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LIBRARY
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University

 

 

 

This is to certify that the
dissertation entitled

LEARNING TO ANTICIPATE STUDENTS’

MATHEMATICAL RESPONSES IN TWO CONTEXTS:

THE CASE OF ONE PRESERVICE TEACHER IN A
UNIVERSITY AND SCHOOL SETTING

presented by

Sarah Elizabeth Kasten

has been accepted towards fulfillment
of the requirements for the

 

 

 

 

Doctoral degree in Mathematics Education
/
/ < / /
l / v 4' .4 ~_ A
ajor Professor ' ignature
/S ISL/LL Zwi
Date

MSU is an Affirmative Action/Equal Opportunity Employer

 

_—¢v -“ m—Dc

_._—._.¢——____.___--.—-._ ~

PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5/08 Kthrolecc8Pres/ClRC/DaleDue indd

 

LEARNING TO ANTICIPATE STUDENTS’ MATHEMATICAL RESPONSES IN
Two CONTEXTS: THE CASE OF ONE PRESERVICE TEACHER IN A
UNIVERSITY AND SCHOOL SETTING
By

Sarah Elizabeth Kasten

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Mathematics Education

2009

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ABSTRACT
LEARNING TO ANTICIPATE STUDENTS‘ MATHEMATICAL RESPONSES IN
TWO CONTEXTS: THE CASE OF ONE PRESERVICE TEACHER IN A
UNIVERSITY AND SCHOOL SETTING
By

Sarah Elizabeth Kasten

Field experiences have been shown to impact mathematics preservice teachers’
beliefs and perceptions about teaching and field experiences, but there is little research
evidence of what preservice teachers learn about teaching mathematics from these
experiences. This lack of research makes the mathematics field experiences a productive
sight for investigation. Further research Shows that mathematics preservice teachers
struggle to consider their students possible reactions to the lesson plans they create as part
of their participation in these experiences.

The purpose of this study was to trace the practices of anticipating students’
mathematical responses of one preservice teacher in the first semester of her field
experience and to examine the factors in the university and school contexts that supported
or constrained her participation in this practice. Using a case study design, lesson plans
and interview and observation data were collected and analyzed to present a picture of the
practices of one preservice teacher working in two contexts. Findings indicated that there
were multiple factors in the university context that promoted the preservice teacher in the
practice of anticipating students’ mathematical responses while lesson planning.
Conversely, there were multiple factors in the school context that constrained her
practices in this area. The findings also suggest that the preservice teacher in this study

struggled to see the relevance of her own preferred teaching and lesson planning

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witness to suppon the

practices, those which aligned with what was advocated by the university, in the lecture-
based school context in which she found herself. This study demonstrates the importance
of preservice teachers having and making use Of authority and resources in field

experiences to support them in enacting their own practices as beginning teachers.

lim grateful to M
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ACKNOWLEDGMENTS

I am grateful to so many of the faculty at Michigan State University for their
support, encouragement, and kindness over the last four years. I am especially grateful to
my committee, Mike Steele, Glenda Lappan, Ralph Putnam, and Sandra Crespo who
helped me make this dissertation a reality and also to Jon Star and Natasha Speer who
provided feedback and encouragement at the early stages. Also, I indebted to Betty
Phillips and Gail Burrill for all that I learned from them that contributed to this
dissertation.

I am so thankful for all of the graduate students I have had the opportunity to
work with over the last four years. I am especially grateful to the members of my writing
group, Lorraine Males and Aaron Brakoniecki, who read and gave feedback on my work
and were always willing to help me with both technical and theoretical issues.

I especially want to thank my parents for being constant models of hard working,
caring, and devoted teachers and scholars. Finally I want to thank my husband, Nick
Pearson, for his constant love, support, and encouragement. This work would never have

been possible without him.

iv

EST OF TABLES .......
LISI OF FIGIRES ......

CHAPTER l
[‘tTRUDI'CIION ........
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TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. ix
LIST OF FIGURES ............................................................................................................. x
CHAPTER 1
INTRODUCTION ............................................................................................................... l
1.1 The Role of Field Experiences in Teacher Education .......................................... 1
1.2 Studying the Role of Field Experiences from the Situative Perspective ............. 5
1.3 Teachers’ Knowledge of Students ....................................................................... 6
1.4 Anticipating Students Mathematical Responses as a Part of Lesson Planning... 7
1.5 Contributions to the Field .................................................................................... 9
1.6 Limitations ......................................................................................................... 10
1.7 Organization of the Document ........................................................................... 10
CHAPTER 2
THEORETICAL FRAMEWORK AND REVIEW OF THE LITERATURE .................. 12
2.1 Theoretical Framework ...................................................................................... 12
2.1.1 The Conceptual Themes of the Situative Perspective ............................... 13
2.1.2 The Relationship between the Situative Perspective and Other
Perspectives on Learning .......................................................................... 16
2.1.3 Examples of Research Employing the Situative Perspective .................... 19
2.1.3.1 A Study of Productive Disciplinary Engagement ......................... 19
2.1.3.2 A Study of SMPTS Learning ......................................................... 23
2.2 Review of the Literature .................................................................................... 25
2.2.1 Secondary Mathematics Preservice Teachers Learning to Teach
Mathematics from Field Experiences ....................................................... 25
2.2.2 Teacher Lesson Planning .......................................................................... 30
2.2.3 Knowledge for Teaching Mathematics ..................................................... 34
2.2.3.1 Shulman’s Framework of Teacher Knowledge ............................. 34
2.2.3.2 Studies Examining the PCK of Secondary Mathematics Preservice
Teachers ........................................................................................ 37
2.2.3.3 Grossman’s Model of Teacher Knowledge ................................... 42
2.2.3.4 Ball and Colleagues Model of Mathematical Knowledge for
Teaching ........................................................................................ 43
2.2.4 Anticipating Students’ Mathematical Responses as a Practice of Lesson
Planning .................................................................................................... 45
2.2.4.1 Anticipating Students Mathematical Responses as Part of a Model
for Facilitating Mathematical Discussion ..................................... 47
2.2.4.2 Cognitively Guided Instruction ..................................................... 52
2.2.4.3 Lesson Study .................................................. 54
2.3 Research Questions and Framework .................................................................. 57

[FATHER 3
I'EIIIOD .......................
3 3.1. Context ...........
3.I.I I'niwrxil)
3.I.I.I'I":1
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3.3 Instruments. Dd
3.3.1 Phase I. I:
3.3.: Phase 3. 1
3.3.3 PILN‘ 3. I
3.3.4 PIId\C\ 4, j
3.3.5 Phase 5, 8
3.3.6 Through .l
3.1. Data AIILII}\|\
3-“ Analyh U

.................

3.41.3 RC.
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3,4

3.4.
3.421(sz

 

 

CHAPTER 3

METHOD .......................................................................................................................... 63
3.1. Context .............................................................................................................. 64
3.1.1 University Context .................................................................................... 64
3.1.1.1 The Senior Year Methods Courses ................................................ 67
3.1.1.2 The Internship Year Methods Course ........................................... 72
3.1.2 The School Context: Logan High School ................................................. 73
3.2 Participants ......................................................................................................... 73
3.3 Instruments, Data, and Data Collection ............................................................. 75
3.3.1 Phase 1. Initial Interviews ......................................................................... 77
3.3.2 Phase 2. Lesson Planning Survey .............................................................. 78
3.3.3 Phase 3. First In-Depth Interview ............................................................. 79
3.3.4 Phases 4, 5, and 6. Lesson Observations and Interviews .......................... 80
3.3.5 Phase 5. Second In-Depth Interview ......................................................... 80
3.3.6 Throughout Fall Semester ......................................................................... 81
3.4 Data Analysis ..................................................................................................... 82
3.4.1 Analysis of Research Question 1: Describing the Practice of Anticipating
................................................................................................................... 82
3.4.1.1 Research Question 1 Data Reduction ............................................ 83
3.4.1.2 Research Question 1 Data Display ................................................ 85
3.4.1.3 Research Question 1 Conclusion Drawing ................................... 90
3.4.2 Analysis of Research Question 2: Factors of the Contexts ....................... 91
3.4.2.1 Describing the Contexts ................................................................ 91
3.4.2.1.] The University Context .................................................. 91
3.4.2.1.3 The School Context. ....................................................... 92
3.4.2.2 Explaining the Factors of the Context That Promoted Anticipating
....................................................................................................... 92
3.4.2.2.1 Research Question 2 Data Reduction. ............................ 93
3.4.2.2.2 Research Question 2 Data Display. ................................ 93
3.4.2.2.3 Research Question 2 Conclusion Drawing ..................... 94
3.5 Presentation of Findings ..................................................................................... 95
CHAPTER 4
THE CONTEXT ............................................................................................................... 96
4.1 The University Context ...................................................................................... 96
4.1.1 The University Methods Course ............................................................... 96
4.1.1.1 Exploring Proportional Reasoning in the University Methods
Course ............................................................................................ 98
4.1.1.2 Creating a Collaborative Teaching Community in the University
Methods Course ........................................................................... 100

4.1.1.3 Encouraging Deliberate Practices in Planning, Teaching, and
Reflecting on Mathematics Lessons in the University Methods

Course .......................................................................................... 104

4.1.2 The Interns .............................................................................................. 107
4.1.3 The University Field Instructor ............................................................... 109
4.2 The School Context .......................................................................................... 1 12

vi

4.2.1 Logan
4.2.2 The .\1
4.2.3 The F.
4.3 Summari/In

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4.2.1 Logan High School ................................................................................. l 12

4.2.2 The Mentor Teacher ................................................................................ 1 14
4.2.3 The Focus Class ...................................................................................... l 17
4.3 Summarizing the Contexts ............................................................................... l 19
CHAPTER 5
FINDINGS ...................................................................................................................... 120
5.1 The Nature of Anticipating .............................................................................. 120
5.1.1 In-Depth Interview 1 ............................................................................... 121
5.1.2 Observation 1 .......................................................................................... 124
5.1.3 Cool Stuff 1 ............................................................................................. 129
5.1.4 Observation 2 .......................................................................................... 136
5.1.5 Cool Stuff 2 ............................................................................................. 142
5.1.6 Observation 3 .......................................................................................... 145
5.1.7 ln-Depth Interview 2 ............................................................................... 150
5.1.8 Summary of Anticipation ........................................................................ 154
5.1.9 Mini-Cases .............................................................................................. 159
5.2 The Interaction between Context and Anticipating ......................................... 173
5.2.1 Zone of Proximal Development .............................................................. 174
5.2.1.1 Skills and Experiences Anticipating SMRS ................................ 176
5.2.1.2 Pedagogical Knowledge for Anticipating SMRS ........................ 178
5.2.1.3 General Pedagogical Beliefs ....................................................... 180
5.2.2 Zone of Free Movement .......................................................................... 181
5.2.2.1 The University ............................................................................. 182
5.2.2.2 The School ................................................................................... 187
5.2.3 Zone Of Promoted Action ........................................................................ 194
5.2.3.1 The University ............................................................................. 195
5.2.3.2 The School ................................................................................... 206
5.2.4 Summary of the Factors .......................................................................... 214
CHAPTER 6
DISCUSSION ................................................................................................................. 223
6.1 Affordances of Making Use of a Situative Perspective ................................... 225
6.2 lrnplications and Recommendations for the Field ........................................... 232
6.2.1 Promoting Intercontextuality through Noticing ...................................... 233
6.2.2 Focusing on the Role of Mentor Teachers .............................................. 234
6.2.3 Connecting with Other Mathematics Education Communities ............... 237
6.3 Future Research ................................................................................................ 240
6.3.1 Investigation Necessary Conditions for Preservice Teachers Anticipating
SMRS ....................................................................................................... 240
6.3.2 Investigating the Role of Experience and Expertise ............................... 241
6.3.3 Investigating the Role of Content Knowledge ........................................ 243
Appendix A. .................................................................................................................... 245
Appendix B. .................................................................................................................... 247
Appendix C. .................................................................................................................... 249

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Appendix D. .................................................................................................................... 251

Appendix E ...................................................................................................................... 253
Appendix F. ..................................................................................................................... 255
Appendix G. .................................................................................................................... 258
Appendix H. .................................................................................................................... 265
Appendix I. ...................................................................................................................... 268
Appendix J ....................................................................................................................... 277
References ....................................................................................................................... 287

viii

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LIST OF TABLES

Table l. The Four Principles of Productive Disciplinary Engagement ............................ 21
Table 2. Data Display for Anticipation Practices ............................................................. 89
Table 3. Summary of Methods Course Assignments ....................................................... 98
Table 4. Schedule of Class Activities during Observation 1 ......................................... 125
Table 5. Schedule of Class Activities during Observation 2 .......................................... 137
Table 6. Schedule of Class Activities during Observation 3 .......................................... 145
Table 7. Summary of Anticipation Activity .................................................................. 156
Table 8. Intern Participation in Specific Anticipation Activities ................................... 204

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LIST OF FIGURES

Figure 1. Grossman (1990) Model of Teacher Knowledge .............................................. 42

Figure 2. Representation of Hill, Ball, and Schilling (2008) Model of Mathematical
Knowledge for Teaching .................................................................................... 44

Figure 3. Shimizu's (2002) Common Framework for Lesson Plans ................................. 55

Figure 4. Representation of Possible Strategies Identified by the Teacher That Students
Might Employ to Find the Interior Angle Sum of a Quadrilateral (Shen et al.,

2007) .................................................................................................................. 56
Figure 5. Anticipation Framework .................................................................................... 59
Figure 6. Schedule of Intern Teaching Responsibilities ................................................... 66
Figure 7. Summary of Data Collection Phases ................................................................. 76
Figure 8. Representation of Miles and Huberman (1994) Components of Data Analysis:
Flow Model ........................................................................................................ 82

Figure 9. Example Anticipation Map ................................................................................ 87
Figure 10. Goos' (2005b) Representation of ZF M, ZPD, and ZPA .................................. 94
Figure 11. Power Point Slide of Directions for Solving the Balloons Task Asking for

Multiple Solutions ........................................................................................... 99
Figure 12. Anticipation Map for Lesson Discussed During In-Depth Interview 1 ........ 122
Figure 13. Anticipation Map for Lessons 1.1 and 1.4 .................................................... 127
Figure 14. Cool Stuff Lesson 1 Anticipation Map .......................................................... 131
Figure 15. A Handwritten Portion of the Cool Stuff 1 Lesson Plan Showing Megan's

Original Solution ........................................................................................... 133
Figure 16. A Recreation of a Page from Teacher Manual of IMP (Fendel, Resek, Apler,

& Fraser, 1998) ............................................................................................. 134
Figure 17. Anticipation Map for Lesson 2.1 through Lesson 2.5 ................................... 138
Figure 18. A Handwritten Portion of Lesson 2.3 Showing Megan’s “Boxing” Example
......................................................................................................................................... 140
Figure 19. Example of Making Use of the Zero Product Property from Lesson 2.4 ...... 141

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Figure 20. Anticipation Map for Cool Stuff 2 Lesson .................................................... 143

Figure 21.

Figure 22.

Figure 23.
Figure 24.
Figure 25.

Figure 26.

Figure 27.
Figure 28.
Figure 29.
Figure 30.
Figure 31.
Figure 32.
Figure 33.
Figure 34.
Figure 35.
Figure 36.
Figure 37.
Figure 38.

Figure 39.

Figure 40.

Figure 41.

Figure 42.

Anticipation Map for Lesson 3.1 and 3.2 ...................................................... 146
A Handwritten Portion of Lesson Plan 3.1 Illustrating Megan’s Language
Choice ............................................................................................................. 148
Student Difficulty Presented in Class Text ................................................... 149
Anticipation Map for Lesson Discussed during In-Depth Interview 2 ......... 152
A Handwritten Portion of In-Depth Interview 2 Lesson Plan ....................... 153
Collection of Anticipation Maps Representing Megan’s Activities over the
First Semester ................................................................................................. 158
Megan’s Handwritten Teaching Note from Lesson Plan 1.4 ........................ 161
A Handwritten Portion of Megan's Lesson Plan from Lesson 1.4 ................ 162
Representation of the Notes Megan Wrote on the Overhead Projector ........ 163
Representation of the Notes Megan Wrote on the Overhead Projector ........ 163
Representation of the Notes Megan Wrote on the Overhead Projector ........ 164
Portion of the Class Text Presenting Plotting Points in Three Dimensions .. 167
A "Box" Model Showing the Factorization of x2 + 1 1x + 24 = 0 ................. 169
Megan’s Boxing Example in Lesson 2.3 ....................................................... 170
Megan’s Boxing Example in Lesson 2.3 ....................................................... 170
Megan’s Boxing Example in Lesson 2.3 ....................................................... 171
Megan’s Boxing Example in Lesson 2.3 ....................................................... 171
Megan’s Boxing Example in Lesson 2.3 ....................................................... 171
Recreation of the Lesson Plan Format Required by Mrs. Stams and Used by
Megan on a Daily Basis ................................................................................ 21 1
A Handwritten Example of the Notes Template Used My Mrs. Stams, Megan,
and the Students ............................................................................................ 213
Venn Diagram of Relationship Between University and School ZF M ......... 215
Venn Diagram of Relationship Between University and School ZPA .......... 216

xi

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Figure 43. Venn Diagram of Relationship between University and School ZFM and ZPA
and Megan’s ZPD ......................................................................................... 218

Figure 44. Experiences, Resources, and Constraints that Impacted Megan’s Anticipating
Practices ........................................................................................................ 220

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CHAPTER 1. INTRODUCTION

The purpose of this case study was to explore one preservice teacher’s use Of a
lesson planning practice that was advocated by her teacher preparation program. This
practice, anticipating students’ mathematical responses to her lessons, was a strategy
introduced and promoted by several Of her university mathematics methods courses. This
study sought to examine how she made use Of anticipating in her field experience
classroom and to identify the features of the university and school contexts that supported
or constrained her ability to participate in this practice. The next sections provide a brief
overview Of the relevant research and a discussion of the contributions and limitations Of

the study.

1.1 The Role of Field Experiences in Teacher Education

Studies show that preservice teachers view the field experience as a major
contributing factor to their learning in teacher education programs. Borko and Mayfield
( 1995) investigated the roles of cooperating teachers and university supervisors in a field
experience in the learning of four preservice middle-school mathematics teachers. A
portion of the study focused on the preservice teachers’ perspectives about how one
Ieams to teach and about their own learning during the field experience. The authors
found that “all four student teachers expressed the belief that a person Ieams by doing——
through experience, practice, and making mistakes” (p. 512). One preservice teacher in
particular stated that in order to learn to teach one needs “experience. I think they can't

teach you how to teach, no matter what anyone says” (p. 512). In addition, all four of the

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preservice teachers reported that the changes they made to the way they taught during
their student teaching were based mostly on these experiences.

The student teachers' comments about factors that influenced these changes

paralleled, to a great extent, their ideas about how a person learns to teach. The

factor mentioned most consistently was their classroom teaching experiences,

"getting out there and doing it myself, seeing the problems, working with the

students" (Ms. Richards). They also mentioned their cooperating teachers and, to

a lesser extent, their university supervisors. (p. 513)

This finding is not true only of preservice teachers. Research conducted in the late 19805
indicated that practicing teachers also viewed experience teaching as the most Significant
factor in their learning. Smylie (1989) surveyed over 1700 teachers and asked them to
evaluate a variety of experiences based on their effectiveness in supporting the
development of knowledge and skills needed for teaching. Although teacher education
field experiences were not among the experiences the teachers ranked, “findings indicate
that, by far, teachers perceived direct experience in classrooms as their most effective
source of learning” (p. 545).

More recent reviews Of literature have supported these earlier findings. Wilson,
Floden, and Ferrini-Mundy (2002) were commissioned by the Office of Educational
Research and Improvement and the US. Department of Education to conduct a review of
teacher education research. In reviewing research concerning the effects of student
teaching the authors found that:

Field experience is a staple of teacher preparation programs. Study after study

Shows that experienced and newly certified teachers see clinical experiences as a

powerful—sometimes the most powerful—component of teacher preparation.

Whether the power [of] field experience enhances the quality Of teacher
preparation, however, may depend on the particular experience. (p. 195)

 

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Even more recently the American Educational Research Association (AERA) Panel on
Research and Teacher Education completed a volume summarizing research on teacher
education. Their conclusions about the role Of field experiences were similarly
compelling.

Field experiences have long been identified by both teacher educators and

prospective and experienced teachers as a major, if not the most important, part of

preservice teacher preparation. It is broadly assumed that field experiences are the
key components of preparation where prospective teachers learn to bridge theory
and practice, work with colleagues and families, and develop pedagogical and
curricular strategies for meeting the learning needs of diverse populations.

(Hollins & Guzman, 2005, p. 493)

The importance of field experiences has been even more elevated in universities
that have chosen to include a fifth-year teaching internship experience as part of their
teacher education program. Darling-Hammond (2000) describes this change in the
structure of teacher education programs as being stimulated by the Holmes Group and
other organizations. This restructuring of the traditional four-year teacher education
program is intended to allow for:

more extensive study of the disciplines to be taught along with education

coursework that is integrated with more extensive clinical training in schools...

because the 5th year allows students to devote their energies exclusively to the
task Of preparing to teach, such programs allow for year-long school-based
clinical experiences that are woven together with coursework on learning and

teaching. (Darling-Hammond, 2000, p. 167)

Studies suggest that increased time in a field experience may be beneficial for
preservice teachers. Wilson, Floden, and Ferrini-Mundy (2002) described a study that
compared a fifth year internship experience to a more traditional four year teacher

education program. They report that findings indicated that the teachers in the fifth year

internship program were more satisfied with their teacher education program, remained in

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the teaching profession longer, and rated their own teaching abilities higher than that of
teachers from four year programs.

A central feature of the field experience that makes it such a powerful venue for
learning for preservice teachers is the constant access to students. “From field experience,
prospective teachers reported acquiring survival skills, learning about students, and
recognizing that their students’ understandings vary, are complex, and differ from
teachers” (Wilson, Floden, & Ferrini-Mundy, 2002, p. 196). Shulman (1986) describes a
teacher’s development of a variety of appropriate representations as one outcome of
experience with students.

Since there are no single most powerful forms of representation, the teacher must

have at hand a veritable armamentarium of alternative forms of representation,

some of which derive from research whereas others originate in the wisdom of

practice. (p. 9)

Field experiences afford preservice teachers the Opportunity to work with students and
gain insight into useful representations, as described by Shulman, as well as other types
of knowledge of students and student interactions with content.

The profound influence of the field experience indicates a critical need for teacher
educators to know what preservice teachers learn from these experiences. A review of
research from 1995 to 2002 by Clift and Brady (2005)I on the outcomes Of field
experiences for mathematics preservice teachers points to the current focus of recent

research on changes in beliefs and not on what preservice teachers learn about teaching

mathematics. Further, Clift and Brady reported that in the studies that investigated field

 

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conducted by the AERA Panel on Research and Teacher Education.

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experiences, very few looked at the use of specific teaching methods within the
experiences. As Wilson, Floden, and Ferrini—Mundy (2002) point out,
The research on clinical experience is weak in several ways. Much of the early
research focused on cooperating and prospective teachers’ attitudes. Although it is
important to know how teachers feel about the benefit of field experiences,
attitude surveys do not answer questions about what prospective teachers actually
learn” (pp. 196-197).

It is important for teacher education researchers to begin to focus their efforts on

what is being learned by preservice teachers in this highly influential experience.

1.2 Studying the Role of Field Experiences from the Situative Perspective

Clift and Brady (2005) argued in their review of research of the impact of
methods courses and field experiences that future studies in these areas must consider
that “frameworks for research should move beyond behavior and cognition, beyond a
limited focus on the individual (alone or in a group), and toward a more sophisticated
knowledge of how practice is shaped by contexts, materials, and other people” (p. 335).
In order to address Clift and Brady’s suggestion, this study investigated the lesson
planning practices of a preservice teacher from the situative perspective. Greeno (1998)
defined practices as “regular patterns of activity in a community, in which individuals
participate” (p. 6). In this study the lesson planning practice was a practice in which the
preservice teachers at the university participated.

Greeno (1998) described one avenue for making use of the situative perspective to
investigate the practices of the individual that also meets the standards for research in
field experiences set by Clift and Brady.

The other available strategy is to begin with the framework of the interactional
studies and work inward. In this strategy, progress will occur by focusing on the

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organization of intact activity systems and analyzing the informational contents of

activity in which people accomplish the goals and functions of tasks they

undertake. (Greeno, 1998, p. 6)

Field experiences are often the first time that preservice teachers have the
opportunity to be part of and engage with a community of teachers as a peer. This
community engagement makes the situative perspective particularly suited to the study of
field experiences, which take place in the complex spaces of schools and universities and
also in which preservice teachers participate in these contexts as both students and
teachers. This perspective allowed this study to focus on the participation of a preservice
teacher in a particular lesson planning practice in the field experience as well as to
recognize and investigate the extensive contextual features of the environments in which

she operated.

1.3 Teachers’ Knowledge of Students

The recent work of Deborah Ball and her colleagues has focused on the
knowledge that teachers need in order to teach mathematics. This work builds on the
framework of teacher knowledge set forth by Lee Shulman in the mid 1980s. Shulman
(1986) asserted that teachers needed not only content knowledge or pedagogical
knowledge in isolation of one another, but a special combination of the two, which he
termed pedagogical content knowledge (PCK). Since that time Ball and her colleagues
have refined Shulman’s knowledge domains into separate sub-domains and further
specified them for mathematics. Among those in the domain of PC K is knowledge of
content and students (KCS). In their work Hill, Ball, and Schilling (2008‘) define KCS

and describe its importance for teaching mathematics:

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We propose to define KCS as content knowledge intertwined with knowledge of
how students think about, know, or learn this particular content. KCS is used in
tasks of teaching that involve attending to both the specific content and something
particular about learners, for instance, how students typically learn to add
fractions and the mistakes or misconceptions that commonly arise during this
process. In teaching students to add fractions, a teacher might be aware that
students, who often have difficulty with the multiplicative nature of fractions, are
likely to add the numerators and denominators of two fractions. Such knowledge

might help her design instruction to address this likely issue. (p. 375)

In addition to field experiences serving as venues for preservice teachers to
participate in a community of practicing teachers, they also serve as an initial and
important opportunity to interact with students. As such, field experiences allow
preservice teachers to learn from and about actual students as well as to make use of what
they may already know. This includes knowledge of content and students as described by
Hill, Ball, and Schilling (2008), who found in a review of literature that when teachers
had access to knowledge about how students learned a particular subject matter that they
changed their practices and student learning improved. The question then arises as to how

preservice teachers learn about and consider for themselves this knowledge of content

and students during their field experiences.

1.4 Anticipating Students Mathematical Responses as a Part of Lesson Planning

A primary focus of field experiences in teacher education is often the design and
implementation of lesson plans. The National Research Council (NRC) (2001) addressed
the important role of lesson planning by stating that “planning needs to reflect a deep and
thorough consideration of the mathematical content of a lesson and of students’ thinking

and learning” (p.424). The NRC recommended that:

 

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rather than simply listing problems and exercises, teachers should plan for

instruction by focusing on the learning goals for their students, keeping in mind

how the goals for each lesson fit with those of the past and future lessons. Their
planning should anticipate the events in the lesson, the ways in which the students
will respond, and how those responses can be used to further the lesson goals.

(p. 425)

Despite the recommendations of the NRC and the attempts made by many teacher
education courses, experience suggests that preservice teachers often do not recognize the
importance of planning in the way the NRC describes, specifically in considering the
possible reactions of their students as they plan. This practice is especially important for
preservice teachers attempting to engage students in mathematics discussion as Stein,
Engle, Smith, and Hughes (2008) explain:

Without solid expectations for what is likely to happen, novices are regularly

surprised by what students say and do, and therefore often do not know how to

respond to students in the midst of a discussion. They feel out of control and

unprepared. (p. 321)

Despite feeling more prepared when they have anticipated students’ likely
responses, considering how a middle- or high-school student will view the mathematics
in a lesson can be difficult for preservice teachers, many of whom are themselves
mathematics majors. Nathan and Petrosino (2003) referred to this problem of preservice
teachers, and others with advanced-subject matter knowledge, as the “expert blind spot.”
Given a possible expert blind spot and their lack of personal experience teaching the

content, it is important for preservice teachers to find ways to consider on their own as

well as to seek out resources to gain access to knowledge of content and students.

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1.5 Contributions to the Field

This study is intended to contribute to the field of mathematics teacher education
by building on prior studies about preservice teacher participation in field experiences.
Specifically, it is informed by prior research and literature on preservice teacher learning
in the context of field experiences (Mossgrove, 2006; Strawhecker, 2005; Vacc & Bright,
1999), the knowledge for teaching needed by a mathematics teacher (Even, 1993;
Grossman, 1990; Hill, Ball, & Schilling, 2008; Kinach, 2002; Shulman, 1986, 1987), and
the practice of anticipating students’ mathematical responses (Carpenter, Fennema, &
Franke, 1996; Carpenter, Fennema, Franke, Levi, & Empson, 2000; Carpenter, Fennema,
Peterson, Chiang, & Loef, 1989; Chokshi & Fernandez, 2004; Fennema, Carpenter, &
Peterson, n.d.; Fernandez, 2002; Hiebert & Stigler, 2000; Lampert, 2001; Leinhardt,
1988; Schoenfeld, 1998; Shen, Poppink, Cui, & Fan, 2007; Shimuzu, 2002; Stein, Engle,
Smith, & Hughes, 2008). Additionally, this study is framed by the situative perspective
on teaming which has been used in the past to study secondary mathematics preservice
teacher learning across a variety of contexts (Broadie, 2005; Engle, 2006; Greeno, 1998;
Greeno, Collins, & Resnick, 1996; Peressini, Borko, Romagnano, Knuth, & Willis, 2004;
Putnam & Borko, 2000).

While this previous work sheds light on teacher learning and knowledge in a
variety of areas, researchers have not explored how experiences help preservice teachers
participate in the practice of considering, or anticipating, their students’ mathematical
responses as they plan lessons in a field experience setting. This study sought to address
this gap by presenting a case of a preservice teacher in a university field experience

which focuses specifically on her development. of the practice of anticipating students

mathematical response
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mathematical responses. Such a picture of preservice teachers learning will contribute to
the field of mathematics teacher education in three important ways. First, this study
contributes to the field by presenting a framework for investigating teachers’ anticipation
practices. Second, this study portrays the journey of one preservice teacher and in doing
so provides a vision of one preservice teacher’s practices of anticipating students’
mathematical responses within various contexts. Finally, by focusing on contextual
features, this study offers insight into the environments and relationships that support this

practice.

1.6 Limitations

A limitation of this study should be acknowledged. Because this case study
provides an in-depth picture of the practices and experiences of one preservice teacher it
is limited in two ways. First, the findings may not be generalizable to other preservice
teachers in other contexts. Second, because this research focuses on a single preservice

teacher, comparisons of practices and school contexts are not possible.

1.7 Organization of the Dissertation

This dissertation is organized into six chapters. The first chapter provided an
overview of the study. Chapter 2 details the relevant literature and lays out the framework
and research questions used in this study. Chapter 3 outlines the methodology of the
study. Chapter 4 describes the two primary contexts of the study, the university and the

school. Chapter 5 presents the findings from the analysis of the data. Finally, Chapter 6

10

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concludes with a discussion of the results as well as implications and next steps for future

research.

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CHAPTER 2. THEORETICAL FRAMEWORK AND REVIEW OF THE
LITERATURE

This chapter outlines the existing educational theory and research as it relates to
this study. The first section introduces a theoretical framework that describes the situative
perspective on learning. This view of learning focuses on the contexts and the contextual
features of learning environments and thus is well suited for studying learning within a
field experience in which a myriad of factors interact to create the preservice teacher’s
experience.

The second section of this chapter reviews the relevant literature. This includes
studies addressing the learning of preservice teachers in field experiences, specifically
those related to learning to teach mathematics. This section also includes research in the
area of lesson planning in order to frame common preservice teacher lesson planning
practices. Additionally several frameworks related to the knowledge that teachers need to
teach are presented along with summaries of three studies that examine the knowledge of
preservice teachers preparing to teach mathematics. Finally, anticipating students’
mathematical responses, a practice of mathematics teachers that is based on their
knowledge of students, is described. The third and final section outlines the research

framework developed for this study and introduces the research questions.
2.1 Theoretical Framework

In 2005 the AERA Panel on Research and Teacher Education released a report on
the state of research in teacher education. In addition to reviewing the current teacher

education research in this report, the panel also set forth recommendations for future

12

 

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research in teacher education. Among these was the recommendation that future research
should be situated in relation to relevant theoretical frameworks. The panel argued that
doing so would aid researchers in explaining their findings about the effects of particular
practices and “potentially contribute to the elaboration and refinement of these
frameworks and to greater understanding of the process of teacher learning in different
contexts” (Zeichner, 2005, p. 741). For these reasons the proposed study will be framed
by a theoretical perspective on learning, namely the situative perspective.

The upcoming sections detail the theoretical framework for this study. The first
section introduces the conceptual themes that underlie the situative perspective are
described. The second section relates the situative perspective to other perspectives
pertinent to this study. Finally, the third section summarizes two relevant studies. The
first is a study of fifth-grade students participating in productive disciplinary engagement,
a notion that closely resonates with the situative perspective. The second study is an
exemplar of how the situative perspective has been used to study secondary mathematics
preservice teachers (SMPTS) learning.

2.1.1 The Conceptual Themes of the Situative Perspective

The situative perspective is one of a variety of lenses that has been employed by
educational researchers to study learning. In contrast to other perspectives, the “situative
perspectives focus on interactive systems that include individuals as participants,
interacting with each other as well as materials and representational systems” (Putnam &
Borko, 2000, p. 4). Putnam and Borko summarize three conceptual themes that describe

the situative perspective: (a) cognition as situated in particular social and physical

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contexts, (b) cognition as social and (c) cognition as distributed across the individual,
other persons and tools.

The first of these themes, cognition as situated in particular social and physical
contexts, asserts that both the activity and the environment are crucial to what is teamed.
Putnam and Borko (2000) explained that the activity must be authentic and provide the
following definition: “1.8. Brown and colleagues (1989) defined authentic activities as
‘the ordinary practices of a culture’ (p. 34)—activities that are similar to what actual
practitioners do” (p. 4). Field experiences in teacher education are intended to be
authentic learning experiences in that they simulate the actual work of teaching by
allowing preservice teachers to teach students in real classroom settings.

Given that the first theme focuses on learning situated within settings and
activities, one issue that arises within the situative perspective is how to characterize the
transfer of learning. In other views that focus on the individual’s experience, transfer is
typically seen as having the ability to apply a particular type of knowledge in different
situations. For example, a student would apply their knowledge of subtraction learned in
mathematics class to a situation of making change at the grocery store. The issue of
transfer becomes problematic from the situative perspective because it is assumed that
learning takes place within the situations in which the knowledge is to be used. Engle
(2006) sheds some light on the issue of transfer from the situative perspective. She argues
that “transfer is more likely to occur to the extent that learning and transfer contexts have
been framed to create what is called intercontextuality between them. Intercontextuality

occurs when two or more contexts become linked to one another” (p. 456). The key to

14

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linking two contexts according to Engle is to frame the learning context and transfer
context as being connected to one another.

The second conceptual theme of the situative perspective is cognition as social.
Just as the activity is central to the learning process, the social interactions that take place
play a major role in what is learned.

Interactions with other people in one’s environment are major determinants of

both what is learned and how learning takes place. The process of teaming, too, is

social. Indeed, some scholars have conceptualized learning as coming to know
how to participate in the discourse and practices of a particular community.

(Putnam & Borko, 2000, p. 5)

This means that all of the learners in a context impact the learning of the group and that
the group is together working towards participating in the practices of a particular
community. Preservice teachers in field experiences take part in a variety of social
interactions in schools and university classrooms. Preservice teachers’ working both of
these contexts is a unique feature of this experience. The field experience serves as a
transitional setting where preservice teachers are moving away from being part of a
community of students toward being part of a community of teachers.

The third conceptual theme that makes up the situative perspective views
cognition as distributed across the individual, other persons, and tools. Learning,
thinking, and participation do not all rest with one individual or one tool; instead, a
variety of resources contribute to the process. Putnam and Borko (2000) cite an example
of this from Hutchins (1990) who describes the workings of a US. Navy ship.

Six people with three different job descriptions and using several sophisticated

cognitive tools were involved in piloting the ship out of the harbor. The

distribution of cognition across people and tools made it possible for the crew to
accomplish the cognitive tasks beyond the capabilities of any individual member.

(1)- 5)

15

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In a field experience a variety of people and resources are part of the preservice teacher’s
experience. In addition to what the preservice teacher brings to the experience, their
mentor teachers, school resources, and university instructors share knowledge and
experience with preservice teachers. Within this conceptual theme Putnam and Borko
(2000) explain the importance of tools:

In the world outside of school, intelligent activities often depend upon resources

beyond the individuals themselves such as physical tools and notational systems

(Pea, 1993). Many of these tools do not merely enhance cognition, they transform

it; distributing cognition across persons and tools expends a system’s capacity for

innovation and invention. (p. 10)

Curriculum and other professional materials are examples of these tools that exist for
teachers and are available to preservice teachers during field experiences.

The situative perspective is not the only perspective that can be used to study
learning. The next section describes how the situative perspective can interact with other
perspectives that have also been used to study teacher learning.

2.1.2 The Relationship between the Situative Perspective and Other Perspectives on
Learning

The tenets of the situative perspective do not necessarily exclude other views of
learning. In fact, in some cases, such as the sociocultural perspective, a relatively large
overlap exists between the perspectives. The sociocultural view of learning, attributed to
Vygotsky, gives priority to social interactions. Palincsar (1998) describes this view of
Vygotsky’s in the following way: “as learners participate in a broad range of joint
activities and internalize the effects of working together, they acquire new strategies and

knowledge of the world and culture” (p. 351-352).

16

 

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In her research with secondary mathematics preservice teachers (SMPT), Goos
(2005b) drew on the sociocultural perspective to extend an existing framework that
allowed her to study preservice teacher learning. Goos based her framework on
Vygotsky’s Zone of Proximal Development (ZPD) and Valsiner’s (1997) extension of
Vygotsky’s work. Palinscar describes the ZPD as “what children can do with assistance”
(Palincsar, 1998, p. 353). Valsiner used the extended framework “to explain children’s
development in the context of their relationships to physical environments and other
human beings” (Goos, 2005b, p. 37) . Valsiner’s framework includes two zones in
addition to the ZPD, the Zone of Free Movement (ZFM) and the Zone of Promoted
Action (ZPA). Goos further adapted this framework by applying it to teacher learning
instead of student learning. Thus within her framework, the ZPD represents the
“symbolic space where the novice teacher’s emerging skills are deve10ping under the
guidance of more experienced people” (Goos, 2005b, p. 37). Included in this
characterization of ZPD are the preservice teacher’s skills and experiences working with
technology, their pedagogical beliefs about technology integration, and their general
pedagogical beliefs. The ZFM “suggests what teaching actions are possible,” and “might
include their students (behavior, motivation, perceived abilities), curriculum and
assessment requirements, and the availability of teaching resources” (Goos, 2005a, p. 2).
Finally, the ZPA “represents the efforts of a teacher educator, supervising teacher, or
more experienced teacher colleague to promote particular teaching skills or approaches”
(Goos, 2005b, p. 38). Goos points out that it is possible for multiple ZPAs to exist if the
actions promoted by the teacher educator are different than the actions promoted by the

supervising teacher. Goos’ framework allowed her to focus on the social and certain

17

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contextual features of the preservice teacher’s experience. In this way the sociocultural
perspective shares with the situative perspective the theme of cognition as social.

The situative perspective can also be integrated with seemingly less related
perspectives, such as the cognitive perspective. The cognitive perspective views
knowledge as individual “understanding of concepts and theories in different subject
matter domains and general cognitive abilities, such as reasoning, planning, solving
problems, and comprehending language” (Greeno, Collins & Resnick, 1996, p. 16). Thus
learning is seen as a “constructive process of conceptual growth” (Greeno, Collins, &
Resnick, 1996, p. 16). Notable within this perspective is the work of Piaget who focused
on the thinking of children. “Although cognitive structures could not be observed
directly, as physical ones could, Piaget tried to reveal the thought processes of children
through a technique of activity-based interviewing” (Resnick & Ford, 1981, p. 156).

According to Greeno (1998), the situative perspective accounts for aspects of
learning that are invisible or ignored in other traditions while maintaining room for those
perspectives’ lenses within the theory. He argues that research embracing the situative
perspective, which he refers to as interactional, and research employing the cognitive
perspective need not operate in complete isolation, as they traditionally have. Greeno
presents a method for taking into account both perspectives: “begin with the framework
of interactional studies and work inward” (p. 6). Doing so could focus research on the
organization of the situation and the community and also look inward at how individuals
within those situations accomplish the tasks that make them part of the community. He

explains then that "at a general level, the situative perspective subsumes the cognitive

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perspective by viewing conceptual understanding, like behavioral skill, as an aspect of

participation in social practice" (p. 17).

2. I .3 Examples of Research Employing the Situative Perspective

In the next two sections three studies are described that make use of the situative
lens to investigate and describe learning. The first section described a study that looks at
the notion of productive disciplinary engagement in a fifth grade classroom and argues
that productive disciplinary engagement falls under the situative perspective by relating it
to the conceptual themes previously discussed. Also in this section is a description of
another study that shows how the principles of productive disciplinary engagement apply
to the case of mathematics teacher learning. The second section presents a study that
Specifically identifies itself as an investigation using the situative perspective to study

secondary mathematics preservice teacher (SMPT) learning.

2.1.3.1 A Study of Productive Disciplinary Engagement

The first study looks at the productive disciplinary engagement (PDE) of a group
of fifth—grade students taking part in a science project. A group of students in the class
was followed as they researched whales and eagles in order to answer an overarching
question about how animals survive. Data was collected on the students’ interactions
within the group via observation and videotape.

The authors describe each part of PDE in turn. First, they explain engagement as
the discourse of the participants in the group. Engagement thus looks at such questions

as: “How are students participating? What proportion of students are participating? And

19

 

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how are students’ contributions responsive to those of other students?” (p. 402). Thus
engagement embodies aspects of the themes of both cognition as social and cognition and
distributed. The authors go on to describe disciplinary engagement in a school context in
the following way: “we mean that there is some contact between what students are doing
and the issues and practices of a discipline’s discourse” (p. 402). This part of PDE
resonates with the conceptual theme that cognition is situated. The productive aspect of
PDE is defined by Engle and Conant as the students actually getting somewhere in their
work. In the case of their study this meant that changes occurred in the way students
argued about science. While this does not address one of the previously discussed three

themes specifically, it does resonate with the students coming to more fully participate in

a particular practice of the domain.
After describing PDE, Engle and Conant (2002) present four principles that have

emerged in their research as fostering PDE in groups of learners. Table 1 presents each of

the four principles and provides brief descriptions of each.

20

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Principle

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Description

“problems do not need to be open from the perspective of experts
in a discipline, but rather open from the perspective of the
students interpreting them.” (Engle & Conant, 2002, p. 404)

“In general, by giving students authority, we mean that the tasks,
teachers, and other members of the learning community generally
encourage students to be authors and producers of knowledge,
with ownership over it, rather than mere consumers of it.” (Engle
& Conant, 2002, p. 404)

“This principle is an expression of the value that each member of
the learning community is not an authority unto himself or
herself, but one intellectual stakeholder among many in the
classroom and beyond.” (Engle & Conant, 2002, p. 405)

“Resources supporting productive disciplinary engagement may
be as fundamental as having enough time to pursue a problem in
depth (Collins, 1998; Henningsen& Stein, 1997) or having access
to sources of information relevant to it.” (Engle & Conant, 2002,
p.405)

U Sing the case of the fifth grade science students researching and presenting an argument

about different animals Engle and Conant (2002) show how these four principles

e rmerged as fostering PDE. They then go on to illustrate how the principles were found in

I:
Wo other examples where PDE was present.

The principles of PDE can be found in examples of mathematics teacher learning

(18 \JVCll those of student learning. Smith (2000) studied the dilemmas of one experienced

ts: .
acher, MS- Henderson, and traced how these dilemmas served as “sprlngboards” for her

1 e - -
<11nrng. Ms. Henderson was in her twenty-seventh year of teaching but was in her first

Q31“ usrng the Visual Mathematics (VM) curriculum. Her style of teaching was focused

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as facilitator of students’ active construction of mathematics and mathematical
explanations as prescribed by the VM curriculum. The study followed Ms. Henderson as
she confronted this dilemma and “began to reorganize her ways of knowing in order to
resolve the situation” (p. 353).
The extended professional development experience that Ms. Henderson engaged
in prompted her to make changes to the way she thought about and enacted her role as a
mathematics teacher. Engagement in this process displayed all four of the principles
SUpporting PDE described by Engle and Conant (2002). First, the content, in this case
mathematics teaching, was problematized for Ms. Henderson through her participation in
the professional development experience. A dilemma “resulted from the tension between
the teacher’s past practice of structuring learning opportunities so that students could
ex perience success and the new view that students needed to engage in complex problem
S()lving that was often accompanied initially by feelings of being unsuccessful” (Smith,
2000, P- 358). Second, Ms. Henderson had the authority as the teacher in her classroom to
make changes to her teaching practices. Additionally, she was supported in this by the
presence of other mathematics teachers and administrators from her school who were also
part Of the professional development experience and responsive to engaging with her
arOUnd the mathematics teaching practices at the school. Third, Ms. Henderson was held
accountable to others and to the new disciplinary teaching norms through her
i rl‘slolvement in the professional development community. After sharing an episode of her
teaching one of the professional development staff “began to question the need to
QQntinue breaking things down for students” (Smith, 2000, p. 362). Finally, Ms.

Ienderson had access to relevant resources in the form of the new curriculum materials

22

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and the professional development staff who were able to help her explore her own

q uestions.

Smith’s (2000) study is an example of a mathematics teacher participating in PDE
of inquiry into her own mathematics teaching and also of the four principles that foster
PDE described by Engle and Conant (2002). Although Smith did not specifically identify
the principles in her study of Ms. Henderson, they were present in ways that contributed
to her learning from her dilemma. The current study also makes use of these four
principles to study the contexts that support or constrain the practices of a preservice
mathematics teacher. The next section describes a study that detailed the learning

ex periences of secondary mathematics preservice teachers (SMPTS).

2 - 1-3.2 A Study of SMPT s Learning

The second study makes use of the situative perspective specifically to study the
learning of SMPTS. Peressini, Borko, Romagnano, Knuth, and Willis (2004) investigated
S MPTS learning to teach over time in a variety of contexts. They described their

application of the situative perspective as follows:

For teachers, learning occurs in many situations of practice. These include
university mathematics and teacher-preparation courses, preparatory field
experiences, and schools of employment. Situative perspectives argue that, to
understand teacher learning, we must study it within these multiple contexts,
taking into account both the individual teacher-learners and the physical and
social systems in which they are participants (Putnam and Borko, 2000). (p. 69)

Their study reports on the use of a situative framework in the Learning to Teach
S
e(zondary Mathematics (LTSM) project that traced the learning of secondary

‘ ‘athematics teachers from their university experiences into their early years of teaching.

23

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Their research framework focused on three domains of professional knowledge:
mathematics content, mathematics-specific pedagogy and professional identity. They
looked at these domains across three settings: mathematics and teacher preparation
courses, preservice field experiences, and schools of employment.

Using data from videotaped lessons, observation field notes, written artifacts and
interviews with preservice teachers, cooperating teachers and university supervisors
collected during the preservice teachers last years in a teacher education program and
first two years in their own classrooms, the authors present two case studies that illustrate
their application of the situative framework. The first case showed the development of
One preservice teacher’s complex notion of proof in her own mathematics learning and
her teaching, tracing her development in the areas of mathematics content and
mathematics-specific pedagogy. The second illustrated the struggle of a preservice
teacher in the area of professional identity after leaving the teacher education program.
The authors stated that the framework applied in this way was useful in tracing SMPT
learning across the various contexts. In fact, when they encountered contradictory
eVidence “the framework enabled us to see these differences as coherent and sensible — as
being integrally connected to norms and expectations specific to the different contexts,

and to the novice teachers’ evolving professional identities” (Peressini et al., 2004, p. 90).
In conclusion, the situative perspective can allow researchers to be concerned
with contexts and environments as well as the participation of individuals within these
Settings. Studies using this perspective have identified principles that support learning
f'\I‘Om this perSpective and have been used with the population that is of interest in this

t‘de. The next section presents a revrew of the literature relevant to the current study.

24

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2.2 Review of the Literature

The review of relevant literature for this study focuses on four main areas of
research. The first is what preservice teachers learn about teaching mathematics from
participating in field experiences. The second area of research is teacher lesson planning
practices, an activity in which preservice teachers participate in field experiences. The
third area of research in this section describes various views of teacher knowledge. The
fi nal area of research looks at the practice of anticipating students’ mathematical

responses, the activity that is the focus of this study.
2.2.] Secondary Mathematics Preservice Teachers Learning to Teach Mathematics
from Field Experiences
Field experiences are a major component of nearly all teacher education
programs. Research has shown that field experiences can change preservice teachers’
beliefs and has also focused on how preservice teachers perceive these experiences (e.g.,
Qady, Meier, & Lubinski, 2006, Philipp et al., 2007). Field experiences are unique in that
they represent the most authentic set of activities available to preservice teachers learning
t0 teach. The following section summarizes three studies that focus on what preservice
teachers, both elementary and secondary, can learn about teaching mathematics from
field experiences,

In the first of these studies, Strawhecker (2005) examined several different

I‘Eparatton treatments to determrne Wthh contributed most to preservrce teachers gain

 

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in Pedagogical Content Knowledge (PCK)2. N inety-six elementary and middle preservice
teachers at a small Midwestern university took part in four different semester-long
preparation treatments. The first combined a content course, a methods course, and a field
experience (CMF) and the second a methods course and field experience (MF). The other
two treatments consisted of only one treatment, either a content course only (C) or a
methods course only (M). Strawhecker used the Content Knowledge for Teaching
Mathematics (CKTM) assessment to measure the PCK of the preservice teachers in each
Of the different preparations. The CKTM "represented a set of survey-based teaching
problems thought to model various components of the specialized knowledge of
mathematics which is needed for teaching" (p. 7). Although the author provided no
details about the exact content of the CKTM, she explained that the survey had 27
multiple-choice items that integrated content knowledge, content knowledge situated in
teaching, and knowledge of students’ thinking ,which provided the researcher an overall
PCK score for each participant. Strawhecker found that overall there were significant
differences between the PCK of the four groups in favor of the two preparations with a
COrnbined methods component and field experience (CMF and MF). Although little detail
Was given about the exact nature of what was learned by the preservice teachers,
Strawhecker’s findings imply that field experiences can have a positive impact on the
mathematical PCK of the preservice teachers.
The second study examined elementary preservice teachers’ application of
QQghitiveb’ Guided Instruction (CGI) methods, learned in a methods course, in their field
Q Xperience classrooms. The authors describe CGI as “an approach to helping ‘teachers

N

lE'CK is a particular type of knowledge for teaching and is described in detail in the next section.

 

26

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.~.t and Bright L‘U'N'
,r‘ml ' Y .
..,.}.1n:,CGl in their ‘2
..~cnattttn\ 0i their '

ix
The first L‘tt\C ‘

rilCT .
. .tehttixet‘iurxe \1'
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use knowledge from cognitive science to make their own instructional decisions’

(Carpenter & Fennema, 1991, p.10)” (Vacc & Bright, 1999, p. 90). Although this
approach has been most often used in the professional development of in-service
teachers, Vacc and Bright’s application with preservice teachers would no doubt also
contribute to their learning to teach mathematics. In fact, the characteristics of CGI
teachers that Vacc and Bright describe could be considered positive outcomes for a

teacher education program.

Teachers who use CGI principles when teaching (a) believe that their

understanding of children’s thinking is a critical component of lesson planning,

(b) facilitate children’s problem solving and discussions of children’s thinking, (c)

listen to their children and question them until the students’ thinking becomes

clearer, and (d) are able to make instructional decisions that are appropriate to the

mathematical needs of their students. (Vacc & Bright, 1999, p. 90)

V ace and Bright constructed two case studies of preservice teachers and their experiences
applying CGI in their field experience classrooms based on data from written work,
Observations of their teaching, and interviews with the preservice teachers.

The first case focused on Helen, a preservice teacher who stated that she began
the methods course with the belief that “the teacher’s role was to model problem
SOlutions for students” (Vacc & Bright, 1999, p. 98). During her time teaching in the field
e Xperience Helen’s beliefs shifted toward a more CGI—oriented approach and the authors
found that to a certain extent she was applying CGI methods:

She was fairly successful in applying some of these [CGI principles] during
student teaching. She planned and implemented instruction that was based on
problem solving, and she facilitated student understanding and critical thinking
through a rather high level of questioning. (Vacc & Bright, 1999, p. 99)

I . . . .
1] contrast, the preservrce teacher Andrea, whose beliefs also shifted to a more CGI

91‘ iented approach, was not able to apply the CGI methods in her field classroom. “She

27

 

,gpcared to focus llll ire
sire-gang" (Vacc & Brt

Vacc and Brigl‘.
amines during the l
heater teachers. Hclc
sent-.gr’s experience '
firstCGl" in. 931_ l~

11

Lemar/e coherent .

;'tr~h1c‘hthe_\ \ttrclcn

on...-

,‘:“;-n])' I
l l:.L.\ _‘lll‘:y ci\l\

h.‘L:‘
t..‘tl
\

...gr~.t\ the itttpw
,JRtOIi‘Li pdellCC\

Tr

it“ ltnal \lutl

eqzrdar} presen it
-1 1' field expericn
t? DTt‘xtnlL‘e teac
_\53 -

“We tou nd t

appeared to focus more on whether the students’ answers matched the responses she was
expecting” (Vacc & Bright, 1999, p. 102).

Vacc and Bright (1999) hypothesized that the differences in learning to apply CGI
techniques during the field experience could have resulted from the influence of the
mentor teachers. Helen’s mentor had “extensive experience with CGI” but Andrea’s
mentor’s experience “was limited to participation in a 2-hour ‘awareness’ workshop
about CGI” (p. 93). From this the authors concluded that “if preservice teachers are to
internalize coherent applications to teaching and learning mathematics, the environment
in which they student teach and the support they receive need to be consistent with the

Pri nciples being advocated in their professional preparation program” (p. 109). This study
h i ghlights the important role of the mentor teaching in the application of university
promoted practices by the preservice teachers.

The final study, by Mossgrove (2006), looked at the planning practices of two
Secondary preservice teachers, Paige and Keith, and their use of instructional tasks in
tI‘leir field experience classrooms. Specifically, she examined the opportunities each of
t‘16: preservice teachers had to learn about student-centered instructional practices.
Mossgrove found that the curriculum used by each preservice teacher contributed to their
learning 0f teaching mathematics:

An analysis of the contextual settings in which Paige and Keith worked point to
key differences in the opportunities that Paige and Keith had during their field
experiences to learn about student centered instructional practices. . .Keith was
greatly influenced by his use of a reform-oriented curriculum, whereas Paige did
not have access to such a curriculum. (p. 5)

The settings for Paige and Keith's field experiences were quite similar, but as the

a IJote above indicates, their exposure to curriculum was not. Paige taught several sections

28

flit‘OlllSB Cdiit‘d lntt‘gt

hiihll‘a‘gill 61h and 7”
512.15ch 3th grade \[llk
lithsmttiu tCMl’t m
tigsiri l.
hittxxgrt'fit‘ ill.“
fist" thtt hix teaching .
Lgsha. Keith chungt‘tl
ttgt'dtsctttcr) lllltl th
r-sfirad to this practice
i533: huh [ti modify
.‘ti‘crti Paige did not
:ii’tiiit'i'g if\\()n_\' er ht‘
it} i-ixs {it 4n “Ppt‘tnu:
{Wilt
Although fiCiLi
imi‘lthtitclt “my

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tutti-get had an i .
Hi I

3; .
k 4h ,
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of a course called Integrated One using a curriculum published by McDougal Littell.
Keith taught 6th and 7th grade mathematics classes as well as an algebra class for
advanced 8th grade students. In his 6th and 7th grade classes Keith used Connected

Mathematics (CMP) as the curriculum and in his algebra class he used Prentice Hall

Algebra I.
Mossgrove (2006) focused her data collection for Keith on his algebra class but

found that his teaching of CMP in the other classes impacted his lesson planning in
algebra. Keith changed the Prentice Hall lessons in order to "let the students make some
sort of discovery and then talk about it as a class" (p. 164-165). Keith's mentor teacher
referred to this practice of modifying lessons as "CMP it." The result was that Keith
learned how to modify lessons from more traditional texts to make them more student

Centered. Paige did not have this opportunity and stayed close to the curriculum in

p l anning lessons for her class. Thus, based on the experience in her field placement Paige

h ad less of an opportunity to learn about creating and implementing student—centered

l e SSons.
Although field experiences play such a major role in the preparation of teachers,

I
I)ere is relatively little evidence of what secondary mathematics preservice teachers learn
9In the experience about teaching mathematics. Strawhecker (2005) found that field

Q
J“lberiences had an impact on the PCK of elementary and middle school preservice

t; -
Q Qchers, more so than methods courses or content courses alone. Although these findings
5 promising, little detail was provrded about the exact nature of the PCK or what

h aIbpened in the field experiences and methods courses that contributed to PCK. The

(Ether two studies (Mossgrove, 2006; Vacc & Bright, 1999) showed the importance of the

29

renter teacher and curt
rating muthemutit‘x t '
ering in authentic e\
it: role eleuntext in it

much around a spet'

tin; field experience

Teacher lew ll“
.1 period at time in '
Student t l983i on t.
21rt1itigxfrttnt researe.‘
reuch utth teachers
3:36..th and though
in .. I
-24; tetehers tend not
Np. 401 t. In tutt-
i'ieurth shtimetl that ‘
itting the mtuct lit.

‘F'hii' .
.‘lng \Iuant\. (-

ii.l'1.x . ‘
L\. \U [hat \IUd'

t“~il'§"t'='-’ I
damn in order I
T .
he \tudte\ rv

“133C? tl: ' .

 

mentor teacher and curriculum in what the preservice teachers were able to learn about
teaching mathematics from the field experience. These studies provide pictures of teacher
learning in authentic experience (i.e., field experiences) and accentuate the importance of
the role of context in what is learned by preservice teachers. The next section summarizes
research around a specific practice of teachers, one that preservice teachers participate in

during field experiences, namely lesson planning.

2.2.2 Teacher Lesson Planning

Teacher lesson planning has been a site of educational research for a relatively
long period of time in the education research community. A review of research by
Shavelson (1983) on teachers’ pedagogical judgments, plans, and decisions sheds light on
findings from research conducted from the late 19605 to the early 19805. This early
research with teachers in a variety of content areas pointed to the importance of well-
developed and thought-out lesson plans. “Plans exert such a strong influence on teachers
that teachers tend not to deviate from them once they have begun teaching” (Shavelson,
1983, p. 401). In addition to the importance of the lesson plans, Shavelson reported that
research showed that teacher planning focused on choosing classroom activities instead
of using the model that many were trained to use: “specifying behavior objectives,
Specifying students’ entering knowledge and skills, selecting and sequencing learning
activities so that students accomplish objectives, and evaluating the outcomes of
instruction in order to improve planning" (p. 402).

The studies reviewed by Shavelson (1983) made use of a variety of methods to

collect data about teacher lesson planning practices. These methods included

 

questionnaires. inten re
@gjtmefuuhngsl“
aereeeneemed \\ llh ‘1
its object mutter. Des
eta-«st: he“ lCN‘n PL1
tamedthatthexe \lULl
mien become si n g l e
Some research
Sidelvitn (1983‘! Int \lt
EztiLMi'lgflttn ( IklMll
esters in three JIL‘thL
Fishes in their stud).
Here inten'iewed belt \l
’55” l‘f‘ti'tiees in th ‘
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questionnaires, interviews, ethnography, simulations, and “think aloud” protocols.
Overall, the findings from the studies suggested that when planning lessons, teachers
were concerned with students and subject matter, although less so with the structure of
the subject matter. Despite their concern with students, two studies were described that
showed how lesson planning could actually detract from students learning. Shavelson
reported that these studies suggested that lesson planning “may be counter productive if
teachers become single-minded and do not adapt their lessons to students needs” (p. 405).

Some research conducted after the studies on lesson planning reviewed by
Shavelson (1983) took the form of expert-novice comparisons. As one example, Borko
and Livingston (1989) examined the differences between expert and novice mathematics
teachers in three areas: planning, teaching, and post lesson reflections. The three novice
teachers in their study, one elementary and two secondary, and their mentor teachers
were interviewed before and after teaching lessons and observed for one week to examine
their practices in these three areas.

Borko and Livingston (1989) found that there were differences in the ways that
the novice and expert teachers planned their lessons. The expert teachers planned on
several different levels including yearly planning, unit or chapter planning, and daily
planning. Although most of the expert teachers’ lesson planning occurred outside of the
formal school day and was not written down, they each described detailed mental plans.

Reported plans typically included a general sequence of lesson components and

content. They did not include details such as timing, pacing, and the exact number

of examples and problems, these aspects of instruction were determined during
the class session on the basis of student questions and responses. In fact, when
asked what would be happening in class each day, [two of the expert teachers]

typically described plans that explicitly anticipated contingencies that were
dependent on student performances. (Borko & Livingston, 1989, p. 480)

31

In contrast. Iht
The authors offered \
Tlti'fitifl teachers were
1112;. planning at 11
Send. one of the n
pas. lt’\_\t"l'l\ me ant t
{saute tedehers felt
3 idi‘l ‘_\ ~tem. l. m

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Ll'dfigda‘ll. “’5‘"- l

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‘lhi‘tl HI] ht)“. Ill
trailed presentat

-._:t: mentor te ael

r3; -
11!; Fr ‘1‘

.tee teaehe
- :.cntu;1\llnlc

as.
-- t. In addtti

’lefi [Cdth-r\ .
~ (

.1
-4:
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{Jilli‘d [b

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In contrast, the novice teachers’ planning was more focused on the short term.
The authors offered several possible explanations for their differing focus. First, the
novice teachers were only in the class for 12 weeks beginning in January. Because of this
timing, planning at the year and even the semester level had already been completed.
Second, one of the novice teachers noted that the amount of time and energy it took her to
plan lessons meant that she was only able to plan for the next day. Finally, another of the
novice teachers felt constrained in this area by her mentor teacher. “Since I’m kind of like
in [his] system, I’m just going by what he wants done. . .He basically has said, ‘Do a page
a day.’ So that is kind of what I’m going on and I haven’t stepped back” (Borko &
Livingston, 1989, p. 486).

Another feature of the novice teachers’ lesson planning was that it primarily
focused on how they would represent the mathematical content to students and creating
detailed presentation plans. The novice teachers mainly drew on the teacher’s manual,
their mentor teachers’ notes, and their own learning experiences to create their plans. For
the novice teachers, “the process of creating a mental script for presenting the lesson
content was time consuming and often appeared inefficient” (Borko & Livingston, 1989,
p.486). In addition to planning their presentation of the material, the two secondary
novice teachers also included working out the problems in their daily planning process.
Although both had access to the problem solutions, one of these novice teachers
explained that “without going through this process it was not possible to ‘own the
problem solution’” (Borko & Livingston, 1989, p. 487).

A final feature of the novice teachers’ lesson planning was that it lacked the

experience that the expert teachers were able to draw on. All three of the novice teachers

32

 

refitted wanting to tr}

A

teething they had lit
experience uith \tutlei
:fziitt there in the ct
Iii“. p. ttfl. ln adtlit
:tgerienee in this area

is: :nta pittltit‘ttls \\ he

l’j’

.t ,_ prepared in udt ant

Based on their
:stTegt‘en. Burke and }
lith Eead them to m.

ital. ‘-‘ ‘ ‘ i' -
“Iii: EXIKYICDL‘C lll

is i' a ,. " .
.llt nut rte lCthllC

meld-“hm? in :lt‘net
SWNI til the
CO“\uming n.1'.
are dit‘ticttltiet
d} 0l‘ kntitt
t e lirxt time t
' ”ln:~'\lun_

 

”i‘k’dt‘tnt ‘. _
hl‘cll‘lun.

[EriltlT
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it CV . ,
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reported wanting to try using less traditional teaching formats but all felt unready to try
something they had little or no experience doing. The novice teachers also lacked
experience with student reactions to the content. “All three reported being unable to
predict where in the curriculum students would have difficulty” (Borko & Livingston,
1989, p. 487). In addition to the novice teachers concern about their own lack of
experience in this area, the authors found that when they enacted their lessons “all three
ran into problems when student questions or comments led them to attempt explanations
not prepared in advance” (Borko & Livingston, 1989, p. 487).

Based on their findings of the novice and expert teachers, planning, teaching, and
reflection, Borko and Livingston (1989) offer several explanations for the difference
which lead them to make recommendations for field experiences and the final student
teaching experience in particular. Borko and Livingston explain that it was the fact that it
was the novice teachers’ first time teaching the content more so than that it was their first
time teaching in general that limited their lesson planning.

Several of the difficulties the novice teachers encountered, such as the time-

consuming nature of their planning and the inability to anticipate student problem,

are difficulties encountered by most teachers the first time they teach a course or
body of knowledge. Any teacher will think and act like a novice, to some extent,
the first time he or she attempts to teach a particular body of knowledge. (Borko

& Livingston, 1989, p. 489)

Based on this explanation and others, Borko and Livingston offer several
recommendations to remedy the situation: (1) novice teachers should take a reduced
teaching load to afford more time for better planning, (2) novice teachers should teach
content they are very familiar with in order to free them to use their time to plan their

presentation instead of learning the content, and (3) novice teachers should be provided

multiple opportunities to teach the same content so that they can immediately make

 

«r: ,r"_‘—’\' [O [hle lC\\‘
Laud-:5"

{networks that UUtl

Discussions t
teeunent r‘seareh 1
its mathematics ct m
aziematies. Tilt ‘ll gl
51563111110 lt‘dt‘hcr
afferent authors. Th1

Fain. and ant and

1}?

.1 Shulman ‘s F,

tlezttn 1 10.36). He 1

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changes to their lessons and try new approaches. The next section describes several

frameworks that outline the knowledge needed for teaching.

2.2.3 Knowledge for Teaching Mathematics

Discussions of what teachers need to know to teach mathematics are prevalent in
the current research literature. Many of these discussions are based on the assumption
that mathematics content knowledge alone is not sufficient to assure effective teaching of
mathematics. Though many share this assumption, the exact knowledge necessary for a
mathematics teacher is not a settled issue and is characterized in different ways by
different authors. This section details the work of Shulman (1986, 1987), Grossman

(1990), and Ball and colleagues in this area.

2.2.3.1 Shulman’s Framework for Teacher Knowledge

One of the most classical characterizations of teacher knowledge is described by
Shulman (1986). He begins the argument for a framework for teacher knowledge that
takes into account content and pedagogy with a historical perspective on the knowledge
deemed necessary to be a teacher. In the 18003 and early 19005 the focus on teacher
knowledge was on content alone, but by the 19805 the emphasis on teacher knowledge
shifted to pedagogy. Shulman argues that both types of knowledge are important for
teachers and that one should not be focused on to the exclusion of the other. To address
this need for a dual focus, Shulman presents a theoretical framework with three types of
knowledge for teaching: (1) Content Knowledge, (2) Pedagogical Content Knowledge

and (3) Curricular Knowledge. He further refined this framework in 1987 to include a

34

 

tafu'll of seven ditlert
.ddiitenally includet
feir characteristics.
std purposes. and
;€:;£|.“‘:lletll content
tutti} the three I} pt
«Eu-used here.

The first I} per
:r::s:tt kntjm ledge. 1
tester requires Illt ire

T0 thtttk’ Phip

lllC‘ lthl\ (if U

\Ubtcct matte;

LiL‘L'i’l‘lt‘d trutl.

pron-Kim,” l\

01th pro” §\i1

praem‘c. (Silt)

a. a teacher of m t

.i'lla ‘
till lnmrlednc t1

 

TLiter ,
WI \8 10 mt: ti

”-31:19" -. . .
Ll kidflfied lhj\
t

   

l"tum,”
{éple\0n[in(y ‘
We ”1 ° “

total of seven different types of teacher knowledge. This formulation of the framework
additionally included: general pedagogical content knowledge, knowledge of learners and
their characteristics, knowledge of educational contexts, and knowledge of educational
ends, purposes, and values. Because this section of the literature review only focuses on
pedagogical content knowledge and characterizations of teacher knowledge that build on
it, only the three types of knowledge included in Shulman’s original framework will be
discussed here.

The first type of teacher knowledge included in Shulman’s (1986) framework is
content knowledge. The content knowledge Shulman describes as being necessary for
teacher requires more than a simple understanding of subject matter.

To think properly about content knowledge requires going beyond knowledge of

the facts or concepts of a domain. It requires understanding of the structure of the

subject matter. . .Teachers must not only be capable of defining for students the

accepted truths in a domain. They must also be able to explain why a particular
proposition is deemed warranted, why it is worth knowing, and how it relates to
other propositions, both within the discipline and without, both in theory and in

practice. (Shulman, 1986, p. 9)

Thus, a teacher of mathematics needs to know how to do mathematics but also how
mathematics as a discipline is structured and functions, both in its content and processes.

The second type of knowledge in Shulman’s (1986) framework is pedagogical
content knowledge (PCK) which is described as going “beyond knowledge of subject
matter per se to the dimension of subject matter knowledge for teaching” (p. 9). Shulman
further clarified this component of his framework:

Within the category of pedagogical content knowledge I include... the ways of

representing and formulating the subject that make it comprehensible to others.

Since there are no single most powerful forms of representation, the teacher must

have at hand a veritable armamentarium of alternative forms of representation,

some of which derive from research whereas others originate in the wisdom of
practice. Pedagogical content knowledge also includes an understanding of what

35

'5

“3" KW“ V;

*' -»:\x_ The ”C

”“ch th’
pr’L‘OnL‘CP
This. PCK immd

the mathematics It

The final C
t;t.--.=.tledge. has re-
issgnhes eurrieul.

the full ran
topics at ti ‘
those prtg'
euntraindre
particular e

EMF-nun pt’tintx nu

{fist-gums at the tin

‘r‘ 11

t. unnerxit-x tenet

Putnam and

2‘51th

on the PC]

enter the cla
their \ttttlen'
of informati
tttle rexeare
understand“

matter L‘ttntt‘

experienced
lid) \lUth \‘C‘b 10 t
17.! .

er . ~
‘Pkl‘lt‘ngc 1C ‘lf

.‘w-
I

XI \Q.

makes the learning of specific topics easy or difficult: the conceptions and
preconceptions that students of different ages and backgrounds bring with them to
the teaming of those most frequently taught topics and lessons. (p. 9)

Thus, PCK includes a teacher’s knowledge of the ways that students may interact with
the mathematics they are teaming.

The final component of Shulman’s (1986) original framework, curricular
knowledge, has received little attention compared to the other two components. Shulman
describes curricular knowledge as knowledge of

the full range of programs designed for the teaching of particular subjects and
topics at a given level, the variety of instructional materials available in relation to
those programs, and the set of characteristics that serve as both the indications and
contraindications for the use of particular curriculum or program materials in
particular circumstances. (p. 10)

Shulman points out that even more than PCK, which was not a focus of teacher education

programs at the time he presented his framework, curricular knowledge was absent from

the university teacher education curriculum.
Putnam and Borko’s (2000) review of literature on learning to teach included

research on the PCK of preservice teachers. They report that in general novice teachers
enter the classroom as a teacher for the first time with little information about who
their students are or what they know about subject matter being taught. This lack
of information affects their ability to design appropriate instruction. . .There is
little research evidence concerning novice teachers’ knowledge of specific
understandings and misunderstandings that children have about particular subject
matter content. Additional research is needed to explore how novice (and
experienced) teachers can be helped to acquire such knowledge. (p. 692)

This study seeks to contribute to this area by examining how one preservice teacher in a

field experience learns to consider what her students know during her lesson planning

process. The next section summarizes two studies that explore the existing PCK of

36

enndary mathemat

iitelopment of PCK
3-3.3.2 Studies Ifxan

De~pite the 11:

:fsen'iee teachers at
tsxgeet to SMPTs \\ e
{the studies conside
f-‘T-Aitlt’ di‘OUl \pt‘t’llilt

Eten t 19931 e
ith in the area of tut
etherxetideneed kr
{iii-Stun. namel} arhit

”it Hbtltart' n

dlUrc Ul-

 

 

secondary mathematics preservice teachers (SMPTS) and in one of these studies the

development of PCK.

2.2.3.2 Studies Examining the PCK of Secondary Mathematics Preservice Teachers

Despite the fact that studies and discussion of PCK are relatively prevalent in the
literature, there are surprisingly few studies that look at PCK in relation to SMPTS.
Currently, there appears to be more inquiry into the mathematical PCK of elementary
preservice teachers and practicing teachers. Three studies that do examine PCK with
respect to SMPTS were conducted by Even (1993), Pitts (2003), and Kinach (2002). All
of the studies consider PCK in terms of the mathematical explanations the SMPTS
provide about specific content.

Even (1993) examined the relationship between SMPTS content knowledge and
PCK in the area of function. Specifically, Even was interested in whether the preservice
teachers evidenced knowledge of the two essential features of the modern definition of
function, namely arbitrariness and univalence. Definitions that did not take into account
the arbitrary nature of functions were characterized as “old.” For this study PCK was
defined as “knowing the ways of representing and formulating the subject matter that
makes it comprehensible to others as well as understanding what makes the learning of
specific topics easy or difficult” (p. 94-95). Data was collected via questionnaires
completed by 152 SMPTS and 10 follow-up interviews.

Even (1993) found that when asked to give a definition of function only 78 of the
152 participants were able to supply a modern definition, while 53 supplied an old

definition, 1 1 other definitions (some kind of a rule, such as the vertical line test, instead

37

 

t definition 1. and l

[55,; the} would prt‘st‘
nation on!) 27 of th
peeripants gate an e
fit-tings indicate that
titunetittn and that n
ifiiftlllt‘n of fund“ ‘ll

item of functions

‘Il‘f .0
~;ll

.iun than meanin:

a

suit .id “hen chum
tnlct‘ and get the rt:
.nzngs indicate that

_rigtz-qtn did not Quart

m definition of 1

iii“; 1. ' 7 '
mngtn dt‘ttnltlt‘n

Another \lU'tl\

r .- v
AJ| l‘._ .
mtlJll was conduct

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of a definition), and 10 provided no response. In contrast, when asked to give a definition
that they would present to a student with difficulties understanding the definition of
function only 27 of the 152 SMPTS came up with a modern definition. Sixty-seven of the
participants gave an old definition, 36 other definitions, and 22 gave no response. These
findings indicate that many of the SPMTs in the study did not have a modern conception
of function and that many of those that did not have sufficient PCK to construct a modern
definition of function for students. Even posits that the lack of attention to univalent
property of functions might lead SMPTs to focus more on procedural understanding of
function than meaning. “That is exactly what many of the prospective teachers in this
study did when choosing to provide the students with the ‘vertical line test’ as a rule to
follow and get the right answers without concern for understanding” (p. 112). Even’s
findings indicate that for the SMPTS in her study, having adequate content knowledge of
function did not guarantee fully developed PCK, because not all of the SMPTS with a
modern definition of function were able to represent it in the modern form or with
meaning in definitions for students.

Another study that looked at content knowledge and PCK of SMPTS in the area of
function was conducted by Pitts (2003). Pitts examined these two types of knowledge
with respect to making translations between algebraic and graphical representations of
functions. Using a two part questionnaire completed by 59 SMPI‘s Pitts gathered data
about (1) how the SMPTS approached problems that called for translations, (2) examined
students’ responses to problems that called for translations, and (3) attempted to address

students’ misconceptions concerning problems that called for translations. To analyze the

 

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responses from the questionnaire Pitts identified the approaches the SMPTS were taking
to translations as process oriented, object oriented, both, or neither.

In looking at the preservice teachers approaches to translation problems in the
first part of the survey, Pitts (2003) found that 90% of the participants in her study had a
dynamic mental model of translations. That is, their mental models of function were
comprised of both process and object orientations and she further found that many of the
preservice teachers were able to move freely between the two perspectives. When
analyzing student work in the second part of the questionnaire the preservice teachers
were asked how they thought the students were thinking about the representations. A5
with the first part of the questionnaire, their responses were coded as process, object,
both, or neither. Pitts found that the preservice teachers’ mental models of the students
thinking were usually accurate when either a process or object orientation was present in
the students’ responses. However, almost 40% of the overall responses were coded as
neither process nor object. Finally, the preservice teachers were asked if they saw any
misunderstandings, or misconceptions, in the students’ responses in the second part of the
questionnaire and if so how they would address these misunderstandings, or
misconceptions. Looking at how the preservice teachers’ explanations of how they would
address the misunderstandings, or misconceptions, Pitts found that more than half of the
explanations, what she referred to as pedagogical approaches, offered made use of neither
the process or object models, even when the preservice teacher had used the model to
solve similar problems themselves. Pitts findings indicate that there are differences

between preservice teachers content knowledge that they use to solve mathematical

 

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problems themselves and their PCK that they use to think about student responses and
construct mathematical explanations.

The final study of SMPTS PCK addresses a content area different than the first
two, namely the subtraction of integers. Kinach (2002) explored the content knowledge
as well as the PCK of 21 SMPTS in terms of Skemp’s (1978) instrumental and relational
understanding. Using three tasks about integer subtraction and discussions and reflections
around those tasks, Kinach challenged the SMPTs’ thinking about their own
understanding of integer subtraction and about how their future students might receive
their explanations of integer subtraction. Each task called on the SMPTS to construct an
explanation about integer subtraction. The first specified no context, the second a number
line context and the third an algebra tile context. As the SMPTS developed and discussed
their explanations they saw that their original explanations, and in fact their own
understandings, were more instrumental than the later explanations in context. Kinach
concluded that her strategy for transforming the PCK of the SMPTS was useful and that it
could be employed for other content.

Beyond providing these specific mathematical insights into changes in the

prospective teachers’ PCK and SMK [subject-matter knowledge] for the example

of addition and subtraction of integers, the results of the present study refine our

understanding of the transformation process itself. (Kinach, 2002, p. 64)

All three of these studies shed light on the relationship between SMPTs’ content
knowledge and their PCK. The authors of these studies all examined PCK using the
preservice teachers’ mathematical explanations. This particular type of PCK has been

characterized by Grossman (1990) as knowledge of instructional strategies". Even (1993)

found that a “modern” understanding of function did not ensure “modern” mathematical

 

3 . , . . .
A complete outline of Grossman s PCK framework can be found rn the next section.

40

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explanations of function. Similarly, Pitts (2003) found that the dynamic mental models of
translations (representing both the product and object perspective) did not guarantee the
explanations of translation were dynamic or in fact would include either a process or
object characterization of function. And Kinach (2002) showed that the type of content
knowledge, either relational or instrumental, impacted the resulting mathematical
explanations and that through carefully designed assignments preservice teachers were
able to recognize the limitations of their explanations and to develop explanations that
were more relational. The next section provides an overview of Grossman’s four forms of
PCK as they appear in her model of teacher knowledge, and highlights one of them as

representing the type of PCK being examined in this study.

41

 

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2.2.3.3 Grossman’s Model of Teacher Knowledge

Grossman’s (1990) model of teacher knowledge builds on Shulman’s (1986)

framework of the knowledge needed for teaching (Figure 1).

 

 

 

 

 

SUBJECT MA'I'I'ER
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Figure 1. Grossman (1990) Model of Teacher Knowledge

The first component of PCK in Grossman’s model comprises of the conceptions teachers
hold about the purposes for teaching a subject at different grade levels. Grossman states
that conceptions are represented by the teachers’ goals for teaching particular subject
matter. The second component, knowledge of students understanding, includes

knowledge of conceptions and misconceptions of particular subject matter. Grossman

42

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explains that “to generate appropriate explanations and representations, teachers must
have some knowledge about what students already know and what they are likely to find
puzzling” (p. 8). This type of teacher knowledge is the focus of this study. The next
section describes how Deborah Ball and her colleagues have also built on Shulman’s
PCK to further characterize this type of teacher knowledge. Curricular knowledge is the
third component of Grossman’s model. This component includes knowledge of available
curriculum materials as well as knowledge of the vertical curriculum, or the topics that
are traditionally taught before an after what the teacher is teaching. The final component
of the model is knowledge of instructional strategies. Included in this is knowledge of
“metaphors, experiments, activities or explanations that are particularly effective for

teaching a particular topic” (p. 9).

2.2.3.4 Ball and Colleagues’ Mathematical Knowledge for Teaching

The work of Deborah Ball and her colleagues also builds on Shulman’s
framework of teacher knowledge. Hill, Ball, and Schilling (2008) describe the knowledge
needed for teaching as mathematical knowledge for teaching which they define as “the
mathematical knowledge that teachers use in classrooms to produce instruction and
student growth” (p. 374). Like Shulman (1986, 1987) and Grossman’s (2000)
characterizations of teacher knowledge, mathematical knowledge for teaching is
comprised of differing types of knowledge that represent both subject matter knowledge

and pedagogical content knowledge (Figure 2).

 

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The type of knowledge in this model that is of particular importance to the proposed
study is knowledge of content and students (KCS) which is defined as “content
knowledge intertwined with knowledge of how students think about, know, or learn
particular content” (p. 375). KCS incorporates elements of Schulman’s PCK and
Grossman’s knowledge of students understanding but also includes knowledge of
content. This addition is important to the formulation of knowledge being examined in
this study because one way that teachers consider how students will interact with the
mathematics is to interact with the mathematics themselves. Teachers’ knowledge of the
mathematics, both at the level at which students engage with it and at higher levels of

understanding, contributes to their ability to consider how students will respond to the

44

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mathematics as they plan lessons. Through the development and testing of measures Hill,
Ball, and Schilling have found that KCS is a type of knowledge held by teachers. “The
factor analysis of multiple forms and interviews with teachers suggest that familiarity
with aspects of students’ mathematical thinking, such as common student errors, is one
element of knowledge for teaching” (p. 395).
A second type of knowledge that will be important for this study is specialized
content knowledge (SCK). Ball, Thames, and Phelps (2008) define SCK as:
knowledge not typically needed for purposes other than teaching. . .This work
involves an uncanny kind of unpacking of mathematics that is not needed—or
even desirable—in settings other than teaching. Many of the everyday tasks of
teaching are distinctive to this special work. (p. 400)
Although SCK is defined by Ball and colleagues as subject matter knowledge rather than
pedagogical content knowledge, it is a type of knowledge that teachers likely use in
preparing lesson plans. In preparing lessons plans and making use of SCK, teachers may
begin to consider their students’ reactions which could lead to making use of KCS. The

next section delves further into these ideas by describing the practice of anticipating

students’ mathematical responses.

2.2.4 Anticipating Students’ Mathematical Responses as a Practice of Lesson Planning

Anticipating students’ mathematical responses during lesson planning is a
practice adopted by some teachers of mathematics that receives its current name from
Stein, Engle, Smith, and Hughes (2008). While using this term to describe this particular
practice is relatively new to some in the field, the practice itself is not. Leinhardt (1988)
described the routines of expert teachers as they related to planning and teaching lessons.

In constructing their lesson scripts, or the set of actions they would use to teach a

45

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particular topic, expert teachers drew on their knowledge of how students engaged with
the mathematics the last time they taught that tOpic. “Teachers seem to construct flags for
themselves that signal material that will cause difficulty as it is being learned, and then
they adjust their teaching of the topic in response to those flags or to past successes of
what ‘worked’” (Leinhardt, 1988, p. 51). Leinhardt also pointed out that novice teachers
lack the knowledge of students needed to do this.

A lesson given by an effective teacher who has been teaching for many years
essentially contains layers of accumulated knowledge about the topic and how to
teach it. . .Without these [lesson] scripts novice teachers face two problems. First,
they are frequently drawn off their focus to follow a particular student in ways
that are not helpful for the rest of the class; and second, they fail to anticipate the
crucial feature or dimensions of what is important or difficult about a tOpic. (p.
52).

Schoenfeld (1998) also addresses a concept that includes anticipating students’
mathematical responses, which he describes a lesson image. A lesson image is:

the teacher’s envisioning of the possibilities and contingencies of a lesson. The
teacher’s lesson image includes knowledge of his or her students and how they
may react to parts of the planned lesson; it includes a sense of what students are
likely to be confused about, and how the teacher might deal with that confusion;
and more. (p. 18)

Schoenfeld goes on to present an example of a lesson image from a teacher named
Nelson. The excerpt of the lesson image provided picks up after Nelson has explained
that his students will have little trouble with problems (a) and (b).

In contrast, Nelson expected some initial student confusion with problem (c). He

planned to begin as with problems (a) and (b), soliciting student answers. There

would, he thought, be some different answers, and some students who said that

didn’t know or understand. To deal with the confusion Nelson planned to work
5 x - x - x - x - x

through the example at the board, expanding £— as , canceling the
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x’s, and obtaining x0 = las a result. (p. 40)

46

 

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Finally, in her book that describes how she has used problems to teach
mathematics, Lampert (2001) provides an example of how she anticipates students’
mathematical responses (SMRS) while planning a lesson around the relationship between
multiplication and division. She began by considering the different strategies that fifth
graders might use to solve the problem she will pose.

With this work, I was anticipating where my students might get stuck or distracted

as well as what might provoke productive work. I needed to think of all the things

they would or could do when presented with the problem. This kind of
preparation showed me what words might be useful in talking about their
solutions, as well as what drawings they or I might use to support their studies. To
respond to their work in a thoughtful way, I needed to be able to anticipate what
they might be able to do independently and where they would need information

from me to proceed productively. (p. 103)

Leinhardt (1988), Shoenfeld (1998), and Lampert (2001) provide evidence that
anticipating SMRS is an explicit practice of some mathematics teachers. The next
sections further describe this practice by detailing a framework for teachers that includes
anticipating SMRS and discussing two widely-used professional development models that
make use of the practice.
2.2.4.1 Anticipating Students Mathematical Responses as Part of a Model for
Facilitating Mathematical Discussion

Anticipating students’ mathematical responses is the first practice in a five-
practice model for planning and facilitating mathematical discussion developed by Stein,
Engle, Smith, and Hughes (2008). Although the authors do not define what they mean by
a practice in this model, their use of the term resonates with Greeno’s (1998) definition of

practices as “regular patterns of activity in a community, in which individuals

participate” (p. 6). The model was developed to guide mathematics teachers in their role

47

 

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as discussion facilitator; a role which is critical in reform-oriented visions of mathematics
teaching. To provide the rationale for such a model Stein et al. begin by describing their
view of the reform movement in mathematics education. The “first generation” (p. 316)
of this movement brought with it a new vision of mathematics classrooms. In these
classrooms students engaged with realistic and complex mathematical tasks and made
sense of the mathematical ideas through their own work with the tasks and class
discussion. Stein et al. point out that in this generation of reform the role of the teacher in
helping students make sense of mathematics through class discussion was “ill-defined”
(p. 316). The result was mathematics classrooms in which discussions were often more of
a “show and tell” (p. 318) of mathematical solutions than a site for building meaningful
mathematical ideas. In response to the lack of direction for teachers the, “second
generation” (p. 319) of reform in mathematics came about. This generation of reform “re-
asserts the critical role of the teacher in guiding the mathematical discussions” (p. 320)
by attempting to identify teacher practices that would support the types of discussions
being advocated for in the first generation. Stein et al. address this goal by presenting five
key practices for facilitating mathematical discussion based on recommendations from
the mathematics education literature that “can make student—centered approaches to
mathematics instruction more accessible and manageable for more teachers” (p. 334).
The authors suggest that these practices are especially suited to teachers who are new to
the idea of facilitating mathematical discussions. “We argue that novices need a set of
practices they can routinely do to both prepare them to facilitate discussions and help
them gradually and reliably learn how to become better discussion facilitators over time”

(p. 321).

48

 

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The five practices from Stein et al. (2008) presented in this section are meant to
aid teachers in the process of planning and implementing strategies to facilitate
mathematical discussion in their classrooms. While only the first practice is the focus of
this study, descriptions of the other practices are also provided to help provide context.
The first of the five practices calls on teachers to anticipate students’ mathematical
responses (SMRs).

Anticipating students’ responses involves developing considered expectations

about how students might mathematically interpret a problem, the array of

strategies—both correct and incorrect—they might use to tackle it, and how those
strategies and interpretations might relate to the mathematical concepts,
representations, procedures, and practices that the teacher would like his or her
students to Ieam (Lampert, 2001; Schoenfeld, 1998; Yoshida, 1999, cited in

Stigler & Hiebert, 1999). (p. 322-323)

As a part of the lesson planning process, anticipating student responses would come after
the selection of a mathematically appropriate and cognitively demanding task. Stein et al.
present a variety of methods teachers could use to anticipate students mathematical

f€SpOUSCSI

' Completing the task themselves and considering a variety of possible solution
strategies from the perspective of the students in their classrooms

' Working with other teachers to consider possible anticipated SMRs
. Drawing on knowledge from the research literature
I Using curricula that provide possible SMRs

' Watching or reading video and written cases of other teachers teaching with
similar tasks

The methods, or sources, of anticipation suggested by Stein et al. are generally available

to teachers. Because anticipation is not necessarily based on teaching experiences, the

49

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practice of anticipating is accessible to preservice and beginning teachers as a way to
begin to develop their skills in facilitating mathematical discussions.

The second practice recommended by Stein et al. (2008) is monitoring students’
responses. Teachers can do this while students are working on the mathematical task by
walking around and observing the different solution strategies and student thinking. The
authors explain that this practice encompasses more than just circulating to keep students
on task; the teacher who is monitoring students’ responses needs to “actively attend to the
mathematics within what the students are saying and doing, assess the mathematical
validity of students’ ideas, and make sense of students’ mathematical thinking even when
something is amiss (Nelson, 2001; Shifter, 2001).” (p. 326). Further, they go on to point
out that anticipating SMRs, the first practice in the model, supports the practice of
monitoring. “Those teachers who have made a good faith effort during initial planning to
anticipate how students might respond to a problem will feel better prepared to monitor
what students actually do during the explore phase (Lampert, 2001; Schoenfeld, 1998)”
(p. 326).

The third practice in the Stein et al. (2008) model is purposefully selecting student
responses for public display. It is during this phase that teachers plan what solutions and
ideas they want to highlight in the upcoming discussion. This practice allows the teacher
to ensure that the desired mathematical ideas and representations are presented and
discussed, that misconceptions are put forth and resolved and that ideas not considered by
any of the students in the class are brought forward. As with monitoring, anticipating

SMRs is helpful for teachers to later select students’ responses to make public.

50

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Like the previous three, the fourth practice, purposefully sequencing student
responses, builds on the others. Stein et al. (2008) explain the value of this practice for
mathematical discussions, “by making purposeful choices about the order in which
students’ work is shared, teachers can maximize the chances that their mathematical
goals for the discussion will be achieved” (p. 329). The authors provide several possible
ways of sequencing student responses including presenting the strategy used by most
students first or beginning with a common misconception. They stress the fact that there
is no best order but that teachers should consider their knowledge of students and their
goals when deciding how to sequence the presentation of student responses.

The final practice for teachers facilitating mathematical discussions in the Stein et
al. (2008) model is connecting students’ responses.

They can help students make judgments about the consequences of different

approaches for the range of problems that can be solved, one’s likely accuracy

and efficiency in solving them, and the kinds of mathematical patterns that can be
most easily discerned. They also can help students see how the same powerful
idea (e.g., there is a multiplicative relationship between quantities in a ratio) can
be embedded in two strategies that on first glance look quite dissimilar...So,
rather than having mathematical discussions consist of separate presentations of
different ways to solve a particular problem, the goal is to have student
presentations build on each other to develop powerful mathematical ideas.

(p. 330)

This final practice is the last step towards creating the kind of mathematical discussion
set forth by the authors at the outset of the article, specifically one in which teachers
“orchestrate whole—class discussions that use students’ responses to instructional tasks in
ways that advances the mathematical learning of the whole class” (p. 314).

As already stated, the practices advocated by Stein et al. (2008) are not new to

mathematics educators; they have just been put together in a novel way in a model that is

approachable and accessible to experienced teachers thinking about teaching mathematics

 

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in a new way and to preservice teachers just beginning to think about how they will teach
mathematics. The practice of anticipating SMRs is also part of two other professional
development activities taking place in the United States and other countries. The
Cognitively Guided Instruction (CGI) model of professional development has been used
in the United States to help elementary teachers make use of their student thinking in
their teaching. The tradition of lesson study has been used in Japan, and is enjoying
increasing popularity in the United States, as a way to help teachers study their own
practices. The following sections outline CGI and lesson study and explain how

anticipating SMRs is an integral part of each.

2.2.4.2 Cognitively Guided Instruction

Cognitively Guided Instruction (CGI) employs a specific strategy of anticipating
SMRs presented by Stein et al. (2008), namely drawing on knowledge from the research
literature. CGI is a professional development program for elementary mathematics
teachers based on the tenets that “1) Instruction must be based on what each learner
knows, 2) Instruction should take into consideration how children’s mathematical ideas
develop naturally, and 3) Children must be mentally active as they learn mathematics”
(Fennema, Carpenter, & Peterson, n.d., p. 15) In working with teachers the CGI program
addresses these tenets by focusing on the development of student’s mathematical thinking
and building on the teachers existing knowledge (Carpenter, Fennema, Franke, Levi, &
Empson, 2000), using research on student’s thinking in particular content areas.

An example of the kind of research on student thinking CGI uses in professional

development can be found in Carpenter, Fennema and Franke (I996). The authors

52

 

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at: Franke expla
archers to under
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provide a variety of research including some about classes of addition and subtraction
problems which they characterize as involving “(a) joining actions, (b) separating actions,
(c) part-part-whole relations, and ((1) comparison situations” (p. 6). Carpenter, Fennema
and Franke explain that their analysis “provides a coherent, principled framework for
teachers to understand children’s development of basic whole-number concepts” (p. 13).
Knowledge of this kind of framework contributes directly to teachers’ ability to anticipate
SMRs: “The CGI framework also provides teachers a coherent basis for identifying what
is difficult and was is easy for students and for dealing with common errors they make”
(p. 14). The knowledge of student learning that teachers in the CGI program have access
to allows them to anticipate students’ mathematical responses by providing them a
research base to make the “considered expectations” to which Stein et al. (2008) refer.
Research focusing on the outcomes of CGI has identified positive results for both
teachers and their students. In their study of 20 elementary teachers enrolled in a summer
CGI workshop, Carpenter, Fennema, Peterson, Chiang, and Loef (1989) found that once
they returned to their classrooms:
CGI teachers more often posed problems to students and more frequently listened
to the process used by students to solve problems. In contrast, in giving feedback
on students’ solutions to problems, control teachers focused more frequently on
the answer to the problem than did the CGI teachers. (p. 520)
In addition, there was a slight difference in student achievement between the students in
the classes of the CGI teachers and those in the classes of the control teachers. “Although

differences in student achievement were modest, the differences found consistently

favored the CGI treatment group” (p. 526).

53

 

12.4.3 Lesson Slut

Thelapanes
nits-2mm from CGI
atcipate SMRs in
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2.2.4.3 Lesson Study

The Japanese practice of lesson study differs as a professional development
program from CGI in that it does not provide information that will help a teacher
anticipate SMRs in the way that Stein et al. (2008) describe. Instead, lesson study expects
that teachers will anticipate SMRs and provides a lesson framework and community that
supports teachers in doing this. Lesson study holds a prominent position in the
professional development of Japanese teachers and dates back to the early 1900s
(Fernandez, 2002; Shimuzu, 2002). “Lesson study brings together groups of teachers to
discuss lessons that they have first jointly planned in great detail and then observed as
they unfold in actual classrooms” (Fernandez, 2002, p. 393). Although the practice
originated in Japan, US. teachers have recently taken up the practice for their own
professional development (Chokshi & Fernandez, 2004). The structure of lesson study
incorporates several features that support teachers in anticipating SMRs.

They begin the process ...by reading about what other teachers have done, what

ideas are recommended by various educational groups, what has been reported on

students’ learning of this topic, and so on. They design several lessons, one group
member tries them out while the others observe and evaluate what works and
what does not, and they revise the lessons. They often base changes on specific

misunderstandings students’ evidence as the lesson progresses. (Hiebert & Stigler,
2000, p. 10)

In addition to taking these steps which are intended to produce increasingly better
lessons, the written lesson plans themselves include what the teachers anticipate that
students will respond. Shimizu (2002) presents the following as a common framework for

lesson plans that are designed in the lesson study process (Figure 3).

54

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Students prohlen
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(Exercise/Extension)

 

 

Figure 3. Shimizu's (2002) Common Framework for Lesson Plans

Notice that the anticipated SMRs are thought out for each phase of the lesson and are
considered before the planned actions of the teacher.

To get a clear picture of what anticipating SMRs might look like in a written
lesson plan, Shen, Poppink, Coi, and Fen (2007) provide an example. Although they
present a lesson from a Chinese teacher and not Japanese teacher involved in lesson
study, the Chinese lesson has many of the same components as the Japanese lesson study
lesson plans and can still serve as an example for those unfamiliar with the practice of
anticipating in a written lesson plan. Shen et al. present steps that are common to the
Chinese teachers’ lesson planning process "(a) specifying cognitive and affective
objectives; (b) identifying key points of the content; (c) anticipating difficult points for
students; and (d) designing the lesson flow" (p. 251). As in Japan, Chinese teachers use
lesson planning as more than just a list of activities to complete in class, instead they
view it as "a practice of professional responsibility and development" (p. 248).

The lesson plan presented by Shen et al. (2007) was designed to allow students to

explore the sum of measures of internal angles of a polygon. In anticipating SMRs the

55

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teacher identifies a possible difficulty students might encounter: "Difi‘icult point: a
student’s understanding that the vertices of a polygon must be on the same plane, a
necessary condition that is diflicult for many students to understand" (p. 252). Also

included in the plan are a variety of possible student strategies (Figure 4).

 

 

 

 

 

 

Figure 4. Representation of Possible Strategies Identified by the Teacher That Students
Might Employ to Find the Interior Angle Sum of a Quadrilateral (Shen et al., 2007)
This lesson plan considers two of the main aspects of anticipating SMRs outline by Stein
et al. (2008), namely student solution strategies and common student difficulties.

The next section presents the research questions for the study and outlines the
framework created by the researcher for use in this study. This framework is based on the
situative perspective and the literature reviewed here about teacher knowledge and the

practice of anticipating.

56

 

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2.3 Research Questions and Framework

This study made use of the situative perspective to investigate a field experience
by describing the situations and communities that made up the field experience and by
“working inward” as described by Greeno (1998) to look at the practices of one SMPT as
she participated in these situations and communities. During field experiences preservice
teachers are influenced by a variety of contextual factors including university courses,
mentor teachers, field instructors, school communities, students, available resources, and
possibly unknown factors. The focus of the situative perspective on environmental
contexts, social interactions, and tools necessitated that the researcher take into account
these contextual factors. Additionally, the researcher was able to focus on a more
cognitive aspect; that is, one preservice teacher and her developing lesson planning
practices.

To narrow the scope of the study within the field experience, this study focused
on a specific lesson planning practice, namely anticipating students’ mathematical
responses. A practice is defined here as Greeno (1998) defines it as “regular patterns of
activity in a community, in which individuals participate” (p. 6). This practice was
selected as the focus of this study for three reasons. First, research has shown that the
practice of anticipating students’ mathematical responses relies on a type of knowledge
that preservice teachers often do not have access to when they enter field experiences.
Second, anticipating students’ mathematical responses represents a practice not yet
studied in relation to SMPTS. Third, this practice was a key practice advocated by the

secondary mathematics teacher preparation program at the university where this study

57

 

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took place. For this study, the practice of anticipating students’ mathematical responses
will be defined as considering how students will react to the lesson, specifically the
mathematics and the mathematical tasks, as the lesson is being planned or any time prior
to the teaching of the lesson. This practice encompasses other practices of teaching that
are described in relation to the upcoming anticipation framework.

Using this definition of anticipating students’ mathematical responses and the
situative perspective to examine both the context of the experience and to narrow in on
the experience of one preservice teacher, the study addressed the following research
questions:

1.. What is the nature of a preservice teacher's practice of anticipating students'
mathematical responses during lesson planning over the first semester of her field
experience?

2. What factors within the contexts of a teaching internship (a field experience and
accompanying university course) promote or do not promote the practices of one
preservice teacher anticipating students’ mathematical responses during lesson
planning?

The research framework employed in this study was developed based on the

previously reviewed literature and the situative perspective. Figure 5 shows the

anticipation framework developed for this study.

58

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Figure 5. Anticipation Framework

59

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The anticipation framework is divided into active/purposeful and
passive/incidental components, both of which have the potential to lead to three types of
outcomes: student solution strategies (SSS), student naive conceptions (SNC), or student
questions and difficult points (SQDP). These outcome categories were based on the
literature that showed that as teachers plan mathematics lessons they anticipate several
types of student responses. First, teachers anticipate students’ solution strategies by
considering various ways that students could solve problems and approach tasks
(Lampert, 2001; Shen, Poppink, Cui, & Fan, 2007; Stein, Engle, Smith, & Hughes,
2008). Second, teachers are mindful of the naive conceptions or misconceptions that
students bring with them to class which will have an impact on how they interact with the
lesson (Carpenter, Fennema, & Franke, 1996; Stein, Engle, Smith, & Hughes, 2008).
Third, teachers identify points in the lesson that may be difficult for students and where
they may need extra help to successfully participate in the lesson (Carpenter, Fennema, &
Franke, 1996; Gaea Leinhardt, 1988; Schoenfeld, 1998; Shen, Poppink, Cui, & Fan,
2007).

The active/purposeful side of the framework represents the actions taken when a
teacher chooses to anticipate students’ mathematical responses (SMRs) as described by
Stein et al. (2008). In other words the teacher “makes an effort to actively envision how
students might mathematically approach the instructional tasks(s) that they will be asked
to work on” (p. 322). After a teacher has made the choice to anticipate SMRs during
lesson planning they can choose to participate in one or more activities that would help
them to do this. Two of the three outside resources listed on the left side of the

framework were suggested by Stein and colleagues. An additional resource, other

60

 

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teachers, was added to this side of the framework based on the situative perspective,
specifically the focus on the distributed nature of knowledge. When the teacher consults
these resources, attempts to complete the task in multiple ways, or takes the position of
students they are drawing on either their own knowledge of content and students (KCS)
or the KCS that is distributed in the mathematics education community.

The passive/incidental side of the anticipation framework is made up of the same
resources and activities that appear on the active/purposeful side of the framework. The
difference is that the right side of the framework represents participation in these
activities or use of these resources for reasons other than the anticipation of SMRs. All of
these resources and activities have purposes in teaching other than anticipation. Despite
the originally intended purpose, participation in these activities or use of these resources
might contribute to a teacher anticipating SMRs if they come across this knowledge and
choose to make use of it. One difference between the elements on the active/purposeful
side and the elements on the passive/incidental side is that when the teacher completes
the activity in one way to prepare for class they are likely not making use of KCS. KCS
includes knowledge of how students respond to particular content, but in this case the
teacher is preparing one strategy to present to students. In this instance the teacher is
likely drawing on their own specialized content knowledge (SCK) to prepare one solution
method for the lesson that is based on their own way of looking at the mathematics.

In addition to the anticipation practices, resources, and outcomes, the anticipation
framework also takes into account the influence of the situated nature of the experience.
The Zone of Free Movement (ZFM) is represented by the resources that appear in the

framework. For example, if the teacher does not have access to research literature they

61

 

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will not have the ability to use it to anticipate or for any other reason. The Zone of
Promoted Action (ZPA) has bearing on which path through the framework a teacher
chooses to take.

In addition to the anticipation framework, this study makes use of a combination
of Goos’ (2005a, b) framework and Engle and Conant’s (2002) principles of productive
disciplinary engagement. In this part of the research framework for this study, the ZFM
and the ZPA are each composed of two of the principles:

Zone of Free Movement

- Relevant Resources

- Authority

Zone of Promoted Action

- Problematizing Content

- Accountability to Others and Disciplinary Norms
ln Goos’ framework the ZFM represents what is possible, which in her characterization
includes teaching resources and curriculum and assessment requirements. Thus, in this
study the ZFM represents the resources available to support the practice of anticipating
and the use of authority that allows for the practice of anticipating. The ZPA represents
the actions of the university or school faculty to promote particular practices. In this
study the ZPA is comprised of the actions or elements of the contexts that problematize
teaching and/or mathematics in ways that lead, or do no lead, to the practice of
anticipating and the norms that the intern is held accountable to that promote or do not
promote anticipating. The next chapter details how the methodology of this study,

including how the anticipation framework was used for data analysis.

62

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CHAPTER 3. METHOD

This study examined how one secondary mathematics preservice teacher
anticipated students’ mathematics responses in her lesson planning during the first
semester of her internship year. This chapter outlines the methods, data sources, and
analytic techniques used in this investigation.

The process of inquiry used in this project was primarily case study. In case
studies, researchers explore “in depth a program, an event, an activity, a process of one or
more individuals” (Stake, 1995 as cited in Creswell, 2003, p. 15). Further, case studies
are defined by “interest in individual cases, not by the methods of inquiry used” (Stake,
2000, p. 435). They are “bounded by time and activity, and researchers collect detailed
information using a variety of data collection procedures over a sustained period of time”
(Stake, 1995, cited in Creswell, 2003, p. 15). This study’s design met these criteria in that
it focused on one individual, Megan, and her practices and interactions during the first
semester of her field experience and used a variety of data to build the case study.

Another process of inquiry present in this study was ethnography. In ethnography,
researchers study “an intact cultural group in a natural setting over a prolonged period of
time by collecting, primarily, observational data” (Creswell, 2003, p. 14) with an intent to
“obtain a holistic picture of the subject of study with emphasis on portraying the
everyday experiences of individuals by observing and interviewing them and relevant
others (Creswell, 2003, p. 200). By studying Megan’s interactions in both of the contexts
of her field experience, as she planned lessons and worked, this study has made use of
ethnographic methods. The researcher in this study was a teaching assistant in one of the

courses taken by Megan and her peers prior to the study and continued to serve as a

63

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resource and source of support to the preservice teachers during the study. In this way,

the researcher participated in participant observation as described by Jacob (1987). “As
the name implies, participant observation combines both participation in the culture being
studied and observation of the patterns” (Jacob, 1987, p. 14).The following sections

further describe the contexts and participants as well as the data collection and analysis

methods employed in this study.
3.1. Context

The two main contexts in which the preservice teacher, Megan4, operated were

sites for analysis in this study. The following two sections describe each of these

contexts.
3.1.1 University Context

The university context was set in the mathematics teacher education program in a
large Midwestern university. Prospective teachers in this program spend five years at the
university. All prospective teachers graduate with a mathematics degree after four years
and obtain a teaching license after their fifth year of structured field experience, referred
to as the internship year. In addition to the education coursework required of all
prospective teachers seeking licensure at the university, secondary mathematics
preservice teachers enroll in four mathematics-specific methods courses in their fourth

and fifth years of study. The first set of courses will be referred to as the senior year

 

4 .
In order to protect anonymity all names are pseudonyms.

64

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methods courses and the second set, which take place during the internship year, simply
as the methods courses.

The university’s internship program is designed as a year-long experience in the
fifth year of the teacher preparation program. During the semester that data was collected,
the program elements remained consistent with what they had been in previous years and
all portions took place as expected. The interns spent the entire school year in a school
teaching mathematics classes alongside their mentor teacher. Figure 6 shows the intended
schedule of the teaching load and university courses, both general and content specific, as

described in the teacher education program handbook.

65

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Figure 6. Schedule of Intern Teaching Responsibilities

66

The experience is designed so that the interns take full responsibility for one class, called
the Focus Class, beginning the first week of the school year and continuing until the end
of the field experience. During the first semester the intern is to have primary
responsibility for one additional class during Lead 1 and two additional classes during
Lead 2. The secondary teacher preparation program handbook provides specific guidance
on the expectations of the intems’ development of lesson planning practices within the
internship experience.

In lesson planning, the intern Ieams to design purposeful and practical

activities that carry a unit forward. She or he learns to anticipate and adapt

to situations that might arise in those activities, to use time efficiently, and

to allow for the unexpected. Again, the Mentor Teacher and field

instructor offer support, feedback and coaching. (Handbook, p. 14)

In addition to teaching in a school during the internship year, the interns
continued to attend classes at the university one day a week for ten weeks of the 15
weeks of the semester. During the first semester the university courses did not meet
during the Lead 1 and Lead 2 weeks. On the days that the interns returned to the
university they participated in two courses, one general education course focusing on
professional roles and teaching practices and the other, the methods course, which was

specifically related to reflection and inquiry in secondary mathematics teaching. The next

sections describe the senior year and internship methods courses.

3.1.1.1 The Senior Year Methods Courses

During their fourth year in the program the secondary mathematics preservice
teachers enrolled in two methods courses, one during the fall semester and one during the

spring semester. Although most of the preservice teachers completed a bachelor’s degree

67

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in mathematics by the end of their fourth year, the senior year methods courses were the
first classes they took in the college of education related specifically to the teaching of
mathematics. An important aspect of the design of the senior year methods courses was
the field component. The preservice teachers were required to spend four hours per week
in a local middle or high school mathematics class and to teach several lessons each
semester. Many of the course assignments and discussions in the senior year methods
courses were based on the preservice teachers’ experiences in this field experience.
Another key component of the fall section of the senior year methods course was the
microteaching lab, in which the preservice teachers collaboratively wrote and
individually taught two to three lessons to their peers and received feedback from peers
and instructors.

The goals of the senior year methods courses as stated in the syllabi were that
preservice teachers would:

I Deepen and connect mathematical content knowledge with student
mathematical understanding.

I Develop a deeper understanding of the mathematics content you are
teaching and how it can be meaningful for your students.

I Analyze from a new perspective what mathematics is and what it
means to Ieam, do, and teach mathematics.

I Learn to listen to and look at students’ work as a way to inform
teaching, using evidence from these to make decisions.

I Learn to design and implement lessons in ways that engage students in
learning (tasks, sequence, discourse, questioning, use of technology)

I Learn to reflect on your practice — both from your perspective as a
teacher, as a researcher, as a learner, and from the perspective of what
you see students learn.

I Recognize what is meant by equity and access to quality mathematics
for students, parents and communities (including attention to policy)

I Develop an understanding of what it means to have high expectations
for all students.

68

 

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I Develop strategies and frameworks for planning and assessment.

I Develop mathematics teaching strategies, such as lectures and class
discussions, cooperative groups, activities, etc.

I Learn about mathematics support systems, including state and national
standards, texts and other teaching materials, mathematics teacher
organizations and conferences, etc.

I Develop knowledge of techniques of assessing students’
understanding.

The senior year methods courses played an important role in this study because
they were the settings in which Megan and the other preservice teachers were first
introduced to the practice of anticipating students’ mathematical responses (SMRs).
Anticipating SMRs was defined for the preservice teachers as thinking about the lesson
from the perspectives of the students. To do this they were asked to consider the various
strategies students might use to engage in the lesson activity, what problems or
misconceptions might arise, and what kinds of questions the students might ask. The
preservice teachers were directed to resources to help them anticipate SMRs, such as
curricula available in the methods course classroom that provided multiple solution
strategies, articles in teacher journals, and their mentor teachers in their field placements.
In addition, preservice teachers were encouraged to solve the mathematical tasks in a
variety of ways on their own and to consult with the instructors of the course and other
interns for ideas. The preservice teachers were expected to make use of the
recommendations to carry out this practice in three different assignments: (l) researching
misconceptions in the National Council of Teachers of Mathematics (NCTM) teaching
journals as part of the microteaching lab, (2) anticipating solution strategies, questions,
and possible misconceptions twice during each semester when they lead the discussion

around a mathematics homework problem, and (3) in writing lesson plans for their field

69

 

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experience. Preservice teachers were further exposed to the practice of anticipating SMRs
during a mathematics education colloquium at the university with Dr. Margaret Smith,
who advocated it as a practice helpful for facilitating mathematical discussions.

The first assignment requiring the preservice teachers to anticipate SMRs took
place in the microteaching lab. As part of the fall section of the senior year methods
course the preservice teachers participated in a weekly microteaching lab in which they
prepared lesson plans on a given content topic and taught the lesson to their peers. Two
times during the fall semester preservice teachers who were not teaching in a particular
week were called upon to research a common student misconception related to the topic
being taught that week. The preservice teachers were directed to research these
misconceptions in the NCTM teaching journals, although many also used other resources
including the intemet and the curricular materials from their field placements. After the
preservice teachers researched the misconceptions they wrote a brief summary of their
findings and brought them to class. The final part of the misconception assignment was to
find a way to pose a question or solve a task that reflected this misconception. Although
this assignment did not broadly call on preservice teachers to anticipate SMRs it was
designed to introduce them to the possible misconceptions associated with specific
content and to begin to build the practice of consulting professional teaching journals for
information about how students might engage with particular mathematics.

The second assignment in which the preservice teachers were asked to anticipate
SMRs was in facilitating discussions of the weekly mathematics homework problems.
This assignment ran through both semesters of the senior year methods courses. Each

week all of the preservice teachers in the course completed a mathematics problem as

70

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homework and brought their solutions to class. Two preservice teachers were assigned to
lead a discussion around their classmates’ solutions and then discuss potential na'i've
conceptions (referred to as misconceptions during the class) that might arise if middle or
high school students were attempting to solve the problem. A written description of this
assignment can be found in Appendix A. The preservice teachers were supported in
considering the possible SMRs in required meetings with instructors that took place
before the class. In these meetings the SMRs that the preservice teachers had thought of
prior to the meeting were discussed. In addition, the preservice teachers were prompted to
consider other SMRs through questioning by the instructors and in some cases the
instructors shared mathematical responses that had come up in their own experiences
using the problems or those like them.

The third assignment that asked the preservice teachers to anticipate SMRs in the
senior year methods course was the lesson plan assignments. During the course of the
year the preservice teachers were asked to write many lesson plans, some for the field
component to teach, some to teach for the microteaching lab, and some for other course
assignments including a lesson study project. The lesson planning format that the
preservice teachers were asked to use can be found in Appendix B. For each lesson plan
the preservice teachers were asked to anticipate student thinking and questions. Each
lesson plan went through several revisions based on instructor feedback and the
preservice teachers’ reflections after teaching the lesson.

Finally, the preservice teachers were re—introduced to the idea of anticipating
SMRs in a colloquium that they attended late in the fall semester. Dr. Margaret Smith,

one of the authors of the article about the five practices for orchestrating mathematical

71

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discussions presented in Chapter 2, spoke to the mathematics education community at the
university, including the preservice teachers enrolled in the senior year methods course.
Dr. Smith discussed the five practices during her talk and gave examples of how teachers
could prepare to do each practice in their own classrooms. A debrief with the preservice
teachers after the colloquium revealed that they recognized anticipating SMRs as an idea
that the instructors had been advocating. Many of the preservice teachers agreed that they
had a better idea of what it meant to anticipate SMRs, how to do it, and why it was
important from listening to Dr. Smith. The next sections describe the internship year

methods course and the school context.

3.1.1.2 The Internship Year Methods Course

In the internship year the preservice teachers, henceforth referred to as interns,
participated in two classes related specifically to the teaching of mathematics, one during
the fall semester and one during the spring semester. The spring section of the internship
year methods course is beyond the scope of this study and will therefore not be described
here. The fall section of the methods course continued the emphasis of many of the
practices introduced in the senior year methods courses, including anticipating SMRs,
and in addition engaged the interns in structured inquiry about the teaching and learning

of mathematics. The activities of the fall semester of the methods course are detailed in

Chapter 4.

72

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3.1.2 The School Context: Logan High School

Megan participated in the teaching portion of the internship experience at Logan
High Schools. Logan is a village located approximately 65 miles southeast of the
university. Megan attended and taught at this school four or five days a week for the
entire 2008-2009 academic year. The responsibility for teaching the mentor teacher’s
classes built gradually over the course of the first semester, as explained in a previous
section. Megan and her mentor teacher, Mrs. Starns, chose an Algebra 2 class as Megan’s
Focus Class. Data for this study was gathered primarily from the Focus Class, and in
some cases from additional sections of the same course when Megan was teaching. The
school, the mentor teacher, and the Focus Class are further described in Chapter 4. The

next sections describe the participants of the study.
3.2 Participants

The main participant of the study was a preservice teacher named Megan. Megan
was several years older than many of the other interns due to the fact that she had
changed her major to mathematics after spending her first two years at the university in
fine arts. Within her peer groups Megan appeared to get along with the other interns. In
her senior year she was seen as a source of help by her peers for creating engaging ways
to begin mathematics lessons. Logan High School was located close to Megan’s home
district and she lived at home with her parents during the internship year. Megan and her

family knew many of the students at the school through social and church relationships.

 

5 Pseudonyms are used throughout to protect the anonymity of all participants

73

 

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Megan was purposefully selected as the participant of the study, as described by
Creswell (2003), because during her senior year methods courses she represented some of
the ideal qualities that a teacher educator might identify for a prospective mathematics
teacher. First, her participation in the senior year methods course showed her to be
interested and engaged in the process of learning to teach mathematics in ways described
by the university. Second, her interest in students as learners was observed by her senior
year mentor teacher and her enthusiasm for working with students in general was
evidenced in her choice to be an assistant basketball coach at a local middle school.
Finally, the level of her lesson preparation exceeded that of many of her peers in the
methods course. Her lesson plans were much more detailed and often focused on
engaging students at the outset of the lesson and finding ways of allowing the
mathematics to unfold as opposed to telling the students what they were supposed to
Ieam.

Megan could be termed “ideal” in that she came to the senior year methods
courses with enthusiasm for mathematics, teaching, and students and that she embraced
the methods and ideas being taught and discussed in the course. Studying the experiences
of an “ideal” preservice teacher afforded the opportunity to trace her development into
her field experience to see how that field experience impacted her participation in
practices of lesson planning, either toward or away from ideas advocated by the
university program. Teacher educators may have experiences with preservice teachers
who are less engaged or less interested in thinking about new ways of teaching
mathematics. Studying these prospective teachers would provide insight into whether or

not the internship changed their beliefs about teaching mathematics and what factors

74

 

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prompted that change. However, this was not the goal of this study. The goal of this study
was to detail how one preservice teacher already on the path of participating in practices
of lesson planning that focused on students, continued on this journey during her final
field experience.

Although the focus of this research was a case study of Megan, additional data
was collected from other group participants. These other groups were important to this
study because they served as elements of the contexts both in general and in the role of
supporting or not supporting the practice of anticipating. The first was the larger group of
secondary interns enrolled in Megan’s methods course. Of the 17 interns in the course, 16
agreed to be part of this study. The second group was the university instructional team
that worked with Megan which was made up of Dr. Rundell, the instructor of the
methods course, and Emily, Megan’s university field instructor. The final additional
participant in this study was Megan’s mentor teacher, Mrs. Starns. Even though these
groups are not the main focus of this study, their interactions with Megan impacted her
participation in practices of mathematics lesson planning. Each participant, or groups of
participants, is further described in Chapter 4. The next section describes the data

collection process that was used in this study.
3.3 Instruments, Data, and Data Collection

Data collection for this study took place in seven phases with an additional phase
of collection that took place throughout the fall semester. Figure 7 summarizes the data

collection phases.

75

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76

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The next sections describe the instruments, data collected, and methods used in each of
the phases of data collection. For each interview conducted, the interview was audio-

recorded and transcribed.
3.3.1 Phase 1. Initial Interviews

Initial interviews were conducted with Megan’s mentor teacher, the methods
course instructor, and Megan’s field instructor. The interview with Megan’s mentor
teacher, Mrs. Starns, took place in early October near the beginning of Megan’s
internship. The interview was adapted from an interview used by Mossgrove (2006) to
collect similar data about the context of a field experience (see Appendix C for the
interview protocol). The interview was designed to elicit response from the mentor
teacher to provide context for the field experience school and classroom, specifically in
the areas of the school environment, the mentor teacher’s expectations of the intern, and
her plans for supporting Megan in meeting those expectations.

The interview with Megan’s field instructor, Emily, was conducted in mid
September (see Appendix D for the interview protocol). The interview was designed to
gather data about the expectations and plans of the field instructor for supporting the

intern during the fall semester in lesson planning as well as other areas.

77

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Finally, the interview with the methods instructor, Dr. Rundell, took place in early
October (see Appendix E for the interview protocol). The interview was designed to
provide context for the methods course by eliciting responses about the goals,
expectations, and assignments of the course, as well as information about how the course
is related to the field experience and whether the intern teachers have opportunities to

collaborate. The interviews with each of these participants resulted in data in the form of

three transcripts.

3.3.2 Phase 2. Lesson Planning Survey

The Lesson Planning Survey (LPS) was designed to gather data to from the group
of interns to address the second research question regarding the factors of the contexts.
The LPS was completed by all 16 of the interns consenting to participate in the study,
including Megan (the survey can be found in Appendix F). This aspect of data collection
generated 16 completed, written surveys. The survey had two parts. The first part asked
the preservice teachers to write a detailed list of what they did during their senior year
methods courses to prepare lesson plans. The second part of the survey asked more
focused questions about the interns past planning and how they think they will plan
differently in the upcoming field experience. Additionally, the survey questions solicited
the intems’ responses as to why the practice of anticipating is important and finally the
interns were asked to anticipate SMRs in a particular content area. The goal of the survey
was to gather information in order to (a) describe what the interns perceived to be their
past practices in anticipating, specifically those that took place in the senior year methods

courses; (b) describe what the interns viewed as the importance and purpose of

78

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anticipating; and (c) illustrate how the interns anticipated SMRs in a content area that was

discussed in the senior year methods courses.
3.3.3 Phase 3. First In-Depth Interview

The In-Depth Lesson Interview 1 was conducted with Megan in mid September
and prior to any school observations (see Appendix G for the interview protocol). The
first part of the interview was adapted from an assignment used by Mossgrove (2006) in
her study of secondary mathematics preservice teachers’ lesson planning practices. The
questions in this protocol included all of the questions from the Pre-Lesson Interview
Protocol, which is described in the next section, as well as additional questions used by
Mossgrove. This part of the interview was designed to provide a much more in-depth
examination of Megan’s lesson planning process than the daily interviews could due to
time limitations. The second part of the interview focused on probing and clarifying
Megan’s responses to the LPS. The goals of the interview were to (1) gather data about
how Megan planned lessons at the beginning of her field experience, including
anticipating SMRs, and (2) to probe further into Megan’s thinking as revealed through
the LPS. Data from this interview consisted of the written lesson plan that Megan brought

to the interview and the transcript of the interview.

79

3.3.4 Phases 4, 5, and 6. Lesson Observations and Interviews

Phases 4, 5, and 6 all consisted of three— to five-day interactions with Megan at
her internship school. During these interactions Megan was observed teaching her Focus
Class and before and after each of these lessons short interviews were conducted (see
Appendix H for the interview protocol). For each lesson observation Megan supplied a
copy of her lesson plan for the day along with any supplemental notes she made. A total
of 15 Pre- and Post-Interviews took place. Phase 4 included five lessons plus a special
Cool Stuff lessoné. Phase 5 included five lessons plus a special Cool Stuff lesson. Phase 6
included three lessons. In each phase the Pre-Interviews asked Megan to describe the
lesson that she was going to teach and how she planned it, including questions related to
her anticipation practices. The Post—Interview asked Megan to reflect on how the lesson
went, including anything that surprised her and whether she would make any changes to
the lesson for the next day based on what happened during the class. The goal of these
interviews was to uncover anything that Megan anticipated that she did not include in her
written lesson plan, to elicit her reaction to events that occurred in the class that she did
not anticipate, and to see if teaching the class prompted her to anticipate for the next

lesson. Data from these interviews consisted of Megan’s written lesson plans and the

transcripts of the interviews.
3.3.5 Phase 5. Second In-Depth Interview

The final interview was conducted in December at the conclusion of the first

semester (Appendix I). As with In-Depth Interview 1, Megan was asked to bring a

 

6 The C001 Stuff lessons were lessons prepared for the university and taught in the school. They are further
described in Chapter 4_

80

 

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completed lesson plan that she would use in her teaching and was interviewed around that
lesson using the same protocol. In addition, Megan was asked to create a list of what
should be included in a lesson plan. The remainder of the semi-structured interview
probed what Megan thought she has learned about lesson planning from the first semester
of participating in her field experience. The goals of the interview were to (l) have
Megan create a list of what belongs in a lesson plan, (2) to inquire into Megan’s
perceptions of the field experience, and (3) to gather data about how Megan plans lessons
at the end of the first semester of her field experience, including anticipating SMRs. Data
from this interview consisted of the written lesson plan that Megan brought to the

interview and the transcript of the interview.

3.3.6 Throughout Fall Semester

The final portion of data collection took place throughout the fall 2008 semester.
All of the methods course meetings were observed and field notes were taken in reference
to how lesson planning in general, and anticipating SMRs in particular, was presented
and discussed during the class. Casual conversations that took place on the side during
classes between the interns were also included in field notes to gather data about the
interns’ perceptions about the course, the internship experience as a whole, and their
lesson planning practices and questions to one another. Finally. during this phase Megan
was asked to provide any lesson plans that she turned in as course assignments. The
purpose of this phase of data collection was to gather data that will provide insight into
the context of the methods course and also into Megan’s lesson planning for the course.

Data from the class consisted of the course syllabus and assignments. observation field

81

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notes, and on-line exchanges and postings on the course website. The next section details

the data analysis methods used in this study.
3.4 Data Analysis

The next sections outline how data was analyzed for this study and are framed by
the two research questions. In general, data analysis was framed by Miles and
Huberman’s (1994) three components of data analysis: (1) data reduction, (2) data

display, and (3) conclusion drawing/verification (Figure 8).
Data Collection Period I

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The application of each of these components is described in the following sections.

3.4.1 Analysis of Research Question 1: Describing the Practice of Anticipating

Research Question I: What is the nature of a preservice teacher's practice of
anticipating students' mathematical responses during lesson planning over the first
semester of her field experience?

82

In order to answer the first research question Megan’s written lesson plans and

transcripts of interviews with Megan were analyzed.

3.4.1.1 Research Question 1 Data Reduction

The analysis of data for the first research question was based on two data sources,
Megan’s written lesson plans and transcripts from lesson interviews (In-Depth Interview
1, In—Depth Interview 2, and Pre- and Post-Lesson Interviews). For this study a lesson
was defined by the classroom activities for a single day, which in most cases represented
one section from the classroom textbook. The exceptions to this were the two Cool Stuff
lessons, each of which served as review lessons, and the lessons in the second set of
observations which were reorganized by Megan and Mrs. Starns and thus did not follow
the sections or sequence of content laid out by the textbook. Only the lessons in which
Megan taught content to the class and the Cool Stuff lessons were included in this part of
the analysis. During the first set of lesson observations three days were used for review of
homework, two quizzes, and a non-mathematical game and so these lessons were not
included in this part of the analysis, though information from the corresponding
interviews was used in the analysis for the second research question. In total 11 lessons
were analyzed to answer the first research question.

Each written lesson plan was paired with the transcripts of the interview or
interviews that accompanied it. In the case of the lesson plan for In-Depth Interview 1 the
lesson plan brought to the interview was paired with the transcript for Pan 2 of the

interview, similarly for the lesson plan for In-Depth Interview 2. Each of the l 1 lessons

prepared solely for teaching in the school was paired with the Pre-Lesson Interview and
Post-Lesson Interview for that day.

The first step in the analysis of the lesson plans and transcripts was the
identification of anticipated student mathematical responses (SMRs). For this study
instances of anticipating were defined as any prediction, either written or verbal, made by
an intern about what students would do during class in response to the lesson. Examples
of such predictive statements are “I think that students will have a hard time with...” or
“they could solve this in two ways. . Each instance was coded using the following
codes: (1) student solution strategy (SSS), (2) student question or difficult point (SQDP),
and (3) student na'i've conception (SNC). Instances coded with SSS represented any
strategy or solution, either correct or incorrect, that the intern anticipated the students
would come up with during the lesson or for assigned homework. The SQDP code was
assigned to any instance that identified a part of the lesson that might be mathematically
difficult for students or questions that students might ask. Finally, The SNC code was
used any time the intern identifies possible na’fve conceptions that might hinder the
learning experience for the students. The interns are most familiar with the word
“misconception” and so these instances were often described using that term. Any
instance that exemplified more than one of these codes was coded with all appropriate
codes.

The second part of the analysis focused on identifying the resources used by
Megan in planning the lesson, and when possible linking the resources to what was

anticipated. Again using the written lesson plans and interview transcripts, resources and

84

 

 

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resource activities that Megan used and participated in were identified. All resources, or
resource activities, were coded with one of the following codes:

- The teacher completes the activity in one way

- The teacher completes the activity in more than one way

- Taking the position of students

- Curriculum materials

- Other teachers

- Research literature

These resources were most often identified by Megan herself in the interviews
when she was asked how she figured out something that she anticipated or when she
explained how she used a resource for another purpose only. After the resource and
resource activities were identified they were then further coded as being part of either
active/purposeful or passive/incidental anticipating7. If Megan made use of a resource,
but not for the purpose of anticipation, it was coded as passive/incidental so that
resources that were used in general were identified. Some of the anticipated SMRs were
indentified as having two sources. As an example, Megan’s mentor teacher shared a
specific SMR that Megan had also considered from completing the problem herself. In

these cases two codes were used.
3.4.1.2 Research Question I Data Display

After the data was coded for each lesson an anticipating map was created to

represent Megan’s anticipation activities within each lesson. These maps were based on

 

A complete discussmn of the distinction between active/purposeful and passive/inCidental antiCIpating
can be found in section 2.3.

mi

 

the original anticipation framework presented in section 2.3, but only included the
activities and SMRs that were anticipated for the specific lesson. The maps represent
lessons and groups of lessons and allow for the viewing of Megan’s practices across time.

Figure 9 shows an example of such a map.

86

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Passive/
Incidental
Anticipating

Active/
Purposeful
Anticipation

  
   

  

The teacher consults
outside resources for
reasons other than
anticipating

     
     
   
   
 

The teacher
completes the
activity in more
than one way

 

 

 

Multiple
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Other
teachers

Research
literature

Curriculum
materials

 

 

SSSs, SNCs, SSSs, SNCs,
SQDPs SQDPs

 

 

 

 

 

 

Figure 9. Example Anticipation Map

The active/purposeful side of Figure 9 represents that in this lesson the intern

actively/purposefully chooses to anticipate multiple student solution strategies by

87

 

completing the task in more than one way. The passive/incidental side of Figure 9
represents that the intern used curriculum materials, such as the class textbook, in
preparing the lesson, but that there is no evidence that doing so provided any information
about SMRs, which is the reason for the dashed line, or that making use of the curriculum
materials was done in order to anticipate SMRs. The passive/incidental side of the figure
also shows that the intern consulted other teachers, again not in order to anticipate, but
that in this case it led them to anticipate SMRs anyway, hence the solid line. Finally, the
figure shows that the intern consulted research literature, perhaps to find an activity for
the lesson, but that there is evidence that no anticipated SMRs came from this source.
Another data display that is used is a matrix in which the codes for each lesson

were entered. An example of that matrix is represented by Table 2.

88

Table 2. Data Display for Anticipation Practices

 

Lessons

Active/Purposeful Anticipating

The teacher consults outside resources in order to anticipate
I Curriculum materials

' Other teachers

' Research literature

Completes activity in more than one way

Takes position of students

Passive/Incidental Anticipating

The teacher consults outside resources for reasons other than anticipating
' Curriculum materials

' Other teachers

' Research literature

Completes activity in only one

way

Takes position of students

Outcomes of Anticipating
SSS

SNC

SQDP

 

Each cell contains counts of the number of codes and categories of each type appearing in
each lesson. Although these counts were not statistically analyzed, they are included to
provide the complete picture of the available data about the nature of Megan’s
anticipation of SMRs by showing the detail of her activities over the entire semester in
one representation.

The anticipation maps and the matrix played an important role in describing the
nature of anticipating. As Miles and Huberman (1994) point out “valid [qualitative]
analysis requires, and is driven by, displays that are focused enough to permit a viewing

of a full data set in the same location, and are arranged systematically to answer the

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research questions at hand” (pp. 91-92). The anticipation maps provide a visual snap shot
of the anticipation practices at different points over the semester and the matrix allows for
comparisons to be made across time.

A final data display consists of two mini—case studies within the larger case study
that detail how various situations of anticipation play out in the intem’s classroom. The
first data analysis for this research question provided a picture of Megan’s practices prior
to teaching her lessons. The mini-cases take a slightly different look at the nature of
anticipating by showing how the practice impacted Megan’s teaching choices. These
cases are based on the lesson plans, Pre-Lesson Interviews, and field notes from the class.

Each mini-case details two distinct outcomes of anticipating SMRs.

3.4.1.3 Research Question 1 Conclusion Drawing

Conclusion drawing for the first research question came from analyzing the
anticipation maps and the matrix for themes and changes in Megan’s anticipation
practices over the semester. This included an analysis of what was anticipated (solution
strategies, na'i’ve conceptions, or questions or difficult points) and the nature of what was
anticipated (e. g. students may not remember procedures), as well as the resources that she
used. Specifically, this included what type of anticipating occurred most over the
semester and which resources she most often made use of and whether these resources
helped her to anticipate SMRs. Additionally, the types of anticipating that rarely or never

occurred were identified as well as which resources were not used.

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3.4.2 Analysis of Research Question 2: Factors of the Contexts

Research Question 2. What factors within the contexts of a teaching internship (afield
experience and accompanying university course) promote or do not promote the
practices of one preservice teacher anticipating students ’ mathematical responses
during lesson planning ?
The analysis of data to answer the second research question took place in two
parts. The first part of the analysis produced a description of the various contexts in
which the intern worked during the semester, which can be found in Chapter 4. The

second part of the analysis resulted in an explanation of the influence of the contexts on

the nature of the intem’s anticipating, which is presented in section 5.2.

3.4.2.1 Describing the Contexts

This study involved a variety of people and settings. This section details the

methods I used to describe each of these people and settings within the two contexts.

3.4.2.1.] The University Context. The university context in this study was made
up of the methods course, the methods course instructor, and the field instructor. Data
used to describe the context came from transcripts of interviews with the methods
instructor, the field instructor, field notes from observations of the methods course,
transcripts of on-line chats from the methods course, and the collection of methods course
assignments. Analytic summaries as described by Miles and Huberman (l994)were
written for each methods course observation and the weekly on-line chats that identified
the general themes of the interactions and how the practice of anticipation of SMRs was

discussed or supported. The description of this context focuses on the general themes of

91

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the course and the assignments, discussions, and models that related to the anticipation of

SMRs.

3.4.2.1.2 The School Context. The school context in this study was made up of
Logan High School, including the field placement classroom, and Megan’s mentor
teacher. Data used to describe this context came from the transcript of the interview with
the mentor teacher, transcripts of formal conversations and notes on informal
conversations between Megan and her mentor, and observations of the school context.
The description of this context focuses on the generalities of working in the school along

with the practices and norms that related to Megan’s ability to anticipate SMRs.

3.4.2.2 Explaining the Factors of the Context That Promoted Anticipating

As with the analysis of the data for the first research question, Miles and
Huberman’s (1994) three activities for data analysis were used: (1) data reduction, (2)
data display, and (3) conclusion drawing/verification. Data for this part of the analysis
consisted of transcripts of interviews (Mentor Teacher Initial Interview, Field Instructor
Initial Interview, Methods Instructor Initial Interview, Pre- and Post—Lesson Interviews,
and In-Depth Lesson Interview 2), field notes from formal and informal conversations
between Megan and her mentor teacher, the field notes from the methods course,
transcripts of on-line chat room conversations, and written assignments from the methods

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3.4.2.2.] Research Question 2 Data Reduction. Within each of the data sources,
including excerpts from conversations and interviews and assignments, influences on
Megan’s anticipation practices were coded using the zones described by Goos (2005a):
the Zone of Proximal Development (ZPD), the Zone of Free Movement (ZFM), and the
Zone of Promoted Action (ZPA). Goos (2005b) explains that the nature of the zones
implies something unique which made coding less straightforward. “As the zones
themselves are abstractions, this analytical process focused on the particular
circumstances under which the zones were ‘filled in’ with specific people, actions,
places, and meaning” (p. 45). Thus, it was often ideas represented by quotations,
discussions, or assignments that were coded within these zones. In addition to being
coded in the zones, the influences on Megan’s practices of anticipating were further
coded according to Engle and Conant’s (2002) principles that foster productive
disciplinary engagements. In the ZFM this included influences that provided relevant
resources and authority and in the ZPA this included influences that problematized
mathematics and mathematics teaching and influences that held Megan accountable to

the disciplinary norms of mathematics and mathematics teaching.

3.4.2.2.2 Research Question 2 Data Display. The data displays for the second
research question includes several Venn Diagrams similar to those used by Goos (2005b)

which represent the relationships between the zones (Figure 10).

 

8 A full description ofthe zones can be found in section 2.1.

93

 

1.

    

school ZPA

Figure 10. Goos' (2005b) Representation of ZFM, ZPD, and ZPA

Figure 10 represents how each zone was related for the novice teacher in Goos’ study.
Another data display is a matrix that displays the influences in the ZFM and ZPA and
identifies each influence as either contributing to, or detracting from the practice of

anticipating SMRs

3.4.2.2.3 Research Question 2 Conclusion Drawing. Finally, conclusions were
drawn about the influences of the contexts on the nature of anticipating based on the
description of the influences and the data displays. Specifically, which context included

practices and supports that would promote an intern to anticipate SMRs and which did

not.

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3.5 Presentation of Findings

The current study aimed to describe the development of the practice of
anticipating SMRs in a field experience using a situative perspective. The situative
perspective as described by Greeno (1998) calls on researchers to begin by exploring the
contexts and then narrowing the focus to investigate the influence of the context on the
activities of individuals. Chapter 4 presents a description of the two contexts relevant to
this study, the university and the school. This is followed by Chapter 5 which addresses

the findings as they relate to the two research questions.

CHAPTER 4: THE CONTEXT

Because the theoretical frame that guided this study was a situative one; primary
considerations include the settings and interactions that occurred during the field
experience of the intern in question. Thus, before the case study findings of the intern
teacher and her experiences are presented, I discuss two of the contexts in which she
operated during the first semester of her internship, the university context and the school
context. This is consistent with Greeno’s (1998) chronology, “begin with the framework

of interactional studies and work inward” (p. 6).
4.1 The University Context

The university context in which Megan operated during the first semester of her
internship was comprised of three major facets. The first was the mathematics methods
course and the instructor of that course. The second facet of the university context was
the group of interns in this course, most of whom had been in classes together prior to
this course and had formed a unique community of their own. The third and final facet of
the university context was the university supervisor who was employed by the university
but met with the interns in their field experience classrooms and supported them in their

specific settings. The next sections describe each of the facets within the university

context.

4.1.1 The University Methods Course

In many university teacher education programs all methods courses are completed

before the final field experience. In this case, a sequence of methods courses were

96

conducted before and during the internship year, providing the preservice teachers four
semesters of secondary mathematics methods experiences, two during their senior year
and two during their internship year. The methods course that took place during the
internship year was taught by a university faculty member, Dr. Rundell. Dr. Rundell was
the leader of the secondary mathematics team at the university and in that role he was
responsible for the coordination of the curriculum for the mathematics methods courses
and the field experiences. Dr. Rundell had taught the course once before and had also
taught one semester of the senior year methods course at the university. He described his
role in the course as, “supporting the interns in their first year of really engaging in
teaching practice over the long term” [Methods Instructor Interview, October, 2008]. He
saw this role as both encouraging the interns in terms of their long-term growth as well as
addressing their immediate needs as they arose in the classroom settings.
Dr. Rundell designed five major assignments as part of the fall semester methods

course which are summarized in Table 3.

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Table 3. Summary of Methods Course Assignments

 

 

Assignment Description

Mathematics The first of three parts of creating a teaching philosophy. Parts

Teaching two and three were completed during the spring semester. Interns

Philosophy described the beliefs and aims that undergirded their mathematics
teaching practices.

Cool Stuff 1 The interns planned, using the lesson plan guide, and taught a
lesson that both they and their students would find cool. For this
lesson the interns were directed to find and modify if needed a
task of high cognitive demand. (See Appendix J for full
assignment description.)

Unit Plan The interns created a plan for teaching a unit of content along

Cool Stuff 2

Researchable
Question 1

with a concept map of the mathematical content in the unit and
an assessment for the unit.

The interns planned, using the lesson plan guide, and taught a
lesson that both they and their students would find cool. For this
lesson the interns were directed to create a task of high cognitive
demand and to make use of technology in their lesson.

Groups of interns decided on a research question based on their
internship classrooms, located and read research on student
learning in that area, conducted a small research project, and
presented their findings during a poster fair.

 

During the fall semester three intertwined foci of the methods course emerged.

They were: (1) exploring proportional reasoning, (2) creating a collaborative teaching

community, and (3) encouraging deliberate practices in planning, teaching, and reflecting

on mathematics lessons. Each of these foci is further described in the following sections.

4.1.1.1 Exploring Proportional Reasoning in the University Methods Course

The first of the foci was the mathematical topic of proportional reasoning. As a

part of this focus the interns worked together on mathematical tasks, read case stories of

98

teaching and research literature in this content area, found instances of proportional
reasoning in the textbooks they were using, and engaged in debates around important
underlying questions of the mathematics such as “What is the difference between a
fraction and a ratio?” These activities provided a venue in which the interns had the
opportunity to both explore their own content knowledge and to discuss issues of
teaching and learning.

Each time the interns worked on a mathematical task that provided the
opportunity for multiple solutions they were asked in solve it in more than one way as is

shown in Figure l l, a presentation slide from one of the classes.

 

The Balloons Task

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Multiple Solutions

99

During one of the problem solving activities, Dr. Rundell also encouraged the interns to
consider incorrect ways to set up the problem and solve it. In addition to solving the
proportional reasoning tasks in multiple ways, the interns also read case studies of
teachers who used the tasks in their classroom or looked at multiple student solutions for
the task. In all of these cases the task was solved in multiple ways, the interns’ multiple
methods were discussed, and multiple student solutions were discussed. While the focus
of these activities was not solely on the possibility of multiple solutions, it was a
recurring theme across the proportional reasoning activities. The next section describes
the second of the focus of the methods course.

4.1.1.2 Creating a Collaborative Teaching Community in the University Methods
Course

The opportunity to share your thinking and have your ideas challenged by others
is a critical part of developing as a teacher, and an experience that is often hard to
come by in the normal course of the school day. This year, we will help you
develop the skills for learning from your colleagues, and, perhaps more
importantly, the disposition to do so, so that you can seek out or create these kinds
of opportunities throughout your career. [Methods Course Syllabus]

The second focus of the methods course was on creating a collaborative
community as described in the excerpt from the syllabus. The design of the methods
course provided multiple opportunities for the interns to collaborate during and outside of
class. During class the interns worked together to solve several mathematical tasks as
described above. In addition to talking about mathematics, the interns discussed the

teaching and learning of mathematics using case studies as well as case stories of their

own teaching. In the case stories the interns shared their design and teaching of two

100

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structured lessons referred to as Cool Stuff Lessons (see Hughes, Smith, Boston, &
Hogel, 2008 for further discussion of case stories).

In addition to the discussions orchestrated during the methods course classes,
there were three on-line opportunities for interns to collaborate, a weekly chat room,
blog, and wiki. The weekly scheduled chat room, referred to as the fireside chat, took
place every Wednesday night for two hours. Dr. Rundell was present at almost every chat
and usually at least one or two course teaching assistants or field instructors were also
present. This opportunity allowed the interns additional space and time to discuss their
experiences outside of class. The blog was accessible by all of the interns as well as the
secondary mathematics interns who had been part of the course in the previous year. Dr.
Rundell described the purpose of the blog in the course syllabus in the following way:

The blog is designed to accommodate narrative accounts of what’s happening in

our classes, so that we can all understand one another’s classroom contexts. This

fosters a sense of a community of practice, where we engage with one another

around issues of teaching. [Methods Course Syllabus]
Current and previous interns had the ability to respond to each other’s comments and
concerns on the blog and some of the postings served as prompts for a weekly in class
discussion in the methods course entitled What’s Happening in Math. The blog allowed
the interns to share their problems and ideas at any time and to receive responses from
others when they had the opportunity to respond, unlike the chat which happened in real
time.

Finally, the wiki was used as a place to share resources and collaborate on class
assignments. While most of the resources, for example websites to find lesson plans and

further information about ideas discussed in class, were posted by the instructors of the

101

course, three interns chose to post their own lesson plans on the wiki to share with the
other interns in the class. In addition to being a space for sharing resources, the wiki also
contained pages for interns to collaborate on a group project and to give each other
feedback on a lesson near the end of the semester. For the project the groups uploaded
their plans for their Researchable Question 1 project as well as several pieces of research
related to their area of interest. The interns also gave substantial feedback to two other
interns on one of their lesson plans on one occasion in the wiki.

These settings often served as a place for the interns to share their “war stories”
with each other and give and receive both support and advice. The interns’ sharing was
not limited to difficulties. In some cases they shared their perceived successes: "I started
making my class student centered this week!" [Fireside Chat, September 17, 2008]. The
topics were varied and often represented traditional teacher dilemmas around assessment,
classroom management, working with other teachers (most often the mentor teacher),
motivating and engaging students, and communicating with parents.

In addition to the sharing of stories, certain aspects of professional collaboration
were modeled for the interns by the instructors in the various settings. This mainly
happened through the sharing of resources and group problem solving. A list of resources
for finding tasks, titled A Treasure Trove for Tasks, was placed on the wiki. In addition,
there were several instances during chats when Dr. Rundell emailed a task or other
resource to an intern in need of help.

The types of problem solving that were modeled were based on the immediate
concerns that the interns brought with them to class or one of the other setting. In one

example from a Fireside Chat an intern, Jane, expressed her frustration about her students

102

understanding of multiplication. Two field instructors, Alex and Emily, helped her clarify

her problem and then offered some suggestionsg. Megan also added what she had heard

about the subject area to the conversation. The following is from the chat log of this

conversation.

Jane:

Alex:

Megan:

Jane:

Emily:

Jane:

Emily:
Jane:
Emily:

Jane:

Emily:
Jane:

Emily:

It is troubling for me, to think that my students don't know or
understand multiplication... I feel they should know this by now. Why
haven't earlier teachers corrected this?. ..

"Don't know" is a bit broad of a statement. Can you clarify what you
mean?

I have heard from various teachers that they have tried to get students
in high school to learn their multiplication tables and they have had
difficulty with it. This is interesting to me

for instance... if I asked them what [three times five] was, it takes them
a minute... most look up at the stupid multiplication posters that are

posted...

What do you think about the fact that the multiplication posters are
posted? Would you post them in your room?

heck no... they post them because students need them to do the math
they are learning today

What do the students do if it’s not on the poster? like [74 times four]
they would want to use a calculator
What do you do then?

it's all about getting the "answer" the quickest way. I say no way jose!
and make them do it longhand...

So they know how to do it longhand?

YUP

Do you ever have opportunities to think aloud (model) other strategies
than the algorithm?

 

9 . .
Names used for interns, instructors, mentor teacher, and school are pseudonyms throughout.

103

Jane: They understand the algorithm for doing it. For instance
74+74+74+74?. I am trying to think another way for them to do it,
because I don't know these other ways.

Alex: 70*4+4*4...(Distributive property)

Emily: 74*2 then *2 again

Jane: I gotcha... no... we never do this in these ways. They just learned the
distributive property in the beginning of the year and they really didn't
get that either.

Emily: This might be a good exercise to help show the students the other ways
that they might breakdown the problem to make sense of it and make it
more reasonable to solve. You could even include one (periodically)
during your warm-up activities.

Jane: okay

Emily: As a way of fitting into your existing routine.
[Fireside Chat, October 29, 2008]

In this interaction the field instructors challenged lane to think more deeply about her
students’ understanding of multiplication and offered her some ideas to try.

In addition to being modeled by the instructors of the methods course, both of the
practices of sharing resources and problem solving were also evident in the practices of
the interns which will be discussed in a later section. The next section describes how the
Plan, Teach, Reflect cycle, which was originally introduced in the senior year methods
course, was carried through into the intern year methods course.
4.1.1.3 Encouraging Deliberate Practices in Planning, Teaching, and Reflecting on
Mathematics Lessons in the University Methods Course

The final focus of the methods course was on participating in deliberate teaching

practices in order to improve student learning of mathematics. The routines that were

104

expected of the interns in this area were modeled by the instructors and discussed in the
class. These routines were based on two frames that were familiar to the interns from
their work in the senior year methods course. The first was the five practices model
presented by Stein, Engle, Smith, and Hughes (2008) and the second was a lesson
planning model based on a cycle of planning, teaching, and reflecting. Dr. Rundell
describes the incorporation of these two models in the class structure in the following
way:
Really everything that we do in class is about these two frames, and the five
practices is one of them that kind of fits at the lesson level, and a lot of what we
do relates to that. And then the planning, teaching, and reflecting cycle is another
one that comes up often and is kind of the larger level up. But those are the
reoccurring structures in the class, both in terms of how we are asking them to
think about lessons and units and investigating their own teaching and thinking
about their own identity as a learner. [Methods Instructor Interview, October,
2008]
The Plan, Teach, Reflect cycle was carried out in both the Cool Stuff 1 and Cool Stuff 2
lessons. The interns were required to use a particular lesson plan format (see Appendix
B) for the plan phase of the cycle and for Cool Stuff 1 they shared their ideas with other
interns in class before teaching in order to elicit additional support during the planning
process. In designing a lesson for the Cool Stuff 2 assignment, the interns posted their
lesson plans on the wiki and received feedback from their peers and an instructor. After
teaching each of these lessons the interns reflected individually on what occurred during
the class and on their own teaching. In addition to individual reflection, after teaching
Cool Stuff 2 the interns spent time in class debriefing with other interns in small groups
about what happened. For both Cool Stuff 1 and Cool Stuff 2 the interns were expected to

make use of educational research literature to analyze their students understanding using

evidence they collected during class.

105

The other frame that focused the interns on deliberate practices of teaching was
the five practices model. The interns were familiar with the five practices model from
their work in the senior year methods course and Dr. Rundell expected them to build on
these practices. Dr. Rundell assigned the article by Stein, Engle, Smith, and Hughes
(2008) for reading to remind interns of the five practices and then focused part of his
feedback to their Cool Stuff lesson plans on whether they had addressed the practices. As
the first of these practices that lays the foundation for all the others, anticipating students’
mathematical responses (SMRs) was a practice that received special emphasis. Dr.
Rundell stated specifically that he looked for evidence of anticipating SMRs, in
particular,

I talked about both considering misconceptions and accurate conceptions, is a way

to think about it. Productive ways of student thinking that may come up. . .if it’s a

task that has multiple solution paths I look for [whether they have] thought about

three or four or five really good solution paths. If it’s a mathematical idea that I

know from research or from experience that has a lot of student misconceptions I

look for did the task really invite particular student misconceptions. [Methods

Instructor Interview, October, 2008]

In addition to early feedback aimed at helping the interns think about anticipating SMRs,
Dr. Rundell also emphasized the importance of the practice in the evaluation rubric for
the Cool Stuff lessons (see Appendix J). Five elements are evaluated in the planning
portion of the rubric: (l) the statement of the mathematical goal, (2) building on prior
knowledge, (3) inclusion of materials and launch, (4) task selection and setup, and (5)
anticipated student questions/thinking. The statement of the mathematical goal was worth
one of the potential ten planning points. Building on prior knowledge, inclusion of

materials and launch, and task selection and setup were each worth two of the potential

ten points. The last category in the planning portion of the rubric, anticipated student

106

questions/thinking, was worth three of the potential ten points. The relative number of
points awarded for anticipating SMRs indicated its importance as a lesson planning
practice in the course.

The focus on deliberate practices was evident throughout the course and was
emphasized in all phases of the Plan, Teach, and Reflect cycle. These deliberate practices
were further spelled out in the five practices model that the interns were expected to
make use of in their lesson planning. Within the cultivating of these deliberate practices
in the methods course, the practice of anticipating SMRs was given special focus as the
foundation of the other practices. In addition, the creation of the collaborative community
and the focus on mathematics throughout provided many opportunities for the interns to
share and discuss mathematical ideas that would benefit them in anticipating SMRs in the
future. The next section describes the intern group and how they participated in their

collaborative community.
4.1.2 The Interns

The interns in the methods course created a unique community of preservice
teachers. Fifteen of the 17 interns in the class were part of a class that completed the
senior level methods course sequence the previous year. The two other interns had
completed the sequence one year earlier. Although these two interns were not part of the
original group, their substantial participation in class and the online chats and blogs could
be interpreted as evidence of their comfort with the larger group of interns.

The group of interns participated in the practices of a collaborative community

modeled by their instructors, including the practices of problem solving and resource

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sharing. In one Fireside chat an intern, Danielle, expressed the need for a math game and

another intern, Brian, responds with an idea.

Brian:

Danielle:

Brian:

Danielle:

I like the factor game

what's that? (and that's perfect, cause we just did our section on
factoring last week)

http://illuminations.nctm.org/activitydetail.aspx?ID=l2. There is a link
to the factor game, its pretty fun.

awesome [Fireside Chat, October 29, 2008]

In another interaction in a Fireside chat Danielle was having problems motivating

students. Another intern, Todd, tried to help her think about her situation.

Danielle:

Todd:

Danielle:

Todd:

Danielle:

Todd:

I think they're lazy...that's the biggest problem...I could give them
another 5 days, and I don’t know how much of a difference it would
make. There are students that put in the effort, come in for help,
participate and ask questions in class...the rest of them...not so much.

That's an interesting assumption you have there What makes you think
they are lazy instead of just say...frustrated and shy?

I think it's a vicious cycle....the more I try to "help" them....by giving
them this practice quiz, or spelling out things in the lesson, the less
work they wanna do... they're definitely not shy... frustrated maybe...
but they don’t like to put in the effort....when we go over homework,
we go through every answer...

They may not be shy about talking in class, they could be shy about
asking for individual help.

some students, if they get one question wrong, they want me to go
over it...but they'll make a little mistake like adding 1+ -2 = 1, and they
don’t even try to see what they did wrong first, they just want me to do
it for them. . .I don’t know...1'm not used to students who don’t have at
least some self motivation it's very different from my high school
growing up

So, didn't get right answer means "out of my hands, help me teacher?"

108

Danielle: yeah, that's probably a good way of putting it [Fireside Chat,
November 19, 2008]

Their conversation continued and Todd asked if it was possible that the students at
Danielle’s school are motivated but they have not had the same experiences learning to
“play school” that many of the interns had. In this interaction Todd challenged Danielle
to reconsider the way she was thinking about the problem with her students.

The interns also often engaged in discussion about the mathematics and student
learning in their classrooms, both when prompted to do so and without being prompted.
One example of two interns discussing student learning on their own occurred early in the
semester. The two interns were writing a mathematical solution to a problem on a white
board near the back of the classroom. As the other interns finished writing their solutions
on other boards around the room the two interns began to talk about what they were
teaching in their classes. Both were teaching students to use rulers in their field
placement classrooms. One of the interns shared with the other a specific difficulty he
noticed that his students had when trying to learn to use a ruler.

Overall, the interns were quite cohesive as a group and though their interactions
with each other demonstrated facets of a teaching community of practice. The next
section describes the university field instructor who worked with Megan during the

semester.

4.1.3 The University Field Instructor

A total of six field instructors worked with the group of secondary mathematics
interns. Megan’s field instructor, Emily, attended each meeting of the methods course

and was present in all of the fireside chats. Although this was Emily’s first year serving

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as a university field instructor, she had previously taught several years of high school
mathematics and thus brought a wealth of experience to the position. The teacher
education program handbook describes the role of the field instructor in the following
way:

Make at least five scheduled observation visits each semester and hold five

conferences with the Intern and Mentor Teacher during the year. They supply

program information, offer an additional perspective on classroom events, and

support interns in meeting the program standards. [Program Handbook, p. 5]
Emily voiced her own perception of her role as a field instructor:

I think it’s my responsibility to facilitate communication between the cooperating

teacher and the university and also to provide support to the intern with respect to

how they're designing lesson, how they're implementing lessons, and give them
constructive feedback on those things. [Field Instructor Interview, September,

2008]

The majority of the conversations between Emily and Megan around Megan’s
teaching took place at the field placement school. When Emily traveled to Megan’s field
placement she always observed her teach a class, usually the Focus Class. After
observing Megan teach, Emily and Megan would debrief about the class and talk more
generally about Megan’s experiences. Emily would ask Megan how she felt the class in
particular went and how things were going more generally. She would also question
Megan about the class activities that were part of the mentor teacher’s regular schedule
that she had adopted and other choices she made on her own. These debriefs served as an
Opportunity for Megan to ask Emily questions about what she was doing and to ask for
advice about how to do things differently. For example, Megan was asked by her mentor

teacher and the school principal to tutor a student during seventh hour who had been

absent for several days in order to catch her up. Megan was concerned that the student

110

was missing the new content in the class and Emily was concerned that Megan was
missing an opportunity to watch her mentor teacher teach. The debrief conversations
offered Megan an opportunity to discuss these types of problems and brainstorm
solutions with an experienced teacher. Many of the conversations between Megan and
Emily resulted in Megan voicing her desire to teach in what she termed a “more student-
centered way.”

In addition to discussing Emily’s observation during these meetings Emily also
commented on Megan’s lesson plans that were collected and organized in her Focus
Binder. Emily noted that in her first meeting with Megan she made a suggestion to her
about her lesson plans.

I gave her feedback about she needs to write down some more questions and stuff

to help prove to me that she's thinking through her lessons. [S]he's doing all of the

math that... the kids are doing, ahead of time... she's solving them all out so [I

suggested that she] spend half the page doing the math and then the other half of

the page writing down things that you think are going to be problematic or
questions that you are going to ask to push student thinking. [Field Instructor

Interview, September, 2008]

Emily had as a goal for all of the interns that she served as a field instructor for that they
would eventually be able “to predict the questions that the students are going to ask and
the misunderstandings that they are going to have” [Field Instructor Interview,
September, 2008]. Emily’s interactions with Megan mirrored those that took place in the
methods course. She attempted to engage Megan in problem solving conversations
similar to those that took place in class and in the chats. In addition to this she served as

an additional support resource by pointing out areas for improvement for Megan,

including anticipating SMRs more often in her lesson planning. The next sections shift in

111

focus from the university to the school context and describe the settings and interactions

that Megan participated in while at Logan High School.

4.2 The School Context

The school context in which Megan operated during the first semester of her
internship was also comprised of three major facets. The first was the school itself, Logan
High School. The second facet of the school context was the mentor teacher with whom
Megan worked during her internship year and whose classes she taught. The third and
final facet of the school context was the Focus Class that Megan taught throughout the

year. The next sections describe each of these contexts.

4.2.1 Logan High School

Logan High School was housed in a large new school building with students in
grades nine through twelve in attendance. It was located in a small rural village in the
Midwest and is the only high school in the district. The hallways were clean and
colorfully decorated with student work from different classes. The staff and faculty of the
school were friendly and always willing to help a visitor find where they needed to be.
The mentor teacher described the other teachers in the building as proactive, positive, and
not afraid to incorporate new things into their classrooms. Students often arrived at
school early and sat in the open cafeteria and talked with each other before the school day
began. Megan’s mentor teacher, Mrs. Starns, explained one positive outcome of the new
building:

It has everything you could possibly want and it’s a beautiful layout. I think it
lends itself for kids to really like to be here because it’s clean and fresh and that

112

kind of thing. We have a beautiful gym, we have a walking track and that’s a nice

feature. And then we have the various groups, like cheerleading has a room and

wrestling has a room and there's a weight room. So it really does for those parties

or distinct groups, they do have their place. So I think people feel they have a

place in the building. [Mentor Teacher Interview, October, 2008]

The school’s principal was young and welcomed the idea of research taking place in his
school.

The students in the high school were also very friendly. In their ethnicity, more
than 94% of the students at Logan High School were Caucasian and according to the
mentor teacher they came from economically diverse backgrounds. Approximately
twelve percent of the students received free or reduced lunch. A relatively large group of
the students lived in different parts of the country during the year depending on where
their parents were needed for farm work. These “migrant students” only attended Logan
High School during the first quarter or semester and then they moved to another location.
Other groups of students live on local farms, ranging from very small to very large and
prosperous, and another group of students live in the town.

The students in the school displayed their school spirit by wearing school tee-
shirts and sweatshirts and creating large banners to display in public places. During the
semester of data collection the football team was having a winning season and the
students in the school were visibly excited at the prospect of the team making it to
tournament games. Megan and Mrs. Starns both attended many of the football games and
they often joined with the students in talking about the exciting things that happened at
the games on Monday morning.

The math department staff at Logan High School was made up of three teachers.

These teachers met once a month through a professional development program sponsored

113

by a local collaborative of school districts. Mrs. Starns noted that if it were not for these
monthly meetings, the three math teachers in the department would not have the
opportunity to interact. She also described their teaching philosophies as being very
different, ranging from very straight forward teaching to trying new activities all of the
time. Megan’s perception of the mathematics department was somewhat different, as she
described it to her peers in the methods class, the “math department is very lecture-

based” [University Methods Course, September 26, 2008].

4.2.2 The Mentor Teacher

Megan’s mentor teacher, Mrs. Starns, had been teaching at Logan High School
for over 15 years. Prior to the beginning of her teaching career she received her
undergraduate degree in mathematics with a minor in physical science from the same
university Megan attended. Several years ago she completed a masters degree in general
science with an emphasis in physics and astronomy from another local university. During
the semester of data collection she was working on a masters degree in educational
administration from a third local university. Working with Megan was her first time
serving as a mentor teacher.

Mrs. Starns appeared to have a warm relationship with most of the students.
Former students stopped by her classroom to talk with her about the events in their lives.
Occasionally at lunch time several students would come to her classroom to sit with her
and eat lunch. Mrs. Starns seemed to be familiar with many of the students in the school

and their families.

114

 

Mrs. Starns’ taught three periods of Algebra 2, one period of Math Analysis (a
class which is comparable to Pre-Calculus), one period of Calculus, and one period of
Physics. Her teaching in the mathematics classes appeared to be based mostly on lecture
while her teaching in the physics class appeared to branch out more, making use of
different technology and other physical examples such as bow and arrows and trebuchets
that students brought in for one of the Physics units. This difference in teaching style
might have been due to her own conception of mathematics which she described to
Megan as “the most boring subject” [Informal Conversation, Lesson 1.4 Observation].

Mrs. Starns created and followed a weekly schedule in all of her classes. The
schedule that she used in Algebra 2 and was posted on her school website was:

Monday: Bellwork, Worksheet, Textbook Homework

Tuesday: Worksheet Due, Vocabulary Quiz, Textbook Homework

Wednesday: Bellwork, Quiz, Class Game

Thursday: Bellwork, Textbook Homework

Friday: Bellwork, Game of Life Mobile, Notebook Check, Every three
weeks there is a test on this day.

Megan and Mrs. Starns had limited time before and after school to work together
to plan lessons. Thus, most of their common planning time took place during fifth hour
which is Mrs. Starns scheduled planning time. The planning conversations between Mrs.
Starns and Megan tended to focus more on what was going to be taught or happen (i.e.
“We will do section 3.2 on Thursday” or “I am going to use this activity”) as opposed to
how it would be taught or how students might respond. On occasion Mrs. Starns shared

some general observations from her experience with Megan about the content in the

Focus Class. For example, she shared with Megan that the chapter on quadratics was

115

traditionally “so confusing for kids” [Methods Course Observation, December 5, 2008].
The opportunity for Mrs. Starns to provide feedback on Megan’s teaching was also
limited to a degree. Mrs. Starns participated in several professional development
programs that occasionally kept her away from the school. In addition, she often worked
with struggling students in another room while Megan was teaching.

Mrs. Starns was very clear about her expectations for Megan’s development

during the beginning of the field experience.

I really want her to focus [during] the first marking period on instruction, [such
as] getting herself to speak well, to speak clear, to get directions out. I think that’s
really important. The little stuff, the nitpicky stuff, it’s important but you have to
do it in layers. So I think the very first layer that she has to get accustomed to is
gaining [students’] attention and being confident in the activities that we are doing
and the material that she is teaching. It’s an overall respect that you have to gain
from the kids and [you need to] be able to project yourself as a strong person and
a strong woman. And that’s my push for her this marking period. And [also] to be
more clear and concise and less 'uhhhh,‘ and that kind of thing. [Mentor Teacher

Interview, October, 2008]

She also Stated that as the year went on she thought Megan would need to spend less and
less time on her lesson planning as she learned how to present material.

MIS. Starns was very vocal with Megan about how she felt about the expectations
0f the field instructor, and by extension the university. She was concerned that the
university field instructor may not have had adequate experience to advise Megan
because she had not taught as long as Mrs. Starns thought necessary. One day after they
had discussed some of the field instructor’s ideas for improvement Mrs. Starns told
Megan “don’t let her bother you too much” [Informal Conversation, Lesson 1.3

Observation], In addition to stating her concerns about the university ideas to Megan,
Mrs, Starns also chose to conduct Megan’s field experience in slightly different ways

th
an the university prescribed. For example, Megan began teaching her focus class in the

116

-,..
‘ '
il—c—ur‘ ~71-

third week of school instead of the first. In another instance, Mrs. Starns had Megan
grade the papers for all of the Algebra 2 classes and part of the Math Analysis course so
that she could get used to the grading load early in the school year instead of only being
responsible for her single Focus Class. Mrs. Starns comments and actions gave the
impression that she had different priorities for Megan’s development than those voiced
by Emily and Dr. Rundell. The next section describes the class that Mrs. Starns and

Megan chose to be Megan’s Focus Class during her internship.
4.2.3 The Focus Class

Megan’s Focus Class was an Algebra 2 class made up of twenty-three students,
eleven males and twelve females. Four of the students in the class received separate
accommodations and were either not physically present in class or were working on their
own material. Two of the students were “migrant students” and usually spend the entire
class period with a special “migrant teacher.” The other two students also attended the
local career center and were usually pulled out by the mentor teacher and worked with
individually during the Class. The students sat at desks that were arranged in pairs and
faced an overhead projector. The class also had an LCD projector mounted to the ceiling
Which was mostly used to display the bell work from a Word document at the beginning
of each class. Along one side of the classroom was a bank of cabinets. One of the doors
Was labeled “Physics CBRs.” This technology was used in Mrs. Starns’s physics class

p Iior ‘0 One observation, but never in the mathematics classes. The main technology used
in the class was the TI—84 calculator that was used individually by the students and at the

OVe . . . . .
rhead prOjector as a presentation tool. The walls were decorated wrth bright posters

1,17

and there were student created mobiles hanging from the ceilings that students took down
and used when playing the Game of Life, a simulation game in which students kept a
checkbook and make deposits and withdrawals each Friday based on the number they
rolled on a dice.

The Focus Class uses the McDou gall Littell Algebra 2 textbook (2004). During
the first semester the Algebra 2 course focused on the following chapters from the
textbook:

Chapter 2. Linear Equations and Functions

Chapter 3. Systems of Linear Equations and Inequalities

Chapter 4. Matrices and Determinants

Chapter 5. Quadratic Functions

Chapter 7. Powers, Roots, and Radicals

Chapter 8. Exponential and Logarithmic Functions
Following Mrs. Starns’s schedule, each chapter was taught and tested in a three week
period.

The Students in the Focus Class mostly were cooperative with Megan, although
on occasion they would conduct short side conversations that hardly ever reach the point
0f disrupting the class. Often before class began the students would ask Megan questions
abOUt their homework or engage her in conversations about the school football game or
Other 0111 of school activities. When Megan moved outside Mrs. Starns’s normal schedule

and structure and asked the students to participate in activities instead of take notes, the

students appeared to do so willingly. For example, Megan asked students to participate in

118

a skit to introduce her Cool Stuff 1 lesson. She had no problem finding students to

volunteer or in keeping the other students engaged during the activity.
4.3 Summarizing the Contexts

The university and school contexts in which Megan participated offered her a
variety of perspectives and opportunities. The university supported her by providing a
collaborative community and experiences that challenged her to think deeply about her
practices. The school supported her by providing a pleasant setting for her to work with
cooperative students and a practicing teacher to give “on the ground” advice. Despite
both contexts providing important opportunities for Megan to Ieam about teaching
mathematics, they presented significantly different perspectives in several areas. The
university context modeled a sharing and problem solving community of teachers. The
school context did not model this type of community, either in the math department or in
the interaction between Megan and her mentor teacher. The other important difference in
the two contexts was their focus on a different set of skills for Megan to develop. The
University context supported the development of deliberate and thoughtful practices of
teaching that revolve around mathematics. The school context supported the development
0f Megan’s general presentation skills and “getting her up to speed” with the daily

responsibilities of a teacher.

119

CHAPTER 5. FINDINGS

The findings of the study are organized into two sections that correspond to the

research questions.

1. What is the nature of a preservice teacher's practice of anticipating students'
mathematical responses during lesson planning over the first semester of her field

experience?
2. What factors within the contexts of a teaching internship (a field experience and

accompanying university course) promote or do not promote the practices of one
preservice teacher anticipating students’ mathematical responses during lesson

planning?
The first section describes Megan’s anticipating practices and presents two mini-cases
that illustrate the impact of Megan’s anticipated students’ mathematical responses
(SMRS) during the lesson enactment. The second section focuses on the interaction
between Megan’s practices of anticipating SMRs and the contexts in which she was
working. This section is organized by each of the zones outlined by Goos’ (2005a)

framework: the Zone of Proximal DeveIOpment, the Zone of Free Movement, and the

Zone of Promoted Action").
5.1 The Nature of Anticipating

Research Question 1. What is the nature of a preservice teacher's practice of
antrctpating students' mathematical responses during lesson planning over the first
semester of her field experience?

The following sections present snapshots of Megan’s practices of anticipating

S . . . . .
MRS during lesson planning at several pornts over the first semester of her internship

Ex ' . . . . . . . .
penence. The primary focus of this section is on the anticrpation practices that occurred

.0\

C
hapief 2 provides a summery of Goos’ (2005) framework with a description of each of the zones.

120

pl'i or to the teaching of each lesson along with the content of what was anticipated. A
Secondary focus of this section is a set of two mini-cases that illustrate the outcomes of
M e gan’s anticipation of SMRs in her enacted lessons. The mini—cases move beyond the
description of what Megan anticipated and the resources she made use of, into how she
enacted the two lessons to account for what she anticipated. The first seven sections that

describe Megan’s anticipation practices are presented chronologically.
5.1.1 In-Depth Interview 1

The first in—depth interview took place in late September during Megan’s second
full week of teaching. During this interview Megan responded to questions about her
answers to questions on the Lesson Planning Survey (LPS) as well as about a lesson she
brought to the interview that she planned to teach the next day. Megan’s responses during
this interview indicated that for this lesson plan she anticipated students’ mathematical
responses (SMRs) both actively/purposefully and passively/incidentally”. The lesson she
described focused on graphing and solving systems of linear inequalities and Megan
CXplained that it would be presented to students via lecture and that they would be

r esPOHSible for taking notes. Figure 5.1 represents Megan’s anticipation of SMRs for this
lesson With callouts that display Megan’s own words. Also included in Figure 12, in the
upper right hand comer, is the entire anticipation framework with Megan’s actions for

‘h' . , . . .
‘8 1eSSOn in grey, and the actions she did not take in white.

.\

1
Ref
r ‘0 section 2.3 for a description of the framework used to classify Megan s anticrpation practices.

121

 

 

Lesson Planning

 

 

  
 
  
 
 
 
 
  
  

   

    
 
 

Passive/

      
    

 

 

Actch/
Purpt iseful

 

InciJental
Anticipating

    
  

Anticipation

"I go ahead and do the

   

”When I first go over individual examples and
homework...l try to think

of the misconceptions as

 
 

the story problems and
the homework and I do all
those"

 
    

I solve"

  
   
  
 
 

 
 
   

Taking

  
 

The teacher consults

    
  
 

The teacher

    
  
 
 
 

 
      
    

the .. .-
positi completes the activny (lLllSldL icsouiccs [or
. on . . . .. .
of ii ,‘ in one way [0 [yrcparc lLdSOllS ()Ihcl liltlll
k“ for class ant1c1patmg
students

  
    
   
  
 

"She has the notes all
ready in a binder for each
class...so for now... I‘m
going to use hers"

 

"Besides getting the story
problems and the
homework problems [I did

not use the textbook]"

    
   

   
 
   
 
  

V
SNCs and
SQDPs

 

Curriculum
materials

 

 

 

Figure 12. Anticipation Map for Lesson Discussed During In—Depth Interview 1

122

 

cal

inii

l .
his

"Br

The left side of Figure 12 represents Megan participating in active/purposeful
anticipation and that those activities lead to possible SMRs. Evidence for this finding is
based on Megan’s responses to why she included certain SMRs in her lesson plan.
“Because you want to [think about] misconceptions, that’s just a given, you want to find
out how to answer [the students] questions. When I first go over homework. ..I try to
think of the misconceptions as I solve” [In-Depth Interview 1, September, 2008].

The right side of Figure 12 represents Megan’s passive/incidental activities in
anticipating SMRs for this lesson. As the figure shows, she indicated that she completed
the task herself, consulted the class textbook (curriculum materials), and used Mrs.
Starns’ notes (other teachers) when planning this lesson. The dashed line from The
teacher completes the activity in one way to prepare for class indicates that this activity
potentially led to Megan’s anticipation of SMRs, but there was no direct evidence that it
did. The next paragraph describes the student questions and difficult points (SQDPs)
Megan anticipated the students would have. While she linked one of the two student
questions or difficult points (SQDPs) specifically with trying to think of misconceptions,
she did not link the other with any specific reason or resource for anticipating. There is
no line connecting curriculum materials or other teachers to anticipated SMRs because
Megan’s own words revealed that she did not use these resources in this way. Megan
indicated that in planning this lesson she only used the classroom mm to “[360 the story
problems and homework problems” [In-Depth Interview 1, September, 2008]. The use of

Mrs. Starns’ class notes from previous years was included in usmg other teachers as

resources since Megan chose some of the examples in her own lesson plan from those

DOIBS.

123

Megan identified two SQDPs during the interview. The first related to solving
inequalities, “I think they might have problems with flipping the sign, like when they
divide by a negative” [In—Depth Interview 1, September, 2008]. Though this particular
SQDP represented content that was integral to what Megan was teaching in the lesson, it
was a1 so content that the students likely learned in Algebra 1. Megan indicated that the
mathe matics that came before this lesson focused on systems of linear equations, thus the
Smdents had likely not reviewed anything about linear inequalities in class. This SQDP
did “Qt represent the new content that students would be learning in the lesson, namely
the graphing and solving of systems of inequalities. Thus, in this case anticipating was
fOC“Sec! on whether the students would remember previously learned procedures in order
‘0 b6 able to make use of them in the current lesson.

The second SQPD that Megan anticipated related to “knowing how to shade
things. like when it's greater than you go above it and if it's less than you do it below”
[In‘Depth Interview 1, September, 2008]. Since Megan indicated that the students had
never before worked with an inequality in two dimensions, this SQDP represented Megan
anticipating SMRS based on the current content of the lesson, but like the first SQDP
foe“ SEC] on a procedure. The next section described Megan’s practices of anticipating

SMRS during the first set of classroom observations and interviews.

5.1.2 Observation 1

The first set of lesson observations and interviews took place over a one-week
period in late September and early October, during the fifth week of the school year. This

Was Megan’s third full week of teaching the Focus Class. During this week, Megan

124

 

lit

followed Mrs. Starns’ weekly schedule and two sections of the class curriculum were
taught from the textbook as was the norm in the class. Table 4 shows the activities of the

week -

Table 4 - Schedule of C lass Activities (luring Observation 1

 

 

Lesson Number and Class Activities

 

 

Monday Lesson 1.1 (Linear Programming)
Lecture and note taking

T“esday Lesson 1.2
Review of previous night’s homework

Wednesday Lesson 1.3
Quiz

Class game

Thursday Lesson 1.4 (Graphing Linear Equations in Three Variables)
Lecture and note taking

Friday Lesson 1.5
Game of Life

Notebook check
Introduction to Election Project

Only Lesson 1.1 and 1.4 were analyzed for the nature of anticipating because the other
classes did not engage students in new mathematics. In preparation for these two lessons
Megan participated in only passive/incidental anticipation, anticipating SMRs related to
past and current content, and making use of resources available at the school.

Lesson 1.1 was taught on Monday and focused on linear Programming and
Lessgn 1.4 was taught on Thursday and focused on plotting single points in three

dlmetlsions and then graphing equations in three variables. Both lessons were taught via

125

a

126

.Mfl i
1'1 bdllikl‘.

 

 

 

      
    
     
 

 

 

Lesson
Planning .
l,
t t i I;
l’assivc/ l 3 .
Incidental 7 it i i T T

 

 

Anticipating

    

 

 

 
    
  

”I go through
and do the
problems"

   

   
    
 
   
     

     
 
 
    
 
  

The teacher consults
outside resources for

The teacher
completes the activity
in one way to prepare
for class

reasons other than
anticipating

     

"I'm just imagining they
[will have a hard time
with it]"

 

 

 
 
 
 

"I have had trouble
with it"

 
 
    
   
 
 
 
  

"That [example]
I got in...the
book."

 
       

 

l
SNCs and
SQDPs

 
   
  

 

Taking the
position of their
students

 
 
   
  

Curriculum
materials

 
    

 

Figure 13. Anticipation Map for Lessons 1.] and 1.4

127

 
   
   
     
   

" I talked to my
mentor about
[students having
trouble with] that
before."

 

 

{7’3“

SI‘J

sh

Figure 13 displays all of Megan’s anticipation activities for Lesson 1.1 and
Lesson 1 .4. The small representation of the entire anticipation frame in the upper right
hand corner of the figure shows that all of Megan’s anticipation activities took place on
the Pi ght, or passive/incidental, side of the framework. Prior to teaching Lesson 1.1
Megan solved all of the problems to be completed in front of the class, discussed the
168801] with her mentor teacher, and used Mrs. Starns’ notes from previous years to find
exaITIIDIes. From speaking with her mentor teacher and “just imagining” Megan
antiCipated a potential SQDP, namely that students would have trouble knowing when to
use a dashed or solid line when graphing inequalities. Megan’s imagining is represented
in Figure 13 as Taking the position of their students because She indicated that She had
not yet had a chance to see the students work in this area and so had no prior knowledge,
She just imagined they struggled with it in the homework. She also anticipated that
Studel‘its might struggle to visualize the constraints when graphing equations and knowing
which side to shade when graphing an inequality. As with the lesson discussed in In-
D epth Interview 1, the struggles that Megan anticipated for this lesson included recalling
past mathematical content that the students should have previously learned.
In preparation for teaching Lesson 1.4 Megan completed all of the examples that
She Would present and students would copy during her lecture, talked with her mentor
teacher about teaching the concepts, looked in the textbook for problems to use, and
reflected on her own learning of the concepts. Her anticipating for this lesson again
focused on SQDPs but also included a possible stUdent na'i've conception (SNC). The
SNC that Megan anticipated dealt with students understanding of what constituted a

s - . . . .
olution in three dimensions. “Because they're used to graphing lines...and now they

128

have to realize that the solution is a plane and not a line and that it equals everything in
the pl ane. And I don't know, sometimes they're like 'oh its just the points' or 'oh its just
the Ii nes with the points' but it’s everything” [Lesson 1.4 Pie-Interview, October, 2008].
In addition to anticipating this possible SNC Megan anticipated that the students would
struggle with the addition of the z-axis to the coordinate plane; “the kids have trouble
visual 1y seeing [three dimensions]” and she noted that she knew this because when she
reflec ted on her own learning she recall that “I have had trouble with it” [Lesson 1.4 Pre-
Imel'View, October, 2008]. She also anticipated that the students would struggle to see z
35 a function of both the variables x and y. As with the previous lesson, Megan’s
antic ipation focused both how students would respond to the new content being taught
that day and whether they would remember the procedures of past content. The next

SBCtiOn describes Megan’s anticipation of SMRs during the planning for her first

university lesson assignment.

5.1.3 Cool Stuff I

The first Cool Stuff lesson took place one week after the first week of
Observations. In preparing for this lesson Megan participated in both active/purposeful
and passive/incidental anticipating. Instead of teaching new content, Megan used the
Cool Stuff 1 assignment to create a review actiVity for an upcoming test. The SMRs that
she anticipated for the lesson reflected this in that some of them focused on past material
the StlJclents should have learned. Unlike PaSt lessons Megan used OUtSide curriculum
resources to create an open—ended task for her students to participate in during class.

Megan Specifically asked for the help of the researcher in choosing an activity from the

129

Interactive Mathematics Program (IMP) Year 2 curriculum (Fendel, Resek, Apler, &
Fraser, 1 998)”. She chose this particular curriculum resource because she had used
activi t ies from it in her senior year field experience and she liked the tasks it provided.

Figu r e l 4 displays the anticipation map for this lesson.

 

“ . . . .‘ ‘ _J _ , 3
For further discussion of the role of the researcher in this study 5“ Chapter

130

 

 

 

   

 

Lesson

Planning

 

  
 
    

Activc/ Purposeful

Anticipation

 
    
 
  

  
  
 
  
  
 
   

"I tried to think of

 
  
 
  
     
  
 

before."
Taking
the
position
of their
students

  
 
  

SNCs and
SQDPs

Fi . ..
gurc 14. Cool Stuff Lesson 1 Anticipation MaP

131

misconceptions and things"

wI‘hcy'vc had trouble with it

completes the activi
in one way to prepare

 

 

  

 

 

I’Llssch/

 

Incidental

  

Anticipating

  
 
 
   
 
  
   
  
   
  
  

 

"I only did it one way to
tell you the truth. because
that was the way I had
done it with the lids

        
    
  
 
    
  
  
   
   
  
 
  
  

earlier The teacher consults

outside resources for

 

   
  

The teacher

reasons other than

    

anticipating

  

for class ,, . .
My peers were givmg me

. ‘ ‘ ‘ H
suggestions.

   
   
   
 

"I started to look in
the IMP book ...and
I found the Big State
U lproblem]"

   
  
  

Dr. Rundell "do understand
that the nature of IMP is going
to be very different for them"

SNCs and

SQDPs

 

SNCs and
SQDPs

 

 

 

 

The right, passive/incidental, side of Figure 14 represents that Megan received
feedback from other teachers about her lesson, in this case other interns in her methods
course and Dr. Rundell, the methods course instructor. Megan shared some of her
original activity ideas with two interns in the methods course during a case stories
activity”. The other two interns recommended adding some scaffolding to the original
activity because they felt it might be too open-ended for students who were used to
lecture and note taking. One of the interns also suggested that Megan complete the task
herself before teaching it, which she eventually did. In addition to consulting these
resources Megan stated that she talked with Mrs. Starns before teaching the lesson.

Although Megan selected the task from the curriculum there is some question as
to whether she consulted the accompanying teacher materials for the activity. This
supposition is based on the fact that in her lesson plan she identified an incorrect final
answer based on incorrect arithmetic even though the correct one was given in the teacher

materials. Figure 15 shows Megan’s own work to find the solution, which she

corrected 1 4.

\

I3

14 See Hughes, Smith, Boston, and Hogel (2008) for further discussion of case stories.

s lAfter completing the lesson post-interview, the reseracher drew Megan’s attention to the fact that the
’SO “"9“ Was (250, 250). She made this change to her work prior to supplying me with a copy, as can be
’ee“ "1 Figure 5.4.

 

132

_ ' ”ire-e J ~
Ari-C, 13’). If) it.) /;)‘ L: 7,19%} {HT-”:31" jet} til, (if/d
_7 [3335-2 ilfiiaaj 7,“,

C._1 M71717 A}: £2.

l” '* . is. 2:» -: s ~ 3 737mg. 275 3*

56 mo
as >

lM’mkf-m errant, it:

 

 

Figure 1 5. A Handwritten Portion of the Cool Stuff 1 Lesson Plan Showing Megan's

Original Solution

Figure l 6 shows a recreated portion of a page from the teacher manual with the correct

solution highlighted by a rectangle at the bottom of the page.

133

 

*— 8000x + 2000_v = 2,500,000

  
   

\ = x —>
Close-up of
the Big State U
( 2 3 5 2 3071) Feasible Region
7’ 7 l l
(283—, 283—)
3 3
(250 250) *— IOOx + 200): = 85,000

- The details of the reasoning

In presenting the problem, students should review how they sketched the graph,

including how they knew which point minimized the cost and how they got its
coordinates.

 

In explaining how they identified the point, students should use the “family of
parallel lines” idea, in which the combinations of in-state students and out—of-

state students that give any particular cost form a straight line, which “slides”
when the cost is changed.

Students need to Choose the “minimum” of this family of parallel lines. If they
sketch One of these lines (for example, 7200x + 6000y = 1,440,000), they Will

see the general direction of the lines in the family. (This is the dashed line in the
fifSt diagram)

    

SIUdents should see that the cost goes up as the line moves up and to the right
and so the minimum is at the point (250, 250).

\

 

 

Figure 16. A Recreation of a Page from Teacher Manual of IMP (Fendel, Resek, Apler,
8‘ Fraser. 1998)

134

Due to this evidence, the anticipated SMRs were not attributed to using the curriculum.
Megan also indicated that she only completed the task herself in one way.

The left, active/purposeful, side of Figure 14 represents the fact that Megan chose
to anticipate SMRs as a part of her lesson planning practices in preparing this lesson.
Megan indicated in the pre-lesson interview that using the lesson plan template (which
was required for the Cool Stuff assignments) and her past experiences with the students
impacted her practice of anticipating. “The lesson plan template helps a lot because it
really gets you through the process and I tried to think of misconceptions and things. I
didn’t research them a lot, but we had been going through things with the students so I
had already thought of misconceptions that might come up” [Cool Stuff 1 Pre-Interview,
October, 2008].

One of the student responses that Megan anticipated for this lesson dealt more
With the structure of the activity than the content. Megan predicted that students might
Struggle with participating in a different kind of mathematical activity since they were so

used to lecture and note taking.

Because this is new to them, as soon as they come into the class I assume they are
going to be having questions as to 'What are we doing? Why are we doing this?‘ It
doesn't have to do with the task, but it does because it has to do with their
motivation. [Cool Stuff 1 Pre-Interview, October, 2008]

This POtential student response to the lesson was brought up in the conversation Megan
had With the other interns as well as by Dr. Rundell in his feedback to Megan.
In addition to anticipating that students might be unsure about the structure of the
lesson, Megan also anticipated SMRs related to the content. As in a previous lesson, she

anticipated that students would potentially have difficulty knowing which side of an

135

inequality to shade which could lead to errors in finding the feasible region. She also
predicted that students would struggle to write equations and inequalities from the

situation that was presented in the task because they were used to being given the

equations and inequalities to work with.

I imagine that they're going to have questions about how to set things up for the
task. I mean everything is in words. So 'Where is the objective function?’ 'Where

are the constraints?‘ and how exactly do they set them up. [Cool Stuff 1 Pre-
Interview, October, 2008]

Finally, Megan anticipated that students might not initially choose an appropriate scale

for their graphs when graphing by hand. The next section describes Megan’s anticipation

of SMRS during the second set of lesson observations.

5.1.4 Observation 2

The second set of lesson observations and interviews took place over a two-week
period in early November during the tenth and eleventh week of the school year. During
these two weeks Megan was in the second guided lead teaching phase of the field
CXperience, which meant that she was teaching all three sections of the Algebra 2 class.
The schedule of observations was adjusted from the first observation to allow for more
observations of Megan teaching and to eliminate the observation of class games and
assessments. During the two weeks Megan followed Mrs. Starns’ weekly schedule. Mrs.
Starns and Megan together rearranged the order of the content in the textbook. This
decision Was based on Mrs. Starns’ experience that students had trouble with quadratics
and the fact that both had recently attended a county professional development workshop

that Eimphasized the importance of students understanding this material. Table 5 shows

the activities of the two weeks.

136

Table 5. Schedule of Class Activities during Observation 2

 

Monday

Tuesday

Wednesday

Monday

Tuesday

Class Activities

Lesson 2.1 (Vertex and Intercept Form)
Lecture and note taking with bridge picture packet

Lesson 2.2 (Vertex and Intercept Form cont.)
Lecture and note taking with bridge picture packet

Lesson 2.3 (Converting between Standard and Intercept Form)
Lecture and note taking with bridge picture packet

Lesson 2.4 (Factoring Quadratics of the Form ax 2 + bx + c)
Lecture and note taking

Lesson 2.5 (Completing the Square)
Lecture and note taking

 

During the first week Megan made use of a packet of pictures of bridges in her teaching.

Mrs. Starns created this packet and gave each student a copy to serve as representations

of parabolas in the real world for use in note taking during the week.

The content over the two weeks focused on quadratic equations and their various

forms (e.g. standard, vertex, and intercept). During this set of observations Megan

engaged in only passive/incidental anticipating during her lesson planning. Her potential

resources for anticipating SMRs during this time consisted of her completion of the

examples for the day and her conversations with Mrs. Starns. Figure 17 presents Megan’s

antiCiI’ation activities for this set of lessons.

I37

 

 

 

 

Lesson

Planning

 

 

 

 
  

  
   

Passivc/

 

Incidental

 
  
 

 

, ,, W, in
, </.\
L \/
V I if 7“, 777 7 7 l
A \
< . /\
. i T
l L \T
' W "M Law—7 _jgs/ ]

 

Anticipating

     

 

 

"I went ahead and made
sure I knew how to do

 
 
   
   
  
 
  

 
  
 
  
   
 
 

The teacher consults

outside resources for

reasons other than
anticipating

everything and I did the

   

The teacher homework that we‘re
completes the activity
in one way to prepare
for class

 
 
 

going to go over.

 
 

   
   
   

"ll only looked in
the textbook] to
find examples."

 

 

 

 

 

 

 

Fi) 1 . . _ ‘
Mira l7. Antietpatlon Map for Lesson 2.1 through Lesson 2.5

 
  
    
  
 
    
    
     
   

  
  
 

”[My mentor] helped
me figure out how to
present it to the kids
and how to look at each
[example] and use it."

 
 
  

 

 

 

 

 

 

As was a primary focus in the previous anticipation maps, this one shows that Megan’s
anticipation activities were passive/incidental.

Lesson 2.1 was the beginning of a new unit on quadratic functions and it focused
on presenting two forms of quadratic equations, the vertex form and the intercept form.
Throughout this series of lessons Megan used the packet of pictures of bridges created by
Mrs. Starns as examples for the class. For this lesson Megan completed all of the
examples herself, looked at her mentor teacher’s notes from the previous year, and

discussed the lesson with her, focusing on how to be more concise in her presentation.
Megan anticipated that students would have difficulty knowing what the focus and
directrix were, understanding what the variable a represented in the equations, and
confusing the forms of the equation. Figure 17 shows that Megan completed the
examples herself and talking with her teacher, in an effort to gain ideas about how to be
more precise, potentially led to the SMRs that she anticipated.

In Lesson 2.2 students continued taking notes using the bridge examples in the
packet and finding the equations of the parabolas they created. Megan again completed
all of the examples and consulted her mentor teacher before teaching this lesson about
how to present the examples though when asked she said, “we didn't solve them ...we
Verbally did it” [Lesson 2.2 Pre-Interview, November, 2008]. One of the examples called

0“ the Students to measure using a ruler and convert to actual measures which Megan
antiCiDated might be difficult for them. “You're given the height, but you're not given the
height h” Om where the vertex is to where the cable is, so we have to use ratios and they

might get a little confused” [Lesson 2.2 Pre-Interview, November, 2008]. Additionally,

I39

Megan thought that the students might struggle to recall the steps in writing the equations

they learned the previous day.

Lesson 2.3 was the final lesson focusing on the vertex and intercept forms of
quadratic equations. In this lesson Megan introduced “boxing” as a method for
converting between the standard form and intercept form of a quadratic equation. (The

boxing method uses an area model to help students multiply binomials.) Figure 18 is a

portion of Megan’s lesson plan that shows the example she used to present the boxing

method.

(a): CX «1th " ‘3")

} a Sfixnwd (LV 4 GOV'éo ' \
L3) '1" Xgfigx .— Cilmtcl QUCLCL ""ht-i'mi )

 

Figure I 8- A Handwritten Portion of Lesson 2.3 Showing Megan’s “Boxing” Example

When preparing this lesson Megan completed most of the examples ahead of time and
her mentor teacher explained the boxing method to her. She anticipated that students
might struggle with the boxing method because they had not seen it in a while and that
they might have some difficulty identifying the correct factor pairs for the coefficient of
the x term when boxing. Megan also thought that students might ask “Why would we
have to switch from standard form to intercept form?” [Lesson 2.3 Pre-Interview,
November, 2008].

In Lesson 2.4 students again applied the boxing method to factor quadratic

expressions, this time with a leading coefficient other than one. Based on her own work

140

doing the examples she found in the textbook and her conversations with Mrs. Starns,
Megan anticipated several SQDPs. First, Megan explained that Mrs. Starns told her that
students have a difficult time knowing to make use of the zero product property when
solving quadratic equations. Figure 19 represents one of the examples Megan presented
in the lesson in which students needed to recall this property to solve after the quadratic

equation had been factored.

 

Ex.l

(5x+7)(x+2)=0
5x+7=0
x+2=0

Solving:

x = -7/5 andx =-2

 

 

 

Figure 19. Example of Making Use of the Zero Product Property from Lesson 2.4

Megan also anticipated that students would struggle with using the boxing method when
the variable a in the standard quadratic equation was a value other than one. Finally,
Megan was concerned that students might not remember the steps in the boxing method
that they were reintroduced to the previous week. “When I [use] the box, maybe they will
forget steps” [Lesson 2.4 Pre-Interview, November, 2008].

The final lesson in the second set of observations was of Lesson 2.5. In Lesson
2.5 Megan showed the students how to complete the square using the boxing method. She
stated that she talked with her mentor teacher about how to teach completing the square
in this way, completed the task herself, and that she used the textbook as a resource to
find examples. Shfi‘ anticipated that students would ask questions about a process used in

Completing the square, Specifically, “Why do you add and subtract the number?” [Lesson

14]

 

2.5 Pre-Interview, November, 2008]. The next section describes Megan’s anticipating in

the second Cool Stuff lesson assignment.

5.1.5 Cool Stuff 2

The second Cool Stuff lesson took place in late November. As with the first Cool
Stuff lesson, Megan participated in both active/purposeful and passive/incidental
anticipating in planning this lesson. Similar to the first Cool Stuff lesson, Megan used this
assignment to review some of the past content in the chapter for an upcoming test. She
also added a portion of new content to the lesson, specifically finding the discriminant of
a quadratic function. Megan and her mentor teacher created the activity for Cool Stuff 2
together and used video of a recent school football game as the context for the activity. In
the activity students watched several clips of the football game in which the football
created parabolas as it traveled through the air. Groups of students were then asked to
write the equations for the parabolas created by the football that they viewed in the video.
Prior to watching the football video Megan presented notes to the class on finding and
using the discriminant of a quadratic equation. Figure 20 presents Megan’s anticipation

activities for this lesson.

I42

 

 

 

Lesson

Planning

 

 

  
         

Activc/ Purposeful
Anticipation

  
   
   
   
    

 
 
 
  
  
  
  
 

Taking
the
position
of their
students

 

 

 
  
     
 

  

Passive/
Incidental
Anttcrpatmg

 

 

< ("

 

 

  
   
  
  
   

The teacher
completes the activity
in one way to prepare
for class

 

 

SQDPs

 

 

 

Figure 20. Anticipation Map for Cool Stuff 2 Lesson

143

 
 
   

"Yes [I did the
task myself]."

 

  
   
  
  

 

The teacher consults
outside resources for

reasons other than
anticipating

      
     
     
  

"I chatted with my
mentor about itjust
verbally and we tried
to think of different
ideas [for the lesson]."

  

 

 

 

The passive/incidental side of Figure 20 indicates that the only resource that Megan
consulted in planning the Cool Stuff 2 lesson was her mentor teacher and only to discuss
ideas for the activity. In addition, she completed the task herself, which potentially led
her to consider some SMRs. The use of the lesson plan format is included in the
active/purposeful side of Figure 20 because it specifically calls on the teacher to
anticipate student thinking and questions. In this section of her written lesson plan Megan
included several SQDPs.

In planning this lesson Megan anticipated only SQDPs. The first was that students
would struggle with the meaning of the variable a in both vertex and intercept form,
something she also anticipated in Lesson 2.1. Megan also considered that the students
might have trouble performing the procedures for completing the square using the boxing
method that they had learned in a previous lesson. In addition, Megan expected that
students would have a difficult time finding the initial location and total distance traveled
by the football and finding the maximum height of the football from the video. Megan
also included in her lesson plan a series of student questions she thought might arise:
Which form do I start with? What is a? How do I get Quad Form on my calculator? What
are the directions? Do I need to write out the vertex? Do I need to write out p and q?
Finally, similar to her Cool Stuff 1 lesson, Megan included in her lesson plan that
students might struggle to engage with the mathematics in ways other than those they
were used to in this lesson. “Students may have trouble answering the group discussion

qUeStions. They are deep thinking questions that students are only used to doing in the

144

 

reading packets.'5” The next section describes Megan’s practices of anticipating during

the third and final set of observations.
5.1.6 Observation 3

The third set of lesson observations and interviews took place in the second week
of December. During this time Megan was only teaching her Focus class. During the
observations Megan followed the sections in the sequence prescribed by the textbook.
Table 6 shows the class activities during the observation.

Table 6. Schedule of Class Activities during Observation 3

 

Class Activities

 

Monday Lesson 3.l (Graphing Square Root and Cube Root Functions)
Lecture and note taking

Tuesday Lesson 3.2 (Statistics and Statistics Graphs)
Lecture and note taking

Wednesday Lesson 3.3
Students complete review worksheet individually

 

Only the first two lessons, Lesson 3.1 and 3.2, were analyzed for the nature of
anticipating because Megan did not teach anything on Wednesday of that week and thus
did not prepare a lesson plan. In preparing the two lessons Megan only participated in
passive/incidental anticipating and drew on her mentor teacher, her own completion of
the tasks, and considerations about student reactions as resources. Figure 2] displays the

anticipation map for these two lessons.

\
M

'5 ' ~ 3 ' 1 * 1 J y .
SOIU‘IODS ‘0 problems from the reading packet were always presented to the students by Megan or Mrs.
Starns

145

   
    
     
   
 

  

lesson

 

Planning

Passive/

  

Incidental
Anticipating

 

 

 

 

 

 

 
  
   

  

"I went
through and
did those

 

  
   

   

The teacher
completes the activity
in one way to prepare
for class

   
     
     
      
      
 

   

 

 
  
 

Taking the

 
 

SNCs and
SQDPs
students

Figure 2]. Anticipation Map for Lesson 3.1 and 3.2

146

notes."

position of their

 
  
     
 
 
   

The teacher consults
< outside resources for

reasons other than
anticipating

"She called it steeper
and I didn't like
that...hecause we
used steeper when

explaining slope."

"I just looked in the
section and tried to
notice if they have any
definitions or theorems
or stuff like that."

 
   
   
 

Curriculum
materials

 

 
  

  
 
    
   
 

"[I consulted]my
mentor teacher and
my teachers
notes."

SNCs and
SQDPs

 

 

 

Figure 2| illustrates that as with many of the past lessons Megan’s anticipation of

SMRs fell on the passive/incidental side of the anticipation framework. However, in
completing the activity to prepare for Lesson 3.] Megan did take the position of her
students in anticipating a particular SQDP. Lesson 3.] focused on graphing square root
and cube root functions. Megan anticipated that the students might have difficulty with
some of the language her mentor teacher suggested using to describe the changes to the
graphs. Mrs. Starns described the cubic functions as becoming “steeper” as the equation
changed- Megan was concerned that this might be confusing for students since they had
used similar language to describe the graphs of quadratic equations and she chose to
describe this differently in her teaching. Figure 22 is an example from Megan’s notes that
shows her use of the phrases “further from” and “closer to” opposed to “steeper” or “less

steep” to describe the differences of the graphs of a cubic equation.

I47

 

00% /.\46 he; Grainy?

wig. earns-M”

Figure 22. A Handwritten Portion of Lesson Plan 3.1 Illustrating Megan’s Language
Choice

This instance on anticipating occurred at a deeper level than much of the past anticipation
Which focused on whether students would remembers procedures. In this case, Megan
reflected on students past experiences with the mathematics in on order to consider their
Possible reactions to her current presentation.

For Lesson 3.] Megan used a worksheet created by her mentor teacher to teach
the lesson. She completed the worksheet herself before teaching the class. The class
Worked through the problems on the worksheet together and Megan pointed out how
Various Changes to the square root and cube root equations impacted the graph of the

equations. This was the first instance during data collection in which the textbook pointed

Out a Possible student difficulty (Figure 23)-

I48

ECOMMON ERROR
EXERCISES 22—39 Students
may confuse the horizontal and ver-
tical shifts or shift horizontally in the
wrong direction. Refer those stu-
dents to the box on page 431 and to
Examples 2 and 3 on page 432

Figure 23. Student Difficulty Presented in Class Text

Because Megan did not make use of the textbook to create her examples, and because she
did not mention this as a potential SQDP, it is possible she did not even see this hint for
teachers in the textbook. In addition to noticing changes to the graph, students were also
expected to find the domain and range of the functions and Megan anticipated that they
would have difficulty understanding what they were, visually seeing how to find them,
and remembering which was which.
For the cube root function the domain and range is all real numbers, so they are
good to go on that. But on the [square root] part they're probably going to have
questions on domain and range. The domain is the x's and range is the y's, so
they'll probably get confused on that... I think looking at it, sometimes visually,
they have trouble knowing that even though the x's go this way, they actually run
this way, even though they're on this axis. So when they look at it they get
confused, visually. [Lesson 3.] Pre-Interview, December, 2008]
She also expected that students might struggle to recall how to enter cube root functions
into their calculators.
Finally, Lesson 3.2 focused on reading and creating box and whisker plots.
Megan did not talk with her mentor teacher about this lesson before she taught it and in
fa“ Wrote her lesson plan while watching the mentor teacher teach the material to the

Other Algebra 2 class first period. "My mentor and I didn't have time to discuss this so I

JhSt recorded her notes that she did during first hour" [Lesson 3.2 Pre-Interview,

I49

December, 2008]. Despite not preparing a lesson plan in advance of the school day,
Megan did state that she looked over the textbook to figure out how this lesson connected
to the previous lesson. Again in this instance the textbook presented a common error that
students might make, but Megan did not reference this error in talking about possible
SMRs.

For Lesson 3.2 Megan anticipated both SQDPs and SNCs. The SNC that she
anticipated was that students might think that the middle line of the box and whisker plot
represented the mean instead of the median. “I think that in the box plot some people
think that the middle line is the average when it's really the median” [Lesson 3.2 Pre-
Interview, December, 2008]. In considering the possible questions of difficulties that her
Students might encounter in this lesson, Megan stated that she thought they might have a
hard time using the calculator to create a box and whisker plot. She also thought that
students might confuse median and mode and that they might not realize that the range of
a set of data is a difference, and therefore not a negative value. The next section presents
Megan’s anticipation practices in a lesson that she prepared to teach near the completion

of data collection.
5.1.7 In-Depth Interview 2

The final interview with Megan took place during the third week of December.
She brought a lesson plan to the interview that she planned to teach that week, which
fOcused on exponential growth functions. Her anticipation of SMRs for this lesson was
Similar to what was found in previous lesson; it was passive/incidental and relied mainly

on her own completion of the examples. For this lesson Megan planned to lecture as the

150

students used their calculators and took notes on the examples she presented. Figure 24

displays Megan’s anticipation activities for this final lesson.

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24. Anticipation Map for Lesson Discussed during In—Depth Interview 2

152

.. ‘fifi-:‘

 

 

Figure 24 displays that, similar to previous lessons, Megan used her mentor teachers’
notes from past years and the textbook to choose her examples and completed all of the
examples herself. However, neither the use of the textbook nor her mentor teacher’s notes

led her to anticipate any SMRs. For this lesson Megan anticipated that the students might

ask a particular question about two of her examples shown in Figure 25.

2. ‘17"

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w-v ‘ Tr—

 

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Figure 25. A Handwritten Portion of In-Depth Interview 2 Lesson Plan

“They might ask me about this fraction in front here and the number here and how that

[difference] impacts exponential growth” [In-Depth Interview 2, December, 2008]. She

also thellght that students might struggle with knowing which way to shift the

exponential graph based on the signs of the constants in the equation. The next section

l53

summarizes the findings related to the nature of Megan’s practices of anticipating SMRs

in her lesson planning.

5.1.8 Summary of Anticipation

The content of Megan’s anticipated SMRs varied from lesson to lesson over the
semester. In ten of the thirteen lessons, Megan anticipated that students would struggle to
recall past content learned in the class and which was needed for the current class. This
often focused on “remembering the steps” that had been presented to them. In nine of the
thirteen lessons, she anticipated that students would have difficulty with the new
materials being presented that day. When she anticipated SMRs around new content, it
dealt primarily with students’ understanding of the material that she herself was
presenting. For example, in Lesson 2.4 when Megan was teaching the students to use the
boxing method to factor quadratic expressions with a non-zero leading coefficient, she
was concerned that students would struggle to know what to do with the leading
coefficient, even though this was the focus of her lesson. The content of what Megan
anticipated also focused mostly on student questions or difficult points (SQDPs) and to a
much lesser degree student naive conceptions (SNC). Noticeably absent from the list of
what was anticipated is multiple student solutions strategies (SSS), which was included in
the anticipating framework. Possible reasons for this absence will be discussed in the
second half of this chapter.

The previous findings also shed light on the resources that Megan made use of in
order to anticipate. Most of the anticipating that she practiced was passive/incidental and

came from her own work of completing the mathematical examples in her way. While

154

Megan often made use of the class textbook she did so exclusively for the purpose of
finding examples, definitions, and homework and quiz problems. Recall that the textbook
on two occasions offered possible SQDPs and that Megan did not report either of these
when discussing the anticipated SMRs for the lesson. On only two instances did Megan
look outside of the class curriculum to find activities, for the Election Project and to find
a task for Cool Stuff 1. Megan also made frequent use of her mentor teacher as a resource
for teaching. Evidence indicated that these interactions rarely included the discussion or
sharing of possible SMRs. Table 7 summarizes the resources that Megan used to

anticipate in each lessons as well as what she anticipated.

Table 7. Summary of Anticipation Activity

 

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156

represent Cool Stuff 1 and Cool Stuff 2 respectively.

The checkmarks in the table indicate instances in which there is evidence that a resource
was used to anticipate, even though it may not have been the original purpose of
consulting the resource. The open circles indicate that a resource was consulted, but that
there is not sufficient evidence to be sure that it contributed to anticipating or that it did
not. The x’s represent evidence that a particular resource did not contribute to
anticipating in any way. The table shows that purposeful anticipating occurred only when
required for a university lesson (as in Cool Stuff I and 2) or in one of Megan’s early
lesson plans (In-Depth Lesson 1). Additionally, all of the instances of both taking the
position of students or interacting with other teachers (mainly her mentor teacher) that led
to anticipating, are found earlier in the semester. Despite this lack of active/purposeful
anticipating and relatively infrequent use of resources to anticipate, the table also shows
that Megan was able to anticipate some kind of SMR for each lesson that she taught,
although they were often focused on students remembering procedures.

Another method of summarizing the results is presented in Figure 26. This figure
shows the entire anticipation framework for each lesson, or set of lessons, with the

shaded areas representing the activities in which Megan participated.

157

 

Cool Stuff 1

Lesson Set 1

In-Depth Lesson 1

 

L

Figure 26. Collection of Anticipation Maps Representing Megan’s Activities over the

First Semester

 

Lesson Set 3

Cool Stuff 2

Lesson Set 2

158

( f

chL

 

 

In-Depth Lesson 2

 

 

Figure 26 again highlights the relative infrequency of Megan’s active/purposeful
anticipation activities, or those on the left of the anticipation framework. Additionally,
the farthest right portion of each map shows Megan’s lack of using research literature or
teaching journals in any way during her field experience. Her use of resources overall
was limited to finding out what content to teach in class and to picking out examples and
problems to make use of to teach that content.

The next section takes a different look at Megan’s anticipating practices by
describing two classroom interactions that took place around the SMRs that Megan
anticipated. While Megan’s anticipation of SMRs prior to teaching was the primary focus
of this research question, these two instances provide insight into how Megan made use
of what she anticipated. These mini-cases are intended to illustrate two possibilities of
what could happen when an intern, or any teacher, anticipates SMRs during lesson

planning.
5.1.9 Mini-Cases

The following two mini—cases represent lessons in which Megan anticipated a
specific student difficulty and the actions that she took based on what she anticipated.
The first mini-case characterizes a situation in which Megan anticipated a student
difficulty and made an attempt to consider ways to address this particular difficulty but
without success. The second mini-case illustrates an example of Megan making a
conscious effort to address the difficulty she anticipated in her teaching, to an arguable

degree of success.

159

The first mini-case comes from Lesson 1.4 and focuses on Megan’s anticipation
of student difficulties related to plotting points in three dimensions. In the interview that
took place prior to teaching her lesson Megan anticipated that students would have
difficulty visualizing three dimensions as they learned to plot points on three axes. She
stated that "when they're graphing things in three dimension[s] sometimes the kids have
trouble visually seeing it" [Lesson 1.4 Pre-Interview, October, 2008]. She went on to
explain why she thought visualizing might be particularly difficult in three dimensions:

I think they will be confused about the z-axis. I think that when we're graphing,

it’s sort of different [than in two dimensions]. Because you would automatically

think if you had a three-dimensional graph that you would go over two, up three.

But then you have to go on your z-axis. And you would think that you would go

up to that line and then count up. [Lesson 1.4 Pre-Interview, October, 2008].
Megan indicated that she too had had trouble in the past visualizing in three dimensions.
She stated that she thought students might have questions about plotting z, specifically
"how do you get 2 in the right spot?"

In addition to the difficulty with the content Megan anticipated, the fact that
students had struggled with other content earlier in the week prompted Megan to further
consider how she would present the material. "I think if I had started Monday with
simpler examples they would have grasped it more, quicker" [Lesson 1.4 Pre-Interview,
October, 2008]. Based on her desire to make her teaching and examples simpler she
talked with her mentor teacher to get some ideas. "From previous lessons I wanted to
make sure that I talked to my mentor teacher before I taught it to see if there was a way to
simplify things or a better way to explain them" [Lesson I .4 Pre—Interview, October,

2008]. Megan talked with her mentor and Mrs. Starns gave her an idea for teaching

which Megan related to me in the interview before she taught. "So we try to say 'start at

I60

the origin, don't pick up your pencil, and then you do the x, and then you do they y, and

then you do the 2. Megan included this technique in her lesson plan by writing next to

her example:

{3.3 5,4, 91W» V“ to
pews 't

Figure 27. Megan’s Handwritten Teaching Note from Lesson Plan 1.4

In addition to talking with her mentor teacher before school Megan made a last
minute change to her lesson plan in order to simplify her first example. "Well the first
one I'm going to do I'm changing to (2, 2, 2)" [Lesson 1.4 Pre-Interview, October, 2008].
When pressed she indicated that this example was simpler than the other one she had
chosen, (3, -l, -5), because all of the values were positive. She planned to follow the first
example with another example which had some negative values. Figure 28 shows a

portion of Megan’s lesson plan.

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SKILL EXAMPLES: . Careful“

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(gash).

Figure 28. A Handwritten Portion of Megan's Lesson Plan from Lesson 1.4

It can be seen from her lesson plan that she did not plot her new example, (2, 2, 2),

before she taught. When asked about the note on the lesson plan that said “Careful!”

 

Megan stated that was in relation to the graphing and was a reminder to her to "just to tell
the kids 'you have to be careful and you have to focus'" [Lesson 1.4 Pre-Interview,
October, 2008]. Thus this was a note to herself to remind the students to be careful about
their work, as opposed to a note in response to an anticipated SMR.

The activities in this lesson were much the same as others in this class, students
came in and collected their work and Megan went over some of the homework questions
from previous day and then began presenting notes to the students. She began the
example portion of the notes with the new example that she added at the last minute. She

wrote the following on the overhead projector:

162

2
Ex. 1 Plot and A

Label ~—
(29 29 2) ""

 

 

X

Figure 29. Representation of the Notes Megan Wrote on the Overhead Projector

As planned, she began by telling the students to "take your pencil and put it on the origin
at (O, O, 0). What ever you do don't lift up your pencil" [Lesson 1.4 Observation Field
Notes, October, 2008]. She used her overhead marker to trace a path to plot the point as

shown in the figures.

2 2
Ex. 1 Plot and A Ex. 1 Plot and A
Label -- Label ‘ --
(2’ 2’ 2) -— (2, 2, 2) --

 

 

 

 

Figure 30. Representation of the Notes Megan Wrote on the Overhead Projector

163

When Megan arrived at the moment to plot the last dimension, the one she had
anticipated that students would have difficulty visualizing, she drew the following:
Ex. 1 Plot and f

Label
(2, 2, 2) *

 

 

X

Figure 3]. Representation of the Notes Megan Wrote on the Overhead Projector

After students finished copying her representation into their notes they began to call out
questions and comments such as "you didn't graph on the z" and "you don't graph the
other points?" Through discussion it became clear that the students think their pencils had
"landed" on a point on the y-axis, which was not the correct location of the point. Megan
attempted to explain where the point actually was but students did not appear to be
convinced that it was not actually on the y-axis. After this example Megan went on to her
next example plotting the point (3, -l, -5). As with the first example students began
calling out questions "I thought you said you graphed the points regular?" "Where would
the y be?" After she completed the two plotting points examples she went on to do three
examples for the class involving graphing equations with three variables.

While reflecting during the interview after the lesson Megan stated that students
"were a little confused," "had trouble understanding," and "had lots of questions" about

plotting points in three-dimensions. She recognized that there were some difficulties with

164

her first example, "that's probably something that didn't go so well...it ended up on the y-
axis and it was really off the y-axis" [Lesson 1.4 Post-Interview, October, 2008]. She
stated that she realized that some students were still struggling with plotting points and
added that "they were a little freaked out at the beginning with just graphing the point.
And like my mentor said, 'sometimes kids get it or sometimes kids don't.‘ So the ones that
didn't, really didn't like not getting it” [Lesson 1.4 Post-Interview, October, 2008]. She
notes she would not be sure if students had met her objective for plotting points until she
looked over their homework the next day.

Megan did reference the specific example of plottingjthe point (2, 2, 2) when
asked if anything about the lesson surprised her.

I don’t know if I graphed it wrong but when they graphed one of the points it

looked wrong...It was just a little off and it sort of threw me and then it threw

them. [Lesson 1.4 Post-Interview, October, 2008]

When asked what she would do differently on Monday based on what happened in class
that day Megan said that she would probably go over a homework problem that asked the
students to plot points in three dimensions. She did not indicate whether she would go
over the example in a different way, although it seemed to be the norm in the class to
model the same procedures in going over the homework problems that were modeled in
the notes.

The preceding is a case of a preservice teacher anticipating a student difficulty
that did occur, but not taking sufficient action in her lesson plan or teaching to address
this potential difficulty. We see in this case that the only resource that she sought out for
helping her deal with this anticipated difficulty was her mentor teacher. Although her

mentor teacher was able to provide her advice for how to simplify her lecture, her advice

I65

did not help Megan address Megan’s concern about the students’ potential difficulty
visualizing in three-dimensional space. Despite not receiving any ideas from her mentor
teacher on her particular concern Megan did not seek out other methods for teaching the
content specifically to help her address the difficulty she anticipated.

One possibility for finding an alternative method could have come from the class
textbook. Megan indicated that she was aware that the text used a different method but
did not explain her pedagogical reason for choosing her mentor teacher’s method over
this one. The textbook method lays the xy-plane on its side and then positions the z—axis
perpendicular to the x- and y-axis through the origin as shown in Example 1 from the

class text (Figure 32)

166

‘7’“? ' Plottina Points in Three Dimensions

 

Plot the ordered triple in a threedimensional coordinate system.

a. (—5, 3, 4) ‘ b. (3, —4, —2)
SOLUTION
a. To plot (-5, 3, 4), it helps to first find b. To plot (3, —4, -2), find the
the point (-5, 3) in the Jay-plane. The (3, —4) in the xy-plane.The
point (-5, 3, 4) lies four units above. (3, --4, -2) lies two units bt

 

 

 

Figure 32. Portion of the Class Text Presenting Plotting Points in Three Dimensions

While this is the same representation of the three-dimensional axes used by Megan, the
way that the text book describes plotting the first two coordinates in the xy-plane and
then moving up or down on the z-axis is different. Megan did not overtly orient the z—axis
relative to the xy-plane in the same way that text book did which may have part of what
caused confusion for the students.

When reflecting on the lesson in the post-lesson interview Megan indicated that
she realized students had struggled with plotting a point in three dimensions as she

anticipated they would. She did not connect the fact: that she did not do the first example,

167

(2, 2, 2), in advance to the fact that it did not play out as she would have wanted or that
failing to do so may have added to the students’ confusion.

She also did not reflect that perhaps she could have sought out a different method
for teaching the content and her mentor teacher’s feedback after the lesson that this is
something that “either you get it or you don’t” would not have prompted her to look
further either. Finally, there was no evidence that Megan planned to demonstrate plotting
points in three dimensions differently the next day when she went over an example of this
from the homework.

The Plan, Teach, Reflect cycle that is integral to the Methods course includes
reflecting on areas of growth after teaching a lesson. In this case this would have included
Megan thinking of a different way to present plotting points in three dimensions for the
next days homework review since she thought that some of the students were still
stmggling with the concept. Although a formal write up of this kind is only required on
two occasions during the semester, the habit of reflecting in this way was also advocated
in the previous year methods courses. This lack of revision to her original teaching plan
for the next day indicates that not only was Megan not seeking additional resources to
help her deal with anticipated student difficulties, but also that she was not consistently
reflecting on how to make her teaching better.

A second case that illustrated an outcome of anticipation was exhibited during an
observation in early November. On this particular day Megan was teaching the students
to factor trinomials using what Megan and Mrs. Starns referred to as the “boxing”

method. This method made use of an area model to factor polynomials. The students

168

were familiar with the model because they had used it to multiply binomials earlier in the

year. Figure 33 shows an example of a completed box.

 

Example: x2 +1 1x + 24 = O

 

 

x 3
x x2 3x
8 8x 24

 

 

 

 

 

 

 

Figure 33. A "Box" Model Showing the Factorization of x2 +1 lx+ 24 = 0

Before teaching the class Megan explained that her mentor teacher told her that
the students would have difficulty choosing factors of the constant term that would sum
to be the coefficient of the x term.

Well they have trouble because they have x squared which goes here, and [they]

have negative ten. But finding that when you add these together that they come up

with this middle number, that can be really hard. Because sometimes you might
end up [with] a negative here and a positive here but then it doesn't work so

maybe you have to flip flop them. [Lesson 2.3 Pre-Interview, November, 2008]
Although she was aware of this difficulty she did not address it anywhere in her written
lessons plans.

Despite not addressing this anticipated student difficulty in her lesson plans,
Megan’s actions during her teaching of the lesson indicated that she did attempt to attend
to the potential difficulty. This occurred when Megan presented a particular example
during the note taking portion of the lesson. Megan began by writing an equation on the
overhead, drawing a part of the box that would be used, and telling the students that there

were some things they could fill in just from looking at the equation, as represented in

Figure 34.

169

 

Example: x2 —4x— 21 = 0

 

 

-2l

 

 

 

 

 

 

 

Figure 34. Megan’s Boxing Example in Lesson 2.3

Megan next tells the students to fill in the two x's on the outside of the box because they

should know what those have to be (Figure 35).

 

Example: x2 —4x-21 =0

X

 

x ,2

 

-21

 

 

 

 

 

 

 

Figure 35. Megan’s Boxing Example in Lesson 2.3

After students copy this into their notes Megan directs them to write out all of the factors
of 21. As students do this Megan also writes out the numbers 1, 3, 7, 21 on the overhead.
She suggests to the students that they try using —3 and 7 as factors of -21 because they
know that one of the factors will have to be negative and she proceeds to write these on

the outside of the box (Figure 36).

170

 

 

 

Example: x2 — 4x — 21 = 0
x -3

x x2
7 -2l

 

 

 

 

 

 

 

Figure 36. Megan’s Boxing Example in Lesson 2.3

She then proceeds to do the rest of the required work to fill in the box (Figure 37).

 

Example: x2 —4x- 21: 0

 

 

x -3
x x2 -3x
7 7x -21

 

 

 

 

 

 

 

Figure 37. Megan’s Boxing Example in Lesson 2.3

To check their work Megan tells the students that they need to add the two x terms to be

sure they sum to -4 as desired (Figure 38).

 

Example: x2 —-4x—2l =0
-3

x
x x2/ +‘W
7 flit/4‘

Figure 38. Megan’s Boxing Example in Lesson 2.3

 

 

 

 

 

 

 

 

 

Megan tells the students, and many appear to notice for themselves, that the sum of -3x
and 7x is not —4x as desired. She then asks the students “what should we do?” As the

students chatted about what to do one student in the class quietly answered that they

17]

could try to switch the negative sign. Many of the students in the class did not quiet down
to hear this idea. Upon hearing the students suggestion but without repeating the idea or
asking the student to say it loud enough for everyone to hear, Megan applied the idea on
the overhead notes by switching the signs and she then moved on to the next problem.

Although Megan did not specifically reference her own selection of these
particular factors in her lesson plan or any of her interviews, other evidence indicates it
may have been purposeful. First, Megan flagged the fact that her mentor teacher shared
with her that students have difficulty choosing factors and in particular knowing what to
do when one negative factor is needed. Second, Megan used these factors in presenting
the example both in her Focus Class and in another class earlier in the day which could
indicate that she was planning this intervention to address the anticipated student
difficulty that Mrs. Starns told her about.

Clearly this mini-case differs from the previous one in that Megan both
anliCipated a student difficulty and found a way to address it during her lesson. Although
her attempt was commendable, the execution during the Focus Class may not have
pFOdUCed the desired effect of addressing the difficulty for all students because not
everyone heard it and the idea itself did not appear to be made public, only the result.
Because the student suggestion to change the sign was done so quietly and executed by
Megan so quickly the other students may not have had time to really consider or
understand this idea. Also, because Megan did not call more attention to the idea and the
outcome of switching the negative signs some students may have missed the method all
together. This was evidenced to a certain extent the next day when students struggled

with “playing” with the factors until they could find the ones that worked. Despite all of

172

this, this case does represent an example of a preservice teacher drawing on another
teacher as a resource, namely her mentor teacher, Mrs. Starns, to anticipate students’
difficulties and devising a strategy to address the potential difficulty during her lesson.
Now that the nature of Megan’s anticipating over the first semester of her internship has
been described, the next section of this chapter addresses the influence of the contextual

factors on her practices of anticipating SMRs.
5.2 The Factors of the Context

Research Question 2. What factors within the contexts of a teaching internship (afield
experience and accompanying university course) promote or do not promote the
practices of one preservice teacher anticipating students’ mathematical responses
during lesson planning?

The second research question addressed in this study focuses on the interactional
aspect of the case. Greeno (1998) describes this aspect as examining the “interactive
systems of activity in which individuals participate, usually to achieve objectives that are
meaningful to their more general identities and memberships in communities of practice”
(p. 6). While the first part of this chapter focused on Megan as an individual and her
lesson planning practices, this part assumes a broader lens to explore the factors present
in the systems, or contexts, in which she participated.

In order to investigate these contexts and their potential effect on Megan’s
practices a sociocultural framework was used which had previously been employed by
Goos (2005b) to examine the contextual factors that influenced one beginning teacher’s

use of technology in his teaching. This framework was well suited to the investigation of

the second research question because it also seeks to explore the effect of different

173

contexts. Goos’ (2005b) framework includes three zones of interest: (I) the Zone of
Proximal Development, (2) the Zone of Free Movement, and (3) the Zone of Promoted
Action'f’. Together, the descriptions of these three zones in this case study provide a
detailed portrait of the ways in which the two contexts, along with Megan’s own skills
and desires for her teaching, interacted to shape her participation in this practice of
anticipating students’ mathematical responses (SMRs) in her lesson planning.

As described in section 2.3, the Zone of Free Movement (ZFM) and the Zone of
Promoted Action (ZPA) will be further subdivided by the four principles identified by
Engle and Conant (2002) as promoting productive disciplinary engagement. The presence
or absence of these principles in the ZFM and ZPA will illustrate the influence of each
context on Megan’s practice of anticipating SMRs. These elements of the contexts,
including Megan’s own choices, influenced her actions as traced by the anticipation maps
in the first part of this chapter. The second part of this chapter is organized using Goos’
(2005) three zones and concludes with a summary of the findings in the first part of the

chapter as they relate to all of the zones.
5.2.1 Zone of Proximal Development

The Zone of Proximal Development (ZPD) is a construct of learning originally
defined by Vygotsky as “the distance between a child’s independent problem solving
capability and the higher level of performance that can be achieved under adult guidance
or in collaboration with more advanced peers” (Goos, 2005b, p. 37). Since the original

development of the idea, the ZPD has been applied to the teacher education context to

 

'6 Each of these zones as characterized by Goos (2005) are described in detail in Chapter 2.

174

examine the learning of preservice teachers. In this context the ZPD is described as “a
symbolic space where the novice teacher’s emerging skills are developing under the
guidance of more experienced people” (Goos, 2005b, p. 37).

Goos (2005b) further articulated ZPD for her study in a teacher education setting
which examined the use of technology, as the preservice teacher’s: “(1) skill/experience
in working with technology, (2) pedagogical knowledge (technology integration), and (3)
general pedagogical beliefs” (p. 40). It can be assumed that the role of the more advanced
peers, or teacher educators and mentor teachers in this case, falls under the other two
zones and for that reason is not included in this characterization of ZPD. Thus, the ZPD
in this study is focused solely on Megan and her skills, experiences, knowledge, and
beliefs as they relate to anticipating SMRs and mathematics teaching in general.

The next sections describe Megan’s ZPD using three parallel elements of ZPD for
anticipating SMRs: (1) skills and experiences anticipating SMRs, (2) pedagogical

knowledge for anticipating SMRs, and (3) general pedagogical beliefs.

I75

5.2.1.1 Skills and Experiences Anticipating SMRs

Megan’s skills and experiences anticipating SMRs were developed in her senior
year methods course. During this time Megan was introduced to the practice of
anticipating SMRs as part of the Stein et al. (2008) model for orchestrating productive
mathematical discussions. Megan participated in this practice as part of creating seven
separate lessons in which she made use of the common lesson planning format used in
the secondary methods courses (Appendix B). Early in her intemship year Megan
reflected on her experiences anticipating SMRs using the lesson planning format the
previous year:

I would go through and do all of the stuff [in the format], like the standards and

the goals... Then I would go and do the [launch, explore, and summarize] And

the big thing with [the format], that I really liked, was that it got you to think what
the students were thinking and then the questions you would want to ask them and
how you'd want to answer them. I feel like I don't get that as much with doing the

lecture style things. [In-Depth Interview 1, September, 2008]

This quotation from Megan is evidence that she had experiences anticipating
SMRs prior to her internship year, and further that she identified anticipating
SMRs as a central practice of the lesson planning format and one that she
enjoyed. This quotation also illuminates the fact that Megan enjoyed using the
lesson plan framework to create all of her lessons during her senior year.

In addition to using the format provided by instructors in her senior year methods
course, Megan twice completed an assignment in which she used teaching journals to
research common student na'ive conceptions. Megan’s responses to the Lesson Planning

Survey (LPS) indicated that during her senior year she also participated in other practices

of anticipating. In the survey she responded that she always consulted the curriculum’s

l76

teacher guide, talked with a methods instructor about her lesson ideas, and completed the
task herself when planning lessons. She also responded that for lesson planning purposes
she almost always read articles about the content she was teaching and sometimes
consulted her senior-year mentor teacher and considered the needs of particular students.
In a later interview Megan clarified her response about sometimes consulting her mentor
teacher by stating that “he would sometimes give me suggestions but he didn't do it all of
the time” [In-Depth Interview 1, September, 2008]. This quotation indicates that it is
likely her interactions with her mentor teacher did not often include the sharing of SMRs,
but instead the giving of suggestions. She also noted in this interview that during her
senior year she participated in interactions with the other preservice teachers in her
methods course around their lesson plans. This occurred through informal discussions;
“discussing what issues you’re having with lesson planning or ideas and to brainstorm
with them. I think brainstorming was a big thing, [just going] through ideas” [In-Depth
Interview 1, September, 2008].

Megan’s experiences in her senior year methods course exposed her to all of the
individual practices of active/purposeful anticipation. Her own responses indicate that as
part of that course she made use of curriculum materials and consulted other teachers in
order to anticipate SMRs, and sometimes took the position of students, when planning
lessons. In addition, assignments in the methods course required her to seek out and make
use of literature on students’ possible naive conceptions and to complete mathematical
tasks in multiple ways. One of Megan’s responses on the LPS indicated that she had

linked at least some of these activities to the practice of anticipating. She responded with

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the following when asked if she thought anticipating SMRs was important and how she
accomplished in her past lesson planning:
Yes, because in a student centered class a teacher needs to be prepared for various
issues and how to provide a solution. In addition, it helps the teacher guide
conversation for student learning. I try to do this by looking up articles on
misconceptions, talking to my mentor teacher, and thinking of misconceptions on
my own. [Lesson Planning Survey response]
The quotation indicates both that Megan sees anticipating as an important practice and
implies that she intends to continue to make use of this practice. Megan’s response also
highlights a final important aspect of her skills and experiences anticipating SMRs during
her senior year methods course. All of the lessons the preservice teachers taught in the
senior year methods course were required to be designed around open-ended tasks. Using
this type of task along with orchestrating student discussion based on the mathematics of
the task was part of what was referred to in that class as student centered. Thus, all of
Megan’s skills and experiences anticipating SMRs during that time developed and
occurred around her understanding of student centered teaching practices. Megan’s own
responses provide evidence that she connected the practice of anticipating to using the
lesson plan format and student centered teaching. These connections appear to be a key

factor in her practices of anticipating SMRs during her internship year. The next section

describes Megan’s knowledge for anticipating SMRs.

5.2.1.2 Pedagogical Knowledge for Anticipating SMRs

As described in Chapter 2, knowledge of content and students, whether based on
distributed or experiential knowledge, plays a role in what SMRs a teacher anticipates.

Megan’s responses on the LPS indicated that she had knowledge of content and students

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for anticipating in a particular content area. One portion of the LPS asked the interns to
anticipate SMRs in the context of teaching students about angles and using a protractor.
Megan indicated in her survey responses that this was content she had discussed during
her senior year methods course and also content that she had taught during that time.
When asked to list all of the possible SMRs for learning about angles and measuring
angles with a protractor Megan produced the following list in her survey:

' Students may confuse acute and obtuse angles.

' Students may have difficulty knowing how to use the protractor.

I Students may not know to extend the lines of an angle in order to use the
protractor.

' Supplementary angles and complementary angles may get confused.

I The protractor has two sets of numbers on them. The students might use
the wrong angle measurements.

' Students may get confused with radians and degrees.

This response exemplifies a portion of the results discussed in the first part of this
chapter, which indicated that Megan’s knowledge for anticipating was often based on her
own teaching or learning experiences and that what she anticipated often dealt with
mathematical procedures. These anticipated SMRs focused on knowing how to make use
of the protractor and distinguishing between two terms (i.e. acute and obtuse) that they
likely had learned already. These SMRs that Megan anticipated early in her internship
experience display similar features of the majority of those she anticipated throughout the
semester.

Although Megan responded in the LPS, and at other times, that she was aware of

other methods for drawing on knowledge for anticipating SMRs, she rarely chose to

I79

make use of them in her internship year teaching. In fact when specifically asked during a
final interview in December, after she had been in the internship experience three and a
half months, if she had considered any misconceptions that might arise she stated: “It's
very interesting the different misconceptions that you stumble upon as you teach the
stuff. So even though you could research stuff, I learn a lot just from the kids [Lesson 3.3
Pre-Interview, December, 2008]. The next section describes Megan’s general

pedagogical beliefs.

5.2.1.3 General Pedagogical Beliefs

Megan’s general pedagogical beliefs were based on a commitment to student
centered practices. Megan defined student centered as:

The students are conversing among one another to get to the answers. And this
allows students to work with each other and stumble upon answers, that’s what I
mean by student centered. They're discovering the math on their own, well on
their own but I am also helping or mediating. [This makes] more meaningful
learning. [Cool Stuff 1 Pre- Interview, October, 2008]

Megan again connected using the lesson plan format to teaching in a more student
centered way in an informal conversation with her field instructor.
So to me student centered learning is doing our whole lesson plan
[format]. So you would do a launch and explore where the kids are doing
everything. And then you do the summary. You know, ALL that. And I
want the kids to be discovering things on their own. [Informal
Conversation with Field Instructor, November, 2008]
Many of Megan’s comments about the teaching she was doing in her internship class
were contrasted with her desire to do student centered teaching.
I would like to see things go from me [lecturing] into more open-ended, group

centered, student centered type things. And that’s where I want to deveIOp over
the year because right now its very lecture based, teacher asking students

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questions, and I want more students asking students questions or more of that kind
of thing. [Lesson 1.5 Post-Interview, October, 2008]

Right now its very much example based, very concrete, setting things out,
writing definitions, writing concept type things. Very concrete. And I'd
like to see it eventually transition into more student centered open tasks...
And more of me being a facilitator instead of the lecturer. [Lesson 1.5
Post-Interview, October, 2008]

Hopefully I'll be able to have some opportunities to do some more student
centered things again. [Lesson 2.3 Post-Interview, November, 2008]

Megan’s comments indicate that for most of the semester she was teaching in ways that
were counter to her beliefs about how mathematics should be taught and learned.
Overall, Megan’s ZPD was comprised of all of the discrete skills needed for
anticipating SMRs as well as the desire to implement student centered pedagogies, which
in her own words included anticipating SMRs. In this way, Megan was poised for further
guidance in the direction of these practices. However, in her past practices of merely
getting suggestions from a mentor teacher and brainstorming teaching ideas with other,
inexperienced, preservice teachers, she did not appear to have had experience in using
other teachers as resources for anticipating. Based on this lack of experience in using
other teachers in this way, the position of the two contexts with respect to making use of
other teachers would become a critical feature that both supported and constrained
Megan’s anticipation of SMRs. The next section describes the Zone of Free Movement in

which Megan worked at both Logan High School and the university.

5.2.2 Zone of Free Movement

The Zone of Free Movement describes that which is possible for the intern (Goos,

2005b). In this way, this zone represents the features of the contexts that enabled or

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constrained Megan’s practices of anticipating SMRs. In Goos’ (2005b) study of a
preservice teacher attempting to use technology in his teaching, the ZFM was comprised
of several key elements:

I access to hardware, software, and laboratories;

I access to teaching materials;

I support from colleagues (including technical support);

I curriculum and assessment requirements; and

I students (perceived abilities and behavior) (p. 40).
As described in Chapter 2, these elements resonate with one of the principles that Engle
and Conant (2002) describe as fostering productive disciplinary engagement for students,
namely providing students with relevant resources. In addition to this, another element of
Engle and Conant’s principles also relates to what is possible for the intern and that is
giving interns authority to address problems. As previously explained, this study applies
Engle and Conant’s principles to intern teachers. These two principles are relevant to this
study because they characterize constraints and affordances relevant to Megan’s practices
in these settings and they are represented by the resources Megan had access to in each of
the settings and the authority she was given. They will be referred to simply as resources

and authority and will frame the discussion in each of the two following sections.
5.2.2.1 The University

The university setting provided Megan with both resources and authority that
contributed to her ZFM while at the university. Megan’s resources while at the university

included her instructors, the other interns, as well as a variety of curriculum materials.

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Both the methods instructor and the field instructor had experience teaching middle and
high school mathematics. Their teaching experience as well as their other experiences in
the mathematics education community made them knowledgeable of both SMRs from
their own experiences and resources for findings tasks and information on SMR, which
was often shared with the intems”. Dr. Rundell noted that he shared his knowledge of
SMRs when providing feedback to the Cool Stuff lessons. “I actually provided all of the
feedback on the lesson plans myself. So they got, to some extent, some expert opinion on
how students might think about the task” [Methods Instructor Interview, October, 2008].
Megan also stated that she received some insight into SMRs from her field instructor
Emily, but that this usually came after her teaching. Although Megan was required to
send her lesson plan to Emily in advance of the observation, there is no evidence that she
asked Emily for help in anticipating SMRs for any of the lessons.

The interns themselves were often put in the position of serving as resources to
one another in the methods course through sharing their experiences and providing each
other feedback. Although this was built into the course in various ways (e. g. case stories,
Fireside chats, blog, and numerous class discussions), Megan rarely made use of the other
interns in this way. “I've known the peOple who are teaching quadratics and I've known
the people are teaching other things but I've never actually discussed it with them” [In-
Depth Interview 2, December, 2008].

The curriculum resources that Megan had access to at the university included
several sets of materials with open-ended student centered tasks and were available in the

methods course classroom as well as the university library. In addition, some of these

 

l7 . . . . .
The practice of sharing resources was a prevalent part of the course culture and IS further discussed In
Chapter 4.

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materials also contained SMRs in their teacher resource materials. On the first day of the
methods course Dr. Rundell reminded the interns that these materials were available to
them in the methods classroom. He again highlighted their availability during the third
class when the interns were brainstorming lesson ideas for their Cool Stuff 1 lessons.
Beyond the resources available to the interns in their university classroom, the required
text for the course was the National Councils of Teachers of Mathematics (NCTM)
Research Companion to Principles and Standards for School Mathematics (Kilpatrick,
Martin, & Schifter, 2003). This compilation of research findings provides a wealth of
information on student learning in a variety of areas, including algebra which was the
mathematical domain of Megan’s Focus Class. Another resource that was provided to the
interns at the university was access to a variety of websites that were posted on the class
wiki for finding tasks and looking up research on the intemet. When asked specifically
about the resources that the interns had access to in order to lesson plan and anticipate
SMRs Dr. Rundell mentioned all of those previously described and added the mentor
teachers to the list.
And they have access to their mentor teachers, they have access to whatever
materials they have in their mentors teachers classroom to draw off of and
certainly their mentor teachers expertise, since most of them have taught the
courses that the interns are teaching in the past, are quite valuable for getting
insights into how students into anticipating how students might make sense of a
particular mathematics idea or mathematics task. [Methods Instructor Interview,
October, 2008]
Although this was a resource of the school and not the university, it is important to note

that Dr. Rundell included it as an available resource for lesson planning and anticipating

SMRs.

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As a student in the methods course Megan was also required to be a member of
the National Council of Teachers of Mathematics (NCTM). This membership allowed her
access to the Mathematics Teacher, a high school teaching journal published by NCTM.
This journal, as well as the middle school teaching journal, Mathematics Teaching in the
Middle School, that was available at the university library, contained both student
centered tasks and SMRs in its pages.

The second element of the ZFM that promotes productive disciplinary
engagement is giving interns authority to address problems. In this context it means that
the interns had the authority to determine the direction of the class discussions based on
their own experiences in their field placements and to negotiate their assignments and due
dates. While in the university setting Megan had all of the authority afforded a beginning
professional in a collaborative community. Like all of the other interns she had the
authority to question ideas and norms during class discussions and even to change the due
dates of her assignments when necessary. Also, Dr. Rundell, the instructor of the methods
course, rarely dispensed knowledge to the interns. He instead allowed them to discuss
their problems and come to consensus on their own when appropriate. In many ways his
teaching strategies mirrored those of Magdalene Lampert, considered by many to be an
expert mathematics teacher, as described by Leinhardt and Steele (2005). Though
Leinhardt and Steele describe Lampert’s role in facilitating mathematical discussions of
grade five students, Dr. Rundell’s actions in the facilitating the discussion of the interns
shared many of the important features.

We see a teacher who facilitated dialogue around a predetermined set of

connected topics, selectively allowing for asides that assisted in developing

conceptual material and occasionally engaging in more bounded explanations
about a particular subskill. Lampert is seen standing to the side of the dialogue;

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she neither led the class through a presentation of material and guided

questioning, nor did she simply pose a problem and step out of the way, allowing

for unbounded exploration that somehow results in students bumping into the
mathematics. She succeeded in standing to the side by creating dialogues that
fulfilled the conditions of an instructional explanation using the exchange of ideas
in ways that chart a sensible, coherent, and complete path through the
mathematics. She stepped into the mathematical community in part as a member

of it and in part as the creator and director of it. (Leinhardt & Steele, 2005, p. 133)
In much the same way, Dr. Rundell was both a member of the community and its
director. In this enacting this role the interns in the class were also invited in as members
of the community, and thus were afforded the authority of a beginning teaching
professional.

Megan’s authority in the university setting had very little to do with her actual
authority in the school setting. The only time that Megan’s authority from the university
impacted Megan’s authority in the school was during the Cool Stuff assignments. On
these two occasions the authority of the university allowed her make changes to fulfill her
university assignments that she would not normally have been able to make.

Overall, the university ZFM afforded Megan the use of a variety of resources as
well as a great deal of professional freedom. Despite the resources available in the form
of curriculum materials and other teachers (including interns, her field instructor, and the
methods course instructor), Megan made limited use of them in her lesson planning.
When these resources were directly provided to her, either through case story interactions
or in the form of instructor feedback on lessons she turned in, she made use of them. But,
she did not on her own seek out any of these resources in order to help her anticipate

SMRs. The next section describes Megan’s ZFM, comprised of the resources available to

her and the authority given her, at Logan High School.

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5.2.2.2 The School

The ZFM of the school context afforded and constrained Megan’s abilities to
anticipate SMRs in different ways than the university setting. Megan had access to a
different set of resources in the school setting that included the class curriculum
materials, her mentor teacher and her mentor teacher’s notes from previous years, and the
larger mathematics teacher community at the school.

The first of the resources that Megan had access to at the school was the class
curriculum materials, which included the student and teacher texts as well as the
publishers set of accompanying worksheets. The textbook supplied very few SMRs and
presented the materials via examples, definitions, and theorems along with numerous
practice problems. Even those SMRs that were supplied in the textbook were not used by
Megan.

I usually don’t read the book. . .We get our homework from the book and use
problems from it for the quizzes. So even though we say we write it ourselves,
that’s where we're pulling things from. And everything else, like our vocab quizzes
are based on the book, our tests are based on the book because we go by the chapter
objectives and pull problems for that. [In-Depth Interview 2, December, 2008]
Megan noted on several occasions that she felt constrained by the lack of availability of
resources to choose tasks from at the school to support her in enacting more student

centered instruction.

If 1 had the right materials, then it will be easier to do more often. [Cool Stuff 1
Post-Interview, October, 2008]

It's hard because my mentor does not already have those tasks available for me to
use. So like I would have to go out and find my own. [Lesson 3.3 Post-Interview,
December, 2008]

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In both of these instances Megan is contrasting her experience with her perception that
other interns were in settings that did provide such resources, either through curriculum
materials or mentor teachers’ personal libraries of outside resources. She did not seem to
view the university resources as being provided for her use in the same way. Given that
Megan appeared to connect the use of student centered tasks to the lesson plan format
and the lesson plan format to the practice of anticipating SMRs, the lack of student
centered tasks may have been a factor that contributed to her lack of
actively/purposefully anticipating SMRs in the field experience.

The second resource available to Megan at the school was her mentor teacher
Mrs. Starns. Throughout the semester Megan made use of Mrs. Starns’ class notes from
previous years.

She has a lot of stuff already done. So she has the notes all ready in a binder for

each class... So for now, just for this first little bit, I'm going to use hers... just to

see where she gets her things from gives me an idea of stuff, so that's nice for

right now. [In-Depth Interview 1, September, 2008]
Megan pointed out that Mrs. Starns’ notes only included the notes and examples she was
going to use. She had not worked out the examples or included any thoughts about
anticipated SMRs. Used in this way Mrs. Starns’ notes were not a resource that helped
Megan anticipate.

Mrs. Starns also served as a resource for Megan through their verbal interactions.
Despite their almost daily communications, the findings presented earlier in this chapter
evidenced the fact that Mrs. Starns was not often a resource for anticipating. The

interactions between Megan and Mrs. Starns often focused on what to teach, that is which

section to do next, as opposed to sharing past experiences with students and content:

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[We focus more on] the schedule, like 'what do we need to do' and 'what sections
are we going over,‘ and 'should we do the reading packet or not'. So I don't sit
down and plan these with her because 1 think she thinks that the math shouldn't
be an issue any more. I mean if it is we'll go over it, but the organizational [part],
she feels that’s how she can help the best right now. [In-Depth Interview 1,
September, 2008]
So when I was just doing the regular lesson plans with my mentor we would just
basically plan out the week and say what we're going to do each day and then I
would take her notes and the book and do things on my own. [In-Depth
Interview 2, December, 2008]
The fact that Mrs. Starns did not often share her teaching experiences in order to help
Megan anticipate SMRs could be attributed to her objectives for Megan during the first
semester. Mrs. Starns stated early in the semester that her goal for Megan was that she
would speak clearly and present herself as a strong person and female in front of the
class. Given this, Mrs. Starns might have viewed their interactions as adequate to helping
Megan progress in this area. It is important to note that Megan rarely sought out Mrs.
Starns as a resource for anticipating.
Available time to discuss lesson plans was another limitation in making use of
Mrs. Starns as a resource. By the final week of observations there was very little
conversation between Megan and Mrs. Starns about the lesson plans at all. Megan noted
before teaching one of the lessons that week, “actually, my mentor and I didn't have time
to discuss this from yesterday and so I just recorded her notes that she did during first
hour” [Lesson 3.2 Pre-Interview, December, 2008]. That same week Megan’s field
instructor, Emily, observed her teaching and asked her afterwards in the debrief a

question about a choice she made during her teaching.

Emily: What made you decide not to read the ant problem before they did it?
What was the ant problem even?

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hdegan:

Emily:

Megan:

Emily:

Megan:

Well, what happened was that Mrs. Starns does the reading packet [in
first hour] and so I wanted to do just what she did. That’s how she
presented it, so I did it like she did.

Okay. Are the students expected to have read it before?

No... The actual question is: Would this ant have a surface area large
enough to meet its oxygen need? If you read it you can actually answer
that question.

Okay. So you only need to do the first [part of the question] because
tomorrow you’re going to answer the oxygen [part of the question]?

Nope, that’s it. . .I don't know if she did it for time's sake that she only
did surface area and volume because then you’re still working with the
roots and different things, you’re still doing those relations. But I
think that she just decided not to do that question because of time and
cause it might confuse them more, that might be why. We didn't have
time to chat. We don't chat about these that much ahead of time. [Field
Instructor Lesson Debrief 2, November, 2008]

When asked specifically about times when she and her mentor teacher discussed possible

SMRs Megan stated that this did not happen often and offered lack of time as the

explanation. “I think sometimes it’s just the flow of the day and it gets busy and we know

what we have to get done so we don't really spend a lot of time [talking about that]” [In-

Depth Interview 2, December, 2008]. The reality of this situation is a stark contrast to Dr.

Rundell’s vision, which he often shared with the interns, of the mentor teacher serving as

a resource.

Finally, the larger community of mathematics teachers at Logan High School was

a possible resource for Megan in anticipating. However, because the interactions between

Megan, Mrs. Starns, and the two other members community were limited to monthly

professional development meetings and occasional conversations in the lunch room,

Megan may not have seen a ready opportunity to get feedback from the other two

mathematics teachers on how students might react to her lessons. In addition, the

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conversations between teachers in the lunch room and the hall were often social in nature
or related to student discipline issues. It seemed to be that it was not the norm in the
school to take part in conversations about teaching with other teachers.

Although not necessarily a resource provided by the school, time commitments
imposed by the school constrained Megan's ability to think very much about lesson
planning and therefore to anticipate SMRs. At the beginning of the year Mrs. Starns and
Megan decided that Megan would grade all of the papers for Algebra 2 and most of Math
Analysis, a total of four classes. This choice was counter to the intent of the program that
expected Megan to gradually take responsibility for the teaching of more classes over the
year but still allow time for the interns to co-plan with their mentor teachers and time for
reflection. Megan found it difficult to prepare lesson plans with all of the extra grading
she was doing for Mrs. Starns along with what she needed to do for her university
courses. When asked early in the semester why she was not making use of outside
resources such as teaching journals to plan lessons, she stated that even though she
wanted to use them she simply did not have the time.

Yes, I would love to [use] them. By the time I come home at night I have a pile

of papers to grade. And that's new to me so I have to get used to that. And want to

get ahead on lesson plans so I have those to do and then if I have anything for the
next day and then I have my homework for my teacher education classes. So by
the time all that’s done I have to have some time for myself, maybe a just little

bit. Getting articles just isn't on the top right now. [In—Depth Interview 1,

September, 2008]

Although the grading situation was corrected in the middle of the semester when Dr.
Rundell intervened, Megan perceived the excessive grading as a major hindrance to her

lesson planning. When asked what impacted her lesson planning over the semester she

stated: “I think the amount of grading that I had this semester really influenced my

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planning because I didn't have as much time to plan, and that was probably three fourths
of the semester” [In-Depth Interview 2, December, 2008].

The second element of the ZFM to be discussed is Megan’s authority in this
context. In this case authority refers to Megan’s ability to make decisions about what was
going to happen in the classes she was teaching. Evidence indicated that Megan’s
authority in the school setting was minimal. First, Megan did not have the authority to
change her mentor teachers’ weekly schedule in any substantive way when she was
teaching. During the two or three days of the week when there was some flexibility
Megan was only given 20 minutes, the allotted lecture time, in the class to try her own
ideas.

She has given me as much freedom as I want to do what ever during that lecture

time. She still wants to have the structure and the routine, but during that chunk of

time I can do what ever I want. [Field Instructor Lesson Debrief 1, October, 2008]
Megan found during the semester that this amount of time was not sufficient to try all of
the student centered activities and ideas that she wanted to.

I have these ideas of what I want to do. I want to have more student centered

learning, I want to have more cooperative stuff. But then once I try do it like with

Spiderrnan [and his] parabolas [I run out of time]. . .but [because of the]

vocabulary quiz and other things you have to do, which are good things, you run

out of time... We don’t have the leniency of doing something where you can take
the whole hour and do what ever you want. [Lesson 2.5 Post-Interview,

November, 2008]

Megan found this to be true in teaching her Cool Stuff 2 lesson when she asked Mrs.
Starns if it would be okay if she continued the lesson into the following day. Although
Mrs. Starns had allowed her to deviate from the schedule to teach the lesson, she would

not allow the lesson to continue into the next day because of the scheduled binder check

and Game of Life. This instance occurred even though Megan was in her second lead

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teaching period and was supposed to be making all of the instructional decisions for her
classes at that time.

Megan also lacked the authority to make decisions about her teaching outside of
the schedule structure. During her two lead teaching periods when Megan was supposed
to be fully responsible for the teaching of her classes, Mrs. Starns would interrupt her
during class and change a due date Megan had just assigned or provide a different
explanation to the students without explaining why an alternate was needed. On one
occasion Mrs. Starns made a change to the plan for the day without consulting Megan.

I had something else prepared... But she decided, I don't know if she was talking

to a kid, but they decided they would have the odds due and then they would have

the evens due the next day. I probably would have just rather had them do the
odds and just go over like one or two problems to start them and get them through
the graphing part and they could solve the rest or something like that. [Field

Instructor Lesson Debrief 1, October, 2008]

Megan accepted her lack of authority in her lesson planning and yielded to her mentor
teacher’s wishes. This was reflected in Megan’s response to a question asking how her
lesson planning had changed over the course of the semester,

This is sort of complicated. I think that my literal lesson planning has changed in

that now it's just a lesson plan lecture style lesson. I think that has changed

because of my mentor and I think that it has changed because of what she wants
for the class. However, my desires for my lesson plans, for [what I want them to

be] has not changed” [In-Depth Interview 2, December, 2008]

A possible reason for Megan being reluctant to try to assert more authority in the
classroom came up in an informal conversation she was having with Emily after the
methods course one day. Megan and Emily were brainstorming possible changes that

Megan might try to make to Mrs. Starns’ schedule in the second semester when she was

to take more control of her classes. Megan noted that Mrs. Starns “has reasons for why

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she does things and she will shoot them at you” [Informal Conversation with Field
Instructor, November, 2008].

Overall, the two contexts varied drastically in the resources and authority they
provided. Within the school context Megan’s authority and access to resources was
limited by lack of time, Mrs. Starns’ strict schedule, and the lack of another teacher with
a similar vision of teaching to share activities and experiences. Given the connection
Megan made between student—centered activities, the lesson planning template, and
anticipating SMRs, the lack of student—centered tasks in the school may have been a
major factor that kept her from anticipating SMRs. However, there were student-centered
tasks available at the university or possibly in her own textbook or by asking other
interns, the university methods instructor, or field instructor that Megan rarely made use
of. Additionally, although the methods course instructor gave her some feedback in the
area of anticipating SMRs, Mrs. Starns did not volunteer this kind of information and
Megan did not seek it out. Megan was not making use of any resources or authority
unless they were provided to her by someone else. In this way, the lack of resources and
authority at the school could be viewed as a failure on Megan’s part to seek out the
resources she needed and assert her own beliefs to teach in the way that she wanted, a
way that she connected with anticipating SMRs. The next section describes the Zone of

Promoted Action in each of the settings.

5.2.3 Zone of Promoted Action

The Zone of Promoted Action (ZPA) describes the particular teaching skills or

approaches that are promoted by teacher educators, field instructors, and more

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experienced teachers (Goos, 2005b). In this case this zone is represented by the actions of
anticipating, or those that relate to and support anticipating, that are promoted in the two
contexts. Because they were such an integral part of the university context the other
interns were also included in the group of those who could promote actions. In Goos’
(2005b) study the ZPA was comprised of three key elements:

I preservice education,

I practicum/beginning teaching experience, and

I professional development (p. 40).
As with the ZFM, certain aspects of the ZPA resonate with two of the principles that
Engle and Conant (2002) describe as fostering productive disciplinary engagement. They
are problematizing subject matter and holding students accountable to others and to
shared disciplinary norms. Applied to the interns, these represent problematizing
mathematics and mathematics teaching and holding the interns accountable to other
teachers and shared norms of mathematics and mathematics teaching. These two
principles will be referred to as problematizing and accountability and will frame the

discussion in each of the two following sections.

5.2.3.1 The University

Within the university context the ZPA comprised a vast array of ideas and
practices. This section will begin with a description of how the subject matter was
problematized in this setting followed by a discussion of how this context held Megan
and the other interns accountable to each other, to their instructors, and to a certain set of

disciplinary norms.

l 95

The university context provided experiences in which mathematics and
mathematics teaching was problematized in almost every class and during many of the
interactions that occurred outside of class. Mathematics itself was problematized through
the semester-long focus on fractions and ratios. Dr. Rundell engaged the interns in tasks
around these two mathematical ideas and allowed them the opportunity to challenge each
other’s ideas. In the second methods class of the semester Dr. Rundell posed the question,
What is the difference between a fraction and a ratio? A relatively heated debate occurred
among the interns that lasted throughout the semester. Further questions were posed: Are
all fractions ratios? Are all ratios fractions? What is the difference between the two? Why
is the difference important for proportional reasoning? The interns would occasionally
turn to Dr. Rundell and ask for a definitive answer, which he would not provide”. In the
last class of the semester the interns were still discussing the issue after having read the
chapter on fractions and multiplicative reasoning in NCTM’s Research Companion to '
Principles and Standards for School Mathematics (Thompson & Saldanha, 2003). Several
of the interns were still passionately engaged in the debate and ended the semester
intellectually frustrated with the questions. This type of interaction problematized one
area of mathematics for the interns, which allowed them to consider it at a different level
and from different perspectives, and provided a model for them to problematize
mathematics for themselves and their students in the future.

The teaching of mathematics was also problematized in the university methods
course. This occurred broadly through Dr. Rundell’s dual focus in the class on what he

referred to as issues of “Now and Forever.” Dr. Rundell saw his role in addressing the

 

18 . . . . . . . ..
Thrs rs also an example of Dr. Rundell promoting mathematical authority wrthrn the group of Interns.

196

Now issues as “responding to their real time needs” [Methods Instructor Interview,
October, 2008]. This view is in accordance with how Engle and Conant (2002) described
problematizing subject matter. “The core idea behind problematizing content is that
teachers should encourage students' questions, proposals, challenges, and other
intellectual contributions, rather than expecting that they should simply assimilate facts,
procedures, and other ‘answers’” (p. 404). Dr. Rundell accomplished this by making
individual intern concerns public to the other interns each week during What’s
Happening in Mathematics (WHIM) time. The individual teaching concerns became
problems for the entire class to tackle. During these discussions Dr. Rundell would ask
the intern to share their story and would sometimes ask other questions in the course of
the discussion. For the most part, he stayed on the sidelines of these discussions and did
not offer “correct answers,” but on occasion did share relevant stories of his own
teaching.

Mathematics teaching was further problematized in the university methods course
through planned class experiences. One example came from a case Dr. Rundell presented
in which a mathematics teacher was teaching content in a new way than she had in the
past. The case detailed the decisions that she made while teaching and included her own
questions to herself about what she needed to do next. Dr. Rundell asked the interns to
identify the teacher moves that contributed to student learning in this case and to explain
why. During these and other conversations around case studies, the interns had the
opportunity view the complexity of mathematics teaching from a perspective other than
their own. While these experiences were not initiated by interns or based on their

experiences, they also represented ways of thinking about teaching that were potentially

197

new to the interns and different than what they experienced in their own learning of
mathematics or what they were seeing in their field experience classrooms. In this way
these experiences problematized the teaching of mathematics for the interns through
alternative vision of mathematics teaching.

Interactions in Fireside chats also problematized the teaching of mathematics. Dr.
Rundell would ask questions and probe for more information from the interns. The series
of questions below were asked by Dr. Rundell in the third weekly Fireside chat when one
of the interns stated that his students had performed very poorly on a test he had given
that day. They are representative of the way he problematized the teaching of
mathematics.

Why do you think students did badly?

So let's think about your test a bit. What was the content?

Okay. Tell me about the instruction that went before.

So tell me, what did you think they knew coming into the test?

What did they do well and what didn't they do well?

Why was memorization tough for them? And what is the value in memorizing

these things? Okay, so how is it important for living in society?

But we all know it, right? When do WE use it?

[Fireside Chat, September 17, 2008]

In this interaction Dr. Rundell pressed the intern to think further about his statements by
questioning, probing, and challenging the intern to articulate more about the experience
and his thinking about it past his initial reactions. Megan’s field instructor, Emily, asked
Megan similar questions during a debrief conversation with her after an observation.

Do you think that’s what all of the students were having problems with? I know

you only worked with those two specifically.

When working with those two. . .what was their misunderstanding?
[Field Instructor Lesson Debrief 1, October, 2008]

198

Finally, mathematics teaching was problematized through the Researchable
Question assignment. For this assignment the interns worked in groups to design a
research project based on their students learning. They carried out their projects and
presented their findings at a poster fair on the last day of the methods course. Dr.
Rundell’s explained how this assignment related to anticipating SMRs.

I think the researchable question assignment is kind of a retrospective on

anticipating in some way. So for that assignment they are asked to identify a

question that they are interested in answering about mathematics and how their

students are dealing with mathematics and most of the time it arises from
something that’s come up in their class. For example, 'my students are having
tremendous difficulty with operations with negative numbers. I wonder why that
is.'... It really feeds back into anticipating and all of it seems to arise from failures
in anticipating. 'I was surprised that my students couldn't do X' and then that leads
to the idea that I might want to use that as my researchable question. [Methods

Instructor Interview, October, 2008]

In both the problematizing of mathematics and teaching mathematics, Dr. Rundell
created a situation in which anticipating SMRs was a reasonable practice in which to
engage for a teacher. Since both the discipline and the act of teaching are comprised of
multiple interpretations and uncertainties, the act of anticipating during lesson planning
becomes an important tool in helping teachers support their students learning in fruitful
ways toward the desired goals. As Dr. Rundell described it,

If they've thought about how a particular way of thinking [represents] a

mathematical idea and what moves they're going to make in response to that...

then there's less kind of a cognitive load right in the moment of teaching and that's
going to help them be more flexible in the classroom. [Methods Instructor

Interview, October, 2008]

The second element of the ZPA in this frame is accountability to other teachers

and to the disciplinary norms of mathematics and mathematics teaching. The university

context created an environment in which the interns were both accountable to their

199

methods course and field instructors as well as the other interns in the class. The methods
course instructor and her field instructor both reviewed Megan's lesson plans and the
field instructor watched her teach. Through their feedback they held Megan accountable
to the disciplinary norms established by the community. There was some evidence that
Megan only felt accountable to her instructors in her lesson planning when writing
lessons for class assignments. When asked what her methods instructor expected of her
daily lesson plans she responded that “he just goes by what my field instructor expects of
me so it would just be what my mentor wants” [In—Depth Interview 2, December, 2008].
Through Dr. Rundell’s creation of a collaborative teaching community in the class
the interns were also held accountable to one another. This accountability was illustrated
in a case stories assignment in which interns shared their ideas for the Cool Stuff 1 lesson
assignment with other interns in small groups. Megan’s group was comprised of herself
and two other interns. As Chuck, an intern, shared his lesson idea he first commented that
for the first part “students can do it two ways” [Methods Course Observation 5,
September, 2008] and then went on to describe what the two methods he foresaw were.
Megan and the other intern, Cathleen, press Chuck to think about how he will grade the
activity and what exactly he will look for in terms of evidence of student understanding.
Next, Megan shared her idea for her lesson and Chuck suggested that she needed to do
the task herself before deciding if it is appropriate for her students. Cathleen stated that
she liked the activity but was concerned it was too open-ended for Megan’s students
since they were used to lecture and note taking. Finally the third intern, Cathleen,
described her ideas and explained that she was planning on using a launch that she saw in

the methods course the previous year. In her explanation of her lesson Cathleen exhibited

200

the already established norm of considering SMRs in thinking about her lesson. She
stated that “hopefully it will work the same as it did last year” [Methods Course
Observation, October 1, 2008]. She went on to further describe how students might
interpret the activity she was considering. As an example referring to how students might
think about the slope of a line she said “they might say 'because it goes up faster' or. .
[Methods Course Observation, October 1, 2008]. In this instance the interns were
accountable to one another to provide helpful feedback on their lesson ideas in order to
improve them and accountable to established norms of anticipating SMRs, which is
discussed further in upcoming sections.

Part of holding the interns accountable in the university setting was holding them
accountable to the disciplinary norms associated with teaching mathematics. The
disciplinary norms of mathematics teaching in this setting reflected the particular view
held within the university community. This view included the use of tasks of high
cognitive demand that engaged all students in discussion around important mathematical
topics. Dr. Rundell described a particular goal of the class relating to this view,
“certainly, working on how to facilitate discussions and how to plan for facilitating a
large scale student discussion is something that we're going to work on this year with
respect to lesson planning” [Methods Instructor Interview, October, 2008].

As previously described, Dr. Rundell often asked the interns to engage with
mathematical tasks and to solve them in multiple ways. Doing so promoted the norm that
approaching mathematical tasks in multiple ways was fruitful and expected. In addition
to doing mathematics in this way, the interns were expected to take multiple solutions

into account in their lesson planning. In explaining his expectations for the Cool Stuff 1

201

and 2 assignments, Dr. Rundell stated, “I'm looking for... complete student solutions to
the task, a variety of representations, a variety of strategies, the ideas related to student
thinking are clear, and it includes both correct and incorrect pathways to thinking about
the task” [Methods Instructor Interview, October, 2008]. Megan mentioned this
expectation when reflecting on her feedback from Dr. Rundell.

When I did my Cool Stuff 1, he wanted me to really pay attention to student work

and how that could impact my teaching and what to do and how to take all those

materials and think about how the kids are thinking to help them. [In—Depth

Interview 2, December, 2008]

Other practices of anticipating SMRs were also part of the norms of the university
setting. Megan’s field instructor, Emily, stated that her own expectations for Megan’s
lesson planning were that “ultimately the goal would be to predict the questions that the
students are going to ask and the misunderstandings that they are going to have” [Field
Instructor Interview, September, 2008]. Similarly, Dr. Rundell noted that in giving
feedback on Cool Stuff lesson plans he also looked for anticipated student questions and
thinking.

Dr. Rundell developed the norm of anticipating SMRs in lesson planning by
building on the practices the interns participated in the previous year. “I went back to the
five practices for orchestrating productive discussion of which anticipating is the first,
and [said] 'if your lesson plan doesn't respond to these five things then there's
something missing from it'” [Methods Instructor Interview, October, 2008]. He also
reflected that in addition to assignments that called on the interns to anticipate SMRs,
there was also a lot of tacit work that took place that supported the practice.

It’s not like we have a session on anticipating student thinking. . .There have been

a couple of opportunities to make my own thinking and pedagogy visible in terms
of how I think about orchestrating their learning experiences... I think seeing

202

some of that in action where I've had opportunities to peel back the layers a little
bit and show them 'hey this is how I prepare for this class' is really helpful in
making it a little bit more real to them and helping them to understand that even
though these practices are difficult to do, and anticipating student thinking in
particular is really difficult to come out of your own head and start to think about,
there is a pay off. [Methods Instructor Interview, October, 2008]
Dr. Rundell modeled this practice on several occasions during the methods course by
presenting alternative solutions to a mathematical task after the interns had shared all of
their solutions. His lesson plans were also held up as a model for the interns. In one class
early in the semester in which Emily the field instructor had already discussed the detail
of Dr. Rundell’s own lesson plans for the methods course, one of the interns asked Dr.
Rundell if he had thought of all of their wrong answers in his lesson plan. Dr. Rundell
responded that his lesson plan conatined all of the incorrect answers he had seen in class.
Also part of the ZPA in the university context were the practices promoted within
the intern community. Evidence from the Lesson Planning Survey (LPS) conducted at the
beginning of the year indicated that many of the interns viewed anticipating SMRs as a
disciplinary norm before beginning the internship year methods course. In the LPS the

interns were asked about the activities they participated in to plan lessons in their senior

year. Table 8 shows their responses.

203

Table 8. Intern Participation in Specific Anticipation Activities

 

Always Almost Sometimes Hardly Never
Always Ever

 

8* 3 2 I
worked the mathematical task or
solved the problem(s) I planned to
give to students

consulted the teacher guide for 3* 1 6 2 2
possible areas where students

might struggle or solution

strategies they might come up

with

read articles about how students 0 2* 9 2 1
might engage with the content I
was about to teach

asked my mentor teacher about 4 3 5* 2 0
their experiences teaching the
content

thought about the learning needs 3 3 4* 4 0
of particular students in the class

(i.e. Will this lesson make sense

for Shannon in my 3rd period

class?)

 

Note. The asterisks indicate a response by Megan
Their responses indicate that many of them were participating in anticipating SMRs
practices at least some of the time.

The LPS also asked the interns to consider why the practice of anticipating SMRs
was promoted in their teacher education program. Most indicated that it was necessary to
be prepared for teaching, which included: addressing misconceptions when they arise,

making use of student solutions, and being ready to answer questions. Two of the interns

204

approached the need for anticipating from another point of view. As opposed to thinking
about how anticipating benefited them, they focused why they would anticipate in terms
of what it would do for their students

It puts us in the students’ shoes. We have to think about what they are doing or

thinking so we can catch that during our lesson. Without thinking about these

things, we may miss important opportunities to discuss ideas and other solution
strategies. [Lesson Planning Survey response]

It is encouraged because if we don’t think about how students process information

then we might as well just have them read something previously written and learn

from that. We need to think about what will confuse them so that we can clear
mistakes up in creative ways without just correcting them but by helping them

discover on their own. For example by leading them to confusion purposely in a

sequence of problems where they eventually clear up their own questions and

make conclusions. [Lesson Planning Survey response]
Overall, the responses to the LPS indicated that the interns were both familiar with the
practice of anticipating SMRs and recognized its importance as they began their
internship year teaching.

Findings indicate that the interns continued to view anticipating SMRs as a norm
of mathematics teaching throughout the semester. They evidenced this in using each
other as resources to anticipate by sharing their own experiences teaching content that
others were teaching. In one discussion about the division of fraction one intern teaching
in an eleventh grade class shared her past experiences with another intern about to teach
fractions: “kids get so confused by fractions. They can do halves and fourths but once
they get to thirds. . .,” then referring specifically to adding fractions she tells the group
"they add the top and the bottom" [Methods Course Observation, October 8, 2008].

This importance of this norm for the interns was also evident on one occasion

when they attempted to elicit experiences from one another to help them anticipate SMRs

when they taught content in the future. Regarding the outcomes of the Researchable

205

Question project, one intern asked “are we going to get to find out what they find cause
I’m teaching that next semester?” [Methods Course Observation, October 22, 2008].
Another intern followed this question by asking “Are you trying to fill in the anticipation
column in the lesson plan?” to which the first intern replied that she was. These
interactions are evidence that not only did the interns view anticipating SMRs as a norm
of lesson planning, but that they had begun to participate in active/purposeful anticipating
by using each other as resources.

Overall, the university context attempted to provide experiences that would both
problematize mathematics and mathematics teaching and hold the interns accountable to
the instructors, each other, and disciplinary norms. The experiences are evidence that
Megan had the opportunity to participate in these promoted practices in this setting. The
next section describes the promoted practices that Megan engaged with in the school

setting.

5.2.3.2 The School

Recall that the ZPA is comprised of the elements of problematizing mathematics
and mathematics teaching and holding the intern accountable to other teachers and
disciplinary norms. This section will begin with a description of how the subject matter
was problematized in this setting followed by a discussion of how this context held
Megan accountable to other teachers and to a particular set of disciplinary norms.

The school context appeared to make considerable effort not to problematize
either mathematics or the teaching of mathematics for Megan. Mrs. Starns advice to

Megan about teaching mathematics was often to “know your content.” There was no

206

evidence during observations or from interviews that Mrs. Starns and Megan had any
discussion with each other about what it meant to know the content. In addition to not
encouraging Megan to problematize the content for herself, Mrs. Starns did not
encourage her to problematize the content for the students. She often advised Megan to
be as simple and concise as possible in her presentation of facts and examples. One day
before class Megan asked her about the kinds of questions she usually asked in order to
engage students in helping her set up a problem during lecture. Mrs. Starns again told her
to be concise and ask concise questions and stated “don’t be too wordy with the Algebra
2ers; they don't like it" [Informal Conversation, Lesson 1.1 Observation].

The problematizing of teaching mathematics was also limited by Mrs. Starns. She
created a weekly schedule for teaching that she used each year that included a test on
every third Friday. Her lecture style of teaching that Megan adopted minimized
unforeseen events in her classroom. Because Megan fully adopted her mentor teacher’s
approach during most of the first semester she also participated in other practices that
minimized potential problems, such as unforeseen student questions. Megan noted this
phenomenon in one of the interview. “Today I did a lot of talking and they didn't do any
work together or any student centered things. So they didn't really have questions”
[Lesson 2.3 Post-Interview, November, 2008]. In an earlier interview Megan was
questioned about one of her stated misgivings that she did not have the opportunity to
explore student thinking as much as she would like to. She explained, “I don't get to
cause the students to think a different way or to scaffold them more in their thinking and
guide their thinking. I think I'm more a dispenser” [In—Depth Interview 1, September,

2008]. Another method used by Mrs. Starns and adopted by Megan to minimize potential

207

problems was to choose homework exercises to review in front of the class instead of
asking the students which exercises they needed help with.

The problematizing of the teaching of mathematics was also limited in the school
context by Megan’s use of Mrs. Starns teaching notes from the previous year. Although
there were some instances in which Megan would stray from her mentor teachers’ notes
and choose different examples for her own reasons, the availability of the resource meant
that Megan was not compelled on a daily basis to make those pedagogical decisions
herself. Mrs. Starns lack of problematizing is perhaps not a surprising finding.
Problematizing is an inherently messy and difficult activity that only adds to the already
complex nature of teaching. Perhaps for that reason Mrs. Starns saw any practice that
problematized mathematics or her own teaching as something to avoid in her own
teaching as well as something from which to shield Megan. This is suggested by her strict
adherence to a weekly schedule and her insistence that Megan follow the same schedule.
The result was that the lack of problematizing of either mathematics or mathematics
teaching could have served as a negative catalyst for Megan to anticipate SMRs. If
mathematics itself is straightforward and the teaching of mathematics is seen as the
teacher clearly laying out what students are to learn without any input from them, then it
would not be necessary to consider alternative student solutions, na’ive conceptions, or
student questions or difficulties because they would not occur.

Accountability to other teachers and to disciplinary norms is the second feature of
ZPA. Within Logan High School, Megan was only held accountable to Mrs. Starns and
her disciplinary norms because of the lack of an instructional community amongst the

teachers at the school. As has already been established, Mrs. Starns disciplinary norms

208

fell under the norms of a more teacher-centered, lecture-based stance on teaching
mathematics. Thus some of her norms, which were also the norms she held Megan to
while teaching at the school, reflected this stance. Several of Mrs. Starns disciplinary
norms impacted Megan’s opportunity to enact student-centered teaching practices, plan
detailed lessons, and perhaps ultimately, to anticipate SMRs.

The first norm of Mrs. Starns’ that influenced Megan’s lesson planning and
enactment was the previously discussed weekly schedule. When asked about Megan’s
daily schedule at the school Mrs. Starns replied that “she puts out the homework, she has
the kids get it, we sometimes have bell work which is like a starter question and we'll go
over the homework and she'll do her notes” [Mentor Teacher Interview, October, 2008].
Mrs. Starns was adamant that she thought following a daily and weekly schedule was
very important and stated that when she was the math department chair for the district
every teacher was on such a schedule and tested their students at the end of every three
weeks, regardless of their position in the curriculum at the time. By holding Megan
accountable to these daily and weekly norms, Mrs. Starns constrained Megan’s
opportunity to enact lessons in her own style that would potentially take more time than
she was allotted.

The second set of norms established by Mrs. Starns dealt with grading papers and
also influenced Megan’s practice of lesson planning and anticipating SMRs in different
ways. Mrs. Starns’s gave primacy to the grading of papers each night, which limited the
amount of time Megan had to plan lessons.

She and I agreed that the homework was really important to get done, grade the

night, the day they turn it in, so that they can receive it back the next day. Because

we go over the homework the next day from that section from the previous day.
So it was important that the kids got their homework back and got feedback right

209

 

away. But the worksheets or the quizzes, we agreed that on Sunday nights is when

we would have everything in, that would be fair. Just the homework was really

important. [Mentor Teacher Interview, October, 2008]

Another aspect of the grading norms promoted by Mrs. Starns was that student papers
were identified with their assigned numbers instead of names. Megan explained in an
informal conversation one day that not having the students names on the top of the papers
when she as grading them made it difficult for her to match the thinking she was seeing
in the work to any particular student. When asked why Mrs. Starns used this method,
Megan replied that she thought it was because it made entering grades into the computer
easier. Finally, Mrs. Starns appeared to use grades to motivate students to participate.
When Megan was getting ready to teach her Cool Stuff lessons, Mrs. Starns advised her
to give students participation grades on their work in the class so that they would
participate in the activity. These norms promoted a focus on grades that appeared to
cause Megan to focus on grades rather than on planning for and trying to understand
students thinking about mathematics.

Mrs. Starns also articulated specific norms for Megan’s lesson planning. While
there was no evidence that these norms prohibited Megan from anticipating SMRs, there
was nothing built in to the structure of the plans that would promote the practice. Megan
explained her mentor teacher’s expectations for her lesson plans early in the semester.

I don't do like the big elaborate lesson plan, but I do one of these for now cause

this is what my teacher wants. So this is what the kids will do and this is what I

will write on the board, it’s pretty straight forward. And then this is what I have to

do before class and this is what we’re going to do during class and I'll go through
that. And then I have stuff that I want to do during prep hour, after school, and at

night. [In-Depth Interview 1, September, 2008]

Figure 39 is recreation of the lesson plan Megan described in the above quotation.

210

 

 

Lesson Plan Week 1.4
Thursday, September 25, 2008
Focus Class: Algebra 2, Fourth Hour

Student Schedule (to write on board)
Algebra 2
Thursday: (Section 3.3)
Bell work
Go over story problems
Notes
Textbook HW - Section 3.3: 21—48 left
column, Story: 55, 56

My Schedule
Before Class

0 Prep materials

0 Overheads for readings

0 Get out bell work on computer, turn
on projector

Set out reading packet

Set out returned student work
Set out note cards

Set out Game of Life materials
Overhead, markers

Get note card box for notebook
check

OOOOOO

Algebra 2
Thursday:

0 Bell work section 3.2
0 Getting started; 5-10 minutes
> Students have
homework and reading
packet to pick up
> Students get calculators
> Attendance
0 Announcements:
> Extra credit due Friday
> Notebook check
tomorrow — will double
count since we didn’t do
it last week
3‘» Game of Life tomorrow

 

O

0000

Prep

Go over story problems 3.1: 57, 3.2:
53, 55

Go over HW 3.2: 17 and 47
Reading packet section 3.2 and 3.3
Notes section 3.3; 10-30 minutes
Textbook HW - Section 3.3:21-48
left column, story 55, 56

Thursday:

0

O
O
O

O

Attendance/Tardies/Discipline
Update Binder

Put homework on schoolnotes.com
Put make—up work in Absent Folder
Bin

Print/create worksheets and answer
keys

Get together quizzes for next week
(15‘ and 4‘“)

After School
Thursday:

0
O

O

O
0

Night

Update Binder

Write student schedule for Friday
on board

Put notes/materials in Friday
folder/book

Make Up Folder for important stuff
Lesson Plans for week 1.5 due

Thursday:

0
0

Grade worksheets if not done
Final prep for Monday, Tuesday
lesson and prep for Wednesday,
Thursday

Send skeleton to field instructor

 

Figure 39. Recreation of the Lesson Plan Format Required by Mrs. Starns and Used by

Megan on a Daily Basis

211

 

Because there is no attention given to students or mathematics content anywhere in the
lesson plan format expected by Mrs. Starns, it can be inferred that she did not promote
anticipating SMRs as part of the written lesson plan. Megan additionally completed the

examples on her own in the notes template used by Mrs. Starns and given to students

(Figure 40).

212

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Figure 40. A Handwritten Example of the Notes Template Used My Mrs. Starns, Megan,
and the Students

213

Again, the format of this document, which was comprised of what was to be learned and
the examples to be presented, did not specifically prompt Megan to anticipate SMRs.

Within the school context Megan was accountable only to Mrs. Stams. This along
with the non-collaborative nature of the mathematics education community at the school
meant that Mrs. Starns was Megan’s one and only resource at the school on a daily basis.
Mrs. Starns norms did not directly prevent Megan from anticipating SMRs, but the nature
of the context did not create the opportunities for her to anticipate that she was used to

having at the university. The final section of this chapter summarizes all of the interaction

findings.
5.2.4 Summary of the Factors

The zones just described represent the complex contexts and interactions that
were part of this intern teachers experience. Goos (2005) presented her findings on the
interactions of these zones using a Venn Diagram to show the common areas between the

ZPD, ZFM, and ZPA. Figure 41 portrays the interactions between the ZFM in each

COIIICXI.

214

University ZFM

 
   
  
   
  
  
   
    
   
  

   
   
   
  
  
   

Resources

- Information on
student learning

- Other interns

- Instructors Megan makes

   
   
   
  
   
  
  
  
  
 

- Teaching use of a
journals university
Internet resource and
resources claims
Task resources authority to

teach in the

     
  
  

Authority
- Professional

authority to
question norms
and make
instructional
choices

based on a
university
assignment

 

 

way she wants

    
  
  
    
 

School ZFM

   
  
  
  
  

Mam
Insufficient time
due to grading
No task resources
Mentor teacher not a
consistent source of
anticipated SMRS
Limited time to discuss
Lack of community of
practice among mathematics
teachers

   
   
    
   
   
  
  
    
   
  

Authority
Lack of authority to

teach in her own way
Lack of authority to
make changes to
the schedule

   
  
  
 

Figure 41. Venn Diagram of Relationship Between University and School ZFM

As shown in Figure 4] , the only overlap between the ZFMs occurred when Megan was

obligated to complete a lesson plan for a university course. Figure 42 adds to the previous

figure the ZPAs in each setting.

215

 

 

                  

                    

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Figure 42. Venn Diagram of Relationship Between University and School ZPA

216

This figure shows that the promoted actions in each context, in terms of anticipating
SMRs, shared no common elements. The overlap of the University ZPA into the school
ZFM again represents the Cool Stuff 1 assignment in which the university lesson
planning norms were applied to Megan’s school lesson plans. This occurred again in
Cool Stuff 2, which can also be represented as part of the overlapping space, even though
university curriculum resources were not used for that lesson as they were for the Cool

Stuff 1 assignment. Finally, Figure 43 adds Megan’s ZPD to the Venn diagram.l9

 

'9 See Chapter 2 for a complete description of the relationship between the zones in the framework.

217

 

 

 

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Figure 43. Venn Diagram of Relationship between University and School ZFM and ZPA

and Megan’s ZPD

218

Figure 43 shows that Megan’s ZPD mostly overlaps the ZPA of the university. The area
within the university ZPA but outside of Megan’s ZPD represents the actions of the
methods instructor to encourage the interns to “bother” their mentor teachers when
needed for help in planning lessons, something Megan did not do. Other than this
exception, her skills, experiences, and desires positioned her well within the range of the
practices for lesson planning promoted by the university, but her day to day physical
presence within the school setting determined her actual practices. As previously

mentioned, the only time during the semester that the possible for Megan in the two

 

settings intersected was during the two Cool Stuff lesson assignments. Only within this 1
space was she free to enact her own ideas for teaching as well as those advocated by the
university.
The next figure displays together all of the contextual features that related to
Megan’s anticipation of SMRs with each labeled according to how it related to the
practice of anticipating. A (P) indicates that the feature promoted Megan to participate in
the practice of anticipating SMRs. The other two labels indicate features that did not
promote participation in the practice of anticipating SMRs. In the first case of this kind
the contextual feature was neutral (N) to anticipating, that is, it did not directly promote
or constrain the practice. In the second case, the presence of the contextual feature

actively constrained (C) Megan’s participation in the practice of anticipating SMRs.

219

ZFM

ZPA

Figure 44. Experiences, Resources, and Constraints that Impacted Megan’s Anticipating

Practices

University

School

 

Resources

I Information on student learning
(P)

I Teaching journals (P)

I Internet resources (P)

I Task resources (P)

I Other interns (P)

I Instructors (P)

Authority
I Professional authority to question

norms and make instructional
choices (P)

Resources

I Insufficient time due to grading
(C)

I No task resources (N)

I Mentor teacher not a consistent
source of anticipated SMRs (N)

I Limited time to discuss (C)

I Lack of community of practice
among mathematics teachers (N)

Authority
I Lack of authority to teach in her

own way (N)
I Lack of authority to make
changes to the schedule (N)

 

 

Problematizin g
I Mathematics (P)
I Mathematics instruction (P)

Accountability to Norms

I Intern community norm of
anticipation (P)

I Requirement of anticipation in
CS] and CS2 (P)

I Modeling of anticipation by
methods instructor (P)

I Field instructor prompting to add
student thinking to lesson plans
(P)

I Collaboration around lesson
plans (P)

I Solving tasks multiple ways (P)

 

Problematizing

I Simplification of Mathematics
(N)

I Simplification of mathematics
teaching (N)

Accountability to Norms

I Lesson planning expectations (N)

I Use of already create lecture
notes that did not include
anticipated SMRs (N)

I Grading norms (N)

I Maintaining a daily schedule (N)

 

 

 

220

La?
Rm
'l

 

This figure makes plain the differences between the two settings in how the practice of
anticipating SMRs was promoted and supported. Within the university context the
practice of anticipating SMRs was promoted by the instructors, the other interns, course
assignments, and other available resources. However, many of the features of the school
context were neutral to, or did not promote, Megan’s practice of anticipating SMRs. For
example, Megan’s mentor teacher’s lesson planning expectations did keep her from
anticipating SMRs, but they also did not call upon her to do so. Finally, some of the
features of the school context, such as limited planning time due to an inordinate amount
of grading, directly constrained her practices of anticipating SMRs by limiting her
resource of time to plan lessons.

Figure 44, along with the findings from the first part of this chapter, illuminates
several interesting occurrences. First, despite the fact that Megan had access to all of the
resources of the university she did not make use of them except in her first Cool Stuff
lesson assignment. These resources would have allowed her the opportunity to choose
tasks and learn about SMRs for those tasks that she often stated she desired. Megan’s
own comments about wishing the school and her mentor teacher could provide her these
resources indicate that perhaps she did not consider making use of the resources from the
university setting in the school setting. It is important to consider along with this that
Megan made little effort to seek out these resources when they were present or to attempt
to assert her authority as a beginning professional in the school setting.

This scenario of wanting a certain type of resource, one that was available to her
at the university, but not making use of it in the school is potentially described by a lack

of intercontextuality as described by Engle (2006). In this case this would mean that

221

 

Megan did not connect the practices of the learning context, the university, to the
practices of the transfer context, the school. By not doing this she did not have the
opportunity to see the continued usefulness of the university resources or the practice of
anticipation of SMRs in an alternative setting with alternative resources and norms.

If over generalized, Figure 44 might indicate that only the university was
providing the experiences and resources needed by Megan to participate in the practice of
teaching mathematics, while the school setting was providing nothing and was in fact
deten‘ing her from learning about teaching. This is not the intention of this figure, or a
finding of this study. This figure only relates to the experiences and resources related to
anticipating SMRs. The school setting certainly provided invaluable resources outside of
anticipating in their students and by allowing Megan to come and make use of an actual
classroom. Megan also had opportunities at the school to engage in other practices of
teaching such as attending professional development meetings, conducting parent
conferences, and working with individual students. In these and many other ways the
experience was fruitful in that it allowed Megan to participate in these other types of
teaching practices. What findings from this study do indicate is that despite all that was
provided by the university and Megan’s own desires to teach and plan lesson in certain
ways, the constraints of the school setting and her own choice not to challenge these
constraints, limited her participation in the practice of anticipating SMRs. The next
chapter discusses the findings of this study in terms of the theoretical framework, the

implications for teacher education, and future directions for research.

222

 

 

CHAPTER 6. DISCUSSION

Chapter 1 described the importance of field experiences in the development of
preservice teachers and the need to know more about what preservice teachers learn as a
result of these experiences. A large part of the work of teaching that preservice teachers

do in field experiences is preparing and enacting lessons. The National Research Council

(NRC) sets clear guidelines for what teachers should consider as they plan for instruction.

[R]ather than simply listing problems and exercises, teachers should plan for
instruction by focusing on the learning goals for their students, keeping in mind
how the goals for each lesson fit with those of the past and future lessons. Their
planning should anticipate the events in the lesson, the ways in which the students
will respond, and how those responses can be used to further the lesson goals.
(National Research Council, 2001 , p. 425)
This study sought to shed light on how one preservice teacher planned mathematics
lessons during a field experience, focusing specifically on how she anticipated her
students’ mathematical responses (SMRs), a practice further detailed by Stein, Engle,
Smith, and Hughes (2008). Additionally, this study examined the extent to which the
preservice teacher anticipated SMRs when planning her lessons, the resources that she
made use of that helped her to anticipate SMRs, and the types of SMRs themselves.
Finally, this study explored how the two contexts, the university and school, supported
and constrained the preservice teacher in participating in this practice.
This study drew on four main areas of research literature: teacher education field
experiences, knowledge for teaching mathematics, lesson planning, and anticipating
SMRs. Results from this study resonate with past studies about field experiences that

identified the mentor teacher and the curriculum as important contextual features in what

the preservice teacher learned about teaching mathematics (Mossgrove, 2006; Vacc &

223

 

 

Bright, 1999). Although no data was collected on the change to the preservice teacher’s
knowledge for teaching mathematics as a result of the field experience, this study did
highlight the sources of knowledge that this preservice teacher drew on in order to
anticipate SMRs. Additionally, the preservice teacher in this study was able to identify
possible SMRs prior to teaching her lessons, unlike the preservice teachers in past studies
(Borko & Livingston, 1989), but most of her anticipating tended to be procedural in
nature.

Findings from this study showed that the constraints of the school context and the
intem’s own inability to challenge these constraints limited the use of the support
provided by the university context in her practice of anticipating SMRs. Megan was
chosen as the participant of this study based on the quality of her lesson plans in her
senior year, which were consistently designed with students in mind. Despite this, during
the first semester of her internship year she showed evidence of actively anticipating
SMRs in only three of the thirteen lessons she planned, two of which were prepared as an
assignment for the university and required her to anticipate. Overall, her anticipation
oscillated between considering that students may not see or remember the way she
presented to do the mathematics and considering that the problems they may have
making sense of the mathematics for themselves, with far greater frequency of the
former. Additionally, she made use of relatively few resources available to her to help her
anticipate SMRs.

The findings also show marked differences in the foci of the two contexts Megan
worked in, the university and the school. While the university focused on communities of

practice; modeled problem-solving and provided resources; and encouraged deliberate

224

 

practices of planning, teaching, and reflecting, the school context represented an isolated
teaching environment focused on developing teaching poise and streamlining the work of
teaching. Lortie (2002) points out that situations like those in Megan’s school context is
not unusual in that teachers often work in “mutual isolation” and that the result is reduced
opportunity “for turning to colleagues and superordinates for informed assistance” (p.
150). The situative perspective that framed this study suggests that the differences in i"

Megan’s anticipation practices may have been impacted by the differences in the

 

contexts. The next section describes the affordances of making use of the situative ' g

perspective in this case study to investigate Megan’s practices of anticipating SMRs.
6.1 Affordances of Making Use of a Situative Perspective

This study made use of the situative perspective to study the practices of one
preservice teacher. As described in Chapter 2, Greeno’s notion of beginning “with the
framework of interactional studies and work inward” (Greeno, 1998, p. 6) was employed
and thus the environments in which the preservice teacher worked as well as her own
actions in these environments were investigated as part of this study. This section
discusses the affordances of being able to investigate a preservice teacher’s anticipating
practices during a field experience from both the context and individual perspectives.

The focus on the contexts provided both background and possible explanations
for the choices Megan was making in her own work. An important finding of this study
was the difference between what Megan professed to want to do in her teaching and
lesson planning, which aligned with university values and expectations of providing

students’ with open—ended experiences in which to explore and discuss mathematics, and

225

what she actually did in the school context. This finding is especially striking given that
Megan was chosen as the participant in this case study because she was so motivated to
participate in and engaged with the university advocated practices during her senior year.
Without knowledge of the school context, which was primarily dominated by the mentor
teacher whose focus was on clear and concise presentation of examples, the contradiction
between Megan’s professed desires and her actions would have been without explanation.
As with past researchers who have made use of the situative perspective, doing so in this
study enable the researcher “to see these differences as coherent and sensible—as being
integrally connected to the norms and expectations specific to the different contexts”
(Peressini, Borko, Romagnano, Knuth, & Willis, 2004, p. 90).

Additional evidence from the contexts is suggestive of why the norms of the
school context might have taken priority over the norms of the university context. When
describing interactions with her methods course instructor about how to change her
lesson plans Megan reflected that,

When I did my Cool Stuff 1 he wanted me to really pay attention to student work

and to how that could impact my teaching and how to take all those materials and

think about how the kids are thinking to help them. And I tried to do that in my

Cool Stuff 2. [In-Depth Interview 2, December, 2008]

This description of her methods instructor feedback is in sharp contrast to Megan’s
account of a conversation with her mentor teacher about making changes to the schedule
and style of teaching. “She has reasons for why she does things and she will shoot them
at you. So that’s the hard part” [Informal Conversation, November, 2008]. These excerpts
from conversations with Megan imply two things about the feedback that she receives in

the different contexts. First, her lack of generalizing Dr. Rundell’s comments may

indicate that she viewed feedback from her methods instructor as pertaining only to the

226

 

 

lessons she prepares for the university. This is supported by another comment Megan
made about her methods instructor’s expectations for her daily lesson plans, those not
prepared for the university. “I think he just goes by what my field instructor expects of
me so it would just be what my mentor wants” [In-Depth Interview 2, December, 2008].
Second, Megan’s description of conversations with her mentor teacher about making
changes to classroom practices implies that she found it difficult to talk with her about

trying different ideas. This difficulty might have kept Megan from wanting to “rock the

 

boat” in the school context. Thus, though it appeared that Megan’s choices to teach and

i

plan lessons on a daily basis were counter to her professed desires, these choices seemed
more reasonable given the complex interactions taking place within the contexts.

Making use of Greeno’s (1998) characterization of the situative perspective also
afforded the opportunity to turn the focus of the study “inwar ” in order to closely
investigate Megan’s practices. As previously discussed, Megan’s practice planning
lessons for the university were drastically different from her practice planning lessons in
the school. Looking inward allowed for additional investigation of the discrepancy, this
time with a focus on Megan.

Within the situative perspective lack of transfer, which in this case is represented
by Megan not applying university lesson planning practices to school lesson plans, is
characterized by Engle (2006) as a lack of intercontextuality.

Intercontextuality occurs when two or more contexts become linked with one

another. When this occurs between learning and transfer contexts, the content

established during learning is considered relevant to the transfer context Thus,
the more related that learning and transfer contexts are considered to be, the more

likely—all other things being equal—that students will transfer content between
them. (Engle, 2006, p. 456)

227

 

The data suggests that there are two reasons for the lack of intercontextuality in this case.
The first is the difference in styles of teaching advocated by each of the contexts, namely
student-centered versus lecture-based. The second reason could have originated with
Megan herself and the way that she attempted to apply university ideas in her field
experience classroom.

As was described in Chapter 5, Megan appeared to link what she viewed as
student-centered teaching to the use of the university lesson plan format and to
anticipating SMRs.

Yes, [anticipating SMRs is important] because in a student centered class a

teacher needs to be prepared for various issues and how to provide a solution. In

addition, it helps the teacher guide conversation for student learning. I try to do
this by looking up articles on misconceptions, talking to my mentor teacher, and
thinking of misconceptions on my own. [Lesson Planning Survey response]

And the big thing with [the lesson plan format] that I really liked was that it got

you to think what the students were thinking and then the questions you would

want to ask them and how you'd want to answer them. [In-Depth Interview 1,

September, 2008]

So to me student-centered learning is doing our whole lesson plan [format]. So

you would do a launch and explore where the kids are doing everything. And then

you do the summary. You know, ALL that. And I want the kids to be discovering

things on their own. [Informal Conversation, November, 2008]

Conversely, it appeared when teaching in a lecture-based style that Megan did not see the
relevance of anticipating SMRs.

I feel like I don’t get [to think about what students are thinking] as much with

lecture style things. . .I don’t get to cause the students to think a different way or to

scaffold them more in their thinking and guide their thinking. I think I’m more a

dispenser. [In-Depth Interview 1, September, 2008]

Today I did a lot of talking and they didn't do any work together or any student

centered things. So they didn't really have questions. [Lesson 2.3 Post-Interview,
November, 2008].

228

 

 

I think that my literal lesson planning has changed [from last year] in that now it's
just a lesson plan, lecture style lesson. I think that has changed because of my
mentor and I think that has changed because of what she wants for the class. [In-
Depth Interview 2, December, 2008]

These responses from Megan could indicate that for her the anticipation practices from

the learning context (the university) were not relevant in the transfer context (the school).

It should be noted that in neither the senior year methods course, nor the fall semester of

run-:1]

the internship methods course was the practice of anticipating SMRs directly linked to

student-centered teaching, but that discussion about practices of teaching, including

 

anticipating SMRs, were framed by this position. Thus for Megan a potential source for i i
the lack of intercontextuality emerged from the different styles of teaching being
advocated in each setting. Past research supports this as a reason for Megan’s actions.
“Research indicated that enacting a desirable practice is more likely when there is
coherence between the methods course and the fieldwor ” (Clift & Brady, 2005, p. 319).
The second reason for the lack of intercontextuality could be related to Megan’s
own views of applying university-leamed ideas outside of the university. Farmer,
Gerretson, and Lassak (2003) describe a framework that they developed to study
mathematics teachers’ appropriation of ideas encountered in professional development
experiences. The first, and most basic, of the three levels of appropriation is described as
Concrete activity and content. At this level,
teachers participating in a mathematics professional development activity
appropriate content such as specific mathematical skills or concepts that they will
actually teach or specific pedagogical techniques to implement in their
classrooms. Furthermore, they look for specific mathematical problems, tasks or
games to use with their students. Participants who appropriate at this level are
focusing on the specific parallel between mathematics activity in the seminar and

mathematics activities in their own classrooms. (Farmer, Gerretson, & Lassak,
2003,p.34l)

229

There is evidence that Megan was operating at this level as opposed to one of the higher
levels in which teachers attempt to integrate ideas, tasks, and theory from seminars into
their specific contexts. When asked about trying to use more open-ended activities in the
spring semester Megan responded, “It's hard because my mentor does not already have
those tasks available for me to use. So I would have to go out and find my own” [Lesson
3.3 Pre-Interview, December, 2008]. In addition, although Megan stated in her final

interview that she never discussed lesson ideas with the other interns during the fall

 

semester of the internship experience, in one of the Fireside chats she did ask one of the
other interns to create a CD of all of the Algebra 2 activities that the other intern had from '
her school”.
These findings regarding Megan’s level of appropriation resonate with previous
findings about preservice teachers. Borko, Eisenhart, Brown, Underhill, Jones, and Agard
(1992) studied preservice teachers teaching grade 6 mathematics.
We found that Mrs. Daniels and the other student teachers came to place more
importance on learning activities that could be directly imported to their student
teaching classrooms than on verbal, theoretical, or conceptual information
covered in their university coursework. (Borko et al., 1992, p. 215)
Operating at the Concrete activity and content level, Megan was likely unable to
adapt university practices, including actively/purposefully anticipating SMRs, to
her own teaching in the school. Instead she was focused on finding what she
considered to be student centered tasks. Thus, Megan’s own level of appropriation

potentially impacted her ability to find the intercontextuality and realize the role

of active/purposeful anticipation SMRs in any teaching context.

 

“0 The other intern was teaching at a school that had the reputation among the interns of havrng superior
activities and resources.

230

Finally, looking inward also afforded the opportunity for glimpses into the
classroom, in the form of mini-cases, to see the accommodations Megan made to
address what she anticipated as well as the identification of different levels of
anticipated SMRs. The mini-cases showed that Megan did not always or was not
always able to make appropriate accommodations within her lesson plan to
address what she anticipate, although she anticipated questions or difficulties, and
on rare occasion naive conceptions, in every lesson. Dr. Rundell noticed at the
beginning of the semester that connecting the anticipated SMRs to teacher moves
in their lesson plans was difficult for all of the interns. This study was able to
provide two in—depth pictures of an intern struggling to make this connection.
Another outcome of looking inward on Megan’s anticipation practices was the
beginning of a discussion of different levels of anticipated SMRs. Much of
Megan's anticipating focused on whether students would remember procedures
that had been taught. This kind of anticipating has been present for quite some
time as evidenced by the mathematical songs and mnemonics such as PEMDAS
that exist to help students remember procedures that they may forget.

There was also another kind of anticipating that Megan evidenced when
she considered that students might be confused if she used the word "steeper" in a
different way than she had in the past. This kind of anticipating considers more
carefully students understanding of concepts, versus their memory of procedures.
This anticipation took place for a lesson that was somewhat different than the
normal presentation of examples that took place in the school content. In this

lesson Megan used a worksheet created by her mentor teacher to guide students

231

 

through the graphing of various square root and cube root functions. Though the
activity was rather directive, in this lesson students graphed all of the functions on
their own calculators and Megan’s role was to help them see how changes to the
function resulted in changes to its graph. This particular activity had a slightly

higher level of cognitive demand (Stein, Smith, Henningsen, & Silver, 2000) than

 

past lessons, which only called on students to make use of the procedures with

which they were presented. This might indicate that the differences in Megan’s

 

anticipating were based, at least in part, on the nature of activities within the .
lessons. Perhaps because this activity was less lecture-based, the anticipation was i
also focused on less procedural aspects of the learning. This study highlighted
these differences in what can be anticipated. The next section presents the
implications of the findings of this study and provides recommendations for the

field.
6.2 Implications and Recommendations for the Field

This study investigated the anticipation practices of one student teacher during
lesson planning in two distinct environments, 3 university and school context. Findings
from this study have implications for mathematics teacher education programs. The next
sections outline three implications from this study: (1) Promoting intercontextuality
through noticing, (2) Focusing on the role of the mentor teachers, and (3) Connecting

with other mathematics education communities.

232

6.2.1 Promoting Intercontextuality through Noticing

As discussed in the previous section, intercontextuality did not appear to exist for
Megan between the two contexts, meaning she did not use what she learned about
anticipating SMRs in the university setting in the school setting. This finding is an issue

for teacher education programs which suggests the programs need to find ways to

' my"

.. em”:

promote intercontextuality between the learning and the transfer contexts. Many of the

experiences in Dr. Rundell’s class likely contributed to the interns’ ability to make this

 

 

connection through the focus on the school contexts which were foregrounded in many of

“(huge m.-.“ ~

the class discussions and activities. Another way to do this would be for university
programs to further develop preservice teachers ability to notice as described by van Es
and Sherin (2002).
Specifically, we propose three key aspects of noticing: (a) identifying what is
important or note worthy about a classroom situation; (b) making connections
between the specifics of classroom interactions and the broader principles of
teaching and learning they represent; and (c) using what one knows about the
context to reason about classroom interactions. (van Es & Sherin, 2002, p. 573)
If preservice teachers were able to make the connections and reason about interactions as
van Es and Sherin describe, this would support them, and teacher educators, in framing
the learning context and the transfer context as being linked. One example of how this
was done in the university context was by making the teaching practices of the university
instructor public. In one instance a field instructor present in the class talked to the
students about the detail of Dr. Rundell’s lesson plans and connected his teaching moves

to his lesson plan and to particular topics already discussed in class. In addition to making

their own teaching moves public, methods course instructors could ask preservice

233

teachers to bring in video examples of their own teaching or to watch examples of other
teachers teaching. Star and Strickland (2008) found that through focusing on noticing in a
methods course the preservice teachers’ were able to improve their ability to notice.
Improving their ability to notice, including making connections and reasoning about the
interactions, could help preservice teachers see the relevance of university practices in
their school contexts, even if they are not immediately obvious to the preservice teachers.
The relevance of anticipating SMRs in the school context was evidenced in the
first mini-case in which Megan anticipated students would struggle to visualize a plotted
point in three dimensions and they did. If Megan had a more developed ability to notice,
she might have recognized the importance of anticipating SMRs as well as her own
failure to address what she had anticipated, both practices that were advocated by the
university. In doing so she might have further linked the practice to the school context
and also learned a powerful lesson and made a more deliberate effort to addressed what

she anticipated in her lessons in the future.

6.2.2 Focusing on the Role of Mentor Teachers

The second implication for teacher education focuses on the role of the mentor
teacher. Although the school context provided Megan opportunities and experiences that
the university could not, many facets of the context would not be considered ideal by
many teacher educators. In fact in this case, particular aspects of the experience,
including the mentor teacher and the lack of a greater school mathematics community,
limited Megan’s ability to participate in a practice promoted by the university. As

discussed in Chapters 4 and 5 Megan’s mentor teacher held particular views of teaching

234

 

students and her role as a mentor teacher. Feiman-Nemser (2001) addresses how the
views of a mentor teacher can impact the experience of the preservice teachers.

[Mlentoring has the potential to foster powerful teaching and to develop the
dispositions and skills of continuous improvement. At the same time, mentors may
also perpetuate standard teaching practices and reinforce norms of individualism and
noninterference. How mentors define and enact their role, what kind of preparation
and support they receive, whether mentors have time to mentor, and whether the
culture of teaching reinforces their work all influence the character and quality of l’
mentoring and its influence on novices’ practice. (Feiman-Nemser, 2001, p. 28) "

1n Megan’s case, Mrs. Starns did reinforce some practices that were counter to what the

university intended, which because of her significant role may have had an indelible [-

 

impact on Megan’s development as a teacher, and specifically a planner of lessons. These
practices included lesson planning expectations that focused Megan on her own thinking,
representations, and delivery as opposed to students’ reactions as well as not following
the recommendations of the university in terms of not asking Megan to do extra grading,
making regular time to discuss lesson plans, and giving her control over her own teaching
in the focus class.

Feiman-Nemser (2006) introduces the notion of educative mentoring in
describing the actions of “an exemplary support teacher.” She describes educative
mentoring in the following way:

Educative mentoring rests on an explicit vision of good teaching and an

understanding of teacher learning. Mentors who share this orientation attend to

beginning teachers’ present concerns, questions, and purposes without losing
sight of long—term goals for teacher development. They interact with novices in
ways that foster an inquiring stance. They cultivate skills and habits that enable
novices to Ieam in and from their practice. They use their knowledge and
expertise to assess the direction novices are heading and to create opportunities

and conditions that support meaningful teacher learning in the service of student
learning. (p. 18)

235

In this case the mentor teacher saw her role quite differently than what was described by
Feiman-Nemser and her actions imply that she held different views of good teaching than
the university or those that Megan professed. Building this shared vision of good teaching
is a complex and difficult issue worthy of consideration. Mrs. Starns’ views are not

indicative of the fact that she, or any other mentor teacher with differing views, does not

 

have the best intentions to support their interns. It does mean that if universities desire the
participating mentor teachers to be educative mentors, as Feiman-Nemser describes, they
may need to develop and support mentor teachers in learning to support their interns in i _

this way.

Feiman-Nesmer (2001) discusses specific activities in the learning experiences of
the exemplary support teacher in her article. Central to these activities was this teacher’s
participation in a professional learning community in which he shared his experiences,
received feedback, and worked with other support teachers. This community took place
within a two-year program, implying that one needs an extended period of time to Ieam
to be an educative mentor. Finally, a specific finding from this study indicates these
professional learning communities would need to support mentor teachers in
relinquishing some of their authority to the preservice teachers in their classrooms so that
preservice teachers can explore and participate in teaching practices that reflect their own

beliefs about teaching.

236

6.2.3 Connecting with Other Mathematics Education Communities

Findings from this study indicate that the community of practice described in the
syllabus was created in the methods course:

The blog is designed to accommodate narrative accounts of what’s happening in

our classes, so that we can all understand one another’s classroom contexts. This

fosters a sense of a community of practice, where we engage with one another

around issues of teaching. [Methods Course Syllabus]
However, this type of community did not appear to exist in the school context. The
existing relationships of the mathematics teachers in the school context meant that for
Megan her mentor teacher likely represented the entire mathematics community in the
school. Despite Megan’s direct access to this community, she did not take full advantage
of her mentor teacher’s knowledge or experience in anticipating SMRs. A possible reason
for this is that she did not know how to engage her mentor teacher in conversations that
allowed her access to this knowledge and experience. This might also have been the
result of Mrs. Starns advice to “know your content,” implying that that was something for

Megan to work through and know on her own, and thus not inviting conversation that

problematized the content or the teaching of the content.

237

 

Meijer, Zanting, and Verloop (2002) addressed this particular issue by asking
preservice teachers in several different content to use a concept mapping exercise and
stimulated recall interview to elicit their mentor teacher’s practical knowledge of
teaching. They found that the concept maps provided the preservice teachers’ access to
their mentor teacher’s ideas underlying their teaching and the ways in which the ideas
were related and the stimulated recall interviews allowed the preservice teacher access to
the mentor teacher’s reasons behind specific teaching actions. These types of activities

and others could help preservice teachers gain access to the knowledge and experiences 1 . ,

 

of their mentor teachers and perhaps even other teachers in their school professional El
learning community.

Megan also did not make use of resources provided by the larger mathematics
education community that could have supported her in anticipating SMRs. In her
responses to the Lesson Planning Survey, Megan indicated that in her senior year, when
she taught four lessons in a classroom and two lesson in lab, she almost always read
professional articles about how students might engage with the content she was about to
teach in planning lessons. This was one of the many factors that distinguished her as ideal
and a participant for this study. Despite this senior year practice, findings from this study
indicate that she never did this in preparing her daily lessons during the fall semester of
her internship year. Evidence also suggests that her mentor teacher also did not make use
of these resources.

These findings indicate the need to continue to support preservice teachers to find
ways to make use of the resources available from the greater mathematics education

community. The teaching journals of the National Council of Teachers of Mathematics

238

 

provide information about student learning based both research and the experiences of
other teachers. Experience working with in-service teachers indicates that teachers

sometimes feel that these resources do not apply to their contexts or their students.

Evidence from conversations between Megan and her mentor teacher support the fact that

her mentor teacher felt this way, at least about the university. Based on a debrief
conversation between Megan and her field instructor, Megan went back to her mentor
teacher and questioned her about some of her practices that Megan was expected to adopt
in her teaching. As one example she asked her mentor teacher why the students played
games during class and why these games were not math games. The mentor teacher
responded to this by rolling her eyes and stating that the field instructor had no idea why
she did it. Further, she went on to say “honestly her criticism of me will not affect what I
do in my class" [Informal Conversation, Lesson 1.3 Observation]. Megan’s stated reason
for not making use of professional articles was a lack of time, based on the fact that she
had so much grading to do. Teacher education programs need to further support
preservice teachers in seeing the relevance of these resources in their own classrooms so
that they see the value of using them, even when it means taking extra planning time.

Teacher education programs need to identify further methods for supporting
preservice teachers in connecting and making use of resources outside of the university
so that they can carry those skills into their future schools and help to create professional
learning communities in those contexts. As Dr. Rundell wrote about in the methods
course syllabus,

Many of our former interns who are now teaching in schools report that the thing

they miss most from the internship is the sustained opportunity to meet with their

colleagues and talk about their teaching. The opportunity to share your thinking
and have your ideas challenged by others is a critical part of developing as a

239

—_vm-_—.——;"

 

teacher, and an experience that is often hard to come by in the normal course of
the school day. This year, we will help you develop the skills for learning from
your colleagues, and, perhaps more importantly, the disposition to do so, so that
you can seek out or create these kinds of opportunities throughout your career.
[Methods Course Syllabus]
An example of a resource and a professional learning community can be found in the
work of Herbal-Eisenmann and Cirillo (2009) who present stories from teachers about
their own learning about their mathematics teaching as part of participation in a larger

professional development program. The next section describes directions for future

research based on the findings of this study.

 

6.3 Future Research

While this study succeeded in shedding light on a particular case of a preservice
teacher anticipating SMRs, the findings also point to areas of need for future research.

The following sections detail the next steps in this line of inquiry for the researcher.

6.3.1 Investigating Necessary Conditions for Preservice Teachers Anticipating SMRs

Findings from this study indicate that the school context negatively influenced
Megan’s practices of anticipating SMRs. Future research should focus on identifying the
conditions needed in a school context to promote anticipating. These studies might take
the form of comparisons. Megan was purposefully selected for this study because she
showed ability and motivation to anticipate SMRs in her senior year methods courses.
Future research might investigate the practices of a preservice teacher who did not show
ability and motivation to anticipate SMRs and was paired with a mentor teacher that both

expected and modeled the anticipation of SMRs in their lesson planning. Studies

240

comparing preservice teachers’ anticipation practices might also include field experience
classrooms with different styles of mathematics teaching. For example, do classrooms in
which students actively and frequently participate and come up with a variety of solution
strategies encourage preservice teachers to anticipate SMRs? An additional feature of the
school context that should be investigated for its impact on preservice teachers is the

availability of resources that provide information to help the preservice teachers to F

4“
_w

anticipate. These resources include curriculum materials and the mathematics teaching

community present at the school. Given the importance of field experiences, future

 

VI-

research needs to focus on identifying the features of school contexts that are necessary

and have the greatest impact on preservice teachers’ anticipation practices.

6.3.2 Investigating the Role of Experience and Expertise

There is no question that experienced and expert mathematics teachers know
different things and teach in different ways than novice teachers. Brown and Borko
(1992) reviewed studies that identified the specific differences between expert and novice
mathematics teachers. They found that:

Expert teachers displayed more pedagogical knowledge, content knowledge, and

pedagogical content knowledge. . .their conceptual systems, or cognitive

schemata, for organizing and storing this knowledge are more elaborate,
interconnected, and accessible...and are more efficient than novices in their
processing of information during both planning and the interactive phases of

teaching. (p. 213)

This knowledge and experience might be viewed as useful in the practice of anticipating
SMRs. However, despite these differences, Borko and Livingston suggest from the

findings of their study that direct experience teaching the content would support a teacher

in anticipating SMRs more than their expertise and more general experience.

241

Several of the difficulties the novice teachers encountered, such as the time-
consuming nature of their planning and the inability to anticipate student
problems, are difficulties encountered by most teachers the first time they teach a
course or body of knowledge. Any teacher will think and act like a novice, to
some extent, the first time he or she attempts to teach a particular body of
knowledge. (Borko & Livingston, 1989, p. 489)

Future research should investigate the role of teaching experience and expertise in

 

anticipating SMRs. Important in this type of investigation is the criteria for the P
identification of expert teachers. The expert teachers in the studies reviewed by Brown

and Borko (1992) were identified as experts in a variety of ways. In one study, expert

teachers were identified as teachers having five or more years of teaching experience, g3

while another defined expert teachers as those whose student test scores showed
consistent growth. Finally, one studied characterized expert teachers as mentor teachers
who had been recommended by building administrators. It will be important for future
research in this area to identify “expert” teachers as those with different knowledge, as
described by Brown and Borko (1992), than the novice teachers in the study. Also crucial
for this type of study would be the choice of participants who must regularly participate
in the practice of anticipating SMRs. Using this, future research needs to investigate the
differences in what is anticipated and the anticipation practices of novice teachers, expert
teachers without experience teaching particular content, and expert teachers with
experience teaching particular content. This comparison will allow for a close look at the
practice of anticipation and enable researchers to tease out the influence of different types

of experience and knowledge on the practice of anticipating.

242

6.3.3 Investigating the Role of Content Knowledge

A final area of focus for future research is the role of content knowledge in
anticipating SMRs. The current research framework based the practice of anticipating
SMRs on teachers’ knowledge of content and students, a primary element of pedagogical
content knowledge. Hill, Ball, and Schilling (2008) define knowledge of content and l“
students as “knowledge of content intertwined with knowledge of how students think .l
about, know, or Ieam this particular content” (p. 375). Future research needs to

investigate the role of other types of knowledge for teaching mathematics, specifically

 

content knowledge, in mathematics teachers’ practices of anticipation.

Findings from the current study suggest that a lack of content knowledge might
prompt teachers to anticipate SMRs. In the Lesson Planning Survey the majority of the
preservice teachers responded that anticipating SMRs was an important practice for
teachers in order to be prepared to teach lessons. Thus feeling unprepared due to lack of
content knowledge might cause a teacher to anticipate SMRs even more than they
normally would.

A different perspective suggests that lack of content knowledge might prevent a
teacher from anticipating because it may prompt them to change how they teach the
mathematics, and therefore the way that they plan their lessons. As Leinhardt (1993)
explains, “as the subject matter becomes more complex teachers tend to reduce their
attention to the individual learner and his or her needs or responses” (1993, p. 6). Future
studies need to investigate the role of mathematical content knowledge, as another type of
knowledge needed to do the work of teaching, in what teachers anticipate and the

anticipating practices in which they participate.

243

 

Appendices

244

 

Appendix A.

Weekly Mathematics Problem Discussion Guidelines

You are responsible for preparing a class discussion of the homework problem
assigned to you. In most cases, the problems will be given as homework for the class to
have ready by the assigned date. One of the pair will be responsible for the discussion,
which should take no more than 30 minutes. The other will organize a 15-20 minute
discussion about the misconceptions students may bring to the mathematics in the
problem. Managing time in a class is important and to help you Ieam about pacing, we
will signal you when you have five minutes left. The instructors will be responsible for
making the problems available to the class and in most cases, the problems will be posted
on Angel.

The “homework time” can be conducted using a variety of instructional strategies
— use your imagination and choose a strategy that seems to make sense for the problem.
To help you think of possible approaches, we ask that you meet with one of the
instructors prior to the class to process how each of the two parts might be conducted and
to send a short description of how you intend to proceed to the instructors on the Monday
preceding the problem presentation. You should include the mathematical objective you
want to make, the questions you want to ask, things you want the class to do, how you are
going to record any discussion, and the summary statement you will make. Your
description should include any materials you might need — flip chart paper, post it notes,
graphing calculators, markers, overhead transparencies, etc.

One of the key ideas in this set of algebra problems is the use of a multiple

strategies to help students develop different ways to think about doing mathematics. It is

245

important to have students share their work and to talk about the strengths and
weaknesses of the different approaches. Try to think of ways to anticipate responses
from the class and how you will process the sharing to maximize the time you have for

discussion. Pay attention to the pacing (remember the 30 minutes).

246

 

Appendix B.

Lesson Planning Format

Lesson Logistics and Setting
Unit Topic:

- Previous Lesson Topic:

- Current Lesson Topic:

- Next Lesson Topic:
Lesson Objectives:

The learner will know/understand/be able to
(choose one and complete the sentence)

- Standards Addressed:

How will I know students have met the objectives?

Lesson Activities (On next page/Attach any handouts you will use)

Homework:

247

 

 

 

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248

Appendix C.
Mentor Teacher Initial Interview Protocol
This is (name of interviewer) interviewing (name of interviewee) on (date). This
is an initial interview.
Thank you for participating in the interview today. The purpose of the interview is
for me to gain insight into how you view certain aspects of the mathematics
education environment at (name of school).

I’d like to talk now about the school.

 

0 Tell me about (name of school). E
o What are the students like?
0 What classes do you teach?
I How long have you been teaching each of the classes?
Is this your first time being a mentor?
What do you know about Megan’s courses at MSU?
I’d like to talk more now about Megan. What will a typical day look like for
hAegan?
Could you talk about how you envision the year proceeding with Megan?
o What do you expect from Megan as your intern?
I In terms of lesson planning?
I Will you plan lessons together? Why or why not?
I In terms of teaching?
I What other kinds of expectations do you have?

0 In what ways do you envision helping Megan meet those expectations?

249

0 How would you describe the philosophy of the mathematics department in your
school?
0 How is it similar to or different from what happens in your classroom
when you are teaching?
o Is there anything else you’d like to say about the math department at
your school?
0 What interactions do you have with other math teachers in your school?
0 How are decisions made about what gets taught?
0 Is there anything else you’d like to say about the environment at (name of

school)?

General prompts for elaboration on ideas that the teacher brings up:
0 Would you say more about [use teacher’s own words here]?
0 Would you say more about what you mean by [use teacher’s own words here]?
0 Would you give me an example [of that/what you mean by teacher ’s own words]?
0 If the teacher uses a word or phrase that you ’re not familiar with: I’m sorry, I’m

not familiar with what is. Would you tell me more about it?

250

 

Appendix D.

Field Instructor Initial Interview Protocol

0 How do you see your role as field instructor?
0 What are your goals for the interns that you will work with this year?
0 If some goals related to lesson planning are mentioned further prompt,
Can you tell me more about your goals with respect to what the interns
will learn about lesson planning from working with you during this

expefience?

 

o If lesson planning is not mentioned, Can you tell me about any goals that
you have with respect to what the interns will Ieam about lesson planning
from working with you during this experience?

0 What are your expectations for the lesson plans that Megan writes?

o What do you know about Megan's internship school?

0 What do you know about Megan's cooperating teacher?

0 How do you plan to work with Megan's cooperating teacher to achieve the goals
you mentioned?

0 How do you plan to work with Megan, either individually or with a group of

interns, to meet the goals and expectations you mentioned?

General prompts for elaboration on ideas that the teacher brings up:
0 Would you say more about [use field instructor's own words here]?

0 Would you say more about what you mean by [use field instructor’s own words

here]?

25]

0 Would you give me an example [of that/what you mean byfield instructor’s own
words]?

I If the field instrzu'tor uses a word or phrase that you ’re not familiar with: I’m
sorry, I’m not familiar with what is. Would you tell me

more about it?

 

 

17
37‘
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252

Appendix E.

Methods Instructor Initial Interview Protocol

0 This is (name of interviewer) interviewing (name of interviewee) on (date). This
is an initial interview.

0 Thank you for participating in the interview today. The purpose of the interview is
for me to gain insight into how you view certain aspects of mathematics education
environment at (name of school).

0 What are your goals for the methods course?

 

o If some goals related to lesson planning are mentioned further prompt,
Can you tell me more about your goals with respect to what the interns
will learn about lesson planning from participating in the course?

0 If lesson planning is not mentioned, Can you tell me about any goals
that you have with respect to what the interns will Ieam about lesson
planning from participating in the course?

0 If it is not yet been mentioned, How does this course relate to the field
expenence?

o What opportunities do interns have to work cooperatively as part of this course?
0 What are the major course assignments for the fall semester?
0 How many lesson plans will be turned in for grading or feedback for this course?

0 What are your general requirements (not specific to any given
assignment) for lesson plans?

0 Do you plan to grade lesson plans that are turned in? If so, how do you

grade them? If not, why not?

253

 

0 Do you plan to provide feedback on lesson plans that are turned in? If
so, what kind of feedback do you provide? If not, why not?
I How will you introduce the idea of anticipating students’ mathematical responses
to the class?
0 When will you introduce anticipating?

o What resources will you direct interns to use to anticipate?

-. .. .3]

o What resources are generally available to interns to use to anticipate?

Where are these available?

 

in:

. . . l
o Are there aspects of assrgnments either in class or out of class, other 3
than lesson planning, that may support the interns in learning to
anticipate SMRs?

- Is there anything else that you would like to tell me about the methods course?

General prompts for elaboration on ideas that the teacher brings up:
0 Would you say more about [use instructor’s own words here]?
0 Would you say more about what you mean by [use instructor’s own words here]?
0 Would you give me an example [of that/what you mean by instructor ’s own
words]?
0 If the instructor uses a word or phrase that you 're not familiar with: I’m sorry,
I’m not familiar with what is. Would you tell me more

about it?

254

Appendix F.

Lesson Planning Survey

Part 1
[To be administered separately from Part 2]

Imagine that you are back in your senior year methods course and you are planning a
lesson to teach in your field experience classroom. What would you do to plan this

lesson? Make a numbered list that details the steps you would take to prepare this lesson

before you teach it. Be as specific as possible.

Part 2

I. When I planned a lesson in my senior year methods course I...

 

Always Almost Sometimes Hardly Never

Always Ever

WWI-fl
”Ml-M

 

a. made sure I had a good
night of sleep so that I

could think clearly about
the lesson

 

Eb. worked the mathematical
\ task or solved the
l
I

problem(s) I planned to
give to students

 

meeting with an
instructor to discuss my
‘ ideas about the lesson

I

\c. arranged a one-on-one

plan

 

sd. consulted the teacher

 

] guide for possible areas

] where students might

I struggle or solution
strategies they might ‘

come up with f

 

 

 

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255

 

 

vim—uk _ -

- wa— . rust—‘1'-

. Always

...-.--.a~

Almost Sometimes Hardly Never

 

Always Ever

.._.—__..

ge - read articles about how

students might engage
with the content I was
about to teach

 

 

 

 

if- videotaped myself i

teaching the lesson and
then watched to see what
I could do better

.g. asked my mentor teacher
5 about their experiences
teaching the content

 

 

h wrote a script indicating
all the words I wanted to
i say in the lesson

 

 

it. thought about the
learning needs of
particular students in the
class (i.e. Will this lesson
make sense for Shannon
in my 3rd period class?)

planned how I would use
the blackboard during
the lesson

_ __... a —
”Mn—«MW -

___ ..— M~u.
I
O

 

 

 

 

 

 

 

 

Consider parts a through j in question 1 above. Do you plan to do any of these
activities differently in your upcoming internship year than you during your senior
year field experience? State which ones you would do differently and why. If you
would do the same things you did during your senior year explain why.

How do you choose tasks, activities, or problems to teach the mathematics of your

lessons? What kinds of tasks, activities, or problems do you tend to choose? Why do

you choose these?

256

4. When you present examples to students in class to teach the lesson, how do you

Choose what examples to use?

5 - (a) The practice of carefully considering students possible solution strategies, nai've

conceptions (misconceptions), difficulties, and questions while planning your lessons

is encouraged by your teacher preparation program. Why do you think the program

encourages this?

(b) Do you agree that this is an important practice? Why or why not? If you do, how

do you try to do this?

 

6. Have you taught a lesson about angle measure before? Cl Yes Cl No

‘14:...1- .-

Have you discussed teaching angle measure in an education class before? Cl Yes C]

No

Imagine that you are going to teach a lesson in which your goal is for students to
Ieam to measure angles with a protractor. What are some possible misconceptions

students might have about the concept of angle measure?
What are some possible problems students might have using a protractor?

What other difficulties, misconceptions, or possibly correct ideas do you think

students might have about angle measure?

257

wifl—u'fl

 

Appendix G.
ln-Depth Lesson Interview 1 Protocol
(Prior to the interview, the preservice teacher will complete the Lesson Planning Survey.

In addition she will be asked to bring a completed lesson plan to the interview that she
has prepared for her focus class but not yet taught.)

INTRODUCTION.
This is (YOUR NAME) interviewing (TEACHER’S NAME) on (DATE) and this

is the first in-depth interview.

Thank you for participating in this interview. The purpose of this interview is for

me to understand your current thinking on lesson planning.

 

 

Time when Interview was started:

 

PART 1. Semi-structure interview based on the intern’s lesson plan”.
0 Could you walk me through the plan you have come up with?
Lesson Background
0 What are your goals for the lesson? What mathematical content and processes do
you hope students will Ieam from their work on this task?
0 What task or activity is the basis for this lesson?
0 Why did you choose this specific task?
0 How will this task or activity help students meet the objective?
0 How will you group students to participate in this activity?

0 Why did you choose this grouping?

 

2' This interview has been adapted from an assignment used by Mossgrove (2006)

258

 

 

0 What will your role be during this lesson?
0 How does this lesson connect to what came before and build toward what comes
after mathematically?
Account of Lesson Planning Process
0 What were the steps that you took in planning the lesson for today?

0 What resources did you consult? (mentor teacher, other preservice
teachers, curriculum materials, journal articles, methods course instructor,
field instructor)

I Why did you search for those resources?
I What did you learn from these resources?
0 How did that impact your lesson?
0 If none, why not?
0 Did anything happen in class yesterday or recently that made you change your
lesson for today?
Selecting and Setting up a Mathematical Task
0 In what ways does the task build on students’ previous knowledge? What
definitions, concepts, or ideas do students need to know in order to begin to work
on the task?
0 Have you solved the task yourself?
0 What are all the ways the task can be solved?
0 Which of these methods do you think your students will use?
0 What misconceptions might students have?

0 What errors might students make?

259

 

 

If the preservice teacher included anticipated student responses ask
I noticed that you have included student (strategies, questions, misconceptions,
difficulties).

0 Why did you include them?

0 How did you think of them? What resources did you use?

0 If you could have had resources available to help you think about student
responses what would you have wanted?

If the preservice teacher did not include anticipated student responses ask
I noticed that you have not included student (strategies, questions,
misconceptions).

0 Why did you choose not to include them?

0 If you were going to include student strategies, questions and
misconceptions in this lesson plan how would you go about figuring out
what they might be?

How will you ensure that students remain engaged in the task?

0 What will you do if a student does not know how to begin to solve the
task?

0 What will you do if a student finishes the task almost immediately and
becomes bored or disruptive?

o What will you do if students focus on non-mathematical aspects of the
activity (e.g., spend most of their time making a beautiful poster of their
work)?

What are your expectations for students as they work on and complete this task?

260

 

 

o What resources or tools will students have to use in their work?
0 How will the students work -- independently, in small groups, or in pairs -
- to explore this task? How long will they work individually or in small

groups/pairs? Will students be partnered in a specific way? If so in what

way?
0 How will students record and report their work? I
l
0 How will you introduce students to the activity so as not to reduce the demands of

the task? What will you hear that lets you know students understand the task?

 

Supporting Students’ Exploration of the Task
0 As students are working independently or in small groups:

0 What questions will you ask to focus their thinking?

0 What will you see or hear that lets you know how students are thinking
about the mathematical ideas?

0 What questions will you ask to assess students’ understanding of key
mathematical ideas, problem solving strategies, or the representations?

0 What questions will you ask to advance students’ understanding of the
mathematical ideas?

0 What questions will you ask to encourage students to share their thinking

with others or to assess their understanding of their peer’s ideas?

26]

Sharing and Discussing the Task
0 Which solution paths do you want to have shared during the class discussion in
order to accomplish the goals for the lesson?
0 Which will be shared first, second, etc.? Why?
0 In what ways will the order of the solution paths help students make
connections between the strategies and mathematical ideas?

0 What will you see or hear that lets you know that students in the class understand

 

the mathematical ideas or problem-solving strategies that are being shared?
0 How will you orchestrate the class discussion so that students:
0 make sense of the mathematical ideas being shared?
0 expand on, debate, and question the solutions being shared?
0 make connections between their solution strategy and the one shared?
0 look for patterns and form generalizations?
o What extensions to the task will you pose that will help students look for patterns,
make connections or form a generalization?

o Is there anything else you would like to say about your lesson plan?

PART 2. Semi-structured interview based on Secondary Mathematics Intern Pre-Field

Experience Survey responses

Now that we have talked about how you planned a particular lesson I would like to

discuss some of your responses to the survey which told me more about how you plan

262

lessons in general. I am going to ask you some questions about your responses on that
survey.
Steps in Writing a Lesson Plan
Reference the list she made for Part I of the survey
0 Can you say more about (anything that is unclear or interesting)?
0 What do you mean by (term they used)?
0 If doing the task or activity is not listed but is included in question I from Part2
ask

0 Are there times when you do the tasks or activities before you teach and

 

times when you do not?
I (If yes) Under what conditions do you choose to do the task or
activity?
I (If no) Why do you think you did not list it as one of the steps in
your lesson planning process?
0 Ask similar questions for (a) consulting the teacher guide, (b) reading articles,
and (c) asking the mentor teacher.
Questions about Responses to Part 2
(General)
0 Can you say more about (anything that is unclear or interesting)?
0 What do you mean by (term they used)?
(Question 2)

0 Can you explain more about why you will do _ activity differently?

263

(Question 8)

0 How do you think the problems, misconceptions, and ideas you have identified

will impact the teaching of a lesson about angle measure?

0 0k, great. Thank you very much for participating in this interview.

 

 

Interview end time:

 

 

- ‘.!’-—. — _fl
3.

-...

 

264

Appendix H.
Pre- and Post-Lesson Interview Protocol
PRE-LESSON INTERVIEW
(Ask that she bring all materials to interview including lesson plan, notes, activities and
tasks)
Lesson Background
0 What are your goals for the lesson? What mathematical content and processes do
you hope students will learn from their work on this task?
0 What task or activity is the basis for this lesson?
0 Why did you choose this specific task?
0 How will this task or activity help students meet the objective?
0 How will you group students to participate in this activity?
0 Why did you choose this grouping?
0 What will your role be during this lesson?
0 How does this lesson connect to what came before and build toward what comes
after mathematically?
Account of Lesson Planning Process
0 What were the steps that you took in planning the lesson for today?
0 If not mentioned as part of the steps, Did you complete this task yourself?
0 What resources did you consult? (mentor teacher, other preservice
teachers, curriculum materials, journal articles)
I Why did you search for those resources?

I What did you Ieam from these resources?

265

 

 

 

o How did that impact your lesson?
0 If none, why not?
Focusing on Anticipating
Refer to the lesson plan and other materials is anticipation has been done in those
documents
0 Do you think that there is more than one way students could approach/solve the
task or activity?
0 (if yes) What are they?

0 (if no) Why not?

 

- What questions do you think students might have during this task or activity?
0 What difficulties do you think students might have during this task or activity?
0 What misconceptions do you think might arise?

General Anticipating
0 How do you think the students in this class will receive the task/activity?
- Did your teaching or student responses yesterday or earlier this week have any

impact on your lesson planning?

POST-LESSON INTERVIEW
I Could you reflect on the lesson and tell me how you think it went?
0 What was the student reaction to the lesson?

0 Do you think the students accomplished the objective(s) you set forth?

 

0 What happened that surprised you?

266

0 Will you change anything about what you were going to do tomorrow as a result
of what happened today?

0 If so, what?

 

267

 

Appendix I.
In-Depth Lesson Interview 2 Protocol

( Prior to the interview, the intern will be asked to bring a completed lesson plan to the
interview that she has prepared for her focus class but not taught yet.)

INTRODUCTION.

This is (YOUR NAME) interviewing (TEACHER’S NAME) on (DATE) and this

is the second in-depth interview.

Thank you for participating in this interview. The purpose of this interview is for
me to understand your current thinking on lesson planning. There will be three

parts to this interview.

 

 

Time when interview was started:

 

 

PART 1. Semi-structured interview about what the preservice teacher thinks should be
in a lesson plan
0 For the first part of the interview, I’d like you to take a minute to write down the

things you believe you should think about when planning a mathematics lesson,
or what you would include in a lesson plan for a mathematics class. Then I’m
going to ask you to tell me about them.

(Give the teacher about a minute or two to write down some thoughts - keep the

recorder running, the purpose is to allow a brief moment for individual think time before

they have to start talking).

0 Ok. I’d like for you to tell me about the things you believe you should think about

268

 

 

when planning a mathematics lesson.

Use the general probes below to ofifer the preservice teacher an opportunity to provide
more specificity or clarify their descriptions.

0 Can you say more about (item that is unclear or brief)?

0 What do you mean by (term they used)?

0 Why did you include (term they used)?
Probe into difi‘erences between the preservice teachers initial numbered list from the
survey to her response to this prompt, specifically differences related to anticipating

students ’ mathematical responses.

PART 2. Discussing the teacher’s lesson planning practices during the first semester of
the internship
0 For the second part of the interview, I’d like to ask some questions that relate to
your lesson planning in general. I’d like you to talk about the things that influence
your planning.
0 What role does your textbook or curriculum play in your planning?
0 How do you use your textbook or curriculum when you plan?
0 Does the textbook or curriculum influence your planning in any way? If
so, In what ways?
0 Do you use your textbook to help you anticipate students’ mathematical
responses?

I What role does your mentor teacher play in your planning?

269

 

What are your mentor teacher’s expectations for your lesson plans?

Do you discuss your lesson plans with your mentor teacher?

Have you planned lessons together?

What kinds of things have you discussed with your mentor teacher, with

respect to lesson planning?

 

Has your mentor teacher shared experiences with you that have helped F?
l

you anticipate students’ mathematical responses when you plan?

Has your mentor teacher helped you find resources to anticipate students’

mathematical responses? If so, what kinds of resources? i

o What role do the other interns play in your planning?

0

0

Do you discuss your lesson plans with other interns?

Have you planned lessons together?

What kinds of things have you discussed with other interns teacher, with
respect to lesson planning?

Have other interns shared experiences with you that have helped you
anticipate students’ mathematical responses when you plan?

Have other interns suggested resources to anticipate students’

mathematical responses? If so, what kinds of resources?

0 What role does your field instructor play in your planning?

0

What are your field instructor’s expectations for your lesson plans?
Do you discuss your lesson plans with your university supervisor?
Have you planned lessons together?

What kinds of things have you discussed with your university supervisor,

270

with respect to lesson planning?
0 Has your field instructor shared experiences with you that have helped you
anticipate students’ mathematical responses when you plan?
0 Has your field instructor helped you find resources to anticipate students’
mathematical responses? If so, what kinds of resources?
0 What role does your methods course play in your planning? T
o What are your methods instructor’s expectations for your lesson plans?

0 Do you discuss your lesson plans with the instructor of this course either i"

 

one-on-one or in class? H

o What kinds of things have you discussed in your methods course, with
respect to lesson planning?

0 What elements of the methods course have helped you find resources to
anticipate students’ mathematical responses?

0 In what ways are the lesson plans you provide for your university
supervisor and methods course similar and different from those you
usually produce?

0 What other things influence your planning?
Move on only after teachers have offered as many factors as they can. (these could
include such things as: time constraints (either in the time they have to devote to
planning or in the time they have to teach something), things they are leaming/doing in
their teacher education program, their beliefs about what it means to Ieam and do
mathematics and about students, resources available, parents, students, etc).

0 Do you believe your planning has changed in any ways, over the course of this

27]

semester?
If yes, then
0 Can you describe the ways in which your planning has changed?
0 Are there any other ways in which you believe your planning has
changed?
If no, then
0 Why not?
0 Were there any areas that you thought about changing but did not?
0 Is there anything else you would like to say about your lesson planning?
PART 3. Semi-structure interview based on the intern’s lesson planzz.
0 Could you walk me through the plan you have brought with you?

Lesson Background

 

- What are your goals for the lesson? What mathematical content and processes do

you hope students will learn from their work on this task?
0 What task or activity is the basis for this lesson?
0 Why did you choose this specific task?
0 How will this task or activity help students meet the objective?
0 How will you group students to participate in this activity?
0 Why did you choose this grouping?

o What will your role be during this lesson?

 

22 . . . . .
This mtervrew has been adapted from an assignment used by Mossgrove (2006)

272

0 How does this lesson connect to what came before and build toward what comes

after mathematically?

Account of Lesson Planning Process
0 What were the steps that you took in planning the lesson for today?

0 What resources did you consult? (mentor teacher, other preservice
teachers, curriculum materials, journal articles, methods course instructor,
field instructor)

I Why did you search for those resources?
I What did you Ieam from these resources?
0 How did that impact your lesson?
0 If none, why not?
0 Did anything happen in class yesterday or recently that made you change your
lesson for today?
Selecting and Setting up a Mathematical Task
0 In what ways does the task build on students’ previous knowledge? What
definitions, concepts, or ideas do students need to know in order to begin to work
on the task?
0 Have you solved this task yourself?
0 What are all the ways the task can be solved?
0 Which of these methods do you think your students will use?
0 What misconceptions might students have?

0 What errors might students make?

273

 

If the preservice teacher included anticipated student responses ask
I noticed that you have included student (strategies, questions, misconceptions,
difficulties).

0 Why did you include them?

0 How did you think of them? What resources did you use?

0 If you could have had resources available to help you think about student
responses what would you have wanted?

If the preservice teacher did not include anticipated student responses ask
I noticed that you have not included student (strategies, questions,
misconceptions).

0 Why did you choose not include them?

0 If you were going to include student strategies, questions and
misconceptions in this lesson plan how would you go about figuring out
what they might be?

How will you ensure that students remained engaged in the task?

0 What will you do if a student does not know how to begin to solve the
task?

0 What will you do is a student finishes the task almost immediately and
becomes bored or disruptive?

o What will you do if students focus on non-mathematical aspects of the
activity (e. g., spend most of their time making a beautiful poster of their
work)?

What are your expectations for students as they work on and complete this task?

274

'lfl“. N. l —.v_—‘

 

 

:5- III
I

 

o What resources or tools will students have to use in their work?
0 How will the students work -- independently, in small groups, or in pairs —
- to explore this task? How long will they work individually or in small
groups/pairs? Will students be partnered in a specific way? If so in what
way?
0 How will students record and report their work?
0 . How will you introduce students to the activity so as not to reduce the demands of

the task? What will you hear that lets you know students understand the task?

 

Supporting Students’ Exploration of the Task
0 As students are working independently or in small groups:

0 What questions will you ask to focus their thinking?

0 What will you see or hear that lets you know how students are thinking
about the mathematical ideas?

0 What questions will you ask to assess students’ understanding of key
mathematical ideas, problem solving strategies, or the representations?

0 What questions will you ask to advance students’ understanding of the
mathematical ideas?

0 What questions will you ask to encourage students to share their thinking

with others or to assess their understanding of their peer’s ideas?

275

Sharing and Discussing the Task
0 Which solution paths do you want to have shared during the class discussion in
order to accomplish the goals for the lesson?
0 Which will be shared first, second, etc.? Why?
0 In what ways will the order of the solution paths help students make
connections between the strategies and mathematical ideas?
0 What will you see or hear that lets you know that students in the class understand
the mathematical ideas or problem-solving strategies that are being shared?
0 How will you orchestrate the class discussion so that students:
0 make sense of the mathematical ideas being shared?
0 expand on, debate, and question the solutions being shared?
0 make connections between their solution strategy and the one shared?
0 look for patterns and form generalizations?
o 'What extensions to the task will you pose that will help students look for patterns,
make connections or form a generalization?

0 Is there anything else you would like to say about your lesson plan?

276

‘7‘1’

.1, .,
I..i'r.‘.fl.~

 

 

Appendix J.
Cool Stuff 1 Assignment Sheet

Cool Stuff 1
Plan, Teach, Reflect for Guided Lead Teaching 1

Executive Summary

You will select and design a lesson for students that is “cool.” (We will discuss what it
might mean for stuff to be cool.) The first part of the assignment involves selecting a
mathematical task of high cognitive demand, identifying the lesson objectives, and
connecting those objectives with evidence of your students’ learning that you will collect.
The second part of the assignment builds on the first and involves seeking information
from educational literature to inform your analysis of student learning.

The products for this assignment will be:

0 The task that you select and implement
A lesson plan for the task using the revised lesson plan template

0 A Case-Stories in-class presentation of your lesson plan and the proposed evidence of
learning

0 Evidence of student learning in the form of written work and/or other classroom
artifacts (e. g., video) .

I A Plan-Teach-Reflect paper summarizing the lesson and analyzing student learning

Purpose
The purposes of the Cool Stuff assignments are to give you an opportunity to:

(1) develop a better understanding of what makes a good instructional task

(2) practice both selecting resources and creating your own resources for students
[For Cool Stuff I you should focus on selecting (and modifying if necessary)
an existing task. For Cool Stuff 2 you should focus on creating a new task or
substantially modifying an existing task.)

(3) develop skills for connecting your lesson’s learning objectives to previous
outcomes

(4) gain experience using educational research resources in lesson planning

(5) Ieam about using formative and summative assessment in ongoing instruction

(6) allow you to share quality instructional materials with your colleagues in
class.

The assignment has two parts and each one emphasizes the assignments goals in different
ways. In the first part, you will focus on the selection of a task of high cognitive demand,
and on connecting the objectives you write with evidence of your students’ learning. In
the second part, you’ll build on what you learned in the first part by creating a new task

277

 

 

 

and seeking out information from educational research literature to inform the analysis of
your students’ learning.

This assignment is called “Cool Stuff” because the stuff you are to select or design
should be, in your opinion, cool. It is likely, however, that we have differing ideas about
what exactly makes stuff cool. By the end of this assignment, though, we should be able
to be much clearer about what we mean when we say something is cool to us. That way,
we can be clearer about what kinds of activities we would like to make part of our
classes, and why we believe these kinds of activities are good.

 

 

 

278

Procedure

(1) Select a task.
Many cool tasks are already in existence and are readily accessible on the
lntemet, as well as in volumes of published materials — perhaps in your very
own textbook. A number of materials available in our math ed classroom (e.g.,
textbooks, the Navigations series) may be useful. For Cool Stuff I, you will
select an existing task to focus on. The task you select for Cool Stuff 1 should
be one that you plan to use during Guided Lead Teach l.

(2) Prepare a lesson plan.
Before teaching the lesson you’ll develop an accompanying lesson plan for
the day you use your task. During one class we will share Cool Stuff Case
stories, during which time you will be able to see the tasks your classmates are
planning to use and give and receive feedback. In preparing your lesson plan,
you should consider what evidence of student learning you will collect. This
tentative plan will be discussed during the Case Stories.

 

(3) Teach the lesson and gather documents of student thinking
After you’ve received comments from your colleagues, revise your task or
plan if needed. Use the task in a lesson you teach. Enlist help from your
mentor or F1 to collect written work, audio, video, photos, and/or field notes
that capture what happened during your lesson. The use of video is strongly
encouraged for this task, assuming you have the appropriate permissions.

(4) Analyze the results, and reflect on what happened

After teaching the lesson using the task, you’ll use all the evidence you
collected to try to construct arguments about what your students know,
relative to your objectives for the lesson. You’ll clarify what you hoped would
be cool about the task and analyze whether the task turned out to be cool in
the end. You will also note things you think you could have done to make the
lesson more successful, in terms of either student learning of the objectives or
other aspects of the “coolness” of the task. For this portion of the task, you
will use the Plan, Teach, Reflect write-up format. You are strongly
encouraged to include artifacts of student work, particularly in the form of
video clips or reproduced (and blinded) student work.

Requirements

As with any assignment in this course, is any of the requirements seem to interfere with
your ability to make use of this assignment in your teaching, please let me know before it
is due and we will negotiate something mutually acceptable.

For Class 3

You will have work time in class to explore the print and online resources we have
available in order to select a task. You may want to bring your own textbook from the

279

 

class in which you intend to use the task. Course instructors will be available to consult
with, along with your peer groups.

For Class 5

Write a plan for your Cool Stuff I using the lesson plan template. The focus for Cool
Stuff I is on developing your ability to use artifacts of students’ work to construct
arguments about whether students have achieved your objectives. In order to prepare for
this, pay very close attention to writing good objectives, and the relationship between
your objectives and the assessment artifacts you will collect.

Bring 6 copies of your task and plan to class for the Cool Stuff Case Stories.
On the day you teach the lesson

Collect documentation of your use of the task in class. This could include student work,
transcribed audio or videotape of the class, and photos of students working on the task.
Your documentation should focus on what students did and thought during the task. A
combination of these things might do the best job of conveying both a sense of your
students’ involvement in the task as well as what your students were thinking during the
work on the task (e.g., student work alone probably won’t tell a rich story). Involve your
mentor of F1 in helping you collect this data.

Record your own reflection on the lesson as soon as possible. This might include making
note of any last-minute or on-the-fly changes to your plan and the reasons for making
them. It might also be helpful to write down anything you saw or heard from students that
might not be captured in your data.

After you teach

Write your analysis paper in the Plan-Teach-Reflect format. This paper should contain
two major themes: your analysis of the extent to which your lesson helped you achieve
the objectives you had for your students, and your thoughts on whether or not this task
was cool.

In the Plan section, you should describe how you selected your task, why the task would
be considered both high-level and cool, and how you planned for the lesson to go. This
should include information on how you set up the task, how you organized students, what
questions you intended to ask and why, and how you intended to bring the lesson to
closure and discuss the big ideas.

The Teach section should be a concise summary of how the lesson went. This is a great
place to include student work artifacts and/or video clips.

The Reflect section should first analyze student learning. This should include the extent to
which your students met the objectives and refer back to evidence from the Teach
section. The next part of this section should describe whether you thought this task was
cool: Was it cool for your students? Was it cool for you as a teacher? What made it cool?

280

 

Use evidence from your documentation to support your claims. Finally, reflect on what
you learned from this experience, again using evidence from your documentation.

Assessment

You will be assessed using the rubric on the next pages. One particular aspect of the
rubric to which we will be paying particular attention to the use of evidence. You should
include actual artifacts of classroom practice — student work, pictures of artifacts on the
board or chart paper, a set of video clips — to help make an argument about what students
learned and whether or not the activity was “cool.”

Your in-class presentation of the case stories will also be assessed as a part of your final
grade (related to the Plan section of the rubric).

28]

 

 

Plan, Teach, Reflect Rubric

Plan
Mathematical Goal 1 Point
1 pt An appropriate math goal is included

0 pts Math goal is inappropriate or not included

Building on Prior Knowledge 2 Points
2 pts Prior knowledge that students will have is identified and connected to the
mathematical task and the mathematical goal

1 pt Prior knowledge that students will have is identified, but connections to
the mathematical task and the mathematical goal are weak or unspecified

0 pts No information about how the task builds on prior knowledge

Materials and Launch 2 Points
2 pts A thorough description of the launch is included, with details about how
students are working and what the teacher will do and/or say to launch the
lesson
Materials for the lesson are listed and appropriate to the lesson and task(s)

1 pt Either launch description or materials information are absent, OR the
materials are inappropriate for the task, OR the launch description is
overly general or vague.

0 pts No information about launch or materials

Task Selection and Setup 2 Points
2 pts The task is a high-level task and is appropriate for the mathematical goal
lnforrnation about how the teacher will set up the task is included
Intended task setup does not reduce the cognitive demands of the task

1 pt The task is a hi gh-level task and is appropriate for the mathematical goal
Information about how the teacher will set up the lesson is included
Intended task setup may the cognitive demands of the task, or the setup
information is overly general

0 pts The task is of low cognitive demand

Anticipated Student Thinking/Questions 3 Points
3 pts Complete student solutions or thinking paths are clearly identified and
represent a range of approaches to the task, varying by representation or
strategy where appropriate
Ideas related to student thinking are fully developed and clear
Solutions include several incorrect pathways/note possible student
misconceptions

2 pts Student solutions or thinking paths are identified and represent a range of
approaches to the task, with some variation by representation or strategy
Ideas related to student thinking are clear but general

282

 

 

lpt

0 pts

Description of Teaching 5 POIIItS l ‘

5 pts

4 pts
3 pts

2 pts
l pt

Solutions include some incorrect pathways/note possible student
misconceptions

Student solutions or thinking paths are very general or unclear OR
Incorrect pathways/note possible student misconceptions are not included
OR

Student thinking represents a narrow (or single) range of approaches to the
task

No information about student thinking

A rich description of the lesson is presented l:-
The description includes detail about how the teacher launched the lesson, ‘i‘
how students worked on the task(s) in the lesson, and how the task(s) were ‘
shared and discussed with the whole class ._
The description provides clear information and examples of the nature of
both student and teacher talk, including the questions asked by the teacher
and the questions asked by students

The description includes a rich body of evidence, which may include but

is not limited to transcription of exchanges, written student work, class
artifacts (e.g., board work), or video clips

Student thinking is well-represented, and it is clear what students did or
did not accomplish in terms of the mathematical goals of the lesson

 

(instructor discretion)

A description of the lesson is presented

The description includes some information about how the teacher
launched the lesson, how students worked on the task(s) in the lesson, and
how the task(s) were shared and discussed with the whole class, but one or
more of these descriptions may be overly general

Examples of teacher and student talk are provided that include questions
asked by the teacher and questions asked by the students

The description includes some evidence, which may include but is not
limited to transcription of exchanges, written student work, class artifacts
(e. g., board work), or video clips

Some student thinking is evident, and it is clear what at least some
students did or did not accomplish in terms of the mathematical goals of
the lesson

(instructor discretion)

A description of the lesson is presented

The description is overly general and one or more aspects of the lesson is
poorly represented or missing completely

The description focuses almost primarily on teacher actions OR on non-
mathematical student behavior (classroom management)

283

 

0 pts

The evidence presented in the description is limited or entirely absent
It is not clear how students were thinking about the task or what students
did or did not accomplish in terms of the mathematical goals of the lesson.

No description of the teaching is present.

Rim—CC!

Analysis of Student Learning 5 Points

5 pts

4 pts
3 pts

2 pts
1 pt

0 pts

Claims are made about what students know and understand
mathematically as a result of their engagement in the task; these claims are
connected back to the lesson objectives

Specific evidence (work, student talk, video, etc.) is presented to support
these claims and is well-suited to the claim

Claims about student learning are not limited to a specific student or small
subset

(instructor discretion)

Claims are made about what students know and understand
mathematically as a result of their engagement in the task; these claims are
connected back to the lesson objectives

Some evidence (work, student talk, video, etc.) is presented to support
these claims, and is suited to the claim but may be limited or
overgeneralized

Claims about student learning are not limited to a specific student or small
subset

(instructor discretion)

Claims are made about what students know and understand
mathematically as a result of their engagement in the task

Claims are not connected back to the lesson objectives OR

Evidence is not presented to support the claim OR

Claims about student learning are limited to a specific student or small
subset

No information is given related to student learning

Reflecting on Practice 2 Points

2 pts

lpt

The teacher reflects back on the lesson, making strong links between the
Plan and Teach sections and using evidence of student learning
Areas of strength and areas for growth in teaching practice are clearly

identified
Reflections make strong connections to issues of student learning

The teacher reflects back on the lesson, making links between the Plan and
Teach sections and using evidence of student learning

Areas of strength and areas for growth in teaching practice are clearly
identified

Reflections make connections to issues of student learning

284

0 pts

No information about reflecting on practice

Analysis of Coolness 3 Points

3 pts

2 pts

0 pts

A description of why the lesson was or was not “cool” is included

The description makes links to at least three of the following: the nature of
the task, student work and talk, the teaching of the task/lesson, students’
prior knowledge, student attitudes and dispositions, use of
technology/manipulatives, classroom management

Arguments are well-supported by evidence

A description of why the lesson was or was not “cool” is included

The description makes links to two of the following: the nature of the task,
student work and talk, the teaching of the task/lesson, students’ prior
knowledge, student attitudes and dispositions, use of
technology/manipulatives, classroom management

Arguments are supported by evidence

A description of why the lesson was or was not “cool” is included

The description makes links to one of the following: the nature of the task,
student work and talk, the teaching of the task/lesson, students’ prior
knowledge, student attitudes and dispositions, use of
technology/manipulatives, classroom management, OR the links are
general and not well-specified

Arguments are supported by minimal evidence

No analysis of coolness is present.

 

References

286

‘WI’I .'__1L

 

. 'if a

References

Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What
makes it special? Journal of Teacher Education, 59(5), 389-407.

Borko, H., Eisenhart, M., Brown, C. A., Underhill, R. G., Jones, D., & Agard, P. C.
(1992). Learning to teach hard mathematics: Do novice teachers and their
instructors give up too easily? Journal for Research in Mathematics Education,

23(3), 194-222.

Borko, H., & Livingston, C. (1989). Cognition and improvisation: Differences in
mathematics instruction by expert and novice teachers. American Educational
Research Journal, 26(4), 473-498.

Brown, C. A., & Borko, H. (1992). Becoming a mathematics teacher. In D. A. Grouws
(Ed.), Handbook of research on mathematics teaching and learning. Reston, VA:
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Carpenter, T. P., Fennema, E., & Franke, M. L. (1996). Cognitively guided instruction: A
knowledge base for reform in primary mathematics instruction. The Elementary
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Carpenter, T. P., Fennema, E., Franke, M. L., Levi, L., & Empson, S. B. (2000).
C ognitively Guided Instruction: A research-based teacher professional
development program for elementary school mathematics. Madison, WI: National
Center for Improving Student Learning & Achievement in Mathematics &
Science.

Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C.-P., & Loef, M. (1989). Using
knowledge of children's mathematics thinking in classroom teaching: An
experimental study. American Educational Research Journal, 26(4), 499—501.

C hokshi, S., & Fernandez, C. (2004). Challenges to implementing Japanese lesson study:
Concerns, misconceptions, and nuances. Phi Delta Kappan, 85(7), 520-525.

C lift, R. T., & Brady, P. (2005). Research on methods courses and field experiences. In
M. Cochran-Smith & K. Zeichner (Eds), Studying teacher education: The report
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