elementary teachers should be equipped to teach RP in meaningful ways. It is crucial,
therefore, to help prospective teachers of elementary grades (PTEs) move from conceptions of
proof as empirical arguments or as having one particular form, towards an understanding of the
role of proof as a process in which one engages to make meaning in mathematics.
To address this need, the Conference Board of the Mathematical Sciences (2012) argues
for undergraduate education across the nation to focus on the development of PTEs’ abilities to
reason and think mathematically. The authors specifically contend that “preparation . . . for
teachers must provide opportunities to do mathematics and to develop mathematical habits of
mind. . . . Teachers should have time and opportunity to reason abstractly and quantitatively, to
construct viable arguments, to listen carefully to other people’s reasoning, and to discuss and
critique it” (p. 16). This need for supporting the development of PTEs’ understanding of
mathematical content and processes related to RP is particularly important when considering
the fact that they will ultimately help students in grades K-8 makes sense of mathematics.
Engaging elementary students in a range of RP processes would lay a strong foundation for
reasoning and sense making upon which later success in mathematics crucially depends (CCSSI,

2

2010; NCTM, 1991, 2000). Providing additional opportunities for students to become more
proficient at creating and investigating conjectures, evaluating mathematical arguments, and
employing various methods of proof in everyday in mathematics classrooms is an important
step in addressing concerns about students’ limited experiences with proof. To do so, however,
places significant demands on teachers’ knowledge and beliefs about proof. As Stylianides and
Ball (2008) explain, “unless teachers have a good understanding of proof, we cannot expect
that they will be able to effectively promote proving among their students” (p. 309). As future
elementary teachers, it is crucial, then, for PTEs to develop substantial understanding of RP to
provide meaningful opportunities for their students to engage with reasoning and proof
(Stylianides, 2007a, 2007b; Stylianides & Ball, 2008).
Mathematics for elementary teachers courses are the primary site for supporting the
development of PTEs’ understanding of mathematical content and processes. In universities
that offer elementary certification, PTEs are typically required to take at least one semesterlong course covering mathematical content PTEs will likely teach in their future elementarygrade (K-8) classrooms (Cannata & McCrory, 2007; Lutzer, Rodi, Kirkman, & Maxwell, 2007, p.
53). Geometry and Measurement is a common content slice for these courses, with explicit
attention to the work of proof (Cannata & McCrory, 2007; McCrory et al., 2008). This attention
on RP is important for developing PTEs’ ability to explain why mathematical relationships are
true, as well as for preparing them to help their students develop their RP-abilities.
1.1 Focus of this Study
In many cases, mathematics for elementary teachers courses are taught by graduate
teaching assistants (TAs), which could have important implications for the development of

3

PTEs’ understanding of RP. As instructors for these courses, the TAs represent a central feature
of PTEs’ experiences learning about reasoning and proving (Philipp, 2007) and teachers’
conceptions are implicated as a major influence on teachers’ practices (Putnam & Borko, 1997;
Thompson, 1992). Further, little research directly explores the role of mathematics TAs in
creating and teaching concepts of RP (Speer, Gutmann, & Murphy, 2005).
In light of these teaching, learning, and research needs, this study examines the nature
of RP opportunities in PTEs’ mathematics courses by focusing on a geometry and measurement
course at a large, public university. At this university, the mathematics-content courses are
taught in the mathematics department with TAs as the sole instructors. To understand the
nature of PTEs’ opportunities for engaging in RP in this course, I investigate TAs’ instructional
decisions when teaching about RP by observing their classroom practices and interviewing
them about their knowledge and beliefs related to the teaching and learning of RP.
The study of TAs’ instructional decisions when teaching about RP is complex due to a
large number of variables that affect teachers’ classroom decisions. To bound this study and aid
in anchoring the examination of TAs’ classroom practices when they are teaching about RP, I
focused on a specific classroom practice, the enactment of mathematical tasks related to RP,
and investigated a subset of factors that influence teachers’ decisions, their knowledge and
beliefs about the content and the teaching and learning of the content (Figure 1). Since
teachers’ knowledge and beliefs are a major factor influencing their instructional decisions
(Putnam & Borko, 1997; Schoenfeld, 2010), TAs’ knowledge and beliefs about RP in
mathematics, teaching, and learning were investigated through the use of semi-structured
interviews. The focus on the enactment of RP tasks was motivated by the fact that enacting

4

mathematical tasks in ways that provide opportunities for students to engage in doing
mathematics (including creating and evaluating mathematical generalizations and arguments) is
a high-leverage (Ball, Sleep, Boerst, & Bass, 2009) teaching practice in secondary mathematics
classrooms (Stein & Lane, 1996). Therefore, investigating the enactment of RP tasks in college
classrooms presents an opportunity to understand how PTEs are given opportunities to
participate in the processes of RP in the classroom community.

Teaching Assistant (TA)
Knowledge &
Beliefs about
Teaching &
Learning RP

Knowledge &
Beliefs about
RP in
Mathematics

Classroom Practice
Curriculum
Written Task
Opportunities
for RP in
textbook

RP-related Classroom Tasks
Task Setup
(Instructional
Design)
RP opportunities
in how task is
introduced

Task
Implementation
How prospective
teachers engage in
RP-processes

Figure 1. Model of major components of the study
(For interpretation of the references to color in this and all other figures, the
reader is referred to the electronic version of this dissertation)
The enactment of RP tasks spans from how tasks are designed in curriculum materials to
how teachers and students interact with tasks in the classroom (Henningsen & Stein, 1997). The

5

opportunities for PTEs to experience RP processes were examined in the textbook and then in
the task set up and implementation during classroom instruction (Figure 1). The extent to which
TAs provided a range of RP opportunities for PTEs to experience and how and why changes in
RP opportunities occurred during classroom instruction was analyzed.
1.2 Contributions to the Field
This study refines the knowledge base about TAs’ classroom practices when teaching RP,
an area that has been under-researched. This study also adds to the understandings of TAs’
knowledge and beliefs about RP, specifically as it relates to teaching prospective teachers about
RP so they understand the processes of RP in particularly robust ways. Furthermore, this
examination of TAs’ instructional decisions when teaching about RP and the enactment of RP
tasks during classroom instruction provides information about professional development needs
of collegiate mathematics instructors. This contribution is particularly timely because the
Conference Board of the Mathematical Sciences (2012, p. 17) recently re-emphasized the
current lack of and need for professional development and support for collegiate mathematics
instructors who are teaching mathematics courses for prospective elementary teachers.
1.3 Limitations
Although this research was carefully designed and executed, it is important to
acknowledge some limitations. First, the study is based on a convenience sample consisting of
graduate students assigned to teach a geometry and measurement course for prospective
elementary teachers at one university. This sample may not be representative of collegiate
mathematics instructors in the United States in general. Second, TAs’ teaching practices were
studied by observing each TA’s classroom for at least six hours of class time to see TAs teach

6

different types of RP-related tasks. This study does not claim to characterize each TA’s teaching
practices in general because the focus was on the enactment of RP tasks. This analysis also
describes patterns in how each TA enacted the observed RP tasks, constituting a subset of the
total number of RP tasks enacted across the semester. Third, this study did not observe TAs
during weekly course meetings to assess any potential effects of this professional development
opportunity on their classroom decisions. This avenue warrants investigation but was beyond
the scope of this study. Finally, the study of classroom practices considered the nature of PTEs’
opportunities to engage in RP processes, but did not include measures of student learning to
assess potential gains in PTEs’ knowledge and abilities related to RP processes. This is a line of
inquiry that also merits further study but was outside the scope of this investigation.
1.4 Organization of the Dissertation
This dissertation is organized into eight chapters. Following this overview chapter,
Chapter 2 reviews relevant literature about studying teachers’ knowledge and beliefs and
classroom practices especially related to the teaching of RP, lays out the frameworks used in
this study, and outlines the research questions. Chapter 3 describes the methods used and the
data collected in this study. Chapters 4 through 7 detail the results: Chapter 4 includes the
results from analyzing TAs’ knowledge and beliefs about RP to outline TAs’ conceptions before
unpacking the classroom observations. Chapter 5 describes the results from analyzing the
textbook for the course, which guided the schedule for the classroom observations. The results
of the analysis of those classroom observations are presented in Chapter 6. Chapter 7 presents
the results from analyzing how TAs talked about their instructional decisions around RP. Finally,
Chapter 8 discusses implications of the study and suggests avenues for future research.

7

CHAPTER 2
LITERATURE REVIEW
This study investigates TAs’ conceptions and teaching practices when they are teaching
PTEs about RP. To frame this examination, I first describe a framework for considering TAs’
decision-making processes in enacting RP tasks. Next, I detail theories of teachers’ knowledge
and beliefs that inform the design and analysis of the study. Then, I describe frameworks for
examining teaching practices, focused on teaching about RP, on which I draw throughout the
study. Finally, I list my research questions.
2.1 Examining Teachers’ Classroom Decisions
Classroom teaching has been depicted as a more demanding and complicated task than
that of a physician determining a diagnosis (Fogarty, Wang, & Creek, 1983; Shulman, 1984). In
contrast to clinical diagnostic settings that usually involve relatively short, one-on-one
interactions between physicians and patients, teaching typically demands daily interactions
with multiple students, simultaneously. During planning and in the midst of instruction,
teachers work through a myriad of interconnected and conflicting decision situations. The
interactive, dynamic nature of teaching gives credence to the importance of studying teachers’
decision making because teachers’ decisions directly impact classroom events which in turn
influence student learning (Munby, 1982). Therefore, examining teachers’ decision making
could shed light on how and why teachers structure lessons and provide learning opportunities
for students, contributing to a better understanding of effective teaching practices.
Mathematics education researchers and policy makers also note the importance of
studying mathematics teachers’ decision making because effective teaching requires teachers

8

not only to have a rich and extensive knowledge base for teaching mathematics, but to
establish challenging and supportive classroom environments and continue to develop their
instructional practices and mathematical knowledge to support student learning (Ball, 1988a;
NCTM, 2000). In the classroom, learning to listen to what students say and then construct
appropriate responses on a moment-to-moment basis places special demands on teachers
(Ball, 1993; NCTM, 2000; Smith III, 1996). In these classroom settings, teachers juggle a variety
of responsibilities, as Ball (1988b) explains:
Teaching mathematics means helping to build bridges between the subject matter and
learners in ways that respect the integrity of both. To do this, teachers explain, ask
questions, respond to students, develop and select tasks, and assess what students
understand. These activities emerge from a bifocal consideration of mathematics and
students, framed by the teacher's own understandings and beliefs about each, and
shaped further by her ideas about learning and her role in promoting learning. (p. 297298)
During teaching, decision situations incorporate a host of factors, including: students’
behaviors, needs, and experiences; teachers’ goals/objectives and lesson plan; teachers’
knowledge and beliefs of the subject matter, the students, and teaching strategies; and time
constraints.
Since teachers’ knowledge and beliefs have been shown to be an important factor
influencing their instructional decisions (Ball, 1988b; Putnam & Borko, 1997; Schoenfeld, 2010),
examining teachers’ knowledge and beliefs about the mathematics content, students, and
pedagogy can help us understand their instructional decisions and potential ways to help them
improve their teaching practices. Thus, this study investigates teachers’ knowledge and beliefs
about mathematics and teaching to understand their instructional decisions made during
classroom teaching. Considering the complexity of studying teachers’ classroom decisions, this
9

study focused on a particular content area, RP, and a specific context, studying TAs when they
are teaching PTEs. The sections that follow, therefore, describe related frameworks and
literature that this study draws upon for examining TAs’ knowledge and beliefs about RP and
their classroom practices around RP.
2.2 Examining Teachers’ Knowledge and Beliefs
Examining teachers’ knowledge and beliefs about mathematics, teaching, and learning
can help one better understand teachers’ instructional decisions within their classroom
practices. Psychological and educational literatures present a range of definitions and labels
(e.g., attributed, professed, goals, conceptions) for beliefs, which is why Pajares (1992)
described beliefs as a “messy construct.” Some of the confusion stemmed from researchers’
attempts to differentiate between knowledge and beliefs (Calderhead, 1996). Although
differentiating between the two domains can be helpful in theory, they are less distinct in
practice (Grossman, 1990). Therefore, in this study the term conception (also used
interchangeably with the term view) is used to refer collectively to teachers’ knowledge and
beliefs. Teachers’ conceptions are defined as how they “think about the nature of mathematics,
teaching, and learning” (Aguirre & Speer, 1999, p. 328).
The influence of teachers’ conceptions of mathematics, teaching, and learning on how
teachers refine mathematical tasks and problems and make other decisions in their
instructional approaches is complex (Calderhead, 1996; Cohen, 1990; Kagan, 1992; Leder,
Pehkonen, & Törner, 2002; Pajares, 1992; Thompson, 1992). Overall, research indicates that
knowledge and beliefs are major factors that shape “teachers’ decisions about what knowledge
is relevant, what teaching routines are appropriate, what goals should be accomplished, and

10

what the important features are of the social context of the classroom” (Speer, 2005, p. 365).
The mathematics education research about teachers’ conceptions and practices, however, has
focused primarily on practicing K-12 teachers (see Leder et al., 2002) or prospective teachers
(e.g., Timmerman, 2004), with only two peer-reviewed examinations of tertiary instructors’
conceptions in mathematics teaching (Gutmann, 2009; Speer, 2008). Gutmann offered insights
into seven beginning mathematics graduate students’ views about the nature of mathematics,
teaching mathematics, themselves as learners of mathematics, and their students as learners,
but did not investigate how their knowledge and beliefs influenced or were evident in their
teaching practices. Speer (2008) conducted an in-depth case study analysis of Calculus TAs’
teaching practices, instructional decisions, and the relationship between their conceptions and
specific teaching practices. By examining the conceptions and teaching practices of collegiate
instructors who are teaching about RP to PTEs, this dissertation study contributes to this
growing and needed area of research (Speer, Smith III, & Horvath, 2010).
2.2.1 Teachers’ Mathematical Knowledge for Teaching
Although a number of factors affect teachers’ classroom practices, this study focuses on
teachers’ conceptions because it is widely acknowledged that the work of mathematics
teaching draws upon a deep and broad knowledge base, including knowledge of mathematics,
pedagogy, and student learning (Fennema & Franke, 1992; NCTM, 1991; Shulman, 1986).
Several studies suggest that the nature, depth, and organization of teacher knowledge
influence teachers’ presentation of ideas, flexibility in responding to students’ questions, and
capacity for helping students connect mathematical ideas (Ball, 1988a; Stein, Baxter, &
Leinhardt, 1990). Mathematics teachers are a special class of users of mathematics; the

11

knowledge they need for teaching mathematics goes beyond what is needed by other welleducated adults, including mathematicians (Ball & Bass, 2000; Ball, Lubienski, & Mewborn,
2001; Ball, Thames, & Phelps, 2008). As described by Hill, Rowan, and Ball (2005):
Mathematical knowledge for teaching [MKT] goes beyond that captured in measures of
mathematics courses taken or basic mathematical skills. For example, teachers of
mathematics not only need to calculate correctly but also need to know how to use
pictures or diagrams to represent mathematics concepts and procedures to students,
provide students with explanations for common rules and mathematical procedures,
and analyze students’ solutions and explanations. (p. 372)
Further, Hill and colleagues link higher levels of MKT to greater student learning gains,
underscoring the importance of teachers’ MKT and its impact on what students learn in the
classroom (Hill et al. 2005).
To ground this investigation of collegiate mathematics instructors’ MKT, this study
focuses on their mathematical knowledge for teaching about the activity of RP. The selection of
RP as a focus was based on the fact that the process of RP is fundamental to the work of doing
authentic mathematics, both in the world of professional mathematicians and in the
mathematics classroom. To be prepared to teach about RP, teachers need a specialized content
knowledge about RP and pedagogical knowledge about the teaching and learning of RP.
Teachers’ MKT about RP would include a robust understanding of these processes of proving
and pedagogical tools for engaging students in these processes (Steele & Rogers, 2012).
Within the context of this study, there are two types of teachers for whom this notion of
MKT about RP applies: (a) the undergraduate students enrolled in mathematics courses for
elementary teachers who are prospective elementary teachers and (b) their collegiate
instructors who are graduate students in mathematics or mathematics education programs.
The notion of MKT in these two settings is elaborated further, next.
12

2.2.2 Teaching about Reasoning-and-Proving in the Elementary Grades
Since the undergraduate students enrolled in the course that is the focus of this study
are PTEs, it is important to consider the knowledge elementary teachers need to teach about
RP in the elementary grades. Despite widespread agreement that RP should be a central
feature of all students’ mathematical experiences, a growing body of research shows that
elementary students and their teachers have difficulties and limited experiences with RP (Healy
& Hoyles, 2000; Stylianides & Ball, 2004; Stylianides, 2005). Teachers often favor empirical
arguments over deductive proofs, finding them more convincing or easier to follow, and are
more likely to choose an empirical argument for classroom use (Steele, 2006; Stylianides &
Stylianides, 2009). In many PTEs’ prior experiences as K-12 students learning about RP, they
were likely often asked to prove fairly obvious statements, and rarely given opportunities to
explore, conjecture, and justify their own conjectures (Herbst, 2002; Usiskin, 1980). These
experiences with proving contrast with the recommendations from the NCTM (2000), the
Common Core State Standards (2010), and the nature of proving in the mathematics discipline.
Preparing PTEs to teach about RP in the elementary context also begs the question
‘what should teachers expect students to learn about proof and proving in elementary grades?’
Research literature is rather sparse concerning proving for young children, with the exception
of the work done by Maher and colleagues at Rutgers (e.g., Maher & Martino, 1996), Lampert
(1990), and A. J. Stylianides (2007c). These works provide evidence that that students are
developmentally capable of generating proof even at the elementary school level, and support
the notion that teachers should create an environment that nourishes this capability. The field
has yet to develop a learning trajectory or more formalized conceptualization of proving in the

13

elementary grades, however, which limits our understanding of what preparing elementarygrade students for RP in middle and high school and beyond entails.
This study makes steps towards such a conceptualization by considering the processes
of RP (including generating and evaluating generalizations and arguments) as key mathematical
habits of mind that are important for all students, starting in the elementary grades. In an
elementary classroom, what would engaging in RP include? In the classroom, proof, as defined
by Stylianides (2007c), emerges as a product of discussions between the teacher and students
where students learn that proof goes beyond convincing oneself and convincing a friend to
convincing the classroom community as a broader audience. By engaging in RP classroom tasks,
students learn to analyze patterns, articulate how they are thinking about mathematics, and
evaluate others’ mathematical ideas and arguments. As elementary children solve open ended
problems and reason mathematically, “they learn to analyze their own, their classmates, and
their teacher’s thinking. . . . It is during these early years that young students lay down those
habits of reasoning upon which later achievement in mathematics will depend (CBMS, 2001,
chap. 7). In alignment with the processes of RP, this study emphasizes the importance of
providing opportunities for elementary students to engage in analyzing mathematical
situations, determining general relationships they think are true, and generating and evaluating
mathematical arguments about these relationships.
To provide such opportunities for elementary students PTEs need a robust
understanding of what constitutes a proof, how to generate and evaluate valid mathematical
arguments, and the various roles of proof in mathematics and in teaching (Steele & Rogers,
2012). As the primary site for developing PTEs’ mathematical knowledge of content and

14

processes, it is important for PTEs’ undergraduate coursework to help them to not only know
and understand mathematics, but also to know and experience the processes of RP themselves
as learners (Copes, 1996; Ernest, 1994; National Research Council, 1996; NCTM, 1991). This
study, therefore, examines the nature of the opportunities to learn about RP in a geometry
course for PTEs. Although TAs assigned to teach courses for PTEs may have robust conceptions
of proof, the lack of clear direction as to what elementary students should be expected to learn
and do with regards to RP may influence how TAs engage PTEs in RP in the classroom.
2.2.3 TAs Teaching Prospective Elementary Teachers about Reasoning-and-Proving
Preparing PTEs with a robust understanding of RP is an important aspect of the
mathematical courses PTEs take and it is not easy for collegiate mathematics instructors (in this
case graduate TAs) to structure a course that accomplishes that goal (CBMS, 2001; National
Research Council, 1996). Structuring a course for PTEs around such a goal could be an
overwhelming endeavor for any mathematics instructor, and the work of teaching these
courses would entail a specialized knowledge for teaching. TAs for these courses will be
expected to make sense of curriculum, design lessons, create and administer assessments, and
facilitate mathematical discussions in ways that encourage PTEs to investigate, conjecture,
evaluate claims and arguments, and produce mathematical justifications. This instructional
setting presents a challenge for TAs teaching PTEs where PTEs need to understand the
mathematical content (as current undergraduate students) and develop the ability to
coherently explain the mathematical ideas to others (as future teachers).
Furthermore, although the elementary mathematics content may seem simple and
straightforward to most graduate students assigned to teach this course, the notion of

15

preparing PTEs with the mathematical knowledge they need to teach elementary mathematics
involves more than just teaching them content. That is, Ball and Bass (2000, p. 98) contend that
in teaching elementary mathematics
one needs to be able to deconstruct one's own mathematical knowledge into less
polished and final form, where elemental components are accessible and visible. We
refer to this as decompression. Paradoxically, most personal knowledge of subject
matter knowledge, which is desirably and usefully compressed, can be ironically
inadequate for teaching. In fact, mathematics is a discipline in which compression is
central. Indeed, its polished, compressed form can obscure one’s ability to discern how
learners are thinking at the roots of that knowledge. Because teachers must be able to
work with content for students in its growing, not finished state, they must be able to
do something perverse: work backward from mature and compressed understanding of
the content to unpack its constituent elements.
This description of the challenges elementary teachers face in connecting to the mathematical
world of their students also holds true for mathematics instructors who are teaching PTEs.
Therefore, “in a college course, prospective teachers' ingenuous questions will require
instructors to ‘decompress’ their mathematical knowledge to find responses satisfying to both
mathematician and teacher” (CBMS, 2001, chap. 7). Teaching PTEs about RP, therefore, calls for
TAs to work from what PTEs already know, understand how PTEs think and misunderstand RP,
and use pedagogical strategies that engage PTEs in a range of RP processes. It is important,
then, to investigate what TAs know and believe about RP.
2.2.3.1 Studying TAs’ Conceptions of Reasoning-and-Proving
An examination of literature related to studying TAs’ conceptions of RP when teaching
mathematics reveals that there are no peer-reviewed empirical studies of TAs’ beliefs about RP
or of teaching about RP to prospective teachers (Speer et al., 2005). Investigating TAs’ beliefs
about mathematics, teaching, and learning without a focus on content, Gutmann (2009)
interviewed seven beginning mathematics graduate students. He found commonalities among
16

his participants in their positive attitudes toward mathematics, views of teachers as role
models, and descriptions of mathematics as having different levels where upper levels were
inaccessible to some students. Studying Calculus TAs’ beliefs of mathematics, teaching, and
learning and their relationship to TAs’ instructional practices, Speer (2008) observed and
interviewed one TA. She found a connection between the TA’s collection of beliefs and his inthe-moment instructional decisions. A common theme in these two studies was the
examination of TAs’ conceptions of mathematics, teaching, and learning. Since TAs are knowers
of mathematics as well as teachers of mathematics, investigating their conceptions along these
dimensions provides a more comprehensive view of TAs’ conceptions that influence their
teaching practices.
Knuth (2002a, 2002b) supports this notion of investigating teachers’ conceptions of
mathematics, teaching, and learning; his work focused on teachers’ views of proof at the
secondary level. More specifically, Knuth examined secondary teachers’ conceptions of the
nature and purposes of proof in mathematics, teaching, and learning. The purposes of proof he
emphasized were drawn from some of the primary functions of proof in mathematics (Table 1)
that have been identified by mathematics education researchers (Balacheff, 1988; Hanna, 1991;
de Villiers, 1999). In this study the terms purpose, role, and function of proof are used
interchangeably to refer to this notion that there are purposes proof can serve in the contexts
of mathematics and mathematics teaching.
These roles are particularly important to note in the K-12 setting because within K-12
school mathematics the primary purpose of proof emphasized has been verification (Herbst,
2002). There have been calls from researchers, however, for more emphasis on proofs that help

17

explain why a mathematical statement is true (de Villiers, 1998). Knuth argues, therefore, for
the importance of teachers having robust views of the purposes of proof in mathematics and in
teaching to provide meaningful opportunities for their students to experience proof in the
classroom. Extending this work to the collegiate context presents an opportunity to investigate
TAs’ salient views of the purposes of RP in mathematics, teaching, and learning.
Table 1. Purposes of Proof in Mathematics and Teaching
(de Villiers, 1998, 1999; Hanna, 1995, 2000; Hanna & Barbeau, 2008; Knuth, 2002a; Steele,
2006; Stylianides, 2009)

1. Verification of truth

Establishing, checking, or confirming the truth of a known idea or
given claim [Note: Direct methods of proof and proof by
contradiction (supposing a claim is false, subsequently drawing a
conclusion that contradicts a claim, thereby verifying the truth of
the given claim) also fit under the verification purpose of proof]

2. Falsification

Establishing, checking, or confirming the falseness of an idea or
given claim. Considering proof as an argument against a claim,
proof by counterexample and “reduction ad absurdum” (stating a
claim and showing that it results in a contradiction—
demonstrating the claim to be false) are captured under this
purpose. (Stylianides, 2009, p. 269)

3. Explanation

Providing insights into why a mathematical claim is valid.

4. Discovery/Creation
of new
mathematical
knowledge

Discovering or developing new mathematical ideas, confirming
conjectures, or building mathematical ideas

5. Communication of
mathematical
knowledge

Helping others understand a mathematical idea; disseminating
knowledge to other doers of mathematics

6. Systematization of
the domain

Imposing a logical structure on the mathematical domain;
organizing and cataloging results with respect to axioms and other
prior knowledge

18

2.2.3.2 What is Known about TAs’ Potential Conceptions of Reasoning-and-Proving
Although mathematics education literature does not yet provide insights into TAs’
conceptions of RP in mathematics, teaching, and learning, based on how mathematicians
engage in RP, I outline an informed hypothesis of how TAs’ may conceive of RP and how that
could compare with PTEs’ possible conceptions. Comparing PTEs and TAs’ experiences learning
about RP suggests potentially different predispositions about RP: PTEs’ may have had prior
experiences learning about RP as a product that should be produced to verify the truth of a
claim, rather than a process of RP in which one engages to make meaning in mathematics
(Herbst, 2002; Usiskin, 1980). These prior experiences may predispose them to viewing proof
more procedurally and with a different purpose than that recommended by the Standards
documents. In a process that is more in line with the Standards, however, mathematicians
typically construct a proof after extensive explorations of mathematical relationships,
determinations of meaningful patterns, formulations of conjectures based on the patterns,
tests and revisions of their conjectures, and the production of informal arguments that
convince them of the validity of the conjectures (de Villiers, 1998; Lakatos, 1976; Stylianides,
2009; Usiskin, 1980). In mathematical research these RP activities (e.g., exploring and
conjecturing) may be more educational than the proof itself.
TAs may experience these process-oriented aspects of proving as they engage in
mathematical research or work with others solving mathematics problems, but these
experiences typically occur outside of the mathematics classroom (Raman, 2003). TAs’ previous
and current experiences as mathematics learners are primarily in classrooms taught in a
traditional, teacher-centered, lecture format (Speer et al., 2010). Based on their apprenticeship

19

of observation (Lortie, 1975), therefore, TAs are more likely to decide to implement more
teacher-centered instructional strategies (e.g., a model that may use primarily lecture-based
instruction with little-to-no student-to-student interactions) because they have experienced
success learning in classrooms that were more teacher-centered (Grossman, 1989). These
potential views of mathematics teaching and learning are hypotheses. To better understand
TAs’ classroom practices, this study uses interviews to investigate how TAs talked about their
conceptions of RP in mathematics, teaching, and learning; thereby refining these theorized
views of TAs’ conceptions.
2.3 Frameworks for Analyzing Teachers’ Classroom Practices
To study TAs’ instructional decisions, this study investigates TAs’ classroom practices
when teaching PTEs about RP. This context is important because many colleges and universities
require PTEs to pass one or more mathematics courses designed for them, but there have been
few empirical studies of collegiate mathematics teachers’ conceptions and teaching practices
when teaching such courses (Cannata & McCrory, 2007; Lutzer et al., 2007, p. 53). In fact, little
research of collegiate mathematics teaching “has focused directly on teaching practice—what
teachers do and think daily, in class and out, as they perform their teaching work” (Speer et al.,
2010, p. 99). The absence of such research restricts the field’s understanding of collegiate
mathematics teaching and the nature of the opportunities for undergraduate students to learn
about mathematics in these courses; a research need this study addresses.
This study examined TAs’ classroom practices by investigating what opportunities they
provided for PTEs to experience RP and how PTEs actually engaged in RP processes during
instruction. Consistent with a learning-as-participating perspective on learning (Sfard, 1998),

20

the examination of TAs’ teaching practices around RP focused on the extent to which PTEs
participated in the creation and evaluation of generalizations and mathematical arguments.
From this perspective on learning, mathematical knowledge is seen as fluid and malleable, a
continually negotiated body of ideas that are each dependent on their context. Knowledge is
not seen as being acquired as personal property, but as the ongoing product of intellectual
doing (Sfard, 1998). In terms of a contemporary mathematics classroom, mathematical
knowledge is constructed and negotiated within the complex classroom space and shared by
students and teachers. Doing and learning mathematics, then, is not the quest for private
ownership; it is the request to participate in an evolving community of mathematical practice
(Lave & Wenger, 1991). To study how PTEs participated in practices of RP during classroom
instruction, the extent to which PTEs were involved with generating and critiquing
generalizations and justifications was examined. The Mathematical Task Framework and then
the RP-framework, which guide the methodological approach, are described next.
2.3.1 Focusing on RP-related Classroom Tasks
Mathematics learners “develop their sense of what it means to ‘do mathematics’ from
their actual experiences with mathematics, and their primary opportunities to experience
mathematics as a discipline are seated in the classroom activities in which they engage”
(Henningsen & Stein, 1997, p. 525, citing Schoenfeld, 1992, 1994). Henningsen and Stein
describe such classroom activity oriented toward the development of a particular mathematical
idea as a mathematical task. Rich, mathematical instructional tasks are central to students’
learning because “tasks convey messages about what mathematics is and what doing
mathematics entails” (NCTM, 1991, p. 24). The nature of tasks in which mathematics learners

21

engage can shape how they think and can serve to limit or broaden their views of mathematics
(Doyle, 1983).The investigation of PTEs’ opportunities to learn about and experience RP in this
study is thus framed by the concept of mathematical tasks.
Drawing upon Henningsen and Stein’s (1997) conceptualization of mathematical tasks, a
segment of classroom activity oriented toward the development of a specific mathematical idea
is considered as a mathematical task. In the textbook used for the course in this study
(Beckmann, 2008), for instance, a question asked students to generate a conjecture about the
measure of vertical angles formed where two lines intersect. The next question in that section
asked students to justify their conjecture by proving the relationship always has to be true.
These questions provided opportunities for students to generate and justify conjectures about
angles formed by two lines, and constitute one task.
Henningsen and Stein’s (1997) framework (Figure 2) depicts the progression of
mathematical tasks from the written textbook or lesson plan, to the way the teacher introduces
the task in the classroom (task set up), and how students and teachers act and interact with the
task (task implementation). Henningsen and Stein focused on how the cognitive demand of
tasks was maintained or changed in how the teacher introduced and interacted with students
around the task. High-level cognitive tasks were typically those that required students to think
conceptually and make connections or do mathematics. During the implementation of such
tasks, however, various factors were identified that contributed to the teachers lowering the
cognitive demand of these tasks, such as time constraints and the participants shifting the focus
onto correct answers instead of process-oriented solutions.

22

Mathematical Task

Mathematical Task

Mathematical Task

As represented in
textbook

As set up by the
teacher in the
classroom

As implemented by
the teacher with the
students

TeacherRelated Factors
Influencing
Task Setup

Student
Learning

Teacher- and
Student-Related
Factors
Influencing Task
Implementation

Figure 2. Framework for studying TAs' implementation of proof-related tasks
(Modified from Bieda, 2012; Henningsen & Stein, 1997)
Henningsen and Stein characterized the cognitive activity of doing mathematics as
“including complex mathematical thinking and reasoning activities” (1997, p. 532) such as
creating and evaluating conjectures, looking for patterns, justifying, explaining, and solving nonroutine problems. Since the processes of RP include exploring mathematical relationships,
generating conjectures, revising conjectures, and producing mathematical justifications, tasks
that engage students in this work of RP would constitute a specific subset of the doingmathematics tasks.
Bieda (2010) applied this notion of proof-related tasks as a subset of the doingmathematics tasks when she extended Henningsen and Stein’s (1997) work. Bieda studied the
nature of teacher-student interactions when middle school teachers enacted proof-related
tasks by observing their classroom practices. Focusing on the shaded portions of Figure 2, her
study depicted the opportunities to prove and characteristics of the pedagogy of proving in the
middle school classrooms observed. Bieda found that time constraints and insufficient feedback
23

from teachers regarding students’ conjectures and justifications were major factors in lowering
the cognitive demand of proof-related tasks. Bieda inferred that these results indicated the
participants needed professional development and curricular support to help them develop
their students’ understandings of justification and proof in more mathematically robust ways.
I extended Bieda’s work by applying the Mathematical Task Framework to examine the
nature of opportunities for RP in a mathematics course for PTEs. To study the classroom
practices, I specifically examined the opportunities for PTEs to engage in RP in the tasks as
written (in the textbook), as set up (that is, as introduced by the TA), and as implemented (how
PTEs engaged in RP as they worked on the task).
2.3.2 Reasoning-and-Proving Framework
Investigating the types of RP opportunities in written, set up, and implemented tasks,
required a framework for studying RP opportunities in a textbook and in classrooms. Gabriel
Stylianides’s (2009) analytic framework (Table 2) was used because he designed this framework
for analyzing RP opportunities in curricula, and suggested its utility for analyzing classroom
practices. This framework was particularly helpful because it identified how students were
asked to engage in RP by considering processes of RP employed in the mathematics discipline
(i.e., a subset of doing mathematics from the Mathematical Task Framework). Since the activity
of RP involves more than generating a proof, examining a textbook by looking for where in the
table of context or index words like proof or prove yielded a narrow view of the opportunities
for students to engage in RP in previous textbook analyses (McCrory et al., 2008). This
framework, contrarily, offered a systematic method for understanding students’ opportunities
to explore mathematical relations, generate conjectures, and produce justifications.

24

Table 2. Reasoning-and-Proving Framework (adapted from Stylianides, 2009)
Reasoning-and-Proving
Mathematical Generalizations
Investigate
Mathematical
Relations
Conjecture
Evaluate Claim
Dimension 1
 Explore
 Make a
 Evaluate
Components &
Provided
Conjecture Mathematical
Subcomponents of
Examples
Claim
RP
 Generate
Examples to
Explore
Dimension 2
Purposes of
Investigations,
Conjectures,
Evaluations, &
Proofs

 Conjecture
 Proof
 Conjecture
Precursor
Precursor
Precursor
 Conjecture
 Proof Non-  Proof Precursor
Non-Precursor
Precursor
 Proof NonPrecursor

Dimension 3:
Meta-level:
Reflecting on proof

Mathematical Arguments
Evaluate
Provide a NonArgument
proof Argument Provide a Proof
 Evaluate a
 Empirical
 Proof
Mathematical Argument
Argument
 Rationale

 Refine a proof
 Refine an
argument
 Justifications
for validity of
an argument
or proof



25

Defining Proof
Identifying Proof

 Explanation
 Verification
 Generation of
New
Knowledge
 Falsification

Applying the RP-framework (Table 2) provided a language by which one could talk about
what specific RP opportunities were in the textbook and in the task as set up in the classroom,
as well as how students engaged in these RP processes when the task was implemented. The
advantage of using this framework was that it integrated the types of RP processes in which
students may engage (Dimension 1) and how students may engage in a range of RP processes
(Dimension 2). By considering the work students would have to do to successfully complete a
task, this framework identified a task as related to RP if it provided opportunities for students
to create mathematical generalizations or mathematical arguments (Dimension 1). That is, the
activities of investigating mathematical relations (through which students may identify patterns
for a given set of data or examples) and generating conjectures were considered under the
overarching category of “making mathematical generalizations”—the transporting of
mathematical relations from given sets to new sets for which the original sets are subsets
(Polya, 1954). The activities of providing nonproof arguments (such as rationales or empirical
arguments) and generating proofs were considered under the overarching category of
“generating mathematical arguments” though which students could have opportunities to
provide support to mathematical claims.
Dimension 2 considered the purposes that investigations, conjectures, and proofs may
serve as students engaged in a particular task, depicting situations where, within a particular
task, a mathematical generalization could give rise to a proof and what roles of proof may be
emphasized. For example, in mathematics, investigating mathematical relations can result in
conjectures, which could lead to the production of proofs (e.g., Polya, 1954). This framework
provided a means for understanding to what extent this structure was found in mathematics

26

textbooks and in classroom instruction. Furthermore, the roles of proof in the framework were
drawn from the same set of primary functions of proof in mathematics (Table 1) upon which
Knuth (2002a, 2002b) relied when studying teachers’ conceptions of proof. Incorporating these
same roles of proof into this framework provided a means of examining how PTEs experienced
different roles of proof in their classrooms, while paralleling the examination of TAs’
conceptions of the nature and purposes of RP.
Since the study’s context focused on TAs teaching PTEs, to analyze the opportunities for
PTEs to engage in RP in the textbook it was important to also consider the specialized content
knowledge PTEs need to develop to be prepared to teach about RP. This specialized content
knowledge included additional aspects about RP to which Stylianides’s (2009) original
framework, which focused on what K-12 students had opportunities to learn about RP, did not
attend. Furthermore, Stylianides (p. 283) suggested that the framework could be used to
analyze classroom instruction and the alignment between the written and implemented tasks
through the lens of RP. Presently there are no peer-reviewed studies that have done so.
In light of using this framework in a different context (i.e., at the collegiate level with a
textbook for PTEs) and extending its use by also examining the task set up and implementation
during classroom practice, the RP-framework used in this study was modified in three ways:
First, the framework was modified to include two additional RP processes (grey columns, Table
2) to be more consistent with the Mathematical Task Framework and with the work around RP
that PTEs would be expected to do as teachers in the future. The activity of evaluating
conjectures needed to be incorporated because it can be part of the RP process and it was one
of the criteria for doing-mathematics in the Mathematical Task Framework. Opportunities to

27

evaluate mathematical claims or arguments could also lead to opportunities for students to
refine or produce conjectures or justifications. Moreover, in their future classrooms, PTEs will
be expected to routinely, often in real time, evaluate mathematical claims or arguments
proposed by students, textbooks, or other outside authorities (Ball et al., 2008). Therefore
categories for evaluating mathematical claims and evaluating mathematical arguments, which
were not explicit in the original framework, were added to the RP-framework.
Second, to be able to apply the framework both in analyzing written mathematical tasks
and their implementation, it was important to focus on how students might engage with the
mathematics of the task when considering the subcomponents of Dimension 1 of the
framework. Specifically, the subcomponents of one of the categories, investigating
mathematical relations, were modified. According to Stylianides (2009), students investigate
when they explore a mathematical situation, example, or data to examine mathematical
relationships. Through such investigations, they determine mathematical relations they think
hold for that situation. Stylianides, however, distinguished between plausible relations and
definite relations as the subcomponents for this category. He coded for questions that
represented situations for which it was not mathematically possible to provide conclusive
evidence for a specific mathematical relation over other relations (plausible) and those for
which it was possible to determine mathematical relations that fit the situation or data given
the information in the problem (definite). Since the focus of the study was on the
implementation of RP tasks, it was helpful to consider how students may engage in these
explorations rather than the type of mathematical relationship they could determine.
Therefore, the RP-framework was modified to distinguish between two ways students may

28

engage in these explorations of mathematical relationships: either (a) investigating provided
examples/data or (b) generating their own examples/data, to identify mathematical relations.
Third, the original framework focused on the production of generalizations and
arguments, but Hanna and de Bruyn (1999) noted that problems or statements can also discuss
characteristics or features of proof and proving instead of asking students to do proof. I added a
third dimension to the framework to identify such opportunities for students to reflect on what
constitutes a proof. These opportunities could occur after students evaluate or provide a
mathematical argument, or they could occur separate from these other dimensions of the
framework. This third dimension added a meta-level layer that was not included in the original
framework, which pointed out opportunities where students could articulate how they know an
argument is or is not a proof. It is important for students and teachers at all levels to be able to
define and identify proofs and nonproofs (Stylianou, Blanton, & Knuth, 2009). The two
subcategories for this third dimension incorporated these notions of characterizing and
identifying proofs and nonproofs:




Characterizing Proof: tasks that focus on discussing the characteristics of proofs, which
could include some or all of the italicized portions of the following characterization—a
proof is a mathematical argument that is general for a class of mathematical ideas and
establishes the truth of a mathematical statement based on mathematical facts that are
accepted or have been previously proven (Knuth, 2002a; Steele & Rogers, 2012;
Stylianides, 2007c).
Identifying Proof: tasks that focus on talking about how one can distinguish proofs from
nonproofs, which could include discussions about features of an argument that help
students recognize proof across a variety of representational forms based on
mathematical criteria (Knuth, 2002a; Steele & Rogers, 2012).
This notion of identifying proof was unique from RP processes of evaluating proofs in

the sense that the later was about doing the evaluating and the former was about recognizing

29

and discussing how to evaluate or distinguish between proofs and nonproofs. This third
dimension offered an opportunity for students to step back from the process of creating
generalizations and arguments so they could think about these proving processes. Providing
opportunities for PTEs to articulate what makes an argument a proof or to reflect on the
proving process could help prepare them for helping others understand what constitutes a
proof. Furthermore, since literature indicated that PTEs are often more convinced by empirical
arguments than by proofs (Martin & Harel, 1989), emphasizing the generality of proof in
situations where PTEs were experiencing the proving process could provide opportunities to
counteract this viewpoint. This dimension, therefore, added a way to categorize if/how the task
in the textbook or classroom instruction provided opportunities for PTEs to reflect on
characteristics of proof and proving instead of to do proof.
This modified RP-framework (henceforth, referred to as the RP-framework) took into
consideration RP processes and ways of thinking about proof that are particularly important to
emphasize with PTEs. By using this framework to study tasks as written and enacted from a
textbook for PTEs, this work contributes to mathematics education research by applying and
refining this framework for use in studying collegiate mathematics teaching.
2.4 Research Questions
This study aims to understand TAs’ teaching practices and conceptions when they are
teaching PTEs about RP. This investigation supports both deeper understandings of the work of
teaching at the collegiate level and the possibilities for improving how PTEs are taught about RP
in their mathematics content courses. One way to approach such a study is to consider a bestcase scenario. That is, where the PTEs’ course is using a textbook with a wide-range of proving

30

opportunities, taught by TAs who have broad conceptions about RP who are also participating
in concurrent professional development. In such a scenario, the instructors for the PTEs’ course
are positioned to potentially make a positive impact on the mathematical preparation of PTEs
because of TAs’ strong knowledge base about RP. The use of a textbook with a range of RP
opportunities and professional development could also help support TAs in implementing
instructional strategies that engage students in the learning process (Wyse, 2010). Within such
a scenario, the aim of this research is to examine how RP-related tasks are enacted in PTEs’
mathematics classrooms and the nature of the RP conceptions that TAs emphasize when they
talk about their enactment of RP-related tasks by investigating the following questions:
1. What are TAs' conceptions about the nature and purposes of RP in mathematics,
teaching, and learning?
2. How do TAs enact RP-related tasks in a mathematics content course for PTEs?
3. In what ways, if at all, does understanding TAs' conceptions about RP in mathematics,
teaching, and learning illuminate their instructional decisions regarding the
enactment of RP-related tasks?
Since teachers’ beliefs have been implicated as a major factor influencing teachers’ practices,
research question one (RQ1) investigates TAs’ conceptions about RP to PTEs. To understand
more about PTEs’ experiences learning RP, RQ2 examines how PTEs' opportunities to engage in
RP were or were not maintained during instruction. RQ3 then investigates potential
relationships between TAs’ conceptions and their teaching practices around RP. This research
informs the pedagogy of collegiate mathematics instruction for PTEs at a time when improving
the RP-related preparation of PTEs is crucial.

31

CHAPTER 3
METHOD
This study focuses on TAs’ conceptions of RP and how they engaged PTEs in RP during
classroom instruction. This mixed-methods study features a textbook analysis and a case study
design, using observations and semi-structured interviews. The multiple case study design
allows for investigation of descriptive research questions where the collective cases are
selected because, together, they help illuminate a more general phenomenon (Stake, 2005).
3.1 Context
TAs in this study were teaching a geometry and measurement content course (referred
to as MTH 2) for PTEs during one semester at a public university. MTH 2 plays an important role
in the elementary teacher preparation program at this university. Approximately 350 students
enroll in MTH 2 every school year, and (in addition to other programmatic requirements) PTEs
must pass MTH 2 to graduate. Therefore, this course represents a significant milestone for PTEs
and an opportunity for TAs to positively impact the mathematical perspectives and knowledge
of numerous PTEs. There are also important features of MTH 2 that support the purposes of
this exploration into TAs’ conceptions and teaching practices: the goals and design of the
course, as well as who teaches the course and how they are supported.
3.1.1 The Design and Goals of the Course
Stated on the course syllabus, one of the three primary goals for the course is to help
PTEs “reason mathematically and do proofs in the context of geometry and measurement”
(MTH 2 Syllabus, p. 1). Moreover, the textbook (Beckmann, 2008) aims to promote the
development of mathematical thinking and reasoning skills in PTEs. It is organized around

32

activities intended to promote engagement, exploration, and discussion. Furthermore,
Beckmann explicitly describes goals of the textbook as intending to help PTEs to “explain why
mathematics works the way it does” (p. xix, italics added), make sense of mathematics, and
carry those abilities into their future classroom. Therefore, the course goals and textbook goals
align with national recommendations for RP and support the focus of this study.
3.1.2 The TAs
At this university, graduate students who are pursuing mathematics or mathematics
education degrees are appointed to teach the course as sole instructors (TAs) for MTH 2. A
mathematics-education faculty member who also coordinates regular professional
development meetings with all the MTH 2 TAs supervises them. These course meetings include
opportunities for TAs to discuss administrative issues, instructional strategies, mathematical
content, and student learning and they are encouraged to collaborate with their colleagues
when designing assessments.
Participating in this study were the six TAs who were assigned to teach the course
1

during one semester: Ace, Laura, Isaac, Peter, Evan, and Kelly. They were assigned to teach
classes that met either for 80-minutes twice a week (Ace, Laura, and Isaac) or for 50-minutes
three times a week (Peter, Evan, and Kelly), a total of three contact hours per week. The TAs’
backgrounds and prior experiences teaching MTH 2 are summarized in Table 3. By investigating
TAs pursuing mathematics and mathematics education degrees, these data may be fairly
representative of the different TAs assigned to teach courses like MTH 2.

1

All names are pseudonyms to protect confidentiality
33

Table 3. Participants' Background Information
Ace

Degree
Pursuing

Laura
Applied
Math PhD Math
Masters

Isaac

Peter

Math PhD

Math PhD

Evan
MathEducation
PhD

Grad School
1
2
3
6
3
Year
International
Domestic Domestic International International Domestic
/Domestic
Prior
Experience
Teaching
0
0
0
0
1
MTH 2
(semesters)

Kelly
MathEducation
PhD
3
International

1

This course setup and textbook presented an opportunity to study a scenario where
these college mathematics instructors are more likely to have opportunities to use innovative
RP tasks in their teaching. Since these TAs are responsible for designing their classroom
instruction (with some guidance from a course supervisor), this course also provides an
opportunity to examine TAs’ classroom practices and conceptions about teaching.
3.2 Instruments and Data Sources
To address the research questions about TAs’ conceptions and teaching practices,
qualitative data were gathered from two series of interviews, six hours of classroom
observations of each TA, and a preobservation survey. A textbook analysis was also conducted,
which guided the selection of classroom observations. The sections that follow describe the
design of the instruments and link the data sources to the research questions.
3.2.1 Data Sources for Studying TAs’ Conceptions of Reasoning-and-Proving
The primary data source for investigating TAs’ conceptions of the nature and purposes
of RP in mathematics, teaching, and learning (RQ1) was a series of interviews, consisting of a

34

pre-semester interview, beginning-of-the-semester interview, and end-of-semester interview
(Table 4). These interviews provided TAs with opportunities to talk about RP in different
contexts. The pre-semester interview protocol (Appendix A.3) was adapted from interviews
used to study secondary teachers’ views of the roles of proofs in mathematics by Knuth (Knuth,
2002a; Steele, 2006; Steele & Rogers, 2012). It was designed to elicit responses about how TAs
learned about RP (Part I) and conceptualized RP in the mathematics discipline (Part II).
Table 4. Data Sources for Research Question 1: What are TAs' Conceptions of the Nature and
Purposes of RP in Mathematics, Teaching, and Learning?

TAs’ Conceptions of:
RP in Mathematics
RP in Teaching
RP in Learning

Pre-Semester
Interview


Beginning-ofSemester Interview





End-of-Semester
Interview




The beginning-of-the-semester interview (Appendix A.4) focused on TAs’ conceptions of
RP in mathematics teaching. The protocol was adapted from interviews used by researchers
examining secondary teachers’ views of the roles of proofs in teaching and learning (Knuth,
2002b; Steele & Rogers, 2012) and tertiary teachers’ beliefs about teaching and learning
(Gutmann, 2009; Wyse, 2010). The questions elicited TAs’ conceptions about the purposes of
proof in teaching and their views of how and why PTEs learn about RP (Table 4).
The end-of-semester interview (Appendix A.5) provided additional opportunities for TAs
to talk about their conceptions of RP in mathematics, teaching, and learning for two reasons.
First, after teaching MTH 2, TAs could have more examples to draw upon to further articulate
their views of RP in mathematics, teaching, and learning than they did during the earlier
interviews. Therefore, a majority of the interview protocol asked questions similar to those
from earlier interviews or asked about TAs’ experiences teaching RP in MTH 2 to elicit
35

responses from TAs about their conceptions of RP. Second, the last two questions of the
interview were included to inquire about how TAs’ views of RP in mathematics, teaching, and
learning relate to the prevalent views of RP from mathematics education literature that
underscore this study. Specifically, TAs described their familiarity with and responded to the
NCTM’s (2000) recommendations about the role of RP in school mathematics, and they
identified which roles of proof they most resonated with, and why. These questions were
included only at the end of the end-of-the-semester interview to reduce the chance that
introducing constructs from mathematics education literature could influence TAs’ decisions or
responses during the study. TAs’ responses to these questions corroborate and further enhance
the data about their views of RP from earlier interviews.
3.2.2 Data Sources for Studying How TAs Enacted Reasoning-and-Proving Tasks
The primary data source for investigating how TAs enacted RP tasks in MTH 2 (RQ2) was
six hours of classroom observations for each TA across the semester. These observations aimed
to understand how TAs set up RP tasks and in what RP processes PTEs engaged during the
implementation of tasks (Figure 3, shaded portions). Additional data sources included an
analysis of the written tasks in the textbook and a preobservation survey.
Applying the RP-framework (Table 2), the textbook analysis characterized the RP
opportunities afforded by the tasks in the Mathematics for Elementary Teachers’ text
(Beckmann, 2008). The main goal of this analysis was to identify tasks that provided
opportunities for PTEs to engage in different aspects of RP processes so that classroom
observations of TAs implementing a range of RP opportunities could be planned. This textbook
analysis also provided groundwork for understanding the enactment of those tasks.

36

Task as Set Up

Task as Implemented

Opportunities for
RP in textbook
(Beckmann, 2008)

Opportunities for
RP when TA set
up task in the
classroom

How PTEs engage in
RP when task is
implemented

Textbook Analysis

Preobservation
Survey

Written Task

Classroom Observations: Video recordings,
field notes, narratives, & classroom artifacts

Figure 3. Data source for Research Question 2: How do TAs enact RP tasks in a
mathematics content course for PTEs?
Prior to each classroom observation, TAs responded to preobservation survey questions
(Appendix A.2). These questions were guided by a preobservation survey modified from Bieda
(2010) to collect data about TAs’ goals and expectations for the lesson and the mathematical
tasks they identified as central for the lesson. This measure was designed to provide context to
the observed lesson and document the extent to which RP tasks were considered by the TA to
be essential components of the observed lesson.
Each classroom observation was video recorded and field notes were taken by attending
to TAs’ classroom teaching related to RP. The field notes were guided by an observation
protocol modified from Bieda (2010) to focus on the TA and PTEs’ actions and interactions
around RP tasks. This protocol (Appendix A.1) documented how each task was implemented
along the dimensions of the Mathematical Task Framework (Stein, Grover, & Henningsen,
1996): task features in the written text, task features as set up by the TA, cognitive demands as
set up by the TA (e.g., how TAs’ instructions compared with the actual problem instructions),
classroom norms evident during implementation, TAs’ and PTEs’ instructional habits during
implementation, task features as enacted, and cognitive demands of the task as enacted.
37

Drawing from Ethnographic research (Erickson, 1986; Goetz & LeCompte, 1981), after
each observation, a narrative account was created to describe what happened during the
enactment by answering the observation protocol questions and referring to the field notes
and classroom video. The narratives documented the enactment of RP tasks as a first step in
organizing the observational data by focusing on TAs and PTEs’ interactions. As outlined in the
observation protocol (Appendix A.1), aligned with the Mathematical Task Framework, specific
questions focused on documenting how TAs set up the task (Questions 2, 2a, 2b, and 2c) and
how they implemented RP tasks (Questions 3 – 10). The narrative attended to how TAs set up
RP tasks and how PTEs engaged in RP processes during implementation.
3.2.3 Data Sources for Studying How TAs’ Conceptions Illuminate Their Instructional Decisions
The primary data for investigating how TAs’ conceptions of RP illuminate their
instructional decisions around RP (RQ3) were a series of interviews conducted after the
classroom observations (Figure 4).
TAs’ Conceptions of
Reasoning-and-Proving

Post-Observation Interviews

Written Task

Task as Set Up

Task as Implemented

Opportunities for
RP in textbook
(Beckmann, 2008)

Opportunities for
RP when TA set up
task in the
classroom

How PTEs engage in
RP when task is
implemented

Figure 4. Data source for Research Question 3: In what ways, if at all, does
understanding TAs’ conceptions of RP illuminate their instructional decisions
regarding the enactment of RP tasks?

38

The semi-structured post-observation interview protocol (Appendix A.6) was adapted from an
interview used by Bieda (2010). It asked TAs to reflect on the lesson, articulating why they
decided to carry out certain pedagogical moves during the implementation of RP tasks, their
learning goals for choosing to implement RP tasks, and how they view RP content from the
lesson as relevant for their students as future teachers.
3.3 Data Collection
Data from the classroom observations, preobservation surveys, and two series of
interviews were gathered at specific points throughout the semester, with the textbook
analysis being completed before data collection. In the sections that follow, I detail the process
for analyzing the textbook, procedures for the preobservation survey and classroom
observations, and procedure for collecting and administering the interviews.
3.3.1 Textbook Analysis
The Geometry and Measurement chapters of Beckmann’s (2008) textbook (Chapters 8,
9, 10, and 11) were analyzed for RP opportunities for PTEs. These chapters addressed geometry
and measurement content, including visualization, angles, two-dimensional shapes, threedimensional shapes, constructions with a straightedge and compass, the geometry of motion
and change, measurement, and understanding relationships for area and volume.
3.3.1.1 Textbook Analysis: Unit of Analysis
Within these four chapters, each separate question was analyzed using the RPframework (Table 2). The term question means that each problem, written question, or “parts
thereof that have a separate marker” (Stylianides, 2009, p. 270) in the textbook was coded,
with the exception of homework problems. Since this study was ultimately concerned with

39

what happened during classroom instruction, the questions listed as practice problems and
questions in the activity manual were analyzed because these questions formed the basis of the
instructors’ classroom teaching. Using the RP-framework, questions were identified that were
designed to offer PTEs at least one opportunity for RP; these questions are referred to as RP
questions (RPQs).
Since a set of questions oriented toward the development of a mathematical idea
constitutes a mathematical task, the questions, by themselves, were not always what one
would consider to be a mathematical task (Stein & Lane, 1996). Therefore, the coding was
aggregated to take this into account. The textbook contained 115 activities in the activity
manual related to the four chapters addressed by MTH 2. The number of questions within each
activity ranged from one to eight, but each activity was oriented toward the development of a
particular mathematical idea and was regarded as a separate mathematical instructional task.
Therefore, if an activity had at least one RPQ then it was regarded as a RP-related task (RPT).
3.3.1.2 Textbook Analysis: Coding Procedure
Questions were analyzed in three stages. First, questions were identified that had RP
potential based on a search for the following words: prove, deduce, conjecture, predict, justify,
verify, show, apply (or use a mathematical fact), convince, and explain why (Bieda, 2010;
Stylianides, 2009). This process was not intended to identify all possible RP-related questions in
the text, but questions where RP was more likely to be a primary mathematical component.
Second, potential RPQs were coded using the categories and subcategories in the RPframework (Table 2). Definitions for each subcategory (detailed further in Stylianides, 2009)
guided the coding, and a single question could be coded with multiple categories from each

40

dimension. Dimension 1 included the activities of investigating mathematical relations,
generating or evaluating conjectures, evaluating arguments, and generating proofs and
nonproofs. Dimension 2 noted situations where explorations into mathematical relations may
serve as precursors for a mathematical generalization, which may then serve as a precursor for
a proof. In Dimension 3, instances for PTEs to reflect on the characteristics and features of
proofs were identified. If a question did not fit the criteria for these dimensions, it was rejected
and coded as not RP-related. For questions that provided opportunities for PTEs to engage in
RP, they were typically coded in order of the dimensions. However, a question could be
identified as providing opportunities for solvers to engage in Dimension 3 only, though this
occurred infrequently. The aggregation of questions by task also identified instances where RP
components served as precursors for other generalizations or arguments within tasks.
Since this was an analysis of tasks that can have various solution paths, the coding
scheme analyzed the questions using reasonable inferences about the expected solution path
for which each problem may have been designed. Factors useful in identifying expected
solution paths included: (a) any approach suggested by the wording of the question in the PTEs’
text, (b) the approach or solution provided in instructors’ supplemental materials, and (c) what
PTEs may be expected to know and understand when encountering the question based on
where it is positioned in the textbook.
In the third phase of the textbook analysis, the activities that were strongly
recommended or encouraged by the course supervisor were used as a reference to identify any
additional RP opportunities in tasks TAs were likely to implement. These recommended
activities were analyzed using the same procedure outlined in the description of phase two to

41

identify any additional RPQs. Throughout Phases 2 and 3, two trained raters conferred to refine
the descriptions and procedure for applying the RP-framework. Both raters coded a random
sample of 31% of the RPQs; agreement was reached on 81% of these codes with all
disagreements resolved through discussion.
3.3.2 Collecting Data about TAs’ Teaching Practices
The textbook analysis identified tasks that embodied different aspects of RP; these tasks
were then located in the course syllabus to identify opportunities to observe TAs enact RP
tasks. The observation schedule was designed to observe each TA’s classroom practices for at
least six hours across the semester, and to see that TA implement a variety of RP tasks.
Table 5. Classroom Observation Details for each TA
TA
Ace
Laura
Isaac
Peter
Evan
Kelly

Class Time
Observed
6 hours 40 minutes
8 hours
8 hours 30 minutes
8 hours 20 minutes
6 hours
6 hours

Class Periods
Observed
5
6
7
10
7
7

Sections of
Textbook Observed
8
9
9
8
9
6

RP Tasks
Implemented
13
13
14
16
13
13

Due to the different time configurations of TAs’ sections, the number of observations of each
TA varied (Table 5) to see their classroom for a minimum amount of time and to observe them
implement at least 13 RP tasks. Prior to each classroom observation, TAs responded to
preobservation survey questions.
3.3.2.1 Preobservation Survey Procedure
Prior to each classroom observation, TAs received a short preobservation survey via
email to elicit information about the extent to which RP was central to their goals and
expectations for the lesson (Appendix A.2). TAs responded in writing or orally. TAs were asked
42

to complete the survey before teaching the lesson in an attempt to more accurately capture
their anticipations for the lesson than would be likely if they were asked these questions after
teaching. Since TAs at this institution often plan right before class (Cervello & Steele, 2009),
asking these questions in an interview setting before class was not feasible.
3.3.2.2 Classroom Observation Procedure
Each classroom observation was video recorded and field notes were taken to
document the enactment of RPTs. TA’s and PTEs’ actions and interactions around RP were the
focus of the observation. After each observation, a narrative account was created to describe
what happened during the implementation by first detailing chronologically how the class time
was used and what mathematical tasks and topics were introduced, and then answering the
observation protocol questions using the field notes and classroom video as a guide.
Consistent with a learning-as-participating perspective (Sfard, 1998), the responses to
the observation protocol questions about implementation of RP tasks considered how PTEs
participated in the creation and evaluation of generalizations and justifications. A PTE making
RP information public to the entire class or a majority of PTEs participating in RP activities was
regarded as evidence of PTEs engaging in RP processes. The observation focused on how the
whole class experienced RP processes and how these thought processes and ideas were shared
among the class. The narrative accounts, therefore, noted what information or aspects of RP
was made available to the entire class and how it was made available to them.
3.3.3 Collecting Interview Data
The procedure for conducting the two series of semi-structured interviews was the
following: interviews were conducted with individual TAs, audio-recorded, transcribed, and

43

verified. The first series of interviews inquired about TAs’ conceptions of RP, and they were not
associated with a specific classroom observation. The three interviews associated with this first
series were named relative to when they were given: Pre-semester (during the days leading up
to the start of the semester), beginning-of-the-semester (during the first three weeks of
classes), and end-of-semester (during finals week or the week following finals week). The first
two interviews were administered before TAs explicitly taught about proof to study their
conceptions of RP before they designed and implemented RP lessons in MTH 2. The end-ofsemester interview was given after the TAs’ primary teaching responsibilities were concluded
for the semester. The first two interviews each lasted an average of 45 minutes and the end-ofsemester interview took an average of 75 minutes with each TA.
The second series of interviews, conducted after classroom observations, investigated
TAs’ instructional decisions related to RP. The variance in number of observations for each TA
led to some variation in the number of post-observation interviews. After observing lessons
where RP tasks were implemented, I conducted a post-observation interview the same day as
the observation (immediately following class, when possible). After consecutive observations of
the same TA, a post-observation interview was conducted following the last observation. Thus, I
conducted five post-observation interviews with both Ace and Laura, and six with each of the
other TAs. The interviews ranged from 22 to 64 minutes, with an average of 44 minutes.
3.4 Data Analysis
The descriptions of the data analysis mirror the structure of the previous section: The
analytical approaches for understanding TAs’ enactment of RP tasks, studying TAs’ conceptions
of RP, and then examining how those conceptions illuminate their instructional decisions.

44

3.4.1 Analyzing TAs’ Enactment of RP Tasks
The narrative accounts written after each classroom observation were compiled and
analyzed to examine TAs’ enactment of RPTs. Enactment of RPTs considered the RP
opportunities for PTEs during classroom instruction of the task by examining both the setup and
implementation phases. RP opportunities were defined as written or verbal tasks that asked
PTEs to engage in one of the RP processes from the framework. The examination of the
enactment of these tasks was guided by the Mathematical Task Framework and RP-framework
as the narrative accounts were analyzed for what mathematical tasks were included in the
lesson and how TAs and PTEs engaged in the processes of RP during the set up and
implementation. The interpretation of the patterns observed in TAs’ classroom teaching was
also informed by considering their preobservation responses.
The analysis of the classroom narratives was modeled after the IQA protocol
(Matsumura et al., 2006), which has established reliability and validity for examining the set up
and implementation of tasks. For each observed lesson, the segments of the lesson that were
considered as the setup phase and the implementation phase for each task were identified.
Then, RP processes for the task-as-written, task-as-set-up, and task-as-implemented were
coded and analyzed. Finally, the enactment of each task was determined by taking into account
the analysis of the setup and implementation phases. In the four subsections that follow,
analyses of the task-as-written, task-as-set-up, and task-as-implemented are first described,
and then the procedure for analyzing the enactment of the task (to address the second
research question) is detailed.

45

3.4.1.1 Analysis of the Task-as-Written
The task-as-written consists of the words, instructions, problems, and diagrams
provided in the textbook or generated by the TA. When the task was taken from the textbook,
the opportunities for PTEs to engage in RP identified in the textbook analysis served as the
written potential for RP for the task. When the task was generated by the TA, the analysis of
the task occurred after the observation but used the same analytical approach as described for
the textbook analysis to determine the RP potential for the task.
3.4.1.2 Analysis of the Task-as-Set-up
The task-as-set-up, describes how the TA introduced the task. The setup phase ended
when PTEs put their pencils to paper to begin engaging with the questions in the task. This
analysis (see Appendix B.4) focused on what TAs said and did when introducing a task and
telling PTEs to begin working on a particular task; it also noted any background information or
examples the teacher may have provided to introduce the content to determine whether the
RP opportunities for a task were modified during the setup. Modifications in the RP
opportunities of a task could occur during setup when a TA increased or decreased the types of
opportunities for PTEs to work on RP. An increase in RP opportunities occurred when students
were expected to engage in an RP category not included in task-as-written. For example, a TA
might explicitly ask PTEs to provide rationales for their answers even though this request was
not included in the written task. A decrease in RP opportunities occurred when RP
opportunities in the task-as-written were omitted during the setup. Consider, for instance, a
task with four questions with only the first two relating to RP. A TA might ask PTEs to work on
the last two questions, thus removing all RP-potential opportunities during the task setup.

46

3.4.1.3 Analysis of the Implementation of the Task
The implementation of the task considered how TAs engaged in RP in their work on the
task. As PTEs engaged with the task, they could work individually, in small groups, as a whole
class, or some combination of these participatory structures. Following the IQA structure, PTEs’
engagement in RP both during individual or group-work time and during whole-class
discussions was analyzed separately, if both participatory structures occurred during the
enactment of a task. The overall analysis of the implementation of the RP task then aggregated
these two analyses to note what RP processes PTEs generated or evaluated across the
participatory structures. The coding schemes for the analysis of the work time and the wholeclass discussions are included in Appendix B (see Appendix B.5 and B.6). To study how PTEs
engaged in RP during individual- or group-work times, how most PTEs worked on RP for most of
the work time was analyzed. Also, how TAs talked with and guided PTEs in their work on RP in
the task was examined to understand how TAs modified or maintained RP demands of tasks. To
study PTEs’ engagement in RP during whole-class discussions (which could include teacher-led
or more discussion-based whole-class participation), how RP processes were discussed and
highlighted by the TA and PTEs was analyzed. This analysis considered how RP processes were
made public for the whole class and who provided the generalizations or justifications (the TA
or PTEs) to indicate how PTEs did or did not engage in the RP processes. The analysis of the
implementation of the task indicated how PTEs did or did not participate in generating or
evaluating mathematical claims or arguments during the class.

47

3.4.1.4 Analysis of the Enactment of the Task
The class activity around the task (i.e., the setup and implementation phases) was
regarded as the enactment of the task. Therefore, for each task the setup and implementation
analyses were considered to see whether or not PTEs engaged in the RP processes from the
task-as-written. This analysis of the enactment of the task could result in (a) an increase, (b) a
decrease, (c) a shift, or (d) no change in RP demands.
First, an increase in RP demands meant that PTEs generated or evaluated an RP process
not included in the RP potential of written tasks, indicated in one of two ways: (a) during the set
up there was an increase in RP opportunities and PTEs were held accountable for producing
that additional RP-category during the implementation; or (b) there was no modification to the
RP opportunities during setup, but during implementation PTEs produced or evaluated a
generalization or argument that went beyond the RP opportunities of the task-as-written.
Second, a decrease in RP demands referred to instances where PTEs did not participate
in one of the six major RP processes elicited in the task-as-written. A decrease in RP demands
occurred as a result of the following two scenarios: One, PTEs had fewer opportunities to
engage in RP than those identified in the task-as-written. In such a situation, during set up or
implementation, RP processes were excluded by a TA or by PTEs. Two, how PTEs participated in
the RP processes differed from the task-as-written because the TA generated RP processes
elicited in the task. In these situations, when introducing the task or during implementation the
TA generated one or more aspects of RP processes called for in the task instead of PTEs.
Third, a shift in RP demands details situations that were not an increase or decrease.
This shift in RP demands referred to instances where during set up or implementation PTEs

48

were asked to or actually did generate generalizations or arguments that were different from
the task-as-written, but the overall RP processes that PTEs engaged in were not different from
the task-as-written. This shift was noted in two types of scenarios: One, during the set up, an
increase or decrease in RP opportunities occurred but during implementation PTEs ended up
engaging in the RP processes expected in the written task. Therefore, the task was shifted in
regards to what PTEs were expected to do related to RP, but during class activity or discussion
PTEs produced the generalizations or arguments elicited in the task-as-written. Two, during
implementation PTEs engage in the same major RP processes identified in the written task but
they did so in a manner that was qualitatively different from what was expected in the written
task. That is, the types of RP processes PTEs engaged in was not increased or decreased, but the
mathematical content or level to which PTEs engaged in RP processes differed from the task-aswritten. For example, consider a task with two questions about properties of prisms. In the first
question, PTEs were expected to generate a conjecture about the shape of the faces of prisms
and provide a rationale for their conjecture. In the second question, they were asked to
generate a conjecture about the relationship between number of vertices, edges, and faces of
any n-sided prism. A shift of RP demands would occur if, during enactment, PTEs did not
provide a rationale for their conjecture in Question 1 but they provided a rationale for their
conjecture in Question 2. In this example, the RP demands were shifted but not in a way that
increased or decreased the aspects of RP in which PTEs engaged.
In the fourth case, no change in RP demands meant that there was no modifications in
the RP opportunities for the task and PTEs engaged in the RP processes that were elicited by
the task. Therefore, when none of the changes that were described in the first three cases

49

occurred, PTEs participated in the activities of RP that the written task called for and the
enactment resulted in no change in RP demands.
3.4.2 Analyzing TAs’ Conceptions of RP and How their Conceptions Illuminate their
Instructional Decisions
To address RQ1 and RQ3, interviews were transcribed and analyzed. To study the
measures for answering RQ1, focusing on TAs’ conceptions of RP in mathematics, teaching, and
learning, I analyzed the first series of interviews (pre-semester, beginning-of-the-semester, and
post-semester). To analyze the data for answering RQ3, considering how TAs’ conceptions
illuminated their instructional decisions, I examined each TA’s responses to the second series of
interviews (post-observation interviews). In light of the exploratory nature of the study, analysis
of interview transcripts followed a constant comparative method (Glaser & Strauss, 1967). This
inductive analytical approach was used as the study sought to promote understanding of TAs’
conceptions, not prove a preconceived theory. The analytical approach is detailed further for
the first series of interviews, then for the analysis of the post-observation interviews.
3.4.2.1 Analysis of TAs’ Conceptions of RP in Mathematics, Teaching, and Learning
The purpose of this analysis was to understand TAs’ conceptions of the purposes of
proof, of methods for teaching about RP, of why it is important to teach about RP, and of what
PTEs should learn about RP in MTH 2. The analysis of the first series of interviews began by first
identifying utterances where TAs talked about their views of RP within mathematics, teaching,
or learning (Appendix C). The goal in this first coding level was to note places where TAs talked
about RP in these general domains to then investigate how TAs talked about the nature and
purposes of RP within the domains. For example, Evan explained proof’s role in mathematics:

50

To establish new results and grow the field. . . . The field’s constantly growing because
of new results that are proven and new objects that can be formed with the results that
we have. And then things can be proven about those. . . . Role of proof in mathematics
so just establish new results and to build onto the field. (pre-semester, para. 72)
This utterance was coded as “mathematics” in the first level of analysis, but TAs did not always
explicitly distinguish between the three general domains. As previous studies of TAs’ beliefs
note (Speer, 2001), TAs made some statements that appeared to reflect their views of teaching
and learning. The intension was to identify these broad areas and faithfully capture TAs’
conceptions, so if a TA talked about teaching and learning together it was coded both as
“teaching” and “learning.”
Within these broad categories (as detailed further in Appendix C) the three transcripts
for each TA were coded for their views of (a) the purposes of proof in mathematics, (b) the
purposes of proof in mathematics classrooms, (c) the educational goal (that is why it is
important) for teaching about RP, (d) the learning goal (that is, what do people need to know)
about RP, and (e) how to teach about RP. The initial categories for items (a) and (b) were
informed by previous literature about the purposes of proof in mathematics and teaching
(Knuth, 2002a, 2002b; Steele & Rogers, 2012), introduced in Chapter 2 (Table 1). The other
aspects of the coding scheme, however, emerged from the data. That is, after initially coding
for these broad-level categories, a constant comparative method (Glaser & Strauss, 1967) was
used as follows: Within statements about RP in mathematics, how TAs talked about the role of
proof in mathematics was coded. The quote above where Evan described a role of proof in
mathematics received a sub-code of “discovery or creating new mathematical knowledge.”
Similarly, statements about RP in teaching were coded for how TAs talked about the role
of proof in teaching, why it was important to teach about RP (educational goal), and methods
51

for teaching about RP. Statements about learning were coded for how TAs talked about their
learning goals for RP (what they thought learners needed to know about RP). Within each of
these categories, utterances where TAs specified who the doer, teacher, or learner of RP was to
which they were referring were coded, as well. Noting the who was helpful because TAs were
teaching current undergraduate students who were also PTEs. This analysis noted to what
extent TAs attended to their students as undergraduate learners or future teachers.
Through an iterative process, the codes and subcodes were applied and refined.
Through this process, when transcript about the nature and purpose of RP could not be
categorized with existing codes and sub-codes, previously analyzed data was re-examined and
sub-codes were modified to reflect the data more accurately. There were a category of
responses that did not fit the original six codes, and as a result, a seventh category was added.
This category was a code about the general purpose of proof where TAs talked about proof’s
foundational role in mathematics or in mathematics teaching; it served as an important feature
of mathematics that PTEs need to understand. Evan, for example, answered questions about
the purposes of proof in mathematics by explaining that:
Proof is doing mathematics. I mean . . . you can’t have mathematics without proof as a
discipline. I mean that would be like you’re gonna go to a culinary arts school and get a
cooking degree and never use food. It’s an integral part of mathematics. (pre-semester,
para. 70)
All the transcripts were coded using the seven sub-categories listed in the coding
scheme in Appendix C. The other sub-codes within the teaching and learning categories
emerged from the cyclic analysis of the data described. The analysis of this first series of
interviews identified prevalent themes (represented in frequencies of the coded categories)

52

detailing TAs’ conceptions of the RP in mathematics, teaching, and learning to describe how
each TA emphasized particular roles of or aspects of proof.
3.4.2.2 Analysis of How TAs’ Conceptions Illuminate their Instructional Decisions
The purpose of this analysis was to examine the relationship between TAs’ instructional
decisions around RP and their conceptions about RP. Using the constant comparative method
(Glaser & Strauss, 1967), TAs’ responses to post-observation interviews were analyzed for how
they talked about their pedagogical choices after implementing RPTs in their classroom. This
analysis focused on TAs’ instructional decisions that directly impact PTEs’ opportunities to learn
about RP: (a) TAs’ selection of RPTs to teach and (b) TAs’ in-the-moment decisions around RP
during classroom instruction.
TAs’ responses about their goals for selecting RPTs for the lesson and their enactment of
RPTs were coded for themes about the extent to which and how RP was emphasized when
talking about their instructional decisions and goals during the enactment of the task. The first
level of the analysis identified where in the post-observation interviews TAs talked about their
selection of RPTs or about their in-the-moment decisions from the class. Within these sections
of the transcripts, the constant comparative method was used to investigate how TAs talked
about learning goals and contextual factors that contributed to their instructional decisions.
Some of the major themes that emerged through this cyclic, iterative coding process were (a)
RP content-related learning goals, (b) RP process-related learning goals, and (c) contextual
factors that informed their instructional decisions. The iterative process of coding, refining
codes, and re-coding was followed through this analysis to reveal patterns in how TAs’
conceptions about RP were related to their instructional decisions.

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CHAPTER 4
TEACHING ASSISTANTS’ CONCEPTIONS OF REASONING-AND-PROVING
This chapter focuses on TAs’ conceptions of RP, important ultimately for informing
professional development strategies for supporting mathematics instructors of future teachers.
Using the constant comparative method (Glaser & Strauss, 1967), TAs’ interview responses
were analyzed to determine their conceptions of RP in the contexts of mathematics, teaching,
and learning (RQ1).
There are two purposes for analyzing TAs’ conceptions of RP: First, although these
courses are a primary means through which PTEs’ knowledge for teaching mathematics is
developed (AMS & MAA, 2012), very little is known about the knowledge and beliefs of the
instructors assigned to these courses (Speer et al., 2010). This analysis describes the range of
conceptions these six TAs have about teaching and learning RP, thus addressing the research
need and first research question. Second, given the major role of teachers’ knowledge and
beliefs in influencing teachers’ instructional decisions (Schoenfeld, 2010), this analysis provides
an important starting point for understanding TAs’ instructional decisions around RP (RQ3);
explored further in Chapter 7. To present these results, this chapter first describes TAs’ views of
RP from the analysis of their responses to the first series of interviews. Then, prominent aspects
of individual TA’s conceptions of teaching RP are presented.
4.1 Analysis of All the TAs’ Conceptions of Reasoning-and-Proving
Based on the influence teachers’ conceptions have on classroom instruction (Putnam &
Borko, 1997), TAs’ success in providing opportunities for PTEs to understand RP during MTH 2
relies in large part on the nature of the TAs’ own conceptions of RP. To understand TAs’

54

conceptions of RP in mathematics, teaching, and learning (RQ1), I examined three aspects that
have particular implications in the context of teaching PTEs about RP: TAs’ conceptions of (a)
the purposes of proof both in mathematics and in teaching, (b) why and what to teach about RP
in MTH 2, and (c) how to teach about RP.
4.1.1 TAs’ Conceptions of the Purposes of Proof in Mathematics and in Mathematics Teaching
I studied TAs’ conceptions of the purposes of proof in both mathematics and teaching to
understand to what extent TAs have conceptions of proof that incorporate a variety of roles of
proof. Such conceptions are important so TAs can help PTEs develop a strong understanding of
multiple purposes of proof; thereby, preparing PTEs to bridge the gap between these roles of
proof and their manifestation in K-12 mathematics classrooms (Balacheff, 1991).
To understand the extent to which TAs articulate a variety of purposes of proof, I
analyzed their interview responses (from pre-semester, beginning-of-semester, and end-ofsemester) for how they talked about purposes of proof. Seven codes were applied (Table 6),
one of which was a general purpose emerging from the data and the others were based on
previous literature detailing purposes of proof (Knuth, 2002a, 2002b; Steele & Rogers, 2012).
For the coding process I attributed a code to phrases, sentences, paragraphs, or sections of the
transcript and TAs’ statements were given no code, one code, or multiple codes. Statements
coded as evidence of purposes of proof were aggregated to portray the degree to which each
TA attended to different purposes of proof. This method assessed how TAs talked about
different purposes of proof in mathematics and mathematics teaching. TAs described proof as
verification, falsification, explanation, communication, discovery, systemization, and in general
fundamental to mathematics (Table 6).

55

Table 6. TAs’ Statements about Purposes of Proof in Mathematics and in Mathematics Teaching
Description

Response about Mathematics / Response about Teaching
“If we can prove that something’s true in a general case, we don’t have to go through that every
Verification: confirming the
single time.”
(Laura, pre-semester, para. 54)
truth of mathematical
“I want [the students] to take away . . . that proof’s a way to be convinced about something
statements
mathematically that is valid.”
(Evan, beginning-of-semester, para. 134)
Proof is used “to prove [a claim] is not true or prove that it is false.”
(Evan, pre-semester, para. 56)
Falsification: establishing
“If they are given a true or false question that they can be like, ‘No, this really can’t happen.’”
when a claim is not true
(Laura, end-of-semester, para. 276)
“Proof can explain something—Why something is true, NOT just a confirmation.”
Explanation: showing why
(Kelly, pre-semester, para. 212)
a claim is true (different
“The students get a feeling for what math is about and it’s not just about guessing or assuming things;
from verifying its truth)
they should provide arguments for WHY it’s true.”
(Isaac, beginning-of-semester, para. 106)
“If I have an idea of mathematics and I want to show this to other people, to do that . . . we need
Communication: sharing
proof. . . . We need a proof to communicate to each other.”
(Peter, pre-semester, para. 179)
mathematical ideas and
“As future teachers . . . they should know the role of proof too . . . [to] explain some facts to other
knowledge with others
people without any logical jumps or gaps.”
(Peter, beginning-of-semester, para. 186)
Discovery: developing new Proof “is the process of building new knowledge.”
(Kelly, end-of-semester, para. 491)
mathematical ideas (no
Students should “learn that you can take things you're given or things you’ve already proven and
sense of communicating
establish new results.”
(Evan, end-of-semester, para. 136)
the ideas needed)
Systematization: imposing
a logical structure (e.g.,
utilizing and classifying
axioms or theorems) on
the mathematical domain
General: reinforcing the
nature of proof as central
in mathematics or learning

Think of it “like a tree; each time you are adding new knowledge, how that knowledge be (sic)
qualified to adding (sic) to the original tree is proof. [Proof] qualifies that it goes into some branch [of
the tree].”
(Kelly, pre-semester, para. 226)
“I still feel [it] is important for the teacher: How they are going organize the knowledge for their
students.”
(Kelly, end-of-semester, para. 513)
Proof “is a fundamental part of mathematics and . . . in a lot of ways it’s synonymous with what math
is.”
(Evan, pre-semester, para. 48)
Students “need to know the idea that proofs are important.”
(Ace, end-of-semester, para. 64)

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These examples provide a means of understanding how TAs generally talked about
purposes of proof in mathematics and in teaching. In both contexts, they talked about the same
types of characteristics for each purpose, but focused on general users of mathematics in the
mathematics discipline or on students in a mathematics classroom setting.
Regarding these seven purposes of proof, mathematics education researchers
conceptualize “an informed conception of proof—one that reflects the essence of proving in
mathematical practice—must include a consideration of proof in each of these roles” (Knuth,
2002a, p. 381, italics added). Like Knuth, I considered a robust or informed conception of proof
to be evidenced by TAs articulating multiple purposes of proof and I studied the TAs’ ability to
articulate informed conceptions of proof. The extent to which each TA talked about the
purposes of proof in mathematics is displayed in Table 7, with the purposes ordered from
greatest to least frequency.
Table 7. Frequency of Purposes of Proof in Mathematics Articulated by TAs

TA (total coded
statements)

Verify
Ace (12)
Laura (5)
Isaac (8)
Peter (11)
Evan (22)
Kelly (16)
a

66.7
60.0
12.5
9.1
18.2
12.5

a

Falsify Discover Communicate Systematize
25.0
20.0
12.5
9.1
18.2
6.2

Explain
General
Why

8.3
20.0
50.0
63.6
13.6

9.1
22.7
25.0

25.0
9.1
31.3

9.1
9.1
18.8

9.1
6.2

Values are percentages out of the total number of coded statements for each TA

The TAs all talked about more than one purpose of proof in mathematics, with some
noticeable differences in their salient views of purposes of proof. The mathematics education
graduate students, Kelly and Evan, articulated the most robust conceptions of proof, with Kelly

57

describing six and Evan describing seven of the seven roles of proof. They also did not
emphasize, in term of a majority of statements, only one role, and talked about the importance
of these various roles in mathematics. Their informed conceptions of proof incorporated all (or
almost all) the purposes of proof, suggesting these two TAs seem more equipped to help PTEs
learn how to talk about and understand each of these roles in mathematics.
The mathematics graduate students, Ace, Laura, Isaac, and Peter, talked about fewer
roles of proof and a majority of their statements were about one of two purposes: verification
or communication. By privileging these purposes of proof in mathematics, these TAs could be
more likely to reinforce PTEs’ narrow conceptions of proof and less likely to articulate and help
PTEs to explain connections between all the roles of proof in mathematics. Among these TAs,
however, Peter was unique in how he explicitly connected a salient purpose to others. Peter
explicitly described communication in a way that encompassed other roles: “Communication
includes everything. Verification of truth, why do you need this? To communicate. Falsification,
why do you need this? To communicate. Explanation? To communicate. Discovery . . . why?
2

Need to communicate” (Peter, end-of-semester, para. 587) . His conception of proof in
mathematics included other purposes of proof that were couched under the primary purpose
of communicating to other doers and users of mathematics. These results do not necessarily
mean that the mathematics TAs do not know or understand all seven roles of proof, rather,
certain roles of proof, or relationships among the purposes of proof, may remain more implicit
than others in their coursework and research.

2

Interview citations include TA’s name, interview name, and transcript paragraph number. A
new paragraph started when there was a change between interviewer and TA talk.
58

Comparing these views about proof in mathematics with the salient views when talking
about teaching, the TAs emphasized different roles and articulated more robust conceptions of
purposes of proof in mathematics teaching. This contrast suggests that TAs conceptualize
purposes of proof in the two contexts differently, supporting the notion that teaching about
proof necessitates different knowledge on the part of the teacher than it takes to use and do
proof in the mathematics discipline. These results are displayed in Table 8, with the purposes of
proof ordered from greatest to least frequency.
Table 8. Frequency of Purposes of Proof in Mathematics Teaching Articulated by TAs
Explain

TA (total coded
statements)

a

Ace (13)
Laura (15)
Isaac (11)
Peter (11)
Evan (18)
Kelly (20)
a

Why
38.4
46.6
27.2
45.4
50.0
15.0

Communicate General Verify Systematize Discover Falsify
30.8
26.7
9.1
36.4
25.0

30.8
36.4
9.1
11.1
10.0

18.2
9.1
22.2
10.0

6.7
9.1

6.7

13.3

5.6
20.0

5.6
20.0

5.6

Values are percentages out of the total number of coded statements for each TA

TAs’ responses about purposes of proof in teaching differed from how they talked about
proof in mathematics; they included at least one of the three roles, explaining why,
communicating, and the general, fundamental nature of proof, contrasting with the salient
purposes of proof talked about in mathematics (verifying, falsifying, and discovering). The three
roles emphasized in the context of teaching are particularly relevant to the work of preparing
PTEs to explain to others about the fundamental nature of proof. The contrasts noted between
mathematics education TAs’ conceptions of proof and the mathematics TAs’ conceptions were
also less apparent in these statements about teaching. Although Evan and Kelly continued to

59

articulate the most robust conceptions of purposes of proof, most of the other TAs talked about
more purposes in this context than when considering the mathematics discipline. Also, the way
TAs talked about the purposes tended to privilege two or three purposes more often in this
context. The way TAs talked more frequently about more purposes of proof in teaching could
suggest a greater likelihood that they will talk about more than one purpose of proof in the
classroom. TAs’ responses about these purposes of proof in mathematics teaching suggest that
their conceptions of proof include multiple purposes that are particularly important in the
context of teaching MTH 2.
4.1.2 TAs’ Conceptions of Why and What PTEs Should Learn about Reasoning-and-Proving
I also studied TAs’ conceptions of why and what to teach about RP in MTH 2 to
understanding how their goals aligned with the RP goals of the course. The course goals and
textbook aim to develop PTEs’ understanding of both RP content and processes so they are
successful in MTH 2 and able to teach others about RP. TAs with conceptions that strongly align
with these course goals could be more likely to develop PTEs’ knowledge of and ability to
articulate RP. To investigate TAs’ goals for teaching RP in MTH 2, I examined TAs’ talk about
why it was important to teach RP and what they wanted PTEs to learn about RP.
To prepare PTEs to understand mathematics in MTH 2 and how their work in MTH 2
relates to their roles as elementary teachers, the course should ideally include learning goals
that attend to both contexts (horizontal axis, Figure 5). To explain why it was important to
teach RP in MTH 2, however, TAs did not all express such a balanced perspective, with two TAs
attending to mathematical learning goals applicable to the MTH 2 context more so than how
those goals related to PTEs roles as future elementary teachers. The mathematical aspects that

60

TAs commonly identified were either RP content or RP processes (vertical axis, Figure 5). Since
TAs talked about RP content or processes goals with respect to the undergrad mathematics or
elementary classroom perspectives, exemplars from TAs along these dimensions are displayed

Processes or Ways of
Thinking

“A kind of habit of doing
mathematical thinking
that they have to have to
become a person I could
say is able to teach math”
(Kelly, beginning-ofsemester, para. 288)

“It would be nice if they
learned . . . that proofs are
necessary and important. .
. . That would be pretty
good and of course how to
do a couple of them here
and there”
(Ace, beginning-ofsemester, para. 228, 230)

“Being able to write a
proof that makes sense. . .
. I think that’s really
important for them to be
able pass on ideas . . . to
whomever they’re
teaching later on”
(Laura, end-of-semester,
para. 52, 54)

College Mathematics
Classroom

What Students Need to Know about RP

“This Geometry is one . . .
[course] that can provide
that kind of logical
thinking. So that’s one
purpose for teaching
proof”
(Peter, beginning-ofsemester, para. 164)

Content or Products

in Figure 5.

Future Elementary
Classroom

Context TAs Focused on When Describing Why It
Is Important to Teach about RP in MTH 2
Figure 5. Examples of how TAs talked about what and why they wanted students
to learn about reasoning-and-proving
All of the TAs made some statements about end products or content related to RP they
wanted MTH 2 students to understand. The quotes from Laura and Ace (Figure 5) depict some
of the common ways RP content was emphasized, including: knowing what a proof is,

61

producing proofs or concise explanations, supporting each statement with a reason, using
logical arguments in writing proofs, and knowing a proof exists for mathematical statements
that could be taught in elementary grades. Most of the TAs also talked about processes of
proving they aimed to teach PTEs. Kelly and Peter, for example, talked about developing PTEs’
ability to think mathematically or logically, a common RP process TAs described. TAs also talked
about providing opportunities for PTEs to “make and investigate mathematical conjectures”
(Laura, end-of-semester, para. 238) and “evaluate the arguments that [their] students [could]
make” (Evan, end-of-semester, para 51).
To prepare PTEs to teach others about both RP content and processes, it is important
for TAs to move back and forth between RP content and RP process goals, as well as to address
both the MTH 2 context and future elementary classroom setting in planning and teaching MTH
2. The patterns in TAs’ conceptions along these dimensions are depicted in Figure 6. The
frequency of statements along the four dimensions guided the placement of the circle in the
figure. The circles were placed to represent each TA’s tendency to focus on particular views of
teaching and learning RP along these dimensions. The intersection of the two axes represents
equally talking about all four dimensions, which would suggest conceptions aligned with the
design and goals of MTH 2.
To understand how data were represented in Figure 6, consider Ace as an example.
Ace’s circle only touches the line between the two contexts along the horizontal dimension.
This placement indicates how Ace rarely talked about why teaching RP was important for PTEs
to be prepared to teach others and predominantly talked about what he wanted them to learn

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and be able to understand in MTH 2. Along the vertical dimension, he only articulated RP

Processes or Ways of
Thinking
Content or Products

What Students Need to Know about RP

content and did not talk about RP processes.

Peter

Evan
Kelly

Isaac

Laura
Ace

College Mathematics
Classroom

Future Elementary
Classroom

Context TAs Focused on When Describing Why It
Is Important to Teach about RP in MTH 2
Figure 6. TAs' conceptions of teaching and learning reasoning-and-proving in MTH 2
The patterns in TAs’ responses suggest that they possess a range of conceptions about
why and what they are teaching about RP. This range of conceptions supports the notion that
teaching about RP to college students who are prospective teachings represents a complicated
context: the TAs are teaching PTEs who need to develop rich understanding of RP to be
prepared to teach others about both RP content and RP processes. To navigate that context,
Peter, Kelly, and Evan articulated conceptions of teaching and learning RP that seemed to align
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with the course goals and design. They focused more on a balance of teaching RP content and
processes and attended both to what they wanted PTEs to understand in the college
mathematics classroom and to prepare them for their future teaching. TAs with these
conceptions could be more likely to engage PTEs in a range of RP processes during classroom
instruction than TAs without such views. The other TAs talked infrequently (Laura and Isaac) or
never (Ace) about RP processes; they focused instead on developing PTEs’ understanding of
mathematical content. Since these conceptions of teaching RP do not align with the textbook
and course goals, they may be more likely to provide opportunities for PTEs to learn RP content
more than processes.
4.1.3 TAs’ Conceptions of How to Teach about Reasoning-and-Proving
I also studied TAs’ conceptions of methods for teaching RP to understand how their
views aligned with the structure of the course. The textbook, course structure, and course
supervisor encourage a student-centered pedagogical approach consistent with how PTEs are
increasingly expected to engage future elementary students in RP (CCSSI, 2010; NCTM, 2000).
Alignment between TAs’ conceptions and this course design could lead to more authentic RP
learning experiences for PTEs.
Based on the stance embodied by the textbook and instructional supports, one might
anticipate a student-centered philosophy to be most resonant for TAs. Their experiences in K16 mathematics classrooms, as learners and instructors, however, are not likely to reflect those
views of teaching. In fact, teaching MTH 2 represented the first experience for all the TAs to
teach mathematics using more student-centered techniques. Isaac explained, for instance:
“I mean everything I taught before this course . . . [has] been a teacher standing at the
blackboard writing and talking for 50 minutes. And now I can see much more how you
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can interact with the students and let the students work instead of [students] just
listening or not listening. . . . I definitely think this is a better way of teaching.” (end-ofsemester, para. 52)
All the TAs had no experience teaching using non-lecture-based pedagogical approaches or
learning mathematics in these ways, but five of the six TAs talked positively about using
strategies to develop PTEs’ conceptual understanding of RP by engaging PTEs in tasks and
discussions in class (Figure 7); strategies encouraged by the text and course supervisor. TAs also
described these strategies as important for helping PTEs understand MTH 2 material better
and/or learn mathematics in ways they will likely be expected to teach it. These results suggest
that all the TAs, except Ace, articulated perspectives that value the pedagogy called for in the
design and structure of the course, talking about it in a way that suggests they aimed to teach
MTH 2 using these strategies to engage PTEs in RP tasks and develop PTEs’ understanding of RP.
The pedagogical strategies these five TAs positively talked about that seemed to align
with the activity-based structure of the course, included: TAs working as discussion facilitators,
Socratic tutors, and monitors of students’ progress; and students working on tasks, working in
small groups, articulating mathematical ways of thinking, and reasoning through mathematics
themselves. Examples of statements about two of these areas are provided to depict how TAs
attended to the importance of PTEs engaging in RP themselves (or with classmates) to develop
a conceptual understanding of the mathematics.

65

TAs

Pedagogical Approaches

Ace Laura Isaac Peter Evan Kelly

Work TAs do:
Facilitate Class Discussions

•

+

+

Monitor Individual/Small Group Progress
Lecture

+

+

+

+

+

Socratic Tutoring
Provide Notes or Example Problems

•

+

•

+
+

•

•

•

-

•

-

+
+

+
+

•

•

+
+
+
+

+
+
+
+

+
+
+

•

Work PTEs do:
Work on Interactive Mathematical Tasks
Work in Small Groups
Articulate Mathematical Ideas & Ways of Thinking
Reason through Mathematics Themselves
Work on Practice Problems
Symbols indicate that TAs:

+

+

•

+

•

Frequently mentioned a pedagogical approach positively
Mentioned a pedagogical approach positively

-

Frequently mentioned a pedagogical approach negatively

Figure 7. How TAs talked about approaches for teaching reasoning-and-proving
Evan, for instance, explained the importance of facilitating class discussions:
By teaching the class the way I do, leading discussions [and] having students come up to
the board, these students are participating in a mathematical discussion in class, which
can get them out of the traditional mindset of a mathematics classroom. So they could
possibly teach in this way in the future. . . . I’m trying to get out of the mode of just
standing at the board and writing things on it for 50 minutes and collecting homework
once a week after I’ve written stuff on the board for two and a half hours over the
course of the week. I’m trying to get people more involved in thinking and sharing their
thoughts. (end-of-semester, para. 74)
Evan described facilitating discussions instead of lecturing to get PTEs “involved in thinking and
sharing their thoughts;” a teaching approach like that could support the development of PTEs’
ability to articulate RP, as well as the ability to hear and critique others’ justifications.

66

Peter also talked about PTEs learning how to reason mathematically and share ideas
during class by leading what he called question-and-answer dialogues. Peter described these
dialogues as involving a whole class discussion he lead by asking questions based on PTEs’
answers to help them think critically and construct arguments. This approach was called
Socratic tutoring because Peter related it to books he read as an undergraduate mathematics
education student: “for example, like, Socrates—the question-answer stuff is from Socrates”
(end-of-semester, para. 312). From Peter’s perspective, he intended to provide opportunities
for PTEs to reason and think mathematically as he questioned their answers and asked them to
think about mathematical ideas from multiple perspectives. He explained that to learn to
reason and think mathematically PTEs “have to experience that question-answer [dialogue]. . . .
You cannot learn that from your teacher. You have to experience that. Your teacher can
encourage that or make them to experience more” (end-of-semester, para. 316-318). The
Socratic tutoring method was a particular approach Peter endorsed for thinking through
mathematical ideas as a class that is similar to the way Evan talked about mathematical
discussions because PTEs have opportunities to articulate, hear, and critique mathematical
justifications and reasoning. Although Peter did not make an explicit connection to PTEs’ roles
as future teachers in his statements about Socratic tutoring (as Evan did about facilitating
discussions) the pedagogical approach and nature of PTEs’ involvement during that strategy still
aligns with the goals and structure of the course.
The pedagogical approaches about which Ace talked positively contrasted with the rest
of the TAs, where he focused on providing notes, lecturing, and asking PTEs to do practice

67

problems; perspectives that tend to be more teacher-centered and encourage passivity on the
part of the students, which is not aligned with the goals of this activity-based course.
Comparing how Ace talked about providing notes for PTEs with the way it was
mentioned by Kelly highlights how Ace’s conceptions of teaching RP in MTH 2 differed from the
other TAs and from the course structure. When talking about providing notes for PTEs, Ace
explained why he often elected to provide proofs as worked out examples, saying that these
proofs are “really just something that [students] hold [their] breath through to get the tools
[they] need to complete the problems at hand” (Ace, beginning-of-semester, para. 204). This
perspective focuses on proof as a product to be memorized or used and does not seem to value
PTEs’ participation in proving processes. Kelly, on the contrary, explained why she provided
notes by writing some main ideas on the board after a verbal discussion of a proof, saying that
the notes could help
some students who lag behind or don’t get what I’m talking about. So leave them a
good notes [sic] that they can refer to after class may be a help for them. . . . I would
rather write more just to make sure that they get the answer but I don’t like to write like
a sentence-by-sentence. That’s kind of stupid. I just write more mathematical
expression and they can reconstruct their sentences from those expressions but I try to
keep the big important steps on the board. . . . So they can use that to recall . . . the
class. (end-of-semester, para. 281)
Kelly used notes to provide some main ideas that she expected PTEs to use to reconstruct the
class discussions, suggesting that (unlike Ace) Kelly endeavored to encourage PTEs’ active
participation in their learning. These examples also highlight how a pedagogical strategy is not
inherently student-centered or teacher-centered. All strategies TAs discussed, from facilitating
class discussions to providing PTEs with notes, may be conceptualized and implemented in ways
that encourage or discourage student engagement in the learning process.

68

In MTH 2, the textbook endorsed student-centered instructional strategies where PTEs
develop their own understanding of and explanations for mathematical ideas by engaging in
mathematical tasks. Most TAs talked about teaching MTH 2 in ways that aligned with the design
of the course and textbook. This alignment suggests that almost all the TAs viewed the use of
mathematical tasks favorably, contributing to their likelihood to enact RPTs during instruction,
which I was then able to examine for the nature of task enactment (task enactment is explored
in Chapter 6). More details about Ace’s contrasting conceptions of teaching RP in MTH 2 are
provided in the next section.
4.2 Case Studies of TA’s Conceptions of Reasoning-and-Proving
A feature of MTH 2 that distinguishes it from general mathematics courses is the
student population: PTEs in the course are current undergraduate mathematics learners who
also need to learn mathematics so they are prepared to explain it to others. Since TAs differed
in how they attended to this dual nature of the course in their conceptions of teaching and
learning RP in MTH 2, four case studies are considered. The goal of these cases is to provide
background information that informs our understanding of TAs’ instructional decisions
(presented in Chapter 6, with relationships discussed in Chapter 7). The first pair of cases
consists of two groups of TAs who attended to the elementary perspective, but have different
conceptions of what RP is important for PTEs to learn. The second pair consists of individual TAs
who attended primarily to the mathematics that MTH 2 students needed to learn, but have
different reasons for holding that perspective.
Recalling general information about the TAs’ backgrounds and prior teaching
experiences provides context for understanding their case descriptions (refer to Table 3). In

69

particular, none of the TAs had experience teaching K-12 mathematics in the US, but they were
expected to teach MTH 2 with the goal of preparing PTEs to engage future elementary students
in RP processes and content. Therefore, the TAs did not have personal teaching experience or
training upon which to base their perspectives of how RP in MTH 2 could relate to PTEs as
future elementary teachers. Furthermore, half of the TAs (Isaac, Peter, and Kelly) did not attend
elementary school in the US. These international TAs, therefore, also approached MTH 2
without any personal experiences upon which to draw to understand the context they were
expected to prepare PTEs to enter. In the first case, however, the two mathematics education
TAs, Evan and Kelly, along with the mathematics graduate student in his sixth year, Peter,
expressed similar conceptions of RP process goals that incorporated the elementary
perspective. In the second case, Laura, a second-year applied mathematics graduate student,
also attended to the preparation of PTEs, but in a way that contrasted with the first case and
the course goals by focusing on RP content. The investigation into these two cases suggests a
commonality among the TAs in the first case that may contribute to the conceptions observed.
The third and fourth cases focus on Isaac’s and Ace’s conceptions of teaching RP in MTH 2,
respectively, as they do not attend to the elementary perspective but for different reasons.
These cases provide an opportunity to understand how Isaac’s background and Ace’s
perspective on pedagogical approaches seem to be significant factors influencing their
conceptions of teaching RP in MTH 2.
4.2.1 Preparing Prospective Teachers to Teach Others about RP Processes
MTH 2 has some unique characteristics in it because you need to build some mathematical
knowledge for undergraduate students and also you need to think about the content
knowledge for future teachers. (Peter, beginning-of-semester, para. 180)

70

It’s the way of thinking mathematics that [the PTEs] should learn, which is more important than
the content that they can directly transfer to their students. (Kelly, end-of-semester, para. 85)
Taking what you’re given, using logic, being able to reason and make an argument about other
stuff that has to be true because of what you’re given and relationships that you have already
established. . . . It’s gonna be important for them . . . when they’re going out as teachers being
able to do that. . . . We just want to establish that thought process. (Evan, beginning-ofsemester, para. 128)
Peter, Kelly, and Evan attended to the dual nature of MTH 2, where the students need
to not only learn mathematics but also be prepared to teach others about it, by talking
extensively about engaging PTEs in RP processes. They specified mathematical processes in
their own ways but incorporated the notion of preparing PTEs to talk about mathematics and
about RP with their future students by developing PTEs’ ability to think and reason
mathematically in MTH 2. Understanding how they articulated these RP processes underscores
the importance for incorporating processes into instructional practices in MTH 2 and suggests a
common experience these TAs share that may have informed their conceptions of teaching RP.
Peter talked about helping students “study proof as a proof in Mathematics . . . [and]
experience more about the role of proof” (beginning-of-semester, para. 186) by engaging them
in the process of proving. To help PTEs experience the proving process, he talked about
providing opportunities for them to construct and refine mathematical arguments in a way that
was similar to the work in which mathematicians engage. That is, he explained how “false
starts,” “logical gaps,” and going the “wrong way” can help mathematicians refine their
mathematical arguments, understand mathematical ideas better through proof, and
communicate those proofs to others (Peter, end-of-semester, para. 226-236). He saw MTH 2 as
an opportunity to help PTEs develop “that kind of logical thinking” (beginning-of-semester,
para. 164). And he thought it was important for PTEs to experience the proving process in MTH
71

2 because “as a teacher if they can experience those things before as their own experience then
they can easily figure out and help their own students” (end-of-semester, para. 250). Peter
talked about how this proving process could help PTEs understand proof in mathematics,
prepare them to engage their future students RP processes, and make sense of their future
students’ thought processes.
Kelly and Evan talked about helping PTEs understand RP as a mathematical way of
thinking as well, specifying the importance of preparing PTEs to engage their future students in
RP. For Kelly, learning about RP helped PTEs develop the ability to reason mathematically, as
she explained:
It’s the idea of thinking, even though they don’t teach those things literally to their
students, they still have that idea that they need to connect the unknown to known . . .
by using . . . valid evidence. Those ideas you have to have them to become a good
elementary teacher. . . . So I think it’s the way of thinking mathematics that they should
learn, which is more important that the content that they can directly transfer to their
students. (Kelly, end-of-semester, para. 85)
Instead of specifying RP content that PTEs might need to rely on in their future teaching, Kelly
viewed the development of “the way of thinking mathematics” as the primary reason for
teaching PTEs about RP. Evan expressed similar sentiments by talking about the importance of
engaging PTEs in thought processes in which they would then engage their future students:
They’re not gonna be teaching about the exact same stuff in their classrooms. Maybe if
they have an eighth-grade classroom sure . . . but if they have a second-grade classroom
they’re probably not. We just want to establish that thought process. If . . . in MTH 2 we
look at proving stuff that you might prove with a second-grade class, . . . [the PTEs] are
gonna understand it but it’s not gonna engage them- like- get them to think. . . . So we
spend most of our time in upper elementary maybe a little bit in high school [content]
just to engage them . . . have them practice building that line of reasoning we want
them to build with their students, even though [in elementary grades] they might be
looking at little bit simpler stuff than what we’re doing. (Evan, beginning-of-semester,
para. 128)

72

These three TAs made explicit connections between RP in MTH 2 and what they wanted
PTEs to understand about RP as future teachers. All three of them also have experience taking
courses as undergraduate and graduate students in mathematics education. Evan and Kelly
used phrases such as “our field of mathematics education” (Kelly, beginning-of-semester, para.
292) and “in a particular mathematics education . . . class” (Evan, end-of-semester, para. 78) to
which they related their conceptions of RP content and processes. Although Peter was not in
the doctoral program for mathematics education, his undergraduate background included
certification to teach secondary mathematics and he was enrolled in a graduate-level
mathematics education course at the time of the study. Peter identified mathematics education
books he read as an undergraduate as prominent influences on his conceptions of teaching
PTEs RP processes (end-of-semester, para. 312). When commenting on the influence of their
coursework or readings, there was not a single, specific class or book to which they all referred
except for the fact that they all expressed familiarity with the National Council of Teachers of
Mathematics’ (2000) recommendations for RP. Therefore, these three TAs identified
mathematics education coursework and the NCTM’s recommendations as helpful for framing
their views of how RP in MTH 2 relates to PTEs’ roles in elementary classrooms. Their views
suggest these TAs have rich conceptions of how the mathematical content and processes in
MTH 2 relate to the PTEs as future teachers that align with the design and goals of the course
and were shaped, at least in part, by their experiences reading and talking about mathematics
education policy and research.
4.2.2 Preparing Prospective Teachers to Produce Mathematical Explanations
I think . . . what [the PTEs] are gonna be doing is explanations of things for students who are
younger and not necessarily doing rigorous proofs. (Laura, end-of-semester, para. 10)
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Laura also attended to the preparation of PTEs to teach others about RP in her goal
statements, but her focus on content over process contrasted with the conceptions in the first
case. This case study helps us understand how and why Laura’s particular focus on an aspect of
RP content contrasts with the goals of the course. Laura talked about helping PTEs produce
concise nonproof arguments, as she explains,
I think the explanations are really probably the most key for [the PTEs] since they’re
gonna be teaching elementary students and really most of them are . . . not gonna be
doing higher-level, like the Grade 8 sort of stuff, where they would actually need to be
using . . . proofs. . . . I think these explanations are really important for them and to be
concise so that that their students can understand . . . this intuitive explanation. (end-ofsemester, para. 162, 164)
Across the interviews, Laura articulated a goal of helping PTEs generate clear, concise, but
informal (or intuitive) explanations for the validity of claims about general mathematical
relationships based on how she remembered learning about RP. Laura also focused on
explanations because (as expressed in the above quotation) she considered formal proofs as
content that only a select few PTEs (possibly “like the Grade 8” future teachers) may ever use
again. Laura’s conceptualized teaching RP, therefore, differently than the NCTM, which
recommended that in all grade levels mathematics classrooms RP should be “a natural, ongoing
part of classroom discussions, no matter what topic is being studied” (NCTM, 2000, p. 342).
Laura, however, seemed unaware of any misalignment between her conceptions and these
recommendations as she expressed unfamiliarity with NCTM’s (2000) recommendations for RP
in K-12 mathematics (end-of-semester, para. 216). Her conception of teaching RP in MTH 2
focused on PTEs producing concise explanations of solutions and informal justifications for
mathematical relationships, which could reinforce PTEs’ misconceptions about what constitutes

74

a proof if the informal justifications are accepted as proofs. In light of the previous case and this
one, TAs like Laura, who aim to prepare PTEs to teach RP but may unknowingly reinforce PTEs’
misconceptions, could potentially benefit from support that incorporates some applicable
mathematics education literature, such as the NCTM’s (2000) recommendations for RP.
4.2.3 Helping MTH 2 Students Understand Formal Proofs
My students . . . should get a feeling for how math works and the best thing is to do proofs in
some kind of formal way. (Isaac, beginning-of-semester, para. 110)
Isaac’s conceptions of teaching RP are diametrically opposed to the previous case
because of his focus on helping MTH 2 students learn how to construct formal proofs and his
difficulty articulating an elementary classroom perspective. Although Isaac’s conceptions of
teaching RP are like Ace’s in how he privileged content, these TAs’ conceptions are presented
as two separate cases because Isaac’s background, as an international graduate student, seems
to be a significant factor in his perspective.
Since Isaac did not grow up in the US, he was unfamiliar with the teaching and learning
of mathematics in the elementary grades. He only talked about how the material about RP
related to his students as future teachers when he was specifically asked about that potential
connection during some interviews. In response to these prompts he frequently asked clarifying
questions about what grade levels PTEs would be certified to teach. He would, then, often
follow up those questions with a statement such as: “I don’t know what they do in grade eight
in US but . . . maybe they would talk about the Pythagorean Theorem for example and maybe
do the proof that was in the last part [referring to a picture proof of the Pythagorean Theorem
in the MTH 2 text]” (Isaac, beginning-of-semester, para. 140, emphasis added). When
prompted, therefore, Isaac tended to specify RP content from MTH 2 that PTEs could
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potentially use in their future teaching. He also expressed some uncertainty about how the
material from MTH 2 prepared PTEs for elementary teaching as he used phrases like maybe and
I don’t know.
Isaac readily attended to the undergraduate mathematics perspective, talking about RP
content he wanted MTH 2 students to know and understand. He articulated purposes of proof
in mathematics teaching by explaining how learning about proof was important content in his
MTH 2 classroom (Int indicates the interviewer, beginning-of-semester, para. 106, 110-112).
106

...
110
111
112

Isaac: I think [proof] serves the purpose that the students get a feeling for what
math is about and it’s not just about guessing or assuming things; they
should provide arguments for why it’s true.
Isaac: My students . . . should get a feeling for how math works and the best
thing is to do proofs in some kind of formal way.
Int:
When you said . . . “do proofs in a formal way”, what does that mean,
“formal”? Can you explain that a little bit?
Isaac: Formal is like, you talk about things like axioms, theorems, definitions
and write things down not just give some kind of talking.

Isaac said that students should understand that proofs are necessary for explaining why
mathematical statements are true (para. 106). He also described the importance of teaching
about proof to help his students understand “what math is about” and “how math works”
(para. 106, 110). He talked about how students needed to understand that math was “not just
about guessing and assuming things,” and that doing proofs is “the best” means to do so (para.
106, 110). This focus on formal proof as a primary content learning goal may also have
contributed to Isaac’s difficulty articulating a connection between the RP work in MTH 2 and
PTEs’ future teaching because formal proofs do not often play a prominent role in elementary
mathematics classrooms. Isaac’s focus on formal proof differs from the goals of MTH 2, which
emphasize PTEs learning to reason mathematically in addition to doing proofs.
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4.2.4 Helping MTH 2 Students Learn RP Content Using Lectures
You mention the proof and [the students] hold their breath while you go through it. . . . You
hope that the students take something away from this and hopefully should be able to
regurgitate it to their students one day. (Ace, beginning-of-semester, para. 220)
Like Isaac, Ace also focuses on helping MTH 2 students understand formal proofs, but
his perspective on methods for teaching RP does not align with the structure of the course,
setting him apart from the other cases. Of all the TAs in this study, Ace also has the least
amount of teaching experience (one semester teaching college algebra prior to teaching MTH
2) and seems to explicitly rely on his experiences as a mathematics learner. Ace ’s perspective
on how to teach RP in MTH 2, by using lecture-based instruction, is unique among these TAs,
and is a significant factor related to his goals for teaching PTEs about RP.
Ace conceptualized his role as the one who “gives [the PTEs] knowledge” (Ace, end-ofsemester, para. 30) in MTH 2. The RP-related knowledge he wanted PTEs to learn included:
what constitutes a proof, how to write proofs, and that proofs exist for elementary school
mathematics content, to ultimately be prepared to respond if elementary students asked a
why-question. The following dialogue typifies how Ace articulated his goals for teaching about
proof in MTH 2 (Int indicates the interviewer, beginning-of-semester, para. 215, 218-222):
215

Int:

Thinking about MTH 2, can you say a little bit about the purpose proof
and proving serves in that classroom?

...
218

Ace:

219
220

Int:
Ace:

It falls under this category: You mention the proof and they hold their
breath while you go through it. And then at the end of the day, they use
the tools at the end of it.
Ok.
You hope that the students take something away from this and hopefully
should be able to regurgitate it to their students one day, and say, “yeah I
did this in class. It comes from so and so.” And you really hope that’s the
case. . . . I’d have go and visit K-8 . . . to really determine how useful it is
gonna be to them in their . . . day-to-day lives which is probably not very.
77

221

Int:

222

Ace:

And you said you “hope they regurgitate this later,” can you say a little bit
about what you were picturing there, what did you mean by “regurgitate
this later”?
Regurgitate is the word I use for memorize- or something like this. It’s not
that you truly learn [the proof] or understand it and you know actually all
the parts of how it fits together and you’ve really explored the idea
completely. But it’s the idea that you believe it and you’ve memorized at
least 90 percent of it so that you can spit it out at someone else.

Similar to Isaac, Ace expressed uncertainty about how proof in MTH 2 related to the K-8
classroom setting (para. 220), but Ace and Isaac differed in their responses to that uncertainty.
Isaac either stated he did not know or (usually after additional presses from the interviewer) he
ventured a guess. In his guess, he often chose a RP task from MTH 2 and described how PTEs
might use it in an upper-elementary-grade classroom (e.g., Isaac, beginning-of-semester, para.
140). Contrarily, as Ace expressed in the statement above (para. 220), without any additional
promptings during the interview Ace quickly evaluated the likelihood that visiting an
elementary classroom would reveal any RP insights as unlikely. The contrast in their responses
is apparent: Isaac seemed open to considering the possibility that MTH 2 proofs can relate to
the elementary context, and Ace seemed certain that they do not relate.
The above dialogue from Ace also indicates another difference between the third case
and this case considering their conceptions how to teach PTEs about RP. To emphasize the
importance of students participating in the construction of proofs, Isaac explained that “if you
never . . . do proofs yourself, you don’t really get a feeling for how . . . to prove things”(Isaac,
end-of-semester, para. 64). A prominent feature of Ace’s conception of teaching RP, however,
was using lectures to teach about RP. He described PTEs’ response during the teaching of a
proof as “they hold their breath while you go through it” (para. 218), suggesting PTEs did not
participate in the proof’s construction while the TA presented it to them. Earlier in the same
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interview, Ace explained his preference for this style of instruction, saying that presenting the
material for PTEs
is much more of a time saving method . . . however it's a trade off because then they
don't understand the material as much. So I mean obviously if they sit there and really
think through these ideas and do all these activities and stuff I think their understanding
is greater however it also takes quite a bit more time. (beginning-of-semester, para. 88)
When describing the “trade off,” Ace said using mathematical tasks (activities from the
textbook) can provide opportunities for PTEs to understand material better. He consistently
talked about the efficiency, however, of using a lecture-style of instruction.
Despite this view, Ace did not ignore the textbook’s tasks and he talked about why he
still asked PTEs to work in groups on tasks, as follows (beginning-of-semester, para. 127-134):
127

Int:

128
129
130
131
132

Ace:
Int:
Ace:
Int:
Ace:

133
134

Int:
Ace:

Before teaching MTH 2, had you ever taught using group work and
activities before?
No. And . . . I would never do it again.
Why do you say that?
I don’t know. It’s not math. It’s not math to me. I don’t know.
What does that mean? Can you help me understand that?
Every single class I’ve had is always lecture-based. And with the exception
of maybe like elementary school or something like this. So the concept of
class work or group work or something like this to help get a problem
solved is unreal. I don’t know. It’s too far out there for me. I’m a lost
cause.
So how did you decide to even use group work then for MTH 2?
I was confused at the beginning. I didn’t understand what was going on
for this course and [the course supervisor] said “these are group work
activities” and stuff like this so then I went with it the first day and have
gone with it ever since.

Although Ace made a statement about how the elementary grades could potentially include
more activity-based instructional strategies (para. 132), it did not seem to influence his
perspective on the utility of this pedagogical approach for PTEs in MTH 2. Ace’s prior
experiences learning mathematics shaped his less-than-favorable view of activity-based

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learning in mathematics teaching. He talked about lecture-based instruction as time saving and
preferable. Ace explicitly devalued the pedagogy called for in the design and goals of MTH 2.
4.3 Conclusions about the Analysis of TAs’ Conceptions of RP
Through this examination of TAs’ conceptions of RP it is evident that TAs’ conceptions of
purposes of proof in mathematics contrast with their conceptions in teaching. The context of
teaching MTH 2, therefore, elicits TAs’ talk about purposes of proof specific to the work of
teaching PTEs about proof, supporting the notion that teachers need a specialized
understanding of mathematical content to teach. TAs may complete their mathematical work in
their courses and research without explicitly articulating the purposes of proof, but, as an
instructor for PTEs, TAs need to know and incorporate the purposes of proof into the classroom
to develop an informed conception of proof in PTEs.
In this context, it is also important for TAs to develop PTEs’ ability to articulate both RP
content and RP processes, and attend to goals specific to the MTH 2 context as well as to more
long-term goals about the future elementary classroom setting. To establish these goals TAs
should take into consideration the mathematical content along with PTEs’ learning needs for
both the undergraduate and elementary contexts; highlighting an aspect of pedagogical
content knowledge called for in this context, described by other researchers as knowledge of
content and students (Ball et al., 2008). To navigate that setting, some TAs (Peter, Kelly, and
Evan) articulated a balance between RP content and RP processes PTEs needed to learn, while
others emphasized RP content (Laura, Isaac, and Ace), varying in their attention to the
elementary context. Their goals for teaching RP in MTH 2 were influenced, in part, by
differences among the TAs regarding their familiarity with and understanding of mathematics

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education theory, national recommendations for teaching and learning RP, and the nature of
teaching and learning of RP in elementary classrooms in the US.
A commonality among the TAs was their lack of experience teaching mathematics in the
K-12 setting. This study did not collect data from any TAs with K-12 teaching experience, and
such an investigation could compare how the RP conceptions of TAs who have experience
teaching in those settings might compare with those in this study. One might expect TAs with
that experience to be more aware of the knowledge and pedagogical skills PTEs need to
succeed in their future classrooms and, therefore, be more explicit about connections to the
elementary context. Research indicates, on the contrary, that it is not automatic for teachers to
learn to teach from their own teaching experiences and they often need outside help to learn
to anticipate students’ potential misconceptions (Grossman, 1989). The study of TAs with K-12
teaching experience could inform our understanding of prominent factors influencing TAs’
conceptions of teaching and learning RP, and whether or not it is important for colleges and
universities to consider assigning TAs to teach courses like MTH 2 who have a variety of
teaching experiences.
The results from this examination of TAs’ conceptions of RP in mathematics, teaching,
and learning have implications for the potential opportunities PTEs will have to learn RP
content and processes in MTH 2. Those who were more aligned with the course goals were
expected to design lessons around these goals, thereby aiming to engage PTEs in RPTs. How TAs
enacted RP tasks is studied in Chapter 6, after first exploring the different types of RPTs in the
textbook, in Chapter 5.

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CHAPTER 5
REASONING-AND-PROVING OPPORTUNITIES IN THE TEXTBOOK
This chapter presents the results of the analysis of the textbook (Beckmann, 2008) used
in MTH 2. The Geometry and Measurement chapters were analyzed to understand the types of
RP opportunities afforded, ultimately guiding the selection of lessons to observe. This
identification of textbook sections wherein PTEs may have different types of opportunities for
RP serves as a resource for understanding how one might structure mathematics courses for
PTEs to provide systematic and varied opportunities to engage in RP.
The RP-framework guided this analysis: questions were coded for the processes of RP
outlined in the framework (Table 2), and questions that were designed to offer PTEs at least
one opportunity for RP were referred to as RP-related questions (RPQs). The questions that
constituted a mathematical task (Stein & Lane, 1996) were then grouped together and a task
with at least one RPQ was regarded as a RP-related task (RPT).
The sections following describe the results along the RP-framework’s dimensions: First,
the overall distribution of opportunities for PTEs to engage in RP is described. Second, the types
of RP opportunities are presented along Dimension 1 to understand how tasks present
opportunities for PTEs to experience the six RP processes. Third, results are detailed from the
Dimension 2 analysis, examining how tasks were designed for PTEs to experience the process
through which generalizations are formed and then justified. Fourth, opportunities for PTEs to
understand features of proof and proving instead of doing proofs are described. Finally, a
summary of these results outlines implications for studying mathematics instructors teaching
PTEs RP from such a textbook.

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5.1 Distribution of Opportunities for Reasoning-and-Proving Across Four Textbook Chapters
Opportunities for PTEs to engage in RP are indicated by how the RPQs and RPTs were
distributed across these chapters of the textbook. The analysis included 565 questions of which
the initial key word search identified 242 as potentially RP-related. Examining the questions
recommended by the course supervisor identified 16 additional questions that were potentially
related to RP. These 16 questions were missed by the key-word search because, although they
focused on investigating mathematical relations, they did not specifically ask PTEs to generate
conjectures or provide justifications. These 258 questions were analyzed for substantive
opportunities for RP. Of the 258 potentially RP-related questions, 77% (198 questions)
represented opportunities for PTEs to engage in at least one RP process from the RP
framework. Therefore, these 198 questions were regarded as RPQs because they were
designed to offer PTEs at least one opportunity for RP. The remaining 60 questions did not offer
any opportunities for PTEs to engage in RP.
Table 9. Distribution of Reasoning-and-Proving-related Questions and Tasks
Number of
RPQs in
Textbook Chapter: Topic of Study Chapter
8: Geometry
86
9: Motion & Change
17
10: Measurement
24
11: More about Area & Volume
71
Total Across these Chapters: 198

Percent of
RPQs in
Chapter
44.3%
15.7%
27.2%
40.6%
35%

Number
of RPTs in
Chapter
21
10
10
25
66

Percent of
RPTs in
Chapter
55.3%
41.7%
62.5%
67.6%
57.4%

The distribution of these 198 RPQs is displayed in Table 9, which indicates that RPQs are
more prevalent in the first and fourth geometry and measurement chapters, but at least 15% of
the questions provide opportunities for PTEs to engage in RP within the middle two chapters, as
well. As described in Chapter 3 (Section 3.3.1.2), the textbook contained 115 tasks, and if a task

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had at least one question that afforded opportunities for PTEs to engage in one or more
dimensions of the analytic framework, it was identified as an RPT. Of these 115 tasks, 57% (66)
contained at least one RPQ. This distribution of RPQs and RPTs (see Table 9) indicates
numerous RP opportunities across these chapters.
11.11
11.10
11.9
11.8
11.7
11.6
11.5
11.4
11.3
11.2
11.1
10.5
10.4
10.3
10.2
10.1
9.4
9.3
9.2
9.1
8.6
8.5
8.4
8.3
8.2
8.1

Reasoning-andProving Tasks

Section of Curriculum*

non-Reasoning-andProving Tasks

0

2

4

6

8

10

Number of Tasks

* The names of these sections of the text are provided in Appendix B.3
Figure 8. Breakdown of RPTs and non-RPTs in the textbook

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12

Since I observed TAs teaching individual lessons, I also considered the RP opportunities
at a smaller grain size by considering sections of the chapters (Figure 8) to examine the
distribution of RPTs and guide the observation schedule. For sixteen of the 26 sections in the
textbook, at least 50% of the tasks were RPTs, suggesting a number of sections for which
classroom instruction would potentially involve tasks with potential to engage PTEs in RP. It is
also important to note that the later sections in some chapters (for Chapters 8, 9, and 10)
tended to have more RP tasks than earlier sections or the chapter had an even spread among
subsections (as noticeable in Chapter 11). It could be problematic if the chapter frontloaded
RPTs in the beginning sections. The sections of the textbook with fewer RPTs were also spaced
out in the textbook so that PTEs could arguably have opportunities to encounter RPTs at various
points throughout a course using this textbook.
5.2 Types of RP opportunities Afforded in the RP Tasks
This analysis also identified the nature of RP opportunities the textbook afforded by
investigating the types of RP processes the RPTs elicited. The distribution of tasks with respect
to the six types of RP processes (Dimension 1 of RP-framework, Table 2) is displayed in Figure 9.
Individual tasks could be coded for more than one RP process because a task fell within one of
these categories if it contained at least one RPQ that elicited the RP process. Two categories in
the framework, evaluating mathematical claims and evaluating mathematical arguments,
accounted for the fewest number of RPTs (7 and 6, respectively). Since these processes were
added to this framework as particularly relevant for PTEs as future teachers, TAs may need to
augment the text or modify tasks to provide additional opportunities for these processes.

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50

Number of Reasoning-and-Proving Tasks

45
40
35
30

25
20

15
10
5
0
Investigate

Conjecture

Evaluate
Claim

Evaluate
Argument

Provide Provide Proof
Nonproof
Argument

Dimension 1—Processes of Reasoning-and-Proving
Figure 9. Number of tasks addressing RP processes in the textbook
The most common category, providing a nonproof argument, indicates that for most of the 66
RPTs PTEs were asked to provide mathematical explanations (i.e., rationales or empirical
arguments) that were not necessarily proofs (Figure 9). Such tasks provide important
opportunities for PTEs to create mathematical justifications, but also have implications for the
implementation of tasks and the extent to which the TA and PTEs may need to talk about
whether or not the justifications produced were or were not proofs and why.

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5.3 Purposes of Explorations of Mathematical Relations, Conjectures, Evaluations, and Proofs
The exploration of mathematical relationships can lead to the generation of conjectures,
which can then be evaluated, revised, or proven. Thus, the RPQs and RPTs were also analyzed
for these purposes (Dimension 2). This analysis investigated how PTEs were asked to engage in
sequences of related RP processes and what roles of proof were emphasized.
5.3.1 Purposes of Investigating, Conjecturing, and Evaluating
Providing PTEs with opportunities to engage in sequences of related RP supports the
development of a richer understanding of RP processes and their classroom use. A major role
that investigating patterns plays in mathematics is they can lead to the development of
conjectures. If such an investigation served the purpose of a conjecture precursor, it was
counted as an opportunity for PTEs to understand the relationship between investigating
mathematical relations and generating conjectures. Conjectures can lead to the development
of proofs, supporting the distinction of whether PTEs were asked to make conjectures that
forerun the development of proofs associated with these conjectures (proof precursors) or not
(proof non-precursors). Similarly, an evaluation of a provided mathematical claim could serve
the purpose of a conjecture precursor or proof precursor and an evaluation of a mathematical
argument could serve the purpose of refining proofs or providing nonproof arguments for why
an argument was or was not valid. These situations were considered as opportunities for PTEs
to understand the trial-and-error and revision processes of generating and refining proofs.
5.3.1.1 Considering the Analysis of the RPQs
The RPQs identified as providing opportunities to investigate mathematical relations led
to opportunities for PTEs to generate conjectures 50% of the time and did not press PTEs to

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generate a conjecture the other 50% of the time. When a question asked for a conjecture, 89%
of the time there was no press to provide a related proof. All instances of evaluating
mathematical claims and arguments did not serve as a precursor for a new claim or refined
argument: PTEs evaluated claims or arguments as valid or not, but were not asked to produce a
revised claim or argument, which could have provided opportunities for them to engage in the
iterative process of generating and refining arguments.
5.3.1.2 Considering the Analysis of the RPTs
At the task level, however, there were additional opportunities for PTEs to experience
these processes of proving that were not captured at the question level. For example, at the
question level, every instance of evaluating mathematical arguments either stopped after
something was identified that was correct or incorrect about the argument or asked PTEs to
provide a rationale (a nonproof argument). There was a task, however, Activity 11H, where
PTEs would evaluate a mathematical argument that served as a proof precursor for the next
question in the task. That is, PTEs were asked to evaluate an argument that incorrectly tried to
prove the formula for the area of a triangle when the height was drawn outside of the triangle
(in 11H.2). Their evaluation of the provided mathematical argument was a precursor for the
development of a proof in the next question in the task (11H.3), asking PTEs to prove the
formula for the area of a triangle for any such triangle.
To capture such opportunities for PTEs to progress through RP processes, the 66 tasks
were grouped based on the types of RP opportunities in tasks, as described in Table 10:

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Table 10. Categories for Types of Reasoning-and-Proving Opportunities in Tasks
Category
Mathematical
Generalizations
(MG)
Mathematical
Arguments-Nonproofs
(MA-N)
Mathematical
Arguments-Proofs
(MA-P)
Generalizations and
Arguments-Nonproofs
(GA-N)
Generalizations and
Arguments-Proofs
(GA-P)

Description
Tasks that focus on investigating mathematical relationships,
generating conjectures, or evaluating mathematical claims (the
generalization side of the RP-framework), but do not prompt PTEs to
generate or evaluate mathematical arguments.
Tasks that focus on evaluating arguments or generating nonproof
arguments (the mathematical argument side of the RP-framework), but
do not press PTEs to provide a proof.
Tasks that only include questions from the mathematical argument side
of the framework, and must contain at least one prompt for PTEs to
generate a proof.
Tasks that include at least one mathematical generalization process
and at least one mathematical argument process, but there are no
explicit prompts for a PTEs to generate proofs.
Tasks that include at least one mathematical generalization process
and at least one prompt for providing a proof.

Grouping the tasks into these five categories provided a way to examine the nature of the types
of RP processes elicited in written tasks (Table 11).
Table 11. Opportunities for PTEs to Engage in RP processes within the 66 RP Tasks
Type of RPT
Generalizations & Arguments: Proofs
Generalizations & Arguments: Nonproofs
Mathematical Arguments: Proofs
Mathematical Arguments: Nonproofs
Mathematical Generalizations

Number of RPTs of this type (% of 66 RPTs)
7 (10.6%)
17 (25.8%)
5 (7.6%)
22 (33.3%)
15 (22.7%)

Considering these last two types of RPTs, approximately 36.4% of the 66 RPTs provided
opportunities for PTEs to generate or evaluate conjectures and then provide some
mathematical arguments for their generalizations. In 17 of those 24 tasks, however, PTEs were
asked to generalize and provide arguments that did not call for a proof, suggesting a prevalence
of non-proof arguments elicited in the text.

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5.3.2 Purposes of Proofs
Aligned with the purposes of proof examined in the interviews, contextualized for a
textbook examination, questions that elicited proofs were analyzed to note if the proof would
serve one or more of the following purposes: explanation, verification, falsification, and
generation of new knowledge. Most of the 27 RPQs that provided opportunities for PTEs to
produce proofs served multiple purposes. A majority of these RPQs served the purpose of
explanation (81%), and about 70% were coded as verification. Approximately one out of ten
(11%) contributed to the generation of new knowledge. Only 7% were coded as falsifications.
These results suggest that PTEs may experience some purposes of proof so infrequently (i.e.,
generation of new knowledge and falsification) that they do not develop a robust
understanding of all the roles of proof in mathematics.
At the task level, 12 of the 66 RPTs provided opportunities for PTEs to produce proofs
and these proofs served multiple purposes. The frequencies of the four codes at the task level
were fairly comparable to those at the question level (83%, 58%, 17%, and 8%, respectively).
The most salient purpose of proof was for PTEs to experience the explanatory power of proof,
which is a role that many researchers have called for more emphasis on in school mathematics
(de Villiers, 1999; Hanna, 2000).
The utility in considering the purposes of proofs at the task-level was particularly
apparent when considering how RP opportunities from one question in a task led to particular
RP opportunities in another question in the same task in a way that represented how
mathematicians make meaning in mathematics. That is, when examining one activity (8J) a
mathematical chain was noticed where the completion of a proof could lead to a conjecture

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about another mathematical relationship. Specifically, 8J.1 (question 1 of Activity 8J) asked PTEs
to generate a conjecture about the measures of alternate interior angles of two parallel lines
and prove their proposed relationship was true. Next, 8J.2 asked PTEs to generate a conjecture
about the sum of the measures of the interior angles of a triangle based on what they
discovered in 8J.1. PTEs were provided with an opportunity to engage in a mathematical
process in which mathematicians frequently partake—stating a conjecture and proving that it is
true (in 8J.1), then using that proven result to generate a new conjecture (in 8J.2).
5.4 Opportunities to Reflect on Characteristics of Proof and Proving Rather than to Do Proof
In the third dimension of the RP-framework, opportunities for PTEs to reflect on
features of proofs and proving instead of doing proofs were examined. Providing opportunities
for PTEs to articulate what makes an argument a proof or to reflect on the proving process
could help prepare them for teaching about what constitutes a proof. In this analysis, however,
only 7 questions were coded as providing opportunities for PTEs to reflect on characteristics of
proofs and proving. The seven questions asked PTEs to consider the importance of the
generality of proofs (4 questions) or how proofs should be logical arguments that were based
on mathematical facts (3 questions), coded as characterizing proofs. Three of the questions
about the generality of proofs also asked PTEs to compare the argumentation form and discuss
whether or not that form of argumentation was a proof and why. These instances talked about
how one can distinguish proofs from nonproofs, and in these particular cases emphasized
whether or not the argument was general in making the distinction. This small number of
opportunities for PTEs to talk about the characteristics of proofs and talk about how to identify

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proofs and nonproofs suggests an aspect of RP that was not readily accessible in these
geometry and measurement chapters of the textbook.
5.5 Conclusions about the Textbook Analysis
This examination of the geometry and measurement questions and tasks indicates
Beckmann’s (2008) text provides a range of opportunities for PTEs to engage in RP. Throughout
these chapters, PTEs are asked to investigate patterns, generate conjectures, and justify
mathematical claims with rationales or with proofs. There are also opportunities for PTEs to
engage in many RP processes within one task, meaning they may experience the iterative
process of generating and refining arguments. This analysis provides a more detailed view of RP
opportunities in Beckmann’s text than previous analyses (McCrory et al., 2008) by identifying
how PTEs could have opportunities to engage in specific aspects of the activity of RP.
These results have a number of implications for how this textbook might be used. First,
the small number of questions addressing Dimension 3 of the framework coupled with the
infrequent opportunities for PTEs to evaluate claims and arguments suggests a potential lack of
opportunities to discuss features of proofs and proving. It was not initially surprising that there
were a small number of questions coded for Dimension 3 (Meta-Level: Reflecting on Proof)
because one might anticipate such discussions being more evident in classroom practices than
in a textbook. For such discussions to occur, however, instructors need to recognize the
importance of having such discussions and these discussions could potentially flow out of
questions designed for PTEs to evaluate claims and arguments. With so few questions about
evaluating claims or arguments, there would not necessarily be opportunities to discuss norms
of proving or to reflect on different types of reasoning explicitly during classroom instruction.

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Second, only seven of the 66 RP tasks provided opportunities for PTEs to prove
generalizations that they themselves had generated (Table 11). This result contrasts with how
mathematicians engage in RP; they typically explore patterns, which can lead to new
generalizations, which can then be tested and revised or proven (Lakatos, 1976; NCTM, 2009).
This finding suggests that when asked to prove mathematical statements in the textbook, PTEs
would more often prove statements that were provided to them and not ones that they had
generated. Admittedly, in mathematics teaching, there is always a tension between how much
to let students discover on their own and when to tell them something (i.e., provide a
mathematical statement) and ask them to do something with that information (i.e., prove that
the statement is valid). In the context of MTH 2, however, PTEs need to develop the ability to
engage in proving processes to also teach others. Providing more opportunities for PTEs to
generate their own conjectures and work through proving and refining their conjectures could
potentially motivate them to explore mathematics and help them experience the process of RP.
Third, as was particularly apparent in Figure 9 and Table 11, a majority of the RP
opportunities were about generating nonproof arguments. This prevalence of nonproof
arguments suggests that the textbook would likely be implemented to provide opportunities for
PTEs to explain why mathematical statements were valid, particularly in the form of informal
arguments. There are positive and negative potential outcomes of the prevalence of that type
of RPT in the textbook: Such a focus could provide helpful opportunities for PTEs to practice
communicating their mathematical reasoning more clearly, important for demonstrating their
mathematical understanding and being prepared to explain ideas to their future students.

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Since many prospective and practicing teachers maintain the misconception that
empirical arguments are proofs (Knuth, 2002b; Steele, 2006), however, when PTEs are taking
mathematics courses for teachers it is important to help them transition from this way of
thinking to a more robust view of the role of proof in mathematics. Although the inclusion of
rationales, explanations, and empirical arguments could help PTEs articulate how they are
thinking about mathematical ideas, without some contextualization and thoughtful
implementation on the part of the instructor, PTEs may walk away from a course using this
textbook with that conception maintained or reinforced. To design instruction without
reinforcing this misconception, for example, an instructor could choose to modify some tasks
by pressing PTEs to provide a proof even when not called for in tasks. An instructor may instead
choose to contextualize the fact that the argument PTEs provided was not a proof and facilitate
a class discussion about features that make arguments proofs and nonproofs. Modifying and
contextualizing tasks from the textbook is an important aspect of instructors’ teaching practice
that needs to happen to broaden PTEs’ experiences with and conceptions of RP, and how RP
tasks from this textbook were enacted is described in the next chapter.

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CHAPTER 6
ENACTMENT OF REASONING-AND-PROVING TASKS
I observed each TA’s classroom instruction for at least six hours to see them enact RPTs
that afforded various opportunities for PTEs to engage in RP processes. I applied the
Mathematical Task Framework to examine how TAs set up and implemented RPTs by
maintaining or modifying RP demands of tasks during classroom instruction. This analysis,
therefore, shows how RPTs were enacted to provide a range of opportunities for PTEs to work
on RP processes, reveals patterns of enactment across different types of tasks and TAs, and
suggests some factors that contributed to modifications of the RP demands of RPTs.
To understand the opportunities for PTEs to engage in RP in MTH 2, the analytical
approach is summarized next. Then, the results are presented by first considering the
enactment of all RPTs observed across all TAs. These results are then broken down by type of
RPT to understand the enactment of tasks that afforded opportunities for PTEs to engage in
different RP processes. Next, the results from individual TA’s classroom practices are presented
to look for patterns in opportunities for PTEs to engage in RP in the individual classrooms. The
final section summarizes the nature of opportunities for PTEs to learn about RP observed in this
study and outlines implications for future research and professional development.
6.1 Analytical Approach for Examining Enactment of Reasoning-and-Proving Tasks
The goal of this analysis was to address the second research question about how TAs
enacted RPTs in MTH 2. I organized the observation schedule by considering the tentative
pacing guide for the course and the opportunity to see TAs implement different types of RPTs.
Based on this observation schedule, I observed the enactment of 35 RPTs, accounting for over

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half of the 66 identified RPTs in the textbook (Table 9). Because TAs occasionally relied on
resources other than the textbook (Beckmann, 2008), I also saw TAs enact 11 RPTs from other
resources. Whenever schedules allowed, I observed the enactment of these 46 RPTs in more
than one classroom. Altogether, I observed 82 enactments of RPTs during the semester.
The 82 observed RPTs entailed engaging PTEs in one or more of the RP processes
detailed in the first dimension of the RP-framework (Table 2). The purpose of this analysis was
to understand how RP demands of these tasks changed or did not change during enactment of
the tasks. The enactment was analyzed by comparing the potential for PTEs to engage in RP in
the task-as-written with the opportunities for PTEs to engage in RP during classroom instruction
(explained in more detail in Section 3.4.1), indicating when RP demands were (a) increased, (b)
decreased, (c) shifted, or (d) not changed. An increase in RP demands means that PTEs
generated or evaluated a process of RP that was not included in the potential RP opportunities
in the written task. A decrease in RP demands refers to instances where PTEs did not
participate in one of the major RP processes that were elicited in the task-as-written. A shift in
RP demands means that during enactment PTEs had opportunities to generate generalizations
or arguments that were different from the task-as-written, but the overall types of RP
processes that PTEs engaged in were not different from the task-as-written. Finally, no change
in RP demands refers to instances where there were no modifications in the RP opportunities
for the task and PTEs engaged in the RP processes that were elicited by the task.
6.2 Overall Results from Analyzing the Enactment of Reasoning-and-Proving Tasks
The analysis of enactments of 82 RPTs compared the RP opportunities in the tasks-aswritten with the RP processes PTEs engaged in during enactment. The enactment of seven tasks

96

and 27 tasks resulted in increases and no changes in RP demands, respectively (Figure 10). That
is, in approximately 41% of the instances when RPTs were observed, PTEs either engaged in
more RP processes than were anticipated for a given task (when RP demands increased) or
engaged in the RP processes that were elicited in these tasks (when there was no change). In
these instances, PTEs worked with their classmates in small groups and engaged in whole-class
discussions by generating and evaluating conjectures and justifications.

No Change
32.93%
Decrease
51.22%

Shift
7.32%

Increase
8.54%

Figure 10. Enactment of RP tasks, results from observing eighty-two tasks enacted
Moreover, for six of the observed RPTs, TAs shifted RP demands without changing the overall
RP processes in which PTEs engaged (Figure 10). That is, TAs changed the order in which
questions were asked or added additional questions to tasks, but PTEs investigated
relationships, generated and evaluated conjectures, or generated and evaluated mathematical
arguments that were expected. During the enactment of 40 tasks, therefore, PTEs had
opportunities to work on RP processes that were called for, or went beyond, the written task.

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When during the enactment of the other 42 tasks, however, the RP demands decreased.
Thus, approximately 51% of the time PTEs did not engage in at least one RP process that was
elicited in an RP task-as-written. Two different types of decreases were observed: (1) RP
aspects from the task were excluded during enactment and (2) TAs generated RP processes
instead of PTEs. These types of decreases are conceptualized in the following way: for type (1),
PTEs were not held accountable for engaging in specific RP processes from the task. For type
(2), during the class the TA, instead of PTEs, generated one (or more) aspect of RP that the task
asked PTEs to determine. The results from observing the enactment of 82 RPTs are displayed in
Figure 11, including the prevalence of these two types of decreases. Accounting for 16 of the 42
RPTs where RP demands were decreased, decrease type (1), where TAs excluded or did not
emphasize RP processes in tasks, was less frequent than type (2).

Decrease (2):
TA Generated
31.71%

No Change
32.93%

Decrease (1):
RP-process
Excluded
19.51%

Increase
8.54%

Shift
7.32%

Figure 11. Enactment of RP tasks including different types of decreases in RP demands

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To understand the nature of the classroom interactions when TAs excluded RP
processes in tasks (decrease type 1), consider the two primary situations when this occurred.
First, during the task set up, PTEs were told to only work on specific questions in a task or to not
focus on parts of questions that provided opportunities for PTEs to engage in RP processes. This
set up decreased what PTEs were expected to do in regards to RP during the task. Second,
during implementation PTEs were not held accountable for producing RP processes elicited in
the task. In these instances, individual work, group work, or whole-class discussions ended
without addressing all RP processes in the task. For example, a task that asked PTEs to generate
a conjecture and explain why it was valid resulted in a decrease of this type when Peter asked
PTEs to move on to the next task after generating the conjecture. Furthermore, these types of
decreases often occurred in Laura’s class when PTEs offered explanations that were not
mathematical or did not explain mathematically why, but Laura and the class accepted these
non-mathematical arguments as justifications and moved on. In these examples, the
conjectures created were the focus and not how to explain its validity or what constitutes a
valid mathematical justification for the conjecture. In general, when this type of decrease
occurred during implementation it was often because TAs affirmed PTEs’ responses even when
they did not fully address all RP processes called for in the task.
For the more common type of decrease, accounting for about 62% of the decreases
observed (i.e., 26 of the 42 RPTs where decreases occurred), TAs generated conjectures,
rationales, or proofs for PTEs. The prevalence of this decrease type 2 (Figure 11) implies a
particular pedagogical choice TAs made frequently. In four of these 26 instances, when setting
up tasks, TAs told PTEs a conjecture (twice), empirical argument (once), or proof (once) before

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PTEs began working on tasks that included opportunities for them to generate these
conjectures and arguments themselves. During the other 22 instances, TAs generated the RP
processes during implementation of the tasks.
There were three prominent ways TAs and PTEs interacted during implementation that
characterized these instances: First, during group- or individual-work time PTEs stopped
working on a task or asked TAs for guidance in generating conjectures or arguments when they
experienced a struggle or confusion. During these interactions TAs were unsuccessful at guiding
PTEs through their struggle without generating the RP process for PTEs. Sometimes TAs’
responses to PTEs’ questions were general (e.g., “just keep trying”, Ace, observation 2, 53mins)
and forward progress on the task did not continue. More often, however, TAs would tell PTEs
an answer, where to find an example explanation in the textbook, or that they (the TA) would
provide an answer for the class in a few minutes. Then, PTEs would read in the textbook or hear
from the TA what an appropriate conjecture, proof, or rationale could be for the task, but PTEs
did not experience the process of generating or evaluating the RP aspects themselves.
Second, TAs led the class through tasks by telling and showing PTEs how to complete
them. In these situations, PTEs often did not have time to think about tasks (individually or with
others) prior to this instruction. During such TA-led discussions, PTEs sometimes provided brief
answers about why approaches shown made sense, but the overall thinking and reasoning was
generated by TAs. PTEs primarily listened, watched, and took notes as TAs generated and
explained mathematical relationships, conjectures, and arguments. Here, again, PTEs were
exposed to ideas about RP and the content in the tasks, but this was regarded as a decrease in
RP demands because PTEs did not participate in the processes of RP the task elicited.

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Third, TAs attempted to elicit PTEs’ ideas during class discussions of tasks but PTEs did
not readily participate and TAs generated the RP processes. These discussions sometimes
occurred after PTEs spent some time working on the task, but not always. To clarify, a class
discussion was not considered to be a decrease in RP processes only because a TA generated a
RP process without input from PTEs. If a majority of PTEs generated RP processes elicited in the
task during individual- or group-work time but they did not share their ideas during a class
discussion that followed the task, the implementation was still regarded as no change in RP
demands. In those discussions, PTEs may not have voiced their ideas because they already
understood them or TAs may have shown examples of the types of answers they had observed
from PTEs’ work. The class discussions that decreased RP demands, however, occurred when
the work of figuring out RP processes associated with the task was still in progress. During these
discussions, TAs often asked for volunteers to share their ideas for specific parts of the task.
Many TAs also used substantial wait time to elicit these volunteers. When TAs were unable to
elicit student participation, or if class time was running out during the discussion, TAs often
chose to generate the RP processes. TAs initiated these class discussions in ways that suggested
they wanted to facilitate a discussion and encourage PTEs to engage in the generation of RP
processes. TAs’ tendencies to eventually lead discussions and difficulty involving PTEs suggest a
need for professional development addressing specific practices (Stein, Engle, Smith, & Hughes,
2008) for facilitating mathematical discussions around high cognitive demand tasks.
6.3 The Types of Reasoning-and-Proving Tasks Observed and How they were Enacted
Parsing these results to see how different types of RPTs were enacted provided insights
into the nature of PTEs’ opportunities to engage in RP in MTH 2. As introduced in the textbook

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analysis, RPTs were grouped according to the type of RP processes in which PTEs could engage.
This grouping provided a means for examining what RP processes were elicited in written tasks
(in Chapter 5) and whether PTEs engaged in those RP processes in class. The five general types
of RPTs, introduced in Chapter 5 (Table 10), were: Mathematical Generalizations (MG),
Mathematical Arguments-Nonproofs (MA-N), Mathematical Arguments-Proofs (MA-P),
Generalizations and Arguments-Nonproofs (GA-N), and Generalizations and Arguments-Proofs
(GA-P). The number of observed RPTs of each type is displayed in Table 12.
Table 12. Types of Reasoning-and-Proving Tasks Observed: Opportunities to Prove in the Tasksas-Written

Type of RPT
Task-As-Written
MG
MA-N
MA-P
GA-N
GA-P
TOTALS

Number
of RPTs
Observed
14
19
24
14
11
82

% of
82
RPTs
17.1%
23.2%
29.3%
17.1%
13.4%
100%

Number of each Type of RPT Observed for each TA
Ace

Laura

Isaac

Peter

Evan

Kelly

2
3
3
2
3
13

3
5
3
1
1
13

3
4
5
2
0
14

1
4
4
4
3
16

3
2
2
3
3
13

2
1
7
2
1
13

The type of tasks observed most frequently (MA-P) provided opportunities for PTEs to
generate mathematical arguments including at least one question that expected PTEs to
generate a proof (Table 12). Considering the GA-P tasks along with MA-P tasks, approximately
43% of RPTs observed explicitly asked PTEs to generate proofs. Examining how these RPTs were
enacted could reveal whether or not TAs were more likely to generate RP processes for PTEs
when the task called for a proof. Furthermore, approximately 40% of the RPTs observed did not
ask PTEs to generate a proof, eliciting nonproof (or informal) arguments instead (in MA-N and
GA-N tasks). For these tasks, did TAs press PTEs to produce a proof, thereby increasing the RP

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demands and emphasizing what constitutes a proof? The examination of the enactment of the
14 MG tasks observed could also provide insights into if TAs often pressed PTEs to generate a
rationale or proof when the written task did not.
Table 13. Enactment of Different Types of Reasoning-and-Proving Tasks
Enactment of Tasks
Type of RPT No Change Increase
MG (14)
4
3
MA-N (19)
7
0
MA-P (24)
9
2
GA-N (14)
4
2
GA-P (11)
3
0
Totals (82)
27
7

Shift
1
0
3
1
1
6

Decrease (2):
TA Generated
3
4
10
5
4
26

Decrease (1):
RP process Excluded
3
8
0
2
3
16

Examining how often TAs maintained or changed the RP demands of RPTs provides
insights into the nature of opportunities for PTEs to engage in RP in MTH 2. These results are
presented (Table 13), from which three notable patterns are highlighted: First, consider the 27
RPTs enacted with no change to RP demands and six RPTs that were shifted during enactment.
These enactments were reasonably spread out among the five different types of tasks. Tasks
that were shifted meant that PTEs engaged in the RP processes that were elicited in the task,
but in different ways or around slightly different content than what was expected from the
written task.
Consider, for instance, a GA-N task with three questions about determining volumes of
prisms and cylinders. In the first two questions, PTEs were asked to investigate mathematical
relationships and generate a conjecture about a formula for the volume of triangular prisms
and parallelogram-base prisms. The third question provided similar prompts for a formula for

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the volume of cylinders but also asked PTEs to justify (informally) why their formula was valid.
During enactment, however, Evan asked PTEs to explain why the formula for triangular prisms
had to be true, and he did not ask them to justify the formula for cylinders. PTEs still
investigated, generated conjectures, and generated a rationale, but the rationale was for a
different conjecture. In general, the way that these shifts and instances of no change in RP
demands were spread out across different types of RPTs suggests that PTEs engaged in a range
of RP processes during MTH 2. For various types of tasks (eliciting generalizations, informal
arguments, or formal arguments) there were a number of observations during which PTEs were
asked to engage in these RP processes and they did so during implementation of these tasks.
Second, TAs did not increase RP demands of MA-N tasks during classroom instruction
(Table 13). These tasks typically asked PTEs to provide nonproof justifications for mathematical
claims, such as explaining why a right pyramid that is 1 unit high and has a 1-unit-by-1-unit
square base has volume 1/3 cubic units. In some MA-N tasks, the textbook specifically stated
that a general proof for the mathematical relationship being explored was outside the scope of
the class, but encouraged PTEs to provide nonproof arguments or demonstrations to develop
intuitive notions about mathematical relationships without providing a formal proof. In the
textbook there were also statements about a proof being outside the scope of the class for
some of these types of tasks. When these tasks were observed, TAs did not press PTEs to
provide a proof. This instructional decision could affect PTEs’ understanding of what constitutes
a proof and perpetuate potential misconceptions that favor empirical arguments over proofs.
Third, when RP demands for MA-P tasks were decreased, it was only because TAs
generated the proofs for PTEs (Table 13). This type of decrease in RP demands was prevalent

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across most of the types of tasks, but it is worth noting that it was most prominent for the MAP tasks. For these tasks, the primary RP focus was on eliciting proofs from PTEs, but during
implementation when there was a decrease it was because the structure, main ideas, and
overall creation came from TAs with little or no student input. There were also no instances
observed where a TA generated the proof and the class had an open discussion to evaluate
aspects of the provided argument. Instead, the provided proof was regarded as a product to
remember and PTEs occasionally asked clarifying questions about why TAs included particular
steps or reasons. This choice could affect PTEs’ understanding of the processes of RP and the
development of their own abilities to construct and evaluate mathematical arguments.
6.4 Examining each TA’s Enactment of Reasoning-and-Proving Tasks
Since RPTs were observed in six classrooms, individual TA’s classroom practices were
studied for patterns in how RP demands were modified during enactment. In Figure 12 and
Figure 13 the results are presented for the observations of each TA’s classroom across the
semester. Each circle in the figures represents one RPT observed. The placement of the circle
indicates when it was observed during the semester (the week of the semester) and how it was
enacted (RP demands decreased, not changed, shifted, or increased). The opportunities for RP
elicited by the task-as-written are indicated by the color. As a reminder, no change or shifted
means PTEs engaged in the types of RP processes elicited by that task. Increased means PTEs
engaged in more RP processes than what was called for in the written task. For either type of
decrease, PTEs did not engage in at least one RP process elicited by that task. From these
figures, I note patterns in individual TA’s enactments of RPTs across the semester (esp., Ace and
Isaac) and of specific types of RPTs (esp., Kelly and Peter).

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Color Represents the Type of Reasoning-and-Proving Task Before Enactment
MG
MA-N
MA-P
GA-N
GA-P

Ace’s Class

Increased
Shifted
No Change
Decreased:
Excluded
Decreased:
TA Generated
1
Increased

2

3

4

5

6

7

8

9

10

11

12

13

14

15 (Semester Weeks)

Isaac’s Class

of Semester)

Shifted
No Change
Decreased:
Excluded
Decreased:
TA Generated
1
Increased

2

3

4

5

6

7

8

9

10

11

12

13

14

15 (Semester Weeks)

Laura’s Class

of Semester)

Shifted
No Change
Decreased:
Excluded
Decreased:
TA Generated
Figure 12. Individual TA's enactments of RPTs. Each circle represents one task observed
during a specific week of the semester: Ace, Isaac, and Laura

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Color Represents the Type of Reasoning-and-Proving Task Before Enactment
MG
MA-N
MA-P
GA-N
GA-P

Kelly’s Class

Increased
Shifted
No Change
Decreased:
Excluded
Decreased:
TA Generated
1
Increased

2

3

4

5

6

7

8

9

10

11

12

13

14

15 (Semester Weeks)

Evan’s Class

of Semester)

Shifted
No Change
Decreased:
Excluded
Decreased:
TA Generated
1
Increased

2

3

4

5

6

7

8

9

10

11

12

13

14

15 (Semester Weeks)

Peter’s Class

of Semester)

Shifted
No Change
Decrease:
Excluded
Decreased:
TA Generated
Figure 13. Individual TA's enactments of RPTs. Each circle represents one task observed
during a specific week of the semester: Kelly, Evan, and Peter

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6.4.1 Individual TA’s Enactment of Reasoning-and-Proving Tasks across the Semester
Considering each TA’s enactment of RPTs across the semester indicates some noticeable
trends in the enactments of RPTs. As presented in Figure 12, Ace and Isaac’s patterns of
enactment across the semester were quite different. Ace had a tendency to decrease the RP
demands of all types of RPTs throughout the semester. As PTEs worked on tasks, Ace would say
things to encourage them to keep working, but in the same sentence remind them that in a few
minutes he was planning on pulling the class back together to tell them what they should have
figured out. PTEs often responded to these comments by slowing or stopping their own
progress on the task. Then, during whole-class discussions, Ace would often generate RP
processes for PTEs with little or no student input. For many of the observed RPTs, therefore,
PTEs in Ace’s class did not work on the RP processes themselves. In Isaac’s class, however, he
tended to enact most of the RPTs without changing the RP demands throughout the semester.
PTEs in Isaac’s class were observed engaging in RP processes across all types of RPTs
throughout the semester. These differences in enactment patterns show variation between TAs
that has implications for PTEs’ opportunities for learning about RP.
The noticeable patterns from Laura’s class (Figure 12) are similar to Isaac’s toward the
beginning of the semester because she tended to shift or not change the RP demands of tasks.
Toward the end of the semester, however, Laura began excluding or generating RP processes
for PTEs more often. Kelly’s and Evan’s enactments of RPTs across the semester (Figure 13)
show this type of trend in a little more pronounced way, as they tended to generate RP
processes for PTEs more from the middle through the end of the semester. These patterns
suggest that there may be some relationship between the week of the semester and some TAs’

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tendencies to decrease RP demands of tasks. Providing instructional support for TAs’ teaching
throughout a semester is therefore important because at different points there could be certain
content, external time constraints (i.e., staying on pace with the syllabus or demands on TAs’
time in their own courses and research), or classroom interactions and dynamics, through
which TAs may need help navigating so they do not decrease the opportunities for PTEs to learn
about RP.
Peter’s enactments of RPTs do not seem to be strongly correlated with time, but a
majority of the tasks observed implemented by Peter were decreased by excluding RP
processes. A tendency for this type of decrease was not realized in the other TAs’ data, and
could suggest that Peter was comfortable modifying tasks and that he often did so in a way that
decreased the RP processes with which PTEs were expected to engage.
Considering these general patterns for the six TAs suggests avenues for continued
research into teachers’ instructional decisions about RP. In particular, all six TAs used the same
textbook, which emphasized nonproof arguments and provided a range of opportunities for
PTEs to engage in RP processes. Despite this commonality, TAs enacted RPTs differently. This
variance suggests the importance of understanding how teachers make instructional decisions
and helping them develop abilities to understand and modify tasks and instructional resources.
Investigating what factors contributed to TAs’ enactment of RPTs could inform the design of
courses and support of instructors for such courses, and is explored further in the next chapter.
6.4.2 How Individual TAs Enacted Different Types of Reasoning-and-Proving Tasks
Another pattern evident in individual TA’s enactments of RPTs was how they enacted
different types of RPTs. Since different numbers of the types of tasks were observed for TAs,

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the data did not indicate specific patterns along this dimension for each participant (Figures 12
& 13). Kelly’s enactment of MA-P tasks (Figure 13), Laura’s enactment of MA-N tasks (Figure
12), and Peter’s enactment of both of these types of tasks (Figure 13), in particular, provide
some additional insights into PTEs’ opportunities to learn about RP in these classes.
In Kelly’s class, tasks that provided opportunities for PTEs to generate proofs for
provided mathematical statements (MA-P tasks) were observed most frequently. During week
eight, Kelly enacted four MA-P tasks: for two of these tasks PTEs generated proofs as elicited in
the task; for the third task Kelly pushed PTEs farther in RP by asking them to investigate
mathematical relationships, generate conjectures, and then prove them instead of proving a
provided statement; and for the fourth task Kelly generated a proof that the opposite sides of a
parallelogram were congruent when PTEs struggled to generate the proof. After week eight,
Kelly enacted MA-P tasks by generating proofs for PTEs during class discussions. These class
discussions occurred after PTEs had worked on the tasks in small groups, but had made little
progress on the task beyond listing some given information or definitions. Kelly often
encouraged the class to work together during the discussion to construct the proof, but PTEs
were not vocal during the creation of these proofs. As the semester continued they explicitly
asked Kelly to just show them the proof because they wanted to see how she would write it
instead of trying to generate the proof themselves. The PTEs seemed to value the end result of
these tasks, the proof that they asked Kelly to generate for them, instead of the process of
reasoning and proving.
In Laura’s class, as the semester continued, she tended to decrease the RP demands of
MA-N tasks by excluding the rationales the tasks elicited (Figure 12). In particular, MA-N tasks

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observed during weeks eight and eleven were designed to elicit rationales. In this study,
rationales were conceptualized as valid arguments for or against mathematical claims that did
not constitute formal proofs. When Laura enacted these tasks, however, PTEs tended to restate
what mathematical claims meant or state theorems or facts they thought might be related to
mathematical claims without explaining why claims were valid. For example, a task asked PTEs
to evaluate an argument and then explain why the provided argument was or was not valid.
The PTEs quickly noticed that a statement in the argument (that one could scale a poster
additively instead of multiplicatively) was mathematically incorrect, but they did not provide
justifications for why other than saying “that’s not the right way to think about scaling.” Laura
did not press PTEs to explain why the additive method was incorrect or note any other features
of the argument that were problematic. In this manner, Laura tended to not help PTEs refine
their understanding of rationales when she enacted MA-N tasks, which could reinforce PTEs’
misconceptions about what constitutes a proof, nonproof, and valid argument in mathematics
and in teaching.
Peter displayed an overall pattern dissimilar from Kelly because he increased or did not
change RP demands of MA-P tasks, but similar to Laura because he decreased RP demands of
MA-N tasks (Figure 13). For the MA-N tasks where he decreased RP demands, however, he did
not emphasize informal explanations as Laura did. Instead, Peter excluded parts of tasks in such
a way that he took away opportunities for PTEs to provide rationales. That is, when Peter
excluded RP processes from MA-N tasks he noted that they had worked on some of the
concepts for the tasks earlier in the week (when I was not observing) and he focused their
attention on the application questions in the task. He therefore diverted PTEs’ attention away

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from the rationale-related questions, but contended that PTEs had talked about these ideas in
previous class periods. Since Peter did not make this pedagogical choice explicit to PTEs, the
way RP aspects of these tasks were excluded could affect how PTEs think about tasks that ask
for informal justifications.
Peter enacted MA-P tasks differently from Kelly in that he enacted all but one of the
observed MA-P tasks by engaging PTEs in the RP demands of the tasks or increasing the RP
demands of the tasks. When Peter increased the RP demands of an MA-P task, he prompted
PTEs to generate conjectures and refine their conjectures by learning from failed attempts to
prove they were true. Since Peter enacted these types of tasks in these ways, PTEs in his class
often generated proofs and engaged a proving process through which they tested and refined
conjectures and arguments during observations of his teaching. By examining how Peter
enacted specific types of tasks, an emphasis on engaging PTEs in generating and refining proofs
emerges as a prevalent theme observed in his practice.
6.5 Conclusions about the Enactment of Reasoning-and-Proving Tasks
These results from analyzing six TAs’ enactment of 82 RPTs during the semester indicate
that PTEs engaged in a range of RP processes for about 49% of the observed tasks because TAs
maintained, increased, or shifted RP demands. Increases in RP demands occurred for three
types of tasks: (1) MG tasks, where PTEs provided justifications for generalizations when the
written task did not call for justifications; (2) MA-P tasks, where PTEs investigated mathematical
relationships and generated conjectures for tasks that only asked them to generate
mathematical arguments, including a proof; and (3) GA-N tasks, where PTEs generated proofs
when the task-as-written only pressed for rationales. Although these were the only types of

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tasks where increases in RP demands were observed, other types of increases were possible.
For MA-N tasks, the task-as-written provided a statement for which PTEs were asked to
generate informal justifications. To increase RP demands of MA-N tasks, for instance, one could
ask PTEs to investigate mathematical relationships and generate conjectures originally provided
in tasks. Such modifications could help PTEs generate justifications based on their investigations
into mathematical relationships and experience the process of generating and refining
conjectures. The lack of increases to RP demands for MA-N tasks suggests the need for helping
TAs understand why and how to modify tasks to help PTEs engage in RP processes.
For the other 51% of observed tasks, TAs decreased RP demands by either excluding
aspects of tasks that elicited some RP processes or generating RP processes for PTEs. These
different types of decreases in RP demands suggest opportunities for supporting instructors
who are teaching about RP. When TAs did not consistently hold PTEs accountable for RP
processes in tasks (as was described with Laura’s enactment of MA-N tasks), PTEs’
misconceptions of RP (e.g., what constitutes a mathematical argument) may have been
reinforced. Reinforcing PTEs’ misconceptions about RP in MTH 2 could lead to less authentic RP
experiences in their future elementary classrooms. TAs who tend to decrease RP demands in
this manner may need help understanding how their instruction around tasks can affect PTEs’
opportunities to learn about RP. A potentially applicable professional development strategy
could include specific supports about designing and implementing instruction in ways that hold
students accountable for RP processes in tasks.
For the other type of decrease, when TAs generated RP processes for PTEs, they did so
as PTEs worked on tasks individually, in groups, or during whole-group discussions. Their

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interactions with PTEs during individual- or group-work did not help PTEs work through
difficulties to experiences success with RP processes. These interactions suggest a need for
helping TAs develop specific teaching practices to monitor PTEs’ work to help them to continue
experiencing RP processes. TAs also decreased RP demands by generating RP processes for PTEs
during whole-group discussions. In these situations TAs often had difficulties facilitating class
discussions around RPTs in ways that encouraged PTEs to participate and develop their own
understanding of RP processes. One noticeable factor common to both types of interactions
(TA-to-small group and TA-to-class) was a sense that PTEs tended to be frustrated or
unmotivated. They either did not stopped making forward progress on RPTs or they did not
readily participate and engage with RP during class discussions.
A better understanding of why PTEs responded in these ways is outside the scope of this
study. It could be motivating to PTEs, however, if TAs explicitly articulated the RP aspects they
wanted PTEs to experience in working through tasks and how the RP processes could prepare
them as future teachers. The RP-framework is one means for helping TAs identify RPTs and
develop TAs’ language for talking about RP processes during instruction so they can
communicate the importance of PTEs engaging in and experiencing these processes in MTH 2.
These results also highlighted how TAs implemented RPTs differently. This variation
suggests that using a common textbook with different opportunities for PTEs to engage in RP
processes does not mean that PTEs across the course sections will have the same opportunities
to experience RP. The different course sections have PTEs with their own experiences with and
views of RP, and TAs approach the course, interpret the text, and plan and implement
instruction based on their own conceptions of RP, contributing to the variation.

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CHAPTER 7
TEACHING ASSISTANTS’ INSTRUCTIONAL DECISIONS
This chapter presents the results from examining the relationship between TAs’
conceptions of RP and their instructional decisions when teaching PTEs about RP. As introduced
in Chapter 1, the study of TAs’ instructional decisions is complex due to a large number of
variables affecting their decisions. In the context of this study, I examined a subset of these
variables: To focus on TAs’ RP classroom practices and conceptions, I examined their enactment
of RP tasks, conceptions of RP, selection of RPTs, and classroom interactions (Figure 14).

TA
Conceptions of
Teaching &
Learning RP

Conceptions of
RP in
Mathematics

Classroom Practice
Curriculum
Written Task
Textbook’s RP
opportunities

RP-related Classroom Tasks
Task Setup
RP opportunities
in how task is
introduced

Task
Implementation
How prospective
teachers engage in
RP-processes

Figure 14. Major components of the study
Considering TAs’ decisions directly impacting PTEs’ opportunities to learn about RP, I studied
two types of decisions: (a) TAs’ selection of RPTs and (b) TAs’ in-the-moment decisions around
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RP during instruction. Through this analysis, I investigated how TAs talked about prominent
factors that influenced these decisions and examined the relationship between TAs’
conceptions of RP and their instructional decisions. These results inform specific professional
development supports for TAs teaching about RP.
Using the constant comparative method (Glaser & Strauss, 1967), I analyzed TAs’ postobservation interviews for how they talked about their pedagogical choices after enacting RPTs.
I examined two broad aspects: First, how did TAs explain their choices to select observed RPTs
for classroom instruction? Second, how did TAs talk about their decisions to implement RPTs in
the ways observed? That is, after classroom observations, TAs responded to questions about
why they enacted particular RPTs and how they made classroom decisions during enactment.
Results indicate how TAs’ conceptions of teaching RP and other contextual factors informed
their selection and enactment of tasks. The case studies of TAs’ conceptions of RP (Chapter 4)
and patterns in TAs’ enactment of RPTs (Chapter 6) provide a lens for interpreting these results
and suggest specific professional development implications. The results about TAs’ selection of
RPTs are described first, followed by the analysis of their in-the-moment instructional decisions.
7.1 TAs’ Decisions to Teach Reasoning-and-Proving Tasks
To understand how TAs talked about their RPT selection and how their conceptions of
RP relate to these decisions, the types of responses are described first, followed by the results.
7.1.1 How TAs Articulated their Selection of Reasoning-and-Proving Tasks
TAs talked about their choices for teaching PTEs RPTs by articulating different learning
goals related to RP and other contextual factors (Table 14). Their statements detailed how they
aimed to use tasks to help PTEs: (a) learn what proof is and how to write proofs, (b) understand

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mathematics content through their use of proof and proving, or (c) understand RP processes by
experiencing them in class. They also talked about contextual factors (i.e., other people,
resources, or assessments) as exerting some authority over their selection of tasks.
Table 14. Categories for How TAs Explained Why they Used Reasoning-and-Proving Tasks
A) Learning about the Nature of Proof and Proving
 Know what a proof is
 Produce clear, concise explanations and proofs
 Understand proving conventions
 Learn a specific proof or way to prove a mathematical statement
 Understand the generality of proofs
Learning Goals
B) Understanding Mathematics Content:
Related to
The mathematical task helps PTEs:
Content
 Understand mathematical relationships in the task
 Make real world connections or solve application problems
 Understand mathematics content better
o Recognize the importance of proof in mathematics
o Understand content needed for teaching
o Prepare PTEs to respond to future students if they ask “why”
C) Engaging in and Developing Understanding of RP processes:
The mathematical task helps PTEs
 Generate conjectures
 Evaluate conjectures
Learning Goals
 Evaluate provided arguments
Related to
 Reason through how to construct proofs
Processes
 Explain why mathematical relationships are true
 Develop mathematical ways of thinking (e.g., critical reasoning,
critical thinking, mathematical thinking)
Contextual
Factors

D) Outside Authorities
 Course supervisor, other TAs for MTH 2, or course meetings
 Textbook, syllabus, or pacing guide
 Assessments (e.g., in-class tests, homework, quizzes, and final exam)

These categories emerged from the data and were characterized in four ways: First, TAs
described features or characteristics of proof they wanted PTEs to learn through their work on
the task (Category A). Ace, for example, talked about his selection of a task that asked PTEs to

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prove the Pythagorean Theorem by explaining that he wanted them to learn “what a proof is or
what a proof should be” (post-observation 4, para. 86). Laura also described why she asked
PTEs to work on the same task, as follows:
In general, I want them to be concise. . . . They always write much more than they need
to and I have told them several times, “I just need the information that I need. Like, I
don’t need this whole paragraph about all this other stuff.” And so I think that it’s
important to emphasize being concise because I think by nature mathematicians are
more concise . . . and so I think that it’s a good thing for them to keep in mind when
they are trying to do a proof. (post-observation 4, para. 54)
In these ways, TAs talked about RP content they wanted PTEs to learn (e.g., a specific proof or
what constitutes a proof) to explain why they used an RPT in their teaching.
Second, instead of specifying RP as a content learning goal for RPTs TAs selected, they
described how RP in tasks helped PTEs understand other mathematical content (Category B).
Isaac, for example, chose a task where PTEs explored and justified conjectures about the area
of objects with a fixed perimeter and explained that by working through the task PTEs should
“remember that . . . given a certain perimeter you can’t say what the area is. . . . And . . .
rectangles with the biggest area is (sic) a square and for arbitrary shapes the biggest area is a
circle” (post-observation 5, para. 24). Isaac’s goal was for PTEs to understand and retain
information about relationships about perimeter and area by engaging in RP in the task. In this
category, TAs sometimes also explained how RPTs could help prepare PTEs with content
knowledge they would need as future elementary teachers. Peter, for instance, said PTEs need
to generate rationales for the area formula for a parallelogram,
because they will be teachers and they will show their students to do that kind of thing;
not just memorize the formula but they can do a lot of different things related with that
formula. That’s what I wanted to teach today. (post-observation 5, para. 10)

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As these examples depict, TAs’ statements coded as Category B described content PTEs should
learn as mathematics learners or explicitly as future teachers. In either case, however, they
articulated RP opportunities in tasks as helping PTEs learn content.
Third, TAs talked less about content and more about RP processes or ways of thinking
the task could help PTEs develop (Category C). TAs made general statements about RP
processes they wanted PTEs to learn as important in mathematics, and sometimes specified
how PTEs could use certain ways of thinking in their future classrooms. For a task where
students investigated the validity of and generated justifications for triangle congruence
criteria, for example, Peter explained, in general, that “this activity is important for all students
in mathematics, not just for . . . future elementary school teachers because this . . . justification
or questioning ability is kind of core idea of mathematics” (post-observation 3, para. 136).
Building off of this idea, that the ability to think critically was important for his students to
develop, Peter also specified how PTEs may apply this ability as future teachers, saying,
as a teacher . . . if [they] have critical thinking ability then [they] can actually . . .
question definitions and information in the textbook and . . . help . . . their future
students to question and think mathematics, too. (post-observation 3, para. 140)
In these types of responses, TAs talked about how RPTs prepared PTEs to understand RP
processes and develop mathematical ways of thinking.
Fourth, TAs talked about contextual factors by noting other people, resources, or
assessments that motivated their decisions to enact tasks (Category D). Statements about
contextual factors included: the course supervisor “told us actually we have to have this one”
(Kelly, post-observation 2, para. 88), “I have a piece of paper [referring to the pacing guide] that
says, ‘Cover these activities’” (Ace, post-observation 5, para. 33), and “so that they can know . .

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. for the exam . . . how to do this or . . . a similar proof” (Laura, post-observation 4, para. 24).
TAs referred to interactions with the course supervisor, use of the text and syllabus, or the
importance of preparing PTEs for assessments as contextual factors guiding task selection.
7.1.2 Analysis of TAs’ Responses about Selecting Reasoning-and-Proving Tasks
This analysis aimed to understand how TAs’ conceptions of RP illuminated their decision
to enact RPTs. The analysis of TAs’ post-observation interview responses yielded the four
aforementioned response types, displayed for individual TAs in Figure 15; the percentage
indicates the frequency for each type of response out of the TA’s total number of coded
statements.
Nature of Proof

Kelly

Evan

Mathematics Content

17.33%

11.84%

24.00%

RP-Processes

Outside Authorities

41.33%

43.42%

28.95%

17.33%

15.79%

2.08%
Peter

2.08%
50.00%

Laura

15.79%

Isaac

47.37%

18.52%

Ace

14.04%

37.04%

21.13%
0%

45.83%

11.11%

39.44%
20%

40%

22.81%

33.33%

39.44%
60%

80%

100%

Figure 15. How TAs talked about their decisions to enact reasoning-and-proving tasks
For each TA, percentages are out of their total number of coded statements
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Kelly’s, Evan’s, and Peter’s talk about their selections of RPTs aligned with their
conceptions of teaching RP (Chapter 4) because they articulated RP process learning goals more
frequently than the other three TAs (Figure 15). Although they also talked about mathematical
content as important to teach PTEs, they articulated RP processes as reasons why they selected
RPTs more often than the other TAs. These three TAs talked about RP processes in their own
ways and in manners similar to how they articulated their conceptions of teaching RP: Peter, for
example, talked about selecting tasks that would provide opportunities for PTEs to construct
and refine mathematical arguments as mathematicians do; explaining,
That’s how . . . you do proofs in mathematics actually. . . . So you have some statement
you want to prove and out of nothing you have to construct step-by-step proof. . . . You
can go into the wrong way and then maybe you can figure out what’s wrong and you
can come back to the original idea or through a different thinking you can take a totally
different route but always you should have a bunch of these different thinking in your
mind together and . . . you maybe connect different segments together to get the whole
idea. So that’s how you build your proof mathematically. . . . So . . . if you really wanted
to study proof then you need to experience that kind of knowledge not just memorizing
the step-by-step procedures. You need to actually do this fixing and correcting type of
activity in your mind to learn proof. (post-observation 3, para. 82, 84, 86)
Kelly and Evan continued to focus on helping PTEs develop their ability to think mathematically,
beyond the specific content in MTH 2. Evan explained his selection of tasks to help PTEs work
on that line of mathematical thinking—coming up with proofs for something; taking
knowledge you’ve applied out of the book and theorems and applying them to a novel
situation where you’re trying to prove something else and build on the mathematics you
already know. (post-observation 3, para. 30)
Kelly explained that “it’s not the content . . . I feel the most important thing is that [PTEs] need
to develop their ability to justify things mathematically” (post-observation 1, para. 122) and
I want to accomplish this goal: “how to build on the old knowledge and develop the new
knowledge.” This kind of thinking is similar to the mathematical- like- we use the known
to prove the unknown. As a teacher [they] have to have a good idea of what the
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students know and what [they] want to build around those things. So . . . the kind of
mathematical thinking may transfer to their teaching practice. (Kelly, post-observation
2, para. 136)
Since PTEs will be expected to teach future elementary students how to think and reason
mathematically, these three TAs considered instruction about RP processes as an important
factor in their decision to teach RPTs. This conception about teaching PTEs about RP processes
was apparent both in their conceptions of teaching RP (Chapter 4) and their responses about
their selection of RPTs. In these responses, Peter, Kelly, and Evan were able to articulate how
certain tasks could help PTEs learn RP processes, suggesting strong pedagogical knowledge for
teaching to be able to analyze textbook tasks for particular instructional goals.
In the earlier analyses of Laura’s and Isaac’s conceptions of teaching RP and this analysis
of their selection of RPTs, they sometimes articulated RP processes, but focused on
mathematical content most often (Figure 15). Isaac occasionally talked about selecting tasks
that provided opportunities for PTEs to generate and evaluate conjectures (post-observation 3
and 5). Laura sometimes explained how a task “challenges [PTEs] to think critically” (postobservation 4, para. 24). Most of their talk about the selection of RPTs, however, attended to
mathematical content (including concise explanations for Laura and understanding specific
mathematical relationships for Isaac) they wanted PTEs to understand. Their articulation of a
range of learning goals suggests they have a foundation of understanding how tasks can
support PTEs’ learning of the nature of proof and mathematical content, but only occasionally
made explicit connections to RP processes.
Ace, however, never talked about RP processes he wanted PTEs to learn from RPTs he
chose to enact (Figure 15), which was also reflected in his conceptions of teaching RP (see

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Figure 6). Ace also talked about outside authorities as often as he talked about mathematical
content as influencing his selection of RPTs. When asked why he enacted a task where PTEs
were expected to explain why there are only five platonic solids, for example, Ace said, “I think
[the course supervisor] really likes this one. So it’s very possible that [the course supervisor] will
like to see a question like this on both the exam and the final” (post-observation 2, para. 148).
In addition to this idea that content in tasks could appear on assessments, Ace also explained
how PTEs needed to know tasks’ content to be “smarter than [their] students and . . . a little bit
ahead of the material that [they’re] actually teaching them” (post-observation 2, para. 162).
Ace’s consistent talk about RP content and contextual factors suggests he has a strong content
knowledge of mathematics and his enactment was heavily influenced by outside authorities.
This analysis shows alignment between TAs’ conceptions of teaching and learning RP
and how they talked about task selection, suggesting a relationship between TAs’ conceptions
of RP content and processes and their ability to select tasks for specific instructional goals. In
addition, TAs who talked less often about RP process-related goals, talked more often about
outside authorities influencing task selection. This relationship implies that TAs who did not
explicitly articulate process-related goals for RPTs still enacted RPTs because they viewed the
course supervisor, pacing guide, textbook, and assessments as supporting the decision to
incorporate RPTs into their teaching. Their tendency to decrease RP demands during
enactment, however, suggests that these outside factors influenced what was taught more
than how it was taught. The selection of curriculum and course supervisor’s recommendations
about what to teach seems to have encouraged the use of mathematically rich tasks, but more
is needed to address TAs’ pedagogical needs, helping them develop PTEs’ understanding of RP.

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7.2 TAs’ In-the-Moment Instructional Decisions
TAs made decisions when orchestrating the enactment of RPTs in the classroom that
maintained, shifted, increased, or decreased RP demands of tasks. As presented in Chapter 6,
TAs occasionally increased RP demands by asking PTEs to generate conjectures, rationales, or
proofs that were not explicitly called for in written tasks; sometimes maintained or shifted RP
demands by holding PTEs accountable and pressing PTEs to engage with RP-aspects of the
tasks; and often decreased RP demands by either excluding RP-aspects of tasks or generating
RP processes for PTEs. To understand TAs’ decisions to enact RPTs in these ways, during postobservation interviews, I asked TAs to talk about how or why they decided to respond to PTEs
and PTEs’ ideas about RP during class. The analysis of their responses is presented next, and it
serves to investigate how TAs talked about their in-the-moment, pedagogical decisions. Then,
to understand TAs’ instructional decisions in light of their conceptions of RP, the cases
introduced in Chapter 4 are revisited and used to interpret these results.
7.2.1 Analysis of TAs’ Responses about In-the-Moment Instructional Decisions
Studying TAs’ responses about their decisions to react to PTEs and PTEs’ ideas about RP
as observed provides insights into these (often quick) decisions TAs made throughout the
lesson. TAs’ talk about their interactions with PTEs focused on their responses to PTEs’
questions and how they evaluated PTEs’ their claims. TAs rely on components of pedagogical
content knowledge (Shulman, 1986) which TAs’ responses illuminate further.
There were two general themes particularly noticeable when TAs talked about instances
where RP demands decreased: TAs talked about time constraints (how class was running out of
time or that there was so much material to cover in so little time) and difficulties eliciting

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student participation in mathematical discussions. When talking about how time constraints
influenced their decision to generate RP processes for PTEs, TAs explained “maybe if there was
more time . . . for them to actually work on it they would have gotten it on their own” (Ace,
post-observation 4, para. 24), “the time limitation is an important factor I consider” (Kelly, postobservation 3, para. 96), or “I knew we didn’t have enough time” (Peter, post-observation 2,
para. 50). All TAs talked about how time constraints drove their decisions, which then
decreased RP opportunities for PTEs.
The TAs also all mentioned student participation during the enactment of RPTs either as
something they did not feel skillful at eliciting or something they chose not to pursue because
they were uncomfortable responding to PTEs’ ideas that could be wrong or not fully formed. On
the one hand, many TAs talked about wanting to encourage more student participation during
mathematical discussions, but were dissatisfied with how that did not happen during the
semester. Kelly and Evan were particularly vocal about a lack of student participation during
discussions; they explained that they wanted PTEs to hear and critique their classmate’s ideas,
but that “people rarely volunteer in this class” (Evan, post-observation 2, para. 44) and they
struggled to encourage “students to participate more” (Kelly, post-observation 4, para. 66).
To explain why he often chose not to engage the class in discussions, on the other hand,
Ace said asking for PTEs’ ideas presents the opportunity for
someone [to get] it wrong. . . . And I don't wanna be like, “Well, no.” I'm not very good
at telling people, “Well, no maybe we could look for another idea.” . . . It's a lot easier
for me to . . . take charge of the class more. (post-observation 2, para. 84 - 90)
Evan and Peter also talked about the challenges associated with eliciting PTEs’ ideas during
mathematical discussions that may be incorrect or only partially formed. These TAs talked

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uneasily about “trying to decide on the spot if [a PTE’s justification] was a proof or not” (Evan,
post-observation 3, para. 52). When faced with this challenge of understanding PTEs’
alternative ways of thinking on the spot, they often chose to generate a RP process for the class
instead of figuring out if the idea raised by the student could potentially be correct. This notion,
of responding to PTEs’ partially formed ideas is a component of pedagogical content knowledge
(Shulman, 1986). Although these TAs understand the content taught in MTH 2 and understand
what RP is in mathematics, their work as teachers also requires pedagogical skills and
knowledge to develop mathematical concepts as they arise from PTEs’ statements. These TAs’
responses imply a need for support that builds their teaching practices, helping them facilitate
mathematical discussions (Stein et al., 2008) in ways that engage PTEs in RP and develop PTEs’
understanding of mathematical RP.
7.2.2 TAs’ Conceptions of RP and their In-the-Moment Classroom Decisions
TAs’ interactive classroom decisions can be influenced by their conceptions of RP in
mathematics, teaching, and learning, their selection of RPTs, interactions and constraints during
enactment, and their interpretation of these classroom events (Figure 14). By considering the
results discussed in the previous chapters, and earlier in this chapter, potential relationships
between TAs’ conceptions and decisions are explored.
7.2.2.1 How TAs who Aimed to Prepare PTEs to Teach others about RP Processes Enacted RPTs
Attending to the dual nature of MTH 2, where PTEs need to both learn mathematics and
be prepared to teach others about it, some TAs described learning goals about engaging PTEs in
RP processes (Figure 6). Peter’s, Kelly’s, and Evan’s conceptions of teaching RP aligned with the
goals and purposes of the course. They also talked about developing PTEs’ ability to think and

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reason mathematically when articulating their task selection. Their conceptions of teaching RP
and learning goals suggest that they wanted PTEs to experience the process of RP and be
prepared to help future elementary students engage in these processes, as well. Their
enactment of RPTs in MTH 2, however, frequently resulted in decreases in RP demands (Figure
13). Peter decreased RP demands of tasks most often by excluding questions or parts of tasks
related to RP, and he sometimes generated RP processes for PTEs. From the middle of the
semester onward, Kelly and Evan tended to generate RP processes for PTEs. During postobservation interviews, after Peter excluded RP processes he often talked about time
constraints and modifying the tasks to try to get back on pace with the course schedule.
Although Peter articulated rich conceptions of RP and how RPTs support process-related goals,
his enactment patterns suggest a need for support about modifying tasks in ways that do not
decrease PTEs’ opportunities to learn RP processes he intended to teach.
When generating RP processes for PTEs, these three TAs did so during class discussions,
typically after PTEs had worked on RPTs, but a majority of PTEs had not generated or evaluated
RP aspects of tasks. The TAs explained difficulties they experienced facilitating mathematical
discussions in a way that would allow PTEs to talk about mathematics and engage in RP
processes. Evan, for example, said why he wanted to have class discussions of RPTs and the
challenges he experienced facilitating these discussions:
I wanted them to explain it just so they’d get experience explaining it. They’d get
experience hearing it from somebody else; seeing how it lines up with what they did.
And the people who don’t have it yet, they can see how it was done. . . . People tell me
to just call on people and I hate cold calling on people because I tend to be a passive
learner and I learn by hearing more than by speaking. And for those people, I appreciate
that. . . . I’ve never BEEN in that situation before where they just- nobody wants to take
responsibility and do anything during class discussion. (end-of-semester, para. 89, 91)

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Kelly also talked about wanting PTEs to talk to one another about mathematics, but during the
eleventh week of the semester PTEs specifically asked Kelly to tell them the proof because they
wanted to see how she would write it instead of generating the proof themselves. As detailed
in the previous section, Peter, Kelly, and Evan articulated their challenges facilitating class
discussions including difficulties eliciting student participation and feeling unprepared to
negotiate class discussions when PTEs shared informal or partially formed ideas. Mathematics
education literature agrees that managing mathematical discussions in the classroom is
challenging work for mathematics teachers (Speer & Wagner, 2009; Stein et al., 2008), which
suggests a need for specific supports for TAs’ pedagogical knowledge for teaching.
7.2.2.2 How a TA who Aimed to Prepare PTEs to Produce Explanations Enacted RPTs
A prominent aspect of many of the TAs’ conceptions of the purposes of proof in
teaching was explanation; they talked about proofs as helping students understand why
mathematical statements are true instead of simply to verify (Table 8). Explaining why was also
stressed in the goals of MTH 2’s textbook and noticeable in the prevalence of tasks that elicited
rationales (Figure 9). In line with this focus, explanation was a salient purpose of proof in
teaching Laura talked about. She also articulated a goal of helping PTEs generate clear, concise,
informal explanations for the validity of claims about general mathematical relationships.
Laura’s talk about informal explanations seemed aligned with how rationales were
conceptualized in this study as valid arguments for or against mathematical claims that did not
constitute formal proofs. In Laura’s class, however, she tended to enact tasks that called for
rationales in ways that did not hold PTEs accountable for generating rationales. During
enactment, Laura accepted PTEs’ responses that included restatements of mathematical claims

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or stated theorems or facts PTEs thought might be related to mathematical claims without
pressing for rationales for why mathematical claims were valid.
Given the fact that many prospective and practicing teachers consider empirical
arguments as proofs (Knuth, 2002a; Steele, 2006), PTEs need to understand the generality of
proof and develop robust understandings of the roles of proof. Although developing PTEs’
abilities to produce concise rationales could be one aspect of helping them understand the
purposes and processes of RP, there is the potential to reinforce misconceptions if the
instruction does not go beyond nonproof arguments. Furthermore, Laura’s tendency to accept
informal, invalid arguments as rationales could reinforce misconceptions PTEs may have about
what constitutes a proof, nonproof, and valid arguments in mathematics and in mathematics
teaching. Since Laura consistently expressed concern for preparing PTEs to teach others, but
she did not talk about how her enactment of RPTs could affect PTEs’ opportunities to learn RP,
it is likely that she was not aware how these instructional decisions alter opportunities to learn.
These results suggest that Laura had difficulty noticing (van Es & Sherin, 2002) classroom
situations that affect PTEs’ opportunities to learn (e.g., recognizing when her decisions to
highlight particular responses can affect RP demands of tasks).
7.2.2.3 How a TA who Aimed to Help MTH 2 Students Understand Formal Proofs Enacted RPTs
Isaac tended to enact most of the observed RPTs without changing the RP demands
throughout the semester, but interpreting this enactment pattern in light of Isaac’s learning
goals and reasons behind his instructional decisions has important pedagogical implications.
Isaac’s learning goals primarily stressed content MTH 2 students should know and understand
as undergraduate mathematics learners; especially helping them understand that proofs are

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necessary in mathematics. He talked about how students needed to understand that math was
“not just about guessing and assuming things,” and that doing proofs is “the best” means to do
so (beginning-of-semester, para. 106, 110). When articulating his selection of RPTs, Isaac also
talked frequently about outside authorities, especially the textbook, explaining,
I have some idea that we should follow the book. . . . I can’t really choose how I want to
do things. The only thing I can do is to follow the book otherwise maybe I just talk about
things that they’re not supposed to learn or something. (post-observation 2, para. 163)
Isaac’s tendency to enact RPTs without changing RP demands aligns well with his focus
on remaining faithful to the text. When Isaac increased RP demands for one observed RPT, he
asked PTEs to justify a conjecture they generated even though the task did not elicit rationales
or proofs, aligning with Isaac’s focus on helping PTEs understand the importance of proofs.
When asked about how he made his instructional decisions, however, Isaac had the most
difficulty out of all the TAs articulating his pedagogical choices. He made statements about how
he made in-the-moment decisions, saying they were “just something spontaneous- I mean- I
didn’t think about it too much” (post-observation 3, para. 72). He expressed uncertainty about
how the material from MTH 2 prepared PTEs to teach others about RP. That is, Isaac only talked
about the relationship between PTEs’ future teaching and what they learn in MTH 2 when
explicitly asked in interviews. In his responses, he also used phrases such as maybe and I don’t
know while articulating RP content PTEs could use in their teaching.
Isaac’s responses about his instructional decisions have particular implications for the
opportunities for PTEs to develop the ability to articulate RP processes to teach others.
Although RPTs in Isaac’s class were enacted in ways that maintained RP demands more often
than in other TAs’ classes, rationales behind Isaac’s instructional decisions were not readily

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expressed during interviews; suggesting that they also likely remained implicit to PTEs during
instruction. Isaac’s hesitancy in responding about how RP instruction in MTH 2 prepares PTEs to
teach about RP also suggests that these connections remained implicit during class. Although
PTEs in Isaac’s class engaged in RP processes during many of the observed tasks, it is unclear to
what extent they could recognize and articulate the work they did as RP. Due to the lack of
explicitness regarding RP, PTEs may be less likely to associate RP in MTH 2 with the elementary
classroom context; thereby, reducing the probability that PTEs will intentionally incorporate RP
processes into their future teaching.
7.2.2.4 How a TA who Aimed to Help MTH 2 Students Learn RP Content Using Lectures Enacted
RPTs
The importance of encouraging TAs to become critically reflective practitioners is
particularly apparent when considering Ace’s conceptions of RP and instructional practices
around RP. Ace’s conceptions of teaching and learning RP differed from the other TAs in how he
focused on the efficiency of using lectures and did not talk about RP processes. Ace’s assertion
that lecture-based instruction can be more efficient than activity-based is valid. His focus on
lecture-based instruction and RP content, however, differed from the particular pedagogical
stance supported by the design and goals of MTH 2. Thus, for a majority of RPTs observed in
Ace’s classroom, RP demands were decreased. He frequently generated RP processes for PTEs,
sometimes while PTEs worked on tasks in small groups or other times before they worked on a
task by first providing a lecture about the material.
In contrast with Ace’s conceptions of teaching RP, he still chose to implement RPTs by
often asking PTEs to work in groups and talk with one another about the mathematics. He

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explained this decision by referring to how the structure of the course and textbook were
designed for the use of mathematical tasks, which he called activities, group activities, or class
work, saying,
In this class there was obviously time built in so that they can do these group activities.
Otherwise I’d be finishing way earlier than everyone else [the other TAs] and I'm sure
[the course supervisor] would yell at me and say, “You should have them do activities
every once in a while.” So I mean if they’re going into teaching this material then it's
important that they actually understand it. So of course that's the reason why they built
in time to this so that you have a better understanding so that when you present it to
your students in the future that you will hopefully actually know it in comparison to
something that you've memorized. (post-observation 2, para. 176)
When Ace interacted with PTEs as they worked, however, he would often tell PTEs an answer or
remind PTEs he would provide them with the final answer in a few minutes. During class
discussions of tasks after PTEs worked on them, Ace primarily lead the class in a lecture-style
format through the major aspects of tasks. In these ways, Ace interacted with PTEs as the one
who “gives [the PTEs] knowledge” (Ace, end-of-semester, para. 30).
Later in the semester, Ace reflected on these interactions with PTEs, explaining that he
wanted to learn how to become better at engaging PTEs in mathematical tasks. During a postobservation interview in week 12, Ace expressed dissatisfaction with PTEs’ interactions with
mathematical tasks, explaining,
For a lot of their activities they just wait for me to give the answers at the end of them. .
. . As soon as activities come out they get out their phones . . . and they just [think],
“well, who cares he always gives us the answers to the activities. I don’t have to work
through it. I can just hear the answers at the end. There’s no sense in me trying.” . . .
Today, I kind of wanted to . . . motivate them to actually try on the activities . . . rather
than just to sit back and wait for me to give them the answers. (post-observation 4,
para. 6, 8)
Ace was, therefore, aware that PTEs were not engaging in tasks, and he addressed this concern
during that class by telling the class they would be expected to do a proof of the Pythagorean
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Theorem on the next exam and on the final before they started to work on the task. Ace did not
maintain this reflective posture or express further dissatisfaction later in the semester,
however. His tendency to telegraph what was going to happen and focus on PTEs learning RP
content seemed to work against the course goals for PTEs to engage in RP and explain why
mathematical ideas make sense (goals which Ace also pontificated occasionally later in the
semester). Ace talked about his instructional decisions by focusing on content objectives with
little consideration of potential mathematical connections between different tasks or
connections to what PTEs already knew or would need to understand as future teachers. These
characteristics are similar to the agendas and routines mathematics education researchers have
described in novice K-12 mathematics teachers (Leinhardt, 1989; Leinhardt, Weidman, &
Hammond, 1987).
Implications of the results from this study are detailed further in the next chapter.

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CHAPTER 8
CONCLUSIONS AND IMPLICATIONS
This study examined TAs’ conceptions of RP and how they engaged PTEs in RP during
classroom instruction. The MTH 2 textbook included tasks providing various opportunities for
PTEs to experiences RP processes, including generating and evaluating conjectures and
mathematical arguments. Results indicate that TAs engaged PTEs in a range of RP processes.
For a plurality of observed tasks, however, opportunities for PTEs to engage in RP were
decreased. A decrease in RP opportunities typically occurred either when TAs generated a
conjecture or justification instead of allowing PTEs to generate them or when RP-related
questions in a task were excluded during classroom instruction.
TAs articulated different conceptions of the goals and purposes of RP in mathematics,
teaching, and learning. The depth and breadth of purposes of proof TAs articulated indicated
that they talked about purposes of proof in mathematics differently than in mathematics
teaching, focusing on different roles of proof in the two contexts. In the context of teaching,
TAs focused on explanation, communication, and the general, fundamental purposes of proof,
suggesting that their conceptions of proof include multiple purposes important in the context
of teaching MTH 2. When articulating learning goals, it was also important in this context to
consider the fact that TAs were teaching PTEs who were current undergraduate students and
future teachers: On the one hand, TAs tended to specify RP content (i.e., what constitutes a
proof) they wanted students to understand as short-term goals focused on the context of the
MTH 2 classroom. On the other hand, TAs’ long-term goals often included talk about their
students as future teachers, identifying both RP content (i.e., generating concise mathematical

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explanations) and RP processes (i.e., developing critical thinking skills) important for preparing
PTEs to teach others. This range of conceptions supports the notion that teaching about RP to
college students who are prospective teachings represents a complicated context, illuminating
some specific professional development needs.
The analysis of TAs’ instructional decisions examined TAs’ selection of RPTs to teach and
their in-the-moment decisions to respond to PTEs about RP as observed. The results indicated
that TAs’ conceptions of teaching and learning RP in MTH 2 aligned strongly with how they
talked about their goals for selecting RPTs. This alignment suggests that there is a relationship
between TAs’ conceptions of RP content and processes and how they select and analyze tasks
for specific instructional goals. When talking about their classroom decisions around RP, TAs
attended to surface level characteristics of classroom interactions (e.g., how to encourage
student participation) and time constraints. In light of these results, implications for supporting
TAs who are teaching mathematics courses for prospective teachers are detailed.
8.1 Implications and Recommendations for the Field
In this study, I considered small groups of TAs and individual TAs as cases because
understanding the conceptions and practices in these cases implies ways to support TAs’
knowledge and pedagogy development. A unique characteristic of the institution at which this
study was conducted is that both mathematics and mathematics education TAs are assigned to
teach mathematics courses for future teachers. This setup provides the opportunity to examine
and learn from case studies of TAs with different backgrounds that could inform collegiate
mathematics teaching in a variety of settings. The four cases in this study were grouped based
on TAs’ conceptions of RP in mathematics, teaching, and learning and how their conceptions

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aligned with the goals and purposes of MTH 2. The ways TAs’ conceptions or classroom
practices did and did not align with aspects of the course’s purposes for RP have particular
implications for PTEs’ opportunities to learn about RP. These four cases inform the field’s
understanding of knowledge needed to teach mathematics at the collegiate level.
Recommendations for supporting TAs who are teaching PTEs about RP are extrapolated from
the study’s results about TAs’ conceptions of RP and their instructional decisions around RP.
8.1.1 Supporting TAs’ Enactment of Reasoning-and-Proving Tasks
Mathematics education researchers have shown that teachers have a tendency to lower
the cognitive demand of mathematical tasks during implementation (Stein et al., 1996), and a
similar tendency was apparent for TAs implementing RPTs in MTH 2. Since maintaining RP
demands of tasks implies there are more opportunities for PTEs to experience RP processes, it
is important to provide pedagogical support that helps TAs maintain RP demands during
classroom instruction. Understanding the two types of decreases observed, and reasons behind
these instructional decisions, informs two primary ways to support TAs’ classroom practices
around RP.
The two types of decreases occurred when RP aspects were excluded or the TAs
generated RP aspects (often proofs) for PTEs. For the first type of decrease, when Peter, for
example, talked about his instructional decisions he was often concerned with time constraints,
without attending to the way his modifications affected RP opportunities. Since Peter
articulated fairly robust conceptions of RP and conceptions of teaching and learning RP that
aligned with the course goals, he seems to know and articulate RP content and processes well.
To support his instruction of RPTs, therefore, I suggest focusing on pedagogical skills and tools

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that could help TAs plan and implement RPTs. These pedagogical supports could include
helping TAs learn how to modify tasks in ways without decreasing PTEs’ opportunities to learn
the RP processes they aimed to teach. The RP-framework could help in this endeavor by
providing a means for TAs to analyze tasks for RP processes, before and after modifying tasks.
Then, TAs could be more aware of the RP opportunities tasks afford before modification and
how modifying tasks affect how PTEs experience RP. Professional development that helps TAs
apply this framework to learn how to analyze tasks could help them plan lessons around RP
processes in tasks and learn how to modify tasks based on RP-learning goals.
Second, the primary type of decrease observed, where TAs generated RP aspects for
PTEs, highlights challenges TAs faced when facilitating mathematical discussions about RPTs.
When describing these enactments, for example, Kelly, Peter, and Evan articulated difficulties
facilitating these discussions in ways that allowed PTEs to talk about mathematics and engage
in RP processes. They also expressed some apprehension about responding to PTEs’ partially
formed or potentially incorrect ideas. Drawing on secondary education mathematics research
(Stein et al., 2008) that describes specific practices for facilitating mathematical discussions
around high cognitive demand tasks is a way to address these needs. Stein et al. suggest it is
particularly important to help novice teachers, those learning how to facilitate mathematical
discussions, to use a set of practices that can prepare them to plan for students’ ways of
thinking or misconceptions. This set of five practices can also “help them . . . learn how to
become better discussion facilitators over time” (p. 321). By anticipating and planning
responses to PTEs’ misconceptions, TAs can feel better prepared “to decide on the spot if [a
PTE’s justification is] a proof or not” (Evan, post-observation 3, para. 52). By learning how to

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monitor, select, sequence, and connect PTEs’ thinking in the classroom, TAs could develop
specific practices for orchestrating rich mathematical discussions that incorporate PTEs’ ideas
and ways of thinking.
As Evan said, “trying to make discovery learning work . . . and get through what you
need to get through- that’s tough. That’s not a simple thing to do” (post-observation 3, para.
140). Incorporating the five practices (Stein et al.) into TAs’ professional development would
not make facilitating mathematical discussions simple, but could provide instructional support
to develop TAs’ abilities to engage PTEs in RP processes and respond to PTEs’ thinking in the
classroom. This case underscores the notion that mathematics instructors need to not only
understand the content and students, but also develop a repertoire of pedagogical tools to use
during instruction to help students develop rich conceptions of RP.
8.1.2 When is an Explanation a Justification? Developing a Reasoning-and-Proving Language
The enactment of RPTs involves TAs interacting with PTEs around RP, and from these
interactions PTEs need to develop an understanding of RP to be prepared to articulate these
processes to their future students. In this complex classroom setting, the way TAs talk about
and design instruction around RP will influence how PTEs experience RP, and ultimately affect
how PTEs will likely articulate and design instruction in their future elementary classrooms. The
study of TAs’ enactment of RPTs and responses about their instructional decisions suggests the
importance for developing TAs’ ability to notice (van Es & Sherin, 2002) instructional situations
that affect PTEs’ RP understanding, as well as developing TAs’ use of a common language for
talking about RP in teaching.

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Laura, for example, often decreased RP demands by accepting PTEs’ responses that
constituted informal explanations (responses that lacked reasons about why mathematical
claims were valid) when the task elicited rationales or proofs. During interviews, Laura could
accurately articulate differences between these types of informal explanations, nonproof
justifications, and formal proofs, indicating that her understanding of the mathematical content
was strong. She did not, however, indicate an awareness of how PTEs may not yet comprehend
the differences between different mathematical explanations and how her instructional
decisions affected PTEs’ opportunities to learn these distinctions. Imprecision about what
constitutes a proof versus a rationale versus an informal explanation in mathematics
classrooms for PTEs is particularly worrisome because such imprecision perpetuates
misconceptions that PTEs could bring into teaching. TAs, like Laura, with strong knowledge of
RP content, may benefit from support that helps them notice or recognize when decisions to
highlight particular responses can affect RP demands of tasks or potentially reinforce PTEs’
misconceptions.
One means for addressing this need is helping TAs use a precise language for talking
about RP content and processes and teaching RP. Although mathematics uses precise language,
K-12 teachers and teacher educators typically talk about teaching mathematics using imprecise
language. Mathematics education researchers have recently stressed the importance of
developing a precise, common professional language to talk about teaching practices (Ball,
2012). Based on the range of conceptions of purposes of proof, RP content, and RP processes
TAs articulated in this study, they did not appear to have a common language for talking about

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these purposes and processes. Research literature presents clear conceptions of purposes of
proof (Knuth, 2002b; Steele, 2006), but it is not clear how well these are shared with TAs.
Providing instruction and support for TAs to use the RP-framework in planning and
teaching is suggested as one way to address this concern. In this study, this framework
provided a lens through which tasks, classroom activity, and TAs’ interview statements could be
analyzed as related to specific aspects of RP or not related to RP. The framework could help TAs
develop a common language for talking about processes of RP (Dimension 1) and purposes of
RP (Dimension 2) in mathematics teaching. This common language is particularly important in
the context of teaching about RP. That is, as users of mathematics, TAs could potentially
complete coursework and do mathematics without the need to articulate various purposes of
proof and aspects of RP processes. In teaching, however, TAs have to articulate RP content and
processes in the classroom, explaining about RP using consistent language to develop PTEs’
understanding of RP. Incorporating consistent RP processes language into MTH 2 could help TAs
model and facilitate ways of talking about proof and proving during instruction that PTEs could
apply in teaching.
8.1.3 Developing TAs’ Knowledge about PTEs and How RP Processes Prepare PTEs to Teach
TAs who have never taken mathematics education courses, who have never taught
mathematics courses for future teachers, who have no K-12 teaching experience, or who
attended primary or secondary school outside the United States are likely to have little or no
basis for knowing how the subject matter for MTH 2 relates to their PTEs as future teachers. It
is well known that in mathematics teaching, it is helpful to make explicit connections to prior or
future courses or subject matter to develop students’ conceptual understanding of

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mathematics. In courses for PTEs, in particular, future connections include, ultimately, the
elementary classroom setting in which PTEs will teach. In MTH 2, it is important to help PTEs
understand how the subject matter (in this case RP content and processes) relates to their role
as future elementary teachers. For TAs to help make these connections explicit for PTEs, they
need a specialized knowledge of RP and of PTEs, calling for outside resources or support. Isaac’s
conceptions of teaching and learning RP help contextualize this aspect of specialized content
knowledge in this context.
In K-12 mathematics, the NCTM calls for teachers to “hold all students in every . . .
mathematics classroom accountable for personally engaging in reasoning and sense making,
and thus lead students to experience reasoning for themselves rather than merely observe it”
(NCTM, 2009, pp. 5–6). To ultimately engage future students in RP, therefore, PTEs need to
articulate the purposes and processes of proof themselves. Isaac, however, focused on RP
content in interviews, describing learning goals that did not connect to the preparation of PTEs
as future teachers. Although PTEs engaged in many RPTs by experiencing RP processes in Isaac’s
class, neither they nor Isaac made explicit the RP processes in which they engaged and how
their work on RPTs prepared them to explain about RP in the future. Engaging in and doing
proofs and proving is not sufficient for preparing PTEs who need to be aware of and able to
teach others about the RP processes in which they engaged. Without explicit connections, PTEs
are less likely to associate the work around RP they do in MTH 2 with the elementary classroom
context.
To develop TAs’ understanding of how RP in MTH 2 relates to PTEs’ future teaching,
additional support materials from textbook authors and mathematics education documents

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(e.g., NCTM, 2009) explaining these connections are one means for addressing this need. TAs
may also better understand what RP entails in K-8 mathematics classrooms if they have
opportunities to watch video or read examples (e.g., Carpenter, Franke, & Levi, 2003) of
elementary students reasoning and generating conjectures. Informing TAs’ understanding of
the context in which PTEs will teach can help TAs contextualize RP instruction, thereby
preparing PTEs to engage elementary students in reasoning and sense making.
8.1.4 Developing Critically Reflective TAs
The first three implications described how the study’s results inform our understanding
of mathematical knowledge for teaching RP at the college level and recommendations for
supporting TAs’ development of aspects of this knowledge. Before attempting to work with TAs
in these ways, however, it is important to develop TAs’ ability to be critically reflective
practitioners—open to identifying and examining their assumptions that influence how they
conduct and perceive their teaching (Brookfield, 1995). Without such an awareness, TAs are
less likely to desire change or input on their teaching practices or conceptions about teaching;
attempts at learning new pedagogical strategies would likely be short-lived if TAs do not
understand their current perspectives and how new strategies or ideas relate to their existing
framework.
The results from examining Ace’s conceptions of RP and teaching practices support this
notion of the importance of developing critically reflective TAs. Throughout the semester, Ace
maintained conceptions of teaching and learning RP that were contrary to the course design.
He taught MTH 2 by combining the style of instruction with which he was familiar, lecturing,
with the instructional strategy endorsed by the course supervisor and textbook, group work on

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mathematical tasks. To do so, he typically lectured about mathematical content related to RPTs
before asking PTEs to work with others on the tasks, usually decreasing RP demands either by
generating RP aspects for PTEs in the notes before they worked on tasks or during
implementation. Trying out new instructional strategies that may be different from one’s
primary views of teaching and learning is not inherently problematic, but calls for thoughtful
integration of new strategies with the intention of continued reassessment and modification to
help students achieve desired learning outcomes. An unexamined, contradictory teaching
practice, however, lends itself to potentially confusing students who receive mixed messages
regarding the purposes of instruction and RP.
There was little evidence that Ace critically examined his teaching practice; at the
beginning of the semester, he said he started incorporating tasks as group work because he
followed the course supervisor’s recommendations, saying, he “went with it the first day and
[has] gone with it ever since” (Ace, beginning of semester, para. 134). A notable instance when
Ace made some inroads reflecting on his teaching practice, however, was during the semester’s
twelfth week: Ace expressed dissatisfaction with how PTEs waited for him to tell them the
answer to tasks instead of working on tasks during group-work time (post-observation 4, para.
6, 8). In response to this dissatisfaction, Ace reminded PTEs about an upcoming exam to try to
motivate them to work on tasks more themselves, without making significant changes to or
assessments of his instructional style. This pedagogical move suggests Ace viewed PTEs’ lack of
engagement as primarily a result of a lack of student motivation, rather than considering any
potential additional factors, such as his instructional decisions or classroom norms. After that
instance, Ace made no other statements that significantly reflected on his teaching practices.

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Continuing to provoke these types of reactions, where TAs experience cognitive
dissonance among their conceptions of teaching RP, how they conducted teaching, and
students’ reactions, could develop a better understanding of their personal teaching
perspectives. Brookfield (1995) encourages this process of discovering one’s assumptions by
viewing one’s practice from four perspectives: (a) writing one’s autobiography as a teacher and
learner, using personal self-reflection and collecting insights into teaching; (b) making an
assessment of one’s self from the students’ perspective by seeking their input, considering the
classroom structure, and learning from their viewpoint; (c) learning from peer review of one’s
teaching, which could include talking regularly with colleagues about teaching experiences and
being open to have colleagues observe and provide feedback about one’s teaching; and (d)
frequently referring to the theoretical literature that may provide an alternative interpretive
framework for a situation. The intention is not that TAs need to apply all four recommended
critical reflection practices before other, potentially more content- or course-specific,
professional development strategies are also incorporated. These are some strategies, rather,
through which TAs could begin to become more self-aware and therefore open to input about
their teaching.
8.2 Future Research
Based on the results of this study, additional studies are recommended to grow the
knowledge base of the teaching and learning of RP, especially at the collegiate level. The
following sections detail next steps in this line of inquiry for the researcher.

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8.2.1 What about the Prospective Elementary Teachers’ Learning?
This study assessed PTEs’ opportunities to learn about RP in MTH 2 by considering how
they engaged in RP processes in the classroom, but did not include measures of student
learning about RP in MTH 2 or potential long-term effects of RP instruction on future teachers’
classroom teaching. Such investigations were outside the scope of this study, but are important
for understanding how PTEs think about and understand RP in college and how they apply that
understanding in their future teaching.
This study’s investigation of TAs’ enactment of RPTs raises the question of how these
different opportunities to learn RP affected PTEs’ understanding of RP. That is, the greatest
number of instances of RP opportunities being maintained was during Isaac’s class, which
contrasts with the tendency to decrease RP demands by generating RP processes for PTEs
observed in Ace’s class. This study was not designed to compare and contrast student learning
between classes, but these different enactment patterns beg the question: How do different
instructional strategies and opportunities to learn about RP in the classroom affect PTEs’
understanding of and retention of RP? Future studies could examine this relationship and may
include test or interview data from PTEs to assess PTEs’ knowledge and abilities related to RP.
Since mathematics courses for PTEs aim to develop PTEs’ knowledge of mathematical
content and processes, preparing them to teach others about mathematics, examining longterm effects of instruction about RP is another potential line of inquiry. In this study, many TAs
talked about how the RP instruction MTH 2 was intended to prepare PTEs to teach. For
example, Evan explained that he included RPTs in instruction to “prepare [PTEs] to evaluate
arguments in their own teaching and make arguments in their own teaching” (end-of-semester,

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para. 51). He endeavored to help PTEs understand RP and develop their ability to teach others
about RP. The study of how, if at all, such instruction affects prospective teachers’ future
teaching is one avenue for understanding long-term effects of RP instruction in college
preparatory programs.
8.2.2 Supporting TAs’ Teaching and Learning to Teach
This study did not collect data from TAs’ weekly course meetings to assess their
potential effects on TAs’ classroom decisions or conceptions about RP, as it was outside the
scope. Understanding whether current support systems are helping TAs grow in their abilities
to teach about RP, represents an important line of inquiry to then determine any necessary
changes to support systems. Professional development strategies to support TAs who are
teaching PTEs about RP were proposed based on TAs’ conceptions and teaching practices in this
study and draw upon mathematics education literature. These recommendations are purposed
to develop TAs as critically reflective practitioners and support their efforts to engage PTEs in
RP in classrooms. Next steps in this research could include examinations of the effects of
providing such supports for TAs teaching PTEs about RP in courses like MTH 2. Such studies
could investigate how professional development affects TAs’ conceptions and instructional
decisions, suggesting strategies that are particularly helpful for developing TAs’ knowledge and
ability to teach.
8.2.3 Collegiate Mathematics Teaching and Learning
Although this study focused on TAs who were teaching about RP in a specific context,
there are recommendations for future research that extend into other collegiate teaching and
learning settings. Collegiate mathematics courses serve to prepare general mathematics

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learners, mathematics majors, and prospective teachers to be able to understand mathematical
ideas. The CBMS (2012) suggests that collegiate mathematics instruction should help learners
“develop not only knowledge of content but also the ability to work in ways characteristic of
the discipline” (p. 8). This notion of preparing learners to “work in ways characteristic of the
discipline” could look different, however, depending on the students to whom that statement
refers. That is, mathematics majors need to have authentic mathematics learning experiences
to be prepared to think like a mathematician. That is, mathematics majors need to have
authentic mathematics learning experiences to be prepared to think like a mathematician. The
work that mathematicians do, however, remains largely behind the scenes because pure
mathematicians typically disseminate a “recorded version of their mathematical work that
differs from their actual thinking, having much greater emphasis on language, symbols, logic,
and formalism” (Thurston, 1994, p. 167). They do not usually need to articulate or make visible
the process through which they did the mathematical work beyond sharing the streamlined
proof (Thurston, 1994). It is important, however, for prospective teachers to have mathematical
learning experiences that prepare them to teach others about mathematics (developing their
specialized content knowledge). To help develop prospective teachers’ specialized content
knowledge, therefore, mathematics courses should provide opportunities for prospective
teachers to talk about how to do the mathematical work, preparing them to explain
mathematics to future students.
The notion of preparing prospective teachers, therefore, to “work in ways characteristic
of the discipline” should look different from that of preparing mathematics majors. Consider,
for example, two mathematics courses both focused on introducing students to proof and

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proving but geared toward different student populations: (a) general mathematics learners and
(b) prospective teachers. One goal in the class for general mathematic learners might be to
“help students learn how to do proofs and important features of proofs.” Instruction that
includes examples of inductive arguments could then help students know what a proof is and
some features of proofs. In the other class, however, prospective teachers need to learn how to
articulate and engage in processes related to RP to teach others about mathematics. They may
need to know “why is induction an acceptable thing to do in the discipline?” The extent to
which the different student populations are expected to explicitly talk about the processes of
RP in the discipline is different and the mathematics instruction should attend to that
difference.
When talking about learning goals for MTH 2, the results indicated that Kelly, Peter, and
Evan talked about this notion of developing PTEs’ understanding of both content and processes
to prepare PTEs to talk about RP with others in their future classroom. Since prospective
teachers may enroll in courses other than MTH 2, how does this notion extend into other
mathematics courses for prospective teachers, into other mathematics courses prospective
teachers may choose to take that also include mathematics majors and general education
students, or to other university settings beyond this study? Future lines of inquiry could
investigate how instructors who teach courses that include prospective teachers do (or do not)
design instruction around learning goals focused on helping prospective teachers “develop not
only knowledge of content but also the ability to” (CBMS, 2012, p. 8) teach the content to
others. TAs who specified RP process-related learning goals for preparing PTEs as future
teachers did not consistently maintain RP demands of RPTs in this study, suggesting that

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preparing prospective teachers is challenging work. Future research could look for exemplary
cases of collegiate mathematics instructors engaging prospective teachers in mathematical
processes. Other collegiate mathematics instructors could learn from these exemplary cases.
Are there best practices for working towards these learning goals in these types of courses?
What resources and supports are needed to help collegiate instructors prepare prospective
teachers to articulate and engage in mathematical processes?
8.3 Conclusion
As current teachers of mathematics and potential future faculty, TAs represent an
important and under-researched group (Speer et al., 2010). By examining TAs’ conceptions and
teaching practices, this study contributes to this growing and needed area of research,
addressing a specific research need by studying TAs who were teaching PTEs about RP. In this
context, the Conference Board of the Mathematical Sciences (2012, pp. 7–8) argues for
mathematical reasoning and thinking to play a more prominent role in mathematics courses for
PTEs. Their recommendations fall short, however, in helping the field understand more about
who is currently teaching these types of courses and how PTEs’ RP abilities are currently
developed.
The TAs in this study articulated fairly robust conceptions of purposes of proof in
teaching and designed lessons around mathematically rich RPTs intending to provide
opportunities for developing PTEs’ understandings of RP concepts or processes. During
classroom instruction around these tasks, however, TAs tended to decrease RP demands by
generating RP processes for PTEs. This type of decrease in RP demands also lowered PTEs’
opportunities to experience proving processes for themselves. The examination of TAs’

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conceptions of RP, their classroom practices, and how they talked their instructional decisions,
uncovered some ways that TAs could benefit from different types of pedagogical supports. TAs,
and other instructors, who desire to engage students in authentic RP experiences could need
support that helps develop (a) pedagogical strategies for anticipating, eliciting, and responding
to students’ thinking during instruction, (b) a consistent and precise language for RP, (c) an
informed perspective on the relationship between the mathematical content and the students’
learning needs, or (d) critically reflective teaching practices. Such supports are particularly
important for instructors teaching prospective teachers, as prospective teachers’ experiences
with and understanding of RP in college courses can ultimately affect the quality of instruction
their future students receive. In light of increasing demands on K-12 teachers’ knowledge for
teaching about RP (CCSSI, 2010; NCTM, 2009), continuing to study and support the
mathematics teaching and learning of prospective teachers in college mathematics classrooms
is vital.

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APPENDICES

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APPENDIX A
PROTOCOLS AND SURVEY INSTRUMENTS
A.1 Observation Protocol
Observation Protocol Dimensions
1.1
2.1
2.2
3.1
3.2
3.3
3.4

Task as designed in the textbook
Task features as set up by the TA
Cognitive demands as set up by the TA
Classroom norms evident during implementation
TA’s and PTEs’ instructional habits during implementation
Task features as enacted
Cognitive demands as enacted

Observation Protocol Questions
1. What was the potential for engaging in reasoning-and-proving for each enacted task? (1.1)
a. List the potential rating for each task by question, also indicating which questions
from the task were specifically addressed during the observation.
b. If an enacted task was not already rated (e.g., a worksheet or TA generated task),
code it for RP potential using the RP-framework after the observation and note those
ratings here.
c. If an enacted task was coded as not related to RP, also record that information here.
2. How did the TA introduce each task? (2.1, 2.2)
a. What instructions were given?
b. What instructions, if any, were given about how PTEs will be expected to
communicate their work?
c. How did the instructions compare with the actual written instructions?
3. What were the PTEs’ responses to the lesson introductions/instructions? (3.2)
a. What questions did PTEs have and what were the TA’s responses?
b. In what ways, if any, did PTEs not follow instructions?
4. How did the TA interact with PTEs during the time when PTEs worked on the task? (3.2)
a. How did the TA assess student progress?
b. How did PTEs respond to the TA’s guidance during task?
c. What changes did the TA make to the lesson during the task?
d. What questions/responses from PTEs prompted changes?
5. What kinds of generalizations or mathematical arguments did PTEs generate? (3.4)
a. Describe the kinds of generalizations or arguments PTEs generated by question.
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b. How were they presented (on the board, verbally)?
c. Were the ideas or explanations discussed, compared, evaluated, etc.? In what way(s)?
6. How were PTEs’ generalizations or mathematical arguments elicited during the lesson? (3.3)
a. Did the TA explicitly ask for generalizations or justifications?
b. Did PTEs explicitly ask for other PTEs’ generalizations or justifications?
7. How did the TA respond to PTEs’ generalizations or mathematical arguments? (3.1, 3.2)
a. Did the TA restate, ask for feedback from the class, or ask for other ideas or
arguments?
b. When did the TA ask PTEs to explain their generalization or justification further?
c. What kinds of information were assumed/“taken-as-shared” during the discussion?
d. Were any responses or explanations highlighted and in what way(s)?
8. Who was responsible for judging the worth of generalizations, justifications, or proofs? (3.1)
a. What elements of PTEs’ justifications were deemed to be good? Who determined
this? How was this established?
b. What aspects of justifying were determined to be not as good as other aspects? Who
determined this? How was this established?
9. What aspects of the lesson discussion seemed difficult for the TA to facilitate? (3.2)
10. What aspects of the lesson discussion seemed confusing for PTEs? (3.3, 3.4)
a. Did the TA address PTEs’ confusion? How?

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A.2 Preobservation Survey
1. What are your goals for the class that I will observe?
2. How will you determine if this class period accomplishes your goals?
3. What problem(s) from today’s class do you think is (are) particularly important?
a. How do you hope students will respond to this (these) problem(s)?

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A.3 Pre-Semester Interview Protocol
TAs’ conceptions about the roles proof in mathematics.
This is (YOUR NAME) interviewing (SUBJECT’S NAME) on (DATE). The purpose of this
interview is to better understand your math background and current thinking on
mathematical ideas related to Geometry.
Part I: Initial Biographical information: To start, I’d like to ask you a few background questions.
 What is your degree program and what stage are you at in your degree?
 Please tell me about your schooling background and the path that led you to your
current status as a mathematics/mathematics-education graduate student?
i) Were there any math classes or concepts with which you struggled as a learner?
ii) Were there any math classes or concepts with which you really excelled?
 Since you’re teaching a Geometry course, what Geometry-related classes have you
taken?
i) And what were your learning experiences like in Geometry?
Part II: TAs’ Conceptions of Proof in the Domain: Since reasoning and proof is one of the goals
of MTH 202, I have some questions about your views of proof
1. How do you think about proof in mathematics?
a) If the TA uses particular language (e.g., formal or informal proof) or refers to students
probe on what the terms mean and what the student population is.
b) If they focus on doing proof, ask can you tell me what a proof actually is?
Use question 2 as follow up if it hasn’t been addressed, yet.
2. What does it mean to prove something in mathematics?
a) Probe any mathematical language
b) If it hasn’t been addressed yet, follow up with, What does it mean to disprove
something?
3. What makes an argument a proof?
a) Intends to get at the social aspects of proof, but do not lead in that direction if they do
not offer that idea. If they use language suggesting a social or community (e.g.,
mathematical community or mathematicians) aspect to proof, probe that language.
b) If it hasn’t been addressed yet, then follow up with, What makes an argument not a
proof?
4. Once a proof is completed, does that proof ever become invalid?
a) Probe for how TAs see proofs as becoming invalid if they do not offer an explanation
5. Considering the discipline of mathematics, and not a mathematics classroom, what
purpose does proof serve in mathematics?
a) Probe any mathematical language.
6. How do you remember learning about proof?
a) Provide some examples of classroom settings to explain
b) Have you ever learned about proof in classrooms that used more hands on or interactive
methods?

155

A.4 Beginning-of-the-Semester Interview
TAs’ conceptions of the roles of proof in teaching and learning
This is (YOUR NAME) interviewing (SUBJECT’S NAME) on (DATE). The purpose of this
interview is to better understand your current thinking on ideas related to the teaching and
learning of reasoning and proving.
The Teaching and Learning of Proof in MTH 2: Please consider your role as an instructor for
MTH 2, as you respond to these questions:
1. Considering your role as an instructor of MTH 2, how do you think about proof?
a) If the TA uses particular language (e.g., formal or informal proof) or refers to students
probe on what the term means and what the student population is.
2. What purpose does proof and proving serve in mathematics classroom?
a) Purpose of proof in K-12 math classes. And probe any mathematical language and ask
for examples.
b) Thinking about MTH 2, can you say a little bit about what purpose proof and proving
serves in that classroom?
3. Considering your current MTH 2 students, what do you want them to learn about proof?
a) Probe any mathematical language.
b) Follow up with: What challenges do you anticipate in supporting their learning about
proof?
i) If it hasn’t been addressed yet: Do you think proof and proving is content they [your
MTH 2 students] will use again?
4. In your opinion, why do you think these students [future elementary teachers] are
required to take this course?
a) Intends to probe for how the TA has a pedagogical focus when teaching this course. If
the TA talks about students as math learners probe any mathematical language there, if
they talk about the students as future teachers, probe any pedagogical language there.
b) Follow up with: Why do you think it is important to teach these students about proof
and proving in this course?

156

A.5 End-of-Semester Interview Protocol
This is Kim interviewing (TA’s Name) on (Date). The purpose of this interview is to better
understand your ideas related to reasoning and proving in mathematics and in teaching
mathematics, at the end of this semester.
1. How do you think about proof?
a) If the TA uses particular language (e.g., formal or informal proof) or refers to students
probe on what the terms mean and what the student population is.
b) If they focus on doing proof, ask “can you tell me what a proof actually is?”
2. In the context of teaching MTH 2, how do you think about proof?
3. In your view, how do the students in your section of MTH 2 best learn about proof and
proving?
a) What purpose does proof and proving serve in the MTH 2 classroom?
b) Do you think proof and proving is content that your MTH 2 students will use again?
i) If so how. If not why not.
4. In your view, what is your role as the mathematics teacher in MTH 2?
5. Have your views about teaching and student learning changed over the course of the
semester?
a) If so, how? If not, why not?
b) Insert a few questions specifically for each TA by providing quotes from earlier
interviews where they expressed a strong view of the nature or purposes of RP in
mathematics, teaching, or learning. Read the quote from their earlier interview and ask:
After your experiences this semester teaching MTH 2, I’m wondering if you have any
reflections on this idea/statement.
6. What people or experiences have been the most prominent influences in your teaching
practice in MTH 2?
a) Probe for details about how they influenced their practice AND for additional influences.
b) Can you describe for me how you typically prepared to teach MTH 2? Do you write a
lesson plan or notes to yourself? If yes, what do they generally include, if not, why not?
Can you tell me what (if anything) you changed about how you prepared to teach MTH 2
and why?
7. If you were to teach this course again, with more autonomy to choose the textbook and
all aspects of the course, what would you do differently and what would you do the
same?
a) Probe for both similarities and differences
b) In this more autonomous situation, please describe your goals for the course.
i) Possibly, probe for student learning goals: At the end of the course, what would you
want your students to be able to do?
c) One of the goals stated on the MTH 2 syllabus is for students to be able to “reason
mathematically and do proofs in the context of geometry and measurement.” In what
ways did your work in MTH 2 address this goal, if at all?
i) Would you include this goal in the MTH 2 course you designed? If not, why not? If
so, why?

157

8. Have you ever seen the NCTM Principles and Standards for School Mathematics
document?
a) If yes, ask: Can you describe in your own words how the authors of the document
describe the role of reasoning and proof in K-12 mathematics.
b) If no, show them the quote from NCTM (2000) that is on a separate page and says the
following:
In the section on reasoning and proof, in the National Council of Teachers of Mathematics
Principles and Standards for School Mathematics (2000) Document, the authors advocate that
students should be able to
 recognize reasoning and proof as fundamental aspects of mathematics;
 make and investigate mathematical conjectures;
 develop and evaluate mathematical arguments and proofs;
 select and use various types of reasoning and methods of proof.
They also contend that in K-12 mathematics classrooms, reasoning and proof should be “a
natural, ongoing part of classroom discussions, no matter what topic is being studied” (p. 342).
c) What do you think about the recommendations for reasoning & proof set forth by the
NCTM?
i) Possibly probe: In what ways do you think this idea for reasoning & proof to be “a
natural, ongoing part of classroom discussions” is or is not applicable in your MTH 2
class?
9. Show them and ask them to read the descriptions of the roles of proof in literature (same
role of proof in Table 1). Of this set, which roles of proof resonate with you and why?
a) Do you have any questions about this list or the descriptions
b) If it is brought up, possibly encourage the TA to answer the question considering their
own uses of proof and proving [General] and then in the context of MTH 2 [Specific].
c) STRESS the WHY!

158

A.6 Post-Observation Interview Protocol
This is (YOUR NAME) interviewing (SUBJECT’S NAME) on (DATE) and this is an interview about
the lesson from today. As a reminder, the point of this interview is not to evaluate you or
your teaching. I want to hear your thoughts and ideas about your teaching.
a) What do you think about how the lesson went today?
i.
If TA asks interviewer for an opinion about how the lesson went say, “I’m trying to learn
about how YOU think about the lesson.” This question could go down a rabbit hole,
don’t stay too long here and consider asking this later in the interview sometimes.
b) [From preobservation survey] You identified Activity ___ as important for today’s lesson.
i.
Can you tell me a little bit more about why you considered it particularly important
for today’s lesson?
ii.
What do you think about how Activity ___ went today?
 What went well?
 What did not go well?
 Describe any difficulties you encountered and what you did to try to overcome
them.
Repeat part (b) for any other activities identified as important.
c) Rereading the goals you wrote before class, in your opinion, did the lesson accomplish
your goals?
i.
Why or why not?
 Provide explicit examples of how you came to this conclusion
d) Did anything surprise you about how students responded to today’s lesson?
i.
Please provide examples from the lesson if you can think of any that illustrate your point
e) Based on specific student responses or questions TAs asked during class: Do you remember
when . . . can you say a little bit more about what you were thinking when that
happened?
Can you say a little bit more about what you were trying to do (or why you asked that
question) in that situation? And why were you trying to do that?
Can you tell me a little bit about why you responded that way?
Why did you decide (not) to write on the board/restate/have someone else respond to
{student’s} explanation?
What kinds of responses were you expecting students to give? Would that be what you
would consider a proof?
What did you think of [student’s name]’s explanation?
f) Additional questions not used every time, but woven in sometimes:
i.
How does the work from Activity/Activities ___ relate to your students as future
elementary teachers?
159




ii.
iii.

TAs may focus on the PTEs as math learners or as future math teachers
Follow up with, in what ways do you see this content as content that these future
teachers will use again in their teaching? How might the content from these
activities impact what the future teachers do in the classroom? What would you
envision their future elementary students doing around this content in the class?
Why was it important to teach this lesson to these particular students?
In earlier interviews, you mentioned you learned about proof _____ , and/but in class
today you used _______. Can you tell me a little bit about why you decided to use this
instructional strategy?
 What do you think about how the use of small groups went today?
 Have you previously ever taught using these types of hands-on strategies? (please
describe)

160

APPENDIX B
TEXTBOOK ANALYSIS AND CLASSROOM OBSERVATION CODING SCHEME

Segment of Task
Implementation

Task as Written

B.1 Coding the Task-as-Written
Top-Level Code
Second-Level Code (with description)
(with description)
IP (Investigate mathematical situation
with provided examples or data)
MG
IG (Investigate mathematical situation
(Task provides opportunities
by generating examples or data)
for students to create
C (Generate a Conjecture)
Mathematical Generalizations)
EvC (Evaluate a given mathematical
claim)
EvA (Evaluate a given mathematical
argument or method)
R (Provide a Rationale—a wellreasoned mathematical argument
that is not a proof)
Ex (Provide an empirical argument or
example)
MA
(Task provides opportunities
Pf (Generate a proof)
for students to generate
Meta_CPf (Discussing or reflecting on
Mathematical Arguments)
a Meta-Level aspect of proof and
proving—in this case Characterizing
Proof)
Meta_iPf (Discussing or reflecting on
a Meta-Level aspect of proof and
proving—in this case Identifying
Proof)
GA-P (The type of argument the task
provides the opportunity for students
GA
to generate is specifically a Proof)
(Task provides opportunities
for students to generate
GA-N (The type of argument the task
Generalizations and
provides the opportunity for students
Arguments)
to generate is a nonproof argument—
an Ex or an R)
NoRP
(Based on the Reasoning-and-Proof framework, this task does not
provide opportunities for students to engage in specific RP processes)

Definitions for some of the second-level codes are included on the next page.

161

B.2 Definitions for Components of Reasoning-and-Proving Framework Used in Task Analysis
Investigate: explore a mathematical situation, example, or data to determine mathematical
relations that fit the given or generated data or examples.
Explore Provided: explore and determine mathematical relations that fit a given set of
data or examples
Generate Examples: involved in constructing, creating, or providing examples or data
that fit the mathematical situation in order to then explore mathematical
relations therein.
Conjecture: a reasoned hypothesis about general mathematical relation based on incomplete
evidence.
Nonproof Argument: an argument for or against a mathematical claim that does not qualify as
a proof
Empirical Argument: an argument that purports to show the truth of a mathematical
claim by validating the claim in a proper subset of all possible cases covered by
the claim.
Rationale: valid arguments for or against mathematical claims that do not qualify as
proofs—it may not make explicit reference to key accepted truths that it uses, or
it uses statements that do not belong to the set of accepted truths of a particular
community.
Proof: a valid argument based on accepted truths for or against a mathematical claim that
makes explicit reference to key accepted truths that it uses.

162

B.3 Sections of Textbook Analyzed
Curriculum Section: Title of Topic of Study
8.1:
Visualization
8.2:
Angles
8.3:
Circles & Spheres
8.4:
Triangles, Quadrilaterals, & Other Polygons
8.5:
Constructions with Straightedge & Compass
8.6:
Polyhedra & Other Solid Shapes
9.1:
Reflections, Translations, & Rotations
9.2:
Symmetry
9.3:
Congruence
9.4:
Similarity
10.1:
Fundamentals of Measurement
10.2:
Length, Area, Volume, & Dimension
10.3:
Calculating Perimeters of Polygons, Areas of Rectangles, & Volumes of Boxes
10.4:
Error and Accuracy in Measurements
10.5:
Converting from One Unit of Measurement to Another
11.1:
The Moving & Additivity Principles about Area
11.2:
Using the Moving & Additivity Principles to Prove the Pythagorean Theorem
11.3:
Areas of Triangles
11.4:
Areas of Parallelograms
11.5:
Cavalieri’s Principle about Shearing & Area
11.6:
Areas of Circles & the Number Pi
11.7:
Approximating Areas of Irregular Shapes
11.8:
Relating the Perimeter & Area of a Shape
11.9:
Principles for Determining Volumes
11.10:
Volumes of Prisms, Cylinders, Pyramids, & Cones
11.11:
Areas, Volumes, & Scaling

163

B.4 Coding the Task-as-Set-Up
Segment of Task
Implementation

Who

Code

Description

T C Push

TA Moves

Task as Set Up

Student
Moves

TA asks students to develop a conjecture
TA asks students to provide an explanation
T R Push
about why their answer makes mathematical
sense without explicitly asking for a proof
TA asks students to develop a proof (general
T Pf Push
argument)
TA asks students to generate examples to
T InvG Push
investigate a mathematical situation
TA asks students to investigate a mathematical
T InvP Push situation using the provided data, examples, or
diagrams/models
T EvA Push TA asks students to evaluate a given argument
TA asks students to evaluate a given
T EvC Push
mathematical claim
TA asks students to explain how or why they
T Proc Q
arrived at a particular answer
TA provides an example of the reasoning why
T ExR S
a mathematical claim associated with the
activity makes sense (with S input or not)
TA asks students to use prior knowledge for
T UsePK
statements
TA leads a discussion about the activity instead
T Lead Diss
of asking students to work through it
T Group
TA asks students to work in small groups
T Ind
TA asks students to work individually
TA does not specify if students should work
T WorkIns
individually or in small groups
TA says, demonstrates, or does something to
T Ch
change the RP-potential of the task
TA does not say, demonstrate, or do
T NoCh
something that noticeably changed the RPpotential of the task
S Ins Q
Students ask questions about instructions
Students do not ask questions about
S NoIns Q
instructions
S Follow
Students, in general, follow TA’s instructions
Students, in general, do not follow TA’s
S NoF
instructions

164

B.5 Coding the Task as Implemented, Focusing on Individual or Group Work
Enactment Who

Code
T Circ
T Front
T CkIn
T Proc Q
T R Push

TA
Moves
Task
Implementation
(GroupWork or
Individual
Work
Time)

T C Push
T Pf
Push
T Meta
Push
T UsePK

T ExA

T Exs
T Und
Push
S Ask
S Ask Pf
Student S InvP
Moves
S Conj
S Exs
S R Gen
S Pf Gen
NA

Description
TA circulates among classroom for a majority of the
group/individual-work time
TA stays near the front of the room for a majority of the
group/individual-work time
TA asks small groups or individual students generalprogress questions, such as “Are you done? Did you get the
answers? Any questions?”
TA asks students to explain how or why they arrived as a
particular answer
TA specifically asks students to explain their mathematical
reasoning and/or why their answer makes mathematical
sense (though they do not specifically ask for a proof)
TA asks students to develop a conjecture—will this work
for all such objects?
TA asks students to develop a proof (general argument) or
if the students know how to prove their claim or answer
TA asks students to consider Meta-Level aspects of proof
and proving
TA asks students to use prior knowledge in developing
statements
TA provides the answer to small groups or individual
students (sub-codes: T ExC, T ExR, T ExPf for TA providing
an example of the desired Conjecture, Rationale, or Proof,
respectively)
TA provides empirical examples to support a mathematical
claim associated with the task
TA encourages students to understand the reasoning or
proof for themselves, or reword it in their own words
Students ask TA to evaluate their work (sub-codes: S AskEv
R, S AskEv Pf, for evaluating Reasoning or Proof,
respectively)
Students ask TA how to prove a statement
Most students investigate mathematical situation (InvP
and InvG for provided or generated examples/data)
Most students generate conjectures
Most students explain claims with empirical examples
Most students explain why their answer makes
mathematical sense that is not a formal proof
Most students develop a proof (general argument)
No time was provided for group or individual student work

165

B.6 Coding the Task-as-Implemented, Focusing on Whole-class Discussions
Enactment Who

Code
T S to S
Push
T R Push

TA
Moves

T C Push
T Pf
Push
T Meta
Push
T Exs
T ExC
T ExR
T ExPf

Task
Implementation
(Wholeclass
Discussion
of Task)

T OutPf
S Stmt
S Reas
S
RepAns
S Inv
Student S Conj
Moves S EvA
S EvC
S Exs
S R Gen
S Pf Gen
S Meta
S Non
NoDiss

Description
TA asks students to evaluate other students’ reasoning
TA asks students to explain why their answer makes
mathematical sense (does not specifically ask for a proof)
TA asks students to tell the class a conjecture
TA asks students to tell the class their proof
TA asks students to consider Meta-Level aspects of proof
and proving
TA generates empirical examples to support a
mathematical claim associate with the task
TA generates example of the conjecture expected (Subcode: attributing to Unnamed S or Specified S)
TA generates example of the rationale expected (Subcode: attributing to Unnamed S or Specified S)
TA generates an example of the proof expected (Sub-code:
attributing to Unnamed S or Specified S)
TA provides an outline of key points or statements-andreasons needed in a proof for a mathematical claim
Students provide statements to use in the construction of
a statement-and-reason type proof
Students provide reasons why a provided statement is
true in the construction of a statement-and-reason type
proof
Students report out answers or results (one-word/short
ans)
Students investigate mathematical situation (sub-codes: S
InvP and S InvG for provided or generated examples/data)
Students provide conjectures
Students evaluate given argument or method
Students evaluate given mathematical claim
Students provide empirical examples as reasons why a
mathematical claim or conjecture is true
Students provide an explanation about why their answer
makes mathematical sense, not a proof
Students develop a proof (general argument)
Students discuss meta-level aspects of proof and proving
Students non-responsive to TA’s request to offer thinking,
reasoning, and ideas
There was no discussion of the task
166

APPENDIX C
INTERVIEW CODING SCHEMES
C.1 Coding Interviews for Conceptions of RP in Mathematics
Initial Second Level:
Coding Specific MathLevel
Related Categories
For what population
/group to know

Third Level: Subcodes within the
level two categories
Teachers
Future teachers
Students
Elementary Students
Undergrad Students
Mathematicians
Ordinary people

Mathematics

Purpose of Proof in
Mathematics

No explicit distinction
Verification

Falsification

Explanation
Communication

Discovery/Create
Knowledge

Systematization

Meta-Level:
Fundamental
167

Descriptions of Sub-codes

Those teaching K-16
Undergraduates in courses (like
MTH 2) who are future teachers
No grade level specified
K-8 students
College-level students
Researchers and professors in
mathematics
For anyone, not necessarily
mathematicians, students, or
teachers
No distinction noted
Establishing, checking, or
confirming the validity of a known
idea or given mathematical claim
Establishing, checking, or
confirming the falseness of an idea
or given mathematical claim
Providing insights into why a
mathematical claim is valid
Disseminating mathematical
knowledge to other doers of
mathematics
Discovering or developing new
mathematical ideas, confirming
conjectures, or building
mathematical ideas
Imposing a logical structure on the
mathematical domain; organizing
and cataloging results with respect
to axioms, major concepts,
theories, and other prior
knowledge
At a broader level, proof is
fundamental to mathematics

C.2 Coding Interviews for Conceptions of RP in Teaching
Second Level:
First
Specific
Level
Teachingof
Related
Coding
Categories

Third Level: Sub
codes within
level two
Self

Who are the
teachers?

College-level
instructors
Other Teachers
Anyone
Verification
Falsification

Teaching

Explanation
Purposes of
proof in
mathematics
teaching

HOW to
Teach about
Reasoningand-Proving

Communication
Discovery /
Create
Knowledge
Systematization

Descriptions

TA being interviewed is talking about their own
teaching of RP
People teaching college courses (but the
participant is not explicitly referring to
themselves and their own teaching)
Code for K-12, Secondary, or Elementary, if
specified.
Either said, teachers at any level, or did not
specify explicitly ‘who’
Establishing, checking or confirming the validity
of a known idea or claim
Establishing, checking or confirming the
falseness of an idea or claim
Providing insights into why a mathematical
claim is valid
Disseminating mathematical knowledge to
other doers of mathematics
Developing/discovering new mathematical
ideas confirming conjectures

Imposing a logical structure on the
mathematical domain; organizing & cataloging
results with respect to axioms, major concepts,
theories, and other prior knowledge
Meta-Level:
Helping students understand that proof is
Fundamental
fundamental to mathematics
Methods that
 Getting students to share their ideas during
Support or
class
Encourage
 Facilitating class discussions
Student
 Importance of using activities or small
Participation
group work
Methods that Do  Using a lecture-style to teach about RP
Not Support or
 Providing well-planned notes and examples
Encourage
for students
Student
 Assigning practice problems for students to
Participation
complete individually

168

Teaching (continued)

Teach
Undergraduate
Mathematics
Students
Educational
Goal:
WHY it is
important to
teach about
reasoning and
proving

Teach
Prospective
Teachers

Outside
Authorities

Focus on RP as important mathematical content
knowledge:
 need to teach this content because this is a
mathematics class
 proof is important in mathematics;
therefore it’s important to know
 students can develop their own ability to
reason and think mathematically
Focus on preparing PTEs to teach about RP:
 prepare 202 students to be able to help
their future elementary students to reason
and think mathematically
 202 students need to know more math than
their future students
Other authorities say it’s important, including
the course supervisor, textbook, and syllabus

169

C.3 Coding Interviews for Conceptions of RP in Learning
First
Level
of
Coding

Second
level:
Specific
learningrelated
categories

Learning

Who are
the
learners

Learning
goals:
WHAT do
people
need to
know
about RP

Third Level:
Sub codes
within level
two

Definitions or descriptions

Sub-codes if TA specifies themselves; elementary,
undergraduate, or graduate students; or PTEs
Mathematicians Research mathematicians
Anyone
Either learners at any level or did not specify ‘who’
Content, structure, or end product students should:
 Know how to write a proof
 Know what a proof is
 Able to regurgitate the same proof later
 Produce clear, concise explanations
Produce a
 Support each statement with a reason
product related  Use logical arguments in writing a proof
to reasoning Know a proof exists for a mathematical
and-proving
statement
o This is important math knowledge
o To respond to future students if they ask
“why”
 Know Generality: examples do not constitute
proof
Students

Engage in RP
processes

Way students build understanding of RP processes:
 Determine “given” info & what is meant “to be
proved” for a mathematical claim
 Investigate mathematical situations
 Generate conjectures
 Evaluate conjectures; Evaluate provided
arguments
 Reason through how to construct a proof
o Build argument from end to beginning &
beginning to end
o Try out approaches that may fail & learn from
these failed attempts
 Be convinced a statement is true by seeing a
proof; not by examples
 Understand RP pervasive in mathematics
 Develop Habit of Mind (e.g., critical thinking)

170

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171

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