DILEMMAS AND TEACHER DECISIONS
IN COLLABORATIVE MATHEMATICS DISCUSSIONS
By
Nicholas Joseph Gilbertson

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Mathematics Education—Doctor of Philosophy
2017

ABSTRACT
DILEMMAS AND TEACHER DECISIONS
IN COLLABORATIVE MATHEMATICS DISCUSSIONS
By
Nicholas Joseph Gilbertson
Over the past few decades, organizations, researchers, and professional developers have
noted the potential for using classroom discussions to improve student learning in mathematics
classrooms. Yet, for teachers, managing the often-competing demands that arise while
attempting to plan for and enact these discussions can be challenging. To better understand these
challenges, this dissertation describes the results of a study characterizing the instructional
decision-making of three experienced middle-grades mathematics teachers as they encountered
instructional dilemmas while planning for and enacting collaborative mathematics discussions.
To study decision-making, I observed teachers across several lessons, interviewed them
after lessons, and analyzed lesson plans. Studying decision-making as it appeared in teachers’
lived classroom experiences provided an authentic and situated perspective on the complexities
of their decision-making. In order to analyze their decisions, I attended to teachers’ instructional
goals, which consisted of lesson-specific content goals, broad content goals, mathematical
practice goals, and goals for establishing a mathematical community. Additionally, I used Herbst
& Chazan’s (2011) theory of professional obligations to understand why teachers made certain
decisions based on their goals, consistent patterns of instruction, and the particular dilemmas
they encountered. This theory argues that teachers’ decisions are a function of their obligations
as representatives of the discipline of mathematics, in developing students as individuals, in
mediating productive interaction between students, and as representatives of the school as an
institution.

The main finding of this study was that all three teachers’ decision-making was largely
explained via their primary professional obligation – with important limitations. Along with this
main finding, there were three subfindings. First, teachers’ decision-making was largely
explained via their primary orientation to their professional obligations. Second, all three
teachers encountered scenarios in which they made decisions that deviated from their primary
orientation to the professional obligations. Third, even though the teachers were experienced,
used instructional resources (e.g., written curriculum materials) that supported implementing
whole-class discussions, and were generally oriented philosophically with the importance of
using discussions, this was not sufficient to avoid challenges one might expect to see while
facilitating whole-class discussions. Most importantly, the types of challenges they encountered
were, by-and-large, independent of the teachers and the nature of their dilemmas (i.e., how the
dilemma was a conflict between two types of goals).
Understanding how the nature of a teacher’s decision-making is influenced by attending
to their professional obligations during instructional dilemmas contributes to multiple areas of
mathematics education. First, this study contributes to the research on teacher decision-making
by expanding teacher decision-making to include many more contributing factors beyond
content-focused decisions. Second, this study contributes to the research on whole-class
discussions by considering teachers’ orientations (via their professional obligations) as a
confounding variable in the nature of their decisions influencing the discussion. Third, this study
contributes to professional development and teacher education by better understanding how
teachers’ professional obligations can influence the opportunities students have to learn
mathematics via their interaction during whole-class discussions.

Copyright by
NICHOLAS JOSEPH GILBERTSON
2017

This dissertation is dedicated to my family that provided countless hours of support throughout
its development. Thank you to Nikki, Josie, Gus, and Ovidia, for your dedication and
encouragement throughout.

v

ACKNOWLEDGMENTS

This dissertation study is a result of many individuals whom I wish to thank. I thank Dr.
Kristen Bieda, my dissertation advisor, for her support throughout the development of this study.
I also thank my committee -- Dr. Jack Smith, Dr. Corey Drake, and Dr. Jeanne Wald -- for their
perspectives and input throughout the process. I also wish to thank Dr. Michael Steele for his
early support in thinking through possible directions for the study. Thank you to Dr. Aaron
Zimmerman and Rani Satyam for their help with data analysis and collection. Finally, this
dissertation study was partially funded through fellowships from Michigan State University’s
College of Education and College of Natural Science. I wish to thank both colleges for
supporting this work.
I would like to thank all those who supported me throughout the dissertation and in my
development as a learner and teacher of mathematics. To my former teachers whom I have felt
are with me every day in a classroom: Carol Hubbard, Dave Coutley, Vernon Daines, Rob Hepp,
Ted Harris, Larry Krajewski, Mike Limberg, and Tom Thibodeau. For my early work in learning
about whole-class discussions as a way to support students in communicating and thinking
mathematically: Fred Rectanus, Nicole Rigelman, Sean Larsen, Luis Saldanha, and Karen
Marrongelle. To those at Michigan State who supported me throughout this process: Kevin
Lawrence, Betty Phillips, Yvonne Grant, and Billie Lozen. Most importantly, to my family:
Nikki, Josie, Gus, and Ovidia, my parents Gary and Judy, and my grandparents. This dissertation
is a direct result of their support and energy throughout the process.

vi

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................................... xi
LIST OF FIGURES ...................................................................................................................... xii
KEY TO ABBREVIATIONS ...................................................................................................... xiii
CHAPTER 1: DISSERTATION OVERVIEW AND INTRODUCTION ......................................1
CHAPTER 2: LITERATURE REVIEW .........................................................................................4
Talk in the Mathematics Classrooms ...................................................................................5
Discourse-Oriented Teaching ..................................................................................6
Collaborative Discussions ........................................................................................8
Teaching Decision-Making Literature ...............................................................................10
Institutional Challenges .........................................................................................15
Disciplinary Challenges .........................................................................................16
Interpersonal and Individual Challenges ...............................................................17
Summary ................................................................................................................19
CHAPTER 3: THEORETICAL PERSPECTIVE AND ANALYTIC FRAMEWORK ...............20
Main Analytic Constructs ..................................................................................................20
Professional Obligations ........................................................................................21
Instructional Goals .................................................................................................22
Consistent Patterns of Instruction ..........................................................................23
Dilemmas ...............................................................................................................25
Decisions ................................................................................................................26
Connecting the Main Constructs ............................................................................26
CHAPTER 4: METHODS .............................................................................................................29
Study Setting ......................................................................................................................29
Recruiting Teachers for the Study .....................................................................................30
Data Collection ..................................................................................................................32
Baseline Data Collection and Analysis ..................................................................33
Initial interview ..........................................................................................34
Curricular experience .................................................................................35
Instructional Quality Assessment ..............................................................35
Lesson Observation Data .......................................................................................36
Deviations in collecting lesson observation data .......................................37
Number of lessons per round .....................................................................37
Identification of lessons .............................................................................38
Lesson plans ...............................................................................................39
Lesson observations ...................................................................................40
Post-lesson interviews ................................................................................42
vii

Video stimulated interviews ......................................................................43
Data Analysis .........................................................................................................44
Goals analysis ............................................................................................45
Consistent patterns of instruction analysis .................................................46
Professional obligations analysis ...............................................................47
Dilemmas and decisions analysis...............................................................48
Background of Teachers’ Instructional Practice while Leading Discussions ....................50
Mrs. Stewart’s Instructional Practice .....................................................................51
Baseline observation data ..........................................................................51
A typical class discussion ..........................................................................53
Goals ..........................................................................................................55
Consistent patterns of instruction...............................................................57
Professional obligations .............................................................................58
Ms. Mitchell’s Instructional Practice .....................................................................59
Baseline observation data ..........................................................................60
A typical class discussion ..........................................................................61
Goals ..........................................................................................................61
Consistent patterns of instruction...............................................................62
Professional obligations .............................................................................63
Mr. Sandberg’s Instructional Practice....................................................................64
Baseline observation data ..........................................................................65
A typical class discussion ..........................................................................66
Goals ..........................................................................................................69
Consistent patterns of instruction...............................................................69
Professional obligations .............................................................................70
Summary of Teachers’ Instructional Practice ........................................................70
CHAPTER 5: INSTRUCTIONAL DILEMMAS AND DECISIONS ..........................................72
Dilemma Type Results .......................................................................................................72
Dilemma Type 1: Lesson Specific Content Goals Conflicting
with Broad Content Goals ........................................................73
Dilemma 1: Mrs. Stewart ...........................................................................75
Dilemma 1: Mr. Sandberg..........................................................................77
Dilemma 1: Ms. Mitchell ...........................................................................78
Summary of Dilemma 1 .............................................................................81
Dilemma Type 2: Lesson Specific Content Goals Conflicting with
Mathematical Practice Goals .....................................................................82
Dilemma 2: Mrs. Stewart ...........................................................................83
Dilemma 2: Mr. Sandberg..........................................................................85
Dilemma 2: Ms. Mitchell ...........................................................................85
Summary of Dilemma 2 .............................................................................87
Dilemma Type 3: Lesson Specific Content Goals Conflicting with
Establishing a Mathematical Community Goals ......................87
Dilemma 3: Mrs. Stewart ...........................................................................89
Dilemma 3: Mr. Sandberg..........................................................................90
Dilemma 3: Ms. Mitchell ...........................................................................91
viii

Summary of Dilemma 3 .............................................................................93
Dilemma Type 4: Broad Content Goals Conflicting with
Mathematical Practice Goals ....................................................93
Dilemma 4: Mrs. Stewart ...........................................................................94
Dilemma 4: Ms. Mitchell ...........................................................................96
Summary of Dilemma 4 .............................................................................97
Dilemma Type 5: Broad Content Goals Conflicting with
Establishing a Mathematical Community Goals........................................97
Dilemma 5: Mr. Sandberg..........................................................................98
Dilemma 5: Ms. Mitchell ...........................................................................99
Summary of Dilemma 5 ...........................................................................101
Dilemma Type 6: Mathematical Practice Goals Conflicting with
Establishing a Mathematical Community Goals ....................101
Dilemma 6: Mrs. Stewart .........................................................................103
Dilemma 6: Mr. Sandberg........................................................................104
Dilemma 6: Ms. Mitchell .........................................................................105
Summary of Dilemma 6 ...........................................................................106
Subfinding 1 .....................................................................................................................106
Subfinding 2 .....................................................................................................................107
Subfinding 3 .....................................................................................................................108
Considering Appropriate Scaffolding ..................................................................109
Handling Student Responses................................................................................110
Managing Time Constraints .................................................................................112
Results Summary .............................................................................................................113
CHAPTER 6: DISCUSSION AND CONCLUSION ..................................................................115
Summary of Findings .......................................................................................................115
Discussion ........................................................................................................................119
Implications for Research and Practice............................................................................121
Future Research ...............................................................................................................125
APPENDICES .............................................................................................................................126
APPENDIX A: Iterations of Consistent Patterns of Instruction Coding ........................127
APPENDIX B: Issues with Recruiting a Second School Site .........................................140
APPENDIX C Initial Interview Questions ......................................................................144
APPENDIX D: Lesson Planning Document....................................................................145
APPENDIX E: Revised Lesson Plan ...............................................................................146
APPENDIX F: Examples of Descriptions for Teachers’ Classrooms (Teacher A).........147
APPENDIX G: Examples of Descriptions for Teachers’ Classrooms (Teacher B) ........148
APPENDIX H: Examples of Key Moments in the Lesson..............................................149
APPENDIX I: Classroom Observation Protocol .............................................................150
APPENDIX J: Combined Observation Document ..........................................................152
APPENDIX K: Teacher Reflections on the Lesson ........................................................153
APPENDIX L: Potential Questions to Ask the Teacher Immediately After the Lesson .154
APPENDIX M: Video-Stimulated Interview ..................................................................155
APPENDIX N: Mrs. Stewart’s Consistent Patterns of Instruction ................................156
ix

APPENDIX O: Ms. Mitchell’s Consistent Patterns of Instruction ..................................160
APPENDIX P: Mr. Sandberg’s Consistent Patterns of Instruction .................................164
REFERENCES ............................................................................................................................170

x

LIST OF TABLES

Table 1 Category Types for Consistent Patterns of Teacher Behavior ..........................................24
Table 2 Observed Lessons by Teacher and Round ........................................................................33
Table 3 Professional Obligations Coding ......................................................................................48
Table 4 Dilemmas Teachers Encountered as a Conflict between Goal Types ..............................73
Table 5 Dilemmas between Lesson Specific Content Goals and Broad Content Goals ................75
Table 6 Dilemmas between Lesson Specific Content Goals and Mathematical Practice Goals ...83
Table 7 Dilemmas between Lesson Specific Content Goals and Establishing a Mathematical
Community Goals ............................................................................................................88
Table 8 Dilemmas between Broad Content Goals and Mathematical Practice Goals ...................94
Table 9 Dilemmas between Broad Content Goals and Establishing a Mathematical Community
Goals................................................................................................................................98
Table 10 Dilemmas between Mathematical Practice Goals and Establishing a Mathematical
Community Goals........................................................................................................103
Table 11 Ms. Mitchell’s Dilemmas and Orientations to Professional Obligations .....................107
Table 12 Mr. Sandberg’s Dilemmas and Orientations to Professional Obligations ....................108
Table 13 Mrs. Stewart’s Dilemmas and Orientations to Professional Obligations .....................108
Table 14 Coding Scheme for Different Patterns of Instructional Behavior.................................137
Table 15 Initial Interview Questions............................................................................................144
Table 16 Frequency of Questions in Ms. Mitchell’s Classroom Discussion ...............................163

xi

LIST OF FIGURES

Figure 1. Phases of implementing a whole-class discussion .........................................................28
Figure 2. Students determining the unit rate for baby chimp food ................................................52
Figure 3. Reversed order of permutations in rows 2 and 3 ............................................................55
Figure 4. Survey data from Internet responses and a grade 7 class ...............................................56
Figure 5. Instructions given to students to justify their thinking ...................................................56
Figure 6. Statement of content and practice goals at the beginning of class .................................61
Figure 7. Mr. Sandberg asks if this table represents a linear relationship .....................................67
Figure 8. Area model for selecting marbles from two buckets ......................................................76
Figure 9. Ms. Mitchell displays walkathon results ........................................................................79
Figure 10. A student draws a graph of a non-linear function ........................................................80
Figure 11. Students try to determine Jun’s 7th test score for a resulting average of 85 .................84
Figure 12. Students fail to come to consensus on determining which graphs are linear. ............100
Figure 13. Classroom Observation Protocol ................................................................................151
Figure 14. Combined Observation Document .............................................................................152
Figure 15. Determining a “fast speed” comparing wood and steel-framed roller coasters .........167

xii

KEY TO ABBREVIATIONS

CCSSM

Common Core State Standards in Mathematics

CMP

Connected Mathematics Project

CMP2

Connected Mathematics 2

CMP3

Connected Mathematics 3

IQA

Instructional Quality Assessment

IRE

Initiate, Respond, Evaluate

LES

Launch, Explore, Summarize

MAD

mean absolute deviation

NCTM

National Council of Teachers of Mathematics

NGA-CCSSO National Governor’s Association and Council of Chief School State Officers
NSF

National Science Foundation

xiii

CHAPTER 1
DISSERTATION OVERVIEW AND INTRODUCTION
With the recent widespread adoption of the Common Core State Standards for
Mathematics (CCSS-M) in the vast majority of U.S. states, the CCSS-M Standards for
Mathematical Practice could act as a much-needed catalyst to shift how mathematics has
historically been taught and learned in the U.S. For example, if students are to “construct viable
arguments and critique the reasoning of others” (NGA-CCSSO, 2010, p. 6), then students should
learn mathematics in classrooms where social interaction around content is encouraged.
However, many U.S. teachers still use direct instruction where teacher-to-student interactions
often involve the teacher initiating a simple fact-based question, receiving a response from the
students, and then providing feedback or evaluation (Cazden, 1988; Chapin, O’Connor &
Anderson, 2009; Franke, Kazemi & Battey, 2007; Mehan, 1979; Sahin & Kulm, 2008; Stigler &
Hiebert, 1999).
It is well documented that teaching using classroom discussions to serve the purpose of
engaging students around rigorous content is quite challenging (Lampert et al., 2013; Tyminski,
Zambak, Drake & Land, 2013). Unlike a more traditional mathematics lesson, which may
involve large sections of teacher lecture with limited student talk, facilitating a classroom
discussion involves being responsive to student thinking while continuing to move the
conversation towards a mathematical goal (Sleep, 2012). The tension between being responsive
to student thinking while simultaneously advancing the mathematical agenda in a lesson has the
potential to create dilemmas for teachers as they grapple with how to attend to competing
instructional goals.

1

If teachers are to teach with ambitious aims (Kazemi, Franke & Lampert, 2009; Lampert,
Beasley, Ghousseini, Kazemi, & Franke, 2010; Lampert, Boerst, & Graziani, 2011) such as
being responsive to students’ thinking while they work on authentic problems and engage in
meaningful discourse, then this necessarily adds complexity to the nature of their decisionmaking. During discussions, teachers are faced with decisions related to the types of questions
they ask students, which student ideas they choose to highlight, and how deeply to pursue a
student response. All these decisions occur within the institutional constraints of schooling with
finite resources, such as time and materials. As Lampert et al. (2010) argued, “Maintaining a
coherent mathematical learning agenda while encouraging student talk about mathematics is
perhaps the most challenging aspect of ambitious teaching” (p. 131).
This study aims to better understand how experienced teachers make decisions when
faced with instructional dilemmas while planning and enacting whole-class discussions.
Specifically, I investigated how teachers made decisions in whole-class discussions where
various students are presenting their ideas while the teacher is managing several instructional
goals in real time. A single decision can influence the direction of a discussion, and has the
potential to send important (implicit and explicit) messages to students about their roles as
learners and doers of mathematics. By studying how teachers manage instructional dilemmas
when planning for and enacting whole-class discussions, this work contributes to the field’s
understanding of how teachers facilitate whole-class discussions (e.g., Chapin, O’Connor, &
Anderson, 2009; Smith & Stein, 2011) by unpacking and explaining teachers’ decisions related
to facilitating discussions.
The primary research question guiding this study is: What is the nature of instructional
dilemmas that teachers face when planning for and enacting collaborative mathematics

2

discussions, and how do teachers respond to those dilemmas through their decision-making? In
order to identify instructional dilemmas and analyze decision-making, I observed and videorecorded three middle grades mathematics teachers, conducted post-lesson and video-stimulated
interviews, and collected lesson plans. To identify instructional dilemmas, I first conducted a
preliminary analysis to identify teachers’ instructional goals and their consistent patterns of
instruction. This preliminary analysis allowed for identifying situations when goals came into
conflict with one another, thus creating an instructional dilemma for the teacher. The decisionmaking related to the dilemma was analyzed using Herbst and Chazan’s (2011) construct of
professional obligations.
This dissertation consists of six chapters. Chapter Two consists of a review of the
pertinent literature related to whole-class discussions and teacher decision-making. Chapter
Three includes the theoretical perspective and analytic framework used in this study. Chapter
Four describes the methods used in the dissertation and provides a description of each teacher’s
instructional practice. Chapter Five describes results pertinent to instructional dilemmas and
teachers’ decisions. Chapter Six includes a discussion of the results, implications of the work,
and ideas for future work.

3

CHAPTER 2
LITERATURE REVIEW
Within the mathematics education research base, the word discussion is used frequently
but is interpreted in different ways. This variance may be because the term connotes a vision of
teachers and students actively talking together, which runs contrary to more historically
prevalent modes of communication in mathematics classrooms, where students have been
passive recipients of teacher talk. Researchers have used descriptors such as “whole-group,”
“whole-class,” and “conceptual” to indicate these different types of discussion (e.g., Cengiz,
Kline, & Grant, 2011; Crespo, Oslund & Parks, 2011; Henning, McKeny, Foley & Balong,
2012). Descriptions of discussions generally fall into two main categories: (a) Discussions are an
approach to teaching allowing students and teachers to interact and talk about their mathematical
ideas (e.g., Moschkovich, 1999); (b) Discussion is a specific portion of a lesson (e.g., Ellis, 2011;
Jackson, Garrison, Wilson, Gibbons & Shahan, 2013). In the first case, discussions can occur at
any point of a lesson with any grouping of students, and this type is more aligned with the view
that discussions are a normative way of interacting in a mathematics class not tied to any
particular portion of a lesson. The second type typically occurs towards the middle to end of a
lesson after students have had an opportunity to work on a task. Although these two
interpretations may be relatively unproblematic, for the purpose of this dissertation clarity about
what is meant by discussion will help indicate which portions of the classroom interaction are
included for data collection.
To provide a clear and consistent way to decide whether a discussion be included, I only
chose those discussions focused on mathematical content or practice that involved the teacher
and multiple students where the class had the opportunity to listen to the conversation (i.e., the

4

teacher is not simply talking to a small group). I refer to the discussions that fit these criteria as
collaborative discussions or, equivalently, whole-class discussions as a way to highlight the
collaborative nature of the interactions that took place with the whole class. As a final note, this
definition omits whole-class talk that may not be a “discussion,” such as clarifying what to do on
a task as well as during instances where a teacher lectured but had minimal student interaction,
such as asking students a rhetorical question with responses that failed to impact the flow of the
teacher’s lecture.
Talk in the Mathematics Classroom
Over time, we have come to recognize normative ways that teachers and students interact
in mathematics classrooms. These ways of interacting within the classroom have notable
differences from talk outside the classroom. Informal peer-to-peer conversations may be less
goal-oriented than the conversations inside a classroom, where there is an explicit instructional
goal by the teacher. As Cullen (1998) noted, while talk in the classroom and talk in the outside
world are dictated by social norms and patterns of interaction, these two types of interactions are
quite different because the main purpose of talking in the classroom is for pedagogical, not
simply social, reasons.
In Western settings, traditional classroom environments rely upon the Socratic Method as
a foundational approach to teaching through asking students questions. Particularly in
mathematics classrooms, many teachers to this day still use the IRE format to talk with students
during a lesson. In this format, teachers initiate a query, a student responds, and the teacher
evaluates the response (Cazden, 1988; Mehan, 1979). An important characteristic of classrooms
where the IRE method is predominant is that the teacher is not the only person talking during the
lesson. This is unlike a lecture-based class where students may engage in little to no verbal

5

interaction. Although researchers have noted the limitations of the IRE method, its historical past
is so deeply rooted in U.S. mathematics education that it is not surprising that this method is still
a predominant form of interaction in many U.S. classrooms, even when teaching for
understanding is a focus of instruction (Franke, Kazemi & Battey, 2007). This predominance
may also be the result of a more information-processing view of learning (e.g., Broadbent, 1958
as cited in Gardner, 1987), as opposed to a constructivist approach to learning mathematics
(Schoenfeld, 1992; Smith, 1997).
This raises the question as to how this more traditional form of the IRE format is different
than the types of collaborative discussions investigated in this study. Over the past few decades,
organizations (e.g., NCTM, 2000) and researchers (e.g., Ball, 1993; Chapin, O’Connor, &
Anderson, 2009; Kilpatrick, Swafford, & Findell, 2001; Lampert, 2001; Nathan & Knuth, 2003)
have described a different vision for how discussions might take place in mathematics
classrooms. There are two main differences between classrooms that predominantly make use of
the IRE format and those that use a collaborative discussion. The first difference relates to the
classroom environment and its normative modes of teacher and student interaction. The second
difference relates to the format of discussion the teacher uses to advance lesson goals.
Discourse-Oriented Teaching
The term discourse-oriented teaching provides a broad description of the type of
classroom environment that supports collaborative discussions. According to Williams and
Baxter (1996),
We introduce the term discourse-oriented teaching [italics in original] to describe actions
taken by a teacher that support the creation of mathematical knowledge through discourse
among students. … [W]e see discourse-oriented teaching as involving not only the

6

establishment of social norms that empower students to discuss mathematics but also a
belief that properly orchestrated discourse among students will result in the production of
useful knowledge. (p. 22)
There are a few key points to this approach in teaching mathematics that are similar and
different from more normative approaches to teaching. One similarity is that both the traditional
approach and discourse-oriented approach are focused on developing student knowledge. A key
difference, however, is that in the discourse-oriented approach the role of the teacher is to
provide opportunities and to support students with talking about mathematics with their peers. In
a more traditional setting, student responses are often communicated through the teacher and are
typically in response to a teacher query. In discourse-oriented classrooms, students are
encouraged to ask questions to the teacher and to their peers, and they respond to these questions
as a member of a mathematical community. Students are encouraged to engage in argumentation,
justification, disagreement, and sense-making. Indeed, letting students talk and describe their
understanding, along with their multitude of misconceptions and underdeveloped notions of
mathematical ideas, makes teaching much more complex.
If teachers are to teach using a discourse-oriented approach, then this necessitates that
teachers have a different type of knowledge in supporting students, such as the knowledge of
student misconceptions and their possible causes, and how to act on those misconceptions in a
productive way. Additionally, if students are to talk with their peers and their teacher about their
thinking, social norms must be in place to help effectively support students’ willingness to share
their thinking. In all, the demands on a teacher are much different in discourse-oriented teaching,
compared to a traditional classroom setting where teaching focuses on procedural fluency,
answering questions from students, and asking students predominantly simple fact-based

7

questions with known answers (Chapin, O’Connor, & Anderson, 2009; Sahin & Kulm, 2008). In
many cases, the “lecture” portion of a more traditional lesson is followed by individual seatwork, thus inhibiting students from having opportunities to engage deeply in the content with
their peers.
For some students this traditional format may work, but for many students, talking
through their underdeveloped ideas about mathematical concepts is essential for continuing to
develop their mathematical knowledge. Perhaps more importantly, the teacher has better access
to students’ conceptions in real time through discourse-oriented teaching, because the students
are describing their thinking at the time they are engaged in the lesson. Assessing student
thinking by only looking at written work at the conclusion of a lesson provides only a glimpse of
a student’s thinking. The main implication for teachers attempting to use a discourse-oriented
approach is that they must structure their lessons differently in order to support student-tostudent interactions as a pedagogical approach to help produce useful mathematical knowledge.
Collaborative Discussions
In contrast to the traditional form of mathematics instruction, a more Standards-based
(NCTM, 1989; NCTM, 2000) approach to a lesson includes three main phases (Jackson & Cobb,
2010). A teacher starts the class by posing a high-demand task (Smith & Stein, 1998); students
then work on the task in small groups, followed by a whole class discussion. These three phases
are sometimes referred to as a teacher Launching the problem by setting up the task for students,
followed by students Exploring in small groups, and then the class coming together as a whole
group to discuss and then Summarizing the ideas in the lesson (Shroyer & Fitzgerald, 1986).
An inherent aspect of this approach to lesson design is that students learn from each other
around rigorous mathematical content (Franke, Kazemi, & Battey, 2007) through collaboration

8

instead of isolation (Lampert & Cobb, 2003). Aspects of high demand tasks include complex or
non-algorithmic thinking, deeply understanding mathematical concepts, and fluently moving
between representations (Smith & Stein, 1998). Students work together in small groups on tasks
that focus on conceptually rich topics, providing opportunities for students to generate multiple
solution strategies and to explore big ideas in mathematics.
In the latter part of the lesson, teachers facilitate students’ work on the task in a
collaborative discussion. During the discussion, students are actively engaged in listening to
others’ ideas about how they solved the task and how their strategies related to the strategies of
their peers. When done effectively, teachers organize student solution strategies to make salient
important ideas embedded in solution methods that correspond to the teachers’ lesson goals. The
teacher’s role in this type of discussion is to be responsive to students while allowing students’
ideas to take center stage, all the while advancing students’ mathematical understanding.
This type of discourse-oriented teaching is quite challenging for experts and novices
alike, because the teacher is constantly attending to ever-evolving ideas while also attending to
individual students’ needs and classroom interaction. Teachers often have several types of goals
from content to communication (e.g., being precise in language use) to process/practice goals
(e.g., justifying one’s thinking). Even if a teacher is only concerned about the coherence of the
mathematical ideas within a lesson; interpreting, connecting, and then building on students’
responses in a way that promotes interaction between students can be daunting. A focus strictly
on content does not take into account teachers’ long-term, dispositional, or social goals that they
may also have for students in the classroom. Teachers often have multiple purposes, both
mathematical and nonmathematical, that they are attempting to satisfy during the same lesson
(Sleep, 2012). As the discussion is occurring, teachers are monitoring the movement towards

9

lesson goals; decisions about when to step in to expand on an idea, when to let students build off
each other’s ideas, and when to stop the discussion to summarize. These considerations occur in
a setting that is time-sensitive, constraint-filled, and dilemma-ridden. The context of whole-class
discussions is prime for studying how teachers make decisions in real time when they encounter
instructional dilemmas requiring them to prioritize one aspect of their instruction over another. In
this study, I investigated the nature of these dilemmas with special attention to teachers’
decision-making during such dilemmas. In the next section, I provide a background of the
teacher decision-making literature with specific attention to decision-making in whole-class
discussions.
Teacher Decision-Making Literature
Studying teacher decision-making dates back at least to the 1970s when different sets of
researchers identified the centrality of decision-making to the improvement of teaching (Borko,
Roberts, & Shavelson, 2008). Bishop and Whitfield (1972) identified a teacher’s values and
experience as contributing factors to a teacher’s decision framework. Later, Shavelson and Stern
(1981) identified other contributing factors such as a teacher’s content knowledge, the nature of
instructional tasks, and the influence of the institutional environment of the school. Recent work
has included studies of science teachers viewing decisions as “argument-driven actions” (Lee &
Lin, 2005, p. 429). Other teacher decision-making studies have focused on a variety of other
aspects of teaching, from metacognition (e.g., Artzt & Thomas, 1998) to reflecting on one’s
teaching practice via video analysis (e.g., Rich & Hannafin, 2008). Put together, all of these
studies in teacher decision-making focused on better understanding the nature of how teachers
make decisions by understanding the underlying contributing factors that influence their
decisions.

10

In mathematics education, two recent contributions to the teacher decision-making
literature provide a foundation for this dissertation. First, Schoenfeld (2010) described modeling
of decision-making and its role in mathematics teaching. He argued that “what people do is a
function of their resources (their knowledge in the context of available material and other
resources), goals (the conscious or unconscious aims they are trying to achieve), and orientations
(their beliefs, values, biases, dispositions, etc.)” (p. xiv). Schoenfeld’s (2010) perspective on
teacher decision-making highlights the cognitive aspects of teacher thinking with strong
emphasis on decisions in relation to mathematical content. He goes on to describe the importance
of understanding teachers’ goals and routines:
If you can understand (a) the teacher’s agenda and the routine ways in which the teacher
tries to meet the goals that are implicit or explicit in that agenda, and (b) the factors that
shape the teacher’s prioritizing and goal setting when potentially consequential
unforeseen events arise, then you can explain how and why teachers make the momentby-moment choices they make as they teach [italics in original]. (Schoenfeld, 2010, p.
10).
Consistent with Schoenfeld’s (2010) perspective on teacher decision-making, the main
focus of this dissertation is to understand decision-making when teachers faced instructional
dilemmas. By understanding their goals and consistent patterns of instruction, one can be better
positioned to identify and explain the nature of their instructional dilemmas. Importantly, I
deviate from Schoenfeld’s (2010) orientation to teacher decision-making as a more rational
nature and his perspective on instructional goals as more specific and content focused.
Others, such as Herbst and Chazan (2011), offer a different perspective on teacher
decision-making. Their perspective considers teachers’ decisions as a product of teachers

11

attending to various professional obligations. Their contribution considers other external factors
(such as the school setting) as being important factors in a teacher’s decisions. These obligations
provide a framework for understanding the reasons for the choices teachers make and for naming
the influences that create instructional dilemmas for teachers. These obligations are described
here:
The disciplinary obligation [italics in original] says that the mathematics teacher is
obligated to steward a valid representation of the discipline of mathematics. This may
include the obligation to steward representations of mathematical knowledge,
mathematical practices, and mathematical applications.
The individual obligation [italics in original] says that a teacher is obligated to attend to
the well-being of the individual student. This may include being obligated to attend to
individual students’ identities and to their behavioral, cognitive, emotional, or social
needs.
The interpersonal obligation [italics in original] says that the teacher is obligated to share
and steward their medium of interaction with other human beings in the classroom. This
may include attending to the needs and resources of shared discursive, physical, and
social spaces within shared time.
And the institutional (schooling) obligation [italics in original] says that the teacher is
obligated to observe various aspects of the schooling regime. These include attending to
school policies, calendars, schedules, examinations, curriculum, extracurricular activities,
and so on. (pp. 450-451).
Much of Herbst and Chazan’s previous work used simulated classrooms (e.g., Herbst,
Chazan, Kosko, Dimmel, & Erickson, 2016), focusing primarily on understanding norms and

12

deviations of those norms through simulated classroom scenarios. In these studies, they used
simulated classroom environments to ask teachers what choices they would make in a particular
situation. There are some limitations to this approach as it can constrain the teacher into choosing
from only a small set of possible choices, which may or may not be plausible or realistic for that
particular teacher. Additionally, it fails to address potential variables that may play out in the real
world of the classroom. For example, a teacher may make two different decisions when posed
with an interesting student question with five minutes left in class: if the question is asked the
day before a chapter test versus the question being asked in the beginning of a unit. An exchange
between teachers and students is inherently embedded within a broader sequence of lessons, so
knowing whether the exchange aligns with a teacher’s multiple instructional goals or not could
make a huge difference as to the path the teacher takes. Finally, an important aspect of
understanding how the interpersonal and individual obligations play out in the classroom is the
teacher’s own understanding of particular students and how they interact with each other.
Studying teachers in their own classrooms has the added benefit of understanding how
teachers enact decisions in a more authentic environment. I deviate from Herbst et al.’s (2016)
use of simulated classrooms by focusing on the inherent “situatedness” of teachers’ cognition as
the teaching occurs as a lived experience in their classrooms with students they encounter day
after day. Consistent with Herbst and Chazan (2011), I use the construct of professional
obligations to understand instructional dilemmas and teacher decision-making, because it affords
the possibility of decisions being influenced by factors that are outside the domain of
mathematical content that likely influence teacher decisions.
The professional obligations construct also provides a mechanism for understanding the
inherent complexity in facilitating discussions while attempting to balance the needs of multiple

13

goals (e.g., Sleep, 2012), the needs of various students, and the nature of classroom interaction.
These conflicts were what made decisions meaningful, in that teachers decided whether to
maintain their typical interaction-consistent patterns of instruction or consciously chose to
deviate from these consistent patterns of instruction. Understanding what is typical and atypical
of classroom behavior for teachers’ facilitation of classroom discussions was important in
understanding when, and eventually why, they made certain instructional decisions. Explaining
teacher decision-making through the lens of professional obligations provided the opportunity to
explain how teachers managed competing demands, based on teachers’ views of who or what
they felt primarily obligated to satisfy.
Balancing the demands of these different obligations may play out in different ways.
Take for example the simple task of selecting which student to call on during a whole-class
discussion. A teacher may choose to call on a student randomly to hold students accountable for
paying attention in the class (institutional obligation). Another teacher might choose to call on a
student because he has an interesting mathematical strategy to share with the class (disciplinary
obligation). A different teacher might choose to call on a particular student because that student
rarely participates, and the teacher wants to build the student’s confidence by having him share
his thinking with the class (individual obligation). Another teacher might choose to call on a
student because she was able to summarize her group’s thinking well, and she they wanted to
talk about the importance of students listening to their peers (interpersonal obligation).
Each of these examples speaks to teachers’ general orientation about their role in the
discussion, and these orientations have important implications for how teachers enact decisions
in the discussion. Experienced teachers particularly may also make decisions that attend to
multiple obligations at the same time instead of prioritizing one over the other when the

14

obligations come in conflict with each other. This construct of professional obligations can
illuminate various challenges teachers face related to implementing whole class discussions. I
describe some of these challenges from the literature as they relate to the four professional
obligations.
Institutional Challenges
One set of challenges that many teachers face is the constraints from the school as an
institution. Institutional challenges include issues that arise from school or district-level policies,
schedules and calendars, mandatory assessments and their potential accountability for student
achievement, pacing guides that require what lessons teachers will be teaching on certain days,
and expectations for managing student behavior, to name a few. All of these challenges and
others like them share one common feature: They typically act to constrain the teacher because
the teacher is often not in a position to change these constraints. Acknowledging these
constraints is important, because as Cobb and Smith (2008) argued, taking the position that
teachers are fully autonomous in their classroom is “flying blind” with respect to reforming
instruction (p. 3).
A main critique of teaching through implementing collaborative discussions is that it is
less efficient. Content coverage is certainly a concern for many teachers given the adherence to
pacing guides by some schools. While the potential usefulness of these guides is to help ensure
that all students have opportunities to encounter all content at a certain grade level, it can also
negatively influence teachers to focus more on simply covering content instead of being
responsive to their students’ learning needs. Making use of collaborative discussions may inhibit
coverage of content, and thus run in opposition to established expectations for pacing.

15

A second related challenge is that teachers are often held accountable for ensuring their
students pass standardized tests, either statewide or department-level (e.g., Michigan Department
of Education, 2012). If the tenet that “assessment drives instruction” is true, then the quality of
these types of assessments likely play an important role in shaping the nature of instruction.
There is evidence from the literature that this interaction between assessment and instruction is
present. McGraw (2002), for example, described the constraints of instructional practice while
facilitating whole-class discussions when she noted that the end-of-semester departmental exam
was an important factor in teacher decision-making while considering ways to handle students’
ideas during whole-class discussions.
Disciplinary Challenges
Students’ learning opportunities to engage in mathematical content are largely
constrained by instructional materials and their teachers’ content knowledge. One might assume
that a teacher’s understanding of content strongly influences their ability to teach in discourseoriented fashion; however, even in situations where the teacher was experienced and had strong
content background, the shift in instructional demands was great. Speer and Wagner (2009), for
example, studied an experienced teacher’s implementation of an inquiry-oriented approach to
teaching Differential Equations at the university level. They noted that even though the teacher
had taught the course previously and had a strong content background, the teacher encountered
challenges in realizing the mathematical potential and relevance of student contributions during a
discussion.
Researchers have also noted that written curriculum materials, such as textbooks, play an
important role in shaping students’ opportunities to learn mathematics (e.g., Stein, Remillard, &
Smith, 2007). A teacher who uses a worksheet of twenty procedural problems provides different

16

opportunities for student interaction than a teacher who is using an open-ended task. This is not
to say that the affordances of an open task are always capitalized. Teachers can readily diminish
the demands of even the most challenging of tasks (e.g., Choppin, 2011). This being said, in a
large-scale study of middle school students, Tarr et al. (2008) reported that students in
classrooms with moderate to high implementation of a Standards-Based Learning Environment
that also used NSF-funded curricula (which often focus on an inquiry approach) achieved
significantly higher in problem solving, communication, and procedural fluency. Although
studies comparing publisher-generated and NSF-funded curricula are not common, Tarr et al.’s
findings are consistent with others that have shown that students using NSF-funded curricula
typically do better on assessments that focus on problem solving or conceptual understanding
(Stein & Smith, 2010). While the teacher certainly plays a vitally important role in implementing
and shaping student’s opportunities to learn, these opportunities are in large part shaped by the
written materials themselves.
Interpersonal and Individual Challenges
Challenges related to Herbst and Chazan’s (2011) interpersonal obligation focus
primarily on the interactions between teachers and their students in the shared social space of the
classroom. A major challenge for teachers using a discourse-oriented approach is that it requires
students to engage and actively participate. In a discourse-oriented approach, students need to
learn how to participate as part of a discourse community, which likely impacts how they view
themselves as learners of mathematics. A teacher can make all the productive teaching moves
they want to, but if it does not promote student interaction it will be difficult to teach using a
discourse-oriented approach. A teacher’s actions are important for the short term (e.g., calling on

17

a student to clarify an idea), but also for the long term, in their role in supporting social norms
and substantiating affective aspects of the classroom environment.
As one example, Jansen (2006) studied 15 middle school students and their motivation
for engaging in classroom discussions. About half of the students in her study actively
participated as speakers more consistently during classroom discussions than the other half,
which tended to be quieter. She found that students who felt “threatened” to participate actively
in the classroom discussions would still participate in the discussions if they felt that their
participation helped their classmates. She later concluded, in Jansen (2008), that student
participation was also related to whether the talk in the classroom was primarily procedural or
conceptual. For a teacher, these realizations are important, because they point to the various
reasons why students choose to participate (or not) in discourse-oriented classroom settings.
In addition to Jansen’s findings, others have noted the importance of student
understanding in relation to how they participate. For example, Webb (1991) found a positive
correlation between student achievement and students “giving content-related explanations” to
their classmates (p. 377). She later argued that the ideal classroom “setting is one in which
students freely admit what they do and do not understand, consistently give each other detailed
explanations about how to solve the problems, and give each other opportunities to demonstrate
their level of understanding” (p. 386).
More recently, Webb et al. (2014) argued that giving students the opportunity to build on
each other’s ideas has the added benefits of encouraging students to monitor their own thinking,
and they may be better positioned to “recognize gaps in their knowledge, misconceptions or
contradictions between their own ideas and those they are hearing” (p. 80). In their study of over

18

100 children ages 8-10, they found a positive relationship between student achievement and
students’ engagement with other students’ ideas.
Summary
Put together, Schoenfeld’s (2010) focus on goals and Herbst and Chazan’s (2011)
construct of professional obligations frame the potential dilemmas teachers can face while
enacting whole-class discussions, particularly when multiple obligations are needing to be
satisfied or prioritized. It also points out the inherent issue when studying teacher decisionmaking as necessarily situated in teachers’ classroom work (Lee & Lin, 2005). This work
includes all the many factors and stimuli that teachers encounter as unique to the real work of
teaching with students they interact with day after day. These factors vary from content specific
(e.g., a student not understanding how to multiply) to those that are less content oriented (e.g.,
student affect).

19

CHAPTER 3
THEORETICAL PERSPECTIVE AND ANALYTIC FRAMEWORK
The guiding theoretical perspective for this study is situated cognition (Greeno, 1989).
The situated perspective places strong emphasis on the interaction between individuals and their
environment. As Greeno (1989) stated, “Cognition, including thinking, knowing, and learning,
can be considered as a relation involving an agent in a situation, rather than as an activity in an
individual’s mind” (p. 134). In this study, teacher decision-making is best understood in the
context of teacher activity in their classrooms; thus, I attended to teachers’ interactions with their
students over multiple lessons, observing behavior, hypothesizing about their decisions, and later
interviewing them about possible decisions. This approach differs from simply interviewing
teachers about a simulated classroom, or asking teachers to state what decisions they would
make in a theoretical situation in their own classroom. The approach I used in this study aimed to
provide a more authentic view of teachers’ decision-making as a cognitive activity necessarily
embedded in their environment.
Main Analytic Constructs
To better understand teacher decision-making during whole class discussions I attend to
five main constructs that individually and collectively support why they made particular
decisions when they encountered a particular dilemma. I argue that a teacher’s decision when
faced with a particular dilemma is strongly influenced by their instructional goals and their
professional obligations (Herbst & Chazan, 2011). Thus, to understand teacher decision-making,
it was necessary to come to understand teachers’ instructional goals, orientations to their
professional obligations, consistent patterns of instruction that emerge when acting upon
instructional goals, instructional dilemmas, and their resulting decisions that surfaced when

20

instructional goals were enacted during whole-class discussions. In the next section, I first
describe each of these constructs in detail, and then describe how these constructs connect.
Professional Obligations
As described in the previous chapter, I use Herbst and Chazan’s (2011) description of
professional obligations as a way to understand teachers’ general orientations to facilitating
whole-class discussions, which in turn influence their decisions during whole-class discussions.
Consistent with this view, I argue that teachers’ decisions are influenced by their obligations to
the institution of the school, to students as individuals, to supporting students in their social
development, and to their role as a representative of the discipline of mathematics. A teacher’s
obligations and goals are connected in that they both orient a teacher’s behavior and their
decisions.
A teacher’s professional obligations differ from their goals in that goal(s) are about
accomplishing a task (e.g., supporting students in effectively communicating their thinking with
the class), whereas the teacher’s professional obligation(s) describe who or what they feel
obliged to satisfy (e.g., the individual student, the school, etc.) during the discussion. For
example, if two students get in a somewhat heated exchange disagreeing about the solution of a
problem during a discussion the teacher could view this situation in very different ways. The
teacher might see the situation as (a) a management issue needing to be addressed (institutional
obligation), (b) an opportunity to re-phrase the crux of the argument and have the class
investigate the mathematical issue further (disciplinary obligation), (c) an opportunity to re-focus
the class on appropriate ways to interact with each other (interpersonal obligation), or (d) a need
to intervene to protect each student’s willingness to participate in the future (individual
obligation).

21

Instructional Goals
The instructional goal is what the teacher hopes to accomplish as a result of enacting a
whole-class discussion. How teachers address their professional obligations influences the
instructional goals they might have. For example, a teacher can have lesson-specific content
goals (e.g., calculate the median of a data set) to address the disciplinary obligation of their role
as teacher or goals for establishing a mathematical community (e.g., students will listen to their
peer’s thinking) to address their interpersonal obligation. For example, a teacher may choose to
pursue an errant line of reasoning because it supports an individual student’s willingness to
participate in class, because it models an important classroom norm related to the value in
sharing mistakes, or because it uncovers an important misconception with possibly long-lasting
effect. These broader instructional goals may support student development but are not
necessarily a content goal tied to a particular lesson. Understanding a teacher’s instructional
goals illuminates what they are intending to accomplish during the whole-class discussion.
I do not make the claim that teachers are always aware of their goals or, if they are aware
of them, that they are able to articulate them clearly. In many cases, teachers may have
instructional goals that are somewhat unknown to them (e.g., a new teacher knowing what they
should be doing without a clear reason why they ought to be doing it). A teacher may be acting
consistent with a goal, without knowing what their goal is. Some teachers, for example, might
simply have a goal to make sure the class runs smoothly, and maintaining order is the most
important aspect with very few mathematical goals. I take as an assumption that the nature of
teaching falls in the category of “goal-oriented activity,” but I am cautious in not assuming that
any discussion necessarily includes all four types of goals. This being said, it is likely the case
that a teacher’s instructional goals are shaped in some capacity by their professional obligations.

22

For example, a teacher with a tendency to orient their instruction towards their individual
obligation may enact lesson-specific content goals differently than a teacher with a tendency to
orient their instruction to the disciplinary obligation.
Consistent Patterns of Instruction
The third construct in this study focuses on teachers’ consistent patterns of instruction.
These could be considered “routines,” but given the small sampling of lessons observed in this
study I use the term “consistent patterns of instruction” to describe characteristics of teachers’
instruction that consistently occurred across several discussions. I am also cautious in not using
the term “routine” to describe these consistent patterns of instruction because of the possibility of
confusing my conception of routine with others in the field (e.g., Kazemi, Franke, & Lampert,
2009). For this study, I define a “consistent pattern of instruction” consistent with other
mathematics education researchers’ views on instructional routines (e.g., Leinhardt & Steele,
2005; Leinhardt, Weidman, & Hammond, 1987) as a socially constructed script of behavior
mutually understood and enacted by members of the classroom community. This definition is
also consistent with other descriptions of routines outside of mathematics education (e.g.,
Becker, 2005; Zisberg, Young, Schepp, & Zysberg, 2007).
A teacher may have particular goals and obligations they aim to attend to, but enact
consistent patterns in their instruction that are incongruent with these actions. For example, a
teacher may want to have students justify their reasoning during the discussion, but the teacher
simply explains the reasons for the students and asks whether they agree with her or not. Thus, a
teacher’s consistent pattern of instruction can help describe how they enact (or fail to enact) their
goals and obligations.

23

A second important aspect related to consistent patterns of instruction is that when
teachers deviate from a reoccurring pattern, there may be greater likelihood that the teacher made
a conscious decision. These deviations from the business-as-usual flow of the discussion can
indicate a teacher’s decision when faced with a conflict between goals, or a conflict between
professional obligations (e.g., a teacher decides not to interrupt a student while describing their
thinking when the bell rings to end class) (Parker, 1984; Sutcliffe & Whitfield, 1976).
I categorized the patterns into the following types: management, instructional support,
activity structure, interaction, and meta-routines, derived from the research base on instructional
routines (La Course, 2011; Leinhardt & Steele, 2005; Leinhardt, Weidman, & Hammond, 1987;
Yinger, 1979). Given below in Table 1 is a list of the definitions and descriptions of each type of
consistent behavior. Greater detail about identifying and categorizing consistent patterns of
teacher behavior can be found in Appendix A. Within each set of patterns, I later identified finergrained patterns of behavior and coded them after conducting several passes through the lesson
observation data.
Table 1
Category Types for Consistent Patterns of Teacher Behavior
Consistent Pattern
Description
Management
Any patterns of behavior related to classroom management,
such as student discipline, grouping students, etc.
Instructional Support

Support teacher-student interaction, such as instructional
resources to be used during interaction, and the location of
where the interaction takes place

Activity Structure

These patterns consist of how the teacher typically starts and
ends discussion, how long the discussion segments last, and
transition points between group/individual work and wholeclass discussion.

Interaction

These patterns specify the ways in which teachers and
students should interact and communicate with each other.

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Table 1 (cont’d)
Meta-Routine

These patterns involve the teacher explicitly referencing a
consistent pattern of behavior with students.

Dilemmas
As teachers enact their goals, they encounter situations where goals conflict with each
other. I refer to this type of conflict as a dilemma. For example, a teacher may want students to
explain their solution strategy to the whole class on how to find the slope of a line. This is done
so (a) the student can engage in mathematical practice (communicating one’s thinking) and (b) to
advance lesson-specific content (calculating slope). But if the student finds the problem way too
challenging to even begin to describe their thinking, the teacher is faced with the dilemma of
managing both goals, which could be in competition with one another based on the teacher’s
interpretation of the situation.
Dilemmas can be observed as deviations from consistent patterns of instruction
(described further in methods) or identified through teacher self-report about the dilemma. An
important limitation of this study is that some data obtained about the dilemmas teachers
encountered during whole-class discussion came from self-report during interviews, and, as such,
some teachers may have been more forthcoming with their own understanding of the types of
dilemmas they encountered. Although these self-reported dilemmas (i.e., a teacher stating, “This
is a dilemma” during an interview that was not observed as a deviation to a consistent pattern of
instruction) may have varied across teachers, omitting this data from the dissertation would
essentially discount several dilemmas that teachers were fully aware of. These dilemmas were to
be expected given the relatively conservative approach I took in identifying consistent patterns of
instruction and using these patterns (and their deviations) to identify dilemmas. Although these
self-reported dilemmas add a layer of variability likely not present in the observed dilemmas,

25

they do provide a fuller picture of the dilemmas that teachers encountered while planning and
enacting whole-class discussions.
Decisions
A dilemma presents an instructional crossroads in the lesson where the teacher must
make a decision about which goals and/or obligations take priority. By decision, I mean those
consciously made choices that teachers make during a lesson. I take a much more restrictive
view on decisions as only those with high likelihood of being consciously made, such as those
that disrupted patterns of behavior that consistently occurred during a discussion or when a
teacher acknowledged during an interview having made a particular decision. The decision is the
end result of the other four constructs; that is, given a teacher’s orientation to their obligations,
goals, and consistent patterns of instruction, one can explain why the teacher made that decision
when they encountered a particular dilemma.
Connecting the Main Constructs
Figure 1 (below) shows a schematic diagram of how the five constructs interact with each
other over the development of a lesson (with special attention to collaborative discussions).
During each of the three phases, dilemmas and their decisions appeared in various ways.
Although lesson planning is not a main emphasis of this study, some dilemmas did appear in
lesson plans and during conversations about future lessons. During planning, teachers consider
their goals for a particular lesson (as well as possibly broad instructional goals) and create a
lesson plan (even if simply a mental script). During the planning phase, they may anticipate
dilemmas that might arise as a result of goals conflicting with each other. Teachers make
decisions based on how they expect the lesson to play out, framed by their goals and obligations.

26

The plan can include both teacher behavior that is consistently present across many lessons, or
behavior that is idiosyncratic to a particular lesson.
While implementing collaborative discussions, teachers make decisions when faced with
instructional dilemmas as a result of goals conflicting with each other. When teachers deviate
from a consistent pattern of instruction, this deviation indicates a decision potentially linked to
some instructional dilemma resulting from these conflicting goals. During the post-lesson phase
(reflecting), teachers describe (via interview) dilemmas they faced during lesson enactment. At
this stage, teachers may evaluate the extent to which they agreed or disagreed with their
decisions related to how they implemented various instructional goals.
Put together, a teacher’s goals and orientations to their professional obligations can help
explain the particular decisions they made as a result of encountering a particular type of
dilemma. It is this relationship between these five main constructs that provides the impetus for
this study: that instructional decisions during whole-class discussions can be explained by their
goals, obligations, and behavior.

27

Pre-Lesson: Planning for Whole Class Discussion(s)
Planning Decisions

Planning Dilemmas
 competing instructional goals

Professional Obligations &
Goals
frame how teachers
make planning decisions

 Prioritizing and balancing
instructional goals

Evidence of Decisions

Evidence of Dilemmas

 explicitly stated in interviews about
planning a lesson
 written lesson plans

 explicitly stated in interviews about
planning a lesson
 written lesson plans

During the Lesson: Enacting Whole Class Discussion(s)
Dilemmas that occur During
the Lesson
 competing instructional goals

Decisions that Occur During
the Lesson
Professional Obligations
frame how teachers
make enactment decisions

 Prioritizing and balancing
instructional goals

Evidence of Decisions

Evidence of Dilemmas

 Researcher hypothesis based
on observation of dilemmas

 deviations from patterns of teacher
behavior
 atypical implementation of goals

Post Lesson: Reflecting on the Whole-Class Discussion(s)
Evidence of Decisions

Evidence of Dilemmas

 teacher acknowledgement / report
during interviews

 teacher acknowledgement / report
during interviews

Figure 1. Phases of implementing a whole-class discussion

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CHAPTER 4
METHODS
In this chapter, I describe the methods used to address the research questions for this
dissertation. In order to address these questions, I collected data from teacher lesson plans,
observations, and teacher interviews. The three teachers in the study were quite different in their
approaches to facilitating discussions and had various professional experiences. At the end of
this chapter, I provide a general overview of each teacher’s instructional practice.
Study Setting
This study took place with three grade seven mathematics teachers (pseudonyms are used
for the teachers, students, and the name of the school). I chose to study middle grades teachers
because they are in a unique position in their districts, in that they are directly affected by
previous curricular experiences within the district (at the elementary level), and they are at least
aware of the obligation to be responsive to the needs of the high school mathematics department.
There is also evidence to suggest that students perceive differences in the nature of the middle
school and the high school curricula, especially when the transition is from reform-oriented
materials to more traditional ones (Star, Smith, & Jansen, 2008). The middle school mathematics
curriculum includes new ideas and presents opportunities for students to think about previous
knowledge in a new way, such as the shift from arithmetic to more complex algebraic thinking
(e.g., Knuth, Alibali, McNeil, Weinberg, & Stephens, 2005; Knuth, Stephens, McNeil, & Alibali,
2006). Put together, the complexities from the potential of competing obligations appearing is, at
the very least, sufficiently possible at the middle grades level.
My initial goal was to select two school sites, with one teacher designated as an “expert”
and one as a “novice.” Expert and novice teachers have been identified in different ways (e.g.,

29

Berliner, 2001; Borko & Livingston, 1989; Sternberg & Horvath, 1995), for example, by the
years of teaching experience or their role (e.g., student teacher or mentor). The literature on
expert and novice teaching practice indicate that expert teachers focus more on student
understanding, tend to plan lessons in relation to long-term plans, have more knowledge about
teaching and content, have deeper connections between their types of knowledge, and are more
adept at implementing their knowledge when faced with situations as they appear in the
classroom.
In contrast, novice teachers tend to have less connected knowledge, tend to plan for the
short-term, have difficulties implementing their knowledge during lesson enactment, and often
fail to capitalize on important mathematical moments in the lesson (Borko & Lingston, 1989;
Leinhardt, 1989; Stockero & Van Zoest, 2013; Westermain, 1991). Thus, one might expect to
see more attention to a variety of instructional goals from expert teachers than novices, and
therefore a greater likelihood of encountering various instructional dilemmas as expert teachers
enact whole-class discussions. This means that if given a choice between studying either expert
or novice teachers, expert teachers would potentially provide more interesting data in
understanding the nature of their decisions while they encounter instructional dilemmas during
whole-class discussions.
Recruiting Teachers for the Study
As a first step in recruiting teachers, I identified schools that used the Connected
Mathematics curriculum (Lappan, Phillips, Fey, & Friel, 2014; Lappan et al., 2005), a Standardsbased (NCTM, 1989) curriculum that is consistent with the view of supporting students to
engage in whole-class discussions to support mathematical understanding (Jackson, Garrison,
Wilson, Gibbons, & Shahan, 2013; Jansen, 2008). Additionally, by selecting a single set of

30

materials, this hopefully mitigated any curricular effect that might have influenced the nature of
decision-making as embedded within whole-class discussions. Two teachers (Mrs. Stewart and
Mr. Sandberg) were piloting the third edition of the materials (CMP3), while Ms. Mitchell taught
using the second edition (CMP2).
To identify schools that used CMP2 or CMP3, I contacted the secretary of the Connected
Mathematics Project, visited school district websites to identify their middle school mathematics
curriculum, and contacted professional development leaders familiar with middle school
mathematics implementation in their regions. I recruited 11 schools for the study, with the end
goal of identifying a pair of grade 7 teachers from each school with various levels of expertise.
Seventh grade was specifically chosen over grade 6 because of the potential issues with students
being new to the curriculum. Additionally, I selected grade 7 because it had many opportunities
for lessons with high-level discussions. Recruiting for school sites started in September 2014.
Only one school agreed to take part in the study.
Highland Junior High School was very enthused about the study. I met with the principal
and the department chair, and they expressed willingness to support the study. They had four
grade seven teachers, and it was hoped at the beginning that all four teachers would be interested
in participating. From our initial conversation, there appeared to be a good chance that the
teachers would be classified with one pair as experts and one pair as novices. This could have
helped with data collection in potentially restricting the study to one site. One teacher, however,
chose not to participate because of other commitments. After unsuccessfully recruiting a second
school site over several months, I decided to collect data with the three teachers at Highland. The
remaining three teachers consented to participate in the study.

31

Due to recruitment issues (described in greater detail in Appendix B), the three teachers
in the study can be best categorized as having different types of expertise. For example, two
teachers (Mr. Sandberg and Mrs. Stewart) had multiple years of teaching the curriculum,
whereas the third teacher (Ms. Mitchell) was in her first year teaching the curriculum. Mrs.
Stewart and Ms. Mitchell had both taught at different grade levels (secondary and elementary,
respectively), which afforded opportunities for understanding the trajectory of mathematical
ideas. Mr. Sandberg had previously taught science before teaching mathematics. I describe later
how I characterized and measured teaching expertise. This was especially important during the
recruitment process of teachers to ensure that the teachers in the study included classroom
discussions as a consistent aspect of their classroom interactions.
Data Collection
I collected data for this study in four rounds. Round 0 consisted of an initial interview
with the teachers (approximately 20-30 minutes) focusing on their beliefs about whole-class
discussions, student learning, and planning. Additionally, there was one observation where I
asked teachers to identify a “typical lesson” (as defined by the teacher) that involved discussion
in the class. I later analyzed observation data using the Instructional Quality Assessment
Classroom Observation [IQA] protocol (Boston, 2012) to determine the level of expertise of the
teachers in facilitating whole-class discussions.
Rounds 1, 2, and 3 involved multiple classroom observations. For the most part, each
observation included a lesson plan and a post-lesson reflection interview. I conducted a videostimulated interview at the conclusion of each round, which was a set of consecutive lessons as
taught by the teacher. There was a break in observed lessons between each round. Later in this
section, I provide greater detail on each aspect of these observations. Table 2 below includes the

32

lessons taught in each round by each teacher. Focus lessons are identified in the yellow cells. I
observed at least one lesson before and at least one lesson after each focus lesson, with the
exception of Ms. Mitchell’s first round as this focus lesson spanned over three class periods.
Lessons that corresponded in the textbook series are colored in blue, green, and red, respectively.
Since Ms. Mitchell used CMP2, some of the lesson numbers as they appear in the materials are
different than the lesson numbers in CMP3. Ms. Mitchell’s third lesson in round 2 was a teachergenerated lesson on expected value.
Table 2
Observed Lessons by Teacher and Round
Teacher
Rnd. Unit
Ms. Mitchell
1
Moving Straight Ahead
(CMP2)
Ms. Mitchell
2
What Do You Expect?
(CMP2)
Ms. Mitchell
3
Data Distributions
(CMP2)
Mr. Sandberg
1
Moving Straight Ahead
(CMP3)
Mr. Sandberg
2
What Do You Expect?
(CMP3)
Mr. Sandberg
3
Samples & Population
(CMP3)
Mrs. Stewart
1
What Do You Expect?
(CMP3)
Mrs. Stewart
2
What Do You Expect?
(CMP3)
Mrs. Stewart
3
Samples & Population
(CMP3)

L1
1-2

L2
1-3

L3
1-4

1-3

2-1

TG

1-1

2-1

4-4

1-2

1-3

1-4

1-2

1-3

1-4

1-2

1-3

1-4

2-1

2-2

2-3

4-1

4-2

4-3

1-2

1-3

1-4

L4
1-4

L5
2-1

L6
2-1

2-1

2-1

2-2

L7
2-1

Baseline Data Collection and Analysis
Baseline data collection consisted of an initial interview and an observation of a typical
classroom discussion. In contrast to the observed lessons in Rounds 1, 2, and 3; there was no
post-lesson debrief between the researcher and the teacher. I gathered baseline data November –
December 2014. To analyze the baseline data, I first transcribed the interviews, then looked for

33

evidence of experience in facilitating whole-class discussions, overall teaching experience,
beliefs about student learning and teaching, curricular experience, and how the teachers think
about instructional dilemmas. Second, I used the Instructional Quality Assessment [IQA]
(Boston, 2014) protocol to score the video-recorded lesson.
Initial interview. I conducted an initial interview with all three teachers with the purpose
of gathering information about the teachers’ beliefs about mathematics teaching and learning,
their overall teaching experience, their experience using the curriculum materials, and their
experience in facilitating whole-class discussions. The goal of gathering this data was (a) to
ensure that teachers used whole-class discussions enough to participate in the study, and (b) to
provide some baseline idea of potential factors (e.g., instructional goals, possible attention to
professional obligations, instructional routines, types of dilemmas they encounter) that contribute
to teachers’ decision-making. The set of interview questions are listed here, with greater detail
provided in Appendix C.


Describe your experience with implementing CMP materials.



Describe a typical lesson in your classroom.



Do you think it’s important that students are active in their math class?



What are some challenges that you encounter when trying to facilitate classroom
discussions?



What do you think you have improved on during your teaching career in facilitating
classroom discussions?

34



Are there resources, organizational structures, or routines that you find helpful for
supporting implementing discussions?
I also asked participants about dilemmas they faced while thinking about planning

collaborative discussions. This data provided a general sense of issues that potentially constrain
or impact how decisions were made during collaborative discussions. For example, teachers
talked about district-mandated tests, concerns for students’ social development, classroom
management concerns, and concerns about preparing students for the next year’s material. These
topics provided insight for thinking about how dilemmas are managed during collaborative
discussions.
Curricular experience. The three teachers varied in terms of their familiarity with the
CMP materials. Ms. Mitchell was new to the materials, whereas Mrs. Stewart and Mr. Sandberg
had previously used CMP2 for at least four years. This being said, both Mr. Sandberg and Mrs.
Stewart were piloting CMP3; therefore, there was some unfamiliarity due to some changes in
lessons and sequencing between versions. Because all teachers used the same curriculum
materials, instructional decisions were less dependent (at least theoretically) on curricular effects.
Instructional Quality Assessment. I used the Instructional Quality Assessment
Classroom Observation Tool [IQA] (Boston, 2012) to determine each teacher’s level of expertise
in facilitating discussions. The IQA tool allows for measuring features of a discussion that
include how the task was implemented as well as teacher and student talk during the discussion
(e.g., the level of questions asked by the teacher). The rubrics on the IQA range from 0 to 4, with
higher scores indicating greater expertise. During Round 0, each teacher chose a lesson that they
deemed “typical” and that they thought had the potential for having a classroom discussion. The
lessons were video-recorded, and both coders used the recordings to evaluate the teachers using

35

the IQA rubrics Student Discussion Following the Task and the Rigor of Teachers’ Question, as
these two rubrics were most appropriate for analyzing teacher discussion practice for this
dissertation. The scores were exchanged and any issues resolved between the two researchers.
This analysis aided in classifying the teachers as expert or novice. There was 100% agreement on
the sets of scores for Ms. Mitchell and Mr. Sandberg, and overall scores differed by 1 point each
on the coding of Mrs. Stewart. Profiles of each teacher are given later in this chapter, based on
the analysis of the interview data and the observed lesson.
Lesson Observation Data
Each set of observations consisted of observing teachers in consecutive lessons. Data
collected from each lesson included a lesson plan, observation field notes taken during the
lesson, a video recording of the lesson, and a post-lesson debrief interview. Additionally, the end
of each set of lesson observations included a video-stimulated interview where I showed clips of
lessons and asked each teacher about their decision-making processes. I observed all the lessons
in person with the exception of two. Data collection took place during the spring semester of
2015. Data collection was planned to begin earlier in the school year, but several issues created
delays (Appendix B).
Each set of lessons consisted of consecutive lessons. The term consecutive lessons used in
this study refers to instructional lessons taught by the teacher that occurred in order Days where
the teacher only gave a quiz, test, homework check, or some other planned non-instructional
activity were not observed. When teachers were absent, these lessons were similarly excluded
from the study. One of the teachers (Ms. Mitchell) had a student teacher who would typically
teach in her absence, and again no observations took place on these days. As one example of
how spaced out these observations could be, Mr. Sandberg’s first round of 6 consecutive

36

observations took 18 calendar days due to various teacher absences, assessments, and days when
students were not present (e.g., parent-teacher conferences).
Deviations in collecting lesson observation data. There were very few deviations from
“typical” observations in terms of collecting data. I was present for all observations with the
exception of two in Mr. Sandberg’s class. These lessons were recorded by another graduate
student. During these lessons there were no post-lesson interviews, but I did later analyze the
video-recordings of the lessons, and questions that seemed to relate to the next lesson were also
included in the next lesson’s post-lesson interview. Additionally, I missed one post-lesson
interview by Ms. Mitchell because she had to leave immediately after class. Out of the 34
recorded lessons across the three rounds, I missed 3 post-lesson interviews.
The second round of all three teachers as well as the first round of Mrs. Stewart’s
observations fell within the state testing window. For Mrs. Stewart and Mr. Sandberg this
impacted the number of students attending their class; in some cases there were as few as 11
students in one class during state testing (typical class size was around 20). Ms. Mitchell’s class
was during the last period of the day, so very few students were absent due to state testing. This
likely had some effect on the types of interactions and planning for those class periods. All three
teachers taught the unit What Do You Expect? (a unit on probability), which was purposefully
chosen by the department to occur during the state testing window.
Number of lessons per round. The lesson observations for each teacher consisted of
three rounds of consecutive lessons (see Table 2). Although most textbook sections were taught
within one class period, some were not, which is the reason why there were more than five
observations in a round in some cases. I originally planned to observe two rounds of five lessons
per teacher. After collecting data for Ms. Mitchell and Mr. Sandberg in the first round, the

37

number of rounds was changed to three rounds, each consisting of three lessons per round (Note:
Mrs. Stewart had not yet been observed when the decision changed, so her observations
consisted of three sets of three lessons). The decision to move to three sets of lessons was
motivated by the following:


There was not much observed lesson variation from day to day in terms of lesson
structure, so adding an additional set of observations could control for content-domain
effects by providing greater variety of topics being taught.



The change to an extra set of lessons would allow for more variance in topics taught,
which would control for decision-making related to specific content domains.



All teachers approved adding an additional round, while shortening the number of lessons
in each round.
Identification of lessons. An important choice in this study was selecting lessons with

high potential for classroom discussions, which ideally could be observed for all three teachers in
the study. Through inspecting the CMP2 and CMP3 materials, I determined several focus lessons
that had the potential for rich discussion. Only lessons that included high cognitive demand tasks
(Smith and Stein, 1998) were selected as focus lessons, because these lessons had the greatest
potential for promoting discussions. Once I selected lessons, I verified their potential for
collaborative discussions by talking with experienced users of the curriculum (who were not also
research participants) and one of the authors of the CMP materials. For each focus lesson, I
observed at least one class session prior to and at least one class session afterwards, in order to
better understand the decision-making across lessons. I determined the focus lesson by the lesson
as written in the materials (e.g., Lesson 2-1), not as the particular material covered in a single
class period. In two cases, focus lessons spanned multiple class periods.

38

A similar reason why the same curriculum series was selected across teachers is that I
had initially wanted to observe all teachers teaching the same lessons. Logistically, this became
quite challenging for several reasons. One issue resulted from one teacher using CMP2 materials
while the other two were using CMP3 materials. Finding lessons that satisfied the focus lesson
criteria and appeared in both materials was also challenging. Second, the observations that took
place at the end of the year consisted of different units that focused on data. Third, issues arose
from scheduling issues, differences in pacing, and teacher absences.
Because observing identical lessons was so challenging, an alternative approach was used
that created three sets of partially overlapping observations. Each teacher had one set of
observations that overlapped with the second teacher, one set of observations that overlapped
with the third teacher, and one set of observations that was unique to the teacher. This
overlapping was done intentionally to limit the possibility that differences across teachers in
instructional decision-making was not a result of highly different content being taught. No set of
lesson observations was consistent across the three sets of teachers. The set of overlapping
lessons can be seen in red, blue, and green in Table 2.
Lesson plans. Before each lesson, I collected lesson plans for the purpose of determining
the extent to which unit-level or long-term goals are present in the plan (see Appendix D for
lesson plan criteria). All teachers were asked to submit lesson plans ahead of time. I asked
teachers to make sure their daily plan included what they would be doing in the class that day,
along with the goals for the lesson and some rationale as to why they thought the lesson could
help attain the goal(s). The hope was that by only asking for a few items, I could balance the
needs of getting quality data without overly burdening the participants. I explained the
expectations of what should be included in the lesson plans before data collection commenced.

39

Two of the three teachers were unable to consistently submit lesson plans ahead of their lessons.
To accommodate this issue, I made some changes between the first and second rounds of lesson
observations (see Appendices D and E for the original and revised versions), which included
revising the lesson agenda and overview with more teacher-friendly language, contacting the
teachers by email, and including questions about the next day’s lesson in the debrief of the
preceding lesson.
In terms of analyzing teacher lesson plans, I paid particular attention to how the teacher
described the goals for the discussion, and whether the goal(s) seemed to be immediately
attainable as well as whether the goals connected to longer-term or unit-level aims. I was
interested in identifying the types of goals they mentioned; for example, whether the goals
focused on content (e.g. students will understand), processes (e.g. students will justify their
solutions), dispositions (e.g., students should be curious about…), or anything else, such as
organizational or classroom norm-specific goals (e.g., students will effectively work in groups). I
provide further detail in the data analysis section on how I coded for goal statements.
Lesson observations. With the exception of the two lessons missed, I arrived shortly
before the start of each lesson to audio record and video record. My interaction with students was
limited (such as when students would accidentally walk in front of the camera). In most cases,
monitoring the video camera and taking field notes was non-problematic given the use of
multiple cameras. Both cameras were in the back of the room, one on a fixed wide-angle shot,
and I operated the second camera so that I could zoom in on work on the board and follow the
teacher. I also used a microphone positioned in the back of the room to record audio.
Prior to lesson observations in Round 1, pilot data helped generate aspects of classroom
interaction during whole-class discussions that could have been important to describe in relation

40

to the research question (see Appendices F and G). These categories were determined through
both identifying aspects that seemed relevant to the interaction during whole-class discussions
via analyzing pilot data and by attending to the relevant literature base on whole-class
discussions (e.g., teacher questioning and student talk). Although this list was not exhaustive
(e.g., it did not include anything related to norms that could have affected the nature of class
discussions, dilemmas, and decisions), it did prove sufficient in being able to identify several
potential consistent patterns of instruction and instructional dilemmas.
During each lesson I wrote field notes for the purposes of noting segments with different
types of interaction (e.g., group work, whole-class discussion, teacher lecture), and situations to
be discussed in the post-lesson discussion. I paid particular attention to those situations that (1)
appeared to deviate from consistent patterns within whole-class discussions across lessons, or (2)
posed instructional dilemmas for the teachers. Identifying these deviations proved somewhat
challenging, as the consistent patterns were necessarily emerging during data collection. The
categories from the pilot data, and the framework for analyzing consistent patterns of instruction,
provided a frame for identifying potential consistent patterns and possible deviations. A list of
situations that potentially showed instructional moments where active conscious decisionmaking occurred is listed in Appendix H.
In the first round of data collection, I took field notes using a classroom observation
protocol (Appendix I). This document proved useful in capturing particular moments, but proved
cumbersome in trying to follow along with the lesson and categorizing decisions in relation to
any of the professional obligations. This issue, combined with the lack of receiving lesson plans
consistently, prompted a change in the structure of collecting field notes. Appendix J shows the
single page observation tool that I used for the final two rounds of data collection. I noted

41

potentially interesting dilemmas, breaks in consistent patterns of instruction, and issues related to
any of the professional obligations (along with a video time code) on the right side of the
document. Additionally, the reflection questions were listed on the top of the document so that
the reflection questions could be more specifically tailored to the content of that day’s
discussions.
While piloting the protocols, I noticed that the Classroom Observation Protocol was
necessary but by itself insufficient to determine the actual decisions made by the teacher. This is
because without the teacher being able to describe their thinking in that moment, there is a large
amount of inference on my part as the researcher as to why they made the decision. Thus, the
Classroom Observation Tool was used to mark moments that were later analyzed on the video
and the video-stimulated interview at the conclusion of each round of lessons.
Post-lesson interviews. At the conclusion of most lessons, the teacher and I met to
debrief the lesson. Whereas all of Mrs. Stewart’s lessons included a post-lesson interview, there
was one post-lesson interview that did not occur for Ms. Mitchell (Round 1, Observation 3), and
two that did not occur for Mr. Sandberg (Round 1, Observations 4 and 5). The purpose of the
post-lesson debriefs was to try to understand their decisions and the dilemmas they encountered
during whole-class discussions while their recollection of the lesson was still fresh.
With this in mind, the field notes and reflection questions were starting points for a
conversation aimed at uncovering the teachers’ decision-making in the lesson and the possible
dilemmas they felt they encountered. The debrief was structured on reflection questions
(Appendices K and L) and field notes from the lesson. I video-recorded the debriefs, but the
teacher and I did not watch any of the video-recorded data from that day’s class period during
the post-lesson debrief. Typically, the debriefs lasted between 5-15 minutes.

42

Looking back on the discussion immediately after the lesson created both benefits and
limitations. The main benefit was that the teachers had a fresh recollection of the decisions made
during the lesson. This hopefully minimized the likelihood of a teacher trying to re-create the
choices they made that led them to a decision were I to have interviewed them days later. The
main limitation, however, was potentially missing an important interaction between the teacher
and the students that was not observed during the class observation. To minimize this limitation,
the video-stimulated interview took place after each round of observations. Lesson videos were
watched multiple times for potential dilemmas and moments potentially influencing teacher
decision-making in the discussions.
Video stimulated interviews. After each round of observations for each teacher, I
conducted a video-stimulated interview. I watched all the video recordings of all the discussions
for each particular round of observations. This included the recordings of the lessons and postlesson debriefs, along with looking back at field notes. The video-stimulated interviews typically
lasted between 20 and 35 minutes. The goal of the video-stimulated interview was similar to the
post-lesson debrief: to better understand teacher’s decision-making during collaborative
discussions. Similar to the post-lesson debrief, I selected video clips based on their potential to
uncover an instructional dilemma or to better understand instructional decisions, especially those
related to deviations from consistent patterns of instruction. I asked teachers about their goals,
the dilemmas they were managing, and their decisions to better understand how their
professional obligations oriented their decisions (see Appendix M for the set of example
questions).
There were other benefits to the video-stimulated interview as well. From an analytic
perspective, watching the videos multiple times allowed me to generate hypotheses of

43

instructional practice, and therefore possible reasons why a teacher made a particular decision.
The video-stimulated interview allowed for checking my own hypothesis with the teacher’s
sense of why they made a particular decision. From a methodological perspective, the videostimulated interview provided teachers with a different perspective on their teaching by watching
an extended segment (anywhere from 30 seconds to 5 minutes).
Data Analysis
A main challenge with studying teacher decision-making during whole-class discussions
is looking for moments when there is observable evidence of a decision. This is challenging
because making decisions is inherently a cognitive process. An outside observer would at best be
able to study the residue of such decisions and infer that the teacher made a decision (Sutcliffe &
Whitfield, 1976). In this study, I take the more restrictive approach to identify those situations
where there is strong likelihood that a teacher made a decision. For example, a teacher explicitly
mentions they made a particular decision during an interview or in their lesson plan. Or, when a
teacher breaks a highly consistent pattern of instruction, one can infer that the teacher made a
decision to break from such a consistent instructional pattern (i.e., the assumption is that such
deviations are less likely to be random occurrences as opposed to purposeful action). An
important aspect of this analysis is describing what was considered typical instruction so that
instruction that deviated from the norm could be identified. In the section that follows this one, I
provide a background of each teacher’s typical instruction in order to provide an analytic tool for
identifying dilemmas and decisions. To determine the latter of these situations, I analyzed the
data to determine consistent patterns of teachers’ instructional practice related to whole-class
discussions, then marked places where deviations from those patterns occurred. In this section, I

44

describe how I analyzed goals, professional obligations, consistent patterns of instruction,
dilemmas, and decisions.
Goals analysis. I identified goals that related to whole-class discussions across all
sources of data: lesson plans, post-lesson interviews, video-stimulated interviews, baseline
interviews, and lesson observations. Most goal statements occurred during interviews, where
teachers were asked directly about their goals -- questions such as, “[W]ere there like a few big
things you really wanted to come out of [this lesson]?” or “Were there other goals, or other
things you wanted…?” The unit of analysis was at the sentence level where teachers talked about
a single goal.
During the first pass through the data, I identified all potential goal statements. There
were two types of goal statements analyzed. Some goals statements were stated explicitly by the
teacher, where they clearly and explicitly identified a goal as such, using the word “goal.” For
example, during Ms. Mitchell’s post-lesson interview (May 1, 2015) she stated, “[M]y goal as I
told you was for them to get a better sense of what a sample space is.” This is an illustrative
example of when the teacher explicitly identified her instructional goal.
Other statements were interpreted as goal statements during analysis because they
described a teaching goal even though the teacher did not explicitly use the word “goal.” For
example, Mr. Sandberg stated, “[T]hey need to be able to understand y-intercept, what that looks
like on a table, graph, and equation” (post-lesson interview, February 25, 2015). I interpreted this
statement as an instructional goal because he identified a topic that he wanted to have his
students understand, and it acted functionally as a goal. Thus, some goals were stated explicitly,
while others were interpreted as goals.

45

After identifying goal statements, I coded goals as to whether the goal was primarily
focused on one of four categories. This first category, lesson specific content goals, focused on
mathematical content to be learned in a particular lesson. The second category, broad content
goals, included unit-level content goals that were broader in nature, such as understanding linear
relationships, a topic taught over several lessons. This category also included long-term content
goals that went beyond the unit, such as learning goals that connected to the high school
curriculum. The third category, mathematical practice goals, are related to NCTM’s (2000)
Process Standards and the Common Core’s Standards of Mathematical Practice (NGA-CCSSO,
2010). This goal type is related to the first two goal types in that it focuses on mathematics, but it
differs in that it focuses not on content but on types of mathematical practices, such as
justification and critiquing the reasoning of others (pp. 6-7). This third category included goals
related to how students learn mathematics within a learning community, dispositions such as
perseverance, and mathematical practices such as justification. The fourth category, goals for
establishing a mathematical community, included statements about developing a classroom that
would support shared exploration and expectations of having students learn in a social
environment. As a final note about goals, these goals primarily focused on students from the
perspective of their mathematical development in the classroom attending to practice, content
(immediate and long-term), and social goals. Goals that were not primarily focused on
mathematical development (e.g., developing into a productive citizen in a democracy) were not
coded for, although these goals may have influenced the nature of students opportunities to learn.
Consistent patterns of instruction analysis. To determine each teacher’s consistent
patterns of instruction, I watched a minimum of five classroom videos in random order. The
order was randomized in order to minimize the effects from watching successive lessons. I

46

determined the maximum number of observations studied for each consistent pattern of behavior
once theoretical saturation (Lewis-Beck, Bryman, and Liao, 2003) of the data occurred. For
each teacher, at least half of their lessons were analyzed. I analyzed the video recording of each
lesson twice. The first pass was focused on identifying interaction codes of teacher talk. This set
of codes was at much greater detail and allowed both coders to attend to this fine-grained
analysis. The second pass included coding for other macro-level consistent patterns of instruction
such as activity structure, management, instructional support resources, and meta-routines, which
was at a much larger grain size. After I completed coding, a second coder coded a minimum 20%
sampling of the discussion sections in each set of teachers’ lessons of the data for the purposes of
inter-rater reliability (85.5% agreement). Once theoretical saturation occurred, I described the
consistent patterns of instruction and deviations and watched the remaining lessons to determine
if there were any major inconsistencies with the reported consistent patterns of instruction. In the
event that inconsistencies had been found, more lessons would have been analyzed until the set
of lesson observations was exhausted or a new consistent pattern of behavior became more
strongly established.
Professional obligations analysis. To understand how a teacher’s orientation to their
professional obligations of mathematics teaching influences their decision-making, I analyzed
teachers’ interview data to determine which of the obligations they tended to prioritize across
lessons. In some cases, the teachers talked about decisions with evidence of prioritizing their
obligations. In other cases, I asked the teachers a question during the interview that posed a
scenario where they may have chosen more than one obligation to satisfy. Based on the interview
data, trends emerged showing tendencies to favor a particular orientation to their professional

47

obligations for each teacher. Table 3 below provides greater detail as to how I coded this
interview data in relation to the teachers’ orientations to their professional obligations.
Table 3
Professional Obligations Coding
Professional
Obligation Type
Disciplinary

Rationale for Decision

Example

Appeal to
mathematics

(When asked if having many students absent for
testing factored into the discussion in any way):
“No, I think I would have probably done it the
same…My focus with that was I wanted them to
realize that this still has to equal up to one…No,
[the discussion] would have looked the same with
the entire class” (Mrs. Stewart, post-lesson
interview, May 11, 2015).

Individual

Appeal to attend to a
particular student’s
needs

“…I was going to just kind of let that go, but then
Larissa got really invested in that, and I was like,
‘Well how would you do that?.’ And then a
couple other people figured that out, okay, we'll
go with this thing too. So I didn't want to shut that
down, I wanted to honor their thinking more so I
was okay letting [that] go…” (Mrs. Stewart, postlesson interview, May 14, 2015).

Interpersonal

Appeal to social goals
/ reasons

“…Something I need to think about, is giving the
think time with the pairs and the communication
time to help them buy in to a simpler social
setting and then move it out” (Ms. Mitchell, postlesson interview, March 2, 2015).

Institutional

Appeal to school
expectations

(When asked about using popsicle sticks as a way
to randomly call on students): “So there's times
when I know everybody should be paying
attention, and every person if they're paying
attention can answer these types of questions
when I use those” (Mr. Sandberg, videostimulated interview, April 17, 2015).

Dilemmas and decisions analysis. This study focused on those dilemmas that pertained
to whole-class discussions. Dilemmas occurred when instructional goals came in conflict with
one another. As mentioned previously, I identified dilemmas across the five data sources: lesson

48

plans, baseline interviews, post-lesson interviews, video-stimulated interviews, and lesson
observations. Some dilemmas occurred during planning (as evidenced by referencing dilemmas
during interviews and/or lesson planning); some appeared during enactment (as potentially
evidenced by a deviation in a consistent pattern of instruction); and others occurred after a lesson
as teachers were looking back on their instruction (as mentioned by the teacher). Some dilemmas
occurred independent of observation (i.e., there was no notable deviation to some pattern of
behavior) because teachers described the dilemmas in either lesson plans or interviews; whereas
other dilemmas were hypothesized during lesson observations (e.g., because a deviation to a
consistent pattern of instruction occurred), and I inquired about these dilemmas during
interviews.
As a first pass through the data, I looked for potential dilemmas by instances where
teachers explicitly mentioned a potential dilemma by either stating a moment in a discussion as a
“dilemma”, “challenge”, or “issue”. I also looked for implicit situations that were instances
where the teacher was describing a potential conflict between goals. In many cases, the teachers
described their decision (and its rationale) and potential other decisions they could have made. It
was not necessary for the teacher to come to resolution, however, as some dilemmas remain
unresolved. It was important however, for the teacher to acknowledge the potential for making a
different decision than the one they made.
After this first pass through the data, I completed a second pass through the potential
dilemmas to identify those situations that involved a conflict between two goals types. I then
used the goals analysis (looking for implicit and explicit references to what the teacher aimed t
accomplish) to determine the specific types of goals that were in conflict. I only noted those

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potential dilemmas as “dilemmas” when they met the requirement that the dilemma was indeed a
conflict between two of the four different goal types.
There are two important limitations to the dilemmas analysis. First, since a dilemma is
defined as a conflict between goal types, the dilemmas were a function of the types of goals
analyzed in this study, (i.e. those focusing primarily on students’ mathematical development).
Second, because one might be able to determine dilemmas between two goals of the same type
(e.g., a long-term mathematical content goal conflicting with some dispositional goal – both
categorized as a broad content goal). While the first case was not analyzed (i.e., analyzing the
data for goals outside the bounds of this dissertation’s analysis), there were very few instances of
the second case which warranted only choosing dilemmas as a conflict between the two goal
types.
Background of Teachers’ Instructional Practice while Leading Discussions
In this section, I provide an overview of each teacher’s instruction, including a
description of consistent patterns of instruction when facilitating a discussion; and an analysis of
a baseline class discussion, their instructional goals, and their understanding of the professional
obligations of mathematics teaching. This description and analysis is intended to frame the
dilemmas and decisions analyzed in the next chapter. In the succeeding chapter, I describe the
types of dilemmas that teachers encountered and the decisions they made as a result of these
particular sets of goals, obligations, and patterns of instruction.
All three teachers included broad instructional goals in either their lesson plans or the
interview data. They all used the four different goal types (long-term content, unit-level content,
mathematical process/practice, being a member of a mathematical community). Additionally, all

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teachers incorporated consistent patterns of instruction, although they were quite varied from
teacher to teacher.
Mrs. Stewart’s Instructional Practice
Mrs. Stewart taught using the CMP2 materials for about four years and attended multiple
professional development workshops for this curriculum series. She previously taught high
school for about five years before teaching middle school. She had a strong sense of the
curriculum materials across grade levels, having talked and worked with other math teachers
beyond seventh grade. She identified two of her biggest challenges in teaching as (1) helping
students connect their mathematical ideas during the discussion to their peers’ thinking and (2)
engaging all students in the discussion beyond the “five or six really confident kids” who always
want to talk.
Mrs. Stewart typically used the Launch-Explore-Summarize structure for organizing
lessons, a feature of the CMP materials. In this format, the teacher sets up the task for the
students (the “Launch”); students then work in small groups (the “Explore”); and finally the
teacher summarizes the main points through a class discussion at the end of the lesson. During
the exploration time, Mrs. Stewart observed students and asked questions to later use in the
whole-class discussion. Mrs. Stewart anticipated what strategies she was going to use before the
lesson so that she could plan for how she was going to order solution methods during the
discussion (e.g., from more concrete to more abstract). She viewed the Summary portion of the
lesson as important in helping students see connections across different strategies.
Baseline observation data. The observed lesson for Mrs. Stewart was Problem 3.3 from
Comparing and Scaling (CMP3) where students explored different ratios of fiber and protein in
chimp food. The students were given two recipes for chimp food: one for babies (40% high-fiber

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and 60% high-protein) and one for adults (60% high-fiber and 40% high-protein). Students were
asked to determine the number of scoops for each mix given three different batches: (a) 100 total
scoops, (b) 1 scoop of high-protein, and (c) 1 scoop of high-fiber. The students’ task was to
determine the missing values in the ratio table (given in Figure 2) and to explain how they
determined those values.

Figure 2. Students determining the unit rate for baby chimp food
Throughout the discussion Mrs. Stewart asked students to share multiple strategies, and at
least three different strategies were given during the class. The connections across the strategies
were primarily teacher-directed, with few opportunities for students to connect strategies during
the discussion. She seemed to have a clear sense of which students she wanted to call on during
the discussion, given that there was no hesitation in deciding who to invite to participate. Each
strategy that was shared included something that had not yet been discussed in relation to unit
rates. This said, after each strategy was shared Ms. Stewart took up the task of justifying and
connecting, and there were few to no instances of probing for each strategy. Mrs. Stewart scored

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a 2 (out of 4) on the Student Discussion Following the Task rubric, and a 2 (out of 4) on the
Rigor of Teachers’ Question rubric.
A typical class discussion. Mrs. Stewart frequently made use of the Launch-ExploreSummarize (LES) format in her classes. In all lessons analyzed, Mrs. Stewart conducted a
summary discussion regardless of how far the class got during that class period. During the
initial interview, Mrs. Stewart mentioned that she tries to stick to the LES format as much as
possible. In her view it is important to give students time to explore the problem and ensure that
a summary occurs. In her words, “I don’t want to just take the rest of the class and work through
all of this and I’ll summarize tomorrow because their brain doesn’t hold on to that.” The
summary discussion, in her view, is where the teacher should be helping students make
connections to previous content, standards, and strategies across problems.
Desks in Mrs. Stewart’s classroom were arranged in groups of 4. In many cases
(especially during state testing) there were fewer than 4 students in each grouping. There were no
observed changes in the seating assignments of the students across all the lessons observed.
Thus, there was no time wasted having students get into groups, since they were in groups the
entire class period. The seating arrangement allowed for a more fluid lesson structure moving
between group work and whole-class discussion throughout the lesson. As compared to the other
teachers in the study, there were very few instances of “teacher lecture” where Mrs. Stewart was
at the board simply talking or explaining an idea to students for a long period of time. Instead,
students were either discussing or working on problems collaboratively in small groups, or
talking/working in front of the whole class.

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Discussions frequently shifted between a whole-class venue and small groups. When
asked about one particular instance of an abrupt move from whole-class to small groups during
the final video-stimulated interview, her response was as follows:
So, for a question like that that I had posed that, I felt like it could have a variety of ways
that kids could think about it or lots of different ways to look at, and lots of different
information to look at. So I wanted to give them some time to discuss it with people in
their group. And then the reason I did it at that point was because I had heard like maybe
there had only been three or four voices sharing. And so often when I do that I can get
more voices sharing, or I'll walk around I'll eavesdrop on groups and purposely call
different kids out to either hold 'em accountable or because I want to make sure they have
actually been listening to their group and can summarize in their own words. And so
that's why I did it at that point. 'Cuz I thought it could have so much discussion behind it
that I didn't want them to just have to either think through it on their own or to think
through it quickly. So I wanted to give them the time to do that with each other. (Mrs.
Stewart, video-stimulated interview, June 2, 2015)
Within each lesson there were between 3 and 5 discussion segments, with discussions
ranging vastly from as short as 1:50 (one minute, 50 seconds) to 22:12. The average discussion
segment was approximately 8 minutes long with the average duration per class period being 30
minutes (half a class period). Some breaks in between segments were lengthy, such as during the
Explore part of the lessons; others were quite short. The shortest duration between discussion
segments was 23 seconds, when students were asked to discuss whether reversing the order of
letters in a permutation was allowable (see Figure 3). Discussions typically started within the
first few minutes of class. There was one exception to this when the discussion did not start until

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about 15 minutes into the lesson. Typically, the beginning of each discussion involved individual
students answering a question posed by the teacher.

Figure 3. Reversed order of permutations in rows 2 and 3
Goals. Mrs. Stewart had all four types of instructional goals. She mentioned broad
content goals such as supporting students in making sense of statistical summaries in comparing
data sets. She also emphasized being a member of the mathematical community. She felt her role
in a discussion was to maximize student-to-student interaction and not to be the focus of the
discussion. She noted this during an interview: “[M]y goal every class is to have the kids just
interact with each other the way they did and almost completely forget the fact that I’m up there,
maybe just as a scribe.” She placed strong emphasis on mathematical practice goals, since a
major theme in Mrs. Stewart’s classroom practice was the focus on justification. She frequently
pressed students to explain their thinking and justify why a claim was true. She mentioned this
repeatedly throughout her post-lesson interviews about the need to cite evidence when making a

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claim. This also played out in one of the lessons from the unit Samples and Populations as
students were investigating survey data about roller coasters (see Figure 4). After a discussion
about the data, students were given the task shown in Figure 5.

Figure 4. Survey data from Internet responses and a grade 7 class

Figure 5. Instructions given to students to justify their thinking

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In addition to mathematical practice goals, she also placed a strong emphasis on broad
content goals, likely influenced by her experience as a high school teacher. She made multiple
references to what students would need to be able to know when they were in high school.
During a lesson on probability, students were asked to determine the probability of selecting
different colored blocks from a bucket with 1 green block, 1 red block, and 2 yellow blocks. Mrs.
Stewart noted in the post-lesson interview that many students used three as the denominator,
“because of three colors.” This stood out to her because as she put it, “And I remember as we get
later in the unit, even when I taught high school, understanding what that outcome was, to put on
a tree chart or a table, they just didn’t have an idea.” Mrs. Stewart noticed this issue of the
denominator and the potentially long-lasting issues related to not understanding the idea of an
outcome.
Consistent patterns of instruction. Three recurrent patterns of instructional practice
occurred consistently across lessons. For Mrs. Stewart these three consistent patterns of
instruction included maintaining a rapid pace of instruction (management), emphasizing
communication and process goals (interaction), and implementing fluid interaction structures
(activity structure). I provide some detail of these three patterns in the next paragraph, and
provide greater detail on Mrs. Stewart’s consistent patterns in Appendix N.
Mrs. Stewart’s rapid pace of instruction was evidenced by the high volume of questions
she asked students. During a typical 30-minute segment of discussion Mrs. Stewart asked about
83 questions, or nearly 3 per minute. On average, Mrs. Stewart made a statement, asked a
question, or responded to students once every 10 seconds. This differed greatly from the other
teachers in terms of the rapid exchange between her and her students. The second consistent
pattern was emphasizing communication and process goals. A main feature of this orientation

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involved students making claims and providing evidence for their claims, both written and
verbally. She often pressed students to justify or clarify their thinking when talking during
whole-class discussions. The third consistent pattern, implementing fluid interaction structures,
describes the back-and-forth nature of moving between small groups and the whole-class
discussion space. Unlike the other teachers in the study, Mrs. Stewart often moved the discussion
between small groups and the whole class several times during a lesson: when students seemed
to struggle with an idea or when she felt the need to support their inquiry.
Professional obligations. The three consistent patterns of instruction in the previous
section all point to a predominant orientation of prioritizing mathematical content as the ultimate
deciding factor in making decisions for whole-class discussions (i.e., the disciplinary obligation).
The rapid pace of interaction provided minimal management issues and therefore maximized the
opportunities for students to engage in mathematical content. During the Caves problem, where
the structure of interaction changed dramatically, Mrs. Stewart continued on managing the class
as best she could. Students were highly engaged, and Mrs. Stewart changed some of her
interaction patterns with the class to support their inquiry and discussion with their peers.
The second and third consistent patterns of instruction supported student understanding
of mathematical content by providing opportunities for students to clearly describe their
mathematical thinking in different interaction structures. On the rare occasion when the nature of
a problem made it difficult for students to justify their thinking, Mrs. Stewart modeled this
thinking for the class, maintaining the opportunity for students to deeply understand
mathematical content. Mrs. Stewart also demonstrated flexibility in how she supported student
inquiry based on the nature of problems and what students were saying in class.

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In addition to the consistent patterns of instruction, interview data also supports this
prioritizing of the disciplinary obligation. For example, when asked during the initial interview,
“[A]re there ways of…organizing the class, or consistent patterns of instruction, or resources that
you use so that you find helpful for implementing discussions?” (a question that could be
answered attending to any of the professional obligations), Mrs. Stewart talked about how she
tries to structure discussions by moving from concrete strategies to more abstract ones. This
careful selection of student work based primarily on content ideas played out during the lessons
as well. During a lesson where students made a listing of the sample space of flipping three
coins, she was asked why she selected the two students to share their work on the document
camera. The first student was chosen because she had an incomplete list that was un-ordered, and
the second student had all eight permutations in an order. Both characteristics (completeness,
ordering) were factors in her choices of sharing the work, showing strong attention to the
disciplinary obligation.
Ms. Mitchell’s Instructional Practice
Ms. Mitchell has taught over twenty years and was somewhat familiar with lessons in the
CMP curriculum, although this was her first year teaching the curriculum and teaching middle
school students (she previously taught elementary and high school). In her words, a typical
lesson involved “an interesting hook” to pique students’ engagement, followed by time for the
students to explore a task. After the students worked on the task, the class engaged in a wholeclass discussion.
Unlike the other two teachers in the study, Ms. Mitchell had a highly varied lesson
structure. In contrast to the other two teachers, Ms. Mitchell’s lesson structure seemed to be a
dependent variable, depending on other factors to determine it instead of the other way around.

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There were no consistent features across lessons in terms of inclusion or order of particular
lesson segments. In one observation (Round 1, Observation 6), the class started with a
discussion, moved to group work, then individual work, back to group work, and back to
individual work. In another lesson (Round 1 Observation 4), the class started with group work
for 33 minutes and was followed by a discussion of 23 minutes. Other observations included
segments of teacher lecture about both mathematical content and non-mathematical content (i.e.,
a 13-minute segment talking to students about behavior). Ms. Mitchell was well aware of this
lack of consistent discussion structure. When asked about this in the initial interview she
explained:
I wouldn’t say there’s too many typical lessons, but what I would hope for, is that there’s
a meaningful, interesting hook with a launch, where students have either their own or
some kind of inquiry that’s leading them to want to explore different mathematical
phenomena either as a group, individually or with a pair. So that they’re thinking about
things from a variety of perspectives and then bringing those together at the end of the
lesson through a class discussion or student sharing either structured or unstructured with
that. (Ms. Mitchell, initial interview, November 25, 2014)
Baseline observation data. In this lesson, the class was working on a modified version
of Problem 3.1 in Comparing and Scaling (CMP2), which focused primarily on percent increase
or decrease in the context of merchandise sales. After working on the task, Ms. Mitchell invited
multiple students to go up to the board and describe their thinking to the class. Ms. Mitchell
consistently asked questions that probed students’ thinking in order to have students provide
more details of why their strategy worked. Based on the observed lesson, Ms. Mitchell scored a 3

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(out of 4) on the Student Discussion Following the Task rubric, and a 4 (out of 4) on the Rigor of
Teachers’ Questions rubric.
A typical class discussion. Ms. Mitchell’s class discussions were highly dependent on
student engagement. Her lesson structure depended on the needs of the lesson. The number of
discussion segments per class ranged from 0 to 3. About half the lessons included summary
discussions, and about half the lessons included discussions in the middle of the lesson. The
average duration of a single discussion segment was 10:37 (10 minutes, 37 seconds), and the
average amount of class time devoted to whole-class discussion was 18:12. The only consistent
feature of Ms. Mitchell’s discussions was a description of the content and/or practice standards
for each particular class period (Figure 6). These standards were stated explicitly at the
beginning of class and sometimes referred to again during the discussion at some point.

Figure 6. Statement of content and practice goals at the beginning of class
Goals. The analysis of Ms. Mitchell’s instruction related to collaborative discussions
included all four goal types. In one of Ms. Mitchell’s lessons, a student was describing the y-

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intercept as the “starting point” on the graph, and during the post-lesson interview she described
how uncomfortable that made her feel. “When I first saw that [the y-intercept] I was like, oh, I
don’t like that one bit, ‘cuz in high school I had to fight that for three years. It’s not a starting
point.” Ms. Mitchell allowed the student to talk about the y-intercept as a starting point as they
were just beginning to understand this important point across representations. This exchange
highlighted Ms. Mitchell’s attention to long-term content goals (broad-content), specifically
related to her experience as a high school teacher.
Ms. Mitchell frequently mentioned Common Core standards (NGA-CCSSO, 2010) at the
beginning of class – both content and practice – and her use of CCSSM practice standards was a
theme throughout her instruction both in describing her decisions and in making these standards
explicit to students during class time. In one interview she mentioned wanting students to engage
in repeated reasoning and thinking about structure. Other practices she mentioned included
having students provide evidence to support a claim and persevering when trying to solve a
problem. These statements reflect mathematical practice goals.
Ms. Mitchell also talked about the importance of protecting students’ think time, and
what students can learn from each other in groups (establishing a mathematical community). She
wanted her students to be able to interact with each other directly in the discussion and for her
role to be minimized in the discussion. In Ms. Mitchell’s classroom this involved a single lesson
where she explicitly changed the norms of interaction by having students talk directly to each
other, taking a less visible role in the classroom dialogue.
Consistent patterns of instruction. Two consistent patterns of instruction stood out in
Ms. Mitchell’s facilitation of class discussions: (a) emphasizing students’ thinking (interaction
pattern) and (b) using teacher mediated discussion structure to facilitate whole-class discussions

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(interaction pattern). This first pattern was highly consistent in that Ms. Mitchell frequently
mentioned wanting to make sure that students’ thinking was valorized during whole-class
discussions. This played out in several ways and, as will be discussed later in results, created
several dilemmas for Ms. Mitchell as she enacted content and practice goals. The second pattern
focused on the talk structure that Ms. Mitchell typically used, which was for students to talk and
then she responded. All student talk was essentially mediated through her. These patterns are
described in further detail in Appendix O.
Professional obligations. Ms. Mitchell’s instructional practice related to collaborative
discussions was predominated by her obligations to her students, both as individuals and
interpersonally. Throughout the interviews she repeatedly mentioned a multitude of specific
students’ needs from understanding content to more personal needs. When asked about why she
called on particular students during a discussion, she responded that it was due to her wanting to
get different voices heard in the discussion. When asked about consistent patterns of instruction
that she uses, a recurring theme in her responses was the use of group work to help students learn
from each other. Even when management issues made it challenging to continue on with group
work, she found value in continuing with it. During one of the post-lesson interviews, Ms.
Mitchell described her commitment to using group work despite the management issues that
arose:
Nicholas J. Gilbertson (NJG): “Yeah, so it's interesting you mentioned that this class is
really productive, that causes some issues about like some teachers would say, like, well
you can't have group work now because you lost the privilege of doing that…”
Ms. Mitchell: “No, I wouldn't.”

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NJG: “...[B]ut there's that sense that you're not losing anything, you're just gaining better
management. And you're not losing anything else, so, it seems to be something you're
paying attention to.”
Ms. Mitchell: “[W]hat worked out so well last time was they worked in pairs first and so
I'm thinking that, that's something I need to think about, is giving the think time with the
pairs and the communication time to help them buy in to a simpler social setting and then
move it out, so. Or to be thoughtful about that, so part of that's how I set it up.”
Mr. Sandberg’s Instructional Practice
Mr. Sandberg has taught math for seven of his nine years teaching, all of which included
the use of CMP2. He has attended multiple professional development workshops on using the
curriculum. Mr. Sandberg noted challenges in getting students to talk to each other, including
explaining their reasoning and connecting to prior knowledge. He attributes this mostly to
student “buy-in” of the class atmosphere, which he described as being “comfortable” and teamfocused. When asked about what it meant for students to be “active” in his classroom, he
responded:
“They’re [students] doing the thinking, connecting to prior knowledge, they’re pulling
those things out and really fending for themselves and wanting, wanting to solve the
problems and not just regurgitating a formula or, or skill-and-drill not just doing the same
thing over and over, but, but really just sticking with it and working through a hard
problem, maybe asking a question of ‘I tried this and I’m not quite sure here how to get
from point A to point B’ as opposed to ‘I don’t get this. I don’t know how to do this, I’m
not doing it because I don’t know how.’”

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According to Mr. Sandberg, during a typical class students start with bell work, which is
usually a review of the previous day’s work. This time is meant for review as well as time for
Mr. Sandberg to take attendance and collect homework. He typically helps students through the
first few problems of a lesson as a whole class before letting each student work with one partner
that he assigns ahead of time. They work on the task for about twenty minutes before a wholeclass summary and discussion takes place. He makes use of the Smart Board and document
camera to help display student thinking during the discussion. More so than the other teachers,
Mr. Sandberg had a highly consistent lesson format, listed below.
1. Bell work: Students work individually on a warm-up problem.
2. Discussion of bell work
3. A description of the task students will be working on for the day out of the textbook.
4. Work time – typically with an assigned partner
5. Reporting work from the day’s mathematical task, with some discussion, or homework
time or housekeeping
Baseline observation data. The lesson observed for his initial observation was Problem
1.1 of Accentuate the Negative (CMP3) where students explored adding positive and negative
numbers based on correct and incorrect answers in a game show. The observed structure of this
lesson was consistent with his description of a typical lesson. Students worked on introductory
problems before going into partner-based group work. When the partner-work concluded, the
whole class talked about what they had learned from the problem. Many of the opportunities for
students to talk in the discussion were based on descriptions of their work. There were no

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instances of students making connections across strategies, nor were there explanations by the
students of why their strategies worked. Although Mr. Sandberg did ask students to explain their
thinking, he often took over the students’ explanations, thereby limiting the opportunities to
probe deeper into students’ thinking. Based on the observed lesson, Mr. Sandberg scored a 2 (out
of 4) on the Student Discussion Following the Task rubric, and a 2 (out of 4) on the Rigor of
Teachers’ Question rubric, which was similar to Mrs. Stewart’s scores and below those of Ms.
Mitchell’s.
A typical class discussion. Within each of Mr. Sandberg’s lessons, there were 1 or 2
segments coded as whole-class discussion. One of these segments usually occurred within the
first five to ten minutes of the class. Discussion segments averaged about ten minutes, but ranged
from as short as 2 minutes to as long as 21 minutes. The average amount of time devoted to
discussion was approximately 16 minutes, and ranged from about 4 minutes to 24 minutes.
All lessons began with bell work, which served many purposes. He presented a problem
to the class, and students worked individually on it for five to ten minutes. From a management
perspective, Mr. Sandberg took care of various housekeeping tasks such as checking homework
and attendance. There were also important content reasons for the bell work. The types of
problems that Mr. Sandberg chose for the bell work usually connected the previous day’s work
with the next lesson. The problems were usually determined by common student questions from
the previous day’s work. Mr. Sandberg described having carefully chosen these problems, in
many cases, so that a common mistake or misconception could be addressed collectively (broadcontent goal). As one example (Figure 7), Mr. Sandberg asked the class to consider a
relationship and determine if it was linear or not. The table presented in Figure 7 shows an

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inconsistent change in x. When asked during the post-lesson interview as to why he included this
example in the discussion he stated,
That's a big misconception I get later in the book…They always see them in the
beginning go zero, one, two, three and then it starts to jump around later. I've seen in
years past where they're completely thrown…So I wanted them to think about that it
could still be linear…It's not just…see if it goes one, two, three or two, four, six, then yes
it's linear. But I wanted them to think a little deeper than that. Seemed to work. (Mr.
Sandberg, post-lesson interview, March 3, 2015)

Figure 7. Mr. Sandberg asks if this table represents a linear relationship.
The bell work served as a way to connect the previous day’s content with the new
material. Discussion of the bell work occurred in all lessons but one, where Mr. Sandberg used
the bell work as a way to introduce box-and-whisker plots, which was mostly new for students.
Discussions of the bell work varied, ranging from as little as two minutes to as long as twenty
minutes. In many respects, the bell work discussions replaced summary discussions from the

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previous day. This work focused on important ideas from the previous day’s work and attempted
to explicitly address student mistakes that were noticed across classes. During the final videostimulated interview, Mr. Sandberg mentioned that he felt that he had student’s attention more at
the beginning of the class (compared to the end) and this was his reason for structuring a
majority of the discussion time at the beginning rather than at the end during a summary.
After the discussion of the bell work, there was an often abrupt transition to starting the
day’s mathematical task. Mr. Sandberg would ask students to get out their notebooks and label
their paper with the appropriate problem number from the CMP materials. This segment of the
lesson could be categorized as “teacher lecture” as it typically involved Mr. Sandberg presenting
new content to students without much dialogue. There was typically some student interaction,
but it was very minimal both in the frequency of students talking and how they responded. These
were generally not coded as whole-class discussion for this reason. In a typical CMP Problem,
there are several parts (A, B, C, etc.). Mr. Sandberg usually worked on part A collectively with
the whole class. When asked about this consistent feature of his lesson structure, he stated,
“Yeah, I don’t have anything set. Seems like this year with CMP3 I’m doing A almost all of the
time…I remember last year…they would just get into partners right away at A.” As a reminder,
Mr. Sandberg had been teaching using the CMP materials for several years, but he was currently
field-testing the third edition materials, CMP3, and was using these particular curriculum
materials for the first time.
After Mr. Sandberg described the mathematical task, students worked with a partner on
the assigned problem. These partners were assigned by Mr. Sandberg, either through a list
provided on the screen or by convenience (i.e., matching up students who were close to each
other in their assigned seats). After partner work time, there was some variance in the final

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lesson segment. A majority of the lessons analyzed included some brief discussion with students
reporting their work. There were also instances of partners moving on to work on homework, or
other management related tasks such as handing out progress reports.
Goals. Mr. Sandberg had a variety of broad instructional goals. During warm-up
problems he would at times focus on student misconceptions in hopes to bring to light issues that
might not easily appear in the day’s lesson but may impact students’ long-term understanding of
an idea. As one example, Mr. Sandberg asked students to consider which of the two equations
had a greater rate of change: y = 1.5x + 2 or y = 2x + 0.5. The second equation exposed students
to situations where the rate of change was an integer and the y-intercept of the graph was a noninteger, different than the more common first equation. This set of equations afforded students
the opportunity to consider the meaning of those values in relation to a context they had
previously investigated and to possibly avoid over-generalizing the types of numbers associated
with the slope and y-intercept in a linear situation.
Similar to both Ms. Mitchell and Mrs. Stewart, Mr. Sandberg also had unit level goals
(e.g., comparing data sets to make statements about them), and mathematical practice goals such
as using technology strategically. In contrast to the other two teachers in this study, there were no
instances of Mr. Sandberg explicitly referencing goals related to being a member of a
mathematical community during interviews, lesson plans, or lesson observations. This may have
been attributed to the fact that data collection did not start until February and these goals may
have been present earlier in the semester.
Consistent patterns of instruction. Mr. Sandberg had two consistent patterns in his
instruction: (a) the consistent use of technology as a lesson resource, and (b) the implementation
of a consistent discussion structure in order to maintain classroom management. In regard to the

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first consistent pattern, he frequently used several types of instructional technology -- from
clickers to graphing calculators to spreadsheets -- to support students in discussing their thinking
during whole-class discussions. This varied use of technology differed greatly from the other two
teachers’ instruction. His second consistent pattern was the use of his lesson structure (described
previously), which differed from the interaction structures that both Ms. Mitchell and Mrs.
Stewart incorporated. His consistent patterns are described in greater detail in Appendix P.
Professional obligations. The predominant obligation influencing Mr. Sandberg’s
decision-making was attention to the institutional obligation because of his overwhelming
emphasis on establishing a well-ordered classroom. Student work and talk were largely
constrained by the need to have consistent interaction and activity patterns. These constraints
allowed for students to have a clear sense of what was expected of them as learners of
mathematics, but they also served as ways to limit the potential opportunities to pursue
interesting ideas that were connected to but outside the exact scope of the lesson flow. In both
consistent patterns of his behavior, there were situations where the need to maintain a consistent
discussion flow and norms of interactions took priority over pursuing student ideas that could
have led to interesting mathematical discussions. These instances, coupled with the lack of
partner/group talk to the whole-class, support the notion that the institutional obligation was Mr.
Sandberg’s primary focus.
Summary of Teachers’ Instructional Practice
All three teachers in this study included the four goal types analyzed as features of their
instruction. They all attended to lesson-specific content goals, broad content goals, mathematical
practice goals, and goals for establishing a mathematical community. All four incorporated
consistent patterns in their instruction, but they were quite varied. While all four had 1 or 2

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predominant obligations in their practice, they all differed; Mrs. Stewart focused primarily on the
disciplinary obligation, Mr. Sandberg on the institutional obligation, and Ms. Mitchell on the
interpersonal and individual obligations. In the next chapter, I discuss the results of how these
teachers’ orientations to their professional obligations acted as a mechanism to frame how these
teachers made decisions when faced with instructional dilemmas as a result of conflicting types
of goals.

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CHAPTER 5
RESULTS: INSTRUCTIONAL DILEMMAS AND DECISIONS
The results for this dissertation addresses the research question: What is the nature of
instructional dilemmas that teachers face when planning for and enacting collaborative
mathematics discussions, and how do teachers respond to those dilemmas through their
decision-making? I remind the reader that dilemmas are defined as a conflict between
instructional goals. Some dilemmas were observed (via a deviation from a consistent pattern of
behavior), whereas others were self-reported (i.e., the teacher identified them as a dilemma
during an interview or lesson plan). Some instructional dilemmas and their ensuing decisions
occurred during planning for a class discussion, while most others occurred during the enactment
of whole-class discussions.
The main finding of this study was that all three teachers’ decision-making was explained
via their primary professional obligation – with important limitations. Along with this main
finding, there were three subfindings. First, teachers’ decision-making was largely explained via
their primary orientation to their professional obligations. Second, all three teachers encountered
scenarios where they made decisions that deviated from their primary orientation to the
professional obligations. Third, even though the teachers were experienced, used instructional
resources (e.g., written curriculum materials) that supported implementing whole-class
discussions, and were generally oriented philosophically with the importance of using
discussions, this was not sufficient to avoid challenges one might expect to see while facilitating
whole-class discussions. Most importantly, the types of challenges they encountered were, by
and large, independent of the teachers’ primary orientations to their professional obligations.

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Dilemma Type Results
Given the four different types of goals, a total of six different types of dilemmas were
possible, considering the set of conflicts between two instructional goal types. This chapter is
organized around these six dilemmas. Although the dilemma types are enumerated, the order is
arbitrary. Within each dilemma, I describe one case from each of the teachers’ practice. I chose
to describe only one representative case per teacher, choosing the case that provided the most
detail for the particular dilemma. Of the 18 possible cases that I describe in this chapter (6
dilemma types, 3 teachers per dilemma), 16 appeared in the analysis. There were 2 cases that did
not occur, which are noted in the tables in each section. Across the 3 teachers, each teacher had
at least 5 of the 6 dilemma types present. Table 4 provides the different dilemmas that occurred
in the data for each of the teachers.
Table 4
Dilemmas Teachers Encountered as a Conflict between Goal Types
Dilemma Type
Teacher
LS-BC
LS-MP
LS-MC
BC-MP
BC-MC
MP-MC
Mrs. Stewart
2
2
1
1
0
1
Mr. Sandberg
2
1
3
0
2
1
Ms. Mitchell
2
5
1
1
2
2
Note: BC: broad content goals, LS: lesson-specific content goals, MP: mathematical practice
goals, MC: establishing a mathematical community goals
I remind the reader that Mrs. Stewart’s instructional decision-making in relation to
collaborative discussions was primarily oriented to the disciplinary obligation. Mr. Sandberg’s
instructional decision-making in relation to collaborative discussions was primarily oriented to
the institutional obligation. Ms. Mitchell’s instructional decision-making in relation to
collaborative discussions was primarily oriented to the interpersonal and individual obligations.
Thus, each teacher represented a case of instruction oriented along a different professional

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obligation. These results provide insight into how teachers who typically favor a particular
professional obligation make decisions when faced with particular dilemma types.
Dilemma Type 1: Lesson Specific Content Goals Conflicting with Broad Content Goals
This type of instructional dilemma took place when teachers encountered situations
where they had to choose between immediate content goals in a lesson with broad content goals.
As shown in Table 4, this dilemma type appeared at the same frequency across all teachers’
observations. Given this dilemma type, one might anticipate a teacher primarily oriented towards
the disciplinary obligation as trying to balance these two types of content goals, with perhaps
more attention to broader content goals because of a possible understanding of the long-term
storyline of certain mathematical ideas. A teacher with a primary orientation towards the
institutional obligation could also favor broad content goals, if they were primarily concerned
with such obligations as departmental or state-wide assessments. They might also focus on
lesson-specific goals, if a deviation from the lesson goals would create more disruption to the
class. A teacher with primary orientations to interpersonal and individual obligations may place
greater emphasis on lesson goals being responsive to student thinking as it appears immediately
in the classroom.
In the subsection below, I provide greater detail on one representative case from each of
the teachers for Dilemma 1. As an overview of this dilemma, all three teachers made decisions
when faced with instructional dilemmas that were consistent with their primary orientation to
their professional obligations. Mrs. Stewart (disciplinary) was able to balance lesson specific
content goals with broader content goals. When Mr. Sandberg was faced with what to do with a
student’s question that might disrupt the flow of the class by pursuing the student’s idea, Mr.
Sandberg (institutional) chose to quickly move on from the student’s question to refocus the

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class on lesson-specific content. When Ms. Mitchell (interpersonal/individual) was faced with a
similar situation, she too chose to refocus the class on the lesson goals, but not before
acknowledging the importance of the student’s insight to the whole class. Table 5 provides an
overview of this first dilemma.
Table 5
Dilemmas between Lesson Specific Content Goals and Broad Content Goals
Teacher (typical orientation
Dilemma
Decision
to professional obligations)
Mrs. Stewart
Complete all parts of the
Complete all parts of the
(disciplinary)
lesson as written in the
lesson, but discuss in depth for
materials vs. Support
a few minutes numerical and
students in connecting
visual equivalence
visual and numerical
equivalence
Mr. Sandberg (institutional)

Continue on with lesson
investigation vs. Pursue a
student’s idea that focuses
on ideas later in the unit

Acknowledge student’s idea,
then redirect class to today’s
lesson

Ms. Mitchell
(interpersonal and
individual)

Understanding how to
interpret a graph in a
context vs. Pursuing a
student’s question about
graphs of functions in
general

State the importance of the
student’s question and tell the
student the class will come
back to that question later

Dilemma 1: Mrs. Stewart. In Mrs. Stewart’s class, students were working on
representing an area model for a compound probability situation. They were given two buckets
with colored marbles in each bucket. In the first bucket there were 3 marbles: red, green, and
green. The second bucket had marbles colored red, blue, green, and yellow. The class discussed
the question, “What is the probability of choosing a marble from each bucket and getting at least
one blue?” A student described his thinking to the whole class, but Mrs. Stewart was unable to
follow the student’s explanation. Had Mrs. Stewart been primarily oriented to the institutional

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obligation (focused on making sure the question was addressed in a short amount of time), she
might have simply rephrased what the student was thinking. Concerned that the class was not
really understanding where the one-fourth was represented in the area model, she posed a
question: “If the probability of that is one-fourth, what size piece out of the whole are we
actually counting?” She called on a student to come up to the board to show how they were
thinking about where the 1/4 showed up in the figure. In Figure 8 the student correctly showed
an area representation of 1/4 (in the lower right corner), but was unable to describe where 1/4
appeared in Mrs. Stewart’s model (on the left).

Figure 8. Area model for selecting marbles from two buckets
The deviation from the consistent pattern of her rapid pace of instruction prompted the
question during the post-lesson interview about why she pressed students about this. If her goals
had been oriented to the institutional obligation, where completing class on time and staying on
pace with the curriculum is a key goal, an alternative decision could have been simply to state
that the two values were equivalent and move on. Mrs. Stewart, however, chose to press students

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to see how the two values were equivalent with the area model. She indicated in the interview
that her reason for doing this was that in a future problem students would be better prepared to
reason through more complicated area models if they understood how the values were
represented directly from the model. She also indicated that she had wanted to possibly go
further with the explanation than she did, but, pressed for time at the end of class, she chose to
press students on connecting the model to the representation on just this one problem.
Given the dilemma of meeting broad content goals with lesson-specific content goals,
Mrs. Stewart found opportunities to support students in connecting symbolic and visual
representations that could support their future work in subsequent lessons. Although the
institutional obligation (time constraints) ultimately limited her ability to go deeper into broader
content goals that address her disciplinary obligation, she was able to identify and facilitate
discussion around fractional equivalence, a topic that shows up in later lessons. Being able to
identify key areas that potentially cause problems in the future and being able to balance shortterm content goals with broader content goals is consistent with Mrs. Stewart’s propensity to
favor the disciplinary obligation.
Dilemma 1: Mr. Sandberg. During the beginning of a unit on probability, Mr. Sandberg
had asked the class to think about flipping one coin and then flipping multiple coins, and to
consider the probabilities of getting all heads. One student had a longer explanation where he
was trying to explain the difference between the probability of an individual coin flips versus
multiple coin flips. The class appeared not to follow the student’s explanation. Mr. Sandberg was
faced with the decision of (a) pursuing the student’s idea on compound probabilities with several
coins or (b) moving on with the simpler lesson goals he had already planned.

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Had Mr. Sandberg been primarily oriented to the disciplinary obligation, he might have
spent time pursuing the student’s idea or, at the very least, restating the student’s idea so that the
class could return to this idea in a future lesson. Mr. Sandberg’s response to the student during
the discussion was, “We have vocabulary for that,” and then he quickly moved on with the
lesson, not returning to the student’s idea. During the post-lesson interview Mr. Sandberg
commented on this exchange: “I had this sense that the other kids were not understanding him; I
was having a hard time understanding him, and it would have taken us in a different direction.”
During the post-lesson interview Mr. Sandberg mentioned that the ideas that the student were
sharing would be brought up later in the unit, but the class (in his opinion) was not ready yet to
investigate those ideas. Mr. Sandberg chose the immediate lesson goals over broader content
goals. Given the particulars of this exchange between Mr. Sandberg and the student, moving
away from the student’s ideas and back towards the flow of the original discussion was
consistent with Mr. Sandberg’s consistent pattern of instruction in line with his orientation to the
institutional obligation.
Dilemma 1: Ms. Mitchell. In one of Ms. Mitchell’s discussions, students were looking
back on their previous day’s work reviewing the graphs of three students who were fundraising
in a walkathon (Figure 9). The problem in the textbook (Moving Straight Ahead, p. 8, CMP2)
gave the fundraising as follows: Leanne $10 regardless of distance; Gilberto $2 per km; Alana $5
plus $0.50 per km. A student commented that the graphs were linear, and Ms. Mitchell asked the
student why they were linear.

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Figure 9. Ms. Mitchell displays walkathon results
Later in the discussion, the student was asked to come to the board and draw a graph of a
non-linear relationship. Whereas the student may have simply drawn a non-linear function (e.g.,
a quadratic or exponential relationship), the student instead drew a graph that was not even a
function (Figure 10). Given this strange graph, a different student asked, “Could a graph actually
go backwards like that? Like could it go like this way and then eventually go back that way
again?” This student was asking a question focused on an important relationship in
understanding a mathematical function that went well beyond the scope of the individual lesson.
Ms. Mitchell was faced with the dilemma of how to pursue this important mathematical insight,
which was beyond the scope of the lesson.

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Figure 10. A student draws a graph of a non-linear function
Had Ms. Mitchell been oriented to the institutional obligation and been most concerned
with maintaining a consistent discussion flow (as was the case with Mr. Sandberg), she too might
have quickly moved on from this student’s thought, minimizing the opportunity for disruption.
Instead, her decision was to write a note on the side of the board and to validate the importance
of the student’s insight. “I’m going to write that right here Charlie because that’s outside what
we want to talk about today. But that’s an extremely important question; could a graph actually
go backwards? And we need to talk about that.”
During the video-stimulated interview after the first round of observations, Ms. Mitchell
commented on this clip:
Charlie’s question was so awesome as it usually is. That is an example where I don't want
to shut down the conversation or, well I do want to shut down the conversation because
the level of thinking it would take based on what they had had by one-two would be too

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great…I do want to push a little bit…Now that thing about going back on itself I still
have that written down and we will come [back] to that.
Although Ms. Mitchell’s decision was to move on from the student’s question, she did
this only after valorizing the student’s question. Ms. Mitchell could have chosen to pursue the
student’s idea directly, essentially moving away from her lesson goals for the discussion. Instead,
she refocused the discussion in line with the lesson goals. Her decision to valorize the student’s
thinking was consistent with her professional obligation to students individually. By supporting
the student’s thinking and publicly acknowledging its importance, she continued to make explicit
the importance of thinking and questioning in her classroom.
Summary of Dilemma 1. In all three cases, teachers encountered dilemmas on how to
manage lesson specific content goals with broad content goals. In all three cases, the teachers
made decisions that were consistent with their primary orientation to their professional
obligations. Ms. Mitchell and Mr. Sandberg encountered very similar dilemmas: A student asked
a question during the course of a discussion that had the potential to move the class beyond the
scope of what the teacher had hoped to accomplish in the lesson. Although both teachers
prioritized lesson specific content goals over broad content goals, they differed in their
interaction with their students. Mr. Sandberg chose to quickly move on from the student’s
question because he felt the class might be confused, which could in turn cause disruption to
accomplishing his lesson specific content goals. Ms. Mitchell chose to acknowledge the
importance of the student’s question before refocusing the discussion on the lesson specific
content goals. These two scenarios contrasted with Mrs. Stewart, whose orientation on the
disciplinary obligation was balanced between broad content goals and lesson specific content
goals.

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Dilemma 2: Lesson Specific Content Goals Conflicting with Mathematical Practice Goals
This second type of instructional dilemma was the most prevalent type of dilemma that
occurred, but it happened more often in Ms. Mitchell’s classroom than in the other teachers’
classrooms. As I will explain in this section, Ms. Mitchell’s orientations around the
interpersonal/individual obligation likely played a role in the frequent emergence of this
dilemma. In Table 6, I provide an overview of these dilemmas.
Given a teacher with a primary orientation to the disciplinary obligation, the teacher may
try to strike a balance between immediate content goals and mathematical practice goals (given
that both focus on doing mathematics). In some cases, the teacher may privilege one goal over
another given the need to co-develop both goals. Given a teacher with a primary orientation to
the institutional obligation, one might expect to see more focus on immediate content goals, as
those are often assessed more readily on standardized assessments, and supporting students in
learning to engage in mathematical practice goals may cause unforeseen management issues.
Given a teacher with a primary orientation to the interpersonal or individual obligation, the likely
result is the teacher prioritizing mathematical practice goals, supporting students to develop their
own understanding through reasoning, communicating, and working with their peers.
As an overview of this dilemma (details provided in Table 6 below), two of the teachers
(Mrs. Stewart, Mr. Sandberg) made decisions when faced with instructional dilemmas that were
consistent with their primary orientation to their professional obligations, while one (Ms.
Mitchell) deviated from her primary orientation. Mrs. Stewart prioritized lesson specific content
goals over mathematical practices when her students became unable to make progress on a
problem consistent with her orientation to the disciplinary obligation. Mr. Sandberg considered
the benefits and limitations of letting students work in groups to support the whole-class

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discussion. His choice to focus on structuring the group work more effectively was consistent
with his institutional obligation. Ms. Mitchell prioritized departmental expectations for content
coverage given the time constraints of teaching during the last few weeks of the year to take
priority over a focus on mathematical practice. This decision was inconsistent with her general
orientation to prioritize the individual and interpersonal obligations.
Table 6
Dilemmas between Lesson Specific Content Goals and Mathematical Practice Goals
Teacher
Dilemma
Decision
(typical
orientation to
professional
obligations)
Mrs. Stewart
Support students in engaging in
Directly model the problem,
(disciplinary)
mathematical practices or provide greater
show the class how to solve it,
access to the content of the problem
and explain how she went about
solving the problem
Mr. Sandberg
(institutional)

Use group work (allows space for students
to discuss their thinking) or do not use
group work (provides extra time for lesson
goals)

Use group work in the lesson,
but structure it more effectively

Ms. Mitchell
(interpersonal
and individual)

Extent to which you can emphasize
mathematical practices when needing to
cover content at the end of the school year

Cover what is expected by the
department as best as possible
during the last few weeks

Dilemma 2: Mrs. Stewart. During a whole-class discussion in Mrs. Stewart’s class,
students were working on a problem from the CMP3 materials, collectively trying to determine
what a seventh test score would be in order for the average to be 85 (shown in Figure 11).
Students were struggling significantly in terms of how they would even begin the problem.
Typically, Mrs. Stewart would press her students to justify their thinking and they would respond
with answers. This problem proved to be highly challenging to students (as evidenced by their
lack of being able to really start the problem in any meaningful way).

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Given the inability of students to describe their thinking in the whole-class discussion,
Mrs. Stewart was faced with the dilemma of having to choose between the lesson specific
content goals (i.e., understanding how to solve the problem) or pressing students in hopes they
would be able to engage in mathematical practices. If Mrs. Stewart’s decision had been focused
primarily on the interpersonal obligation, she may have chosen to have students do a similar,
easier problem that would have allowed them access to the more challenging problem presented.
Usually, Mrs. Stewart provided time for her students to persist when challenged with a problem.
In this instance, Mrs. Stewart directly modeled the procedure to the class in terms of how to
solve the problem. The use of direct explanation (or modeling) was rare for Mrs. Stewart’s
instruction. This scenario highlighted that when Mrs. Stewart was compelled to decide between
mathematical practice and lesson-specific content goals, Mrs. Stewart would choose lessonspecific content goals. This choice is consistent with Mrs. Stewart’s general orientation to favor
the disciplinary obligation.

Figure 11. Students try to determine Jun’s 7th test score for a resulting average of 85.

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Dilemma 2: Mr. Sandberg. In Mr. Sandberg’s class, students often had the opportunity
to work with a partner on a problem to prepare for a whole-class discussion. Although this
dilemma did not occur directly during the whole-class discussion aspect of the class, it did allow
students to engage in discussing their thinking with each other in order to prepare their ideas for
the ensuing discussion. During a post-lesson interview, Mr. Sandberg acknowledged this
dilemma of whether allowing students to share their thinking with a partner prior to a discussion
was productive or not for the discussion. Given the somewhat long sections of teacher
explanation that frequently occurred during discussion segments, not allowing students to work
in groups would have saved (in his words) about ten minutes of class time. The value of letting
students work in groups is that they are potentially better prepared to discuss their thinking when
the discussion begins. Both of these choices are focused on maintaining efficiency to support the
flow of the lesson and discussion.
If Mr. Sandberg had been primarily oriented to the interpersonal obligation, he likely
would have seen the value in students being able to share their thinking with a partner prior to
sharing their thinking with the whole class. It would have been much less problematic than it
was. His choice was to allow students to work in their groups but to be more structured in how
he set up the groups. Creating more structure was consistent with his focus on management and
structure and his orientation to the institutional obligation.
Dilemma 2: Ms. Mitchell. This second dilemma type for Ms. Mitchell was the most
prevalent for any dilemma and teacher, appearing in five different situations. She frequently
grappled with the need to meaningfully engage students in mathematical practice while also
attending to lesson-specific content goals. This dilemma’s frequency may have been a result of
Ms. Mitchell’s greater comfort with focusing on mathematical practice (given her general

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orientation to learning, her role as a teacher, and her experience in teaching several different
grades) than her comfort with specific lessons (being in her first year teaching middle school and
using the CMP2 series).
Although many instances of this type of dilemma occurred during enactment of
collaborative discussions, one of the most telling explanations of Ms. Mitchell’s second dilemma
type was described in her thinking about planning for discussions. During an interview she
described the dilemma of how to fit in as much content as she is expected to (by the department)
over the last few class periods remaining in the year while still trying to focus on mathematical
practices.
Well there's a dilemma, because on the one hand I would love to have all the ideas for
them to have access to; on the other hand there's the bigger idea of the mathematical
practices, like what do I do when I don't know what to do?...And, so I've got this
treadmill keeping us moving toward the end at the same time as I'm trying to incorporate
as much of the thinking stuff that I can. (Ms. Mitchell, post-lesson interview, May 19,
2015)
She later referred directly to how this dilemma impacted that day’s lesson and the
previous day’s lesson, wondering if students saw the mathematical purpose of what they were
doing. This dilemma mostly went unresolved, but she conceded the pressure of having to get
through material as best she could before the end of the year. Although Ms. Mitchell’s primary
orientation to her obligations attended to students individually and interpersonally, in this case
Ms. Mitchell’s institutional obligation (being a member of mathematics department with
expectations of coverage) seemed to override her primary orientation to the individual and

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interpersonal obligations. This is noteworthy because it shows the limitations of the
implementation of her orientation to the interpersonal and individual obligations.
Summary of Dilemma 2. All three teachers encountered dilemmas where they had to
make instructional decisions about lesson content goals and mathematical practice goals. The
three teachers encountered very different scenarios in this second dilemma. In Mrs. Stewart’s
discussion, she encountered a scenario that compelled her either to continue pressing
mathematical practice or to model directly and focus on lesson-specific content goals. Her choice
was illustrative in showing her attention to mathematical content, when her students collectively
seemed to be unable to access a problem. Mr. Sandberg’s attention to maintaining an orderly and
structured classroom environment presented itself in his second dilemma type. He grappled with
the benefits and limitations of using group work as a mechanism to support students prior to a
discussion. His choices all pointed to efficiency and structure, consistent with his orientation to
the institutional obligation. Ms. Mitchell struggled to maintain goals for engaging students in
mathematical practice, given the time constraints and coverage expectations at the end of the
school year. Her decision, to subvert mathematical practice over content coverage, was the first
instance of a teacher making a choice that was inconsistent with her primary orientation to her
professional obligations. This choice showed the strength of the institutional obligation, and how
time constraints can factor greatly into a teacher’s decision.
Dilemma 3: Lesson Specific Content Goals Conflicting with Establishing a Mathematical
Community Goals
This third type of instructional dilemma occurred when teachers encountered situations
where they had to choose between lesson specific content goals and goals focused on
establishing a community of mathematical learners. Given a teacher with an orientation towards

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the disciplinary obligation, one might expect to see the teacher choosing lesson specific content
goals over goals for establishing a mathematical community, taking into account this
orientation’s focus on mathematical content. In contrast, a teacher with obligations to the
interpersonal or individual obligations may tend to prioritize establishing the community aspects
of the classroom. A teacher who orients primarily toward the institutional obligation may
prioritize lesson specific content, given the expectations of student learning and being assessed
on content.
As an overview of this dilemma (details in Table 7 below), all three teachers made
decisions when faced with instructional dilemmas that were consistent with their primary
orientation to their professional obligations. Mrs. Stewart and Ms. Mitchell encountered this
dilemma when a student asked an interesting question (in Mrs. Stewart’s case) or posed an
incorrect answer (in Ms. Mitchell’s). In both cases, the teachers were careful about how they
responded, mindful of how their responses might impact their goals for establishing a
mathematical community. Mr. Sandberg encountered this dilemma during the last few minutes of
class as he chose how to best use the few minutes remaining, balancing student participation and
the need to summarize the lesson.
Table 7
Dilemmas between Lesson Specific Content Goals and Establishing a Mathematical Community
Goals
Teacher
Dilemma
Decision
(typical orientation to
professional obligations)
Mrs. Stewart
How to appropriately
Describe how context matters
(disciplinary)
respond to a question about (probability or general case) for
representing probability
whether that representation
with non-whole numbers
would be appropriate
Mr. Sandberg
(institutional)

How to include student
participation and having
enough time to summarize

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Limit participation to ensure
adequate time to summarize

Table 7 (cont’d)
Ms. Mitchell
(interpersonal and
individual)

How to respond to a student
that gives an incorrect
answer during the
discussion

Have other students explain
their thinking to the student and
the class

Dilemma 3: Mrs. Stewart. During one of the first problems involving probability, Mrs.
Stewart asked her students to determine the probability of choosing an individual block given 3
blue blocks, 6 yellow blocks, and 9 red blocks. One student stated 1/18. Another student asked if
she could state the probability as 0.5/9. One could imagine several choices about how to proceed
when asked this question, ranging from a brief answer and moving on to deeply pursuing the
student’s question. Mrs. Stewart paused for a few seconds to think about this question before
responding, deviating from the typical rapid pace of instruction that often occurred in her
instruction.
What is interesting about this particular scenario is its similarity to the first dilemma that
Mr. Sandberg and Ms. Mitchell encountered. Specifically, a student asks a question that has the
potential to deviate the conversation into possibly interesting mathematics. A teacher who
generally oriented to the institutional obligation may choose to quickly move the focus of the
discussion back to lesson specific content goals to make sure that the flow of the discussion is
not unduly interrupted. A teacher who focuses on the interpersonal or individual obligation may
stop the class to have students discuss the question in their groups before resuming the
discussion.
Given Mrs. Stewart’s general orientation to the disciplinary obligation, her response was
to tell the student she could write 0.5/9 but that in the context of probability they should use
integer values. Her response also included how they would use these values in the class versus

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how a mathematician might respond. This response attended to both the lesson specific goals of
understanding how to appropriately represent probability numerically, while also communicating
to the student the value in their thinking – an important aspect of establishing a mathematical
community in the classroom. In the video-stimulated interview after the first round of
observations Mrs. Stewart stated,
There’s still some weird things about that [the question]. And I don’t know if she’s
thinking in terms of a probability ‘Could that happen?’ or just in general in math world,
but I did want to make sure we tied it back to the probability of what that would mean.
(Mrs. Stewart, video-stimulated interview, May 7, 2015)
Her reasoning for why she chose to handle the question during the discussion was
consistent with her general orientation to focusing on the disciplinary obligation.
Dilemma 3: Mr. Sandberg. Mr. Sandberg encountered this dilemma during the end of a
class period, when he was trying to get students to participate in sharing their solutions to a
compound probability problem. The problem asked students to collect data to determine the
likelihood of getting a “match” (heads-heads or tails-tails) when flipping two coins. Students had
earlier in class predicted if a match was more, less, or about as likely as no-match (one each of
heads and tails as the result). Mr. Sandberg was hoping to have students discuss their thinking
about what they thought about their results compared to their predictions.
A teacher who prioritized the disciplinary obligation might not care how the data was
presented on the board, leaving as much time as possible for a discussion of the results. Mr.
Sandberg originally had students filling in the data on the board, but later called on students
directly so that he could more efficiently put the values on the board for them. Running short on

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time, he described to the class why they might expect to see about 50% of the resulting coin flips
as a match. He then quickly summarized the main points of their inquiry.
During the post-lesson interview I asked Mr. Sandberg about this exchange and he said,
“I want them to be able to do that, just one tiny thing like that [putting numbers on the board];
they love it, and they buy in more, sometimes you have to cut ‘em off, but it was simply taking
too long.” In this case, time constraints created the dilemma of whether to make an instructional
decision that would support the community aspect of the classroom versus using the time to
summarize what the class had determined and ask questions about the data. In this case, Mr.
Sandberg chose to ensure he got through the end of the lesson as prescribed in the materials. This
decision was consistent with his general orientation to prioritizing the institutional obligation,
because of the need to complete the lesson and continue to stay on pace with the unit.
Dilemma 3: Ms. Mitchell. Similar to Mr. Sandberg’s third dilemma, this dilemma took
place as students in Ms. Mitchell’s class explored a problem on compound probability. In this
lesson, students rolled two six-sided dice and found the product of the two values. They did
several trials. One student won if the product was even; the other student won if the product was
odd.
Ms. Mitchell asked the class if they thought the game was fair. One student, Jonathon,
said the game was fair because there are 3 odd outcomes and 3 even outcomes per die. This is
incorrect since the outcome is based on the products of the two values, not just the individual
values appearing. Ms. Mitchell was faced with how to appropriately respond to the student given
that this particular student often has difficulties in the class. On the one hand, Ms. Mitchell
needed to advance lesson-specific content goals, helping students see the inherent unfairness to
this game. On the other hand, if her response was to be consistent with trying to establish a

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mathematical community, she would need to be careful about how she responded to a student
who typically has difficulties in the class. A teacher with a general orientation to the disciplinary
obligation may have pressed the class (or Jonathon) to see why he was incorrect. She also could
have simply stated that Jonathon was incorrect, or called on a higher-status student in the class to
explain why he was incorrect.
Instead, Ms. Mitchell supported Jonathon’s thinking, and acknowledged how he was
reasonable (in some way) in thinking that the game could be fair. During the discussion she had a
student (Josh) sitting near by talk to Jonathon about the odd number times odd number case, and
later asked for other students’ thinking about the problem. She described this scenario during the
post-lesson interview:
Jonathon, you know, he flat out fails; he doesn't have a whole lot of perseverance
intellectually. He thinks of things sometimes really in strong ways, but I was happy that
he was participating, and is in fact that half the numbers are even and half the numbers
are odd. So I wanted to, you know, honor that he said, and then, but also try to build on
that and then I felt like Josh kind of jumped ahead and wasn't real clear about odd times
odd. So rather than just go there, and have him or someone explain how he knew that, I
felt like the whole group would kind of get lost, so I just stated it very simply after he
said it so people could think about it. Seemed to me there was still a question about the
odd times odd. And they were all just kind of letting it go… (Ms. Mitchell, videostimulated interview, May 11, 2015)
The decision Ms. Mitchell made when faced with an incorrect student solution was
consistent with her attention to the individual obligation when faced with how to attend to
lesson-specific content goals and goals for establishing a mathematical community.

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Summary of Dilemma 3. All three teachers encountered dilemmas where they had to
navigate between lesson-specific content goals and goals for establishing a mathematical
community. In all three cases, the teachers considered how they would continue to meet the
needs of students to ensure that participation continued during whole-class discussions. This
dilemma highlights the importance that teachers placed on establishing an environment where
students felt welcome to contribute to the whole-class space, something that does not simply
automatically occur but must be carefully cultivated. Mrs. Stewart was able to contextualize
representing numbers by bringing attention to different communities and content ideas. Ms.
Mitchell attended strongly to the individual and interpersonal obligations by allowing other
students in the class to support a student with an incorrect answer. Mr. Sandberg ran into time
constraints, and limited student interaction in order to summarize the main ideas from the lesson.
All three teachers attended goals that were consistent with how they were each generally oriented
to their professional obligations.
Dilemma 4: Broad Content Goals Conflicting with Mathematical Practice Goals
This fourth type of instructional dilemma occurred when teachers encountered situations
where they had to choose between broad content goals and mathematical practice goals. This
dilemma only occurred with two of the three teachers. Given a teacher with a primary orientation
to the disciplinary obligation, one might expect to see this teacher attempting to balance these
two goals, with content possibly being favored over practice. Given a teacher with a primary
orientation to the individual or interpersonal obligation, one might expect to see this teacher
prioritize mathematical practice as these practices often include aspects of communication and
students working with their peers. Given a teacher with a primary orientation to the institutional

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obligation, one might expect to see prioritization of broad content goals, given the institutional
expectations of content based standardized assessments.
As an overview of this fourth dilemma (more details in Table 8 below), only two of the
three teachers encountered this dilemma. Mrs. Stewart encountered this dilemma while she
considered how much scaffolding to provide students via instructional resources in preparation
for their collaborative discussion. Ms. Mitchell encountered this dilemma during planning while
she considered how standardized assessments interfered with opportunities to support students in
being more proficient in mathematical practice. Both teachers’ decisions were consistent with
their primary orientation to their professional obligations.
Table 8
Dilemmas between Broad Content Goals and Mathematical Practice Goals
Teacher
Dilemma
Decision
(typical orientation to
professional obligations)
Mrs. Stewart
How much scaffolding to
Standardize scale & axes on
(disciplinary)
provide students via
graph paper so that students can
instructional resources
communicate with each other
(handouts) prior to a
more effectively
discussion
Mr. Sandberg
(institutional)

No Dilemma Present

No Decision Present

Ms. Mitchell
(interpersonal and
individual)

How to balance practice
goals with expectations to
prepare students for
standardized assessments

Focus on practice but try to
prepare students for
assessments as best as possible

Dilemma 4: Mrs. Stewart. This fourth dilemma type occurred for Mrs. Stewart while
she was planning for a class discussion on a problem where students analyzed data from different
roller coasters. The lesson focused on the differences between categorical and numerical data,
and measures of central tendency. Part of the data to be used in the discussion was a comparison

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of different data sets (responses from Internet data and a fictitious seventh-grade class). Mrs.
Stewart was faced with the dilemma of having students make their own bar graphs from scratch
(i.e., just provide graph paper), or to provide more consistent scaling and axes on the graph. On
the one hand, providing students with very little support provides students with the opportunity
to ask important questions about graphing, such as which variable is independent (and
dependent), and what scale to choose. Providing less information may prove useful to students’
long-term understanding of how to create a graph on their own. On the other hand, providing
consistent scales and axes to all students limits students’ opportunity to consider these questions
about graphing, but provides students with greater opportunity to clearly communicate their
mathematical ideas with each other about the questions in the lesson. Choosing the graph paper
with scales and axes would support students in engaging in mathematical practice.
A teacher who is generally oriented to the institutional obligation might have seen the
value of attending to the broad-content goal of supporting students in knowing how to correctly
set up a graph. In contrast, Mrs. Stewart chose the handout with the scale and axes provided. In
the post-lesson interview, she noted that she had provided this handout because (a) her students
are “notoriously slow” in making certain types of graphs, and (b) having consistent scales and
labelled axes would allow students to compare and contrast solutions from different groups more
readily. One of her goals in facilitating a discussion during this lesson was for students to
understand the difference between categorical and numerical data – a topic that spans across
several lessons (hence the categorization of this as a “broad content goal”). Mrs. Stewart rarely
handed out supporting worksheets during her lessons, but felt that they were necessary for this
lesson even though it effectively limited students engaging in thinking about how to best
represent the data, how to label the axes of the graph, and which scales to use. Mrs. Stewart

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chose not to let students engage in the practice of modeling (representing) to serve a broadcontent goal that she identified as problematic for students. Given her primary orientation to the
disciplinary obligation, it was likely that any decision she made would have been consistent with
this obligation. Her decision to focus on content over practice was consistent with her general
orientation.
Dilemma 4: Ms. Mitchell. This fourth dilemma was discussed during a post-lesson
interview with Ms. Mitchell, during the first of her observations. Thinking about her planning for
discussions, she talked about the institutional expectations of preparing her students for
standardized assessments (both departmental and state-wide). These assessments factor into
broad-content goals because in many ways they describe the content that various stakeholders
view as important for students to learn. Given the high value that Ms. Mitchell placed on
mathematical practice, these standardized assessments created a recurring dilemma as she
planned for collaborative discussions.
During the interview she talked about how her vision of effective collaborative learning
differed from many other administrators whom she had known. In her view, students discussing
mathematics and “putting up with lots of disequilibrium” and “being very patient” were positive
traits, but other administrators viewed a teacher negatively for “not scaffolding appropriately.”
While she did not question her focus on process standards, she often felt constrained by not
being able to pursue interesting student ideas as deeply because this choice might not prepare
students adequately for departmental or state-wide tests.
Given absolute freedom to teach as she felt best, she would have tailored much more of
her instruction (including discussions) to individual students’ insights to further encourage their
engagement in mathematical practice. This differed with a teacher who might have felt more

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oriented to the institutional obligation, as that teacher might feel the need to make decisions on
how students will best meet expectations for standardized assessments. This dilemma highlighted
Ms. Mitchell’s primary orientation to the individual obligation, yet also showed how the
institutional obligation played an important role in constraining her decision-making via the
tension between teaching for content as mandated by outside voices and trying to encourage her
students to engage in mathematical practice.
Summary of Dilemma 4. Overall, both Mrs. Stewart and Ms. Mitchell encountered this
dilemma of how to ensure students are engaging in mathematical practices while also attending
to either long-term or unit-level content goals. Whereas Mrs. Stewart’s dilemma appeared in
considering scaffolding via instructional resources, Ms. Mitchell’s occurred more in planning,
thinking about the broadness of assessment and its impact on being student-centered in class
discussions. Both teachers navigated these dilemmas consistent with their typical orientations to
their professional obligations.
Dilemma 5: Broad Content Goals Conflicting with Establishing a Mathematical
Community Goals
This fifth type of instructional dilemma occurred when teachers encountered situations
where they had to choose between broad content goals and mathematical practice goals. Given a
teacher with a primary orientation to the disciplinary obligation, one might expect to see this
teacher prioritize broad content goals since the disciplinary obligation primarily focuses on
content-based understanding. In contrast, given a teacher with a primary orientation to the
individual or interpersonal obligation, one might expect to see this teacher prioritize goals for
establishing a mathematical community considering the focus on social interaction. Given a
teacher with a primary orientation to the institutional obligation, one might expect to see

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prioritization of broad content goals, taking into account the institutional expectations of content
based standardized assessments.
As an overview of this dilemma (more detail provided in Table 9 below), only two of the
three teachers faced this dilemma. Ms. Mitchell encountered this dilemma when her class failed
to reach consensus on determining whether a set of graphs represented linear relationships. Mr.
Sandberg encountered this dilemma as he wrestled with the benefits and limitations of
supporting student talk by first letting them work in groups. Both teachers made choices
consistent with their interpersonal obligation, which was consistent with Ms. Mitchell’s primary
tendency, but inconsistent with Mr. Sandberg’s.
Table 9
Dilemmas between Broad Content Goals and Establishing a Mathematical Community Goals
Teacher
Dilemma
Decision
(typical orientation to
professional obligations)
Mrs. Stewart
No dilemma present
No dilemma present
(disciplinary)
Mr. Sandberg
(institutional)

Use or not use group work
to support student thinking
prior to a discussion

Continue on with group work
so that students feel better able
to share their thinking in the
discussion

Ms. Mitchell
(interpersonal and
individual)

What to do when a lack of
consensus occurs on an
important topic spanning
multiple lessons (linearity)

Let students state their ideas,
and press them to describe their
ideas more clearly

Dilemma 5: Mr. Sandberg. Mr. Sandberg identified this fifth dilemma during an
interview, regarding how he grouped students to prepare for a whole-class discussion. He
mentioned that when students get into a group for the first time, it often takes them a few days
before they are willing to talk with each other about prior content knowledge. His overall goal
with group work was to help establish a mathematical community by supporting students in

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being able to discuss their thinking more effectively (both in group work and the discussion). He
acknowledged, however, that an important aspect of group work that was missing was being able
to connect back to prior work – an aspect of broad content goals. A teacher with a general
orientation to the disciplinary obligation would likely see group work (in service of preparing
students for a whole-class discussion) as supporting students in engaging in mathematical
practice. Although Mr. Sandberg’s primary orientation to the institutional obligation (namely
maintaining a well-ordered classroom) likely would have resulted in not pursuing group work,
his choice to structure group work as best he could is more consistent with a strong orientation to
the interpersonal obligation.
Dilemma 5: Ms. Mitchell. This fifth dilemma occurred during a discussion in the
walkathon problem, where Ms. Mitchell was asking students which of the three graphs were
linear. The graphs (Figure 12) represent money that fictitious students (as provided in the
textbook) earned in a fundraiser for how many miles they walked. Note that all three are linear;
two have non-zero slopes, one of which intersects at the origin while the other does not. Students
discussed which (if any) of the graphs represented linear relationships. Understanding linearity is
a broad-content goal, one that spans multiple lessons. The dilemma took place when she asked
the students whether the graphs were linear and very little consensus occurred.

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Figure 12. Students fail to come to consensus on determining which graphs are linear
A teacher primarily oriented to the institutional obligation (concerned primarily with
preparing students for standardized assessments) may have chosen to be more direct in
highlighting the differences between the graphs and focusing the conversation on what makes a
graph linear. Instead, Ms. Mitchell chose to let students come to their own understanding of
linearity by restating different students’ ideas of whether they thought each of the graphs was
linear or not. Although this choice may have confused some students during the lesson, it was
consistent with her focus on mathematical practices and, by presenting non-evaluative
comments, she supported her goals for establishing a mathematical community. During the
video-stimulated interview, I asked Ms. Mitchell about this part of the discussion and she
commented:
Yeah, I recognize that moments like those are always difficult because what I want to do
as a teacher of course is just clarify it. You know, just say, oh no these are all linear, see.
And then, let's make this table and okay, let's make this table, let's make this one, and see
they're all going up even; they're all linear. But I don't think, I mean I have multiple goals

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in the classroom; one of them is that they become a learning community and they listen to
each other. And that one of the ways that they learn is by comparing their thinking
against each other so really whether, I mean think, whether something's linear or not is
very, very simple concept. And so I think I can easily kind of let them go with a lot of
misconceptions about that for quite a while and practice this, well we have some
uncertainty and I'm not always going to tell 'em which is important. (Ms. Mitchell, videostimulated interview, April 1, 2015)
In this case, Ms. Mitchell chose to structure the discussion in a way that emphasized the
class becoming more of a mathematical community rather than simply making the unit-level goal
of understanding linear relationships be the main goal. Her decision was consistent with her
general orientation to her interpersonal obligation.
Summary of Dilemma 5. Mr. Sandberg and Ms. Mitchell both encountered dilemmas
related to broad content goals conflicting with goals for establishing a mathematical community.
Both teachers made decisions consistent with their interpersonal obligation, highlighting the
importance of establishing a community of learners as a somewhat necessary condition to be able
to facilitate whole-class discussions. This tendency to favor the interpersonal obligation was
consistent with Ms. Mitchell’s general orientation, but was inconsistent with Mr. Sandberg’s
general orientation. This inconsistency in Mr. Sandberg’s practice highlighted the limitations of
his tendency to favor the institutional obligation over the others.
Dilemma 6: Mathematical Practice Goals Conflicting with Establishing a Mathematical
Community Goals
This sixth and final type of instructional dilemma occurred when teachers encountered
situations where they had to choose between mathematical practice goals and goals for

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establishing a mathematical community. All three teachers encountered this type of dilemma.
Although all of the lessons observed took place during the second half of the school year, all
three teachers continued to describe situations where establishing a mathematical community
was important for enacting whole-class discussions. Given a teacher who is primarily oriented to
the disciplinary obligation, one might expect to see this teacher focus on mathematical practice
goals, as these goals are closer to mathematical content. Given a teacher who is primarily
oriented to the interpersonal or individual obligations, one might expect this teacher to try to
balance mathematical practice goals with establishing a mathematical community considering
that both likely include the teacher focusing on students clearly communicating their
mathematical thinking with their peers. Given a teacher who is primarily oriented to the
institutional obligation, one might expect to see a focus on mathematical practice goals if these
goals are an expectation of the school or department (which was the case at Highland Junior
High School).
As an overview of this dilemma (more detail provided in Table 10 below), all three
teachers encountered this dilemma. Mrs. Stewart was faced with the dilemma of what to do with
an overly quiet class. Mr. Sandberg encountered a student who forgot what he was going to say
and had to decide how to respond and when to move on with the discussion. Ms. Mitchell
attempted a new talk structure that created management issues. Both Ms. Mitchell and Mr.
Sandberg made decisions consistent with their primary orientation to the professional
obligations, whereas Mrs. Stewart prioritized the interpersonal obligation over her normal
tendency to the disciplinary obligation.

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Table 10
Dilemmas between Mathematical Practice Goals and Establishing a Mathematical Community
Goals
Teacher
Dilemma
Decision
(typical orientation to
professional obligations)
Mrs. Stewart
How to get a generally quiet Avoid random selection as
(disciplinary)
class involved in a
much as possible by calling on
discussion
students purposefully during
discussion
Mr. Sandberg
(institutional)

How long to wait for a
student when the student
forgets his answer

Wait for a few seconds, then
move on with the discussion as
planned

Ms. Mitchell
(interpersonal and
individual)

How to support
mathematical practice when
a student receives harsh
criticism from other
students based on their
contribution to the
discussion

Stop the discussion and remind
students about classroom norms

Dilemma 6: Mrs. Stewart. This sixth dilemma occurred as Mrs. Stewart was planning
for a class discussion. In one of her lesson plans, she described the difficulties she considered
about how to make sure students stay involved in the class discussion given the general quietness
in her first period class. It was important for Mrs. Stewart to get as many students as possible
involved in the discussion, yet the overall quietness of the classroom created the dilemma of how
best to call on students.
If she had been primarily oriented to the institutional obligation, she might have
considered calling on students randomly as a management technique to ensure that students were
paying attention in the discussion. She mentioned explicitly that she “waffled” on calling
students randomly. She was hesitant in using this technique and talked about several other ways
she might get students involved without resorting to random call. Her main goal was to make

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sure that students participated in communicating their mathematics, but she wanted this to come
from the students’ willingness to participate in a safe environment instead of feeling like she was
putting them on the spot. Her decision to avoid putting students on the spot via random selection
during discussions was more consistent with the individual obligation than her primary
orientation to the disciplinary obligation.
Dilemma 6: Mr. Sandberg. This final type of dilemma occurred in Mr. Sandberg’s class
during the discussion of his bell work exercises on distance-rate-time relationships. During the
discussion, a student wanted to ask a question; he was called on and then forgot what he was
going to say. The situation created the dilemma of how long to wait for the student to respond.
On the one hand, giving the student ample time likely communicates to students that their
thinking is valued (supporting the goal for establishing a mathematical community). On the other
hand, waiting for the student to remember their thought can interrupt the flow of the discussion
due to management issues, thereby limiting others’ opportunity to engage in mathematical
practices such as communicating their thinking.
Had Mr. Sandberg been primarily oriented to the individual obligation, he might have
pressed the student by prompting him so that the student could remember what he was thinking.
Instead, given his attention to the institutional obligation and maintaining a structured lesson
flow, Mr. Sandberg responded, “We’ll come back to you.” There was no evidence of returning to
the student’s thought during subsequent discussions in that lesson. In the video-stimulated
interview after Mr. Sandberg’s first round of observations, he described this dilemma.
I try to develop a good atmosphere to keep them, you know raising their hands and
answering questions...You don't want to just cut him off and act like it wasn't important,
because then he wouldn't raise his hand for a week, or ever; you know they're very

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concerned about their social activities right now, it's huge in seventh grade. (M.
Sandberg, video-stimulated interview, April 17, 2015)
His acknowledgment of the dilemma played into his decision to move on to the next
question to the class. This choice was consistent with his general orientation to the institutional
obligation of feeling the necessity to maintain the lesson structure (staying consistent with his
management and interaction structures) as opposed to pressing the student further. This decision
prioritized mathematical practice over establishing a mathematical community.
Dilemma 6: Ms. Mitchell. This last dilemma occurred in one of Ms. Mitchell’s lessons
where she purposely chose to change the structure of the discussion format to be more studentcentered. She was generally dissatisfied with the teacher-to-student interaction pattern that
occurred in many of her discussions. Before the discussion occurred, Ms. Mitchell explained to
students that they would be interacting directly with each other and that they could talk after the
speaker called on the next student or once the speaker had completed their turn. This was much
different than the typical structure in her classroom up to this point, where most turns alternated
between teacher and student. The reason for this was to support the class in being a more
autonomous learning community, as well as for the class to better communicate with each other.
The dilemma took place when, according to Ms. Mitchell, this new structure became
unproductive. She felt that two students were “ganging up” on another student by “using a
mocking tone.” She wanted to avoid the speaker being “fed to the sharks,” as she put it, so she
intervened in the discussion after being mostly silent. On the one hand, Ms. Mitchell felt the
need to support students in being less dependent on her voice to have a discussion (goal for
establishing a mathematical community). On the other hand, if students were unable to
accomplish this goal it would hinder students’ opportunity to engage in mathematical practice.

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The dilemma was about how long to support students in learning to interact with each other
directly as a mathematical community while ensuring that students felt protected individually in
their thinking.
A teacher who is primarily oriented to the institutional obligation might have viewed the
desire to help the class be less dependent on the teacher as inefficient, unnecessary, or
problematic since this perspective views the teacher as a necessary component to a discussion.
Ms. Mitchell’s decision was to stop the discussion at the point where she felt the students were
no longer talking to each other productively. She described this decision during the post-lesson
interview: “[T]hat made me very uncomfortable, so that was what I was sorting through how
long to let that go ‘cuz the conversation was great; the tone of it was not.” Later, she commented
that “…if people were being kind I would have let that muck around for a long time…But I
would really hold on as long as I could if I felt like everyone was being safe…emotionally and
all that.” Her decision to end the discussion was consistent with her primary orientation to the
individual obligation, prioritizing the need to establish a mathematical community over her
mathematical practice goals.
Summary of Dilemma 6. All three teachers faced dilemmas related to the tension
between having students share their thinking with the class and feeling that they are a member of
a mathematical community of learners. This tension shows up at different times based on the
student and the type of response they provide. All three teachers tended to prioritize the
importance of establishing a mathematical community based on their decisions. While Mr.
Sandberg and Ms. Mitchell made decisions consistent with their primary orientation to their
professional obligations, Mrs. Stewart did not.

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Subfinding 1
As evidenced by the dilemma results from the previous section, the first subfinding was
that teachers’ primary orientation to their professional obligations was highly influential in how
they made decisions related to whole-class discussions. By-and-large teachers’ decisions when
faced with instructional dilemmas was consistent with their primary orientation to their
professional obligations (see tables 11, 12 and 13 below). In some cases this involved teachers
prioritizing one type of a goal over another, in other cases, the decision involved teachers
balancing the needs of multiple goals.
Subfinding 2
The second subfinding was that each teacher encountered one dilemma where they made
a decision that deviated from their primary orientation to their professional obligations (see
tables 11, 12 and 13 below). These exceptions are noteworthy as they highlight the complexity
with which teachers managed their goals and obligations. The subfinding also shows the
limitations to how the teachers managed and prioritized their professional obligations in relation
to whole-class discussions.
Table 11
Ms. Mitchell’s Dilemmas and Orientations to Professional Obligations
Dilemma
Professional obligation
Was the professional obligation consistent or
emphasized in decision
inconsistent with teacher’s primary orientation?
LS – BC
individual
Consistent
LS – MP
institutional
Inconsistent
LS – MC
individual
Consistent
BC – MP
individual
Consistent
BC – MC
interpersonal
Consistent
MP – MC
individual
Consistent
Note: BC: broad content goals, LS: lesson-specific content goals, MP: mathematical practice
goals, MC: establishing a mathematical community goals

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Table 12
Mr. Sandberg’s Dilemmas and Orientations to Professional Obligations
Dilemma
Professional obligation
Was the professional obligation consistent or
emphasized in decision
inconsistent with teacher’s primary orientation?
LS – BC
institutional
consistent
LS – MP
institutional
consistent
LS – MC
institutional
consistent
BC – MP
no dilemma present
no dilemma present
BC – MC
interpersonal
inconsistent
MP – MC
institutional
consistent
Note: BC: broad content goals, LS: lesson-specific content goals, MP: mathematical practice
goals, MC: establishing a mathematical community goals

Table 13
Mrs. Stewart’s Dilemmas and Orientations to Professional Obligations
Dilemma
Professional obligation
Was the professional obligation consistent or
emphasized in decision
inconsistent with teacher’s primary orientation?
LS – BC
no dilemma present
no dilemma present
LS – MP
disciplinary
consistent
LS – MC
disciplinary
consistent
BC – MP
disciplinary
consistent
BC – MC
disciplinary
consistent
MP – MC
individual
inconsistent
Note: BC: broad content goals, LS: lesson-specific content goals, MP: mathematical practice
goals, MC: establishing a mathematical community goals
Subfinding 3
The third subfinding of this study was that teachers encountered several challenges that
spanned across teachers and dilemma types. Although these challenges were identified more or
less in the literature base provided earlier, this dissertation found that these challenges did not
depend on a teacher’s primary orientation to their professional obligations nor on their particular
instructional goals. These three challenges for facilitating whole-class discussions included (a)
considering appropriate scaffolding, (b) handling student responses, and (c) managing time
constraints. In this section, I provide greater detail on each of these three challenges. Taken
together, these three aspects of facilitating discussions provide insight into the various ways that
teachers (with different primary orientations) made decisions and thereby provided different

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opportunities for their students when these challenges presented themselves in whole-class
discussions.
Considering Appropriate Scaffolding
The first challenge that all three teachers encountered focused on how to appropriately
scaffold the discussion. Indeed, learning to talk and listen for the purpose of understanding
mathematics can be an unnatural experience for many students. Many students are unwilling to
share their thinking with their classmates; and for those who are willing, understanding how to
make sense of another student’s ideas that differ from one’s own ideas can be difficult.
All three teachers encountered various situations of how to appropriately scaffold the
discussion. In one case this dealt with instructional resources. Mrs. Stewart considered what type
of handout to use in her fourth dilemma (broad content goals vs. mathematical practice goals).
She chose to support easier communication between students by providing graph paper already
made with scales and axes. Teachers likely encounter this type of dilemma frequently, knowing
that there are certain skills students should do proficiently (e.g., knowing how to determine the
scale and axes on a graph given a set of data). Yet these skills might go undeveloped exactly for
the reason that was viewed through Mrs. Stewart’s choice: that there was a more compelling
immediate goal that took precedence over the opportunity for students to develop a long-term
skill.
Another type of scaffolding challenge took place as teachers considered the structural
aspects of whole-class discussions – that is, how one goes about supporting student talk through
various organizational structures. Mr. Sandberg encountered this while he considered broad
content goals and mathematical community goals (Dilemmas 2 and 5). He wrestled with the
benefits and consequences of using group work prior to a discussion. Mrs. Stewart’s “quiet”

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class posed a dilemma (Dilemma 6) of how best to call on students to adequately engage them in
mathematical practice while still supporting the development of a mathematical community. She
also encountered this challenge (Dilemma 2) when students became perplexed with a problem
and showed little ability to start it. Ms. Mitchell encountered this dilemma (Dilemma 6) when
she grew dissatisfied with the discussion being filtered through her role as the teacher. She
changed the talk structure to be more student-driven, yet ran into issues when students began
talking negatively to another classmate.
In all these cases, the teachers considered how to most appropriately scaffold the learning
to take place in a discussion. It is not surprising that all of these dilemmas involved mathematical
community goals or mathematical practice goals (or both). This shows that while teachers are
considering the mathematical content to be learned during a discussion, they are sensitive to
helping students learn to talk and listen to others, whether this is done through group work prior
to the discussion, through how they call on students, or through the talk structure of the
discussion itself.
These scaffolding challenges highlight the fact that these teachers felt it necessary to
support students prior to and during whole-class discussions independent of their goals and
primary orientation to their professional obligations. This being said, all three teachers made
choices that focused on their interpersonal obligation by supporting students in developing
mathematical practice and the community aspects of the classroom. Scaffolding the discussion
was a necessary component to enacting whole-class discussions.
Handling Student Responses
Another important component to facilitating whole-class discussions is how teachers
make decisions when students respond during the discussion. Unlike more traditional math

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classes where the classroom discourse may involve large sections of teacher lecture with
minimal student talk (i.e., possibly just one- or two-word responses), facilitating whole-class
discussions requires the teacher to listen to students and use what they are saying in some
productive way to move their agenda forward. This has the potential to create problems, such as
when a student poses an interesting question that goes beyond the scope of the lesson, when a
student states an incorrect answer, or when a student simply forgets what he is going to say. This
challenge occurred across all three teachers, and across several dilemma types, with no particular
pattern in the types of goals across dilemmas.
All three teachers encountered situations where they made decisions when students posed
an interesting question. In Ms. Mitchell’s class (Dilemma 1) a student considered the nonlinearity of another student’s graph where the graph was not even a function. In Mr. Sandberg’s
class (Dilemma 1), a student posed a question that the class would eventually be looking at
during a lesson on compound probability. Mrs. Stewart encountered an interesting question as a
student wondered whether a probability could be represented using non-whole numbers
(Dilemma 3). All three teachers made decisions that were consistent with their primary
orientation to the professional obligations, which presented different opportunities for students to
engage in mathematics.
During Ms. Mitchell’s discussions she encountered situations where students provided
incorrect answers. In one situation (Dilemma 3), a student gave an incorrect answer during a
compound probability lesson looking at the product of the values on two dice. In a different
situation (Dilemma 5), the class failed to come to consensus on which graphs were linear. What
is noteworthy about this set of data is that only Ms. Mitchell encountered dilemmas where
students provided incorrect answers. This raises the question as to whether Ms. Mitchell’s

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primary orientation to the interpersonal or individual obligation might have supported
uncovering misconceptions and incorrect ideas in contrast to the other two teachers. A competing
hypothesis might be that Ms. Mitchell’s lack of experience with the curriculum and general focus
on mathematical practice could also have contributed to the unmasking of student errors.
Finally, Mr. Sandberg encountered a situation (Dilemma 6) where a student simply forgot
what he was going to say. Mr. Sandberg quickly moved on with the lesson, but not before
considering the implications for moving on without adequate wait time. This highlighted both his
tendency to maintain a consistent discussion structure while trying to make sure the student felt
comfortable with participating in the future. This situation and the others described in this
section all highlight that when teachers open up the discourse space of the classroom, they need
to make decisions as students pose interesting questions, share incorrect answers, or simply
forget what they are going to say. In all these situations, teachers made decisions that impacted
the nature of how students learned mathematics in the classroom, in many ways consistent with
the teachers’ primary orientations to their professional obligations.
Managing Time Constraints
The third type of challenge that teachers encountered focused on how teachers managed
time constraints. Mrs. Stewart (Dilemma 1), Ms. Mitchell (Dilemma 2), and Mr. Sandberg
(Dilemma 3) all dealt with how to manage time constraints while attending to their instructional
goals and their professional obligations. Mr. Sandberg and Mrs. Stewart encountered this
challenge towards the end of lessons when they were running short on time. Ms. Mitchell
encountered this challenge as she considered what content she needed to cover during the final
weeks of the school year. In all three of these situations, one of the types of goals was lesson-

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specific content. All three teachers made decisions consistent with their primary orientation to
their professional obligations.
This challenge highlights the importance of considering decision-making with the very
real and constraint-filled dynamics of teachers in their classrooms. These challenges might not
have been observed had this dissertation been focused on depictions of other people’s
classrooms, or if the decision-making had only focused on content-focused reasons. Indeed, the
very real nature of these teachers’ decision-making was necessarily embedded in their daily
interaction with their students with all of the institutional constraints that go along with that
interaction, such as managing time both within and across lessons.
Results Summary
All three teachers had instructional goals of the four types analyzed, and all teachers had
at least five of the six theoretical permutations of conflict between pairs of goal types. In some
cases, teachers acknowledged a dilemma and had no resolution for it. In many other cases they
acted to resolve the dilemma. As teachers encountered instructional dilemmas while planning for
and enacting whole-class discussions, their decisions could be determined (for the most part) as a
function of their professional obligations. However, each teacher made at least one decision that
deviated from their primary orientation. Even though the three teachers had different primary
orientations to their professional obligations, they all encountered challenges of how to handle
student responses, how to appropriately scaffold the discussion, and how to manage time
constraints. In the next chapter, I discuss the importance of these results and their value for
research and practice. All teachers encountered various challenges to implementing discussions
that, for the most part, were independent of the types of dilemmas. For example, all three

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teachers considered how to appropriately scaffold the discussion in a productive way. In the next
chapter I discuss the relevance and implications of the findings of this dissertation.

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CHAPTER 6
DISCUSSION AND CONCLUSION
For decades now, researchers, practitioners, and organizations have advocated for the
potential benefit of using whole-class discussions in teaching mathematics, yet this vision has not
been fully implemented. This raises the question as to why this is. In this dissertation, I analyzed
the decision-making of three experienced middle grades teachers to better understand the nature
of dilemmas they encountered while planning for and enacting whole-class discussions.
Summary of Findings
The main finding of this study was that all three teachers’ decision-making was explained
via their primary professional obligation – with important limitations. This is important because
the opportunities for students to learn mathematics in their classrooms are determined in large
part by the decisions that teachers make. These decisions are influenced by their orientations to
their professional obligations. Along with this main finding, there were also three important
subfindings.
First, all three teachers’ decision-making were mostly explained via their primary
orientation to their professional obligations. In the vast majority of dilemmas, teachers made
decisions which were consistent with their primary orientation to their professional obligations.
This subfinding may not be surprising as one might have anticipated before this study took place
that a teacher’s orientations to the professional obligations would explain their decision-making,
but the extent to which this was the case was insightful in understanding the relationship between
decision-making and the multiple potential factors (e.g., teachers’ professional obligations) that
contributed to their decision-making.

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Second, the three teachers in the study, at times, made decisions which were inconsistent
from their primary orientation to their professional obligations (Subfinding 2). Each teacher
encountered one dilemma where they actually deviated from their primary orientation. This
deviation is noteworthy because it highlights the complexity with which teachers manage their
goals and obligations, and because it also shows the limitations to how a teacher manages their
professional obligations in the classroom. It also illuminates that while one of the obligations can
generally explain how a teacher might make decisions in general, it cannot fully capture the
complexity with which teachers may make decisions in their situated experience of classroom
instruction. In general, these deviations might occur because atypical situations occur in the
classroom (e.g. the class is interrupted multiple times) which may lead a teacher to atypically
prioritize one of their obligations over another. More specifically, these deviations may occur for
multiple reasons. In the next few paragraphs, I describe these inconsistencies and provide
potential reasons why they may have occurred.
Ms. Mitchell was generally oriented to the interpersonal and individual obligations. Yet
when she encountered a dilemma between lesson-specific content goals and mathematical
practice goals (Dilemma 2), she made an instructional decision more consistent with the
institutional obligation. In this scenario, Ms. Mitchell considered how she could incorporate
mathematical practice given the content coverage expectations at the end of the year. This
decision may have been attributed to (a) Ms. Mitchell’s inexperience in teaching the curriculum
ensuring that she met departmental expectations, and (b) as someone experienced in teaching
within the district recognizing the importance of making sure her students have similar content
exposure to their peers in other classes. Ms. Mitchell was in many ways an experienced novice in
teaching the curriculum, having taught several years, but not at this specific grade level. Thus,

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we see in her teaching a confluence of her experience and inexperience in her decision-making.
Overall, Ms. Mitchell’s dilemma showed the importance of considering institutional constraints
while studying decision-making; in this case, they became too great, and Ms. Mitchell found it
necessary to cover what was expected instead of focusing more on mathematical practice.
Mr. Sandberg was generally oriented to the institutional obligation. He encountered a
dilemma between a broad content goal and establishing a mathematical community (Dilemma 5).
In this scenario, Mr. Sandberg was considering how group work was used to support students in
the whole-class discussion. Mr. Sandberg felt it was necessary to have students talk in small
groups prior to the whole-class discussion, which favored the interpersonal obligation. Even
though Mr. Sandberg generally made decisions consistent with maintaining an orderly classroom
and discussion flow, in this case, the potential disorder of group work was seen as necessary for
students to be able to think through their ideas prior to a discussion.
Mrs. Stewart was generally oriented to the disciplinary obligation. She encountered a
dilemma between mathematical practice goals and establishing a mathematical community goals
(Dilemma 6). Her dilemma presented itself with how best to call on students, given a very quiet
class. Her unwillingness to put students on the spot through random selection was consistent
with supporting the individual obligation, making sure students are feeling supported in the
classroom environment.
Both Mr. Sandberg and Mrs. Stewart primarily oriented themselves to obligations that
were not the individual or interpersonal obligations, yet they made decisions aligned with those
orientations. These decisions speak to the importance of establishing a mathematical community
where students felt comfortable and capable in discussing their mathematical thinking. In both
cases, the teachers may have made these decisions because they felt it was necessary to support

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the students more directly in supporting them learn mathematical content via whole-class
discussions. One could not expect students to share their thinking openly in the class (especially
at the middle school level) without feeling at least minimally supported to engage in the public
discourse space. While the teacher may try to establish some norms explicitly, many other
messages can be sent to students implicitly – based on how a teacher responds to questions or
specific decisions they make in the class. Both teachers made decisions that supported
developing a classroom atmosphere that would encourage students to this public discourse space.
In contrast, Ms. Mitchell primarily made decisions consistent with the interpersonal and
individual obligations. Although her students appeared comfortable in talking with Ms. Mitchell
and each other, she was not able to avoid dilemmas, which caused her to deviate from her
primary orientation. In the dilemma described above, the institutional demands became too great
and she chose to deviate from her typical student-centered decision-making approach. Thus, one
conclusion from this subfinding is that attention to social norms for interacting during wholeclass discussions was a necessary but not sufficient condition for ensuring that students were able
to engage in mathematical thinking during whole-class discussions.
Third, even though the teachers were experienced, used instructional resources (e.g.,
written curriculum materials) that supported implementing whole-class discussions, and were
generally oriented philosophically with the importance of using discussions, this was not
sufficient to avoid challenges one might expect to see while facilitating whole-class discussions.
The types of challenges these teachers encountered were, by-and-large, independent of the
teachers’ primary orientations to their professional obligations (Subfinding 3). One might have
anticipated prior to this study that particular challenges would be more prevalent given a
particular orientation to an obligation, or to a particular type of goal (e.g., that managing time

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constraints might be less challenging for someone primarily attending to their institutional
obligation or lesson-specific content goals). Yet all three teachers encountered situations where
they had to consider appropriate scaffolding, how to handle student responses, and how to
manage time constraints. Put together, these findings point to how students’ opportunities to
learn mathematics -- and what it means for them to be thinkers and doers of mathematics -- were
impacted to some degree by their teachers’ decisions related to whole-class discussions. In the
next sections, I describe the implications of these findings for research and practice.
Discussion
Prevailing models of mathematics teachers’ decision-making (e.g., Schoenfeld, 2010)
have focused strong attention on teachers’ content-based decisions and how they impact the flow
of discussion. In this dissertation, I found that teachers make decisions for a multitude of reasons,
but those reasons can be consistently described via the professional obligations (with some
limitations) situated in the authentic experience of their classrooms. By understanding teachers’
primary orientations to their professional obligations, one may be able to explain why certain
behavior is occurring (or not) during instruction. This may be helpful for supporting teacher
learning both pre-service and in professional development. This dissertation contributes to our
understanding of teacher decision-making through the investigation of three different (with
respect to their primary orientations to their professional obligations) teachers in the highly
complex and situated environment of whole-class discussions.
This dissertation reinforced others’ perspectives in the field that the teacher’s role in
facilitating discussions is centrally important for moving along mathematical content and
motivating students (Chapin, O’Connor, & Anderson, 2009; Lampert, 2001). Similar to
Lampert’s (2001) explication of her teaching practice using whole-class discussions, the teachers

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in this study encountered many pedagogical demands simultaneously, such as keeping the
discussion on track, engaging non-verbal students, deciding which students to call on, and
choosing when to move on from an idea and when to press further.
As teachers facilitate discussions, the dilemmas they encounter and the decisions they
make send implicit messages to students about what is valued as thinkers and doers of
mathematics. These messages are important because as Jansen (2006, 2008) pointed out, students
participate in discussions for various reasons, yet student participation is a necessary component
of teaching using whole-class discussions. All three teachers in this study encountered dilemmas
that required them to consider the importance of establishing a mathematical community as a
necessary prerequisite to accomplishing other content-specific goals. Although some research
(e.g., Sleep, 2012) has noted how non-mathematical aspects of instruction could potentially
detract from accomplishing mathematical goals, in this study all three teachers found it necessary
to support students in what Williams and Baxter (1996) referred to as analytic (scaffolding
students’ engagement in mathematical ideas) and social scaffolding (scaffolding students’
expectations regarding norms of social behavior and talk).
Perhaps more importantly, Skott (2004) pointed out in reference to reform-minded
instruction that there are “expected classroom practices and learning outcomes…but there is no
set of well-defined methods for the teacher to carry out and only vague hints as to what kind of
practice a certain situation may require” (p. 239). While the last decade has seen support grow
(e.g., Smith and Stein’s 5 Practices) for this discourse-oriented approach to teaching, there is still
much work to be done. The findings of this dissertation point out the largely unavoidable
challenges that teachers are likely to encounter while planning for and enacting whole-class
discussions. Filling the void that Skott pointed out may include methods for how best to support

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teachers in understanding the benefits and limitations in handling student responses, scaffolding
the discussion, and managing time constraints while planning for and enacting whole-class
discussions. Another well-defined method might include helping teachers understand their
tendencies when making decisions during instructional dilemmas in favoring a particular
professional obligation and goal type. Supporting teachers in making their practice explicit to
themselves might result in both better teaching practice and better support for novice teachers
should they be mentored by these teachers.
Implications for Research and Practice
This study informs research on whole-class discussions and teacher decision-making.
Although the results of this study are limited by the focus on teacher decisions related to wholeclass discussions (and thus not the entire lesson cycle), there are some notable implications from
this study for research. In relation to whole-class discussions, the findings of this study indicate
that there were several challenges teachers encountered that were independent of their
orientations to their obligations and their goals. Studying teachers as they navigate discussions in
the realness of their classroom practice provided the opportunity to understand instructional
dilemmas and the challenges they encountered. Research on whole-class discussions has yet to
consider how teachers’ various orientations to their obligations influence the learning
experiences of their students in their classrooms. This dissertation found that the ways teachers
scaffolded the discussion, structured the discussion, handled student responses, and managed
time constraints were all generally explained via their orientations to their professional
obligations. This study provides a first step in developing a theory of how teachers meet the
demands of the challenges of facilitating whole-class discussions as influenced by their
orientations to their professional obligations.

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In relation to teacher decision-making, this dissertation highlighted the importance of
teachers making decisions as necessarily embedded in the institution of the school with students
they interact with daily. Prevailing models of teacher decision-making (e.g., Schoenfeld, 2010)
have focused on the cognitive aspects of teachers’ content-focused decisions. Current decisionmaking research and support for facilitating whole class-discussions ignore this aspect of
teaching: that teaching is fundamentally an interaction between human beings in a social space
over long periods of time, which therefore impacts the ways in which people interact.
Facilitating collaborative discussions is complex, even when experienced teachers are
generally oriented to teaching using this method of instruction, they have classroom resources
(e.g., curriculum materials) that support this type of instruction, and they are generally
knowledgeable about this teaching approach. Yet if teachers are going to effectively implement
collaborative discussions, the question then is how does one learn to do this?
For decades now, the NCTM and others have advocated for the use of whole-class
discussions. Improvement has happened no doubt, but many teachers still teach in a directinstruction way with limited student talk, or talk that is not consistent with the vision of
discourse-oriented instruction. I argue that what is necessary to support this vision is a better
understanding of the real constraints, dilemmas, challenges, and concerns of real teachers in the
complex space of teaching mathematics in today’s classrooms. This dissertation highlighted
these characteristics of practice. Although these three teachers are not representative of the
population of mathematics teachers across the United States, their stories provide some insight as
to why supporting teachers in teaching this way is so challenging while suggesting potential
pathways to move the field forward.

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Before describing the implications of the results in terms of practice, I remind the reader
that the purpose of this study was not in any way to measure effectiveness of implementation, or
to make claims about how one teacher “got it right” and the others were somehow deficient. This
perspective would be unproductive for understanding how different teachers make sense of their
role in facilitating discussions. What this study did illuminate was that teachers’ decisions could
be understood via the types of dilemmas they encountered and their orientations to the
professional obligations. Given any set of teaching practices that one would consider
“productive” in supporting students to more deeply understand mathematics via a discussion,
these teachers provide some insight as to the potential obstacles and challenges one might expect
to see.
As one example, an outside observer may see the mathematical importance of pressing
students to justify a mathematical claim they make during the discussion. Assuming this fails to
happen, a teacher educator or professional development coach might wonder why this was.
While a simple explanation is that the teacher missed the opportunity, this study illuminates
several alternative reasons. Perhaps the teacher felt the student would be unable to make a
justification, and they are considering the implications for that individual student’s development
(attending to the individual obligation). Perhaps the teacher wanted to leave the question open
and they plan on circling back to the question later (attending to some broad-content goal).
Perhaps the teacher felt pressed for time and unable to pursue this idea further (trying to manage
the challenge of time constraints). Indeed, a teacher’s decision not to press students during a
lesson (even if they are convinced it is useful during planning) could be influenced by their
goals, obligations, or one of the challenges described earlier in this chapter. By considering these
other possibilities, and by trying to uncover the root of the decision, the field may be better

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positioned to support teachers in making effective decisions during whole-class discussions via
understanding the reasons why they acted the way they did.
By better understanding teachers’ these underlying reasons, teacher educators and
professional development providers may be able to counteract any potential mismatch that
happens as the teacher is learning in a professional setting. This mismatch may occur when a
teacher is oriented primarily one way (e.g., to the institutional obligation) and the PD
provider/teacher educator is primarily oriented another (e.g., to the disciplinary obligation). One
could imagine a setting where a PD provider is describing the benefits of justification to teachers,
while the teachers are considering when they would have time to incorporate this into their
classroom. By understanding these different orientations, PD providers and teacher educators
may be better positioned to effectively change teachers’ practice and possibly orientations to
their practice.
As one such specific example of how these results might be useful on a large scale,
consider Smith and Stein’s (2011) Five Practices for Orchestrating Productive Discussions.
These practices include anticipating student strategies, monitoring students as they work in
groups on a task, and finally selecting, sequencing, and connecting student strategies throughout
the discussion. Taken together, these practices can support novice teachers in reducing the
cognitive load of a lesson by shifting much of the thinking to the planning phases. However,
these strategies are designed to support teachers in addressing their disciplinary (contentfocused) obligations. For example, selecting and sequencing practices could involve having
students consider multiple representations (e.g., table, graph, symbolic) and how one might order
certain strategies to create a coherent storyline, or whether to select an underdeveloped strategy
for presentation compared to a more abstract and generalizable one.

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The results in this study highlight the different types of instructional goals that teachers
had, and the inherent challenges that go along with facilitating such discussions. Thus, the
choices teachers make when implementing the 5 Practices (Smith & Stein, 2011) likely go
beyond just addressing content-focused goals. For instance, a teacher’s selecting and sequencing
practices could be influenced by a teacher’s orientations to the interpersonal and individual
obligation, affecting who they called on and in what order.
Future Research
In terms of future research, it is still an open question as to the extent to which teacher
decision-making during whole class-discussions is different or not based on a teacher’s
experience. The teachers in this study were all experienced teachers who used instructional
resources (i.e., written curriculum materials) that generally supported whole-class discussions,
and who were generally aligned philosophically with this approach to instruction. This raises the
question, however, about how this study might have been different had it included novice
teachers. As one example, when Ms. Mitchell considered how to deal with meeting the
expectations of content coverage during the final few weeks of the school year, she deviated
from her primary orientation to the interpersonal obligation and instead made a decision
consistent with her institutional obligation. As novice teachers are learning their role and
professional identity in the classroom, they too might bow to institutional expectations.

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APPENDICES

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APPENDIX A

Iterations of Consistent Patterns of Instruction Coding
First Pass through the Classroom Observation Data
As a first pass through the data, I watched 1 set of videos from the three teachers with the
goal of determining a general sense of their decision-making during whole-class discussions. To
do this, I first watched the set of videos that were independent of the other teachers: Mrs. Stewart
- Round 1, Ms. Mitchell – Round 3, Mr. Sandberg Round 2; that is, there was no overlap
between any of the three teachers in these particular lessons. Although this means that the first
pass through the data is not in chronological order (except for Mrs. Stewart), it is assumed that
due to the years of experience of each teacher their decision-making processes did not evolve
drastically from one set of observations to the next. Doing this also ensured that three videos
each would be watched for this first set. Attention was only given to segments of the video that
were considered “whole-class discussion.” To help mitigate issues of analytic contamination
across teachers, the lessons were watched in chronological order (lesson before focus lesson,
focus lesson, lesson after focus lesson) but rotated across teachers. Given below is the order I
analyzed this first set of videos:
1. Mrs. Stewart – Round 1, Lesson 1 of 3 – What Do You Expect (CMP3) Problem 2.1
2. Mr. Sandberg – Round 2, Lesson 1 of 3 – What Do You Expect (CMP3) Problem 1.2
3. Ms. Mitchell – Round 3, Lesson 1 of 3 – Data Distributions (CMP2) Problem 1.1
4. Mr. Sandberg – Round 2, Lesson 2 of 3 – What Do You Expect (CMP3) Problem 1.3
5. Ms. Mitchell – Round 3, Lesson 2 of 3 – Data Distributions (CMP2) Problem 2.1
6. Mrs. Stewart – Round 1, Lesson 2 of 3 – What Do You Expect (CMP3) Problem 2.2

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7. Ms. Mitchell – Round 3, Lesson 3 of 3 – Data Distributions (CMP2) Problem 4.4
8. Mrs. Stewart – Round 1, Lesson 3 of 3 – What Do You Expect (CMP3) Problem 2.3
9. Mr. Sandberg – Round 2, Lesson 3 of 3 – What Do You Expect (CMP3) Problem 1.4
Pacing and Lesson Structures
A consistent feature across the 3 teachers was that each used an instructional
technique/consistent pattern of behavior to keep the flow of the discussion going (i.e., there was
a productive “rhythm” to the lesson flow). These consistent patterns of behaviors differed in their
form but served similar purposes – to keep the lesson moving while allowing students’ thinking
about the mathematics to be made visible for the whole class.
In Mr. Sandberg’s classroom, there were four consistent major lesson segments. When
students arrived to class they typically worked on “bell work,” which consisted of problems
related to the previous day’s lesson. The problems were typically formative in nature. Problems
often targeted common mistakes or misconceptions students would make related to the previous
day’s Problem. After students had an opportunity to work independently on these problems, the
second lesson segment was a class discussion of the bell work. The next lesson segment was
more focused on direct instruction that launched the problem. In contrast to Mrs. Stewart, Mr.
Sandberg typically worked through the first part (i.e., “Part A”) of the Problem with the class.
The final segment of the class involved students working on the task in groups. In contrast to
both Mrs. Stewart and Ms. Mitchell, Mr. Sandberg infrequently summarized the day’s lesson
during the same period. Instead, the summarizing of main ideas usually occurred during the next
day’s discussion of the bell work. Whole-class discussions almost exclusively occurred during
the second lesson segment. Occasionally, a brief summary occurred at the end of class, but these
discussions were generally shorter.

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One main advantage to Mr. Sandberg’s lesson structure was that it provided more
processing time between identification of key issues and understandings and when these ideas
were discussed in class. This approach also provided opportunities to share key insights across
all classes investigating the same problem. Additionally, by waiting until the next day to discuss
the previous day’s work, the early discussion in the class served as a way to create coherence
across lessons. The main limitation was that the discussion generally focused on problems from
the bell work, which tended to be more procedural in nature. While Mr. Sandberg employed
different techniques to get a variety of students involved in the discussion, the discussion format
was generally in the scope of IRE or brief response to a procedural type problem. While Mr.
Sandberg was able to get many students involved in the discussion, the involvement seemed to
be more typically at a low level of conceptual engagement.
The class observed for Ms. Mitchell was the final class of the day. Ms. Mitchell
mentioned how different this class was in terms of its makeup of students (more racial/ethnic and
SES diversity and issues related to “fitting in” with school culture) and in relation to students’
ability to stay on task – perhaps due to the class being the final class of the day. Ms. Mitchell was
also in her first year teaching middle grades, which when combined with other factors may have
contributed to her daily lesson structure being more varied (i.e., it appeared somewhat
“experimental” at times) than Mr. Sandberg’s and Mrs. Stewart’s. This being said, discussions
typically took part during two well-defined lesson segments, at the beginning of the lesson –
either to wrap up or review the previous day’s lesson -- and towards the end of the lesson to
summarize key ideas from the lesson’s exploration.
In contrast to the other two teachers, Ms. Mitchell tended to rely on a small group of
students to keep the conversation going across lessons. This core group of students engaged in

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conversations mediated through the teacher, as well as with each other in the whole-class setting.
This is not to say that students were excluded from the conversation; rather, she capitalized on
strong students who were willing to participate and take on a leadership role for the collective
learning. This small group of students showed evidence of willingness to share their
underdeveloped thinking with the class, and to defend their reasoning. She was generally
supportive of students’ thinking, whether this was by explicitly stating she valued their thinking
or by using instructional moves that demonstrated this (e.g., using wait time while students were
processing). This general orientation toward valuing thinking and relying on a small group of
students as “discussion leaders” proved beneficial in maintaining the flow of whole-class
discussions across lessons.
Whereas Ms. Mitchell and Mr. Sandberg both had more well defined lesson segments,
many of Mrs. Stewart’s lessons had discussions occurring throughout the class period. Students
would work on parts of a problem individually or in small groups, and the class would discuss
these portions. There usually was a “summary” portion of each lesson that occurred at the end of
the lesson, but this differed only minimally from the discussions occurring throughout in terms
of the types of student-to-student interaction. Often there were times when student responses
seemed to occur simply to keep the conversation going in the class with minimal comment from
Mrs. Stewart. For example, multiple students shared their strategy for solving a particular
problem, and the teacher response was simply “Okay” before moving on the next student.
There was generally a quick pace to both her questions and what she was trying to
accomplish during the lesson, perhaps due to wanting to get through as many subparts of each
problem as possible – something that seemed less pressing for both Ms. Mitchell and Mr.
Sandberg. There were cases where Mrs. Stewart would cut students off mid-thought and fill in

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their thinking for them. This being said, there were instances of Mrs. Stewart slowing down her
pace, especially during key conceptual parts of the lesson. At these times, her comments were
longer, questions were more specific and poignant, and the discussion seemed to be more a
conversation and less of a rhetorical interchange. For example, during these points of the lesson
Mrs. Stewart would ask for clarification by asking, “What do you mean by…?” and re-stating a
student’s strategy or approach to thinking about the problem. Similar to Mr. Sandberg’s class,
there were many students who spoke during discussions; however, in contrast to Mr. Sandberg’s
class, the contributions tended to vary between procedural and conceptual aspects. Given the
consistent discussion in her classroom, the variable pacing of discussion allowed Mrs. Stewart to
move through many parts of the lesson, but this may have contributed to masking the big idea of
the lesson.
Put together, the three sets of lesson videos (one from each teacher) were useful in
providing a general sense of the nature of the discussions, but more importantly they highlighted
the individual characteristics within and across the teachers’ discussions. This data provided the
foundation for developing a coding scheme that aimed to more precisely describe the nature of
teachers’ interactions and inevitably their decision-making. The next few sections describe how
this process of developing a coding scheme was responsive to both the data and the research
literature pertinent to decision-making as embedded in classroom discussions.
Second Pass through the Classroom Observation Data
As a reminder, the first pass through the data was focused on all three teachers. The
second pass of the data focused just on Mrs. Stewart’s videos because they were the longest
discussions and seemed to be more structurally varied both in terms of showing up in different
parts of the lesson as well as in having intermingled group work with whole-class work. The

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process of determining consistent patterns of behaviors for Mrs. Stewart was a primarily
grounded process, although many categories were consistent with research on classroom
discussions, consistent pattern of behaviors, and teacher decision-making. The process of
creating a coding scheme was iterative; first constructing a coding scheme through watching
observed lessons, refining the scheme, watching more lessons, and so on until the point of
theoretical saturation occurred. The goal of developing a reliable coding scheme was for the
researcher (me) to develop a complete coding scheme by the end of the 2nd round of observations
so that reliability coding could be determined in the 3rd round by a second coder. This process of
creating the coding scheme is described in greater detail in the next few paragraphs.
To start the second pass, I watched all (n = 9) of Mrs. Stewart’s videos consecutively,
noting consistent patterns of behaviors across the dimensions given in the bulleted list below. A
short list was generated at the beginning, but as more aspects were found they were added to the
list. As I progressed from one lesson’s video to the next, I noted which consistent patterns of
behaviors seemed to change from the previous lessons and which consistent patterns of
behaviors seemed to be maintained. I only watched the observation videos for this second pass
through the data (i.e., not looking at interview or lesson plan data) because the focus was on
macro-level consistent patterns of behaviors at the lesson level. The following is a list of
categories that were determined.


Pacing



How discussion was started



How discussion ended



The number of discussion segments



Individual versus group participation in the discussion

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

Role of content, specifically how do students show or explain their mathematical thinking
and use mathematical resources?



Explicit attention to establishing or maintaining norms



Role of group talk in the discussion



Pressing students for justification



Connecting back to previous content



Supporting students being careful about language



Re-voicing students’ work
Once these types of consistent patterns of behaviors were noted, the coding scheme was

refined in order to code the first round (3 lessons) of Mrs. Stewart’s observations. Coding
included the following:


Pacing
o How interruptions were handled
o Did the teacher quickly move on to other student comments quickly (with a
possible press to the original student)?
o Did the teacher get through all subparts of the problem?
o Was the teacher overly patient in fully listening to a student?



Question and Student Participation
o Did the teacher call on a volunteer?
o Did the teacher call on a non-volunteer?
o Did the teacher probe the student response with questions?



Teacher sense-making of Student talk
o Did the teacher clarify what the student meant, including revoicing?

133

o Did the teacher press for justification?


Role of content
o Did the teacher describe conceptual ideas?
o Did the teacher clarify vocabulary?
o How do students explain or show their thinking? (e.g., in their seats)



Norms
o What type of explicit talk of norms took place in the classroom?



Group Talk
o When does group talk occur during the lesson?
o What tasks are students working on in groups?
o Possible intention of group work



How does the discussion start?
o Did the discussion start by a volunteer?
o Did the discussion start by a non-volunteer (e.g., directly after group work or by
teacher monitoring)?



How does the discussion end?



Other
o Notes about what was happening during the interaction in the classroom around
the mathematical content
o Questions and comments about coding
o What part of the lesson was being taught (e.g., Launch 2.1 or Summary 2.2)
o Possible deviations of consistent pattern of behaviors
o Influences on disciplinary obligations

134

After coding these items, clusters of instances appeared around the types of activity that
the teacher was engaged in. The following items were generated primarily from the data, but as a
second check, the literature base was reviewed around classroom discussions, teacher decisionmaking, and instructional consistent pattern of behaviors. From the perspective of professional
obligations, teacher decisions are a function of at least one of the following: disciplinary
(content), individual, interpersonal, institutional. Because decision-making is inherently a
cognitive process, one way to study decision-making is to look for activities that may indicate
that decisions had been made. (Note: Another way is to interview individuals about their
decision-making, which also took place). From the situative perspective (Greeno, 1989),
studying decision-making as it occurs during the lesson is important; thus, it is necessary to
determine the types of teacher activities that likely correspond to teacher decisions. In many
cases decisions are a result of consistent patterns of behaviors; in other cases, teachers make
carefully thought-out, conscious decisions during a discussion. The purpose of this revised
coding scheme was to more effectively describe teacher activities related to decision-making and
to better operationalize how these activities might relate to the influence of professional
obligations (Herbst & Chazan, 2011) on a teacher’s decision-making process.

135

Activity Codes
Student Activity Decisions
How does the teacher start the discussion?
General description
How does the teacher end the discussion?
General description
Teacher chooses to change the structure of student interaction
Types: Small group, individual, whole class
Teacher gives students a task to work on
Ex: A problem or question to work on in small groups or individually
Type of Student Interaction Decisions
Is the student the teacher calls on during the discussion a volunteer or non-volunteer?
Is student discussion filtered through the teacher or student-to-student?
When does group talk occur during the discussion?
Types: before, during, after
How does group work support the discussion (i.e., possible intention for group work)?
Ex: Less student interaction than typical; teacher asks students to discuss in
groups.
How do students discuss their thinking in a whole class?
Types: Physical location (in seats, at board)
Resources: e.g., Using their own work on document camera, re-creating work at
board
When does the teacher use wait time?
Attending to interaction norms
Ex: Teacher explicitly points out how to interact, talk, learn or think in the
classroom.
Pacing
Types: Short answer - moving on quickly, giving students time to think
Teacher Talk Moves (These could be in the form of a teacher statement, a question, or a prompt
for students to engage in this type of activity.)
(Pimm, 1987; Sfard 2008; Chapin, O’Connor, & Anderson, 2009)
Re-voicing, (i.e., closely restating what a student said)
Clarifying / Expanding (restating what a student said with the same intended meaning,
but said differently)
Pre-voicing (i.e., cutting a student off to describe their thinking for them)
Questioning
Connecting back
Ex: Describing how an idea connects to previous work
Connecting forward
Ex: Describing how an idea connects to future work
Connecting across
Ex: Describing how two student strategies are similar or different
Connecting beyond

136

Ex: Describing how mathematics connects to contexts outside of the mathematics
classroom
Pressing for justification (could be a “how” or a “why”)
Simple response
This may be used to maintain the flow of the lesson. One example might be in
having students share their answers with the class quickly.
Talking about vocabulary
Providing an explanation
Evaluating a student response
Ex: “I liked that you said…”
With thi new set of codes, I coded the third lesson of the first set of observations and
created a description of Mrs. Stewart’s consistent pattern of behaviors to be later refined in
subsequent lessons. The goal was to continue using this iterative process; however, it was
determined that the construct of what exactly is meant by the term consistent pattern of behavior
was sufficiently problematic, as identifying deviations from a consistent pattern of behavior
necessitates defining the existence of a consistent pattern of behavior to begin with.

Table 14
Coding Scheme for Different Patterns of Instructional Behavior
Type of Pattern
Description
Guiding Questions to Identify Patterns
Patterns
in
this
category
include
student
 How are disruptions typically
Management
discipline routines, norms for interaction,
and how the teacher handles disruptions.

managed by the teacher?

Instructional
Support
Resources

Patterns in this category are connected to
Leinhardt et al.’s (1987) support routines,
with attention to the types of materials and
technologies that students and teachers use to
support their interactions.



Activity Structure

Yinger (1979) identified 7 dimensions of
instructional activity. For this category, the
first four will be the focus as the others are
described in other categories. These are:
1. Location
2. Structure and sequence
3. Duration (how long the activity
lasts)











137

What technologies does the teacher
incorporate to support instruction?
What instructional materials does the
teacher use to support instruction?
How does the teacher start the
discussion?
How does the teacher end the
discussion?
What is the task students are working
on?
Duration of activities
How are students grouped when they
work on the task (independent, size of
groups)?
Transition points when the teacher
chooses to change the structure of

Table 14 (cont’d)
4.
5.
6.
7.

Interaction

Participants (specifically how they
are grouped)
Acceptable student behavior
Instructional moves
Content and materials

This category is based off Leinhardt et al.’s
(1987) exchange routine, and Yinger’s
(1979) instructional routines. It describes the
ways in which teachers and students interact
with each other. It includes the way in which
teachers and students ask questions and
communicate with each other in the
classroom.

student interaction (e.g., move from
small group to whole class)

Specific Codes:
 Is the student the teacher calls on
during the discussion a volunteer or
non-volunteer?
 When does the teacher use wait time
(Question they asked, possible
purposes, duration)?
 What is going on when the teacher
has the class move to group work
(e.g., after Launching, during the
discussion to clarify or further
investigate, etc.)?
General description
 Pacing
 Is student discussion filtered through
the teacher or student-to-student?
 How does group work support the
discussion (i.e., possible intention for
group work)?
 How do students discuss their
thinking in a whole class (i.e., small
group, partners, individually; where
they present their work)?
Teacher Talk Moves (These could be in the
form of a teacher statement, a question, or a
prompt for students to engage in this type of
activity.)
(Pimm, 1987; Sfard 2008; Chapin, O’Connor,
& Anderson, 2009)
Restating moves:
 RR: Re-voicing, (i.e.,
closely restating what a
student said)
 RC: Clarifying / Expanding
(restating what a student
said with the same intended
meaning, but said
differently)
 RP: Pre-voicing (i.e.,
cutting a student off to
describe their thinking for
them)

138

Table 14 (cont’d)
Connecting moves:
 CK: Connecting back (e.g.,
describing how an idea
connects to previous work)
 CF: Connecting forward
(e.g., describing how an
idea connects to future
work)
 CA: Connecting across
(e.g., describing how two
student strategies are
similar or different)
 CY: Connecting beyond
(e.g., describing how
mathematics connects to
contexts outside of the
mathematics classroom )
Questioning
 QG: Asks any type of
question
 QJ: Pressing for
justification (could be a
“how” or a “why”)
General Discussion Moves
 GS: Simple response to
answer
 This may be used
to maintain the
flow of the lesson.
Ex: have students
share their answers
with the class
quickly.
o GV: Talking about
vocabulary
o GX: Providing an
explanation
o GV: Evaluating a student
response (e.g.,“I liked that
you said…”)

Meta-Routine

This is connected to Yinger’s (1979)
executive planning routine. Meta-routines
are when a teacher makes explicit the routine
during instruction for students in order to
reshape, reinforce, or break the routine.

139



The teacher explicitly refers to a
routine or norm for the purpose of
changing students’ actions in
relation to that routine (e.g., teacher
explicitly points out how to interact,
talk, learn, or think.)

APPENDIX B

Issues with Recruiting a Second School Site
At the same time that I was preparing for data collection at Site 1, I was trying to arrange
a second site for the study. After several unanswered e-mails and non-returned phone calls, I
simply showed up one day to talk to the principal at potential site 2A; but unfortunately, he was
out of the building that day. I stopped in a few days later to talk with him and instead ended up
talking to the assistant principal, which was quite productive. This school had three seventhgrade teachers. One was excluded as a potential participant because I knew the teacher through
previous encounters during professional development. One teacher was quite active in CMP
work and had been teaching the curriculum for a while. Due to other research obligations in
which she was already taking part, she declined participation. The third teacher did not respond
to inquiries via e-mail.
Potential Site 2B had used CMP previously, but after communicating with the principal
via e-mail, I learned that the school was not currently using the program.
For Potential Site 2C, I contacted the department chair via e-mail after e-mailing the
principal. They very quickly decided not to be included as a school site.
Potential Site 2D originally seemed promising. This site had a principal who was highly
supportive of CMP and university research. There were two seventh grade teachers at this site,
one of whom only taught grade 7 while the other taught one section of advanced seventh grade
students. I met with the main seventh grade teacher, and although she was initially quite excited
about participating in the study, she decided to not participate. It seemed like the two main issues
were (1) being video recorded, (2) not being an ideal classroom, and (3) hosting a student-

140

teacher in the spring. In regard to (1), I addressed this by telling her that if she did not want the
classroom recordings used in future presentations I would ensure this would be the case. I found
issue (2) to be the most interesting. She told me that the other teacher would be better to study
because her students were more advanced. As with all the other teachers I talked with, the main
thing I told this teacher was that my goal was to learn as much as I could from teachers in
“typical” settings, and to study how these teachers go about thinking through the challenges that
are presented to them. The study is not about evaluating the quality of instruction, but rather to
help paint the picture (especially to PSTs) of the complexity of classroom interaction and to
support new teachers in how to process this complexity. It seemed that the combination of the
three factors were enough for the teacher to decline participation.
Potential Site 2E was chosen because they have four grade seven teachers, and three of
them teach multiple subjects. Additionally, the entire department was new to teaching CMP, so
many of them could be considered novices with the curriculum though they all had at least seven
years of teaching experience. I met with the principal shortly before Winter Break. Teachers
were contacted shortly after they returned from break in January. While this site seemed ideal, I
was not able to visit any of the teachers before receiving an e-mail from the principal that
indicated none of the teachers were interested in participating. During the meeting with the
principal, I found out that all four of the grade 7 math teachers had at least five years of
experience, which was mostly due to drastic budget cuts that the district had faced in the last few
years. This may have contributed to a lack of willingness to do extra work given the pressures
that these teachers most certainly face.
Potential Site 2F was a small, rural school district. The middle school has two grade 7
teachers, and both teachers co-teach with special education teachers for certain periods of the

141

day. Again, I contacted the principal and had a face-to-face meeting. The principal was on-board
with the study, appreciated having interest from the university in conducting a research study,
and as a former middle school math teacher who had contact with people at the university she
seemed very optimistic about including the school as a site. I met with both grade 7 teachers to
talk about the study. Neither teacher had taken part in a research study before. After a few days,
they let me know via e-mail that they would not be able to commit to the study given other
professional development obligations that were currently taking place in the district.
Potential Site 2G was contacted multiple times via e-mail about the possibility of being
used as a school site, but the principal never replied. During the second exchange, I offered to
meet with an assistant principal or department chair as a first point of contact. Again, there was
no reply. Given the experience at potential site 2A, I decided to move on to other districts.
In order to broaden the pool of potential CMP schools, I contacted a math specialist who
is the coordinator of professional development at the middle school level in mathematics at a
large Intermediate School District on the east side of the state of Michigan. She was quite helpful
and sent along many contact people and potential school sites. Potential Site 2H was contacted
because they currently have two grade 7 teachers. The principal forwarded me the teachers’
contact information. After a few e-mails, both teachers decided not to participate.
The principal at Potential Site 2I talked to two of his teachers about the study. One was
interested; one was not. Because the study requires two teachers at each site, I informed the
principal that I would let him know if that requirement changed in the future, but that at this
point I would not be interested in using that school as a potential school site.

142

The math coach for Potential Site 2J was contacted, and she checked with the appropriate
people to see if their school could be a site for the study. The teachers she talked to in the
building were not interested in receiving any information or being potential research participants.
The math coach for Potential Site 2K met with the curriculum director to see if this
district could be used as a school site, but I was unable to reconnect with the people at this school
site.

143

APPENDIX C

Initial Interview Questions
Table 15
Initial Interview Questions
Question
1. Describe your experience with
implementing CMP materials.

Follow-up Questions

Notes

How long have you used these materials?

The purpose of this question is to gauge
the types of curricular experiences the
teacher has had in enacting the
curriculum, learning about the
curriculum, and having a sense of how
the materials are connected within and
beyond a specific grade-level.

What grade levels have you taught?
Have you had experiences with attending
or presenting professional development
specifically focused on the curriculum
materials?
2. Describe a typical lesson in your
classroom.

In a typical lesson, what percent of class
time is devoted to teacher-to-student
interaction? …student-to-student
interaction? …student work time?
…going through homework?

The purpose of this question is to better
understand to what extent teachers
typically make use of classroom
discussions in their instruction. This
first question is purposefully open so
that the teachers are able to talk about
their instruction. If they do not mention
interaction, the follow-up question will
hopefully address that issue.

3. Do you think it’s important that
students are active in their math class?

If no, have the teacher talk more about
that position.
If yes…describe some ways you think
students should be active as learners of
mathematics.

This question will hopefully provide
some insight to the teacher’s
orientations toward student talk and
how they might use it in class.

4. What are some challenges that you
encounter when trying to facilitate
classroom discussions?

What are some important constraints for
your planning of discussions? What
dilemmas do you typically encounter
when you are enacting or planning to
have a discussion?

This question is intended to have the
teacher talk about factors that contribute
to enacting discussion in their class.
Some of these factors act as external
constraints, and others relate to how the
teacher thinks about the costs and
benefits of their choices.

5. What do you think you have
improved on during your teaching
career in facilitating classroom
discussions?

For example, has your ability to
implement or your beliefs about the role
of discussions changed over your career?

This question may provide some insight
into the differences between the experts
and novices in terms of how their
practice or beliefs have changed over
time.

6. Are there resources, structures, or
routines that you find helpful for
supporting implementing discussions?

If the teacher is having a difficult time
with this, mention some examples, such
as the use of technology (e.g., document
camera, Smart board, etc.), grouping
strategies, how they structure the lessons,
etc.

This question provides some context for
the classroom environment, specifically
what resources the teacher is making
use of to support the implementation of
discussions.

144

APPENDIX D

Lesson Planning Document
Teachers will be submitting lesson plans before each lesson. Listed below is the initial and
revised set of questions used in the lessons.
Original:

Problem 1.3 – What Do You Expect
Lesson Goals/Objectives:

General flow of lesson (best estimate of major pieces of the lesson, e.g., homework collection,
review of yesterday’s work, Launching the problem, Exploration time, Summarizing, etc.)



What do you expect to happen in the whole-class discussion?
o Are there ideas/strategies you anticipate showing up that are productive?

o Are there ideas/strategies you anticipate showing up that will be less productive?

o Are there misconceptions you anticipate showing up?

o Are there particular examples/solutions/strategies you want to make sure show up in
the discussion?

o Are there any non-mathematical aspects of the discussion you are preparing for (e.g.,
participation, management, etc.)?

o Is there anything atypical about how you might structure the discussion (e.g., a new
grouping strategy)? If yes, please list.

o Is there anything else about the discussion you are anticipating occurring that will be
important for students interacting in the discussion?

145

APPENDIX E

Revised Lesson Plan

Lesson Overview and Agenda
Problem 1.2 – What Do You Expect
What do you hope students understand/be able to do as a result of this lesson?

What ideas/strategies do you anticipate showing up in the lesson? Are any of these particularly
problematic?

Are there particular examples/solutions/strategies you want to make sure show up in the
discussion?

Is there anything else about the discussion you are anticipating occurring that will be important
for students interacting in the discussion (e.g., non-mathematical aspects such as participation,
management issues, grouping issues)?

146

APPENDIX F

Examples of Descriptions for Teachers’ Classrooms (Teacher A)














Classroom arrangement
o Students are working in small groups of approximately 4 students.
Managing disruptions
o There were very few if any noticeable disruptions in the class.
Typical flow of lesson
o Describe the problem to students as a whole class
o Ask students for clarifying questions
o Give students time to work on the problem in small groups, actively talk with
groups of students about the problem in their groups
o Whole-class discussion
Teacher’s Positioning in the classroom
o During “Launch,” teacher is in the front of the room.
o During “Explore,” teacher is actively talking with students in their groups sitting
down with them.
o During the whole-class discussion, teacher is on the outside of the room,
sometimes in the front, but mostly behind students near the wall.
Typical flow of the discussion
o Have students introduce their strategy to the class, have them talk about why they
solved the problem that way. Sometimes, students are addressing the class from
their seats; other times they are in the front of the class showing their solution
strategy. Many students share their strategies for solving the problem, while the
teacher actively connects the strategies for the students.
Types of Question being asked by the teacher during the discussion (e.g., Boston, 2014;
Boaler and Brodie, 2004)
o Teacher asks lots of clarifying and connecting questions/comments, such as
 “Why did your group solve it this way?”
 “If you look at what this group did, they solved it this way, which is
similar to how this other group solved it.”
Student Talk during the discussion
o The discussion seems very open and respectful. Students listen to the person who
“has the floor,” and if other students have questions for the speaker, the students
can ask the speaker directly instead of asking the teacher. This shows some
evidence that students have a certain level of authority in the classroom in relation
to learning the content collaboratively.
Types of student responses shared during the discussion (Boston, 2014)
o Teacher seemed to have a strong sense of the different types of representations
that she wanted to have come out during the discussion. She asked particular
groups to share their strategies.

147

APPENDIX G

Examples of Descriptions for Teachers’ Classrooms (Teacher B)















Classroom arrangement
o Students are working in small groups of approximately 4 students.
Managing disruptions
o Redirecting students individually using gestures and language, e.g., saying, “Hold on…” and
putting a hand up.
o Reminder to class to give students at the front of the room their attention while they are
presenting.
Typical flow of lesson
o Describe the problem to students and give them a chance to talk with each other about some of the
big ideas in small groups.
o Reconvene as a whole group, occasionally ask students to come to the board to work on a problem
from a previous day’s lesson that will connect to the current lesson’s work.
o Describe the problem for the whole class and have the students start working.
o Give students time to work on the problem in small groups, actively talk with groups of students
about the problem in their groups.
o Whole-class discussion
Teacher’s Positioning in the classroom
o During “Launch,” teacher is in the front of the room. A student gives their solution strategy to the
class, teacher recounts the strategy by asking straightforward questions to the class, such as, “what
fraction does this diagram represent?”
o During “Explore,” teacher is actively talking with students in their groups sitting down with them.
o During the whole-class discussion, teacher is on the outside of the room, and frequently moved to
the front to summarize students’ arguments for the class.
Typical flow of the discussion
o Variety of strategies are shown to the class to help structure the discussion
o Student presents their solution to the class
o Questions are asked by students to the presenter
o Teacher summarizes the presenter’s mathematics in the front of the class
o Next group presents
Types of Questions being asked by the teacher during the discussion (e.g., Boston, 2014; Boaler and
Brodie, 2004)
o Ask whole class if they agree with a student’s solution strategy before they sit down. The teacher
frequently uses the question, “Does everyone agree with this?”
o Teacher presses students for justification. When a claim is made, she asks, “Why?” to cue them to
explain more.
o Lots of recalling questions about what strategies had already been solved, e.g., “How did we get to
the value here in the denominator?”
Student Talk during the discussion
o The discussion seems more teacher directed; however, there were multiple times when students
had things they noticed, such as patterns, where they raised their hand and after being called on
shared their thinking with the class.
o Lots of redirecting the students while speaking to talk to their peers
o Clarifying questions were asked by students to other students during whole class discussion.
Description of the types of student responses shared during the discussion (Boston, 2014)
o Teacher ordered the responses in a way to connect one idea to another, although many of the
responses seemed quite similar.

148

APPENDIX H

Examples of Key Moments in the Lesson













BREAKING THE FLOW OF THE LESSON
When a student asks an unanticipated question, which might be a key insight or
something mostly unrelated to the problem being discussed
When there is at least five seconds of silence before or after the teacher talks
When the teacher talk moves beyond re-voicing/clarifying
o E.g., the teacher is being more directive about a point, they are “planting” a point.
o E.g., the teacher makes a connection to previous student work/conversations
When the flow of the discussion is student-to-student for a period of exchanges, but then
the teacher eventually joins back in the conversation after multiple turns; the reason why
the teacher inserted herself at that point in the conversation.
MAINTAINING THE FLOW OF THE LESSON
Why did the teacher choose particular students to share their thinking with the class…
o During the Launch of the problem to clarify what was going to be solved in the
problem?
o During the Summary portion of the lesson, where certain examples of student
work were put on display for the class to see at the onset of the discussion. What
did the set of different strategies afford?
When the teacher decides to summarize the lesson why was it done at that point?
o What ideas/strategies to talk about during the summary?
o What ideas/strategies that didn’t show up in the discussion to put into the
summary?
The teacher follows up a students’ response by asking a similar question.
The teacher clarifies a students’ response in an atypical way. For example, a teacher
typically lets students talk through their explanations, but for a response they go up to the
board and show the class the students’ thinking.
Press for justification, asking students to explain their reasoning (when this is a normative
expectation).

Key Moment Type 2: Talk that explicitly references the discussion or to a decision that
must be made before moving on.
Examples of these include:
 “I think we need to decide what to do with the equation before we move on…”
 When a student talks to the “wrong audience,” for example, the student is describing a
solution strategy and they talk to the teacher, and the teacher tells them, “You need to talk
to the class; they are the ones you are trying to convince.”

149

APPENDIX I

Classroom Observation Protocol
The purpose of this protocol is to capture key moments that occur during the discussion.
The chart will be filled out initially during the classroom observation, and will be added to after
watching the videos of the lesson. As many of the key moments as possible will be analyzed
through the video, through follow-up discussions with the teacher (including after lesson
discussions, post-lesson reflections, and the video-stimulated interview). As a first pass, the last
four columns focusing on the four professional obligations will act as a running hypothesis of
why each decision was made. For example, interpersonal decisions might include aspects of
communication and participation; individual decisions would include aspects pertaining to
student motivation; institutional decisions are related to classroom management and homework;
and disciplinary decisions include content related decisions pertaining to lesson goals, the use of
academic language, and developing productive dispositions as learners of mathematics. The next
page shows an example chart of the Classroom Observation Protocol.

150

Time / Part of
Lesson

Teacher
Action

Teacher
Talk

Student
Action

5:03 Launch

Describing
the task for
students

Listening,
asking
clarifying
questions

12:14 Explore

Monitoring
work in
small groups

17:25 Summary /
Whole Class
Discussion

Redirecting
class after
multiple
students off
task

17:54 Summary /
Whole Class
Discussion

Teacher asks
Susan to
share her
strategy

“Today we
are going to
focus on
parts A and
B”
“So how do
you know
this always
works?”
“I need to
make sure
that
everyone is
paying
attention.”
“Susan can
you show
your
strategy?”
“Think
about Joe
and Susan’s
strategies”

47:00
Asks class to
Summary/Whole- think about
Class Discussion difference
across
solutions

Key
Moment:
(Type #)
Description

Professional
Obligations
Attending to

Listening to
presenters

(2)
Redirecting
class’s
attention

Institutional:
Teacher
acted to
redirect focus
of the class.

Susan
presenting
solution

(1)

Listening

(1) Having
students
look across
strategies

Individual:
Susan was
eager to
volunteer her
solution with
the class.
Disciplinary:
Having
students look
across
strategies

Working on
task in small
group

Institutional:
2 minutes
left in class
tie
Individual:
Validation of
student
solution
strategies
Figure 13. Classroom Observation Protocol

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APPENDIX J

Combined Observation Document:

Figure 14. Combined Observation Document

152

APPENDIX K

Teacher Reflections on the Lesson
Listed below are some examples of prompts that teachers will be asked to answer after each
lesson.


If you taught the lesson again, would there be anything you would have done differently?



If you taught the lesson again, would you change anything in the discussion?



Were there questions you would have liked to ask, or student ideas you would have liked
to pursue but did not get a chance to?
o What were some of those questions/ideas?
o What were some possible reasons why you were not able to pursue them as you
had wished?



How do you feel about how the lesson set up the next day’s lesson?



How do you feel about the goals of the lesson in relation to how the lesson played out?



Were there moments in the discussion that highlighted something important to you about
student learning?



Were there moments in the discussion that highlighted conflict in which direction to go
with the discussion?

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APPENDIX L

Potential Questions to Ask the Teacher Immediately After the Lesson
1. I noticed you called on [NAME] first in the discussion. Why did you think that student
might be a good person to start off the discussion?
2. At the beginning of the lesson I saw that you put up groups 1, 4, and 5’s work? Was there
any reason for why you chose those three groups?
3. This student asked a really interesting question, and you said that the class was going to
look into that later on in the unit. Were you thinking about pursuing his thinking further
during this class?
a. Alternative: Did you hear any interesting questions during the discussion today?
4. I noticed that there was a time in the discussion where you listened to students go backand-forth with each other. Can you talk a little about when you decide to jump back into
the conversation?
5. What helps you determine where you position yourself during the discussion?
6. I noticed that you asked [NAME] quite a few follow-up questions during the discussion.
Do you have a sense of the types (and how many) questions you feel comfortable with
asking [NAME]?

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APPENDIX M

Video-Stimulated Interview
Listed below are some examples of the types of questions that will be asked during the videostimulated interview. Clips will be selected for reviewing, and then I will ask the teacher to talk
about their thinking at that moment during the lesson.
Example Questions:
 Did you have a sense of how the discussion was going to unfold before the lesson?
o How well did it follow your vision?


During this day’s discussion, the first student to talk was Franklin. Was there any reason
why you wanted to have Franklin start the discussion?



Before Sally’s description of how she solved the problem with the table, Joe’s group used
a graph. Did the ordering come up more naturally as part of the discussion, or was there
anything purposeful about having Sally’s group present before Joe’s?



I noticed at this point in the lesson, a student asks a question, and there’s a long pause
before asking the student to re-state their thinking. Were there other paths you were
considering at this point?



At the end of the discussion, you inserted a point about solving the problem using an
equation. Was there any reason why you decided to put that in the discussion even though
none of the students had really talked about that?

155

APPENDIX N

Mrs. Stewart’s Consistent Patterns of Instruction
Although there were many aspects of Mrs. Stewart’s instruction observed across the
lessons, only a few rose to the threshold of being sufficiently consistent (i.e., they were present in
either all, or all but one, lessons analyzed) to be deemed a consistent pattern of instructional
behavior. These three patterns are (a) maintaining a rapid pace of instruction, (b) emphasizing
communication and process goals, and (c) implementing fluid interaction structures. I provide
greater detail of these three patterns in this section.
Pattern 1: Maintaining a Rapid Pace of Instruction
The first consistent pattern of instruction for Mrs. Stewart was how she maintained a
rapid pace of instruction across lessons and was categorized as an interaction pattern.
Maintaining a rapid pace of instruction may have been attributed to a need to get through as
much of the lesson as possible within the class time. In many cases, Mrs. Stewart was able to get
through the entire lesson in order to learn as much as possible from the piloting of the CMP3
materials. Although this rapid pace limited the potential for disruptions and management issues,
the consistent pattern of behavior was primarily about how students interacted with Mrs. Stewart
during the class discussion. Most discussions could be described as a somewhat rapid exchange
between the teacher and students individually, with not much idle time in between speakers.
During each discussion, Mrs. Stewart maintained a pace that supported a high volume of
questions, statements, and responses by students. In a typical 30 minutes of discussion time in
class, Mrs. Stewart asked about 83 questions, or nearly 3 per minute. These consisted of
questions that probed students to justify or clarify their thinking, or for students to share their

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work or solution to a problem. On average, Mrs. Stewart made about 1 statement, question, or
response to students every 10 seconds.
Wait time was occasionally present during discussions. In some cases, wait time appeared
intentional, waiting for students to respond to a particular question or providing space in the
discussion for students to think about a question. For example, in multiple instances, Mrs.
Stewart asked, “Tell me what questions you have on…” about the particular topic they were
studying before moving on to something slightly new. In other cases, the wait time seemed less
intentional, as they occurred as a result of a student question.
With the exception of one lesson, Mrs. Stewart never had to refocus the class more than
once. In some cases, this involved simply waiting and then continuing on with work at the board.
Additionally, there was only one instance where Mrs. Stewart explicitly reminded a student to be
on task during the discussion, which involved a student not taking notes when she asked the class
to do so.
Pattern 2: Emphasizing Communication and Process Goals
This second pattern of instructional behavior was categorized as a type of interaction
pattern. A main feature of this orientation involved having students make claims and provide
evidence for their claims. Throughout the lesson observations, Mrs. Stewart consistently pressed
students to explain their thinking. At the surface, this could be noted by the amount of time and
opportunities for students to share their thinking with other students. Mrs. Stewart described her
own orientation towards focusing on communication and process goals during a video-stimulated
interview.
[W]hat seventh graders will explain versus write out is completely different. And they’re
way better verbal. And I want to make sure in terms of our math practice standards, I

157

don’t want ‘em to be able to just write out and be precise with their details but I want ‘em
to be precise verbally too. (Mrs. Stewart, video-stimulated interview, May 7, 2015)
In an average class period (with discussion being 30 minutes of that time), she averaged
11 questions asking students to justify their thinking (e.g., “Why is this the case?”) and 11
questions for students to clarify their thinking (e.g., “What do you mean by that?”). One reason
for this orientation was described by Mrs. Stewart in the initial interview:
[E]ven in your classwork you have to show me how you got there because when you
raise your hand and tell me it’s thirty, and I say okay, how did you get it? It like takes you
five minutes to figure it back out and it’s not going to serve you any purpose when you
leave this room, like you’re not going to be able to tell what you did, you’re not going to
be able to reproduce it, see how it applies to something else...” (Mrs. Stewart, initial
interview, November 25, 2014)
During the initial interview Mrs. Stewart shared, “[I]deally I would love a kid to ask a
question even if they’re looking at me and have four kids trying to answer that question, and me
never having to initiate that.” During the observations, there were no cases of two students
speaking consecutively, having one student build their response directly off another’s thinking in
the whole-class space. Since students reported their work individually and not as a group this
also limited the opportunity for students to build directly off each others’ work without first
being mediated by the teacher. In an average 30-minute section of discussion (the average
amount of time spent per lesson), there were approximately 31 restatements of student thinking,
or about one per minute. Of those, 21 were a clarification, expansion, or exact restatement of a
student’s idea; and 10 were simply a short restatement such as a student’s answer to a question.

158

While Mrs. Stewart devoted a large portion of class time to whole-class discussions, they were
largely a product of students talking through the teacher instead of to each other.
Pattern 3: Implementing Fluid Interaction Structures
Throughout her class discussions, she often prompted students to talk with the students in
their groups as a way to scaffold the discussion. An overwhelming majority of class time
involved students talking about their mathematical work either in small groups or whole-class.
Mrs. Stewart fluidly moved between group talk and whole-class discussion, sometimes planned,
other times based off of what was occurring at the moment in the discussion. In almost all
instances, students shared their thinking and work from their desks individually (i.e., there was
no “group reporting” or “group voice”), even when their work was displayed for the class via the
document camera. Only one exception occurred to students staying in their seats to address the
whole class, when one student came up to the board to draw a visual representation of an area
model. These fluid shifts between small group work and whole-class discussion differed greatly
from the other teachers’ interaction structures and provide the third and final consistent pattern in
Mrs. Stewart’s instruction.

159

APPENDIX O

Ms. Mitchell’s Consistent Patterns of Instruction
Of the few consistent patterns of instruction that were observed in Ms. Mitchell’s class,
two stood out for how consistently they occurred. The two consistent patterns of instruction were
(a) emphasizing students’ thinking (interaction pattern), and (b) using teacher mediated
discussion structure to facilitate whole-class discussions (interaction pattern). These patterns are
discussed in further detail below.
A main emphasis of Ms. Mitchell’s teaching approach is on the interaction aspect of the
classroom. One of her goals for classroom interaction was “that they become a learning
community and they listen to each other.” The use of partner work was tied to the Standard for
Mathematical Practice #3 (NGA-CCSSO, 2010), generating and critiquing the reasoning of
others. She felt that students should have a voice in the classroom and that she should protect
students’ think time to let them process their ideas. In her view, the discipline of mathematics
was one that was mostly generated by human beings as a creative endeavor. She was the only
teacher to share her broad view of mathematics as a field. This belief, along with wanting her
classroom to avoid “social reproduction,” created an environment that, on the one hand, ensured
students’ voices were heard and their thinking was honored, and, on the other hand, often created
multiple disruptions.
Consistent Pattern 1: Emphasizing Students’ Thinking
A good example of this propensity to emphasize student thinking was evident in an
interview question about a student’s rather long response during a lesson observation. In such
situations, a teacher could make multiple decisions, based on (1) how long to let the student go

160

with their response, and (2) what to do after the response. The decision for (1) can indicate
whether a teacher is worried about management issues (i.e., other students getting restless), the
individual as a creator of mathematical knowledge, or whether the description accurately
describes the mathematical idea. When asked about this type of situation -- “Do you have a sense
of how long do you wait for a student to complete their thoughts before you move on?” -- Ms.
Mitchell responded,
I try to stay as long as I can…because I’m often surprised…I think I would be very
frustrated if I was trying to say something and a teacher cut me off, even if I were wrong.
Because then how do I know what I think and how do people respond to me if I even had
my voice heard. (Ms. Mitchell, video-stimulated interview, May 11, 2015)
A main feature of Ms. Mitchell’s teaching was what she referred to as “honoring
students’ thinking.” This involved many different aspects of her teaching practice. For example,
Ms. Mitchell consistently used wait time after students described their thinking to the class to
ensure that students had space to complete their thoughts, and when asking a question to the
class she also consistently provided wait time. When disruptions did occur during the discussion,
Ms. Mitchell would remind students of the expected norms that were typically focused on
listening to each other’s thinking. This consistent pattern to allow students space to share their
thinking by “honoring thinking” was Ms. Mitchell’s first consistent pattern of instruction.
Consistent Pattern 2: Teacher Mediated Discussion Structure
During whole-class discussions, students were typically seated at their desks describing
their thinking to the whole class. During work time, students often worked with a partner, as she
mentioned the difficulties of having the students work in larger groups during the last period of
the day.

161

More so than the other teachers, Ms. Mitchell made attempts to make connections
between the content they were learning and other sources. In an average class period, Ms.
Mitchell made 2.6 connecting statements. These connections ranged from statements connecting
back to previous knowledge, to those connecting forward (e.g., what they would be expected to
do in high school), to those connecting their work to other classes’ work. In one lesson in
particular, Ms. Mitchell made multiple references to outside contexts. When trying to describe
the meaning of a data cluster, she talked about why someone owning a hat store would care
about data clusters.
At various times, students were invited to describe their work in front of the class
individually. Ms. Mitchell freely allowed her students the opportunity to share their thinking at
their desks or at the front of the room. A vast majority of class discussion was mediated by the
teacher; that is, after a student talked Ms. Mitchell would either re-state the student’s answer,
provide an explanation, ask a question, or evaluate the student’s response. More so than the other
teachers, Ms. Mitchell evaluated student responses (generally positively), occurring about two
and a half times per class period (18:12 of discussion). The interaction in the discussion typically
involved Ms. Mitchell asking questions students had not yet had a chance to think about,
students offering an explanation, and Ms. Mitchell either asking students to clarify their thinking
or herself providing an explanation for the class (Table 16). Although Ms. Mitchell emphasized
mathematical practice goals and generally attended to the individual and interpersonal
obligations, the vast majority of her discussions were mediated by her (i.e., the turns in talk were
teacher-student-teacher-student).

162

Table 16
Frequency of Questions in Ms. Mitchell’s Classroom Discussions
Frequency per class period
(18:12 of discussion time)

Question or Statement Type
“New” Question – students had not had the opportunity to
work on the specifics of the question.

7.6

Questions asking students to clarify their thinking
Questions asking students to justify their thinking
Questions asking students to report their work on a previously
solved problem

3.9
1.7
2.0

163

APPENDIX P

Mr. Sandberg’s Consistent Patterns of Instruction
Mr. Sandberg had two consistent patterns of instruction noted in the data. These two
patterns were (a) the consistent use of technology as a lesson resource (resource pattern), and (b)
implementing a consistent discussion structure in order to maintain classroom management
(interaction pattern). In regard to the first consistent pattern, he frequently used several types of
instructional technology -- from clickers to graphing calculators to spreadsheets -- to support
students in discussing their thinking during whole-class discussions. His second consistent
pattern focused on his consistent use of his lesson structure, which differed from the interaction
structures that both Ms. Mitchell and Mrs. Stewart incorporated.
Consistent Pattern 1: Consistent Use of Technology as a Lesson Resource
Besides the use of CMP3 curriculum materials, a consistent feature of Mr. Sandberg’s
lessons involved the use of various technologies to support the teaching and discussing of
mathematical content. Mr. Sandberg used technology in various ways to support specific lesson
goals. It was not used simply as a way to keep students engaged, but rather as a way to support
students’ thinking about mathematical problems. As one example, Mr. Sandberg used a
spreadsheet during a lesson on mean absolute deviation. When asked during the post-lesson
interview why he chose to use this technology, he stated that he had actually not used it during
his first period class, but “it took too long and I felt like I already gave them sufficient amount of
time to calculate it on their own…I wanted them to do it quicker so I just opened Excel and
threw the numbers in there real quick.” This was a good example of Mr. Sandberg’s strategic use

164

of technology, and his decision was influenced by both the need to efficiently use class time and
to help students attend to the mathematical idea, not just calculations.
Mr. Sandberg used his Smart Board daily, both as an interactive white-board and as a
display device from his computer and document camera. He used spreadsheets, an app-based
graphing calculator, instructional videos (how to make a box-and-whisker plot), a cartoon video
(presenting a scenario for students to think about), and clickers. Although the particular type of
technology was lesson-dependent based on the specific demands of a lesson, he consistently
implemented technology across several lessons. This consistent use of technology as a lesson
resource was Mr. Sandberg’s first identifiable consistent pattern of instruction.
Consistent Pattern 2: Consistent Use of Lesson Structure during Discussion
The interaction in whole-class discussions in Mr. Sandberg’s class typically involved
students sitting at their desks and individually responding to questions posed by Mr. Sandberg.
There were no instances of a student being asked to share their thinking at the board or to share
their thinking with their partner; all questions were answered by students in their seats. Long
sections of teacher explanation often dominated whole-class discussions. On average in every
two minutes of discussion Mr. Sandberg spoke three times, with student talk in between each of
his turns. There were no instances of one student following another student’s talk directly
without first being called on by Mr. Sandberg. Students were typically prompted to respond to
questions that provided short answers with little or no explanations required, such as collectively
filling out a table or making an ordered list.
In an average discussion segment (i.e., the coding was averaged to represent a 9-minute,
31-second segment), there were just under 2 questions asked for students to justify their thinking,
less than 1 question to have students clarify their thinking (with 80% of these occurring within

165

one lesson), and no instances of questions asking students to explain their work in how they
solved a problem. There were, however, over nine questions asking students to provide their
answer for the class, and eleven instances of Mr. Sandberg providing an explanation of the
mathematics for the class. There were only two questions across lessons that asked students to
make a prediction.
In many lessons, there were potential opportunities for Mr. Sandberg to press students’
thinking or justification, but often these situations involved simply getting student answers into
the whole-class space. For example, in the last observation of Round 3, students analyzed roller
coaster data (Figure 15) and were asked, “How do we find a fast speed?” Mr. Sandberg received
several responses from averaging the data, finding the fastest speed overall, to creating a boxand-whisker plot. As was typical, students responded but were not asked to support their
reasoning as to why they chose a particular strategy, or to explain possible benefits of their
strategy over those of their classmates.
In a different lesson, Mr. Sandberg asked students what they thought the longer whisker
represented in a box-and-whisker plot. After a student gave a short answer, Mr. Sandberg
followed by providing an explanation of what it represented. This again was typical of a possible
opportunity for a whole-class discussion that failed to transpire, and was instead replaced with
direct explanation by Mr. Sandberg.

166

Figure 15. Determining a “fast speed” comparing wood and steel-framed roller coasters
There were very few disruptions in Mr. Sandberg’s class. When disruptions did occur,
they were handled quickly and students remained on task. There were no instances of Mr.
Sandberg having to explicitly describe expectations to students, which should not be surprising
given the late nature of the lesson observations during the school year. After analyzing the lesson
videos and interview data, the main reason for the lack of disruptions likely was the result of two
aspects of his classroom management: first by solving problems through implementing
consistent patterns of instruction, and second by constraining the nature of student interaction
(especially in classroom discussions) to minimize the potential for disruptions.
Attention to managing interpersonal interactions and organizing work for a class of
students seemed to be a high priority for Mr. Sandberg. For example, when asked in the initial
interview about consistent patterns of instruction or resources that he finds productive or useful
for facilitating discussions, he talked about how he groups students by partners, and how he ends
class with having students put their calculators and desks back. What is notable of this response

167

is that he could have answered the question by talking about consistent patterns of instruction of
dialogue (e.g., how he structures classroom talk), or how resources could support discussion in
the class as opposed to more logistical aspects of classroom interaction.
In a similar light, during the last observation in Round 2, Mr. Sandberg walked around
and talked to students at the beginning of partner work time. When asked during the post-lesson
interview about monitoring students working on the problem -- “When you were walking
around…was there anything that stood out as unusual, or someone going in the wrong
direction?” -- he responded, “No, not at all. I was just getting them started…giving them their
own individual invitation to get pencil on paper.” In this scenario, Mr. Sandberg was monitoring
his student’s progress, but it was more focused on management aspects rather than attending to
individual students’ needs or helping students with understanding mathematical content.
It was also during partner work that a Smart-Board app was displayed on the screen that
kept track of how loud the class was working. If the noise-level increased over a certain
threshold a certain number of times per class, then the class had to stay after. If the class worked
quietly they earned a letter and were rewarded with a video at the end of the unit. This reward
and consequence system was implemented consistently across lessons. In one lesson Mr.
Sandberg was mid-conversation when the bell rang, and he let them go immediately. When
asked about this in the video-stimulated interview as to why he did not continue on with his
explanation he referred to the value of his tally-system and that the consequence of staying after
worked only if he kept them after class for being too noisy.
In addition, the nature of student interactions in the discussion also allowed for
maintaining a lesson flow that minimized the potential for disruptions. Not pursuing student
ideas deeply, or opening up the conversation significantly, allowed the lesson to progress mostly

168

as planned with minimal deviations. Having many students involved in providing short answers
allowed for accountability to make sure students were doing their work by possibly needing to
share their work with the class. Mr. Sandberg frequently used random calling on students via
drawing sticks with their names on them for students to provide short answers. The
predominance of Mr. Sandberg filtering the discussion through him meant that at any time any
student might be called on to talk in front of the class without support from a partner or group.
The second consistent pattern in Mr. Sandberg’s instructional was his systematic
approach to classroom management, related to maintaining a strong lesson flow and structure.
When asked about his classroom management, he shared that the location of his Smart Board,
the direction the class faces, and the location of the pencil sharpener amongst other things, were
all thoughtfully located to minimize disruptions. He also mentioned that he considered outside
distractions (e.g., eighth-grade students walking by the door heading to lunch) as a factor in
potentially changing his lesson structure to minimize potential interruptions.
Students appeared to have a strong sense of what was expected of their interaction in the
discussions as there were very few disruptions during this time. The nature of Mr. Sandberg’s
questions was such that wait time was rarely used (i.e., students either had a short response or did
not). When wait time was used for a students’ response, generally Mr. Sandberg waited a few
seconds and then he moved on to another student or to an explanation. Limiting student
opportunities to provide in-depth answers allowed for a consistent flow between teacher and
students, with little opportunity for deviations.

169

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