examining classroom interactions & mathematical discourses...
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Examining Classroom Interactions & Mathematical Discourses
Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy
in the Graduate School of The Ohio State University
By
Melva R. Grant, M.Ed.
College of Education and Human Ecology
The Ohio State University
2009
Dissertation Committee:
Patricia A. Brosnan, Advisor
Sarah Anderson
Diana B. Erchick
Azita Manouchehri
Copyright by
Melva R. Grant
2009
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Abstract
This investigation examined interactions in three classrooms to determine how
they influenced Discourses related to mathematics learning and teaching. Mathematics
education literature suggests that effective mathematics instruction includes mathematical
Discourses. However, effective mathematical Discourses within mathematics classrooms
vary widely and effective Discourses in one classroom may be ineffective in another. The
purpose of the investigation was to gain insights for developing effective Discourses or
classrooms that exhibit reform oriented cultures (ROC). The primary research question
addressed was how do classroom interaction influence the Discourses related to
mathematics learning and teaching in MCP1
MCP supported classrooms are within buildings were an MCP instructional coach
is assigned. The MCP coaches’ role within the building includes: a) developing his or her
knowledge and understanding of research-based mathematics education reform; b)
providing sustained classroom-embedded professional development to small groups of
mathematics teachers; and c) offering school-based professional development for all
mathematics teachers in the building.
supported classrooms?
1 Mathematics Coaching Program
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The site of the investigation was a large urban school district in the Midwest
United States of America. According to the school district’s website for the school year
2008-2009 they reported 23,850 students enrolled and 70 % qualified for free-reduced
lunch. Data were collected in four MCP supported classrooms within this district, but
only three grade six classrooms were used in this report.
The final report includes input from 52 students, 3 teachers, and 2 MCP
instructional coaches. Data was collected using interviews, classroom observations, and
surveys. The teachers responded to two instruments to provide data about their
perceptions related to mathematics education reform (TCMER2) and teaching efficacy
(TSES3). The student survey4
Qualitative analyses were organized and managed with NVivo 8 (QSR
International, 2008) software and used to develop three case studies, which examined
classroom interactions and Discourses related to mathematics learning and teaching. This
interpretive research investigation was framed by a theoretical model based upon a
relational perspective (Cobb & Hodge, 2002; Cobb & Yackel, 1996/2004) with three key
constructs including: a) classroom culture; b) Discourse or community; and c)
captured data about attitude and identity perceptions.
2 TCMER - This survey is an adapted form of the TCMER from Ross, J. A., McDougall, D., Hogaboam-Gray, A., & LeSage, A. (2003). A Survey Measuring Elementary Teachers' Implementation of Standards-Based Mathematics Teaching. Journal for Research in Mathematics Education, 34(4), 344-363. 3 TSES - This survey is an adapted form of the TSES from Tschannen-Moran, M., & Woolfolk Hoy, A. (2001). Teacher Efficacy: Capturing an Elusive Construct. Teaching and Teacher Education, 17, 783-805. 4 Students Survey - This survey is an adapted form of that used in Sullivan, P., Tobias, S., & McDonough, A. (2006). Perhaps the Decision of Some Students Not to Engage in Learning Mathematics in School Is Deliberate. Educational Studies in Mathematics, 62(1), 81.
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relationships that support learning opportunity. A cross-case analysis was used to
compare the classrooms and to further articulate the findings related to the research
questions.
There were two primary findings from this investigation including: a) some
classroom interactions are on the boundary and b) conceptualization of a Reflective
Triad. The interactions on the boundary refer to sociocultural elements of classroom
interactions that can either enhance or hinder mathematical Discourses based upon
extenuating circumstances related to learning and teaching. The Reflective Triad is a
practical framework or tool for use by teachers and those who support their
implementation or study of ROC.
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Dedication
This work is dedicated to the students at The Ohio State University and anyone
else who is able to leverage this investigation to advance mathematics learning, thinking,
or teaching.
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Acknowledgements
My partner, my friend, and the love of my life, Michael, thank you for being there
in every way imaginable and then some. I also want to acknowledge my children, Iesha,
Michael Jr., and Rashida for their patience, love, and indulgence; and Nana’s treasures
Jamir, Michael “O”, Jasmine, and Michael III “Tres” for their youthful joy and
unconditional love. Although, not my children they are near enough and worth
acknowledging for their comfort and support, Mike and Tonya. Additionally, I recognize
my family, including parents, Barbara and Bruce; siblings, Walter, Helen, and Jothi; and
many others including in-laws (especially Wayne, Bridgette, and the boys for keeping me
sane during visits up north), aunts and uncles, cousins, nieces and nephews (especially
Jovon for his flare with words on a day when the going was tough ).
I want to thank Patti, my advisor and long-time mentor and friend, without her
steadfast support I do not believe I would be where I am today. When the journey was
most challenging, she emerged with a gentle hand to guide me. Also, instrumental in this
dissertation process was Azita, who read tirelessly and offered critical feedback at times
when it was most needed, whether I knew it or not. Another mentor who must be
mentioned is Greta, my professional sounding board and personal friend who always has
and continues to constantly encourage and support me. I look forward to working with
these great women in the future.
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Finally, I want to acknowledge my friends, who stood by me especially after I
became a self-absorbed dissertating fool. Thank you, Cathy, Alicia, Vanessa, and Emma
for being there for me and then sticking around. I must also acknowledge Tim, a peer
turned faculty, a man who during my doctoral process always showed up with the right
support at the right time. He gave me an inscribed bound Bird that was never in a bush,
but helped inspire a finished dissertation. In the end, a finished dissertation is the best
dissertation of all!
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Vita
June 1980 . . . . . . . . . . . . . . . . . ... .Northwestern Sr. High School, Baltimore, MD
1983 . ... . . . . . . . . . . . . . . . . . . . . . . . B.S. Mathematics, Coppin State University
1984 . . . . . . ... B.S. Electrical Engineering, University of Maryland, College Park
1989 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.Ed., The Ohio State University
2005 to 2009 . . . ... Graduate Research Assistant, Mathematics Coaching Program
2007 to present . . . . . . . . . . ... . Educational Consultant, MMG EdServices, LLC
Field of Study
Major Field: Mathematics Education
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Table of Contents
EXAMINING CLASSROOM INTERACTIONS & MATHEMATICAL DISCOURSES ....................II
ABSTRACT ..................................................................................................................................................II
DEDICATION .............................................................................................................................................. V
ACKNOWLEDGEMENTS ....................................................................................................................... VI
VITA ......................................................................................................................................................... VIII
FIELD OF STUDY ...................................................................................................................................... VIII
TABLE OF CONTENTS ........................................................................................................................... IX
LIST OF TABLES .................................................................................................................................... XV
LIST OF FIGURES ................................................................................................................................ XIX
LIST OF CLASSROOM SNAPSHOTS .................................................................................................. XX
CHAPTER 1: INTRODUCTION ............................................................................................................ 1
OVERVIEW .................................................................................................................................................. 1
PROBLEM .................................................................................................................................................... 2
PURPOSE ...................................................................................................................................................... 5
Research Questions ................................................................................................................................ 5
Definition of Terms ................................................................................................................................ 6
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RATIONALE AND SIGNIFICANCE .................................................................................................................. 7
MCP Classroom-Embedded Teacher Support ....................................................................................... 8
METHODOLOGICAL OVERVIEW ................................................................................................................... 9
Data Collection ...................................................................................................................................... 9
Theoretical Model ................................................................................................................................ 10
ASSUMPTIONS AND LIMITATIONS OF THE STUDY ...................................................................................... 10
INTRODUCTION SUMMARY ........................................................................................................................ 11
CHAPTER 2: LITERATURE REVIEW .............................................................................................. 12
OVERVIEW ................................................................................................................................................ 12
LITERATURE REVIEW RATIONALE ............................................................................................................. 13
RELATIONAL PERSPECTIVE FOR LEARNING AND TEACHING ...................................................................... 15
Relational Perspective – A Framework for Studying Classroom Interactions .................................... 16
An Elaborated Relational Perspective ................................................................................................. 18
Theoretical Model & Synthesis of the Relational Perspectives ........................................................... 22
CLASSROOM DISCOURSES AND MATHEMATICS PRACTICE ........................................................................ 25
1. Learning and teaching Related to Classroom Discourses ............................................................... 25
2. Tasks and Curriculum Related to Classroom Discourses ................................................................ 27
3. Conversations and Cooperative Groups Related to Classroom Discourses .................................... 31
4. Developing Classroom Discourses .................................................................................................. 35
CLASSROOM CULTURE(S) AND MATHEMATICS PRACTICE ......................................................................... 43
1. Argument and Proof and Classroom Culture .................................................................................. 43
2. Problem Solving and Mathematical Modeling and Classroom Culture .......................................... 46
3. Discussion and Talk and Classroom Culture ................................................................................... 53
CLASSROOM RELATIONSHIPS AND MATHEMATICS PRACTICE ................................................................... 57
1. Relationships that Support Learning Opportunities ........................................................................ 58
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2. Authority that Supports Learning Opportunity ................................................................................ 59
3. Identity that Supports Learning Opportunity ................................................................................... 63
LITERATURE REVIEW SUMMARY ............................................................................................................... 66
CHAPTER 3: METHODOLOGY ......................................................................................................... 68
OVERVIEW ................................................................................................................................................ 68
RESEARCH DESIGN & RATIONALE ............................................................................................................ 69
Introduction ......................................................................................................................................... 69
Interpretive Research Paradigm .......................................................................................................... 70
Case Study Methodology ..................................................................................................................... 71
RESEARCH PARTICIPANTS & SITES ............................................................................................................ 73
Teachers, Classrooms, and Schools ..................................................................................................... 73
The MCP Coaching Model, Coaches’ Training, and Teachers’ Support ............................................ 76
RESEARCH PROCEDURES ........................................................................................................................... 80
Research Overview and Timeline ........................................................................................................ 80
DATA COLLECTION ................................................................................................................................... 80
Purposeful Sampling ............................................................................................................................ 81
Data Sources & Purposes .................................................................................................................... 83
Observations and Interviews ................................................................................................................ 84
Surveys and Instruments ...................................................................................................................... 85
DATA ANALYSIS ....................................................................................................................................... 88
Analysis Coding ................................................................................................................................... 89
Analysis of Instruments & Surveys ...................................................................................................... 91
Analysis Process and Tools ................................................................................................................. 94
VALIDITY AND RELIABILITY ..................................................................................................................... 95
METHODOLOGY SUMMARY ....................................................................................................................... 97
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CHAPTER 4: RESULTS & FINDINGS ............................................................................................... 98
OVERVIEW ................................................................................................................................................ 98
DISTRICT AND SCHOOL DESCRIPTIONS ...................................................................................................... 99
West Middle School ........................................................................................................................... 101
East School ........................................................................................................................................ 102
MCP Support ..................................................................................................................................... 103
CASE STUDY 1: ADA’S CLASSROOM ........................................................................................................ 107
Ada’s Perceptions about the Class .................................................................................................... 108
Case Data for Ada’s Classroom ........................................................................................................ 113
Findings from Ada’s Classroom Interactions .................................................................................... 122
Summarizing Matrix for Ada’s Classroom ........................................................................................ 135
CASE STUDY 2: EVA’S CLASSROOM ........................................................................................................ 137
Classroom Overview .......................................................................................................................... 137
Eva’s Perceptions about the Class .................................................................................................... 138
Case Data for Eva’s Classroom ........................................................................................................ 143
Findings from Eva’s Classroom Interactions .................................................................................... 152
Summarizing Matrix for Eva’s Classroom ......................................................................................... 170
CASE STUDY 3: KIA’S CLASSROOM ......................................................................................................... 172
Kia’s Perceptions about the Class ..................................................................................................... 173
Case Data for Kia’s Classroom ......................................................................................................... 178
Findings for Kia’s Case ..................................................................................................................... 187
Summarizing Matrix for Kia’s Classroom ......................................................................................... 202
RESULTS & FINDINGS SUMMARY ............................................................................................................ 204
CHAPTER 5: DISCUSSION ................................................................................................................ 205
OVERVIEW .............................................................................................................................................. 205
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CROSS CASE ANALYSIS: SUMMARIZING THE DATA AND ANALYSES ....................................................... 205
Teacher Surveys: Reform Orientation and Teaching Efficacy ........................................................... 205
Student Surveys: Attitude and Identity Perceptions ........................................................................... 206
Analysis Summary of Coding ............................................................................................................. 208
Summary of Classroom Interactions by Type and Focus ................................................................... 209
CROSS CASE ANALYSIS OF THREE CLASSROOMS .................................................................................... 211
Cross Case Analysis for Classroom Culture ...................................................................................... 211
Cross Case Analysis for Discourse or Community ............................................................................ 228
Cross Case Analysis for Relationship Opportunity ........................................................................... 238
REVIEW OF THE INVESTIGATION .............................................................................................................. 248
Summarized Response to the Research Questions ............................................................................. 249
Classroom Interactions on the Boundary – A Result of this Investigation ......................................... 250
Reflective Triad – A Result of this Investigation ................................................................................ 252
LIMITATIONS OF THE INVESTIGATION ...................................................................................................... 254
Factors that may have Affected the Results ....................................................................................... 254
IMPLICATIONS AND RECOMMENDATIONS ................................................................................................ 255
Implications for Mathematics Practice .............................................................................................. 256
Implications for Teacher Support ...................................................................................................... 257
Recommendations .............................................................................................................................. 257
SUGGESTIONS FOR FUTURE RESEARCH ................................................................................................... 258
DISCUSSION SUMMARY ........................................................................................................................... 259
APPENDIX A: VIDEO TAPING PROTOCOL .............................................................................. 270
APPENDIX B: INTERVIEW PROTOCOL .................................................................................... 271
COACH INTERVIEW QUESTIONS ................................................................................................................. 271
TEACHER INTERVIEW QUESTIONS .............................................................................................................. 271
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APPENDIX C: CODE BOOK ........................................................................................................... 272
APPENDIX D: RESEARCH TIMELINE ........................................................................................ 277
RESEARCH TIMELINE ............................................................................................................................... 278
APPENDIX E: STUDENT SURVEY ............................................................................................... 283
STUDENT SURVEY ..................................................................................................................................... 284
APPENDIX F: TEACHERS’ SENSE OF EFFICACY SCALE .................................................... 285
TEACHERS' SENSE OF EFFICACY SCALE ................................................................................................... 286
APPENDIX G: TEACHERS’ COMMITMENT TO MATHEMATICS EDUCATION REFORM
287
TEACHERS' COMMITMENT TO MATHEMATICS EDUCATION REFORM ....................................................... 288
APPENDIX H: ANALYSIS GRID USING THEORETICAL CONSTRUCTS ............................ 289
APPENDIX I: ADA’S CLASSROOM INTERACTION SUMMARY ......................................... 295
APPENDIX J: EVA’S CLASSROOM INTERACTION SUMMARY ......................................... 309
APPENDIX K: KIA’S CLASSROOM INTERACTION SUMMARY .......................................... 319
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List of Tables
Table 1.1: Comparing MCP supported and non-supported schools' percent of students at
or above proficient. ............................................................................................................. 3
Table 2.1: Theoretical construct comparison among model constructs and theoretical
perspectives. ...................................................................................................................... 24
Table 2.2: Cooperative group interactions correlated to achievement. ............................ 34
Table 2.3: Strategies for developing cooperative groups. ................................................. 35
Table 2.4. Summary of teacher levels of developing math-talk community (adapted from
Table 1 in Hufferd-Ackles et al., 2004, pp. 88-90). .......................................................... 37
Table 2.5. Summary of student levels of developing math-talk community (adapted from
Table 1 in Hufferd-Ackles et al., 2004, pp. 88-90). .......................................................... 38
Table 3.1: An analysis summary. ...................................................................................... 84
Table 3.2: Nodes (categories) depicted in a tree structure organized by theoretical
constructs. ......................................................................................................................... 90
Table 3.3: Emergent codes for generalizing classroom interactions ................................ 91
Table 3.4: Interpretation of teacher instruments using the theoretical constructs. .......... 92
Table 3.5: Student survey questions and associated codes. .............................................. 93
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Table 4.1: Student demographic data for the school district that the investigated schools
are a part. ........................................................................................................................... 99
Table 4.2: School-based state achievement test scores for mathematics and reading
comparing data collection sites, the district, and the state for grade six. ........................ 100
Table 4.3: A summary of student survey responses (n=10) for Ada’s classroom. ......... 113
Table 4.4: Interpretation of Ada’s responses to the TCMER and TSES teacher surveys
and summary of coding densities for Ada’s case data organized by theoretical constructs.
......................................................................................................................................... 115
Table 4.5: Summary of coding densities for interaction types for Ada's case data. ....... 117
Table 4.6: Summary of coding densities for interaction focus categories for Ada's case
data. ................................................................................................................................. 118
Table 4.7: Summary of coding densities for the most frequently referenced categories for
Ada’s case data organized by theoretical constructs. ...................................................... 120
Table 4.8: Interaction summaries used in Ada's case study. ........................................... 121
Table 4.9: A summary of the analysis of Ada’s case study through the lens of the
theoretical model. ............................................................................................................ 136
Table 4.10: A summary of student survey responses (n=22) for Eva’s classroom. ........ 144
Table 4.11: Interpretation of Eva’s responses to the TCMER and TSES teacher surveys
and summary of coding densities for Eva’s case data organized by theoretical constructs.
......................................................................................................................................... 146
Table 4.12: Summary of coding densities for interaction types for Eva's case data. ...... 148
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Table 4.13: Summary of coding densities for interaction focuses categories for Eva's case
data. ................................................................................................................................. 149
Table 4.14: Summary of coding densities for the most frequently referenced (>10)
categories for Eva’s case data organized by theoretical constructs. ............................... 150
Table 4.15: Interaction summaries used in Eva's case study. ......................................... 152
Table 4.16: A summary of the analysis of Eva’s case study through the lens of the
theoretical model. ............................................................................................................ 171
Table 4.17: A summary of student survey responses (n=15) for Kia’s classroom. ........ 179
Table 4.18: Interpretation of Kia’s responses to the TCMER and TSES teacher surveys
and summary of coding densities for Kia’s case data organized by theoretical constructs.
......................................................................................................................................... 181
Table 4.19: Summary of coding densities for interaction types for Kia's case data. ...... 183
Table 4.20: Summary of coding densities for interaction focus categories for Kia's case
data. ................................................................................................................................. 184
Table 4.21: Summary of coding densities for the most frequently referenced (>15)
categories for Kia’s case data organized by theoretical constructs. ................................ 185
Table 4.22: Interaction summaries used in Kia's case study. .......................................... 187
Table 4.23: A summary of the analysis of Kia’s case study through the lens of the
theoretical model. ............................................................................................................ 203
Table 5.1: Percent favorable responses to teacher surveys (TCMER and TSES) for each
teacher. ............................................................................................................................ 206
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Table 5.2: Students’ survey responses about attitude and identity perceptions by class. 207
Table 5.3: Summary of coding density of theoretical construct group categories as a
percent by classroom. ...................................................................................................... 208
Table 5.4: Summary of interaction type categories by classroom. ................................. 209
Table 5.5: Summary of the interaction focus categories by classroom. ......................... 210
Table 5.6: Classroom culture summarizing matrix for each classroom. ......................... 212
Table 5.7: Comparing Ada’s and Kia’s classroom Discourses related to cultural
influencer, collaborative sense making. .......................................................................... 216
Table 5.8: Comparing mathematical Discourses among the three classrooms related to
mathematical practice, communication. ......................................................................... 219
Table 5.9: Excerpts that depict typical social norms within Ada’s classroom. .............. 222
Table 5.10: Excerpts that depict typical social norms within Kia's classroom. .............. 224
Table 5.11: Excerpts that depict typical social norms within Eva's classroom. ............. 226
Table 5.12: Discourse or community summarizing matrix for each classroom. ............ 228
Table 5.13: Relationships that support learning opportunity summarizing matrix for each
classroom. ....................................................................................................................... 239
Table 5.14: Comparing beliefs about mathematics and teaching actions by classroom. 247
Table 5.15: Classroom interaction concepts by theoretical construct on the boundary for
either enhancing or hindering mathematical Discourses. ............................................... 251
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List of Figures
Figure 2.1: A graphic model of the relational perspective of a classroom microculture .. 17
Figure 2.2: A graphic model of the interpretation of the elaborated relational perspective
........................................................................................................................................... 19
Figure 2.3: A theoretical model depicting key sociocultural constructs of mathematics
classrooms used for this empirical study about how interactions influence Discourses
related to mathematics learning and teaching ................................................................... 23
Figure 2.4. A mathematical model-eliciting sample problem ........................................... 51
Figure 5.1: Kia’s Classroom – a graphical representation for adding or subtracting
decimal numbers ............................................................................................................. 233
Figure 5.2: A practical framework for supporting teacher reflection ............................. 252
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List of Classroom Snapshots
Classroom Snapshot 4.1: Students Selecting Partners .................................................... 111
Classroom Snapshot 4.2: Acceptable approaches for estimating fractions .................... 123
Classroom Snapshot 4.3: Evaluating written responses ................................................. 126
Classroom Snapshot 4.4: Student sharing homework at overhead ................................. 128
Classroom Snapshot 4.5: Student at overhead during problem solving ......................... 130
Classroom Snapshot 4.6: Defining procedure for multiplying mixed numbers ............. 133
Classroom Snapshot 4.7: Evidence of a Changed Structural Norm ............................... 140
Classroom Snapshot 4.8: Reflecting on a Problem and Learning about Students .......... 141
Classroom Snapshot 4.9: Fun, Mathematical Focus, and Back to Work ........................ 142
Classroom Snapshot 4.10: Listening and Revoicing ...................................................... 154
Classroom Snapshot 4.11: Managing Discourse Participation ....................................... 156
Classroom Snapshot 4.12: Student B Shares Authentic Thinking .................................. 158
Classroom Snapshot 4.13: Reflecting and Evaluating a Mathematical Procedure ......... 160
Classroom Snapshot 4.14: Teacher Actions that Support Learning ............................... 164
Classroom Snapshot 4.15: Using Student Responses for Teaching Support .................. 165
Classroom Snapshot 4.16: Eva Assesses Understanding before Continuing ................. 166
Classroom Snapshot 4.17: Too Much Student Support??? ............................................. 174
Classroom Snapshot 4.18: Kia Balances Objectives and Student Input ......................... 177
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Classroom Snapshot 4.19: Thinking and Ideas ............................................................... 189
Classroom Snapshot 4.20: Informer Teacher Behavior .................................................. 191
Classroom Snapshot 4.21: Students Thinking and Ideas to Interpret Meaning .............. 193
Classroom Snapshot 4.22: Limited Participation ............................................................ 197
Classroom Snapshot 4.23: Teacher explaining how to compare estimates .................... 201
Classroom Snapshot 5.1: Teacher explaining how to compare estimates ...................... 213
Classroom Snapshot 5.2: Student at overhead during problem solving ......................... 214
Classroom Snapshot 5.3: Reflecting and Evaluating a Mathematical Procedure ........... 217
Classroom Snapshot 5.4: Adding fractions with calculators. ......................................... 229
Classroom Snapshot 5.5: Developing fraction representations. ..................................... 231
Classroom Snapshot 5.6: Kia's classroom engaged in contributive Discourse. .............. 233
Classroom Snapshot 5.7: Expert informer teacher behavior .......................................... 236
Classroom Snapshot 5.8: Student challenges expert informer teacher behavior ............ 237
Classroom Snapshot 5.9: Defining procedure for multiplying mixed numbers ............. 240
Classroom Snapshot 5.10: Limited Participation ............................................................ 242
Classroom Snapshot 5.11: Contributive Authority and Positive PAID .......................... 244
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CHAPTER 1: INTRODUCTION
Overview
Mathematics education reform has been under way for over twenty years. One
must wonder if the implementation of reform is progressing as anticipated by those who
initially conceptualized reform. Mathematics reform-based practices suggest the need for
instruction with less focus on procedural practice and skill building and more on problem
solving that requires reflexive and authentic student thinking (T. P. Carpenter, Fennema,
Peterson, Chiang, & Loef, 1989; Fennema et al., 1996; Hiebert & Carpenter, 1992).
Additionally, and most critical is the need for discourses related to mathematical concepts
that reveal students’ thinking and understanding (Ball, 1993; Ball & Bass, 2003; Cobb,
Yackel, & Wood, 1992; Yackel, Cobb, & Wood, 1991). Many have suggested that
mathematics education reform as described puts teachers in the unenviable position of
leading the implementation and owning the responsibility for transforming traditional
practices to reform-based practices (e.g., Prawat, 1992; Woodbury, 2000).
The consensus for some time has been that in order to implement reform-oriented
practices successfully, teachers need to change the way they think about and their beliefs
related to mathematics teaching and learning (e.g., Borko, Davinroy, Bliem, & Cumbo,
2000; Megan L. Franke, Carpenter, Fennema, Ansell, & Behrend, 1998), use problem-
based tasks for instruction (Driscoll, 1999; Hiebert et al., 1997; Schoenfeld, 1994), and
listen to and use student thinking to inform instructional decisions (T. P. Carpenter &
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Romberg, 2004; Lampert, 1990; Steinberg, Empson, & Carpenter, 2004). More simply,
mathematics education reform is a complex endeavor that involves teachers changing
classroom environments, beliefs, and the pedagogical foundations that underlie them
(Ball, 1993). More recently, Franke, Kazemi, and Battey (2007) posited that mathematics
practice requires more than selecting appropriate tasks and listening to students,
specifically “we need to consider relationships between particular classroom practice and
opportunities for students to engage” (p. 226); and this is the primary impetus for this
investigation.
The types of practical changes called for by mathematics education reformers are
neither simplistic nor easily achieved through authoritative mandates. Even so, today
standardized test scores are used to measure the effectiveness of teaching and teachers
(U.S. Department of Education, n.d.), and I posit, mathematics education reform. That is,
standardized test scores may indicate the presence of change, but close scrutiny of
classrooms and practice enable us to determine if increased test scores are a result of
reform or some other element of teaching and learning. However, if a change in test
scores is related to a change in practice, we need to understand the nature of the practice
and compare it to the vision of reform in order to label the practice as reform (Spillane &
Zeuli, 1999). In the end, increased test scores alone tell us very little about practice,
teaching, or learning.
Problem
Preliminary results from the Mathematics Coaching Program (MCP, Brosnan &
Erchick, 2008) suggest MCP supported schools have experienced substantial growth in
3
student achievement as measured by the mathematics component of the state
achievement test. Further, Brosnan and Erchick have shown that MCP supported schools
have been consistently outperforming similar non-supported schools as indicated by the
percent of students at or above proficient as illustrated in Table 1.1.
Grade ‘05-‘06 ‘06-‘07 ‘07-‘08 Non-
supported MCP
Supported Non-
supported MCP
Supported Non-
supported MCP
Supported
5 23% 49% 29% 52% 26% 51%
6 31% 63% 38% 66% 37% 66%
Table 1.1: Comparing MCP supported and non-supported schools' percent of students at or above proficient.
State achievement test scores do not explain the nature of mathematics learning and
teaching in any of these schools. I believe the achievement gains may be related to the
classroom Discourses related to mathematics learning and teaching within MCP
supported classrooms.
Mathematics education reformists have been calling for mathematical learners’
engagement in thinking and participation in classroom interactions (NCTM, 1991, 2000).
Specifically, both mathematical participation and thinking have been deemed essential
for developing mathematical dispositions and understandings (NCTM, 1991); Yackel and
Cobb (1996) referred to this as developing intellectual mathematical autonomy.
Elaborating further, according to Gutstein (2007), research is needed to understand how
mathematics can be taught so that students’ develop mathematical autonomy.
4
Additionally complicating matters, teachers appear to be unwilling or unable to
change teaching practices (Patrick & Pintrich, 2001; Senger, 1999; Smith, 2000). Given
the state of affairs, how are students to develop mathematical autonomy through
mathematics instruction? This conundrum suggests the need to research mathematics
classroom interactions to understand how they influence mathematical Discourses that
support students’ development of mathematical autonomy and learning. My research will
look closely at classroom interactions in several MCP supported classrooms to see how
engagement and participation play out related to mathematics learning and teaching.
Reform-based practice stories and videos show classrooms as spaces where
teachers facilitate learning, students engage mathematically, and actions/interactions are
available for all (T. P. Carpenter, Fennema, Franke, Levi, & Empson, 2000; T. P.
Carpenter & Romberg, 2004; Hiebert et al., 1997; L. West & Staub, 2003). The problem
is that teachers lack understanding about how to create such environments (Hufferd-
Ackles, Fuson, & Sherin, 2004) and there can be no specific steps prescribed for doing so
because of the dynamic and complex nature of the sociocultural elements of classrooms
(Megan L. Franke et al., 2007).
Teachers and researchers want and need guidance for creating classroom
interactions that support student development of mathematical ideas (McClain & Cobb,
2001) and mathematical autonomy (Gutstein, 2007; Yackel & Cobb, 1996). This lack of
know how perpetuates mathematical classroom interactions that are more procedural
(Spillane & Zeuli, 1999). There is a need for helping teachers and others to understand
5
the relationship between classroom interactions, mathematical Discourses, and student
learning opportunities (Megan L. Franke et al., 2007).
Purpose
The purpose of this investigation was to see how reform-oriented culture (ROC)
manifested within several mathematics classrooms and how several sociocultural
elements compared. To that end, we examined how classroom interactions influenced
Discourses related to learning and teaching in mathematics classrooms. A second purpose
was to see what meaning emerged about classroom Discourses related to mathematics
learning and teaching from classrooms within a school district with improving student
mathematics achievement as measured by state tests.
The investigation was designed to examine Mathematics Coaching Program
(MCP) supported classrooms and how the nature of classroom interactions influenced
classroom Discourses that supported (or hindered) student development of mathematical
ideas. Because MCP supported schools have experienced an increase in mathematics
achievement test scores and MCP coaches received sustained training related to reform-
oriented culture (ROC), MCP supported schools were selected as the site for the
investigation.
Research Questions
The specific questions governing this investigation include:
1. How do classroom interactions influence the Discourses related to mathematics learning and teaching in MCP supported classrooms?
a. What is the nature of Discourses or community in each classroom? b. What is the nature of classroom culture in each classroom? c. What is the nature of relationships that support learning
opportunity in each classroom?
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2. How do classroom interactions influence Discourses related to mathematics learning and teaching compare among the classrooms?
Definition of Terms
Classroom interactions – sociocultural encounters among two or more persons
within a classroom
Culture – socially accepted ways by a group that governs expectations and
actions of individuals within the group
Discourses – “different types of communication that bring some people together
while excluding others” (Sfard, 2007, p. 571); more than verbal communication
Discourse encompasses beliefs and values; “requires a combination of both reflection and
action” (Manouchehri & St. John, 2006, p. 545); denoted with a capital “D,” implies
engagement and participation beyond talk
MCP – the Mathematics Coaching Program, a large-scale, multi-year research
and development project interested in the relationship between classroom-embedded
professional development and student achievement (Erchick & Brosnan, 2005)
Reform Oriented Culture (ROC) – sociocultural elements of classrooms related
to mathematics education reform; ideas that innately imply sociocultural elements, such
as developing student mathematical power (NCTM, 1991) or using student-centered
pedagogy (Cornelius-White, 2007); but not curriculum, pedagogy, or teaching strategies
that do not possess an implied sociocultural element.
7
Rationale and Significance
This is a study about classroom interactions and the influence on the Discourses
in mathematics classrooms. The initial rationale for this study was to contribute to the
mathematics education body of research literature. My contribution is significant because
researchers have called for support for emergent understanding related to the
sociocultural aspects of mathematics classrooms (Megan L. Franke et al., 2007). Recall in
this investigation, the sociocultural elements of mathematics education reform of interest
are referred to as ROC.
In general, teachers appear to not understand how to create or develop effective
ROC in their classrooms. Specifically, teachers do not create effective mathematical
learning communities (Hufferd-Ackles et al., 2004) or classroom interactions (McClain &
Cobb, 2001) within their classrooms as described by the mathematics reform literatures.
Thus, another contribution of this investigation is to offer teachers and those who support
them a nuanced perspective for considering education reform that may lead to different
teaching and support that lead to new outcomes related to students’ mathematics learning.
A second rationale for this study is to understand how the nature of different
classroom interactions within a mathematics classroom contributes to, or hinders,
Discourses related to mathematics learning and teaching. Further, classroom interactions
tend to be sticky, resistant to change, because they are reified by repeated behaviors that
reinforce classroom culture (Hufferd-Ackles et al., 2004; Patrick & Pintrich, 2001). The
significance of this is creating awareness for both teachers and those who support them as
they endeavor to transition toward reform-based practices, especially for those engaged in
8
classroom-embedded support situations. Awareness or new understanding enables
reflection on old problems in new ways and perhaps lead to teacher or cultural changes
related to mathematics learning and teaching.
The final rationale for this study was establishing a clearer connection between
effective teacher support and student achievement, which is one of the goals for the MCP
research (Erchick & Brosnan, 2005). Unfortunately, this investigation was not able to
show such a connection because of circumstances that are articulated more fully in the
next chapter.
MCP Classroom-Embedded Teacher Support
The development component of the Mathematics Coaching Program (MCP)
provides sustained training and support for the MCP mathematics instructional coaches
(Erchick & Brosnan, 2005). During the school year, MCP coaches participate in four or
more days each month of professional development. Their training is geared toward
reform-oriented pedagogy (Ball, Hill, & Bass, 2005; T. P. Carpenter et al., 2000; T. P.
Carpenter & Romberg, 2004; Cobb et al., 1992; Fennema, Franke, Carpenter, & Carey,
1993; Hiebert et al., 1997; Steinberg et al., 2004); teaching mathematics content using
reform-oriented pedagogical approaches (Ball, 1993; Lampert, 1990/2004); and
mathematics instructional coaching (L. West, 2007; L. West & Staub, 2003).
The MCP mathematics instructional coaching support for teachers is intended to
be sustained and classroom-embedded (Erchick & Brosnan, 2009a). MCP coaches are
expected to work with a set of three to four mathematics teachers at the coach’s assigned
building. For every set of teachers, the MCP coaching occurs about four days per week
9
for about six weeks, and takes place within the teachers’ classrooms, with the teachers’
students, and using district adopted curriculum. Collaboratively, the coach and teacher
plan, teach, and monitor student learning, with their focus trained on student
understanding, they reflexively adjust instruction.
Methodological Overview
The central theme of this investigation was how classroom interactions influenced
mathematical Discourses in MCP supported classrooms. A thorough investigation of my
research questions required both qualitative and quantitative methodological approaches.
The analysis for this study was primarily qualitative using an interpretive epistemology,
case study methodology, and cross case analysis used to develop a comprehensive
response to the research questions. The quantitative analysis was used to situate the case
study classrooms among the schools within the district and then to describe the data and
summarize the analyses.
Data Collection
The goal of data collection for the investigation was to identify classroom
Discourses, cultures, and relationships to uncover how classroom interactions influence
the Discourses related to mathematics learning and teaching in MCP supported
classrooms. This study was a sociocultural study situated within four mathematics
classrooms supported by MCP coaches. Specifically, MCP mathematics instructional
coaches provide support for the participating teachers.
10
Theoretical Model
The analysis examined and interpreted the classroom interactions using Cobb and
Hodge’s (2002) elaborated relational perspective of mathematics classroom interactions.
This perspective emerged through researchers’ engagement in developmental research
focused on improving classroom-based mathematics learning and teaching. The original
theoretical perspective conceptualized by Cobb and Yackel (1996/2004) underlies many
research studies about sociocultural aspects of mathematics education (e.g., Cobb &
Yackel, 1998; Empson, 2003; Lampert & Cobb, 2003; Sfard & Kieran, 2001). The
evolution and more specific details about this theoretical framework are discussed in
Chapter 2.
Assumptions and Limitations of the Study
The primary assumption of this study was that the relationship among
mathematics classroom interactions and Discourses related to mathematics learning and
teaching will emerge from the data collected and analysis methods employed. The
sociocultural constructs that are the focus for this investigation are not simplistic and may
prove elusive under scrutiny; however, I will rely on interpretive methods and research
perspectives to guide my actions (Denzin, 1994; Erickson, 1986; Janesick, 1994).
A second assumption was that within the classrooms there existed mathematical
Discourse related to learning and teaching. That is, the teacher’s relationship with
mathematics education reform had progressed beyond, discourse (just talk); and students
were afforded opportunities to participate mathematically within the mathematics
classroom. The selected classrooms were each supported by a trained MCP mathematics
11
instructional coach who had completed one year of sustained training related to, among
other things, mathematics education reform. Thus, I believed that the purposeful
classroom selection processes would yield classroom teachers who were actively engaged
in implementing a reform-oriented mathematics practice and ROC. However, if not, then
I looked forward to seeing what emerges from the investigation.
One limitation of the investigation was not being able to generalize the findings
because of the chosen methodology, case study, and the associated limitations on sample
size. Another limitation of this study was the elusive nature of the constructs under
examination or the inadequacy of theory to address the analysis needed to respond to the
research questions. On the other hand, the nature of qualitative research, is often not
bounded by a priori constructs or theory (Lather, 2006; Somekh & Lewin, 2005). The
breakdown of theory, in qualitative research rarely results in failure, but opportunity for
divergent conclusions to be revealed or evolved theories to emerge.
Introduction Summary
In this chapter, the problem that inspired this research was articulated and the
research purpose and questions were shared. The rationale and significance for this
investigation were set forth. An overview of the methodology was presented, including a
preview of data collection, analysis, and an introduction of the theoretical model, which
guides this investigation. The chapter is concluded with a discussion about the
assumptions and limitations of the investigation.
12
CHAPTER 2: LITERATURE REVIEW
Overview
This review of the literature provides the foundation for this investigation by
exploring research and other scholarly writings related to mathematics learning and
teaching. The chapter begins with an introduction that articulates the rationale for the
selecting the four major sections that comprise this literature review. The four major
sections of the review are: a) Relational Perspective for Teaching and Learning; b)
Classroom Discourse(s) and Mathematics Practice; c) Classroom Culture(s) and
Mathematics Practice; d) Classroom Relationships and Mathematics Practice. Each
section has several subsections inspired by the literature and essential elements related to
mathematics practice.
The first of the major sections, Relational Perspective for Teaching and Learning,
introduces and explains, while the concluding subsection, Theoretical Model & Synthesis
of the Relational Perspective, further articulates the theoretical model that governs this
investigation. Within the first major section, three key constructs, Discourse, culture, and
relationship opportunity, emerge prominently and define the theoretical model. The
remaining major sections examine the literature through the lens of the three key
constructs of the theoretical model to complete the review.
13
Literature Review Rationale
To introduce the four major sections of this review, consider a comprehensive
discussion about mathematics teaching and learning by Franke and colleagues (2007).
Three key features emerge from mathematics teaching and classroom practice literature
that are central for understanding mathematics learning and teaching: “(a) creating
mathematical classroom discourse, (b) developing classroom norms that support
opportunities for mathematical learning, and (c) building relationships that support
mathematical learning” (Megan L. Franke et al., 2007, p. 226). Franke and colleagues
contend that the concept of good teaching remains unresolved and disputed among
researchers and may never be resolved. Even so, classrooms are rich sites for studies
intent on developing understanding about mathematics teachers, teaching, and learning,
which is the intent of this investigation. Franke, Kazemi, and Battey’s synthesized
meanings derived from the literature for both teaching and learning align well with this
investigation.
Further, inspired by the literature, Franke, Kazemi, and Battey (2007) describe
mathematics teaching as relational, multifaceted, interpretive, and generative. More
simply stated, teaching is sociocultural and complex. Their proffered descriptors for
mathematics teaching need clarification, and in that effort, Franke and colleagues’ ideas
are synthesized throughout the remainder of this rationale. Teaching that is relational
occurs when teachers, students, and content are interwoven; knowledge and
understanding are not found in isolation. This notion of relational teaching provides the
foundation for the theoretical model used for this study and is the topic of the first section
14
of this literature review, Relational Perspective for Teaching and Learning. The first
section articulates the theoretical grounding for the investigation. The three remaining
sections emerge from the theoretical discussion and serve as the lenses for analyses and
interpretation during the investigation.
Two examples of the multifaceted nature of teaching are a) diverse learning
opportunities created by different contexts, and b) students participate in classrooms in a
variety of ways. These multifaceted concepts are topics in the fourth and final section
entitled classroom relationships. The interpretive nature of teaching can be depicted as
instructional decisions that are informed by students’ actions, understandings, and
learning. It is unlikely that two teachers’ interpretations of a single instructional event
will be the same. Additionally, teaching actions are influenced by teachers’ knowledge,
beliefs, and goals. Thus, teachers’ individual interpretations make teaching interpretive
and this is a topic included in the second section entitled Classroom Discourse(s) and
Mathematics Practice.
Teaching is generative when teaching results in learning for multiple constituents,
like teachers, parents, and administrators in ways that support students’ learning. Simon
(1995) describes teaching as a continual process of fine-tuning as a co-learner with
students collectively engaged in mathematical activities. The generative aspect of
learning and teaching is a topic in the third section entitled Classroom Culture(s) and
Mathematics Practice.
With respect to learning, Franke and colleagues (2007) posit that the research
community is moving toward consensus about student learning as opportunity for
15
developing both mathematics concepts and skills, understanding mathematics in ways
that are adaptive and flexible, and creating mathematical identities. In their own words,
they put it this way, mathematics “understanding is more than a set of knowledge and
skills; understanding includes one’s perception of oneself in relation to the mathematics”
(p. 228). The research community has linked the idea of student identity with
mathematics understanding, learning. In other words, mathematics learning is a
sociocultural activity and this investigation uses a sociocultural lens to explore
mathematics learning and teaching. Identity is a focus in the final section entitled
Classroom Relationships and Mathematics Practice.
The structure of this literature review, while directly related to the theory which
grounds this study, one can see that many key ideas emerged from the comprehensive
discussion of mathematics teaching and learning by Franke, Kazemi, and Battey (2007).
In deference to Franke and colleagues, they are cited consistently throughout this
literature review to purposefully highlight significant aspects from their comprehensive
work that are germane for this study. For example, several sections are introduced using
Franke and colleagues; however, care was taken to develop a comprehensive and specific
review of the literature for this investigation that is expansive and not a regurgitation of
the perspectives of Franke and colleagues.
Relational Perspective for Learning and teaching
Drawing on cognitive and sociocultural perspectives, Franke, Kazemi, and Battey
(2007) conclude, “Teaching is relational. Teachers, students, and subject matter can only
be understood in relation to one another” (p. 227). The social context became apparent to
16
Paul Cobb and Erma Yackel (1996/2004) during initial research attempts to develop
individual psychological explanations for student’s activities and learning within
mathematics classrooms. The researchers’ individual accounts proved inadequate for use
in developmental research and the consequence was emergence of a primary theoretical
objective, “exploring ways to account for students’ mathematical development as it
occurs in the social context of the classroom” (Cobb & Yackel, p. 210). These theorists
offer a framework for formulating explanations in terms of classroom level processes –
“individual interpretations and actions, and face-to-face interactions and discourse” (p.
210). These classroom level processes are a focus for this investigation and thus rationale
for adopting a theoretical perspective inspired by their work.
Relational Perspective – A Framework for Studying Classroom Interactions
This interpretive framework was initially outlined during the course of
developmental research comprised of in-class instructional development and classroom-
based research (Cobb & Yackel, 1996/2004). A graphical representation of this
perspective is shown in Figure 2.1. The researchers posit that the framework offers a
relevant sociocultural perspective for an observer interested in interactions within the
context of a situated community such as a mathematics classroom. This vantage point
enables one not of the classroom community, such as a researcher, to view the actions of
an individual, such as a teacher or student, who is situated within the sociocultural
practices of the classroom community.
17
Figure 2.1: A graphic model of the relational perspective of a classroom microculture
In this situated context, learning is characterized as a process influenced by the
culture, relationships, and discourse of the community. Thus, the researcher is able to
observe classroom interactions for insights into learning and teaching from a
sociocultural perspective using this framework. That is, the relational or social
perspective offers a point of view that is beneficial for interpreting or making sense of the
complexities of a mathematics classroom (Yackel & Cobb, 1996). This framework also
offers a psychological perspective for those interested in an individual psychological
perspective; however, that is beyond the scope of this investigation and will not be
considered further.
The relational perspective defines a classroom microculture and consists of three
dimensions or constructs; a) social norms, b) sociomathematical norms, and c)
mathematical practices (Cobb & Yackel, 1996/2004). These constructs represent three
facets of classroom microculture that constitute shared or collective classroom processes
worth taking account of for empirical studies of classroom interactions. Social norms are
Classroom Microculture
Social Norms Sociomathematical Norms
Mathematical Practice
18
the accepted ways of social interactions within a specific context, such as a classroom.
Sociomathematical norms are the accepted ways of interacting mathematically within the
context of learning and teaching. Classroom mathematical practices are the routines or
processes used within a mathematics classroom related to mathematical learning and
teaching.
An Elaborated Relational Perspective
Cobb and Hodge (2002) elaborated on the relational framework for interpreting
classroom interactions, their conceptualization extends to include authority and identity
constructs. A graphical interpretation of this perspective is shown in Figure 2.2. This
graphic is not an explicit depiction of the elaborated relational perspective, but is a
simplified conceptualization of the implications of effective classroom interactions using
the constructs defined by Cobb and Hodges’ theory.
According to the elaborated relational perspective, authority and identity overlay
the classroom microculture; a key element of the theory not shown in the graphic. Cobb
and Hodges’ (2002), relational perspective describes when pluralistic power exists, where
teacher authority in the classroom is shared with students, learning and teaching extends
to the core constructs of classroom microculture – social norms, sociomathematical
norms, and classroom mathematical practices and cultural capital emerges.
Identity is conceptualized as a relational construct and ways of participation
within communities (Hodge, 2006). The authority and identity themes were present
within some of the earlier work that preceded this elaborated perspective, but their
influence on the theory was not explicit (McClain & Cobb, 2001; Yackel & Cobb, 1996).
19
Figure 2.2: A graphic model of the interpretation of the elaborated relational perspective
Additional constructs of this elaborated relational perspective are communities of
practice and cultural capital. One or more communities of practice reside within every
mathematics classroom (depicted by the circles labeled COP in the graphical model
shown in Figure 2); and cultural capital is the manifestation of social residue from the
mathematics classroom and is valued by the community (Cobb & Hodge). The cultural
capital is depicted in Figure 2 as the outcome from effective combining of authority and
identity and classroom microculture. An example of cultural capital is an accepted way of
proving a mathematical argument. An assumption of this perspective is that the
classroom is an environment for students’ thinking and ideas, a space for developing
mathematical culture or disposition (Hodge).
Communities of practice are characterized by three interrelated dimensions “a)
mutual engagement; b) negotiated enterprise; and c) repertoire of negotiable resources
accumulated over time” (Cobb & Hodge, 2002, p. 260). Communities of practice are
collectively dynamic and benefit from communal learning as follows (citing Wenger,
1998 in Cobb & Hodge, p. 260). Cobb and Hodge stated, “a community of practice
Authority & Identity
Classroom Microculture
Cultural Capital
COP
COP COP
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constitutes the local social context of its members’ learning” (p. 260). Communities of
practice develop culture among its members, who share socially negotiated
understandings.
Communities of practice are local, within the mathematics classroom, but
Discourses or communities, denoted with a capital “D,” is suggestive of a broader
community and implies wider meaning (Cobb and Hodge). The Discourses of
mathematics refers to a community of practice of mathematics that develops a
mathematical culture, but also the members develop shared “. . . linguistic practices,
beliefs, values, and ways of acting” (Cobb & Hodge, 2002, p. 262). The Discourse(s) or
community of practice related to mathematics learning and teaching offers a lens for
examining classroom interactions for this investigation.
Cultural capital is comprised of core aspects of the dominant Discourses within a
society (Cobb & Hodge, 2002). From the relational perspective, those who opt to
participate from within the classroom mathematics community negotiate the dominant
Discourses of the mathematics classroom. This is very different from traditional
classroom Discourses that rely on expert mathematicians, texts, and tradition to define
what counts as knowledge, ways of knowing, and appropriate actions. Using traditional
mathematics Discourses offers a vision of mathematics learning and teaching that
includes dominant authoritative teachers transmitting mathematics as they know it to
passive student receptors who are presumed void of ideas (Apple, 2004; Freire, 1970;
hooks, 1994).
21
Hodge (2006) offered examples of cultural capital for mathematics as negotiated
norms related to: a) the use of mathematical language; b) collaborative reasoning and
argumentation; and c) what constitutes mathematical knowing. Cultural capital for
mathematics is characterized as students’ active participation in and contributions to
classroom community through collective development of ways of reasoning, speaking,
and acting (Cobb & Hodge, 2002). One might posit that cultural capital is held by those
with sufficient agency and motivation to participate in negotiating the classroom
microculture within the dominant classroom Discourses. The constructs, agency and
motivation, are of sufficient interest, but may prove to be beyond the scope and access of
this investigation; analyses and interpretation will be the final arbiter for inclusion.
Related to cultural capital, but also classroom microculture is identity. From a
relational perspective, Hodge (2006) defined identity as the way “. . . individuals perceive
that others are perceiving them in interaction” (p. 380). Lave & Wegner (1991) defined
learning as a process for forming identity; substantial participation within a community
forms identity (as cited in Hodge). Both perspectives relate identity to participation.
Hodge calls this a “micro sense” that is not a personal attribute, but an element of
classroom microculture (p. 380).
In summary, communities of practice, including Discourse communities,
influence and define a set of interrelated constructs, namely mathematics cultural capital,
identity, and microculture within every mathematics classroom. The mathematics
classroom microculture is comprised of three additional constructs, social norms,
sociomathematical norms, and classroom mathematical practices. In total, these
22
constructs and their interrelatedness form an interpretive framework and using a
relational perspective allows empirical investigation of classroom complexities through
observations of classroom interactions to interpret meaning related to mathematics
learning and teaching. This relational perspective forms the theoretical foundation for this
investigation.
Theoretical Model & Synthesis of the Relational Perspectives
This theoretical model was inspired by the elaborated relational perspective of
Cobb and Hodge (2002), the initial framework and research of Cobb and his colleagues
(Cobb & Yackel, 1996/2004; McClain & Cobb, 2001; Yackel & Cobb, 1996), and
informed by the synthesis of the mathematics teaching and classroom practice literature
(Megan L. Franke et al., 2007). This model, depicted in Figure 2.3, is a pictorial
representation of the constructs used to interpret meanings related to interactions within
the classroom guided by the relational perspective framework. Within the graphic model,
the key constructs for this study are Discourse or community, classroom culture, and
relationship opportunity.
The theoretical model is comprised of three-sectors organized in a circular region
with arrows traversing the circumference because the key constructs are comprised of
several underlying sociocultural concepts that sometimes overlap when considering the
context of mathematics classroom interactions. The underlying concepts are organized
hierarchically in tree structures by construct and are discussed within the next chapter in
the analysis process discussion.
23
Figure 2.3: A theoretical model depicting key sociocultural constructs of mathematics classrooms used for this empirical study about how interactions influence Discourses related to mathematics learning and teaching
The yellow background beneath the thee-sectored circular region represents that
which emerges through and from the constructs, such as cultural capital that influences
Discourses related to mathematics learning and teaching (Hodge, 2006). Naming what
emerges is dependent on what exists and in what combinations within the three-sectored
construct regions and is beyond the scope of this investigation, thus that space remains
purposefully unnamed within the graphical theoretical model. This theoretical model
represents a synthesis of the relational perspectives theory by Cobb and colleagues (Cobb
& Hodge, 2002; Cobb & Yackel, 1996/2004, 1998) and understandings from the
Relationship that supports
learning opportunity
Classroom Culture
Mathematical Discourses or Community
24
mathematics learning and teaching literature and offers the theoretical grounding for this
investigation.
The relationship among the theoretical model constructs and the constructs from
the two relational perspectives are shown in Table 2.1. Notice the inclusion of authority
and identity attributed to the elaborated relational perspective, a theory grounded in an
interpretive framework for studying diversity and equity (Cobb & Hodge, 2002). This
study is about how classroom interactions influence Discourse related to mathematics
learning and teaching, and not about diversity, equity, or power; however, if these
constructs emerge from the data in a significant way they will not be overlooked.
Constructs from theoretical model
Constructs from original relational perspective5
Constructs from the elaborated relational perspective 6
Mathematical Discourses or Community
Classroom Communities of practice Discourses (i.e., cultural
capital emerges from)
Classroom Culture
Social norms Sociomathematical norms Classroom mathematical
practices
Social norms Sociomathematical norms Classroom mathematical
practices
Relationship that supports learning opportunity
Collective or communal learning
Authority (e.g., distributed, shared, etc.)
Identity (e.g., participation, opportunity, etc.)
Table 2.1: Theoretical construct comparison among model constructs and theoretical perspectives.
5 (Cobb & Yackel, 1996/2004) 6 (Cobb & Hodge, 2002)
25
The final three sections of this chapter are devoted to articulating findings and
understandings derived from the mathematics teaching and learning literature related to
the key constructs, Discourses or community, classroom culture, and relationship
opportunity.
Classroom Discourses and Mathematics Practice
As described in the previous section, mathematical Discourse community is a
construct of the theoretical model that grounds this investigation, and developing
classroom discourse was highlighted by Franke, Kazemi, and Battey (2007) as important
for understanding mathematics learning and teaching. In this section, the literature is
reviewed in search of various research trajectories related to discourse and mathematics
teaching, learning, and classroom practice. Presented are three big ideas related to
discourse: a) learning and teaching; b) tasks and curriculum; and c) conversations and
cooperative groups. The fourth and final part of this section considers the ways educators
and researchers have approached development of discourse within mathematics practice.
We turn our attention now to this discussion about Discourse beginning with the first
section about learning and teaching.
1. Learning and teaching Related to Classroom Discourses
Reform-oriented culture (ROC) for mathematics learning and teaching, is not the norm
for most classrooms in the United States (Stigler & Hiebert, 1999). One reason for slow
reform related to classroom discourse was offered by Spillane and Zeuli (1999), they
concluded from their mixed-method research project that changing “epistemological
regularities of instruction” (p. 19) was hard for teachers; they were not able or willing to
26
challenge their conceptions about knowledge and knowing, which made it hard to
fundamentally change their instruction. In this study, the researchers administered
surveys from the Third International Mathematics and Science Study (TIMSS) to all
science and mathematics teachers of grades three, four, seven, and eight from nine urban
school districts with significant ethnic and economic diversity. The 25 selected teachers
were among the top 10% of high scoring TIMMS survey respondents, indicating that
their practices were reform-oriented. Their findings indicated that fewer than four of
these reform-oriented teachers were able to change epistemological regularities of
instruction. That is, fewer than four teachers pressed students to explain and defend their
ideas, thinking, and solution approaches. Additionally, of these, few teachers nurtured
student-student discourse about mathematical ideas and concepts. The vast majority of
the teachers did not change their normal instruction patterns, such as pressing students for
explanations or nurturing peer discourse. A conclusion drawn from these findings by the
researchers is the need for policy that supports teacher learning related to reforming
practice.
This investigation is about examining how classroom interactions influence
Discourses and is situated in MCP supported classrooms and the MCP advocates teaching
as a learning process (Erchick & Brosnan, 2009c), but that is not sufficient for this
investigator to anticipate the presence of reform-oriented Discourses, in part because of
findings from studies like Spillane and Zeuli (1999). In fact, many studies since have
found similar findings, that teachers are challenged to change teaching to more reform-
oriented practices (e.g., Borko et al., 2000; Flores, 2003; Sfard, 2007; Sherin, Mendez, &
27
Louis, 2004). Given the literature, this investigator will be pleased to find any evidence
of reform-oriented practice within the observed classrooms.
The study by Spillane and Zeuli used the top 10% of high scoring survey
respondents and even though the researchers attempted to find a sample more likely to
show change, the data and analyses suggested that little change resulted. One may
conclude that when looking at classrooms Discourse, learning, and teaching, intentional
sample selection does not always yield anticipated results, but the intentional sampling
approach of Spillane and Zeuli, strengthened their finding about teachers not easily
changing “epistemological regularities of instruction” (p. 19).
Changing what a teacher does is only one aspect of reform, just as important are
the tasks or curriculum used for instruction. Reform efforts initially seemed to focus
almost entirely on this aspect of practice, for many years mathematics reformers
promoted curricula as the panacea for changing practice. Even though curriculum is
beyond the scope of this investigation, given the historical significance of curriculum
with respect to mathematics education reform, no literature review about reform-oriented
practice would be complete without mention of curriculum.
2. Tasks and Curriculum Related to Classroom Discourses
The challenges teachers face for reforming mathematics teaching are complex.
“Many believe that teaching and learning would be improved if classrooms were
organized to engage students in authentic tasks, guided by teachers with deep disciplinary
understandings” (Ball, 1993, p. 373). However, reform-oriented curriculum and
conceptually based tasks are not always sufficient to establish the spirit of reform within
28
practice (Spillane & Zeuli, 1999). Recall Spillane and Zeuli’s mixed-method study
described in the previous section. They found that often conceptually based tasks led to
procedural-bound discourses, that is, the instructional emphasis was on computation and
finding right answers, and not on thinking and explaining.
Ball’s (1993) often cited research investigation of issues that emerge in teaching
“mathematics in the spirit of current reforms” (p. 375) has much to offer as it is
theoretically framed by the intersections and interrelations of three elements of
mathematics teaching – mathematics content, discourse, and classroom community.
During a teaching experiment within a racially, ethnically, and economically diverse third
grade class, the teacher researcher engaged in reform-oriented teaching, and three
dilemmas of practice emerged: a) representing content, b) respecting children, and c)
creating and using community for mathematics teaching and learning. The focus for
Ball’s study was examining dilemmas of practice7
The teacher researcher for this study, Deborah Ball (1993), was engaging students
in making sense of negative numbers. Her learning goal for students was developing
understanding of negative numbers for use beyond the classroom. Although not Ball’s
theme for this study, arguably the representations she selected were a central element of
the resultant tasks from which the Discourse for mathematical learning and teaching
and classroom interactions offered a
site for teacher researcher reflection and analyses. For this discussion, the focus is on
findings from to the first dilemma and several tasks used for developing discourses
related to mathematics learning and teaching.
7 Dilemmas of practice occur when beliefs and reality clash related to implementing new approaches to practice.
29
emerged. The emergent Discourses also exemplify the interrelatedness of the three
elements of mathematics teaching – mathematics content, Discourses, and classroom
community. “The things that children wonder about, think, and invent are deep and
tough. Learning to hear them is, I think, at the heart of being a teacher.” (Ball, 1993, p.
374).
Reflecting on these ideas from Ball’s 1993 research, several implications emerge
for examining Discourses related to mathematics learning and teaching. First, listening to
students is a critical element of discourse and opportunities afforded for students’ voices.
Second, interpreting mathematically naïve thoughts of children is an art form teachers
develop and requires deep mathematical content knowledge. Lastly, developing tasks
using representational models or other process-based pedagogical approaches requires
reflexive contemplation before, during, and after instruction if mathematical Discourse is
an instructional goal.
Teaching mathematics using tasks alone can be problematic for developing
mathematical connectedness. Specifically for Ball’s (1993) study, curriculum was a
coherent organizer of the mathematics content for teaching that enabled a teacher
researcher opportunity for reflection about mathematical connections with the potential
for catalyzing Discourse.
Contemporary standards-based curriculum calls for mathematical pedagogies that
support organizing students for communal thinking and learning and for teachers to not
usurp thinking on behalf of their students (NCTM, 2000). In other words, reform-oriented
curriculum calls for development of Discourse for mathematics learning and teaching. A
30
challenging situation, Lloyd (2008) stipulates teachers’ efforts to implement
contemporary curriculum often face dilemmas of practice related to “teaching actions,
pedagogical strategies, and student engagement in mathematical explorations” (p. 164).
Using a different trajectory than Ball (1993) with respect to developing classroom
discourses, Lloyd’s (2008) investigation focused on a mathematics teacher’s
implementation of contemporary curriculum and analyzed emergent discourses along the
way. She articulated three types of discourses related to curriculum and pedagogical
approaches to implementation a) authoritative or univocal; b) contributive; or c) dialogic.
Authoritative or univocal discourse is a one-sided presentation from an expert or
known perspective with an expectation that what is given is correct and complete; no
need for outside contributions. This type of discourse is representative of traditional
mathematics teaching; the teacher and text serve as the one and only voice and final
arbiter for knowledge (e.g., Resnick, 1988; Schoenfeld, 1994).
Contributive discourse is an interactional discourse, but in the context of
classroom interactions can be procedurally bound (also see Spillane & Zeuli, 1999). For
example, teachers offer opportunity for students to share ideas and answers for the
purpose of clarifying mathematical procedures or skills (Lloyd, 2008).
Dialogic discourse is generative with respect to developing meaning for
mathematical ideas and concepts. “[I]n dialogic classroom discourse, the teacher
explicitly solicits and attends to students’ differing ideas and points of view” (citing
Peressini & Knuth, 1998 in Lloyd, 2008, p. 166). Worded differently, this type of
discourse is collaborative sense-making. Lloyd posits the need for including other
31
discourses beside univocal discourse in teaching practices given contemporary curricula
designed to support discourses that are more dialogic. This study while focused on
curriculum and is informative, the findings leave one to wonder whether reform-oriented
Discourses are comprised of multiple discourse types or are learning and teaching
maximized when all discourses are dialogic. There may be times when one discourse type
or combinations of discourse types are more effective within a classroom Discourse
community.
Curriculum and discourse types are not the primary focus for this investigation,
but Discourse is the focus as we examine classroom interactions related to mathematics
learning and teaching. Classroom interactions are nothing without conversations and
cooperative groups; the discussion about Discourse continues with this focus.
3. Conversations and Cooperative Groups Related to Classroom Discourses
Curriculum designed to influence teacher discourse was found to be ineffective
(Lloyd, 2008; Spillane & Zeuli, 1999). Conversely, according to Webb, Nemer, and Ing
(2006), teacher discourses readily influence student discourses within mathematics
classrooms. In a multi-phased teaching experiment in six ethnically and racially diverse
seventh-grade classrooms, researchers studied effects of classroom discourse resulting
from implementation of a cooperative learning program focused on altering student
behaviors related to discourse; changing teacher behavior related to discourse was not
part of the study. They found that teachers8 and students9
8 whole-group context
discourse patterns in grouping
situations were eerily similar. Specifically, if the teacher consistently used a recitation
9 Cooperative small-group context
32
style discourse during whole-class instruction, then the students discourse patterns were
similar. For example, the teacher verbalized his or her thinking about strategy, set up
procedural steps for finding answers to problems, wrote down numbers and calculations
without embellishment (e.g., like labeling), and requested students provide calculations.
The teacher almost never asked students to explain or describe, teacher questioning was
designed to correct mistakes and not uncover misconceptions, and help giving was in the
form of unembellished numbers or calculations (e.g., hints to the answer).
Unsurprisingly, the student cooperative group discourses followed the same patterns as
their teachers. Students asked only low-level procedural-based questions and offered
similar type help, provided no peer monitoring for understanding, and demonstrated
limited willingness to test help provided against their own thinking or understanding (i.e.,
help was accepted without question).
On the other hand, when cooperative groups are effective, their Discourses create
learning opportunities (Yackel et al., 1991). Yackel, Cobb, and Wood discuss the second
year of a three-year research and development project related to models of early number
development and constructivist learning theory. This second-year study was set in one
second-grade classroom of 20 students for one school year. The research team was
responsible for ensuring that all district learning objectives were addressed, while
attempting to follow their a priori theories and learning models. Unlike Webb’s research
team (2006), who decided to teach cooperative strategies using direct instruction,
Yackel’s research team decided to limit group size to two until the students and teacher
had negotiated cooperative group norms for effective collaborative work. However,
33
similar to Webb (1991), Yackel’s team found that in most instances groups comprised of
members that are more homogenous, or with a narrow range of variability, with respect to
mathematical development, ability, and social compatibility are more collaborative.
A goal of the project was to facilitate student learning through small group
collaboration (Yackel et al., 1991), which can be interpreted based on the research report,
as a more dialogic type discourse (Lloyd, 2008). Yackel’s research team articulates
“collaborating to learn” as “problem solving, consensus seeking, and genuine attempts to
communicate” is different from the type of learning seen in most traditional classrooms
during collaborative activities (p. 401). Further, they posit there are two types of problem
solving happening when cooperative groups are collaborating to learn: a) the expected
learning that deals with working toward solving mathematical problems, and b) a
sociocultural-embedded type of problem solving that entails figuring out how to work
together productively. Working productively includes individual group members
verbalizing their thinking, explaining or justifying their solutions, and asking for
clarifications.
Once groups resolve sociocultural challenges, there are opportunities for learning
through cooperative group interactions – creating shared meanings through conversation,
analyzing and evaluating errors and explanations, and working collectively to reach
consensus. The literature does not evidence a clear consensus about how to create
effective cooperative groups that are able to engage in productive work through
Discourse. However, Webb (1991) in a meta-analysis literature review linking task-
related verbal interactions and learning in small groups in mathematics classrooms, offers
34
insight into how cooperative group interactions are correlated to achievement and the
summary of that analysis is shown in Table 2.2. The table reflects how cooperative group
interactions correlate to student achievement; the first column describes group behaviors
that correlate positively and the second column shows the behaviors that correlate
negatively. Additionally, inspired by Webb, indirect and direct strategies for developing
cooperative groups that increase Discourse is articulated in Table 2.3.
Cooperative Group Interactions Correlated to Achievement (Webb, 1991) Positive Correlation Negative Correlations
Group members giving content-related explanations
Receiving less or different help than requested
Group members listening to each other
Group members ignoring or being non-responsive to help requests from members within the group
Group members providing explanations
Off-task discussion and behavior
Table 2.2: Cooperative group interactions correlated to achievement.
Discourse is certainly a characteristic of reform-oriented culture (ROC), and
developing classroom Discourses are essential. In this investigation focused on classroom
interactions and how they influence Discourses, we will explore organizing structures
used for instruction in hope of gaining insights related to mathematics learning and
teaching. Organizing structures such as cooperative groups are not sufficient for
developing mathematical discussions or Discourses, a specific search into developing
classroom Discourses within the mathematics teaching and learning literature offers
insights.
35
Indirect and Direct Strategies for Developing Cooperative Groups for Increased Discourse (Webb, 1991)
Indirect Strategies Direct Strategies Assign introverted students to
groups with members with lower ability
Provide students with direct instruction about group behavioral norms
Assign groups with a narrow range of abilities (e.g., medium and high or medium and low, but not low and high) and equal gender mix
Offer opportunities for students to practice cooperative group strategies (e.g., help-giving, explaining, asking questions)
Implement a group and/or individual reward structure
Offer feedback to students about their implementation of cooperative group strategies
Table 2.3: Strategies for developing cooperative groups.
4. Developing Classroom Discourses
The literature has much to offer related to teaching and learning related to
developing discourse. From a reform perspective, the purpose of mathematics classrooms
is to develop mathematical meaning through Discourse and both teachers and students
share this responsibility. Sfard (2001) described discourse as communication for thinking
that leads to learning; she wrote, “. . . there is more to discourse than meets the ears, and
that putting communication in the heart of mathematics education is likely to change not
only the way we teach but also the way we think about learning and about what is being
learned” (p. 23). This perspective suggests that making sense of the mathematics is
learning; mathematical sense making and learning are synonymous. If communication is
placed at the center of mathematics education then, when a group, like teacher and
students, are interrogating meaning, it is insufficient for only one person, the teacher, to
36
be the interrogator or holder of the meaning. Rather, all participants need bare
responsibility for meaning making and learning, which may be a different perspective
related to learning than that held by most. This perspective of learning is a communal
perspective.
Hufferd-Ackles, Fuson, and Sherin (2004) generated a case study of one third-
grade classroom within a small urban private school with a very large Latino population
who were not native English language learners. Four teachers from this school
participated in transitioning their traditional practices to practices centered on developing
math-talk community over a school year. The researchers described math-talk community
as teachers and students collaboratively supporting their own and others’ learning
through participation and engagement in mathematical discourse. Three themes emerged
for analysis for studying math-talk community development: a) evidence of community,
b) teacher actions, and c) student actions. The researchers suggested that the nature of
these themes was relational and they were key for their analysis; “mathematical
community development was linked to specific teacher and/or student actions” (p. 87).
For example, a particular teacher action would set the stage for student actions and the
discursive nature of the community would emerge from the teacher-student interaction.
Further, they identified four distinct actions related to math-talk community
development: a) questioning, b) explaining mathematical thinking, c) source of
mathematical ideas, and d) responsibility for learning. These distinct elements
independently within a mathematics classroom do not constitute an effective mathematics
Discourse as described by reform literature (NCTM, 1991; 2000). However, Hufferd-
37
Ackles and colleagues (2004) were able to define four distinct levels for describing
teacher and student progress toward implementation of effective math-talk for learning
within classrooms using the actions related to developing math-talk collectively.
Summaries of the four levels for teachers are in Table 2.4 and for students in Table 2.5.
Actions Questioning Explaining thinking
Source of ideas Responsibility for learning
Level 0 Traditional teacher
Teacher directed and used to maintain order and attention
Teacher may tell answers, student responses focus on answers only
Teacher shows and tells students how to do math
All responses are to the teacher to validate or refute with correct way
Level 1 Teacher- centered math-talk
Teacher led focused on student thinking with follow up
Teacher probes slightly or offers the explanation, limited sharing
Teacher is main source of ideas, may ask class for some ideas
Teacher sets up limited peer support, teacher feedback only
Level 2 Teacher models new student role
Teacher led probing and encourages student-student discourse
Teacher probes deeper to get at student ideas and thinking from multiple students
Teacher probes student ideas and builds on them, errors used for learning
Students held responsible for understanding and evaluating ideas of peers
Level 3 Teacher peer learner
Teacher guided, expects students to probe peers
Facilitates student thinking, encourages rich descriptions
Allow student ideas to guide instruction before or during
Students teach and co-evaluate peers ideas and thinking
Table 2.4. Summary of teacher levels of developing math-talk community (adapted from Table 1 in Hufferd-Ackles et al., 2004, pp. 88-90).
Synthesis of additional literature suggests that developing reform-oriented
classroom Discourses related to mathematics learning and teaching requires transitioning
the teacher’s role, using student ideas for the development of significant mathematical
38
concepts, and establishing social scaffolds10
(Ball, 2002; Empson, 2003; Lampert,
1990/2004; Nathan & Knuth, 2003; Sherin, 2002; Sherin et al., 2004). However,
transitioning teacher roles within the classroom are problematic for both teachers and
students.
Actions Questioning Explaining thinking
Source of ideas Responsibility for learning
Level 0 Passive student
Students give short answers to teacher only
Answers only, no thinking required
Students mimic teacher ideas and procedures
Teacher is responsible for all learning
Level 1 Passive student participant
Students respond to teacher questions, and listen or wait passively for their turn
Students respond to teacher probing with little elaboration about their thinking
Some student ideas are shared, but not lead to new thinking trajectories
Students support peer learning by teacher request, student show how they did the math
Level 2 Passive student engagement
Students ask questions of peers with teacher urging
Students respond to probing from teacher and begin to defend their ideas
Students show confidence and offer ideas and thinking even when different
Students listen more intently to peers and mimic teacher probing of peers
Level 3 Student-regulated learner
Students initiate peer discourse without teacher urging
Students defend and justify their conjectures without teacher urging
Students believe ideas are valued and offer them without being asked
Students listen to peers critically and together work to clarify and extend meaning for all
Table 2.5. Summary of student levels of developing math-talk community (adapted from Table 1 in Hufferd-Ackles et al., 2004, pp. 88-90).
10 Social scaffolds are help-giving behaviors between peers or within social networks
39
From a teacher’s perspective, Sherin (2002) described a dilemma faced by one
middle-school mathematics teacher intent upon developing a mathematics Discourse
related to learning and teaching within her classroom. Her dilemma was finding the right
balance between creating space for student sharing of mathematical ideas and learning
significant mathematics. While similar to Sfard’s (2001) perspective discussed
previously, that mathematical sense making and learning are synonymous, the teacher
dilemma described by Sherin is one step removed from Sfard’s. Specifically, Sfard’s
perspective assumes that sharing student strategies and solutions is sense making, while
Sherin’s teacher is trying to develop sharing through discourse in ways that create
meaning.
This dilemma described by Sherin is considered a process-content dilemma11
The teacher described by Sherin (2002) used a filtering approach for developing a
mathematics Discourse. Filtering begins by soliciting student ideas, the teacher filters
and
is faced by many teachers attempting to establish mathematics Discourses within
classrooms. Vignettes have been written by experienced teacher researchers (e.g., Ball,
1993; Lampert, 1990/2004; Sfard, 2001) to shed light on what these types of classrooms
might look like and are critically important models. These models cannot be taken as
procedures that one follows because every classroom is different with its unique set of
complexities; the models establish a vision for teacher reflection and framework for
establishing professional development trajectories for teacher support.
11 The process aspect of the Discourse are specific student-centered norms related to learning mathematics, such as knowing what counts as proof or sociomathematical norms; the content aspect of the Discourse is the mathematical substance, such as using appropriate mathematical vocabulary or mathematical practice.
40
(i.e., selectively incorporates) student ideas with her own to develop significant
mathematical concepts, and then students are encouraged to reflect on the ideas by
comparing and evaluating their thinking with the corpus of ideas. Sherin encourages
professional developers to consider filtering as an approach for supporting teachers
attempting to transition their practices to be more reform based.
Another approach for developing mathematics Discourse related to learning and
teaching was articulated by Nathan and Knuth (2003) as social scaffolding. The notion of
social scaffolding from the teacher’s perspective is to decrease the teacher role as analytic
center to increase opportunities for student participation without unduly compromising
the learning of precise mathematical concepts. Establishing social scaffolding can be
likened to a negotiation for establishing sociomathematical norms (McClain & Cobb,
2001).
The teacher dilemma within this research is similar to the process-content
dilemma discussed earlier; the researchers described this teacher’s dilemma as trying to
balance when she should participate and how with creating opportunities for student
participation. Nathan and Knuth found that when the teacher decreased her participation
as the analytic center, student-student talk increased, but the quality of the student-student
talk lacked mathematical substance. The teacher in this study was successful in
implementing social scaffolding and her students were able to engage in sustained
41
mathematical discussions. However, they were unable to sustain mathematically rich
discussions or engage in analytic scaffolding12
Empson’s (2003) research related to developing Discourse related to mathematics
learning and teaching focused on using revoicing to establish participant frameworks.
“Participant frameworks can be thought of as a unit of activity that defines relationships,
roles, and. . . the use of domain-specific content in teacher-student interactions”
(Empson, p. 308). Establishing participant frameworks involves two processes: a)
creating general classroom norms and sociomathematical norms through classroom
experiences; and b) establishing identity and position related to mathematics learning and
teaching. Revoicing supports these two processes.
.
Revoicing is a pedagogical approach for developing participation frameworks.
Simply stated, revoicing is restating student ideas orally, in writing, or symbolically for
the purpose of clarifying, expanding, or focusing significant mathematical ideas within a
collective environment. One aspect of revoicing that is significant for development of
classroom Discourse related to mathematics learning and teaching is animating; “a verbal
mechanism used by speakers to position participants” (Empson, p. 309). Teachers
animate students’ ideas by positioning students as mathematical authorities, claim makers
and justifiers, or contributors and helpers within the context of classroom collaborative
situations. For example, a teacher may explicitly remind the class to recall Rashida’s13
12 Analytic scaffolding for mathematics is helping or providing support for a learner to understand or create meaning for a mathematical concept; arguably, in order to support a new learner the scaffold must hold sufficient mathematical content knowledge to provide support.
13 Rashida is a fictitious student name created for this illustrative example of animation.
42
definition of triangles from the other day during a discussion about isosceles triangles.
Similarly, students, within the participation framework, may position themselves and
others using animation.
Challenges to developing Discourses include: a) time limitations related to
instruction; b) student participation norms; c) student and teacher beliefs and identities; c)
teacher pedagogical and content knowledge; and d) other challenges explicated within the
literature. Each of these approaches for developing Discourses for mathematics learning
and teaching is different, but a common element of these approaches is that change is
incremental and requires iterative reflective teacher attempts to develop Discourses where
both students and teachers are learning.
Additionally, each of these approaches shows how teachers struggle in balancing
their role when using student ideas and developing meaningful and precise mathematical
concepts. Even so, teachers continue to attempt to develop Discourses related to
mathematics learning and teaching. This Discourse development can be interpreted as
enculturation, or a reform-oriented culture – teaching mathematics learners to be
authentic communicators as they engage as mathematical thinkers and mathematics
doers. Thus far, we have thoroughly examined key elements of the mathematics teaching
and learning literature with respect to Discourses or community, the first sector of the
theoretical model that grounds this investigation. Next, we examine the literature in terms
of the second theoretical construct, classroom culture.
43
Classroom Culture(s) and Mathematics Practice
Returning to the mathematics teaching and learning comprehensive literature
review, Franke, Kazemi, and Battey (2007) concluded that opportunities for
mathematical learning were related to the development of classroom norms if
understanding is the focus for mathematics learning and teaching. In this case, arguably,
one can consider classroom norms as classroom culture. In this section, the third major
section of this literature review focused on developing reform-oriented classroom culture
related to mathematics learning and teaching. Culture is a key construct of the articulated
theoretical model and will be a focus for analysis within this investigation. The flow for
this section includes an exploration of classroom culture development related to: a)
argument and proof; b) problem solving and mathematical modeling; and c) discussion
and talk.
1. Argument and Proof and Classroom Culture
A Discourse-based culture may manifest as argument and proof within a
mathematics classroom. Researchers have recommended that argument and proof be the
focus for mathematics classrooms (e.g., Ball & Bass, 2003; NCTM, 2000; Schoenfeld,
1992) where students construct knowledge and engage in sense making (Hiebert et al.,
1997). Constructive engagement in mathematical argumentation and proof create
opportunities for students to clarify their thinking, make sense of diverse views, and
develop conceptual understanding; and when student thinking and ideas become the
focus for mathematics instruction, diverse ideas emerge creating opportunities for
44
students to disagree, experience cognitive dissonance, and transform their thinking
(Wood, 1999).
Arguments and proof in K-12 classrooms are different from those of professional
mathematicians; even so, when teachers use student-centered participative approaches to
create opportunities for argument and proof centered Discourses, rigor need not be
sacrificed (Sfard, 2007; Stylianides, 2007; Wood, 1999). Stylianides offers a
conceptualization for the meaning of proof in K-12 mathematics classrooms:
Proof is a mathematical argument, a connected sequence of assertions for or against a mathematical claim, with the following characteristics:
It uses statements accepted by the classroom community (set of accepted statements) that are true and available without further justification;
It employs forms of reasoning (modes of argumentation) that are valid and known to, or within conceptual reach of, the classroom community; and
It is communicated with forms of expression (modes of argument representation) that are appropriate and known to, or within the conceptual reach of, the classroom community. (p. 291)
This conceptualization of K-12 mathematical proof aligns with the relational perspectives
upon which this investigation rests. Translating Stylianides’ definition of proof into the
relational perspective language, proof is a coherent mathematical argument consisting of
statements that conform to classroom social and sociomathematical norms using
standards for reasoning consistent with classroom mathematical practices.
Stylianides offers a heuristic for teachers developing instructional practice that
cultivates a culture of proof (p. 317). Nevertheless, Wood (1999) concludes that
developing a culture for mathematical argumentation and proof within a mathematics
classroom is not simple because it is hard to change classroom interaction routines. Many
45
teachers attempt to develop mathematical Discourses through argument and proof but
they struggle to connect the Discourses to meaningful mathematical learning (Ball, 1993;
Ball et al., 2005; Empson, 2003; Stylianides, 2007). This type of mathematics classroom
Discourse does not emerge without strategic intentional effort by the teacher and in
collaboration with participative students.
Wood (1999) studied one second-grade teacher’s development of argument within
her classroom for over a year; the teacher had been engaged in a larger professional
development project for establishing reform-based practice for several years. At the start
of the school year, the teacher created a context for argument by establishing social
norms for engagement and expectations for participation. For example, to establish social
norms for argument, the teacher offered explicit statements to students about how
important it is during group work to communicate their mathematical thinking. This
teacher’s approach established explicit expectations for student participation. In addition,
to prepare her students for disagreements, she explicitly made clear the difference
between personal criticism and mathematical criticism; followed by opportunities for the
students to practice mathematical criticism.
Wood (1999) claims that “children are involved in learning what others expect of
them in terms of participation as well as learning the content of lessons” (p. 174). The
social contexts of classrooms influence student ideas related to their beliefs about
mathematics, the nature of mathematics, and how it is learned (Carraher, Carraher, &
Schliemann, 1985; Lave, Smith, & Butler, 1988; Schoenfeld, 1992). This suggests that
for students in traditional classrooms, where argument and proof Discourses are not the
46
central theme for instruction will hold different mathematical beliefs and understandings
related to the nature of mathematics, and how it is learned. The path is neither prescribed
nor clear, but the work of Wood (1999) and Sylianides (2007) offers insights and
guidance for developing Discourse through argument and proof within a mathematics
classroom. Argument and proof is not the only path to developing mathematics classroom
culture that result in effective Discourses related to mathematics learning and teaching,
problem solving and mathematical modeling offers another path. The literature is from
mathematics education, problem solving or mathematical modeling, and classroom
culture.
2. Problem Solving and Mathematical Modeling and Classroom Culture
Mathematics education reform literature has recognized problem solving and
mathematical modeling as a path for developing student understanding (NCTM, 1989;
1991, 2000; National Research Council, 1989) and reform-oriented mathematics teaching
and learning emerges from effective problem solving (Schoenfeld, 1994; Schroeder &
Lester, 1990). Teaching problem solving in classrooms is not always consistent.
Schroeder and Lester describe three problem solving teaching approaches: a) teaching
about problem solving – teaches problem solving steps, like Polya’s (1973) model; b)
teaching for problem solving – teaching focuses on acquiring mathematical knowledge
and creating opportunities for applying acquired knowledge to selected problems; and c)
teaching via problem solving – when problem solving is used as the mechanism for
learning mathematics and constructing understanding. Problem solving that engages
students in mathematical sense making and understanding is effective problem solving
47
and was alluded to in the literature. While most agree that problem solving should be a
focus for mathematics instruction, exactly what and how to achieve a problem-centric
practice is unclear (Lester & Kehle, 2003). Further, complicating consistent approaches
to classroom problem solving is the different ways that teachers conceptualize problem
solving (Wilson, Fernandez, & Hadaway, n.d.).
Teachers’ beliefs about the nature of mathematics influences their interpretation
and implementation of problem solving (Schoenfeld, 1992). For example, if mathematics
is conceived as a set of facts and procedures, problem solving is trivialized; Schoenfeld
likens this impoverished mathematics to English being taught with a focus entirely on
grammar. Conversely, mathematics “conceptualized as the ‘science of patterns,’ an
(almost) empirical discipline closely akin to sciences in its emphasis on pattern-seeking
on the basis of empirical evidence” (Schoenfeld, p. 335). Using this conceptualization
problem solving becomes a social activity that involves systematic observation, evidence
gathering, and experimentation in pursuit of generating rules (e.g., axioms, theorems,
etc.) models, and generalizations.
Teachers in pursuit of mathematical Discourses related to the latter
conceptualization of problem solving create opportunities for students to see
“mathematics as an exploratory, dynamic, evolving discipline rather than as a rigid,
absolute, closed body of laws to be memorized” (National Research Council, 1989, p.
84). These types of problem solving and mathematical modeling classroom Discourses
are cultures that support students’ development of mathematical power – students who
are quantitatively literate, capable of making mathematical value judgments using their
48
own interpretations, able to apply mathematics in practical ways, and flexible thinkers
who examine their own and others’ mathematical arguments analytically (Schoenfeld,
1992).
Lave and colleagues (1988) propose developing effective problem solving
Discourses through an apprenticeship model. Their apprenticeship model is similar to a
craft apprenticeship where the craft is doing mathematics as mathematicians.
Mathematics learners or apprentices, in this model, learn skills related to the mathematics
craft like understanding processes for resolving complex tasks, recognition of
opportunities that can be solved mathematically, understanding when and how to use
mathematical tools, and knowing how to exploit “properties of the presenting situation”
(Lave et al., 1988, p. 62). The researchers point out that the apprenticeship model
becomes problematic for classrooms because teachers’ mathematics content knowledge is
sometimes fragile and ill fitted to enable them to serve as master crafts persons. Even so,
they are enthusiastic about a model that supports and encourages learning by teachers and
students engaged in doing mathematics.
Considering Lave and colleagues’ (1988) apprenticeship model, this approach to
problem solving for Discourse development suggests mathematics apprentices would
likely develop understanding of mathematical principles through engagement in
mathematical practice. In their words, mathematical apprenticeship learning “. . . assumes
that knowing, thinking, and indeed, problem-solving activity, are generated in practice, in
situations whose specific characteristics are part of the practice as it unfolds” (Lave et al.,
1988, p. 64). Implicit in this perspective is the need for curriculum and instruction that is
49
contextually rich enough to incite engagement, sufficiently informal enough to create
space for children’s unique and naïve mathematical strategies, yet concrete and specific
enough to enable understanding from diverse perspectives.
Further, Lave and colleagues discuss several benefits related to mathematical
apprenticeship, two are highlighted specifically because of their significance for
classroom Discourses. First, apprentices typically possess some means for evaluating
their mathematical skills or ideas, which is critically important if student-student analytic
scaffolding is a goal of classroom Discourse as described by Nathan and Knuth (2003).
Second, apprentice knowledge and skills are measured holistically in context, no formal
tests are used to legitimate or screen learners’ ways of knowing, and the apprentice’s
expertise is judged by the fruit of his or her labor and weaknesses are perceived as
opportunities for additional instruction. That is, assessment is diagnostic, formative, and
never punitive; and apprentice knowledge is not constructed from a deficit perspective.
Finally, there is an inherent and shared commitment to the craft among master crafters
and apprentices and a goal of apprenticeship is for the apprentice to surpass his or her
master’s level of artisanship (Lave et al.).
Similar to problem solving, but slightly more specific and articulated in the
context of learning mathematics, is mathematical modeling. Lesh and Jawojewski (2007),
conceptualize mathematical modeling as an alternative perspective for problem solving
for developing understanding of any mathematical concept or process. They suggest that
a more traditional problem solving perspective is that problem solving is independent of
concept or context and is for practicing learned skills and procedures; and problem-
50
solving strategies are taught from a procedural perspective. They conclude that this
traditional approach segregates problem solving from important mathematical ideas.
Their integrated (problem solving for mathematical learning and understanding)
conceptualization of mathematical modeling occurs when “a problem situation is
interpreted mathematically, and that interpretation is a mathematical model” (Lesh &
Jawojewski, p. 765). Embedded in this definition, is the existence of a particular class of
problems, model-eliciting activities – realistically complex situations that engage learners
in mathematical thinking in order to “produce, refine, or adapt complex artifacts or
conceptual tools that are needed for some purpose and by some client” (Lesh &
Jawojewski, p. 784).
Solutions to model-eliciting activities are complex artifacts or tools related to a
mathematical process in a specific context. A sample problem is shown in Figure 2.4 to
assist your conceptualization about the nature of mathematical modeling problems and
solutions versus other types of mathematical problems. The complex artifacts or solutions
generated from mathematical modeling activities need to be flexible and adaptable for the
given situation, but also sharable and reusable for other situations with different data or
by different problem solvers (Lesh & Jawojewski). This aspect of mathematical
modeling is much like professional mathematical or scientific practice in that these
professionals are not finding isolated answers to problems, but rather they are developing
interrelated webs of knowledge for themselves and others within their mathematical
communities to build upon (Kuhn, 1996).
51
Mathematical modeling is cyclical and iterative and student mathematical
thinking, reasoning, and ideas evolve and change during the process (Lesh & Jawojewski,
2007). These cognitive changes are in part due to the nature of model-eliciting activities,
but also because of how they differ from conventional story problems, both routine and
non-routine. Conventional story problems are often pre-mathematized for students and
result in short mathematical answers that are either right or wrong. When students
encounter mathematical model-eliciting situations they must establish their own set of
assumptions and constraints, which evolve as the process proceeds. Additionally, they
must test their models and articulate a series of support materials like graphs, table,
procedures, and explanations sufficient to meet the demands of the situation. With respect
to the sample problem in Figure 4, the students must decide and articulate for whom their
solution is designed, such as state or federal legislators or community members interested
in starting a grass roots campaign to change the electoral process. This is very different
Elections
It is almost election time and it is time to revisit the electoral vote process. The constitution and its amendments have provided a subjective method for awarding electoral votes to states. Additionally, a state popular vote, no matter how close, awards all electoral votes to the winner of that plurality. Create a mathematical model that is different than the current electoral system. Your model might award fractional amounts of electoral votes or change the methods by which the number of electoral votes are awarded to the states. Carefully describe your model and test its application with the data from the 1992 election (in the attached table). Justify why your model is better than the current model. (COMAP, 2006)
Figure 2.4. A mathematical model-eliciting sample problem
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from most types of problem solving situations encountered in typical mathematics
classrooms in the United States.
Mathematical modeling is a type of problem solving, but seems to be more
specific, yet more open in terms of the mathematical skills and knowledge requirements
needed for student engagement. Students can participate at varying knowledge and skill
levels and are able to apply their understanding in ways different from that required by
more traditional story problems. In addition, mathematical modeling offers an element of
mathematical community similar to that experienced by professional mathematicians and
scientists. This type of problem solving environment arguably approaches that conceived
by Lave and colleagues (1988) in terms of the apprenticeship model described
previously.
Problem solving research has been underway in earnest since the 1970s and much
has been learned (Lester & Kehle, 2003). The earlier research used primarily quantitative
methods in search of problem solving techniques, classifying types of problems, and
heuristic-based problem solving methods; later research has used more qualitative
methods comparing successful and unsuccessful problem solvers and related strategies,
cognition and metacognition, beliefs and affect variables related to problem solving, and
situated sociocultural influences (Lester & Kehle).
In recent years, problem-solving research has been less prevalent; Lester and
Kehle propose one reason for this trend is the complexity due to the integrated nature of
problem solving with sociocultural and cognitive factors. Even so, this integrated style of
problem solving called mathematical modeling, creates an interesting trajectory for
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problem solving when investigating classroom culture and Discourses related to
mathematics learning and teaching. Implicit to mathematical modeling and other types of
problem solving is the need for sharing solutions, and more often than not, sharing will
include some form of communication. With that in mind, we explore the mathematics
teaching and learning literature related to discussion or talk and classroom culture.
3. Discussion and Talk and Classroom Culture
This section has discussed the literature related to mathematics classroom culture
from different mathematical contexts, including argument and proof, and problem solving
and mathematical modeling, culminates with this final section about discussion and talk,
which has been a persistent theme in this literature review about Discourse. Mentioned
earlier and a recurrent idea in the literature is that discussion and talk are intricately
related to learning (e.g., Brown & Hirst, 2007; Sfard, 2001; Sfard & Kieran, 2001;
Weber, Maher, Powell, & Lee, 2008; Zevenbergen, 2000).
Among others, Lampert (1990/2004) and Ball (1993) demonstrated in their
research the significance of respecting children’s ways of knowing, that is articulated
through their talk, and a means for children to construct mathematical meaning and
understanding. From a different perspective, learning and teaching from a constructivist
epistemology situates discussion and talk as a central element of the learning process
(Zevenbergen). Cazden (1988) suggests that the main purpose of schools, learning and
teaching, would not occur without communication; and essential for school
communication is talk, which connects the cognitive and social elements of learning.
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Further, discussion and talk are the medium for sharing, negotiating, and mediating
meanings for ideas and concepts (Zevenbergen).
Zevenbergen (2000), posits that because of a mismatch between teacher and
student talk, peer talk among students may improve mathematical access for students in
mathematics classrooms. Zevenbergen articulated three communication approaches used
in discussion or talk within mathematics classrooms: a) structured mathematical language
like that found in mathematics texts or tests; b) mathematical discourse that is governed
by implicit rules; and c) legitimate mathematical knowledge determined by specific
contexts embedded within the mathematical tasks used within the classroom. Each of
these communication approaches are used as a priori modes of mathematical speech in
classrooms; however, everyone within the classroom may not know or share these
conversational norms, thereby access is limited for those who are unfamiliar
(Zevenbergen); thus, challenging classroom equity with respect to access to mathematical
learning.
Different classroom cultures result in different patterns of discussion and talk
(Lave et al., 1988; Lesh & Jawojewski, 2007; Sfard, 2007; Sfard & Kieran, 2001; Wood,
Williams, & McNeal, 2006). Wood, Williams, and McNeal analyzed mathematics
lessons of 7- to 8-year-olds within classrooms from three types of classroom cultures,
including: conventional textbook; strategy reporting; and inquiry/argument. In this study,
the researchers investigated how the types of classroom cultures influenced Discourses
related to mathematics teaching and learning and the level of student cognition.
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They found that within a conventional textbook culture the discussion and talk
were largely comprised of regurgitation of facts with little Discourse and teachers efforts
to clarify often led to decreased mathematical rigor and cognitive effort for students. The
strategy reporting classroom culture discussion and talk was comprised of dyadic teacher-
student interactions with quality student cognition, but student collaboration was not
fostered. The inquiry/argument classroom culture discussion and talk created opportunity
for collective sense making and shared understandings through collective cognitive
processes. These collective cognitive processes include strategy development, evaluative
analyses about solution reasonableness, and reflective consideration of alternative
perspectives as a means for strengthening mathematical arguments.
Sfard and Kieran (2001) studied interactions between a pair of 13-year-old boys
over two months. They were interested in understanding the boys talk from two
perspectives, the level of mathematical content and the meta-messages inherent within
their conversations. The meta-messages are the elements of a discussion that are
unspoken and allow the researchers to examine things, such as the level of engagement
and participation, which reveals evidence related to the effectiveness of discussions and
talk. That is, how effective are mathematical discussion or talk between students as
related to mathematical learning and understanding.
The researchers realized early on during their study that “learning-by-talk cannot
be taken for granted” (p. 42). This realization altered the initial framework for their study;
they no longer held the notion that “all” talk related to mathematics within classrooms is
effective for learning. Additionally, they came to consider the nuanced nature of talk and
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the implications to communication and effective learning. This study made clear the need
to look beneath the spoken words for additional meanings. For example, in their analysis,
they recognized that one of the boys regularly used techniques to be a non-participant
through avoidance, uninitiating, and unresponsive contributions during discussions with
his partner. The non-participative boy’s talk on the surface appeared innocent, but the
analysis of his spoken words coupled with his actions and behaviors showed camouflaged
intentions to disengage, which contributed to ineffective communication between the two
students.
Cazden (1988), a researcher, left the academy for one year to teach in an
elementary school for a year to study communication and to test ideas she had been
researching. Her study was premised on three assumptions: a) talk is the primary means
of communication in schools; b) teachers control the flow of talk within classroom,
including that which enhances and hinders learning; and c) talk is a critical element of
identity. She found much in her study related to socio-linguistic topics, including a
caution worth highlighting related to classroom culture; beware, within classrooms
teachers are natives and students are immigrants. The implication of such a metaphor
likely resonates very strongly with researchers belonging to cultures outside the dominant
culture, and she points out that classroom culture should be a community where everyone
learns and grows and the environment should support growth for all.
This brings to mind another metaphor for classroom culture that applies to both
teacher-student and student-student interactions, the United States, a country full of many
immigrants and few natives and power relations are relative. Are we a melting pot, where
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everyone contributes to the whole and the resultant culture represents contributions from
the many; or are we to assimilate and the resultant culture represents a dominant powerful
few? Inherent in this metaphorical dilemma and our classrooms are issues of
relationships, authority, and identity. Again, we turn to the mathematics teaching and
learning literature, and this time our focus is on classroom relationships, identity, or
authority and mathematics classrooms. This discussion about relationships that support
learning opportunity is the final key construct of our theoretical model and for this
literature review.
Classroom Relationships and Mathematics Practice
The final focal point for this investigation from the comprehensive literature
review by Franke, Kazemi, and Battey (2007) is establishing learning supportive
relationships. The relationship construct is the final element of the theoretical model that
grounds this investigation. Arguably, classroom relationships are essential for creating (or
hindering) learning opportunity and opportunity is essential when learning for
understanding is the goal, which is the crux of mathematics education reform.
This portion of the literature review is about relationships that are essential for
Discourses related to mathematics learning and teaching. Wood, Williams, and McNeal
(2006) concluded that classroom cultures create different interaction patterns and
Discourses. By extension and synthesis using a relational perspective, one might posit
that from different classroom cultures emerge different relationships, authorities, and
identities. This section will consider mathematics classrooms with respect to supporting
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learning opportunities for learning and teaching in terms of: a) relationships; b) authority;
and c) identity.
1. Relationships that Support Learning Opportunities
Learning mathematics in a traditional sense is knowledge acquisition, and all that
is needed is an individual learner and the content; a view Sfard (2007) describes as an
acquisitionist perspective. From this perspective, according to Sfard, context and
situation have little baring and research is not hampered by the messy constructs of
sociocultural elements. In contrast, she describes a participationist perspective that is a
sociocultural perspective, where relationships and interactions matter greatly and
influence mathematics learning and teaching. Discourses emerge from relationships and
interactions, and create the impetus for concern about who is and who can participate in
learning within mathematics classrooms; hence, learning opportunities are afforded or
denied based upon relationships.
Opportunity for learning is limited or not afforded when different Discourses
(e.g., diverse ideas, thinking, and ways of knowing) are not valued and treated inclusively
within the dominant classroom mathematical Discourses (Ball, 1993; Hill, Sleep, Lewis,
& Ball, 2007; Sfard, 2007). Creating opportunity for learning is not simplistic, cannot be
prescribed, and varies from one classroom to the next, and there are numerous examples
in the literature (e.g., Empson, 2003; Hufferd-Ackles et al., 2004; Manouchehri & St.
John, 2006; Nathan & Knuth, 2003; Webb et al., 2006; Yackel et al., 1991).
One specific opportunity for learning and of interest for this study involves
teachers creating opportunities for student engagement in constructing meaning of
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mathematical concepts and ideas (T. P. Carpenter et al., 1989; Lampert, 1990/2004).
Another interesting avenue for this study is the opportunity for learning related to if and
how teachers create opportunities for students to struggle in their pursuit of mathematical
understanding (Hiebert et al., 1997; Smith, 2000; Stigler & Hiebert, 1999). The big ideas
for opportunity for learning are for teachers to recognize the need for and to relinquish
control over classroom learning interactions, and to trust that student contributions will
substantially enhance the learning of mathematics. However, these ideas are not about
relationship, per se as much as they are about teacher-student power relations, often
thought of in terms of authority. We will examine the mathematics teaching and learning
literature related to authority.
2. Authority that Supports Learning Opportunity
Power relations among teachers and students are present in every classroom and
influence teaching and learning (Ares, 2006; Gutstein, 2007; Hodge, 2006), or
opportunities for learning. “For classroom learning to take place, teachers must persuade
students to cooperate, and students must be willing to give their assent to what is
‘deliberately taught’ (Erickson, 1987)” (as cited in Pace & Hemmings, 2006, p. 4). This
quote exemplifies the notion that authority is not power, but perhaps a negotiation
process in some classrooms.
The words power and authority are often used interchangeably. In situations
where one perspective is persistently privileged over another (Hodge, 2006), which is
often the case in classrooms among teachers and students, it is easy to see why authority
and power become interchangeable. With respect to this study about classroom
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interactions, how classroom authority is negotiated and how those negotiations influence
mathematics learning and teaching will be analyzed.
Authority is a social relationship where some are granted legitimacy and others
consent to follow. In classrooms, authority is continuously negotiated among teachers
and students and the balance between legitimacy and consent teeters in response to
teacher and student actions (Pace & Hemmings, 2006). Students decide if and how they
will participate in classroom interactions, sometimes participation can hinder learning
and teaching; an example is students with negative identity perceptions may opt to act in
ways that hinder classroom learning and teaching through resistance, alienation, or
avoidance (Hodge, 2006). Pace and Hemmings concluded that teachers’ authority options
in classroom interactions are used for meeting instructional goals and maintaining social
control within classrooms.
Authority Types
Pace and Hemmings (2006) articulated several authority types including:
traditional, charismatic, legal-relational, and professional. They described a traditional
authority classroom with a teacher who uses directive-teaching strategies and students
comply with obedience because the teacher is the teacher. A charismatic authority
teacher garners legitimacy through personality and charm; students comply because their
needs and interests are met. The legal-rational authority teacher gets legitimacy because
of hierarchical rankings, rank infers the responsibility to reward and punish, and
compliance is assumed. The professional authority teacher has strong mathematical
content and pedagogical knowledge and his or her legitimacy for authority is due to that
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expertise. Teachers typically do not adopt one authority type, but rather a hybrid of
several authority types for interactions with students in classrooms.
The charismatic authority teacher would likely use a distributed leadership
approach. This leadership style enables followers to participate in leadership (Bass,
1990). With respect to the classroom, teacher leadership is about learning and teaching,
thus, the distributed teacher might support student participation in mathematics learning
and teaching in authentic and significant ways. The legal-relational authority type uses
approaches similar to those used by transactional leaders. According to Bass (1990),
transactional leaders use extrinsic rewards and punishments as persuasive tools for
garnering control. The transactional teacher would be a master of negotiation and rewards
for participation. The professional authority type is similar to the transformational leader.
This type of leader attempts to work collaborative, supports followers, and creates
opportunities for followers to meet their own goals and to lead (Bass). The
transformational teacher is the exemplar for teaching and capable of developing reform-
oriented culture and practice.
This investigation is about how classroom interactions influence Discourses
related to mathematics learning and teaching and in that vein, we pursued teacher
authority and leadership types. However, the discussion would be incomplete if we were
not to discuss social control, but note well that this aspect of the classroom power
relations is beyond the scope of this investigation.
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Classroom Social Control
A cultural norm of teaching held by many people is that teachers are judged, in
part, by their ability to maintain orderly classrooms (Pace & Hemmings, 2006). They
articulated several strategies used by teachers for maintaining social control including
exchange, influence, coercion, and structural. These strategies, according to Pace and
Hemmings, are sometimes used to strengthen or substitute for authority. Exchange was
described as transactions, within a classroom, give students something they value in
return for compliance. Some teachers use personal influence, like charm, good looks, or
humor to influence student cooperation. An alternate approach from the other end of the
spectrum is coercion, teachers use punishment, removal, and other negative treatments to
gain student compliance. Structural approaches used by teachers for social control
include room organization, and classroom routines, or strategic partner selections. As
teachers make decisions related to social control strategies and authority types they
influence classroom interactions and the Discourse of mathematics.
Authority Influencers
Pace and Hemmings (2006) identified several influencers of authority that affect
teachers and students. For teachers, they enact authority types and control strategies and
additionally their conceptions about learning and teaching influences classroom power
relations. For students, among other things, their orientations to schooling and
participation influence classroom power relations (Hodge, 2006).
Recall cultural capital is a value-laden construct related to environments; cultural
capital is the accepted ways of participation (e.g., speaking, arguing, proving, etc.) within
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a community (Hodge). She proposes both teachers and students participate to continually
regenerate mathematics cultural capital.
The issue of how students participate in the cultural capital of the mathematics classroom is of primary importance since it has consequences for their mathematical learning . . . students’ membership in the classroom as a mathematical community and . . . relates directly to their access to opportunities to learn significant ideas in mathematics and to develop an appreciation of mathematics. (p. 378)
This suggests students’ participation or lack of participation, identities (mathematical and
personal identities), and access to cultural capital influences their mathematical learning.
Implied by the passage are teachers’ roles and responsibilities related to students’
learning and also their identity constructions. Specifically, teachers must create and
facilitate inclusive classroom environments that nurture productive student identities.
Productive student identities allow for community where students and teachers
collectively construct mathematical ideas and understanding through Discourses. Further
elaboration follows from the mathematics teaching and learning literature related to
identity and learning opportunity.
3. Identity that Supports Learning Opportunity
Considering a reform perspective of mathematics community with teachers and
students as collaborative doers of mathematics requires an appropriate conception of
identity. For this study, a relational perspective of identity is adopted; identity is an
ongoing process linked to sociocultural interactions, is not a personal characteristic, and
changes situationally (Hodge, 2006; Nasir, 2002). Hodge, inspired by Lave and Wegner,
1991, described identity development in terms of participation in a particular community
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– as participation becomes more substantial, identity evolves. Nasir, inspired by Wegner,
1998, suggests identity develops through agency and participation. They all agree that
identity and learning are linked; “learning is about becoming as well as knowing” (Nasir,
p. 219). Further, students possess a myriad of identities and participate in multiple
communities, and they must find ways to reconcile their participation when their
memberships conflict (Hodge). Reconciliation can result in ways that encourage or hinder
learning in mathematics classrooms. Recall, some of the less productive ways to
reconcile conflicting memberships are resistance, avoidance, and alienation (Hodge).
According to Sullivan, Tobias, and McDonough (2006), classroom culture is
another explanation for student identity that is non-participatory. These researchers
studied 50 students about age 13 from low-socioeconomic Australian communities, their
teachers reported having difficulty engaging them mathematically, and many students
experienced difficulty learning mathematics. This study sought to understand how
“students’ perceptions of the extent to which their own efforts contribute to their success
in, and enjoyment of, school in general and mathematics in particular” (Sullivan et al., p.
82). Sullivan and colleagues used Dweck’s contrasting theories of intelligence, fixed
versus incremental (growth) intelligence, to analyze student identities.
People with fixed intelligence perceive themselves as possessing a set level of
intellect and require extrinsic sources of approval, shy away from challenges, and their
mathematical self-image and esteem are linked to achievement. Fixed intelligence
persons focus on “performing and looking smart” (citing Dweck, 2000 in Sullivan et al.).
Conversely, people with incremental intelligence perceive themselves as controlling their
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capacity to learn through effort, embrace struggle as a path to understanding, and do not
see failure as a personal indictment. Persons who embrace this identity endeavor to
increase understanding and learn new things.
Sullivan and colleagues (2006) consider implications for student identity positions
in terms of classrooms. Teachers and other adults who overly value student achievement
and positive feedback may be positioning fixed identity students to adopt beliefs that are
counterproductive for learning. For example, students feel that learning should come
without effort and teachers are supposed to simplify tasks that are challenging or tell
students how to get right answers. Teachers and students working collaboratively to build
this type of culture may be a breeding ground for rote memorization and limited or no
understanding.
Sullivan’s team found that students described as non-participatory did not exhibit
such behavior in the one-on-one research settings. The data supported the conclusion that
most students in their study identified as fixed intelligence persons, but they engaged and
persevered through challenging tasks. From this, the researchers concluded that many of
the students in their study opted not to engage or participate in mathematics learning
activities. Even so, students were able to offer significant insights and suggestions related
to ways that both teachers and students could improve students’ effort, engagement, and
participation to learn mathematics. Thus, this study shows the intricate relationship
between identity and learning opportunities through participation and concludes our focus
on relationships that support learning opportunities. A comprehensive summary of this
literature review follows.
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Literature Review Summary
This literature review was designed to support our investigation about
understanding how classroom interactions influence Discourses related to mathematics
learning and teaching within a classroom supported by an MCP mathematics instructional
coach. Our assumption is that the support from an MCP instructional coach influences
mathematics classroom interactions, but the specifics of that influence are unknown. This
literature review brings clarity and defines the scope for this investigation by defining
specifics areas for focus given broad research questions and messy sociocultural
constructs (i.e., Discourse and community, classroom culture, and relationships that
support learning opportunities).
The Discourses or community focus areas included teacher and student behaviors
and Discourse types and influencers. The empirical studies revealed a theme that
changing behaviors is challenging. From the theoretical and empirical literature, one
specific aspect of Discourse influencers to explore are the types of tasks used and their
affect on enhancing or inhibiting Discourses (or community) because they are directly
related to mathematics learning and teaching. The focal areas for the classroom culture
construct include sociomathematical and social norms, mathematical practice, and
cultural influencers. One significant and specific cultural finding is mathematical practice
comprised of regurgitating facts results in diminished rigor (Wood et al., 2006). Finally,
the relationships that support learning opportunity focal areas include identity
perceptions, authority, and learning and teaching influencers. An overarching theme
throughout both the empirical and theoretical literature related to this construct was that
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participation and identity influence learning and teaching by way of learning
opportunities. That is, opportunities for learning are not available to all learners, are often
transparent or hard to see, and are subject to issues of equity. Thus, looking at
participation, authority, and identity may reveal the nature of learning opportunities.
The theoretical model articulated in this literature review created focus for the
investigation, analysis, and writing of descriptive case studies. Additionally, the
theoretical model guided the cross case analysis and subsequent interpretative responses
to the research questions. The theoretical constructs from the model and this literature
review included: a) Discourses or community; b) classroom culture; and c) relationships
that support learning opportunity. The scope of this investigation was informed by the
findings highlighted within this literature review. The next chapter describes the
processes, procedures, and methodology used for understanding how classroom
interactions influence Discourses related to mathematics learning and teaching.
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CHAPTER 3: METHODOLOGY
Overview
The purpose of this study is to understand how classroom interactions influence
the Discourses related to mathematics learning and teaching in MCP supported
classrooms. To that end, the plan was to use qualitative methods and descriptive statistics
to develop a comparative case study of three MCP supported classrooms. Descriptive
statistics were used to situate the three focus classrooms among the population of schools
in the participating school district and other MCP supported schools from other school
districts. The cross case analysis served to summarize, compare and contrast, and draw
conclusions from the findings articulated within the case studies. The goal of this
investigation was to examine how classroom interactions influenced the Discourses,
cultures, and relationships using the lenses described by the theoretical framework and
the following research questions:
1. How do classroom interactions influence the Discourse related to mathematics learning and teaching in MCP supported classrooms?
a) What is the nature of classroom culture in each classroom?
b) What is the nature of Discourse or community in each classroom?
c) What is the nature of relationships that support learning opportunity in each classroom?
2. How do classroom interactions that influence Discourses related to mathematics learning and teaching compare among the classrooms?
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Research Design & Rationale
Introduction
The research questions that guide the investigation were best addressed using
qualitative approaches, but some quantitative methods were useful for establishing
context. According to Glesne (2006) and others, qualitative research is most effective in
understanding and making sense of people within social settings using complex and
interrelated variables that are hard to measure (e.g., Erickson, 1986; Stark & Torrance,
2005). Clearly, classroom communities, cultures, and relationships are each complex
interrelated constructs that influence Discourses in mathematics classrooms, and
independently or in combination are not easily measured. Further, learning related to
practice offers explicit challenges for many reasons that have been well documented in
the literature (e.g., Benzie, Mavers, Somekh, & Cisneros-Cohernour, 2005; Torff &
Sternberg, 2001; Woolfolk Hoy, Davis, & Pape, 2006).
A synthesis of several researchers’ findings suggested traditionally, scientific
research was quantitative and was geared towards building upon prior findings to
establish “truth,” but qualitative research is geared toward expanding understanding and
knowledge (Crook & Garratt, 2005; Erickson, 1986; Kuhn, 1996; Lather & Moss, 2005;
National Science Foundation, 2003). This investigation is best described as a case study
including comparative case study analysis governed by an interpretive research paradigm.
In this section, the discussion begins with a description of the interpretive research
paradigm and the implications for this investigation. Then, case study methodology and
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the implied practices are discussed in relation to the research questions. Finally, case
study is described and the rationale is presented for its use in this investigation.
Interpretive Research Paradigm
The interpretive research paradigm goes beyond meaning making. Many have
described research paradigms as sets of commonly held beliefs by researchers about
epistemology, axiology, and ontology (e.g., Erickson, 1986; Kuhn, 1996; Somekh &
Lewin, 2005). Overly simplistic definitions for epistemology, axiology, and ontology are
philosophical ideas about the nature of knowledge and truth, values, and existence,
respectively. Interpretive research paradigm can be articulated as a process for
expanding, emerging, and refining understanding and knowledge for multiple purposes
through a chorus of voices and perspectives (e.g., Kuhn, 1996; Lather, 2004; Lincoln,
1995).
This definition is articulated in terms of epistemology (knowledge and truth),
axiology (values), and ontology (existence). Interpretive research from an
epistemological stance suggests that knowledge and truth are not fixed based upon
multiple perspectives; they each will emerge, expand, and be refined, perhaps over time
and more than once. The axiological stance for interpretive research suggests that things
appear variable or changing because of the implied inclusion of multiple people. The
underlying assumption is that no two people share the exact same values and values will
be privileged based upon the group’s power dynamics. Interpretive research paradigm
from an ontological perspective for what “is” will be mediated and then emerge from the
collective; it is the group’s recognition that brings a thing to existence. The culmination
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of these perspectives represent the stance of a researcher governed by the interpretive
paradigm, but note well that according to Lincoln (1995), interpretive inquiry is not and
may never be well defined, and continues to emerge.
The implication of using an interpretive research paradigm for this study means
that one must expect and anticipate that the methodology, theory, and/or the
understandings will emerge, expand, and need to be refined for the duration of the
project. Qualitative researchers engaged in an interpretive study must be flexible, but also
be knowledgeable enough to position the research within the field and consider building
upon applicable theories. Thus, decisions and choices about methodology and theory are
made before, during, and after the data collection and analysis processes and if they
change, are referred to as emergent design; and that is fine and acceptable from an
interpretive perspective.
Case Study Methodology
Case study can be used for both qualitative and quantitative studies (Stake, 1994).
This investigation will use qualitative case study to address the research questions for
reasons described previously. According to Stark and Torrance (e.g., 2005), a qualitative
case study is an ‘approach’ to research that “stresses social interaction and the social
construction of meaning in situ” (p. 33). They also suggested that this approach was
useful in examining the complexities of sociocultural situations in the process of
representing meaning in context, which was of import for this research. Case study was
the approach used for analysis and write up. That is, articulating the understandings and
insights that emerged from the data and analysis throughout this investigation.
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Stake (1994) categorized three case study types: a) intrinsic, a case that is
intrinsically interesting; b) instrumental, a case that advances understanding; and c)
collective, a number of cases when taken together advance understanding and theory. The
comparative case approach used for this investigation can be characterized as both
instrumental and collective. Stake warns that researchers should take care to not allow
theory creation to overshadow development of the case(s); understanding that emerges
from well designed cases was the primary impetus for deciding to use a case study
approach for this investigation. The research intent of this study may lead to theory, but
describing ill-defined and complex sociocultural aspects of classroom interactions that
influenced Discourses related to mathematics learning and teaching was the primary
focus for case study development and cross case analysis.
Implications and Limitations of Case Study
A significant implication of using case study was deciding which cases to include
and how deep to delve; often these decisions cannot be made reliably until after data
collection and analysis has begun (Stark & Torrance, 2005). Data were collected from
four MCP supported classrooms and after analysis the decision to use only three cases
was made in part because of the limited number of parental consented students, the
overall small class size, and the dynamic attendance; the rationale behind the decision is
describe in more detail later.
Given the ambiguity and elusive nature of case study design and development,
validity is a concern especially for case researchers; triangulation is recommended as a
good defense (Stake, 1994). Stake suggests triangulating data sources, analyses, and
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interpretations. For this investigation, triangulation was used extensively, especially
during the analysis phase. Another implication of this investigation was the decision to
use comparative case study. According to Stake, comparative case “readers learn little
from researcher-provided cases as the basis for comparison” (p. 242). He argues that
researchers engaged in comparative case studies typically do not choose intrinsic cases
because the purpose is the comparison and not the individual cases themselves. The
researcher for this investigation struggled with questions related to coverage and depth
for each case; but the theoretical framework governed each case and the subsequent case
comparison.
Research Participants & Sites
Teachers, Classrooms, and Schools
The purpose of this study is to understand how classroom interactions influence
the mathematical Discourses related to learning and teaching in MCP supported
classrooms. Four teachers were selected as primary informants for this investigation and
their students were primary, but indirect informants, as their actions (and inactions) were
the focus for close observation and analysis. The MCP mathematics instructional coaches
supported learning and teaching within the classrooms were secondary informants and
provided another perspective.
Participating Teachers and Classrooms
The selected teachers worked within a building assigned an MCP coach who had
participated in the Mathematics Coaching Program (MCP, Erchick & Brosnan, 2005) for
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at least one year, worked in an urban school district, and taught mathematics to middle
childhood students, grade five or six.
The middle childhood grades were selected for this study for several reasons.
First, MCP for the last three years has targeted the elementary grades, kindergarten to six.
The researcher is a high school (grades 7 – 12) certificated mathematics teacher,
possesses high school classroom teaching experience, and grades kindergarten through 12
coaching experience. Also, grade five has been consistently underperforming across the
state according to the state report card over the last three years since the inception of state
testing at that grade level (ODE, n.d.). However, MCP supported schools had been
consistently outperforming similar schools across the state that were not supported by
MCP (Brosnan & Erchick, 2008). Finally, grades four and five students last year, were
very likely the grades five and six students this year. Thus, grades five and six were
anticipated to and did offer insights with the greatest potential for informing other
underserved and/or low performing schools.
The MCP mathematics instructional coaches were predominantly elementary
certificated or licensed teachers who no longer have classroom responsibility. On
average, the MCP coaches have been out of the classroom for three years, and have been
serving within their districts in leadership roles since leaving their classrooms (Forrest &
Douglass, 2007). The vast majority of MCP instructional coaches work full-time with the
mathematics teachers at one elementary school, but a few coaches are assigned to two
schools or they teach part-time. Most MCP mathematics instructional coaches provide
support within and outside the classroom for mathematics teachers teaching within the
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coaches’ assigned building(s), but the number of teachers teaching mathematics within
each building ranges from 6 to more than 25 teachers.
Participating Schools
Additionally, some buildings have all self-contained classrooms – one teacher
teaches all subjects. Other buildings use a variety of teacher configurations for instruction
at various grade levels for specific subjects, like mathematics. For example, one teacher
configuration is one teacher teaches mathematics and science and another teacher teaches
English and social studies for two classes, and these two person teacher teams exist for
each grade level three through five. Another example of teacher configuration is one
teacher teaches mathematics for four classes that span grades three through five. Many
different teacher configurations exist within the MCP supported schools often driven by
the number of students in the building, scheduling special classes such as art or gym, and
other structural criteria.
The building sizes vary from small, about 200 students, to large, over 900
students. The building grade-level configurations vary also; for example, a building may
be an elementary building comprised of grades kindergarten to five or an intermediate
building comprised of grades five and six. Grade configurations within classrooms vary,
also. There may be split classes, made up of students in grades three and four, for
example, and the teacher is required to teach all students at their appropriate grade level.
The environment for teaching and coaching in middle childhood schools is multifaceted
and complex.
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Before schools can choose to participate in the MCP, they must be deemed
eligible for participation by the state. The selection process is set up to privilege schools
that are consistently low performing in mathematics as measured by the state
achievement test. These eligible schools’ student bodies typically are comprised of larger
than average representations of students from families with lower socioeconomic status
and belong to ethnic or racial minority groups. The MCP invites all schools identified by
the state to participate in the comprehensive teacher support and coaches’ training
program. For schools that choose to participate in the MCP, they receive state funding to
offset coaches’ salaries or other costs associated with a coaching model of teacher
support (e.g., coaches are typically teachers not assigned to classrooms, coaches’ health
benefits, stipends for teachers attending after school workshops, etc.).
The MCP Coaching Model, Coaches’ Training, and Teachers’ Support
The Mathematics Coaching Program (MCP) advocates that the MCP mathematics
instructional coaches support all mathematics teachers in the school who want to be
coached. The intent for MCP coaching is for schools to receive sustained, high-quality,
classroom-embedded professional development for teachers. The MCP coaching
approach was built upon an existing coaching model (Erchick & Brosnan, 2009a), but
coaches are not constrained to use the one model that they advocate.
The MCP Coaching Approach
There are several coaching models, such as cognitive coaching (Costa &
Garmston, 2002) and the heart of coaching (Crane & Patrick, 2002) used for instructional
coaching. The MCP purchased the book for coaches, provided training, and support for
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one coaching approach, Content-Focused Coaching (CFC), chosen because it was
specifically designed for mathematics instructional coaching, offers video examples of
coaches and teachers working collaboratively, and advocates a simplistic approach to
coaching (L. West & Staub, 2003). Using the CFC coaching approach, coaches are
encouraged to engage in three phases of interaction with teachers including: a) pre-
conferencing for planning for instruction; b) observing the teacher teaching, co-teaching,
or coach modeling collaboratively planned lessons; and c) post-conferencing for
collaborative reflection and practical problem solving related to instruction and student
learning.
MCP Coaches’ Training
The MCP developed mathematics instructional coaches through sustained
professional development over one and up to three years, based upon their schools
willingness to support their participation. Unfortunately, over time as mathematics
achievement improved, schools no longer qualified for state funding, and financially
strapped districts were not always able and/or willing to fund a coaching model for
teacher support.
The coaches’ development focused on introducing coaches to mathematics
education researcher’s best practices related to student-centered methodologies governed
by the MCP conceptual framework (Erchick & Brosnan, 2009b). The three key elements
of the MCP conceptual framework are learner responsive mathematics education with
respect to mathematics content, pedagogy, and sociocultural elements. The coaches’
training program designed to deepen the coaches’ professional knowledge for teaching
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and coaching affords them the potential for sharing their understandings related to
research-based mathematics learning and teaching approaches with teachers in their
schools.
The MCP mathematics instructional coaches provided classroom-embedded
professional development and support to mathematics teachers within teachers’
classrooms, while working with the teachers and their students, and using the school
adopted mathematics curriculum. To prepare coaches for this work, they received
training and support through several venues and structures including: a) regular large
group (i.e., 30 to 50 coaches from across the state) professional development in a central
location; b) professional learning with an MCP trained facilitator and a small group (four
to eight MCP coaches organized geographically); and c) personal learning experiences
gained from supporting teachers engaged in mathematics teaching and learning in a
variety of classrooms spanning multiple grade levels.
The large group professional development occurred two days each month during
the academic year, and the professional learning small groups met twice during the time
between large group meetings. Coaches were expected to work with classroom teachers
in their classrooms about four days per week and in pre-/post-conferences at least once
per week.
MCP Teacher Support
The MCP suggests that coaches focus their coaching effort on no more than three
to four teachers over a period of four to six weeks and then switch to a new set of
teachers. During the four to six weeks that an MCP mathematics instructional coach
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works with a teacher, the MCP expectation was that coaches were in teachers’ classrooms
all of the time during mathematics instruction unless they were attending one of the MCP
sponsored workshops or support meetings. While visiting classrooms, coaches observed
instruction, co-taught, or modeled instruction (sparingly) dependent upon teacher needs
and coach-teacher negotiations. These negotiations occurred during pre- and/or post-
conference meetings related to instruction.
Coaches optionally decide at the end of the four to six weeks whether to continue
working with one or more teachers for longer periods as warranted. Sometimes principals
influenced which teachers coaches supported, and in these situations coaches typically
abided by the principal’s request; however, the MCP coaches were not obliged to discuss
the specifics of their work with teachers with the principal. Even so, the MCP
mathematics instructional coaching goal was to positively influence mathematics learning
and teaching for students and teachers.
The MCP coaches, while not the primary informants, offered much to this
investigation directly and indirectly. From a direct perspective, the MCP coaches were
members of the school community and knew the teachers, they recommended potentially
good research informants, were able to provide support and access in establishing rapport
among informants, teachers, students, and the researcher. The coaches were also called
upon to support this investigation indirectly with limited logistical support and directly by
offering historical perspectives about internal classroom communities, cultures, and
relationships.
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Research Procedures
Research Overview and Timeline
An extensive research timeline was prepared and is included in Appendix D. The
timeline includes the following types of research related activities:
Start up tasks, such as information sharing, recruitment of strategic personnel, and garnering consent;
Site visits for information exchange, survey/instrument administration, pre-video set up, and video captures
Data collection through observation, interviews, and document analysis Data analysis processes, such as reviewing video/audio, selective
transcription, and developing claims and warrants Establishing validity through member checks and ERP reviews Write up, both informal and formal dissertation formatted
Start up tasks spanned from autumn 2008 through winter 2008, followed by video
captures beginning in February 2009 and continuing until March 2009. Data analysis
started during the data collection phase and continued until the final report was drafted in
September 2009. Note well, analysis was depicted as ending in the timeline only because
a dissertation must conclude. Preliminary writing began before December 2008, edited
and used to facilitate reviews by members of the expert review panel (ERP), and was
transitioned into the formatted dissertation for an autumn 2009 defense. This was a
research plan and as all plans are concerned was subject to change, and several tasks were
marked to indicate their critical nature to the timeline.
Data Collection
The informants needed for the development of a comparative case study about the
influence of classroom interactions on Discourses related to mathematics teaching and
learning included four mathematics teachers, three MCP mathematics instructional
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coaches, and 42 children (with parental consent) from the four teachers’ mathematics
classrooms.
Purposeful Sampling
All efforts were made to ensure that the four teachers did not teach at the same
school, but two teachers were from the same school. After teachers consented, the
coaches who supported them were invited to participate. The MCP mathematics
instructional coaches, interested in participating in this study had to be second year MCP
participants. This restriction significantly reduced the number of eligible coaches and
school districts eligible for participation, but second year coaches had experiences that
made the decision worthwhile. These coaches understood their roles as MCP coaches and
participation in the investigation was not overwhelming. Also, they were complicit in the
prior year’s successful student achievement gains, were familiar with MCP tenets and
expectations, were likely implementing the MCP coaching and teaching approaches, and
they had or were experienced in establishing rapport with the teachers in their buildings.
The number of coaches selected to participate was directly related to the number
of teachers. Coaches did not become eligible to participate until after one teacher
consented to participate from their assigned building. The recruitment began by soliciting
MCP coaches who were interested in participating in this investigation. The MCP
coaches were approached in writing during on of their monthly MCP workshops, and
interested coaches were asked to meet the researcher during dinner. If a coach expressed
interest in participation, an informational e-mail message about the research was sent to
all grade five or six mathematics teachers in the interested coaches’ school(s). The
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informational message described the research. The requirements for teacher participation
included: a) the teacher teaches within an MCP coach supported building; b) teaches
mathematics to grade five or six; and c) the teacher is interested in the study. The
informational e-mail message was followed by face-to-face visits with teachers who
responded and an invitation to participate. Four teachers were recruited.
Following data collection and during the analysis phase, the decision was made to
exclude one of the classrooms from the investigation for several reasons. First, this was
the only grade five classroom of the four classrooms. Second, the overall class size was
small (9) and the number of consenting students was very small (5) compared to the other
three classrooms. Finally, the student attendance patterns were too fluid because of other
mathematics interventions underway within the school. For example, there was a pull out
program, and on certain days, one or more students would leave during mathematics
instruction to attend small group instruction with a representative from a community
partner organization. The teacher would send assignments and worksheets with the
students. Another example was students leaving for mathematics intervention because of
Individualized Education Plans (IEP) developed for them to address learning challenges
through special education services. Thus, the three factors that led to the exclusion of this
classroom, included: a) the only grade five classroom; b) the small number of students
with parental consent; and c) dynamic student attendance. The latter two factors were
deemed critical influencers for an investigation about the nature of Discourse and
community, culture, and relationships. More simply, as the only grade five classroom this
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classroom would not be sufficiently representative of other grade five MCP supported
classrooms.
Data Sources & Purposes
Following the tradition of qualitative research, data was collected via observation
and interview (Somekh & Lewin, 2005). Additional data sources included a research
journal (i.e., notes about observations, interviews, informal conversations, and
reflections) and artifacts from the field (i.e., instructional handouts) (Altrichter & Holly,
2005). The research journal was officially started the day after the IRB approval was
received and continued throughout the research process.
The data corpus was comprised of the following data: videotaped mathematics classroom
observations, audio-taped teacher interviews, teacher responses to survey instruments,
audio-taped coach interviews, student responses to a survey instrument, research journal,
instructional handouts, and state achievement test data for MCP supported schools. Each
of the data sources were selected because each contributed to insights into and enabled
adequate responses to the research questions governing this investigation. For example,
state achievement test data were used for background information and site description.
The purposes for the data related to the research questions were summarized by data
sources in Table 3.1. The numbers of data captures or instances of data collection was
included beneath each data source.
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Data Source (# of captures)
Data Purpose in Terms of Research Questions (RQ) Data Formats
Mathematics classroom observations and corresponding instructional handouts (5 obs. per classroom)
RQ # 1, 2 1. How do classroom interactions
influence the Discourse related to mathematics teaching and learning?
2. How do classroom interactions influence mathematics teaching and learning compare among the classrooms?
Narrative, documents, and multimedia data: Video tapes Selective transcriptions
Teacher & coach interview(s) (1 each)
RQ # 1. a., b., c What is the nature of: a) Discourses; b) classroom cultures; and c) relationships that support learning opportunity within mathematics classrooms?
Narrative and multimedia data: Audio tapes Selective transcriptions
Teacher surveys & instruments (2 per teacher)
RQ # 1, 2 Categorical data
Student survey (1 per student)
RQ # 1. a., b., c, 2.
Categorical and narrative data
Table 3.1: An analysis summary.
Observations and Interviews
Observations were videotaped using a protocol that is documented in Appendix
A. The purpose for capturing classroom observations by video tape was to ensure
sufficient data were available for analyzing classroom interactions in ways that enabled
thorough interrogation of the research questions and in the end offered reasoned and
critical response. The research plan for this study called for 24 video segments of
mathematics instruction, but after the fifth videotaping, the teachers decided to forego the
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final taping. Thus, there were only 20 video tapes created. The videotaped observations
were the primary data source used for analysis and all other data were used to support
what was learned from or experienced within the classrooms.
Teacher and coach semi-structured interview questions are in Appendix B. All
interviews were limited to 30 minutes or less. The set of questions for both teacher and
coaches are very similar, each designed to extrapolate their perspectives about the
teachers’ practice related to teaching, students, and classroom environment. The coaches’
interview asked coaches about coaching. The teachers’ interview asked teachers about
teaching and coaching support. A unique and interesting part of the interview was the
request for a memorable story related to their work – a teaching moment for teachers and
a coaching moment for coaches. Of those interviewed, no one provided a story related to
classroom interactions. The goal of the coach and teacher interviews was to gain insight
into their perspectives related to Discourses, culture, and relationships without asking
directly.
Surveys and Instruments
The survey that was used with teachers is included in Appendix G, the Teachers’
Commitment to Mathematics Education Reform (TCMER, Ross, McDougall,
Hogaboam-Gray, & LeSage, 2003). This survey, developed and extensively tested for
validity and reliability by Ross and colleagues (Ross et al., 2003) has much to offer
professional developers intent upon influencing teacher beliefs and practices, such as
MCP coaches. Prior to creating the survey, the research team investigated mathematics
standards-based teaching by examining National Council of Teachers of Mathematics
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(NCTM) documents and reviewing the literature spanning from 1993 to 2000 to generate
an informed conceptual model, “blueprint for standards-based teaching” (Ross et al., , p.
347). The blueprint describes nine dimensions for elementary mathematics reform, listed
and described very briefly as:
Dimensions (D) of Elementary Mathematics Reform
D1 Program Scope: mathematics topics clumped together with more attention paid to less popular concepts, like probability, access to all mathematics for all student
D2 Student Tasks: complex, open-ended, real-life contexts, multiple solutions, not algorithmic, drill, and practice, decontextualized, single solution problems
D3 Discovery: Instruction that facilitates student construction of mathematical ideas through exploration, not transmission
D4 Teachers Role: co-learner and creator of mathematical community, not the sole expert
D5 Manipulatives and Tools: mathematical problems are solved with access to and aided by manipulatives and tools
D6 Student-Student Interactions: environment is organized to promote peer interactions, this is not off task behavior
D7 Student Assessment: authentic assessments, formative – integrated with instruction, assesses multi-levels of performance, not all paper-pencil, end of unit, summative, regurgitation of memory
D8 Teacher’s Conceptions of Math as a Discipline: a dynamic subject, not a fixed body of knowledge
D9 Student Confidence: teacher strives to raise student self-confidence in mathematics, neither impede nor repress it
(adapted from Table 1 in Ross et al., p. 348)
These dimensions look very similar to the critical features for classrooms
designed for student understanding (Hiebert et al., 1997), which was expected given the
MCP coaches’ training. This instrument was selected for this investigation because of its
alignment with the goals and purposes underlying the MCP coaching and teaching
approaches (Erchick & Brosnan, 2009a, 2009c). The instrument used for this
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investigation includes all items except for those offered for dimension five (D5),
manipulative and tools. These questions were removed because they did not correlate to
the overarching sociocultural focus of this investigation. Additionally, this exclusion
eliminated two questions, which may have increased teachers’ ability to respond to all
survey questions within the 20 minutes allotted during the introductory classroom visit.
The instrument that was used with teachers is in Appendix F, the Teacher Sense
and Efficacy Scale (TSES, Tschannen-Moran & Woolfolk Hoy, 2001). Tschannen-Moran
& Woolfolk Hoy defined teacher efficacy as a judgment of one’s ability to create desired
outcomes for student engagement and learning, and including students who are
unmotivated and behaviorally challenging.
The TSES is available in a long or short form version, and is comprised of three
subscales related to teacher efficacy including student engagement, instructional
strategies, and classroom management. For this investigation, the student engagement
and instructional strategies subscales aligned best with the theoretical framework and the
key constructs of interest. For example, one item from the instructional strategies
subscale reads, “How much can you use a variety of assessment strategies?” and another
from the student engagement subscale reads, “How much can you do to motivate students
who show low interest in schoolwork?” (Tschannen-Moran & Woolfolk Hoy, p. 800).
The instrument requested teachers respond using a 9 point Likart scale, with 1 (nothing),
5 (some influence) and 9 (a great deal). A researcher or professional developer, such as a
coach, interested in improving mathematics instruction may find great interest in such
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teacher responses. The developers of this tool statistically vetted it for validity and
reliability.
Student Survey
The student survey used in the investigation is included in Appendix E. It was
created using a subset of questions posed in a study by Peter Sullivan and colleagues
(2006). The purpose of their study was to interrogate the idea that some students choose
not to participate in mathematics learning. Sullivan and colleagues’ study was carried out
in Australia, and some of the language used for the questions was altered to conform to
American language patterns. Additionally, they performed tasks-based inquiries that were
beyond the scope of this investigation, and those types of activities and questions were
not included.
Data Analysis
A comparative case study of three of the four mathematics classrooms was
developed. The purpose of this investigation was to understand the nature of the
Discourses, cultures, and relationships related to mathematics learning and teaching and
to that end case studies were created for each classroom individually and then the cases
were compared in a cross case analysis. The three classrooms included in the case studies
were sixth-grade classrooms and each was described individually and then compared to
articulate how classroom interactions influenced Discourses related to mathematics
learning and teaching. The analysis followed the constant comparative analyses process,
iteratively reviewing the data through the theoretical perspective in an effort to construct
meaning from the data, and in the end allowing the findings to emerge (Huberman &
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Miles, 1994; Merriam, 2002b; Wolcott, 1994). More simply stated, the analysis was
comprised of repeatedly reviewing the data, looking for patterns and anomalies, making
inferences or claims, and using specific data as evidence to warrant the validity of claims
(Erickson, 1986).
Analysis Coding
The data analysis began after the first data were collected. The video tapes were watched
repeatedly and then coded using a priori established codes (categories) derived from the
literature. Additional codes (emergent codes) were added to articulate what was being
evidenced by the data. All of these codes are described in the Code Book in Appendix C.
Some of the emergent codes were added to the theoretical construct organized tree
structure and others were not as they were codes used for categorizing the types of
interactions episodes. Tree structures shown in Table 3.2 were created and organized by
theoretical construct (i.e., Discourse or community, classroom culture, and relationship
opportunity). The emergent codes added to the tree structure are displayed using an
italics font in Table 3.2. This structure shows relationships among constructs, codes, and
emergent codes.
The other emergent codes that were not added to the construct organized tree
structure were designed to generalize the classroom interactions. They offered a high-
level view of the interactions and all classroom interactions were coded using these
codes, shown in Table 3.3. This enabled the classroom interactions for each classroom to
be compared by type and focus.
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Discourse or Community (DC)
Teacher Behaviors (TB)
Discourse Types (DT) Student Behaviors (SB) Discourse Influencers (DI)
• Catalyst • Connector • Enforcer • Expert • Informer • Reflective • Revoicing • Validator
• Authoritative • Contributive • Dialogic
• Collaborations • Compliance • Discourse challenging • Efficacy • Helping & supporting • Persistence • Self-regulating • Thinking & ideas
• Answers only • Connected math • Exercise tasks • Problem-based tasks • Static math
Classroom Culture (CC)
Sociomathematical Norms (SMN)
Social Norms (SN) Mathematical Practice (MPr)
Cultural Influencers (CI)
• Questioning • Student conjectures • Student explaining • Teacher explaining
• Listening • Low risk
environment • Negative SN • Respect
• Argument & proof • Communication • Making connections • Multiple
representations • Problem solving
• Collaborative sense-making
• Dyadic talk • Fact/process
reproduction • Powerful other • Ownership for
learning • Teacher expectations
Relationships that Support Learning Opportunities (RO)
Identity Perceptions (IP)
Learning Influencers (LI)
Authority/Leadership (AL)
Teaching Influencers (TI)
• Chosen non-participation
• External identity • Math autonomy • Math identity • Negative PAID • Positive PAID
a
• Teacher action
a
• Student action o Peer pressure
• Charismatic • Distributed
(transactional) • Professional
(transformational) • Traditional (autocratic)
• Fixed intelligence • Growth intelligence • Math as chaos • Math as procedural • Math as science
a
Participation & Agency Identity
Table 3.2: Nodes (categories) depicted in a tree structure organized by theoretical constructs.
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Code Category Emergent Codes for Categorizing Interaction Episodes
Interaction Type • Student-student • Teacher-class • Teacher-group • Teacher-student
Interaction Focus • Interpreting meaning
• General explaining • Reflecting and/or evaluating • Mathematical problem solving • Social problem solving • Student explaining mathematical idea • Teacher explaining mathematical idea
Table 3.3: Emergent codes for generalizing classroom interactions
Analysis of Instruments & Surveys
Data was gathered using three instruments, two for teachers and one for students.
For teachers the Teachers’ Sense of Efficacy Scale (TSES, Tschannen-Moran &
Woolfolk Hoy, 2001) and the Teachers’ Commitment to Mathematics Education Reform
(TCMER, Ross et al., 2003) to offer insights about teachers’ efficacy and commitment to
reform implementation, respectively. These data were analyzed qualitatively first and
then quantitatively.
Analysis of Teacher Survey and Instrument
All questions, except two, one question from each the TSES and TCMER, were
coded (i.e., summarized into short phrases that captured their meaning relevant to the
interests of this investigation) and categorized (i.e., grouped by the theoretical constructs
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that govern this investigation). The summary of the coding and categorizing was captured
in Table 3.4.
Discourse and Community
Classroom Culture Relationships that Support Learning Opportunities
TCM
ER
1. Rich tasks 2. Relevant tasks 5. Learn from students 9. Cooperative student
learning 10. Answer focused 13. Teaching not telling
3. Share strategies 4. Merge standards 6. Independent student
work 8. Integrated assessment 11. Assessment rubrics
7. Student math efficacy 14. Accept math as is 15. Facts first 16. Teach explaining
TSES
4. Creates good questions
6. Offers alternate explanations
9. Supports critical thinking
11. Assesses students understanding
1. Motivates 5. Varies assessment 8. Varies teaching
approach 13. Differentiates
instruction 15. Challenges students
2. Influences students beliefs
3. Influences students to value learning
7. Supports students thru family
10. Supports difficult students
12. Supports failing students
Table 3.4: Interpretation of teacher instruments using the theoretical constructs.
Counting the total number of questions (items) from both instruments by
construct turned out to be evenly distributed Discourse and community (10 items),
classroom culture (10 items), and relationships that support learning opportunities (9
items). Refer to Table 3.4 for the list of coded questions (i.e., short phrases) and the
corresponding question numbers. The actual questions can be found in Appendices F
(TCMER) and G (TSES) using the question numbers included in the table for each
instrument.
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The Lickert-scaled scores for the items by construct were analyzed using
descriptive statistics (i.e., mean and standard deviation). This approach of the TSES
instrument and TCMER survey analysis preserved teachers’ response anonymity and
offered additional insights from the teacher’s perspective that further warranted claims,
thereby strengthening the findings.
Analysis of Student Survey
The student surveys were comprised of two parts, six Lickert-scaled questions and
four open response questions. The student survey is in Appendix E. The four questions
that required a narrative response from students were not used to inform the case studies
because in most instances the student responses were incomplete phrases or single
sentence responses, not sufficiently substantive on their own to draw meaningful
inferences.
Survey Question Code
I feel confident that I can learn most math topics. Math Autonomy
I can learn anything in math if I put my mind to it. Math Efficacy
If I find the work hard, I know that if I keep trying I can do it Math Agency
My friends say that I keep trying when math gets hard Math Identity
You are either good at math or not and you cannot get better by trying. Fixed Intelligence
If I can’t do the work in math I give up. Persistence
Table 3.5: Student survey questions and associated codes.
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On the other hand, the Lickert-scaled responses from the student surveys were
analyzed using the same type of combined quantitative and qualitative approach as
described for the teacher’s instrument and survey. The student survey had six questions
with a four-point Lickert-scale: strongly agree (4); overall agree (3); overall disagree (2);
and strongly disagree (1). Table 3.5 shows how each question was coded to one of the
qualitative codes (i.e., categories). Then for each classroom the survey responses were
analyzed by code (i.e., mean and standard deviation), inferences were made, and related
to the qualitative findings.
Analysis Process and Tools
The analysis process began with the theoretical framework and the literature
review, which manifested as a detailed Analysis Grid presented in Appendix H. In the
analysis grid, the theoretical constructs and concepts are outlined in the first column,
guidelines for the analysis tasks are listed in the second column, and the relationship
between analysis and research question responses comprise the final column. The
analysis process as depicted by the Analysis Grid can be summarized as follows. The
analysis process began with coding the data using the codes defined by the literature and
organized by the theoretical constructs. Additional codes emerged as the analysis
progressed. Then patterns were identified by looking at the numbers of categorical codes
assigned to the data. The next step was drawing inferences and finding specific examples
in the data, thereby creating claims and warrants. The final step was to triangulate the
findings and consulting with members of the expert review panel.
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The more evidence available to support a claim, the more strength can be
attributed to the claim (Erickson, 1986). With respect to coding, the idea of strength is
related to the number of times a specific code is associated with units of data (e.g., small
segments of video-taped observations, transcribed interview data, etc.). Thus, the greater
the number of coding references, the stronger the claim derived from the specific code.
NVivo 8 (QSR International, 2008) is a qualitative analysis software program
used to facilitate the analysis process with respect to identifying strong claims. In each
case, the strongest claims were found using NVivo 8 queries to determine the codes
(categories) with the greatest number of coding references. Summary tables were created
within each case to exemplify this type of analysis. NVivo 8 was instrumental in
determining which claims to make and then locating specific data associated with the
claim from which to select examples to share in the write up. Another aspect of analysis
included using descriptive statistics. Means and standard deviations were used to describe
teacher and student survey data. Percentages were used to describe data analysis and
coding results.
Validity and Reliability
This investigation was designed to ensure validity and reliability of the findings
through triangulation of theory, data sources, and analyses (Lather, 1986, 1993, 2006).
According to Lather, triangulation is achieved when two or more entities are used to
inform a research process. The theoretical triangulation was manifested through the
merging of the relational interpretive framework for mathematics classroom interactions
(Cobb & Yackel, 1996/2004) and the elaborated relational perspective (Cobb & Hodge,
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2002) and from the two theoretical perspectives emerged the interrelated theoretical
constructs – Discourses or communities, classroom cultures, and relationships that
support learning opportunities which govern this investigation.
The data source triangulation was accomplished through data collection, which
yielded a variety of data sources including: a) observations; b) teacher and coach
interviews; and c) teacher and student surveys. The data sources offered diversity of
perspectives from the variety of informants who contributed including three different
classroom teachers, students from each classroom, and two MCP mathematics
instructional coaches. Additionally, data sources were collected using different formats
including video, audio, documents, categorical, and statistical.
Analyses triangulation was accomplished by including both quantitative and
qualitative approaches and interpretations by the researcher and review by an expert
review panel (ERP). Efforts were made to offer only data-backed claims that emerged
from multiple data sources. Recall, claims are strengthened when they are warranted by
multiple coding references, the greater the number of coding references the stronger the
claim. The strongest claims emerged by examining NVivo 8 queries that highlighted the
number of coding references and identified them by data sources (e.g., observations,
interviews, etc.). The purpose of the ERP was to evaluate the strength of claims and
warrants by examining the evidence and then concurring or challenging the claims.
For this investigation, data sources include observations, interviews, and surveys.
Thus, an example of analysis triangulation might be a claim warranted by data from one
instructional episode in one classroom that showed student helping behavior and a second
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warrant of the same behavior evidenced within a second instructional episode from the
same classroom but on a different day. Another example might be a claim made about
student math efficacy found within an instructional episode that was also evidenced by
student survey responses. NVivo 8 (QSR International, 2008) was used to ensure analysis
triangulation, such that no claims were made that were not supported by the data (i.e.,
heavily coded) and each was warranted by at least two different data sources (e.g., two
different video segments, a survey and a video segment).
Methodology Summary
This chapter began by describing the research design and rationale. The research
design was governed by the interpretive research paradigm and the methodology used
was case study. Then, the research participants and sites, where the research was carried
out was portrayed and included information about the teachers, classrooms, and schools.
Additionally, the Mathematics Coaching Program (MCP) was explained, including an
overview of its coaching approach, coaches’ training, and expectations for teacher
support. Next, the research procedures, data collection, and analyses were portrayed in
some detail, including purposeful sampling; data sources and purposes; interviews and
observations; instruments and surveys; and analysis coding, processes, and tools. The
chapter concluded with a discussion about validity and reliability. In the next chapter, the
data and the three classroom case studies are presented.
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CHAPTER 4: RESULTS & FINDINGS
Overview
This chapter describes the findings from this investigation. The discussion begins
by setting the context for the investigation through brief introductions about the
classrooms that were the focus for this investigation. Then descriptions including some
details about the school district, the schools, and the mathematics coaching program
(MCP) coaches assigned to the schools are discussed. Next case studies for each
classroom are presented followed by a cross case analysis comparing the cases.
Additionally, the research questions guiding this investigation are addressed within the
case descriptions and cross case analysis.
The classroom teachers are never associated with the schools or the coaches in an
effort to maintain anonymity for the participants. The rationale for this decision was that
there were only two coaches, Ann and Lyn (pseudonyms) and two buildings; two
teachers teach in the same building and work with the same coach. Thus, by associating
teachers with coaches and/or buildings, those familiar with the investigation, schools, or
teachers may be able to infer teacher identities. The teachers’ classrooms were always
referenced by pseudonyms Ada, Eva, and Kia. Even so, this investigation was about
classroom interactions and not about teachers. Students were referred to by “Student”
followed by a letter, for example, Student A or Student C. The single letter may or may
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not be associated with the students’ name, but the letters are used consistently to refer to
the same student.
District and School Descriptions
The investigation was situated in a large urban school district that encompasses
62.47 square miles and over 5 million square feet of building space according to the
school district’s website. For the 2008-2009 school year the school district’s website
reports 23,850 students enrolled, 70 % qualified for free-reduced lunch, a graduation rate
of 76.4%, and attendance rate of 92.9%. The student demographic data for the district are
Race Gender Mobility
Asian/Pacific Islander 2.0% Male 51% 38%
American Indian 0.1% Female 49%
Black/African American 48%
Hispanic 1.4%
White/Caucasian 42.4%
Multi-racial 6.0% Table 4.1: Student demographic data for the school district that the investigated schools are a part.
displayed in Table 4.1. The student population enrolled in the district is primarily Black
and White with the majority being Black. There are slightly more boys than girls. The
school district was rated at the third level of five levels for district ratings assigned by the
state department of education for the 2007-2008 school year. This school district is one of
the 10 largest urban school districts in the state.
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The configuration of the school district was comprised of seven clusters, one for
each high school. There were 39 total schools, 8 middle schools, and 24 elementary
schools in the 2008-2009 school year according to the school district’s web site. I
collected data in two different schools, each within a different cluster in the school
district designated as cluster 1 and 2. The participating school sites were referred to as
West and East in Table 4.2 and both schools were designated as middle schools with
grades six through eight. Each of the schools was supported by a different coach.
Mathematics/Reading Scores School name (pseudonym)
Cluster Total # Schools
MS ESa 2005-2006 (%)
b 2006-2007 (%)
2007-2008 (%)
West 1 11 2 8 37.4/67.6 46.0/64.7 60.8/68.7 East 2 6 1 4 47.0/76.2 48.0/64.2 47.2/54.5 District 6th 1-7 grade 39 8 24 49.1/73.7 52.0/60.9 59.4/66.5 State 6th grade 44.4*/65.9 74.0/77.7 * 76.6/79.7 * State test scores from a state department of education designated similar school district. a MS – number of middle schools in the associated cluster b
ES - number of elementary schools in the associated cluster
Table 4.2: School-based state achievement test scores for mathematics and reading comparing data collection sites, the district, and the state for grade six.
Table 4.2 illustrates this urban school district’s trend of improvement in grade-six
mathematics as measured by the state achievement test. The district’s reading
achievement dropped after the first year but began to rise again for the final year reported
in the table. With respect to mathematics achievement at the participating school sites,
East has been constant with little change, while West has been steadily rising over the last
three years. Similarly, the reading scores at West stayed constant. However, at East there
was evidence of steady decline for the past three years. At the time of data collection, the
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Mathematics Coaching Program (MCP) had been established within this school district
for two years. The achievement test scores for both mathematics and reading for all grade
six classrooms across the state have risen for each of the three years reported in Table
4.2. Thus, across the state, grade six classrooms were experiencing sustained
improvement in mathematics achievement, but the gains were more tenuous within the
school district and for the school sites where this investigation was situated.
West Middle School
West was spread across two temporary buildings while a new school was being built.
The sixth grade classrooms were in one building and the seventh and eighth grade
classrooms were in a different building several miles away. A portion of the temporary
building where the sixth grade classrooms reside was a community center according to
the signage on the outside of the building. I never observed people going in or out of the
community center during school hours when I visited. There was a vice-principal
assigned to West, who managed the daily operations of the building. The school campus
(sixth through eighth grade) principal worked primarily at the other building, but he
influenced what went on at West according to the teachers and MCP coach.
The building was clean, the hallways were dark, but most of the classrooms
appeared to be bright and functional. The very limited wall space in the hallways boasted
an interesting array of student work. Outside one of the mathematics classrooms,
students’ descriptive “I Am …” poems hung all over the limited wall space above the
lockers. I do not recall the specifics of any of them, but I do remember feeling that they
would make a parent proud to see one from her child. The classrooms were spacious with
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lots of large windows and shades that appeared to be functional. Walking through the
hallways and looking into classrooms one could see technology throughout the building
including calculators, overhead projectors, SMART boards, document cameras,
computers, data projectors, and televisions with DVD players. The teachers and coaches
indicated that they used technology in this building to enhance instruction.
The building’s staff were always very friendly and helpful and teachers seemed to
have well established rapport with one another. The site of two or more teachers huddled
in conversation between classes in the hallway or during breaks was a common
occurrence. There was a coffee pot in the teachers’ lounge. Coffee drinkers pitched in to
keep coffee, condiments, and cups in supply. During visits to the school, I met and talked
with teachers in the lounge as I enjoyed the wonderful coffee on more than one occasion.
There was evidence of many empty coffee cans, which suggested some level of
participation and perhaps community among the staff. The sense of community extended
beyond the school. There was a church near the school and one day a tray of homemade
cookies was left in the teachers’ lounge for the staff with a thank you note from the
church. According to one of the teachers I met in the lounge, the church supported the
school by doing things such as this on occasion.
East School
East was housed in a very large and old building with many stairwells and
hallways leading to various points within the building. The principal was very present
within the school; one could always tell when she was around because she was visible.
According to staff members through casual conversations, nothing happened at East that
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was not sanctioned by the principal. For example, this research could not begin until she
met with me and asked questions. Then she insisted on speaking with the teachers prior to
me about this investigation. After meeting with her, it was clear that what motivated her
high level of involvement was concern for those within her school.
There were many classrooms in East. The sizes of the classrooms seemed to vary
based upon their location, but most seemed to be of average size, able to accommodate
classes as large as 30 students. The principal seemed to spend significant time out of her
office observing and monitoring the running of the building. There were usually student
workers working in the office doing a variety of tasks such as escorting visitors to their
destinations or carrying messages to classrooms. On the day I brought pizza to celebrate
with the teacher and students, one of the office workers helped me carry pizza and pop
from my car to the classroom. He was very happy to help and responded gratefully when
offered pizza for his assistance.
MCP Support
The coaches did not work with the participating teachers in ways that I had
anticipated during the design of the investigation. The expectation of the Mathematics
Coaching Program (MCP) for MCP coaches was that they work with a group of four
teachers for one mathematics period or block about four days each week for six to eight
weeks. During this sustained one-on-one classroom-embedded professional development,
the coach and the teacher collaboratively plan, teach, and assess student understanding
(Erchick & Brosnan, 2009a). Using what they learn about students’ understanding, they
collaboratively problem solve around issues of practice related to mathematics learning
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and teaching. Their shared objectives included trying research-based teaching approaches
and addressing issues that emerge related to implementing new instructional strategies,
and their ultimate goal was increasing student understanding and learning. Using this
approach, an MCP coach for a typical school year offers 12 to 16 teachers individualized
and focused support, assuming four 6-week coaching segments.
According to the coaches through interviews and casual conversations, there were
several reasons why teachers participating in this investigation did not receive the
prescribed type of MCP coach support. One reason was the coach was assigned so many
teachers she was unable to work with all of the teachers who wanted to work with her.
Another reason was in this era of accountability, the campus administrator has directed
the coach to work with particular teachers all day for intensive on-the-job training or
intervention. A third reason was the MCP coach was doing other school related things
(e.g., cafeteria or bus duty, class coverage, tutoring, etc.) and lacked time to support
teachers per the MCP prescribed way. The down side of such redirection from the actual
activity of coaching was that it diminished the opportunity for teachers to benefit from
focused individualized MCP coaching support.
Prior to and during the data collection phase, participating teachers described
MCP coaches’ support as infrequent classroom visits, resource support, responding to
questions via e-mail, and facilitating workshops at the district office during the summer
and on occasion during the school year. None of the participating teachers described their
coaching experiences as sustained one-on-one classroom-embedded professional
development as prescribed by the MCP (Erchick & Brosnan, 2009a).
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Several of the teachers indicated that they had attended workshops run by the
MCP coaches at the district office. The teachers who talked about the workshops
described them as very beneficial and they indicated that they had learned a lot from
attending. When pressed, no teacher could recount specifics of what was learned. One
teacher recalled a feeling of learning something significant, but she was unable to recall
what it was specifically. Another teacher talked more generically about using
manipulatives and questioning, but nothing more specific was recalled. Even so, the
teachers viewed the workshops as helping them in their practices. They also spoke
specifically about timely e-mail and resource support coaches provided.
On a day when I was videotaping in one of the classrooms, the coach co-taught a
lesson with the teacher and based on comments made during the observation it was
apparent that the two had spent some time planning for the lesson. Based on follow up
questions from the coach, she and the teacher worked collaboratively for several days
focusing on two reform-oriented teaching strategies. This was one-on-one and classroom-
embedded professional development, but not the sustained model advocated by the MCP.
On two other occasions, coaches observed lessons during videotaping. Following each of
these observation periods, I did not stay around to observe whether the coach and teacher
discussed the instruction. However, that was beyond the scope of this investigation and I
typically attempted to quickly pack up the equipment after videotaping and leave so as
not to disturb the next class or quickly get to the next taping session.
In summary, the MCP support was less than anticipated during the investigation,
but that does not diminish the purpose or the opportunity to examine how classroom
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interactions influence Discourses related to mathematics learning and teaching in MCP
supported classrooms. MCP coaching support was still present, albeit not as anticipated.
More importantly, each of the classrooms observed possessed a form of Discourse or
community, classroom culture, and relationships that supported learning opportunities.
This hindered the ability to discuss implications related to extensive MCP coaching
support within the cases, which was never a primary purpose of this investigation.
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Case Study 1: Ada’s Classroom
Ada taught three 90-minute blocks of sixth grade mathematics every day during
this investigation. I observed a total of about 450 minutes of mathematics instruction
during Ada’s last afternoon block. She described her last block class as challenging and
fun during our first meeting (Jan. 27, 2009). The observations were scheduled so that the
first two occurred on consecutive days, a Tuesday and Wednesday. The third was on the
following Monday and the final two occurred on the Monday and Tuesday of the
following week. The reason for clustering the observations was to see interactions over
time and to see if there were connections from day to day related to mathematics content,
learning, teaching, and/or Discourses.
The class was comprised of mostly boys (10) and fewer than half were girls (4), a
diverse representation of students by race and ethnicity including Blacks (4), Whites (7),
and others (3). There were at most 14 students present on observation days, but on
average, there were 12 students present.
During the observation phase of the investigation, Ada’s key topics for
mathematics instruction included a) isometric drawings of three-dimensional block
objects and estimating fraction operations; b) representing three-dimensional block
objects with orthogonal drawings; c) adding and subtracting fractions with unlike
denominators; d) finding exact and estimated products for mixed numbers; and e)
division with mixed numbers. I called these key topics because there were other things
taught on each of the days, but the listed topics appeared to be the primary focus for each
day; Ada typically did several different things each day.
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Ada’s Perceptions about the Class
In this section, Ada’s perceptions are shared about the observed class’
environment, her instructional style, classroom interactions, and student engagement.
These perceptions came from Ada via her pre-observation interview (Feb. 23, 2009) and
supported using observation data. Ada’s perceptions included structured diversified
instruction, avoiding failure, discipline, and respect.
Classroom Environment
Ada described the classroom environment during the pre-observation interview as
very strict (Feb. 23, 2009). As she described the environment there were clues about the
types of classroom interactions found in this classroom. For example, she stated, “I tell
them the rules and I expect them to follow it, I like them to be pleasant and polite. I like
them to stay on task and I definitely like structure” (Feb. 23, 2009). Ada’s classroom
environment and interactions, in general, during the observation period, were very
teacher-centered and directed and students had limited opportunities to construct their
own understandings but they were encouraged to participate in teacher prescribed ways.
Classroom management and discipline were also important to Ada. She shared a
story and the rationale for sharing it with students and the comments exemplified her
commitment to discipline and perhaps motivated her expectations for learning and
teaching.
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I try to give them [students] ah examples of things that I've done dumb in my life. For example, like in college I always told 'em [students] the story that I was very stubborn and I ended up having to take a class over again because I refused to get along with the teacher. So, I try and give 'em lots of stories like that that will help them along their way and especially with math. I bring in shopping alot and credit cards and real life strategies that I've learned along the way that relate to math… and I try to say please and thank you and not to yell. I try and keep my voice more calm and quiet most of the time.
(Pre-observation interview, Feb. 23, 2009)
These comments illustrated Ada’s sincere desire to support students by giving them the
benefit of what she has learned and by treating them well (i.e., not yelling and keeping a
calm voice). The story stressed the importance of compliance and implied a relationship
between compliance and success.
Instructional Style: Structured and Diversified
Ada described her instructional style as “diversified, I get bored easily so I figure
students do too, so I try to do about four different things during an instructional period"
(Feb. 19, 2009). Her observed instruction reflected “diversified instruction” as she
articulated. The 90-minute block was subdivided into three or four very distinct
instructional segments. The daily mathematics focus was never singular because of this
approach.
She articulated several instructional elements used regularly, including: a) whole-
class problem solving using the overhead; b) student-to-student conversations; c)
individual student practice; and d) hands on activities or technology (e.g., calculators,
smart board, etc.). Of the four instructional elements, the one she elaborated on was
individual practice. She posited that mathematics was about "doing the work ‘cause I feel
that's very important" (Feb. 23, 2009). Ada regularly gave students individual work,
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which she reviewed, collected, and implied would be graded. She also spent significant
time during instruction ensuring students wrote things on papers and worksheets in
particular ways. She also, in many instances, made sure the answers were correct; this
practice included homework and most in class assignments that were collected. This
action suggests that Ada appeared to want guaranteed success for students.
Reflecting on all observations, there was no regular pattern of instruction as Ada
alluded to in her pre-observation interview (Feb. 23, 2009). She did; however, tend to do
some of the same types of things, albeit not in a prescribed way or order, such as review
answers to homework and/or class work or to show students how to do something before
allowing them to puzzle with it on their own.
At the beginning of most observation days, Ada announced to the students what
they would be doing for the day and she often used the opportunity to advertise specific
parts of the lesson that she thought were especially challenging or fun. Ada encouraged
students’ compliance by saying, "...behaving students will get to do model building with
partners others will need do it by yourselves with teacher assistance" (Interaction 3.1,
Mar. 2, 2009). Working with the teacher appeared less desirable, while the offer for
students to interact and work collaboratively appeared more fun and inviting.
Classroom Interactions: Most Fun with Peers
Working with peers was very desirable for students in Ada’s classroom. During
the observation period, I noticed that most students’ attitudes or demeanors consistently
brightened whenever they were given an opportunity to work with peers. One day there
was an interaction (3.10) that exemplified students’ desire to work together. In this
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classroom interaction episode, Ada was giving directions and she mentioned students
would need to find partners to work with; the students got mobile, started calling names,
pointing, and gesturing, and the room felt energized and charged.
Classroom Snapshot 4.1: Students Selecting Partners Line 1: Ada: I want you to clip your calculator closed and bring me your calculator. Line 2: Student: Yes, ma'am [students begin to move] Line 3: Ada: Stop! [movement stops] As you're doing that, grab a partner. You need
one… Line 4: Students: [students begin moving again, one or two are several paces from
their seats in pursuit of specific partners; talking ensued, motioning, and gesturing followed]
Line 5: Ada: Student S, sit. No, no, no! Everybody sit down. Let me finish my directions.
(Interaction 3.10)
Ada finished the directions and clarified by restating some portions more than once.
When she was satisfied that the directions were adequately communicated, she allowed
the students to go about finding and working with peers.
Interestingly, on this occasion, and most of the observed instances of cooperative
group work, the students were significantly more engaged and boredom was a nonissue.
Most of the student-student interactions in this classroom were observed during partner
work. During the observation period, there was never an activity with collaborative
mathematical objectives initiated for cooperative groups with greater than two members
and students were always allowed to opt out and work independently.
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Student Engagement: Small Group Better than Whole-Class
When asked about student engagement during the pre-observation interview (Feb. 23,
2009), Ada’s response was reflective. She recognized that few students participated
during whole-group instruction and the best way to get students engaged was through
small groups, learning centers, and/or technology. Describing whole-group participation,
Ada said, “I get the very least amount of participation and I get kids zoning out and not
paying attention” (Pre-observation interview, Feb. 19, 2009). During the observation
period, Ada planned for and implemented at least one activity each day that required
small groups (usually pairs) and/or technology (calculators, overhead projector,
individual white boards). When small group work was the structure or technology was
used during the observation period, more students participated and engagement was
improved compared to whole-group instruction, just as Ada reflexively described.
Ada’s commitment to diversified instruction and supporting students was
articulated in her interview and substantiated during observations. Also, of import for this
classroom was a belief that good student behavior, meaning students do what they were
asked to do, resulted in successful learning. After observing Ada’s classroom
interactions, these themes were manifested in the analysis as traditional teaching and
compliance. Specific examples from Ada’s classroom related to Discourse/community,
classroom culture, and relationship/opportunity are described while addressing the
research questions that guide this investigation, which follows after the presentation of
the data.
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Case Data for Ada’s Classroom
The case data that was analyzed and used to construct this case study included
450 minutes of video-taped classroom observations, Ada’s responses to teacher surveys,
students’ responses to a student survey, and pre-observation interview with Ada. The
analysis process for both teacher and student surveys was described in Chapter 3.
Student Survey Data
The student survey had six questions with a four-point Lickert-scale: strongly
agree (4); overall agree (3); overall disagree (2); and strongly disagree (1). The student
responses for the Lickert-scaled questions were assigned to interpreted categories and
averaged, then summarized in Table 4.3. The interpreted categories are phrases that
capture the meaning of each question. As data summarized in Table 4.3 illustrates, the
students in Ada’s classroom on average responded to the first three questions in a way
Questions Interpreted category
Responses (mean, �̅�𝑥)
Standard deviation (𝜎𝜎)
I feel confident that I can learn most math topics. Math confidence 3.3 0.9
I can learn anything in math if I put my mind to it. Math efficacy 3.8 0.4
If I find the work hard, I know that if I keep trying I can do it. Math agency 3.5 1.0
My friends say that I keep trying when math gets hard Math identity 2.7 0.7
You are either good at math or not and you cannot get better by trying.*
Fixed intelligence
2.8 1.3
If I can’t do the work in math I give up. Persistence * 1.3 0.7
*
Table 4.3: A summary of student survey responses (n=10) for Ada’s classroom. Low scores are most favorable for these questions
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that suggests they perceived themselves as confident (�̅�𝑥 = 3.3,𝜎𝜎 = 0.9) and possessed a
high level of math efficacy (�̅�𝑥 = 3.8,𝜎𝜎 = 0.4) and agency (�̅�𝑥 = 3.5,𝜎𝜎 = 1.0).
Interpreting the fourth question using Hodge’s (2006) definition of identity as the way
“…individuals perceive that others are perceiving them in interaction” (p. 380), Ada’s
students did not on average perceive themselves appreciably with respect to math identity
(�̅�𝑥 = 2.7,𝜎𝜎 = 0.7) because the mean was so low. There was no consensus by the class
about intelligence (�̅�𝑥 = 1.9,𝜎𝜎 = 1.2) from the student survey data given that the mean
was close to and well below the median and the standard deviation was large given the
small number of students (n=10). Students choices were split evenly between the strongly
agree (4) and the two disagree options (1 or 2). Selecting one of the disagree options
suggests a growth intelligence, the perception that one can get better at math by trying;
and selecting strongly agree suggests a fixed intelligence, the perception that one is born
able to do math or not and trying does not help. The last question on the student survey
suggests the students in Ada’s classroom were persistent (�̅�𝑥 = 1.3,𝜎𝜎 = 0.7).
Teacher Survey Data
The teacher responses were interpreted using the theoretical model to preserve
anonymity for Ada and summarized in Table 4.4. The columns in the table represent the
theoretical constructs. The first two rows of the table represent Ada’s responses to the
teacher surveys14
14 Refer to Appendix G for the Teachers’ Commitment to Mathematics Educational Reform (TCMER, Ross et al., 2003), Appendix F for the Teacher Self-Efficacy Scale (TSES, Tschannen-Moran & Woolfolk Hoy, 2001)
. The specifics about the alignment were detailed in Chapter 3.
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The first row of Table 4.4 compares Ada’s highly favorable responses to the
survey, analysis within construct group. For example, there are 10 questions from the two
surveys categorized as Discourse/community and of those 10 questions, Ada responded
with highly favorable responses to 7 of those questions, hence 70% highly favorable
responses is recorded in the first row, third column in the table. The second row
compares Ada’s favorable responses to her overall favorable responses, analysis between
construct groups. Specifically, Ada’s total for highly favorable responses across all three
constructs was 20 and 7 of them were categorized as Discourse/community, thus 35% of
her favorable responses were to questions categorized as Discourse/community; written
in the table in row two column three.
Classroom
Culture % ( N)
Discourse Community
% ( N)
Relationships that Support Learning
Opportunities % ( N)
Teacher Survey Data
Highly favorable responses within 80 (10) 70 (10) 56 (9)
Favorable responses between 40 (20) 35 (20) 25 (20)
Analysis Summary of Case Data
Coding densities 34 (742) 33 (742) 33 (742)
Table 4.4: Interpretation of Ada’s responses to the TCMER and TSES teacher surveys and summary of coding densities for Ada’s case data organized by theoretical constructs.
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The actual survey responses resulted in the following conclusions: a) Ada’s
commitment to mathematics education reform was greatest for classroom culture
(TCMER); and b) Ada was most efficacious with respect to Discourse/community
(TSES). However, the overall survey analysis from Table 4.4 suggests the most favorable
responses were related to classroom culture and the classroom interaction analysis
concurred, suggesting agreement between the two analyses and analysis triangulation.
Analysis Summary Data
The last row of Table 4.4 summarizes the coding done during analysis for all the
data associated with Ada’s classroom interactions. The coding process described in an
overly simplistic way is assigning selected pieces of data to categories. The data in this
row was derived from queries run against the coded data using the qualitative analysis
software program, NVivo 8 (QSR International, 2008). The software program’s query
function counted and reported the number of times categories were coded or connected to
specific data (e.g., a segment of video or a transcribed paragraph) related to Ada.
The percentages shown in row three of Table 4.4 were found as follows. The
categories were grouped by theoretical construct, the counts totaled, and percentages
calculated by dividing the total of coded references for each construct by the total number
of coded references. The percentages recorded for the coded references do not include the
emergent codes related to interaction type and interaction focus because only the
theoretical construct related categories (codes) were included in the calculation. The
emergent codes were defined during the analysis process and emerged as questions arose
about the overall nature of interactions and in the quest to seek patterns in the data.
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For Ada’s classroom interactions, the analysis phase resulted in coding densities
that were about the same across all three construct groups, classroom culture (34%),
Discourse/community (33%), and relationship/opportunity (33%). However, classroom
culture was coded slightly more densely than the others, there were more instances when
categories from classroom culture were coded or connected to specific data from
classroom observation or other data sources (e.g., videotaped episodes, transcribed
interviews).
Interaction Types and Focuses Data
The emergent categories (codes), interaction type and interaction focus are
summarized for the classroom observation data in Table 4.5 and Table 4.6, respectively.
The interaction types have literal meanings. For example, interactions coded as
interaction type15
student-student, the meat of the interaction was between students.
Similarly, the teacher-group interaction type were between the teacher and a small group
of students (i.e., not the whole class).
Interaction types Coding references % Coding (n=55)
Student-student 18 33%
Teacher-class 29 53%
Teacher-group 1 2%
Teacher-student 7 13% Table 4.5: Summary of coding densities for interaction types for Ada's case data.
15 The code book describes each of the codes used in this investigation in Appendix C.
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The most prevalent interaction type coded in the analysis of observations from
Ada’s classroom interactions were teacher-class (53%) interactions and the least coded
were teacher-group (2%) interactions illustrated in Table 4.5. The second most coded
classroom interaction was student-student (33%) interactions.
Interaction focus Coding references
% Coding (n=67) Interaction focus descriptions
General explaining 5 7% Explaining that is not mathematical
Interpreting meaning 1 1% Developing clarity or shared understanding
Math problem solving 12 18% Mathematical problem solving
Reflecting or evaluating 13 19% Cognition about prior event, activity, or action
Social problem solving 13 19% Resolving sociocultural situations
Student explaining math idea 10 15% Students revealing mathematical cognition
Teacher explaining math idea 13 19% Teacher revealing mathematical cognition
Table 4.6: Summary of coding densities for interaction focus categories for Ada's case data.
It was beneficial that student-student interactions were the second most prevalent
interaction type for Ada’s classroom because of concerns about data quality that arose
early during the data collection phase. As an amateur videographer, the challenges related
to recording students became evident; they speak softly, tend to be shy on camera, and
capturing their voices from afar was challenging, especially when the classroom was in
an animated working state, the point when capturing interactions would be most
desirable. The two-camera tripod video set up worked well; on several occasions, a video
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of an interaction captured on one camera, could be heard on the second camera, thus
enabled the interaction capture, in spite of the novice videographer.
Interaction focuses are overarching categorical themes that were derived to
capture the essence of all of the classroom interaction summaries for Ada16
Analysis Coding Summary by Construct Groups
. More
precisely, every classroom interaction that was analyzed was assigned an interaction type
and at least one interaction focus; some interactions have two interaction focuses, such as
teacher and student explaining math idea. For all of Ada’s classroom interactions, the top
four interaction focuses coded during analysis included: a) reflecting or evaluating
(19%); b) social problem solving (19%); c) teacher explaining math idea (19%); and d)
math problem solving (18%).
Table 4.7 to see across all coding grouped by the theoretical constructs, Discourse
or community, classroom culture, and relationships that support learning opportunities. In
this summary table, the most referenced three or four categories coded during analysis
were selected by construct group. As indicated in the table, Discourse or community
codes were selected if they had greater than 20 coded references. For example, as data
(e.g., videotapes, interviews) were reviewed and analyzed, situations arose on 24 separate
occasions that offered evidence of thinking and idea (see row three column two), and that
category emerged among the most referenced for those related to Discourse or
community. The third column of the table is the percent of coded references within each
construct group and the last column is the percent between the construct groups. Thus,
16 Interaction summaries for Ada’s classroom are in Appendix I.
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thinking and ideas accounts for 32% of the frequently coded references within the
Discourse or community construct of the 74 frequent coded references within this
construct. Similarly, 9% of all frequently coded references of the 275 frequent coded
references between or among all constructs.
Frequently coded categories grouped by theoretical construct
Coding references
% coding within
% coding between
Classroom Culture (> 21 coded refs)
Teacher explaining (sociomathematical norm) 30 31% 11%
Questioning (sociomathematical norm) 24 24% 9%
Low risk environment (social norm) 22 22% 8%
Negative Social Norms (social norm) 22 22% 8%
Total 98 100% 36%
Discourse or Community (> 20 coded refs)
Informer (teacher behavior) 29 39% 11%
Thinking and ideas (student behavior) 24 32% 9%
Authoritative (discourse type) 21 28% 8%
Total 74 100% 27%
Relationships that Support Learning Opportunities (> 25 coded refs)
Traditional (authority type) 37 36% 13%
Positive PAID a 37 (identity perception) 36% 13%
Math as procedural (teaching influencer) 29 28% 11%
Total 103 100% 37% a
PAID - participation and agency identity
Table 4.7: Summary of coding densities for the most frequently referenced categories for Ada’s case data organized by theoretical constructs.
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Ada’s Case Study Interaction Summaries
Finally, Table 4.8 summarizes the specific interactions discussed within Ada’s
case study and is a subset of all interactions from Ada’s classroom17
Table 4.8
. Several of the
interaction episodes summarized in have been used to elaborate points made
within this case thus far, and the other interactions will be used to expound on the
findings as the research questions are addressed in the next section.
Interaction Identifier Interaction Type and Focus Interaction Description (Snapshot #)
1.4 Teacher-class; Reflecting or evaluating
Evaluating written explanations for multiple choice selections (Snapshot 4.3)
1.10 Teacher-class; Teacher explaining math idea
Modeling acceptable approaches for fraction estimation (Snapshot 4.2)
2.1 Student-student; Students explaining math idea
Students share homework solutions (Snapshots 4.4)
3.10 Student-student; Social problem solving
Student enthusiastically select partners (Snapshot 4.1)
4.2 Teacher-class; Teacher explaining math idea
Teacher articulates procedure for multiplying mixed numbers (Snapshot 4.6)
5.1 Teacher-class; Teacher and Student explaining math idea
Students sharing mathematics homework and collaboratively explaining (Snapshot 4.5)
Table 4.8: Interaction summaries used in Ada's case study.
17 All classroom interactions coded were summarized in Appendix I. The summaries in the appendix also
include notes about the teacher and student actions.
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Findings from Ada’s Classroom Interactions
The primary research question addressed in this section is: How do classroom
interactions influence the Discourse related to mathematics learning and teaching in
Ada’s MCP supported classroom? This question was addressed by attending to the three
sub-questions derived from the theoretical model that focuses this investigation. Those
questions include:
a) What is the nature of classroom culture?
b) What is the nature of Discourse and community?
c) What is the nature of the relationships that support learning opportunity?
The discussion will begin by addressing the nature of classroom culture first
because the data and analysis in Table 4.4 showed that classroom culture (34%)
categories were coded slightly more densely than the others, Discourse/community (33%)
and relationships/support (33%). Subsequently, these two construct groups will be
discussed together following the classroom culture discussion. The case is completed
with a summarizing matrix related to all three constructs and the major research question
about how classroom interactions influence Discourses related to mathematics learning
and teaching.
Classroom Culture
Analysis of Ada’s classroom interactions yielded frequently referenced classroom
culture categories of teacher explaining math idea (31%), questioning (24%), low risk
environment (22%), and negative social norms (22%) (Table 4.7). These classroom
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cultural elements influence Discourses related to mathematics learning and teaching.
Consider Interaction 1.10, coded interaction type teacher-class with interaction focus
teacher explaining math idea. The classroom interaction episode is about Ada setting
expectations for learning and informing students about one way to compute with fractions
using estimation.
Classroom Snapshot 4.2: Acceptable approaches for estimating fractions Line 1: Student: [reading from text] Line 2: Ada: I'm gonna show you three different ways to do this and you may pick
the way you like to do this best. OK. You must do all three of my different ways, after I'm done teaching, you can pick your own. OK. Alright, number 1: the first way you need your calculator and what I want you to do on the first way is to think of 0 as no cents, 50 cents as being your half, and a dollar as being your whole. So, in your calculators put in 7/8 and get a decimal, please.
Line 3: Students: [several students talking at once]…7/8?... zero point eight seven five… dude
Line 4: Ada: When we are talking about, Line 5: Student: [interruption by several voices]… point 88… Line 6: Ada: [interruption] When we're talking about cents, Line 7: Students: [interruption by several voices] … one dollar… Line 8: Ada: Student K, you're yelling out again. [pause] When we're talking about
cents, do we wanna go to the tenths, hundredths, or thousandths place? Line 9: Students: Hundreths [from multiple students] Line 10: Ada: Hundreths, so that would be two [referring to decimal places] so, 7/8
would be like having 88 cents, so put 88 cents above 7/8 [writes it down as .88 above the 7/8 written in the original problem]. Student M, can you do 1/16?
Line 11: Student M: I don't have a calculator [walks over to get one]
(Interaction 1.10)
In Classroom Snapshot 4.2, Ada sets the expectation that students need to be able to do
each of three methods for computing with fractions using estimation, and that they should
choose one of the given methods for their own. She also shows students how to record
this approach on their paper. Students were not asked for different approaches for
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estimating, which may have led to mathematical Discourse for comparing or evaluating
the best or most efficient approach. During the course of the instructional period, Ada
shared two other approaches for students to consider, but they were not asked to compare
the approaches for their mathematical efficiency or clarity, but were tasked to select the
approach they liked best. The students were not asked to explain or justify why they liked
one method over another.
Ada was explaining a math idea, adding fractions using estimation. The students
were given very specific steps to achieve this such as, “think of 50 cents as a half” (Line
2) or “write the .88 above the 7/8” (Line 10). These types of classroom interactions
suggest that mathematical knowledge was procedural and learning mathematics was
about fact or procedural reproduction, indicative of transmission learning (Freire, 1970;
Gur-Ze'ev, 1998; Hasbrook, 2002; hooks, 1994). In this snapshot, clear expectations were
established for students’ mathematical thinking, doing, and writing when estimating
fraction addition. Mathematics learning and teaching were potentially hindered when
students were not afforded opportunities to ‘do’ mathematics for themselves (Empson,
2003; Resnick, 1988; Schoenfeld, 1994).
Another aspect of the Classroom Snapshot 4.2, were the social norms; many of
the norms did not enhance learning and were coded negative social norms (22%), a
frequently coded category in Table 4.7. Negative social norms occurred regularly, such as
people interrupting one another, but only one student was reprimanded. The dynamic
between this student and the teacher arose on several occasions during the observations; I
suspect there was some history between the two that is beyond the scope of this
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investigation. There were other times when several people talked at once, which suggest
that listening was challenged; it is likely that students did not hear peers or the teacher
and the teacher was unable to hear students. Thus, collaboration was compromised
because listening and hearing was unsynchronized which could result in ideas not being
built upon or inspired by others. Further, teacher and peers were potentially deprived of
opportunities to question one another about ideas shared.
Finally, evidenced in Classroom Snapshot 4.2 was the type of questioning, skill or
knowledge level questions mostly. For example, “. . . put in 7/8 and get a decimal,
please” (Line 2); “When we’re talking about cents, do we wanna go to the tenths, . . . ?”
(Line 8); and “. . . can you do 1/16?” (Line 10). In this snapshot and throughout the
observations, most questions asked of students were skill and knowledge level questions
making them a sociomathematical norm for Ada’s classroom. These types of questions
hinder Discourses related to mathematics learning and teaching because they do not
require explanatory or expounded responses that lead to critical thinking (M. Carpenter,
Matthews, Krump, Whitesell, & Pena, 2003).
Student explaining was evidenced often during the observation period; however,
the explanations were often short or perhaps compromised by low cognitive demand; a
natural consequence of the skill and knowledge questioning. To illustrate this idea,
consider another classroom interaction (1.4) coded interaction type teacher-class and
interaction focus reflecting and/or evaluating. This snapshot is about sharing written
explanations about multiple-choice selections, evaluating response quality, and revising.
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Classroom Snapshot 4.3: Evaluating written responses Line 1: Ada: Student B, Would you read me your sentences, please? [B did not
volunteer] Line 2: Student B: C [refers to an answer choice], because if you take the four sides
and switch them around then it looks the same as the other figure Line 3: Ada: Student A and I just had this conversation, that's an OK answer; but,
it's kind of like a ‘because I said so answer,’ and how do you feel when mom tells you, ‘because I said so?’
Line 4: Students: [students talking at once] I hate it...I don't like it... Line 5: Ada: OK. So it's like you really didn't get an answer. So, I want you to just
add a little to that by using like; what could you add to that for C? Student A?
Line 6: Student A: Um, [inaudible]... it went vertical? Line 7: Ada: How many are vertical? Line 8: Student A: Two? Line 9: Ada: Two. So you could say, two are, two cubes are verticle and two cubes
are? Line 10: Student A: Horizontal Line 11: Ada: Horizontal. OK
(Interaction 1.4)
Students were sharing their written responses when asked and they were responding to
questions. Students, whenever called upon, independent of whether they volunteered or
not, during the observation period, freely and willingly participated. This behavior
evidenced the presence of a low risk environment (22%); one of the frequently coded
categories for Ada’s classroom interactions in Table 4.7. Low risk environments enhance
Discourses because students are not stymied by fear or other prohibitive emotions
(Hiebert et al., 1997).
The types of questions in this snapshot were not of the skill or knowledge level
type questions highlighted in Classroom Snapshot 4.2. The questions in this snapshot are
higher-level questions, such as: a) comprehension question – what could you add… (Line
5); and b) analysis question – “how do you feel when. . . “ (Line 3). Even so, the
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sociomathematical norm for questioning was well established as described in relation to
the previous snapshot, students’ responses or explanations were short and suggestive of
low cognitive demand. For example, “I hate it” (Line 4) or “it went vertical” (Line 6).
Follow up probing questions were not used to elicit deeper or more clearly
articulated responses. Instead, students’ responses were interpreted and meanings
proffered through teacher explaining (31%), a frequently coded category in Table 4.7. In
this snapshot, the teacher offered meaning for ‘because I said so’ and offered the
elaborated statement to be added to the students’ brief initial responses (Line 4). The
students were not the critical thinkers in this snapshot.
These snapshots captured the nature of Ada’s classroom interactions for the
observation period. They evidenced the four most frequently coded categories for the
theoretical construct classroom culture, namely: a) teacher explaining math ideas (31%);
b) questioning (24%); c) low risk environment (22%); and d) negative social norms
(22%). Each of these classroom culture elements, excluding negative social norms, in
many instances would likely enhance mathematics Discourses, but as observed they
could have been hindering the Discourses related to learning and teaching mathematics.
Discourse/Community and Relationship Opportunity
Analysis of Ada’s classroom interactions yielded frequently referenced Discourse
or community categories in Table 4.7 of informer (39%), thinking and ideas (32%), and
authoritative (21%). Similarly, analysis of Ada’s classroom interactions yielded
frequently referenced relationships that support learning opportunity categories also in
Table 4.7 of traditional (36%); positive participation and agency identity (positive PAID)
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(36%); and math as procedural (28%). These two theoretical construct groups, Discourse
or community and relationships that support learning opportunity, will be discussed
together.
Consider a classroom interaction (2.1), coded interaction type student-student
with interaction focus student explaining math idea. The snapshot is about a student
explaining his homework solutions to the class.
Classroom Snapshot 4.4: Student sharing homework at overhead Line 1: Ada: The first thing we are going to do is go over your homework. If
you've gotten anything wrong, feel free to correct it before I look at it. Student J, would you come up and do 1 and 2?
Line 2: Student J: [goes to the overhead; prepares to do the assigned problems] Line 3: Ada: When you do um, when you show your work and when you come up
to the overhead, please talk and tell us what you are doing as you do it for those people who need a little help with understanding it again.
Line 4: Student J: So, for number 1, I just added up all the boxes and I got 5; shaded and added up the rest of them and got 8; then I put it in my calculator to see what the percentage or decimal would be [pointing to the work he is writing on the overhead].
Line 5: Ada: OK. Thank you. Line 6: Student J: and 5/8 would be 1/2 [estimating fraction] and I did the same
with number 2. I added them all up, 9 and 1 is 10 and that was one whole. Line 7: Ada: OK. One thing that has helped me is that when I read a fraction I read
the line that splits the denominator from the numerator as "out of" so I'm reading 9 out of 10 and that gives me a visual picture in my head just because of the way I am reading the words. So, who wants to do 3, 4, and 5?
(Interaction 2.1)
As the interaction snapshot begins, Ada makes clear her expectations for student
presentations and she selects the first student to present. Her discourse type was
authoritative (Lloyd, 2008), meaning her expectations were not suggestions to be
questioned and she does not solicit input or contributions from others. This approach
could also be considered traditional teacher authority or autocratic leadership (Bass,
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1990; Pace & Hemmings, 2006). These discourse and authority/leadership styles were
used almost exclusively in Ada’s classroom and likely hindered the Discourses related to
mathematics learning and teaching (Lloyd, 2008). Additionally, these
authority/leadership styles align with a strict classroom environment; the description
offered by Ada for her classroom environment during her pre-observation interview (Feb.
23, 2009).
As illustrated in the snapshot, Student J was afforded the opportunity to explain
what he had done. His explanations were not interpreted or preempted by teacher
explaining as evidenced by Classroom Snapshot 4.3, but this type of opportunity was not
the norm during the observation period. At the end of his explanation, neither teacher nor
peers asked questions; perhaps because his explanation was well articulated and
understood by those listening. However, asking questions after explanations was not a
sociomathematical norm for this classroom. Also absent from this interaction were
negative social norms, such as interruptions or multiple people talking at once; all
appeared to be listening to Student J’s explanation.
In general, when students were explaining, in most instances other students
listened and treated one another with respect, albeit sixth grade student type respect.
Another classroom episode illustrates this aspect of Ada’s classroom interaction. This
next interaction was coded using the same interaction type and focus coding as the
previous Classroom Snapshot 4.4: Student sharing homework at overhead (i.e., type
student-student and focus student explaining math idea).
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Classroom Snapshot 4.5: Student at overhead during problem solving Line 1: Student D: [Reading] Tomorrow is the fourth of June. What day will it be
three days from today? Line 2: Ada: OK. How would you do that? Line 3: Student D: [~ 2 second pause] Line 4: Ada: [interrupting] Remember, you can ask… Line 5: Student D: [end pause] 4, 4th
Line 6: Ada: [Interrupting, over talking] You know you can ask people?
Line 7: Student K: [Immediately raises hand, makes a sound, Student D is looking down] Bing!
Line 8: Student D: [abandoning his thinking and idea] Student K. Line 9: Student K: Four plus three equals seven, so it'd be of June. So, it would be
the 7th
Line 10: Ada: OK. Then, can you write it up there? Like put 1... up higher…very good.
of June.
Line 11: Student D: [Writing per Ada's directions] Line 12: [Students chatter softly] Line 13: Student B: [speaking above the other students, chatter stops] I don't get it
because if it’s the 3rd and tomorrow is the 4th, and you add three days, wouldn't it be the 6th
Line 14: Student K: Nooo! ?
Line 15: Ada: [Thinking aloud] ...because if tomorrow is the 4th
Line 16: Student B: If it's the third...
of June, what day would it be? 4..., you're right! Student B, can you explain what you did?
Line 17: Student D: [interrupting] I'm wrong? Line 18: Student B: [Continuing his thinking and explanation] If it’s the third of
June, you add three more days it’s going to be the 6th
Line 19: Ada: So, it’s not the 4 of June.
th. It says if tomorrow is the 4th. So, it's actually the 3rd
Line 20: Students: Nice job, Student B! . Did you see that word there?
Line 21: Ada: Very good Student B! [More accolades from other students]
(Interaction 5.11)
A key difference between the last interaction episode and this one was the task
required slightly more cognitive demand a mathematical exercise versus a word problem
in this new snapshot. The word problem had to be interpreted before mathematics could
be used to solve it, but neither problem offered significant mathematical cognitive
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demand for grade six students. In this interaction (5.11), students were volunteering to
come to the overhead to explain an achievement test item the class had been working on
for practice during class.
This interaction snapshot evidenced one of the best examples of collaboration
during the observation period within Ada’s classroom because multiple students
contributed to the final solution to the problem. Further, the Discourse in this interaction
emerged from multiple contributors making it one of the few interactions coded as
discourse type contributive (Lloyd, 2008). Student D offered an idea. “ 4, 4th
Student K offered a second idea and explained it (Line 9). Ada accepted the idea
as a valid solution; an established sociomathematical norm for this classroom. After
reflecting “I don’t get it because…” (Line 13), Student B questioned what had been
accepted as the answer, June 7
” (Line ,5),
unfortunately it was not acknowledged and he did not have the opportunity to explain
what he meant by it.
th. He revised the answer from June 7th to June 6th
In addition to collaboration, two other Discourse enhancing student behaviors
were evidenced within this interaction snapshot, student thinking and ideas (32%) and
peer helping and support. Two or three examples of thinking and ideas were described
above, the contributions made by the multiple students, each was an example this (Lines
and
explained his rationale to Ada. She reflected aloud about his input and accepted the
revised answer, and Student B was asked to restate his rationale to the class (Line 15). An
implication of Ada’s request is that she had not expected students to have listened to or
heard what Student B had said.
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5, 9, and 13). The peer helping and support included students responding by raising hands
after Ada invited Student D to call on other students for help (Line 6). The second
example was students’ enthusiastic support of Student B with accolades and
congratulations for his mathematical thinking after he explained the revised the answer to
the class (Lines 20, 21). More importantly, they did not chastise or tease Student K or D
for their incorrect ideas.
There were also two elements of relationships that support learning opportunities
evidenced in this episode related to identity perceptions, positive participation and agency
identity (positive PAID), and mathematical autonomy (confidence). Positive PAID (36%)
was one of the most frequently coded categories (Table 4.7) for Ada’s classroom
interactions and student participation was a key element for the interaction depicted in
this snapshot. Had Student B not participated with sufficient agency and/or confidence,
the class would likely have continued and never recognized the incorrect answer.
During observations, Student B had not been an unsolicited contributor to the
class Discourse; he never volunteered to speak unless someone asked for input or a
response. This suggests, in this instance, he possessed sufficient mathematical autonomy
(confidence) to participate with sufficient agency to challenge an idea offered by one of
the most vocal students in the class, Student K. This student was a regular unsolicited
contributor to the classroom Discourse. He was a student Ada had acknowledged on
multiple occasions as “smart” or “intelligent” to the class.
Pointing out students as intelligent or smart could be interpreted as an intelligence
perception, which has been recognized as a teaching influencer related to thinking and
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learning (Dweck, 2006). An example of this was evidenced in interaction (4.2), coded as
interaction type teacher-class and interaction focus teacher explaining math idea. In this
classroom interaction episode, Ada is explaining a mathematical procedure for
multiplying mixed numbers.
Classroom Snapshot 4.6: Defining procedure for multiplying mixed numbers Line 1: Ada: Take the mixed number and the first step is change to improper
fraction Line 2: [Writes out step 1; a student writes then starts beating softly on the desk] Line 3: Student A: Do we need to write what you're writing? Line 4: Ada: No, I'm just putting it up here for a reminder for you. OK. Step 2 is
going to be to simplify. Line 5: [Writes out step 2] Line 6: Ada: You're gonna simplify in an 'X' ... Line 7: [Describes and shows reducing by factors the numerators and denominators
of two fractions separated by a multiplication symbol] Line 8: Ada: Is there anything that will make 4 and 8 smaller by dividing? Line 9: Students: [multiple students responding] yes...2... 2 Line 10: Ada: Are there any bigger numbers? Line 11: Students: [multiple students responding] 4... 24... 4 Line 12: Ada: 4, alright. So, you take 4 and you go 4 divided by 4 is [pause] Line 13: Students: [multiple students responding] 1 Line 14: Ada: [Writing] so you cross it out and put a 1. So all these steps I want you
doing on your paper. and then you take the 8 and you divide it by 4 again and you get [pause]
Line 15: [Simplifying continues, using the same teaching strategy] Line 16: Ada: The second step is to simplify. The third step is gonna be to multiply
and I put it like that because you're gonna multiply your numerators and you’re gonna multiply your denominators. [emphasis on the word multiply]
Line 17: [Ada writes the multiplication, while asking students for computation facts] Line 18: [Several students look bored, begin mumbling, heads on hands, sighing,
yawning, and making other noises] Line 19: Ada: Not done yet. Last step is to simplify again. Line 20: [Converts improper fraction to mixed number, while asking students for
computation facts] Line 21: Ada: ...and your answer is 3 and 1/3. Line 22: Students: [students speak softly, but one says] I know another way to do
that. Line 23: Ada: So, that is the way you do that. And if you notice up here, I have to go
through all those steps. If I just would have multiplied...[multiplies across the mixed number numerators and denominators as proof] So you see, if I
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would have just done it my own way it just does not work. You have to follow the steps.
(Interaction 4.2)
The teaching approach in the snapshot limited both student input and the
multiplication algorithm. The students’ participation was limited to note taking and
computation facts only, which was interpreted as teaching influencer that limited learning
opportunity related to fixed intelligence, that is, students cannot contribute to knowledge
building until they have been taught and fixed intelligence learners require knowledge in
comprehendible chunks (Dweck, 2006). The multiplication algorithm was presented as a
static procedure exemplified the meaning of mathematics as procedural (28%), a
frequently coded category. In other words, mathematics is done by a series of steps and in
extreme cases can only be done one way (Baroody, Feil, & Johnson, 2007). These two
elements, fixed intelligence and math as procedural were teaching influencers found
often during the observation period in Ada’s classroom. These teaching influencers
coupled with this being an urban school district with high free and reduced lunch
population suggests the Discourses related to mathematics learning and teaching were not
enhanced (Anyon, n.d.; Haberman, 2002).
One last element of Discourse or community to attend to from this classroom
interaction snapshot was a teacher behavior and one of the most frequently coded
categories, informer (39%) as indicated in Table 4.7. In the snapshot, the informer
teacher behavior may have been a contributor to students’ participation being limited to
computation facts. Also, in other snapshots in this section there were examples of this
behavior that may have contributed to limited student participation. For example,
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Classroom Snapshot 4.2: Acceptable approaches for estimating fractions, the informer
behavior limited students’ opportunity to share their estimation strategies; Classroom
Snapshot 4.3: Evaluating written responses, the informer teacher behavior may have
stifled students’ from understanding why ‘because I said so’ was an ineffective
mathematical response; or Classroom Snapshot 4.5: Student at overhead during problem
solving, the informer teacher behavior may have hindered Student D from explaining
what he meant when he offered the idea 4th
Summarizing Matrix for Ada’s Classroom
. Each of these suggestions was speculative,
but the point was that student learning and thinking potential may have been dampened
by the informer teacher behavior.
The nature of classroom culture, Discourse or community, and relationships that
support learning opportunities have been articulated through the lens of the theoretical
model and supported with the data and analysis. The primary research question: How do
classroom interactions influence Discourses related to mathematics learning and teaching
in Ada’s MCP supported classroom? and key elements from the preceding discussion are
depicted in Table 4.9. This table represents a summarized response to the primary
research question for Ada’s classroom interactions.
Table 4.9 was organized to illustrate the case study through the lens of the
theoretical model. The matrix columns represent the three theoretical constructs: a)
classroom culture; b) Discourse or community; and c) relationships that support learning
opportunity. The rows of the matrices depict the four concepts that define the construct
group. The theoretical construct classroom culture the four concepts include: a) cultural
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influencer; b) mathematical practices; c) sociomathematical norms; and d) social norms.
The theoretical construct Discourse or community the four concepts include: a) Discourse
influencer; b) Discourse types; c) student behaviors; and d) teacher behaviors. The
theoretical construct relationships that support learning opportunity the four concepts
include: a) authority type or leadership style; b) identity perceptions; c) learning
influencer; and d) teaching influencer. Specific elements of this table are discussed in
detail within the cross case analysis in the next chapter.
Classroom culture Discourse/community Relationship/opportunities
Cultural influencers included fact or procedural reproduction and powerful other
Discourse influencers were static mathematics that manifested in practice via exercises and an emphasis on answers
Authority types (and leadership styles) varied and were often traditional (autocratic) with some charismatic
Mathematical practices often included communication
Discourse type was overwhelmingly authoritative
Identity perceptions were overwhelmingly positive participation and agency identity
Sociomathematical norms included a lot of teacher explaining and low-level questioning
Student behaviors included: individual efficacy, thinking and ideas, and peer help and support
Learning influencer included both students and teacher actions
Social norms included a low risk environment, but often did not enhance the learning environment
Teacher behaviors included informer, expert, and validator
Teaching influencers included perceptions of mathematics as procedural and fixed intelligence
Table 4.9: A summary of the analysis of Ada’s case study through the lens of the theoretical model.
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Case Study 2: Eva’s Classroom
Classroom Overview
Eva taught mathematics for 90 minutes three times each day during this
investigation. I observed a total of 450 minutes18
During the observation period, all of the mathematics topics in Eva’s classroom
were related to fractions and included: a) comparing fractions using word, pictures, and
symbols; b) comparing fractions using mathematical approaches, no pictorial
representations; c) using fractional understanding to represent decimal numbers on grid
paper; and d) applying fraction knowledge for understanding rulers, the markings and
measuring.
of mathematics instruction in Eva’s
classroom during an afternoon block. The observations were scheduled so that the first
two occurred on consecutive days, a Tuesday and Wednesday. The third was on the
following Monday and the final two occurred on the Monday and Tuesday of the
following week. The reason for clustering the observations was to see interactions over
time and to see if there were connections from day to day related to mathematics content,
learning, teaching, and/or Discourses.
Eva’s class included a diverse group of students. The class was comprised of
slightly more males (12) than females (8), a diverse representation by race or ethnicity
18 Only 360 minutes of observation were included in the analysis because Eva was not present for one day
due to a family emergency and the classroom interactions were significantly different from those observed
when Eva was present.
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included a balance of Black (8) and White (8) students, and there were biracial (2) and
other (2) racial or ethnic students. On average there were 20 students present on each of
the observation days.
Eva’s Perceptions about the Class
In this section, Eva’s perceptions are shared about the observed class, her
instructional style, student engagement, and classroom interactions. These perceptions
came from Eva from the pre-observation interview (Feb. 20, 2009) and supported using
observation data. Eva’s perceptions included organized and structured instruction, high
expectations for student engagement, interactions that were fun and focused on
mathematics.
Classroom Environment
Eva described this class as a good class, but not her best. During the initial pre-
observation interview, Eva described her instructional style as one that was “organized”
and “structured” (Feb. 20, 2009). During the observation period, instruction appeared
well organized and structured, following a pattern. First, students completed bell work19
19 Bell work lasted 30-45 minutes each day, consisted of about two or three problems related to the
previous day’s mathematics learning and teaching. .
as Eva circulated observing student work, answering questions, and taking notes
(sometimes written) of who did what and how. Next, Eva usually collected the bell work
before reviewing it in a whole-class format using 30-45 minutes of the 90-minute
instructional period. Then, students engaged in activities, usually in small cooperative
groups of two to four followed by them sharing solution strategies. Lastly, during the last
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5 to 10 minutes of class, Eva articulated a summary review of the days’ mathematics or
presented new mathematical ideas. The students’ role in this instructional format included
note taking and responding to skill and knowledge level questions. This instructional
pattern was consistent with little variation on the observation days.
Instructional Style: Organized and Structured
Eva’s organized and structured planning was evidenced when she ushered
children through planned activities with one flowing to the next. Further, I rarely
observed students finishing learning tasks early causing Eva to manufacture things to fill
time or give “free” time; the class worked from bell to bell on the days I observed.
Given Eva’s admission of being structured and organized, it was not surprising
that the normal desk configuration in her classroom was straight rows facing the front of
the room. Each day I observed her classroom, prior to children entering the room, she
spent time straightening the rows and preparing supplies for children’s ready access or
for easy distribution at the appropriate time during instruction. However, when she
wanted students to work cooperatively, she did not hesitate to allow students to
reorganize desks as needed for participation. The movement of desk was done with what
I would characterize as organized and structured chaos as only kids can facilitate, but Eva
ensured that the process was always completed in a timely fashion. At an appropriate
time following a group activity or before leaving Eva’s classroom, students returned
desks to their original positions. I observed several transitions during observations for
this investigation.
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As stated earlier, Eva started class with bell work, a formative assessment. It was
apparent that students knew what to do upon entering Eva’s room based upon their
behavior, they moved with purpose collecting things and preparing for work. I suspect
Eva made an adjustment to accommodate my visits. The following classroom snapshot
illustrates this point.
Classroom Snapshot 4.7: Evidence of a Changed Structural Norm On the first day of videotaping, students were looking for bell work handouts in the front corner of the room where I had set up video recording equipment, field notebook, and other miscellaneous things associated with data collection. Eva and I had discussed where in the room I might set up on a prior visit. Thus, on that day, Eva had written the bell work on the chalkboard. As students entered the room, several of them greeted me and excused themselves passed me and the video equipment to rummage through the tray that sat on a table in the corner behind me and my things. One after another, they came, greeted me, and rummaged until finally someone went out into the hallway to ask Eva. She announced that the bell work was written on the chalkboard. After Eva’s single announcement the word spread from student to student.
(Observation, Feb. 24, 2009)
This classroom snapshot described the student behaviors that signaled a structural
norm of Eva’s classroom had changed. Eva reinstated normalcy with an announcement.
After the tardy bell rang, students settled down and got to work. On most observation
days, after tending to hall duty between classes, Eva entered the room. She reminded
students what they were expected to do (i.e., pick up supplies and begin bell work
independently), how much time remained before she would collect their work, and
encouraged them to read if they finished early. This exemplified Eva’s commitment to
and implementation of organized structure infused with her expectations for the learning
environment.
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Eva was reflective about the bell work she assigned to students. The following
classroom snapshot offers some insight into her reflection about a problem she posed to
students:
Classroom Snapshot 4.8: Reflecting on a Problem and Learning about Students I probably shouldn't have put those fractions up like that up there. um, but I didn't realize that you would see it that way and I guess, bravo! Because you're understanding fractions. I'm lookin over here because you two had the same thought process. You didn't work it out. Student B and J, you didn't work it out, you had good logic behind it. Alright 3/5, 3/4, and 3/12 and I said that I wanted you to compare them and of course show your work. In the event that you two boys would have written out a good explanation of what you did you would have been fine [referring to constructed response type questions on the state achievement test]. They didn't show a picture, they didn't find the least common multiple, they just rewrote the fractions. Would you guys tell me, from least to greatest?
(Interaction 2.4)
Eva announced to the class that she had made a mistake in assigning a fraction
comparison problem because she had not anticipated students using fractional
understanding to order the three fractions with the same numerators. She congratulated
the students for their mathematical thinking, and then explained that it was problematic
because they were able to compare the fractions without showing any work. Her apparent
goal for assigning the problem was to see who could use the approaches for comparing
fractions using one of the two methods the class had been working on over the past
several days. Eva adjusted her instruction to accommodate the students’ approach. The
mistake paid off well because Eva was able to see students apply conceptual
understanding for comparing fractions and an opportunity was created that exposed the
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other students to a conceptual approach. This classroom episode is elaborated more fully
in other snapshots later within this case study.
Classroom Interactions: Fun, Yet Serious
Eva described classroom interactions with students as fun, yet serious. She
elaborated on her interactions with students during our pre-observation interview, “we
laugh and joke a lot and we try to have fun, but we're also pretty serious and we know
how to bring it back to math” (Feb. 20, 2009). The following is one example of the class
having fun and then getting back to work.
Classroom Snapshot 4.9: Fun, Mathematical Focus, and Back to Work Line 1: Eva: With 16 students, can we make groups of 3? Line 2: [Several students offered ideas] Line 3: Student B: There would be 5 groups of 3 with 1 left over. Line 4: Eva: So, what are we going to do with you, Student B? Line 5: Student: Throw Student B out!
The class erupted with laughter, including Eva and Student B, but within seconds after the laughter started, Eva asked Student B if he would select three others to form a group of four and adding that he could choose his friend, Student A for his group.
(Interaction 1.2)
The affect of Eva Student B selecting his group signaled the other students to quickly
settle down and refocus on the task of grouping. After a good, albeit brief laugh, the class
was back on task and in fewer than 5 minutes, with limited assistance from Eva, the
students had selected their own groups of three, rearranged their desks, and were ready to
begin the group activity.
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Student Engagement: High Expectations and Caring
Eva also talked about student engagement during our pre-observation interview,
“There's always that choice few that you have to call on them whether their hand's up or
not, and I try to give them the eye so they know I'm gonna call on them next so they're
not surprised or shocked or get withdrawn” (Feb. 20, 2009). This statement evidenced
Eva’s concern about students and their feelings in her classroom. After observing
classroom interactions in Eva’s classroom, her effort to make children comfortable and
nurture them as participants in mathematics learning and teaching emerged as an
important aspect of Eva’s classroom interactions as evidenced by her responses to the
teacher surveys and analysis of the classroom observations. Specific examples from this
classroom related to the theoretical model that guide this investigation were used to
address the research questions following the presentation of the data.
Case Data for Eva’s Classroom
The case data that was analyzed and used to construct this case study included
360 minutes of video-taped classroom observations, Eva’s responses to teacher surveys,
students’ responses to a student survey, and pre-observation interviews with Eva and the
MCP coach assigned to her building. The analysis process for both teacher and student
surveys was detailed in Chapter 3.
Student Survey Data
The student survey had six questions with a four-point Lickert-scale: strongly
agree (4); overall agree (3); overall disagree (2); and strongly disagree (1). The student
responses for the Lickert-scaled questions assigned to an interpreted category and
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averaged, then summarized in Table 4.10. The interpreted categories are phrases that
capture the meaning of each question. As data summarized in Table 4.10 illustrates,
Questions Interpreted category
Responses (mean, �̅�𝑥)
Standard deviation (𝜎𝜎)
I feel confident that I can learn most math topics. Math confidence 3.7 0.5
I can learn anything in math if I put my mind to it. Math efficacy 4.0 0.2
If I find the work hard, I know that if I keep trying I can do it. Math agency 3.5 0.6
My friends say that I keep trying when math gets hard Math identity 2.8 0.9
You are either good at math or not and you cannot get better by trying.* Fixed intelligence 2.3 1.3
If I can’t do the work in math I give up. Persistence * 1.4 0.6 *
Low scores are most favorable for these questions
Table 4.10: A summary of student survey responses (n=22) for Eva’s classroom.
the students in Eva’s classroom on average responded to the first three questions in a way
that suggested they perceived themselves as confident (�̅�𝑥 = 3.7,𝜎𝜎 = 0.5) and possessed
a high level of math efficacy (�̅�𝑥 = 4.0,𝜎𝜎 = 0.2) and agency (�̅�𝑥 = 3.5,𝜎𝜎 = 0.6).
Interpreting the fourth question using Hodge’s (2006) definition of identity as the way
“… individuals perceive that others are perceiving them in interaction” (p. 380), Eva’s
students did not on average perceive themselves appreciably with respect to math identity
(�̅�𝑥 = 2.8,𝜎𝜎 = 0.9). There was no consensus about fixed intelligence (�̅�𝑥 = 2.3,𝜎𝜎 = 1.3)
from the student survey data. The large standard deviation indicated that students chose
either strongly agree or strongly disagree. Selecting strongly agree, suggests students
perceived their intelligence as fixed, you can either do math or not; and strongly disagree
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suggests growth intelligence, you can get better at math by trying. The moderate average
for fixed intelligence (�̅�𝑥 = 2.3,𝜎𝜎 = 1.3) on a 4-point scale indicates the student
responses were split evenly with very few opting for the moderate responses (i.e., 2:
disagree or 3: agree). The last question on the student survey suggests that the students in
Eva’s classroom were persistent (�̅�𝑥 = 1.4,𝜎𝜎 = 0.6).
Teacher Survey Data
The teacher responses were interpreted using the theoretical model to preserve
anonymity for Eva and summarized in Table 4.11. The columns in the table represent the
theoretical constructs. The first two rows of the table represent Eva’s responses to the
teacher surveys20
The first row of
. The specifics about the analysis were detailed in Chapter 3.
Table 4.11 compares Eva’s highly favorable responses to the
survey, analysis within construct group. For example, there are 10 questions from the two
surveys categorized as classroom culture and of those 10 questions, Eva responded with
highly favorable responses to 8 of those questions, hence 80% highly favorable responses
is recorded in the first row, second column in the table. The second row compares Eva’s
favorable responses to her overall favorable responses, analysis between construct
groups. Specifically, Eva’s total for highly favorable responses across all three constructs
was 19 and 8 of them were categorized as classroom culture, thus 42% of her favorable
responses were to questions categorized as classroom culture; written in the table row
two column two.
20 Refer to Appendix G for the Teachers’ Commitment to Mathematics Educational Reform (TCMER, Ross et al., 2003) and Appendix F for the Teacher Self-Efficacy Scale (TSES, Tschannen-Moran & Woolfolk Hoy, 2001),
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Classroom
Culture % ( N)
Discourse Community
% ( N)
Relationships that Support Learning
Opportunities % ( N)
Teacher Survey Data
Highly favorable responses within 80 (10) 50 (10) 66 (9)
Favorable responses between 42 (19) 26 (19) 32 (19)
Analysis Summary of Case Data
Coding densities 39 (308) 34 (308) 27 (308)
Table 4.11: Interpretation of Eva’s responses to the TCMER and TSES teacher surveys and summary of coding densities for Eva’s case data organized by theoretical constructs.
The actual survey responses resulted in the following conclusions: a) Eva’s
commitment to mathematics education reform was greatest for classroom culture
(TCMER) and b) she was most efficacious with respect to relationships that support
learning opportunities (TSES). However, the overall survey analysis from Table 4.11
suggests the most favorable responses were related to classroom culture and the
classroom interaction analysis concurred, suggesting agreement between the two analyses
and analysis triangulation.
Analysis Summary Data
The last row of Table 4.11 summarizes the coding done during analysis for all the
data associated with Eva’s classroom interactions. The coding process described in an
overly simplistic way is assigning selected pieces of data to descriptive categories. The
data in this row was derived from queries run against the coded data using the qualitative
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analysis software program, NVivo 8 (QSR International, 2008). The software program’s
query function counted and reported the number of times categories were referenced or
connected to specific data (e.g., a segment of video or a transcribed paragraph) related to
Eva.
The percentages shown in row three of Table 4.11 were found as follows. The
categories were grouped by theoretical construct, the counts totaled, and percentages
calculated by dividing the total of coded references for each construct by the total number
of coded references. The percentages recorded for the coded references do not include the
emergent codes interaction type and interaction focus because only the theoretical
construct related categories (codes) were included in the calculation. The emergent codes
were defined during the analysis process and emerged as questions arose about the
overall nature of interactions and during the quest to seek patterns in the data.
For Eva’s classroom interactions, the densest coding occurred for classroom
culture (39%) and the least dense coding occurred for relationship/opportunity (27%).
Thus, during the analysis phase of Eva’s classroom interactions, there were more
instances when categories from classroom culture were coded or connected to specific
data from classroom observation or other data sources (e.g., videotaped episodes,
transcribed interviews). Similarly, there were fewer instances when
relationship/opportunity categories were coded or connected to specific data.
Interaction Types and Focuses Data
The emergent categories or codes, interaction type and interaction focus were
summarized for the classroom observation data in Table 4.12 and Table 4.13,
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respectively. The interaction types have literal meanings. For example, interactions coded
as interaction type student-student, the meat of the interaction was between students21
The most prevalent interaction type coded in the analysis of observations from Eva’s
classroom interactions were teacher-class interactions and the least coded were teacher-
group interactions illustrated in
.
Similarly, the teacher-group interaction type were between the teacher and a small group
of students (i.e., not the whole class).
Table 4.12. The second most coded interaction type was
student-student interactions. It was beneficial that student-student interactions were the
second most prevalent interaction type for Eva’s classroom because of concerns about
Interaction types Coding references % Coding (N=39)
Student-student 7 18%
Teacher-class 27 69%
Teacher-group 2 5%
Teacher-student 3 8% Table 4.12: Summary of coding densities for interaction types for Eva's case data.
data quality that arose early during the data collection phase. As an amateur
videographer the challenges related to recording students became evident; they speak
softly, tend to be shy on camera, and it was hard to capture their voices from afar,
especially when the classroom was in an animated working state, the point when
capturing interactions would be most sought after. The two cameras on tripods set apart
21 The code book which describes each of the codes used in this investigation are in Appendix C.
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from each other video setup worked well; on several occasions video captured on one
camera could be heard better on the second and enabled interaction captures, in spite of
the novice videographer.
Interaction focuses summarized in Table 4.13 are overarching categorical themes
that were derived to capture the essence of all of the classroom interaction summaries22
.
To be precise, every classroom interaction that was analyzed was assigned an interaction
type and at least one interaction focus; some classroom interactions have two interaction
focuses, such as teacher and student explaining math idea. For all of Eva’s classroom
interactions, the top three focuses coded included: a) student explaining math idea (22%);
b) interpreting meaning (20%); and c) reflecting or evaluating (16%).
Interaction focus Coding references
% Coding (n=49) Interaction focus descriptions
General explaining 2 4% Explaining that is not mathematical
Interpreting meaning 10 20% Developing clarity or shared understanding
Math problem solving 6 12% Mathematical problem solving
Reflecting or evaluating 8 16% Cognition about prior event, activity, or action
Social problem solving 6 12% Resolving sociocultural situations
Student explaining math idea 11 22% Students revealing mathematical cognition
Teacher explaining math idea 6 12% Teacher revealing mathematical cognition
Table 4.13: Summary of coding densities for interaction focuses categories for Eva's case data.
22 Eva’s classroom interaction summaries are in Appendix J.
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Analysis Coding Summary by Construct Groups
Table 4.14 summarizes all coding densities grouped by the theoretical constructs,
classroom culture, Discourse or community, and relationships that support learning
opportunities. In this summary table, only codes were selected if they had greater than 10
coding references. That is, in an effort to understand classroom interactions, case data
Frequently coded categories grouped by theoretical construct
Coding refs
(>10)
% coding within
% coding between
Theoretical Construct: Classroom Culture
Collaborative sense making (cultural influencer) 14 21% 9%
Teacher expectations (cultural influencer) 15 23% 10%
Communication (mathematical practice) 11 17% 7%
Student explaining math idea (sociomathematical norm) 26 39% 18%
Total 66 100% 45%
Theoretical Construct: Discourse or Community
Thinking and ideas (student behavior) 16 38% 11%
Catalyst (teacher behavior) 14 33% 9%
Connector (teacher behavior) 12 29% 8%
Total 42 100% 28%
Theoretical Construct: Relationships that Support Learning Opportunities
Positive PAID (identity perception) 13 32% 9%
Teacher action (learning influencer) 12 30% 8%
Math as science (teaching influencer) 15 38% 10%
Total 40 100% 27% Table 4.14: Summary of coding densities for the most frequently referenced (>10) categories for Eva’s case data organized by theoretical constructs.
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coding was reviewed and for the categories listed in the first column, the number of
coded references was greater than 10. For example, as data (e.g., videotapes, interviews)
were reviewed and analyzed, situations arose on 14 separate occasions that offered
evidence of collaborative sense making (see row two column one), and that category
emerged among the most coded for those categories in the construct group Classroom
Culture.
The third column of the table is the percent of coding within each construct group
and the last column is the percent between the construct groups. Thus, collaborative
sense making accounts for 21% of the frequently coded references within the classroom
culture construct of the 66 frequent coding references within this construct. Similarly,
collaborative sense making accounts for 9% of all frequently coded references of the 148
frequent coding references between (among all) construct groups.
Case Study Interaction Summaries
Finally, Table 4.15 summarizes the specific interactions discussed within Eva’s case
study and is a subset of all interactions from Eva’s classroom23
Table 4.15
. The summaries in the
appendix also include notes about the teacher and student actions. Several of the
interaction episodes summarized in have been used to elaborate points made
within this case thus far, and the other interactions will be used to expound on the
findings as the research questions are addressed in the next section.
23 All classroom interactions coded during analysis were summarized in Appendix J.
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Interaction Identifier Interaction Type and Focus Interaction Description (Snapshot #)
1.2 Teacher-class; Mathematical Problem solving
Determining the numbers of groups and students in each group (16 kids groups of 3) (Snapshot 4.9)
1.10 Teacher-class; Reflecting or Evaluating
Reflect on student work and offering suggestions for improvement
2.4 Student-student; Students explaining math idea
Using reasoning to compare three fractions (an approach Eva had not anticipated) (Snapshots 4.8; 4.10)
2.5 Teacher-class; Teacher explaining math idea
Explaining a mathematical (procedural) way to compare three fractions 3/4, 3/5, and 3/12 (Snapshot 4.11; 4.12; 4.13)
4.6 Teacher-class; Interpreting meaning and reflecting or evaluating
Teacher addresses student misconceptions using iterative formative assessment (Snapshots 4.14; 4.15; 4.16)
Table 4.15: Interaction summaries used in Eva's case study.
Findings from Eva’s Classroom Interactions
The primary research question addressed in this section is: How do classroom
interactions influence the Discourse related to mathematics learning and teaching in
Eva’s MCP supported classroom? This question was addressed by attending to the three
sub-questions derived from the theoretical model that focuses this investigation. Those
questions include:
a) What is the nature of classroom culture? b) What is the nature of Discourse or community? c) What is the nature of the relationships that support learning opportunity?
The discussion will begin by addressing the nature of classroom culture first because the
data and analysis in Table 4.11 showed that classroom culture (39%) was the most
densely coded construct group in the analysis and the other two elements Discourse or
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community (34%) and relationship/opportunity (27%) were less densely coded.
Subsequently, these two construct groups will be discussed together following the
classroom culture discussion. The case is concluded with a summarizing matrix
describing the nature of Discourse for each of the three constructs and an initial response
to the major research question about how classroom interactions influence Discourses
related to mathematics learning and teaching.
Classroom Culture
Analysis of Eva’s classroom interactions (Table 4.14) yielded frequently
referenced classroom culture categories of students explaining mathematical ideas (39%),
teacher expectations (17%), and collaborative sense making (21%). Each of these cultural
elements were perceived as strengths by Eva according to her responses to the teacher
surveys about commitment to mathematics education reform and teaching efficacy (Table
4.11). Further, these classroom cultural elements influenced Discourses related to
mathematics learning and teaching.
Consider an interaction mentioned earlier in this case, Interaction 2.4 (Classroom
Snapshot 4.8) coded as interaction type teacher-class and interaction focus student
explaining math idea. The classroom snapshot is about two students who used reasoning
and understanding of fractions to compare three fractions 3/4, 3/5, and 3/12, and order
then from least to greatest. This following descriptive vignette illustrates how
sociomathematical norms influence Discourses related to mathematics learning and
teaching in Eva’s classroom.
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Classroom Snapshot 4.10: Listening and Revoicing Eva began the class discussion of the bell work by inviting two boys to share their thinking. Eva discovered the two boy’s approach as she circulated the room assessing student work and understanding. The boys explained their thinking and approach without Eva interrupting or correcting errors in their explanations. At the end of each explanation and throughout the mathematical Discourse related to the bell work, Eva congratulated each boy. She revoiced what each boy attempted to explain following explanations from both.
(Interaction 2.4)
Two students compared the three fractions using conceptual understanding of fractions
and logic to order the fractions. They used the fact that each fraction had the same
numerator and different denominators, to compare the relative sizes of the pieces if the
whole were the same for the three fractions. Their approach demonstrated conceptual
understanding of fractions and did not require the use of the less efficient procedural
approaches the class had been practicing over the past several days.
Eva anticipated students would variations of the procedural approach (i.e., find
least common multiples, create equivalent fractions with like denominators, then compare
fractions with like denominators) they had been practicing for comparing fractions. Eva
had become aware of the boys’ conceptual approach because she observed the students’
work as she circulated the room while they completed the bell work, then she used what
she learned to inform the bell work review and discussion.
Student explaining was a sociomathematical norm of Eva’s classroom and was
evidenced by the second highest coding density (18%) derived from the qualitative
analysis as reported in the analysis summary Table 4.14. Consequences of this
sociomathematical norm included opportunities created for a) students explaining their
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unique ways of thinking and doing mathematics; b) Eva learning from her students; and
c) deeper mathematical discussions emerging.
The two students’ explanations about how they compared the fractions
conceptually led to a broader class discussion and analysis than perhaps would have
otherwise emerged had the discussion that day been limited to comparing fractions using
the two procedural approaches the class had been accustomed to using. Instead, the class
discussion included students coming up with generalizations about relationship between
the magnitudes of denominators and the size of the pieces that comprised fractional
wholes.
After the two boys explained their strategy, Eva used revoicing (Empson, 2003)
for adding clarity, building mathematical language and vocabulary, and correcting minor
mathematical errors or omissions introduced by the boys’ explanations. Her approach to
revoicing always attributed credit to the original student author and never usurped the
credit for herself. The mathematical practice of communication (17%) was a common
element of classroom culture in Eva’s classroom (Table 4.14) and revoicing was a
frequently coded teacher behavior that accompanied communication.
Teacher expectations (15%) were frequently coded in the analysis and influenced
the Discourses related to mathematics learning and teaching in Eva’s classroom. During
all three explanations of the same problem, the other students listened quietly, and
without interruption. Students appeared to listen to explanations; they were quiet when
others spoke, often looked in the direction of the speaker, and were not typically engaged
in off-task behaviors. Eva expected this from the students and they appeared to
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understand the expectation and participated, thus it was a norm and a cultural influencer
of Eva’s classroom culture.
Eva had established several cultural norms that encouraged participation and
engagement. One way she developed these student behaviors was by sometimes asking
students questions repeatedly or rephrasing the same question in different ways. In an
interaction coded as interaction type teacher-class and interaction focuses student
explaining math idea and reflecting and evaluating. This interaction episode begins as the
discussion about comparing 3/4, 3/5, and 3/12 and ordering from least to greatest
transitions from conceptual approaches to mathematical (procedural) approaches. The
following classroom dialogue illustrates one example of how Eva encouraged Discourse
participation and engagement.
Classroom Snapshot 4.11: Managing Discourse Participation Line 1: Eva: There are of course a bit more of a show your work mathematical way
to do it. Um. Let's look at these fractions and just say, alright, um How could we compare them without drawing a picture? without just knowing, a good understanding of fractions? I'm really proud of you guys. How coud we do it? How could we make these fractions so that we could look at them, student J, and say you know which one is bigger? Student A? [~40 seconds elapsed]
Line 2: Student A: We could use least common multiples. Line 3: Eva: You know you can. Do you remember way back when and we did least
common multiples and they threw that curve ball at you and they said now let's find least common multiple of three numbers. That is why we do that. and now let's find the least common multiple of 4, 5, and 12. What are we gonna do? What are we gonna do? Student A.
Line 4: Student A: count by 5's Line 5: Eva: Sure, we can do that. count by 5's
(Interaction 2.5)
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During this classroom snapshot, Eva repeatedly asked the same questions and placed
parameters to guide students’ responses. Eva talked at length (Line 1) during this
interaction episode, and as she spoke, her speech took on a slower cadence and she
watched the students. Typically, when she used this approach, as time went on, more
students raised their hands and some showed signs of urgency with hand/arm waving
and/or soft utterances (e.g., oh, oh, oh). Otherwise, the environment was peaceful, the
students were quiet and Eva’s voice was low and rhythmic, a created space for thinking,
about 40 seconds. The pedagogical approach created wait time, which appeared to
support students’ thinking.
Eva used the repeated questioning approach regularly, and students responded
predictably, which suggests they understood Eva’s expectation. The observed student
response to repeated questioning on most occasions in Eva’s classroom was increased
potential for participation, as more hands would go up indicating students desire to
respond; and interpreted as hand-raising as positive participation and agency identity
(positive PAID) because they rarely shouted out. Therefore, Eva’s expectation was
student participation and engagement in mathematical Discourse.
A closer examination of the language Eva used with her students in Classroom
Snapshot 4.11 (Interaction 2.5), such as, “You know you can” (Line 3) or “Sure, we can
do that” (Line 5). This language encouraged and motivated students to be mathematically
confident, agentic, and efficacious during mathematical Discourse. The confirming
evidence for this claim were the looks on their faces when she spoke this way and the
way students spoke up during mathematical class discussions. This encouraging language
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and the repeated questioning are each elements of communication, a well established
mathematical practice within Eva’s classroom that was frequently referenced during the
analysis phase of this investigation.
One example of a student speaking up mathematically without solicitation from
Eva was shown in the following classroom episode that picks up where we left off in
Classroom Snapshot 4.11, the mathematical (procedural) approach for comparing three
fractions.
Classroom Snapshot 4.12: Student B Shares Authentic Thinking Line 1: Students: 5, 10, 15, … [Counting by 5's up to 65] Line 2: [Students’ enthusiasm wanes toward the end, Eva records the multiples on
the overhead] Line 3: Eva: I ran out of room. [B has his hand up] Um. Student B. Line 4: Student B: Well this is how I did it. Line 5: Eva: I liked how you did it! Line 6: Student B: I said that well 4 can't go into. . . [spoke for ~ 60 seconds
without interruption] The specifics of what he said were sufficiently inaudible to prevent transcription, but he spoke about his reasoning related to finding a least common multiple (LCM). He used divisiblity rules and multiplication facts. From his reasoning, he concluded that the LCM had to be a multiple of 10 if 4, 5, and 12 were to divide into it; then he considered 20 + 40 = 60; and he knew 𝟓𝟓 × 𝟏𝟏𝟏𝟏 = 𝟔𝟔𝟔𝟔 ; therefore, the LCM was 60.
Line 7: Eva: Student B, remember those rules of divisibility that you hated so much?
Line 8: Student B: [shakes his head affirmatively] Line 9: Eva: You just used them whether you realize you used them or not. Now of
course if you are asked to show your work, I don't know how you do that, but it makes sense to me.
(Interaction 2.5)
After the class had counted multiples of 5 up to 65, Student B offered his thinking about
an alternate way to find the least common multiple that did not involve counting all of the
multiples for each denominator. This was an example of a student initiated mathematical
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discourse, a reform-oriented student behavior (NCTM, 1991; 2000). Student B offered an
authentic mathematical idea and demonstrated ownership for learning by sharing his idea
with the class; Hufferd-Ackles and colleagues suggested that these cultural influencers
enhanced Discourses related to mathematics learning and teaching (Hufferd-Ackles et al.,
2004).
Eva, as pointed out previously, in most observed interactions when students were
explaining mathematical ideas displayed patience, did not correct small errors, and
listened carefully without interrupting. Additionally, Eva’s listening approach likely
influenced the way that students listened to one another (Webb et al., 2006).
Making connections was a mathematical practice in Eva’s classroom. Teacher and
students made mathematical connections. Student B made several mathematical
connections among factors, multiples, and rules of divisibility within his explanation as
he reasoned his way to finding the least common multiple for 4, 5, and 12. Then, Eva
pointed out explicit mathematical connections when she responded to Student B’s
thinking in Classroom Snapshot 4.12 (Interaction 2.5); she recalled something from
Student B’s mathematical past (Line 7) and pointed out how he used prior mathematical
understanding to his present mathematical idea (Line 9). She created a connection to
relevancy for mathematics learned in class, even mathematics students profess to not like.
These examples and others made a compelling case for positing that mathematical
connections were a mathematical practice that enhances classroom culture in Eva’s
classroom.
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The final element in interaction 2.5 to consider with interaction focus reflecting
and evaluating was the classroom episode when the class considered the mathematical
efficiency of their procedural approach. Again, we pick up from Classroom Snapshot
4.12 where we left off following Student B’s idea, and Eva encouraged the students to
evaluate their approach for comparing three fractions.
Classroom Snapshot 4.13: Reflecting and Evaluating a Mathematical Procedure Line 1: Eva: by 4's Line 2: Students: 4, 8, 12, … [Counting by 4's up to 48]. Line 3: [Students are unenthusiastic, and lose synch. Eva writes on the overhead
sighs, and then runs out of space] Line 4: Eva: Do we have to count like this? Seriously guys, what would be the
easiest way to do this? We'll be counting forever. What's an easier way to do this Student A?
Line 5: Student A: You know how we put the numbers at the bottom and circle them? Instead of going through the whole thing.
Line 6: Eva: OK. OK. Student G? Line 7: Student G: Factor tree Line 8: Eva: Guys? Factor tree. Awesome. I would say factor tree. You're probably
gonna spend less time than if you do it the other way. Let's try it? Let's try factor tree.
(Interaction 2.5)
Eva appeared to have contrived the situation that caused students to reflect and evaluate
whether or not their mathematical procedure was efficient. As the class counted multiples
of five in Classroom Snapshot 4.12 and again as they counted multiples of four in this
snapshot, Eva was communicating beyond just talk. Specifically, during the multiples of
fives counting, she commented about running out of space on the overhead she was
writing on, in a matter of fact way. Then, while recording the multiples of four on the
overhead, had the Academy heard the exasperation in Eva’s voice, I think they would
have awarded her an Oscar. Finally, Eva followed up the award winning exasperated
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writing of multiples of four, with an intentional and strategic question posed to students.
She asked in Line 4 if there was a better way to find the least common multiple than
counting ALL of the multiples of a number. Eva’s behavior of exasperation was out of
the ordinary, which suggests that her acting contributed to students recognizing the need
to consider other options. The result was several students raised their hands to offer ideas;
significant potential participation and agency.
The other interesting point to note about Classroom Snapshot 4.13 was that the
first student could not name the approach but described how to record the alternate
method, an approach Eva had modeled for the class on a number of different occasions.
The second student was able to name the approach using mathematical language, prime
factorization. This was one example of collaborative sense making; one student builds
upon an idea of another. Eva and the students completed the problem using prime
factorization to find the least common multiple for the three fractions.
Eva called on 15 of the 19 students present in the class to offer input to finish the
problem using prime factorization. Students had to listen to their peers and follow along
with the process in order to contribute the next input. A second of many examples of
collaborative sense making (21%) or analytic scaffolding (Nathan & Knuth, 2003), a
cultural influencer that occurred as a cultural norm and frequently referenced element of
Eva’s classroom culture in Table 4.14 that enhanced mathematics learning and teaching.
During the collaboration, some students volunteered and others were volunteered
when Eva called them by name. All who were called upon provided input into the process
except for one student who appeared sleep, but Eva called upon him again shortly after
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and he contributed. Again, if Eva called on a student who was not ready to respond
immediately, she waited and let him or her think. The result was student input and
teacher praise for good thinking. This was strong evidence of the extent and result of
teacher expectations for student participation and engagement in mathematical
Discourses.
In summary, Eva’s classroom interactions have evidenced signs of emergent
reform-oriented classroom culture and specific interaction episodes were described
through classroom snapshots selected as exemplars of Eva’s classroom culture.
Specifically, there were examples of social (e.g., listening and respect) and
sociomathematical (e.g., student explaining) norms, mathematical practices (e.g.,
mathematical connections and communication), and cultural influencers (e.g., teacher
expectations and collaborative sense making) that positively influenced mathematics
learning and teaching in Eva’s classroom. The connections among these types of reform-
oriented cultures and improved mathematics learning and teaching are well established
by the literature (e.g., Cobb & Hodge, 2002; Empson, 2003; Hufferd-Ackles et al., 2004;
Wood et al., 2006; Zevenbergen, 2000). In addition to classroom culture the theoretical
model includes two other constructs that influence Discourses related to mathematics
learning and teaching, Discourse or community and relationships that support learning
opportunity; they are discussed next.
Discourse/Community & Relationships/Learning Opportunities
The analysis of Eva’s classroom interactions summarized in Table 4.14 depicts
three frequently referenced Discourse or community categories including: a) student
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behavior, thinking and ideas (38%); b) and two teacher behaviors, catalyst (33%); and
c) connector (29%). Similarly, the three frequently referenced relationships that support
learning opportunities categories included: a) identity perception, positive participation
and agency identity (positive PAID, 32%); b) teaching influencer, mathematics as science
(38%); and c) learning influencer, teacher action (30%). Each of these elements of
classroom interactions influence Discourses related to mathematics learning and teaching
in Eva’s classroom.
The mathematics thinking and doing in Eva’s classroom was not driven by
answers only. To illustrate this, consider classroom interactions 2.4 and 2.5, described
extensively in the previous section in several classroom snapshots about comparing three
fractions with the same numerators and different denominators using both conceptual and
procedural approaches. The answer of how to order the fractions had emerged within the
first two minutes of the Discourses that spanned more than 15 minutes. The majority of
the time was spent reflecting on, thinking about, and finding different ways to compare
the three fractions and most students appeared to be listening, engaging, and/or
participating in some way at various times. This suggests persistent students interested in
mathematics beyond answers, the types of student behaviors documented in the literature
as contributing to effective Discourses related to mathematics learning and teaching (e.g.,
Ball, 1993; Hodge, 2006; Lampert, 1990/2004; Sullivan et al., 2006; Webb et al., 2006).
Recall also that students perceived themselves as persistent based upon their responses on
the student survey in Eva’s classroom.
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A second example of classroom interactions that enhanced Discourse was teacher
action (30%), a learning influencer frequently referenced and categorized as a
relationship opportunity in Table 4.14. Interaction 4.6 was coded interaction type
teacher-class and interaction focuses interpreting meaning and reflecting or evaluating.
This interaction episode was about Eva’s teacher actions, she used iterative cycles of
formative assessment to inform teaching and support student learning.
Classroom Snapshot 4.14: Teacher Actions that Support Learning Eva displayed an enlarged copy of a ruler on the overhead and used colored pens to mark positions on the ruler or color spaces. Each student had a ruler, enlarged paper copy of the same ruler, 10” X 12” white board, dry erase marker, and calculator.
Eva asked students twice to represent the position marked on the overhead ruler using fractions. Students held up their white boards.
Line 1: Eva: Put them [the white boards] down. Do not erase. From zero to one, let's count how many little slashes we have.
Line 2: Students: We have 1, 2, 3, 4, 5, 6, 7, 8. [Eva colors in the spaces between the ruler’s hash marks as she and the students count]
Line 3: Eva: And we've counted from zero to one. One out of, one out of how many?
Line 4: Students: 8 [the class answers in unison] Line 5: Eva: 8, Please erase. [Eva drew another mark on the ruler between 0 and 1]
What is the fraction?
(Interaction 4.6)
Neither of the two video cameras were positioned to capture the student white
boards as they held them up, but it was obvious by Eva’s demeanor and teaching moves
that several of the student responses were incorrect. She seemed to be using what she
learned from observing the students’ responses on the white boards. Eva counted with the
class the number of spaces between the 0 and 1 on the ruler; perhaps, the students were
writing incorrect denominators on the white boards.
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Eva‘s instructional strategy for correcting the student misconceptions was to make
an explicit mathematical connection between what the students did not understand, the
ruler markings and something the students had demonstrated significant mathematical
understanding, pictorial fraction models. This was an example of teacher behavior that
positively influenced Discourse and community, coded as connector; using student ideas
to develop understanding (Hufferd-Ackles et al., 2004) or analytic scaffolding (Nathan &
Knuth, 2003).
Eva’s initial attempt to correct the students’ thinking was tentative and apparently
did not correct the problem for all students because she asked the students to represent a
different ruler marking with a fraction and she again had them set the white boards down
and not erase. In this next classroom episode, a continuation of interaction 4.6, Eva
intervenes a little more explicitly.
Classroom Snapshot 4.15: Using Student Responses for Teaching Support Line 1: Eva: This ruler is broken up into, each inch is broken up into how many
pieces? Student D? Line 2: Student D: 8 Line 3: Eva: For now, let's make every denominator 8, every denominator until I
tell you otherwise is going to be an 8. Line 4: [Eva writes a fraction without a numerator and a denominator of 8] Line 5: Eva: When we count from zero, we don't count this slash, the one that is
marked as 0. We don't count it, we count after that and we're gonna count 1, 2, 3 [As she speaks, Eva used her pen to illustrate what not to count and how to count the spaces on the ruler up to where she had drawn the mark] and we're going to use the denominator as 8.
(Interaction 4.6)
Eva’s intervention was very teacher directed in this snapshot. She told students what the
denominator would be (Line 8), and she showed them how to write the fraction using the
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given denominator (Line 9). Then she counted the number of spaces for the numerator
and showed them explicitly (Line 10) how to represent the ruler markings as fractions,
but before continuing, she stopped to assess students’ understanding.
Classroom Snapshot 4.16: Eva Assesses Understanding before Continuing Line 1: Eva: Why am I telling you that we have to use the denominator of 8?
Student A? Line 2: Student A: Because that's the number of lines that are in between. Line 3: Eva: That's right, that's how many lines, how many pieces this inch is
broken in to so we counted 3, so it should be 3/8. Ok, everyone follow me. [Revoicing and showing what Student A said on the overhead ruler and coloring spaces with the pen]
Line 4: [Most students nod their heads to signify understanding] Line 5: Eva: So get ready, let's try another one. Uncap your marker. Ready? What
fraction is it? Line 6: [Students thinking and doing for ~5 seconds] Line 7: Eva: You should be done in 3, 2, 1. Hold 'em up. Ahhh. That's much better.
(Interaction 4.6)
Eva was the reflector/evaluator in this interaction episode and her teaching behavior
influenced student understanding. She used formative assessment to check and recheck
what students understood, she offered students support a little at a time until they
demonstrated understanding as evidenced by written responses on white boards. This is
one of many examples of a teacher action (30%), a frequently coded category in Table
4.14, and in this situation contributed to student participation. Eva was a catalyst (33%)
for student participation, another frequently coded category in Table 4.14. There was
100% participation using the white boards and represented an explicit example of
positive participation and agency identity (positive PAID), an identity perception that
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influenced Discourses related to mathematics learning and teaching (Hodge, 2006; Nasir,
2002).
Additionally, this interaction described a task that required students to think about
mathematics and their resulting ideas were conveyed via their white boards, an example
of the category named thinking and ideas. This category was grouped in the
Discourse/community construct and has been identified in the literature as a student
behavior that enhances Discourses related to mathematics learning and teaching
(Choppin, 2004; Sfard & Kieran, 2001; Sherin, 2002).
In addition, in this interaction, Eva’s reflective teacher behavior, coded as a
category in Discourse or community. The classroom snapshots that elaborated interaction
4.6 about Eva’s reflective teacher behaviors – she assessed and reflecting on students’
understanding; intervened instructionally using evidence of student understanding; and
repeated the process iteratively until students provided evidence of understanding. This
type of reflective teacher behavior influenced Discourses related to mathematic learning
and teaching in Eva’s classroom.
Considering interaction 4.6 when Eva was giving teacher directed interventions, it
was clear that she was in control of the Discourse flow, her control was strategic and
intentional. This controlled Discourse flow was characteristic of the type of Discourses in
Eva’s classroom and helps explain why most interaction types during the observation
period were teacher-class (69%) (Table 4.12). The tightly structured environment
benefited from the well-established norms and effective classroom interactions related to
mathematics learning and teaching.
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Eva’s classroom, though managed at times traditionally, she used combinations of
authority types and leadership styles to facilitate relationships that support learning
opportunity. There were times when her authority or leadership relationship with students
was contributive or dialogic (Lloyd, 2008) as she assumed the attributes of a charismatic
or distributed leader (Bass, 1990). To illustrate these authority types and leadership
styles, consider what happened in interaction 4.6 in Classroom Snapshot 4.15: Using
Student Responses for Teaching Support. In Line 8 she decreed that the class would use 8
as the denominator for all fractions until she said otherwise, this was an example of
traditional authority or authoritative discourse type. However, she appeared to have
changed her approach because shortly after her decree, she created an opportunity for
student thinking and ideas in Line 11 when she asked, “Why am I telling you that we
have to use the denominator of 8?” (Interaction 4.6), which suggested a contributive
discourse type and characteristic of a charismatic leader (Bass, 1999). This shift from
authoritative to contributive discourse type was one example of many for Eva, and
similarly her leadership styles shifted from autocratic to charismatic.
Another example of Eva using contributive discourse type was during Interaction
1.10, while sharing student work Eva invited students to make suggestions for
improvement. The students responded respectfully and constructively with ideas for
improvement. In summary, teacher behaviors that contributed to Discourse/community
included using combinations of authority types and leadership styles, reflective behaviors
during instruction, and facilitating mathematical and other types of connections during
learning and teaching.
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The focus thus far has been on teacher behavior, but there is much to say about
student behavior. The students’ behaviors within the aforementioned classroom
interaction episodes enhanced the Discourses related to mathematics learning and
teaching. With respect to Discourse/community there was evidence of student
engagement, mathematical thinking and ideas, self-regulation, and helping and
supporting behaviors. Also, there were many examples of students possessing
confidence, agency, and persistence during mathematical activities. All of which
contribute to effective mathematics learning and teaching (e.g., Ball, 1993; Lloyd, 2008;
Sullivan et al., 2006). For example, in interaction 1.10 students offered ideas and used
self-regulating behaviors as they offered constructive feedback for peers, such as, “they
need to explain” or “they need to include a symbol” between two fractions being
compared. Within this section and the previous section there were many examples of
student behaviors that contributed to Discourses related to mathematics learning and
teaching.
Similarly, many of the interaction episodes included in this section had elements of
mathematical connections. This connectedness among the lessons exemplified how Eva
and the students viewed mathematics as a science; mathematics was a coherent,
connected study to be explored, and the class spent significant time focused on
understanding mathematical concepts and procedures. The mathematics were not
presented as unrelated or chaotic collection of facts and procedures to be memorized.
Clearly, in most instances Eva’s classroom interactions enhanced Discourses related to
mathematics learning and teaching.
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Summarizing Matrix for Eva’s Classroom
The nature of classroom culture, Discourse or community, and relationships that
support learning opportunities have been articulated through the lens of the theoretical
model and supported with the data and analysis. The primary research question: How do
classroom interactions influence the Discourse related to mathematics learning and
teaching in Eva’s classroom? and key elements from the preceding discussion are
depicted in Table 4.16. This table represents a summarized response to the primary
research question for Eva’s classroom interactions.
Table 4.16 was organized to illustrate the case study through the lens of the
theoretical model. The matrix columns represent the three theoretical constructs: a)
classroom culture; b) Discourse or community; and c) relationships that support learning
opportunity. The rows of the matrices depict the four concepts that define the construct
group. The theoretical construct classroom culture the four concepts include: a) cultural
influencer; b) mathematical practices; c) sociomathematical norms; and d) social norms.
The theoretical construct Discourse or community the four concepts include: a) Discourse
influencer; b) Discourse types; c) student behaviors; and d) teacher behaviors. The
theoretical construct relationships that support learning opportunity the four concepts
include: a) authority type or leadership style; b) identity perceptions; c) learning
influencer; and d) teaching influencer. Specific elements of this table are discussed in
detail within the cross case analysis in the next chapter.
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Classroom culture Discourse/community Relationship/opportunities
Cultural influencers included teacher expectations and collaborative sense making
Discourse influencer connected mathematics and math thinking and doing not focused on answers only
Leadership styles (and authority types) varied and were often either charismatic or distributed
Mathematical practices often included and math connections and communication
Discourse types varied with contexts and contributive was the most frequently coded
Identity perceptions were overwhelmingly positive participation and agency identity
Sociomathematical norms included student explaining
Student behaviors included: mathematical thinking, ideas, self-regulating, persistence, and efficacy
Learning influencer included teacher actions
Social norms included listening and respect for peers and ideas
Teacher behaviors included reflective and connector
Teaching influencers included teacher and student perceptions of mathematics as science
Table 4.16: A summary of the analysis of Eva’s case study through the lens of the theoretical model.
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Case Study 3: Kia’s Classroom
Kia taught three 90-minute blocks of sixth grade mathematics every day during
this investigation. I observed a total of about 450 minutes of mathematics instruction
during Kia’s late morning block. The observations were scheduled to accommodate Kia’s
busy district schedule and to allow for consecutive days of observation. Observation over
consecutive days could potentially allow for observing day-to-day connections related to
mathematics content, learning, teaching, and/or Discourses. Additionally, the principal
did not want data collection to interfere with preparation for or administration of the state
achievement test. The first two days were not on consecutive days; they occurred on
consecutive weeks, on a Tuesday and a Thursday, respectively. One week later, the final
three days were scheduled on consecutive days beginning on a Monday.
During the observation period, Kia’s topics for instruction included a) using
pictorial representations to perform addition and subtraction with decimal numbers; b)
estimating addition and subtraction using decimal numbers; c) creating histograms to
represent data; and d) reviewing histograms; and e) problem solving with decimals and
fractions using estimation.
Kia’s class included a diverse group of students. The class was comprised of near
equal numbers of males (10) and females (9), a diverse representation by race and
ethnicity with more Blacks (12), and including Whites (4), and other (3) races or
ethnicities. There were on average 18 students present on each observation day.
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Kia’s Perceptions about the Class
In this section, Kia’s perceptions are shared about the observed class, her
instructional style, student engagement, and classroom interactions. These perceptions
came from Kia in a pre-observation interview (Feb. 18, 2009) and supported using
observation data. Kia’s perceptions included a somewhat chaotic classroom that is not
overly structured, with students who are involved in mathematics learning, and teaching
that is focused on supporting students for success.
Classroom Environment
Kia described this class as challenging during an informal conversation (Feb. 6,
2009). I did not ask her what she meant by this comment, but the observation data and
analysis provided insight that is discussed later. During the interview, on two occasions
Kia mentioned that students interacted negatively toward one another, but she did not talk
about how she intervened or responded in those situations. Specifically, she said, “…they
yip at each other…” while describing classroom interactions; and then while describing
student engagement, she ended by saying, “…besides when they are interacting
negatively with each other” (Feb. 18, 2009).
On the other hand, two themes appeared to be very important to Kia because she
reiterated them in response to most questions: students’ participation and comfort. When
Kia spoke of student comfort, “I want kids to feel comforatable enough to interact” (Feb.
18, 2009). Upon entering Kia’s classroom, on the walls were student work, and a variety
of posters, including; inspirational, character-building, literacy strategies, and
mathematical facts, formulas, concepts, and processes. Her room was bright, colorful,
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and full of instructional stuff, such as, stacks of papers, books, shelves with many baskets
with supplies and manipulatives. Kia’s comments and her room decor further evidence
the value she placed on students’ participation and comfort.
Kia’s comments and observations suggested that Kia was also concerned about
student success and their having a good sense about themselves. In her responses to
questions in the pre-observation interview, she said, “I don't want kids to feel bad about
their answers.” (Feb. 18, 2009). A commitment to student success and their having
positive feelings could lead to being too supportive in some instances. A brief classroom
interaction episode to illustrate this point follows. Kia selected a number for the class to
factor, 36, and then she started soliciting input from students for creating multiple factor
trees.
Classroom Snapshot 4.17: Too Much Student Support??? Line 1: Kia: Ok, give me a start. Line 2: Student C: Six times six Line 3: Kia: OK and someone else over here had something else. Line 4: Student J: two. . . [slight pause, less than 1 second] Line 5: Kia: Times? What's half of 36? Line 6: Student J: Ah. [~2 second pause] 18. Line 7: Kia: Does someone else have a way to start this?
(Interaction 2.2)
Notice in this interaction how very quickly the student was offered help for what he was
thinking about factors for 36 (Lines 4-6). Student J paused ever so slightly and he was
presented two intervention ideas. Perhaps had he been given two or three seconds to
complete his thinking he may not have needed intervention or he may have been able to
craft a question. After offering the initial intervention idea, an operation, “times” (Line
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5), the student had no time to think before the question was changed into a fact question,
“What is half of 36?” (Interaction 2.2). What if the student was thinking about halving or
doubling and not thinking about the situation as multiplication or division; the
intervention potentially interrupted and usurped the student’s approach rendering his
thinking invalid or abandoned to consider the new question. In either case, it was unclear
what the student thought because he was not afforded an opportunity to elaborate or ask a
question that might have revealed his thinking.
Instructional Style: Teacher Introduction and Student Practice
Kia described her instructional style in terms of lesson flow, “I like to introduce a
concept and then let the kids try… we do a lot of group work and then at some point we
come back together and review and… I like them to draw, to make, to create something
that has to do with what we’re doing…” (Feb. 18, 2009). The observed pattern of
instruction included: a) passing out supplies, such as pencils; b) teacher-led instruction
using the overhead; c) student practice, either individually or in small groups; and d)
reviewing answers together throughout the instructional period.
Classroom Interactions: Sometimes Chaotic and Active
Kia described classroom interactions during the pre-observation interview as,
“sometimes it’s chaotic because everyone's moving around, building and creating” (Feb.
18, 2009). The observations supported the notion that Kia’s classroom was not overly
structured and at times students were moving about the room freely, usually in pursuit of
supplies or support for the things they were doing related to mathematics learning.
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As mentioned earlier, she also stressed her desire that all students participated and
be involved in the learning process. Kia said, “I want everyone's answers. I like to
interact with kids. I want them interacting with each other; that's why I do group work….
I want the kids to be part of the class” (Feb. 18, 2009). From these comments, it appears
that Kia values correct answers, participation, and active. Additionally, these comments
implied creating opportunities for students to interact with her and their peers was valued.
With respect to interacting, Kia said, “we're constantly yipping stuff out and alot
of times they yip at each other while they're yipping at me or helping me or helping us”
(Feb. 18, 2009). The chaotic or less organized instructional structure was implied as she
suggested students constantly shouted out (yipping); raising hands did not appear to be
the norm during observations or based upon these comments. Also, suggested by the
language used in the quote, there was an implied issue of perhaps listening or respect
related to how people interacted – yip or yipping ‘at’ as opposed to yip or yipping ‘to’ or
‘with.’ These comments and the classroom observations attested to Kia’s challenge with
communications, specifically in the ways that she and some students’ communicated with
one another.
Student Engagement: High Expectations
Kia perceived herself as committed to student participation and comfort, as
mentioned earlier; she was not dissuaded by the unstructured nature of her classroom,
which sometimes presented with an ambiance of informality and bedlam. She described
engagement as, “…I like to think that I can get my kids to be engaged … and they feel
comforatable enough to be shouting stuff out...” (Feb. 18, 2009). There were many
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observed instances where students shouted out freely and others when Kia called on
students, both those who volunteered and those who did not. However, offering input
upon request, one could argue, does not always equate to engagement.
Kia was sometimes challenged by moving the lesson toward teaching objectives
and making space for student input. An example of Kia’s challenge to balance between
lesson objectives and student input was captured in the following classroom interaction
episode when the class was going over answers to achievement test practice questions.
The class had been discussing what 43
Classroom Snapshot 4.18: Kia Balances Objectives and Student Input
meant. Kia appeared to have a teaching objective
to mathematically connect exponential representations and prime factoraization.
Line 1: Student J: So like . . [Kia spoke over the student making her comment inaudible].
Line 2: Kia: What is the 4? I'm sorry sweetheart. I don't want to dismiss you, but I'm gonna show you what we're gonna relate it to. She's absolutely correct that we did this, didn't we? We did this all the way back in the fall, but I want to continue it.
(Interaction 2.1)
In this brief interaction snapshot, it appeared that Kia heard the student because she
apologized for interrupting the student. Kia was likely animating or giving the student
credit for her idea in Line 2 when she said, “…She’s absolutely correct…” (Interaction
2.1). Giving the student credit, likely did not assuage the student, if her feelings were hurt
for not being heard. Kia did acknowledge and gave her credit for her contribution; but,
she did not use the student’s contribution to further the teaching objective. Nonetheless,
Kia moved on and pursued the intended objective of connecting exponential
representation and prime factorization (i.e., exponential representation is used to show
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prime factorization). Kia’s comments showed remorse for interrupting the student and
evidenced Kia’s challenge for balancing learning objectives and creating space for
student input.
After observing Kia’s classroom interactions, supporting and engaging students in
mathematics learning and teaching were important to her. She also stated that she valued
student input; however, communications within her classroom among herself and
students were challenged as she endeavored to encourage participation and support
learning. Specific examples from Kia’s classroom related to the theoretical model that
guide this investigation were used to address the research questions following the
presentation of the data.
Case Data for Kia’s Classroom
The case data that was analyzed and used to construct this case study included
450 minutes of video-taped classroom observations, Kia’s responses to teacher surveys,
students’ responses to a student survey, and pre-observation interviews with Kia and the
MCP coach assigned to her building. The analysis process for both teacher and student
surveys was detailed in Chapter 3.
Student Survey Data
The student survey had six questions with a four-point Lickert-scale: strongly
agree (4); overall agree (3); overall disagree (2); and strongly disagree (1). The student
responses for the Lickert-scaled questions assigned to an interpreted category and
averaged, then summarized in Table 4.17. The interpreted categories are phrases that
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Questions Interpreted category
Responses (mean, �̅�𝑥)
Standard deviation (𝜎𝜎)
I feel confident that I can learn most math topics. Math confidence 3.3 0.8
I can learn anything in math if I put my mind to it. Math efficacy 3.7 0.5
If I find the work hard, I know that if I keep trying I can do it. Math agency 3.4 0.9
My friends say that I keep trying when math gets hard Math identity 2.5 1.1
You are either good at math or not and you cannot get better by trying.* Fixed intelligence 1.9 1.2
If I can’t do the work in math I give up. Persistence * 1.3 0.8 *
Low scores are most favorable for these questions
Table 4.17: A summary of student survey responses (n=15) for Kia’s classroom.
capture the meaning of each question. As data summarized in Table 4.17 illustrates, the
students in Kia’s classroom on average responded to the first three questions in a way
that suggested they perceived themselves as confident (�̅�𝑥 = 3.3,𝜎𝜎 = 0.8) and possessed
a high level of math efficacy (�̅�𝑥 = 3.7,𝜎𝜎 = 0.5) and agency (�̅�𝑥 = 3.4,𝜎𝜎 = 0.9).
Interpreting the fourth question using Hodge’s (2006) definition of identity as the way “. .
. individuals perceive that others are perceiving them in interaction” (p. 380), Kia’s
students did not on average perceive themselves appreciably with respect to math identity
(�̅�𝑥 = 2.5,𝜎𝜎 = 1.1). The mean was close to the median and the standard deviation was
large for such a small sample (n=15) suggesting that the data was evenly distributed
across the four choices. There was some consensus about intelligence; many students
leaned toward growth intelligence (�̅�𝑥 = 1.9,𝜎𝜎 = 1.2); these students perceived they
could get better at math by trying. The data showed students overwhelmingly (3:5) chose
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this option by selecting strongly disagree (1). Conversely, selecting one of the agree
options (3 or 4), suggested students perceived their intelligence as fixed, one was born
with a set amount of math ability and trying would not help. Fewer students (1:3) chose
one of these options. The last question on the student survey suggested students in Kia’s
classroom perceived themselves persistent (�̅�𝑥 = 1.3,𝜎𝜎 = 0.8).
Teacher Survey Data
The teacher responses were interpreted using the theoretical model to preserve
anonymity for Kia and summarized in Table 4.18. The columns in the table represent the
theoretical constructs. The first two rows of the table represent Kia’s responses to the
teacher surveys24
The first row of
. The specifics about the analysis were detailed in Chapter 3.
Table 4.18 compares Kia’s highly favorable responses to the
survey, analysis within construct group. For example, there were 10 questions from the
two surveys categorized as Discourse/community and of those 10 questions, Kia
responded with highly favorable responses to 7 of those questions, hence 70% highly
favorable responses was recorded in the first row, second column in the table. The second
row compares Kia’s favorable responses to her overall favorable responses, analysis
between construct groups. Specifically, Kia’s total for highly favorable responses across
all three constructs was 15 and 7 of them were categorized as Discourse or community,
24 Refer to Appendix G for the Teachers’ Commitment to Mathematics Educational Reform (TCMER, Ross
et al., 2003), Appendix F for the Teacher Self-Efficacy Scale (TSES, Tschannen-Moran & Woolfolk Hoy,
2001)
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thus 47% of her favorable responses were to questions categorized as Discourse or
community; written in the table in row two column two.
Classroom
Culture % ( N)
Discourse or Community
% ( N)
Relationships that Support Learning
Opportunities % ( N)
Teacher Survey Data
Highly favorable responses within 40 (10) 70 (10) 44 (9)
Favorable responses between 26.5 (15) 47 (15) 26.5 (15)
Analysis Summary of Case Data
Coding densities 29 (575) 41 (575) 30 (575)
Table 4.18: Interpretation of Kia’s responses to the TCMER and TSES teacher surveys and summary of coding densities for Kia’s case data organized by theoretical constructs.
The actual survey responses yielded the following conclusions: a) Kia’s
commitment to mathematics education reform was greatest for Discourse or community
(TCMER); and b) she was most efficacious with respect to Discourse or community
(TSES). Thus, both survey analysis of teacher’s perception and qualitative analysis (the
third row of Table 4.18) yielded high percentages for Discourse or community, which
suggests agreement among analyses and analysis triangulation.
Analysis Summary Data
The last row of Table 4.18 summarizes the coding done during analysis for all the
data associated with Kia’s classroom interactions. The coding process described in an
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overly simplistic way is assigning selected pieces of data to categories. The data in this
row was derived from queries run against the coded data using the qualitative analysis
software program, NVivo 8 (QSR International, 2008). The software program’s query
function counted and reported the number of times categories were referenced or
connected to specific data (e.g., a segment of video or a transcribed paragraph) related to
Kia.
The percentages shown in row three of Table 4.18 were found as follows. The
categories were grouped by theoretical construct, the counts totaled, and percentages
calculated by dividing the total of coded references for each construct by the total number
of coded references. The percentages recorded for the coded references do not include the
emergent codes interaction type and interaction focus because only the theoretical
construct related categories (codes) were included in the calculation. The emergent codes
were defined during the analysis process and emerged when questions arose about the
overall nature of interactions and in the quest to seek patterns in the data. These
categories or codes are discussed in the next section.
For Kia’s classroom interactions, the densest coding occurred for Discourse or
community (41%) and the least dense coding occurred for classroom culture (29%).
Thus, during the analysis phase of Kia’s classroom interactions, there were more
instances when categories from Discourse or community were coded or connected to
specific data from classroom observations or other data sources (e.g., videotaped
episodes, transcribed interviews). Similarly, there were fewer instances when classroom
culture categories were coded or connected to specific data.
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Interaction Types and Focuses Data
The emergent categories (codes), interaction type and interaction focus were
summarized for the classroom observation data in Table 4.19 and Table 4.20,
respectively. The interaction types have literal meanings. For example, interactions coded
as interaction type student-student, the meat of the interaction was between students25
The most prevalent and almost exclusive interaction type coded for Kia’s
classroom interactions was teacher-class illustrated in
.
Similarly, the teacher-group interaction type were between the teacher and a small group
of students (i.e., not the whole class).
Table 4.19. The second most coded
interaction type was student-student. It was beneficial that student-student interactions
were coded because of concerns about data quality that arose early during the data
collection phase. As an amateur videographer the challenges related to recording students
became transparent; they speak softly, tend to be shy on camera, and it was problematic
capturing their voices from afar, especially when the classroom was in an animated
Interaction types Coding references % Coding (N=55)
Student-student 5 9%
Teacher-class 48 87%
Teacher-group 1 2%
Teacher-student 1 2% Table 4.19: Summary of coding densities for interaction types for Kia's case data.
25 The code book which describes each of the codes used in this investigation are in Appendix C.
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working state, the point when capturing interactions would be most desired. The two
cameras sitting on tripods on two sides of the room worked well; on several occasions,
video watched on one camera could be heard on the second camera and enabled
interaction captures in spite of the novice videographer. Interaction focuses are
overarching categorical themes that were derived to capture the essence of all of the
classroom interaction summaries for Kia included in Appendix K. More precisely, every
Interaction focus Coding references
% Coding (n=72) Interaction focus descriptions
General explaining 2 3% Explaining that is not mathematical
Interpreting meaning 18 25% Developing clarity or shared understanding
Math problem solving 12 17% Mathematical problem solving
Reflecting or evaluating 14 19% Cognition about prior event, activity, or action
Social problem solving 9 12% Resolving sociocultural situations
Student explaining math idea 5 7% Students revealing mathematical cognition
Teacher explaining math idea 12 17% Teacher revealing mathematical cognition
Table 4.20: Summary of coding densities for interaction focus categories for Kia's case data.
classroom interaction was analyzed and assigned an interaction type and at least one
interaction focus; some classroom interactions have tow interaction focuses, such as,
teacher and student explaining math idea. For all of Kia’s classroom interactions, the top
four interaction focuses coded included: a) interpreting meaning (25%); b) reflecting or
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evaluating (19%); c) math problem solving (17%); and d) teacher explaining math idea
(17%).
Analysis Coding Summary by Construct Groups
Table 4.21 summarizes all coding densities grouped by the theoretical constructs,
classroom culture, Discourse or community, and relationships that support learning
opportunities. In this summary table, only codes were selected if they had the greatest
Frequently coded categories grouped by theoretical construct (top 3 or 4 per construct)
Coding refs
(>15)
% coding within
% coding between
Theoretical Construct: Classroom Culture
Collaborative sense making (cultural influencer) 17 26% 6%
Teacher explaining (sociomathematical norm) 21 32% 8%
Negative social norms (social norm) 27 42% 10%
Total 65 100% 25%
Theoretical Construct: Discourse or Community
Discourse challenging (student behavior) 30 24.5% 11%
Thinking and ideas (student behavior) 35 29% 13%
Expert (teacher behavior) 21 17% 8%
Informer (teacher behavior) 36 29.5% 14%
Total 122 100% 46%
Theoretical Construct: Relationships that Support Learning Opportunities
Traditional/autocratic (authority/leadership) 33 43% 13%
Fixed intelligence (teaching influencer) 21 28% 8%
Math as procedural (teaching influencer) 22 29% 8%
Total 76 100% 29% Table 4.21: Summary of coding densities for the most frequently referenced (>15) categories for Kia’s case data organized by theoretical constructs.
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coding references within each construct group of categories. That is, in an effort to
understand classroom interactions, case data coding was reviewed and for the categories
listed in the first column, the number of coded references was among the highest three or
four by construct group. For example, as data (e.g., videotapes, interviews) were
reviewed and analyzed, situations arose on 21 separate occasions that offered evidence of
teacher explaining (see row three column two of Table 4.21), and that category emerged
among the most referenced for those categories in the construct group Classroom Culture.
The third column of Table 4.21 depicts the percent of coding within the construct
groups and the last column is the percent between the construct groups. Thus, thinking
and ideas accounts for 29% of the frequently coded references within the Discourse or
community construct of the 122 frequent coded references within this construct group.
Similarly, thinking and ideas accounts for 13% of all frequently coded references of the
263 frequent coded references between (among all) construct groups.
Case Study Interaction Summaries
Finally, Table 4.22 summarizes the specific interactions discussed within Kia’s
case study and is a subset of all interactions from Kia’s classroom26
Table 4.22
. Several of the
interactions summarized in have been used to elaborate points made in this
case thus far, and the other interactions will be used to expound on the findings as the
research questions are addressed in the next section.
26 A summary of all interactions for Kia’s classroom are in Appendix K. The summaries in the appendix include notes about the teacher and student actions.
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Interaction Identifier Interaction Focus(es)* Interaction Description (Snapshot #)
2.1 Social PS and General explaining
Kia interrupts student input to redirect the instructional focus (Snapshot 4.12)
2.2 Teacher and student explaining math idea
Kia guides students to find prime factorization of 36 (Snapshot 4.11)
2.1 Social problem solving and General explaining
Teacher refocuses lesson in lieu of student input (Snapshot 4.18)
2.2 Teacher and Student explaining math idea
Explaining how to find prime factorization of 36 (Snapshot 4.17)
2.6 Interpreting meaning and Student explaining math idea
Developing understanding for a story problem (Snapshot 4.19)
2.7 Reflecting or evaluating Selecting the best multiple choice option and explaining (Snapshot 4.20)
3.3 Interpreting meaning Whole class identifying pertinent information for histogram creation (Snapshot 4.21)
4.2 Interpreting meaning Introducing a worksheet that guides histogram creation (Snapshot 4.22)
5.5 Teacher explaining math idea
Estimating fractions using a pizza analogy (Snapshot 4.23)
*
Table 4.22: Interaction summaries used in Kia's case study.
Interaction type is teacher-class for all interactions in this table.
Findings for Kia’s Case
The primary research question addressed in this section is: How do classroom
interactions influence the Discourse related to mathematics learning and teaching in Kia’s
MCP supported classroom? This question was addressed by attending to the three sub-
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questions derived from the theoretical model that focuses this investigation. Those
questions include:
a) What is the nature of classroom culture? b) What is the nature of Discourse or community? c) What is the nature of the relationships that support learning opportunity? The discussion begins by addressing the nature of Discourse or community
because the data and analysis in Table 4.18 showed that Discourse or community (41%)
was the most densely coded construct group in the analysis and the other two construct
groups, classroom culture (29%) and relationship/opportunity (30%) were less densely
coded. Subsequently, these two construct groups will be discussed together following the
Discourse or community discussion. The case is concluded with a summarizing matrix
describing the nature of Discourse for each of the three constructs and an initial response
to the major research question about how classroom interactions influence Discourses
related to mathematics learning and teaching.
Discourse or Community
Analysis of Kia’s classroom interactions yielded frequently coded Discourse or
community categories of student behaviors, thinking and ideas (29%), and Discourse
challenging (Discourse or community) (24.5%) and teacher behaviors, expert (17%) and
informer (29.5%) (see Table 4.21). Discourse challenging behaviors as depicted in Table
4.21 represents student behaviors only; however there were Discourse challenging
teacher behaviors coded, including informer and expert. The Discourse challenging
student behaviors were mostly carried out by a subset of students (approximately 6 of 19)
in the class. The subset of students varied some, but consistently included three students,
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one male and two female students. The Discourse challenging behaviors of teacher and
students in general hindered Discourses related to mathematics learning and teaching.
More specific and contextualized examples of Discourse challenging behaviors are
described as Kia’s classroom interactions are examined.
Conversely, student thinking and ideas (29%) was a student behavior frequently
coded during analysis of Kia’s classroom interactions. Interaction 2.6 illustrates student
thinking and ideas related to mathematics learning and teaching in Kia’s classroom. The
interaction type was teacher-class and the interaction focuses were interpreting meaning
and student explaining math idea. In this interaction, the class was developing
understanding about a story problem. The interaction episode began with a student
reading the problem followed by the class interpreting the meaning of range (a measure
of spread).
Classroom Snapshot 4.19: Thinking and Ideas Line 1: Kia: When we were doing our [work] the other day, did we work out the
range? of what? Line 2: Student X: Ah, ah, numbers. Line 3: Kia: What do we call those numbers when we're working out range? When
we're working out mean, median, and mode? Line 4: [Kia moves to the word wall where definitions are posted and begins
pointing to words] Line 5: Student K: The data? [. . . Several lines of dialog were skipped here. . . ] Line 6: Kia: The question is asking you to decide which of these [answer choices]
explains what the range represents. Now look up here [pointing at the word wall]. What do we call range?
Line 7: Class: The measure of spread. Line 8: Kia: So, it’s from umm to umm? Line 9: Students: [multiple students talking at once]. . . one end to the other . . .
from high to the low (Interaction 2.6)
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This interaction continued, students thinking and ideas supported the class’
articulation of how to find the range and how it was used in the problem, interpreting
meaning. In the snapshot, the students were thinking and coming up with ideas about
range (Lines 2, 5, 7, and 9). One could argue that the type of questioning in this
interaction (Lines 1, 3, 10) diminished the cognitive demand related to the students’
thinking and ideas, and that may be so; however, analyzing cognitive demand was
beyond the scope of this investigation.
The questions used to elicit the students’ ideas were very specific and leading
rendering the interaction an authoritative Discourse type (Lloyd, 2008) governed by
traditional authority type (Pace & Hemmings, 2006). The traditional authority type was
coded often for Kia’s classroom interactions. The Discourse type most coded for Kia’s
interactions were contributive, a Discourse that includes contributions from more than
one person.
The informer teacher behavior was also present in Classroom Snapshot 4.19. The
informer teacher behavior manifested through hints, such as pointing at answers on the
word wall (Lines 4 and 6) and leading questions, such as “So, it’s from umm to umm?”
(Line 8). These instructional preemptive supports were offered before students were
given an opportunity to think, which exemplified the informer (29.5%) teacher behavior,
a frequently coded category for Kia’s classroom interactions (Table 4.21).
This interaction also included several examples of Discourse challenging (24.5%)
behaviors, some were evident in the dialogue (Classroom Snapshot 4.19), but others were
not. While several students were thinking and offering ideas, there were others not
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focused on the Discourse because they were engaged in activities such as, digging inside
a purse, looking at fingernails, and writing. These student behaviors were not evident
from the snapshot dialogue. The behaviors may have been encouraged by the task, an
exercise to define and tell the steps to calculate range. Was this task sufficiently engaging
for a class of 18 students? A response to this question was beyond the scope of this
investigation, but was considered and perhaps influenced the writing of this case study.
Another Discourse challenging behavior that was evident from the snapshot
dialogue and occurred frequently in Kia’s classroom were people talking at the same
time. When this happened, it was coded as a student behavior; however, there were times
when students and the teacher talked simultaneously or one person talked over another.
The frequency of talking at the same time warranted the behavior as a social norm and in
most instances a negative social norm that may have prevented some from hearing ideas
from others. However, in at least one instance of Classroom Snapshot 4.19 talking at the
same time may have been collaborative as one student refined another’s ideas (Line 9).
The next interaction (2.7) was coded with interaction type teacher-class and
interaction focus reflecting or evaluating. The class was evaluating the best of four
answer choices and explaining why they made their selection. The mathematics in this
interaction was an exercise that was potentially insufficiently engaging for the class, and
may have encouraged Discourse challenging behaviors.
Classroom Snapshot 4.20: Informer Teacher Behavior Line 1: Kia: Hey. Is letter A what you consider to be the best definition? For what
the definition of range would be in this situation? [Kia reads answer choice A]
Line 2: Students: [Several students talking at once] No... no... noooo...
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Line 3: Kia: OK. I heard one person, a couple people say they would eliminate letter A. Does range have to do with how many are in class?
Line 4: Students: [several students talking at once] No...no... noooo... Line 5: Kia: No. Letter B. [Kia reads answer choice B] Line 6: Students: [several students talking at once before Kia finishes reading]
That ain't got nothin' to do wit it...that's right...no... Line 7: Kia: I mean as I said in my last class that's one of those where I go, huh. I'm
not quite sure. So, I'm gonna put a question mark next to it. We definitely eliminated A, but that's a huh?
(Interaction 2.7)
In this interaction students were talking at the same time (Lines 2, 4, and 6), but
notice in Line 6 there appeared to be student collaboration, when one student built upon
another student’s statement by saying, “that’s right” and a third student said, “no.” Also,
students were making individual contributions of “no.” I attempted to show a cacophony
of voiced no’s through spelling and capitalization, “No...no... noooo...” (Lines 2 and 4).
In the recorded video, several different voices formed the ‘no’ chorus; several individual
voices could be heard offering a variety pitches and tones creating several renditions of
no; individual students appeared to want to be heard. The effect was collaboration at best,
but at a minimum, it was collective student behavior.
After the students said no to answer choice B, the teacher behaviors informer and
expert were evidenced. The students responded clearly about what to do with answer
choice B, “That ain't got nothin' to do wit it...that's right...no...” (Line 6), but in Line 7 a
different choice and rationale were offered for handling choice B. The teacher behavior
expert was coded because student input was replaced with the expert’s idea. The teacher
behavior informer was coded because the informer supplied or informed the students of
the rationale for the response. The rationale was switched away from students’ thinking
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to informer’s thinking. The disadvantage for Discourse was that students’ input was not
acknowledged or used to further the mathematical Discourse related to learning and
teaching (Hodge, 2006).
This dialog continued and the class determined that the best explanation was
given in answer choice C. Once the answer was stated and validated by Kia, the students’
participation decreased and Kia completed the task mostly independently. This situation
suggests that in Kia’s classroom the focus was on answers only, that is, mathematical
Discourses revolved around answers more than process. On more than one occasion, the
students’ participation appeared to be answer focused and this connection likely hindered
mathematical Discourses in Kia’s classroom.
In this next interaction example, the teacher was acknowledging student input and
using questions to gain meaning. The teacher in this episode was the MCP coach.
According to the coach, she and Kia were co-teaching to practice using accountable talk
(L. West & Staub, 2003) with students (Informal conversation, Mar. 16, 2009). The
lesson focus was preparing students for the state achievement test and they were solving a
problem about representing data using a histogram. The interaction focus was coded
interpreting meaning. This interaction illustrated student behavior thinking and ideas as
they determined what things were important for solving the problem.
Classroom Snapshot 4.21: Students Thinking and Ideas to Interpret Meaning Line 1: Coach: Can you tell me one important thing that you think that I should pay
attention to in this question? Line 2: Student D: Numbers? Line 3: Coach: Alright, numbers. Which numbers are you thinking of right now? Line 4: Student D: um... Line 5: Coach: You just have to tell me one thing. Line 6: Student D: uh, 49.
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Line 7: Coach: OK. So you're looking at these numbers aren't you [pointing at the table on the overhead].
Line 8: Student D: [nods assent] Line 9: Coach: Alright, so he's looking at this table. . . How about you? Tell me one
thing that we should be paying attention to. [waiting, ~3 seconds, students raising their hands] Can you see one thing up there? [more waiting, ~3 seconds]
Line 10: Student O: The number of intervals. Line 11: Coach: OK. Can you tell me how many intervals it is? Line 12: Student O: 5 Line 13: Coach: OK. I think that would be important. 5 intervals and we'll have to
think about what that means.Ok how about you? Line 14: Student J: Um, the um that sentence about like the rate of the um, the rate
of the, on the scale is 0 to 50. Line 15: Coach: Oh. So there's that word scale and it tells you the rate has to go from
0 to 50. Restaurant ratings on a scale from 0 to 50. I think that would be important. . . .
(Interaction 3.3)
In this interaction, the MCP coach (teacher) was soliciting ideas from students to
develop shared meaning about the problem they were trying to solve. In the earlier part of
this interaction, the students responded tentatively (Lines 2, 4, and 6), but as the
Discourse progressed, they appeared to respond more certainty (Line 14), and the teacher
revoiced what the student said to add clarity (Line 15).
The teacher behavior in this interaction was coded as catalyst. This teacher
behavior occurred when teacher talk, questioning, and other teacher moves positively
influenced student mathematical Discourse. For example, the teacher’s revoicing (Line
15) added clarity for others who may not have understood Student J’s contribution.
Another aspect of the catalyst teacher behavior in this example was the teacher’s
questions; they were persistent and she waited for students to respond. In other words,
students were not let off the hook for giving input. For example, Student D said he
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noticed numbers, the teacher responded with two questions, one asked for specifics (Line
3) and the other afforded access, “You just have to tell me one thing” (Line 5). The
student was hesitant in his initial response (Line 2), so the second question (Line 5) may
have lowered the threshold sufficiently that allowed him to contribute (Line 6). The
teacher revoiced (Line 7), but this time she interpreted meaning and animated, gave credit
to the student in a way that enhanced the Discourse (Empson, 2003).
One student speaking in this interaction, Student D, often engaged in Discourse
challenging student behaviors that resulted in his being reprimanded. Up until this
interaction, he had never been asked to contribute to mathematics Discourses.
Additionally, he and Kia’s relationship appeared to be tenuous at best. As Student D was
speaking, Kia moved to stand beside him; however, it was unclear if or how that teacher
move influenced the interaction. However, and worth noting, was that Student D
contributed positively to the mathematical Discourse and he refrained from Discourse
challenging student behaviors.
In summary, Kia’s classroom interactions sometimes enhanced and other times
hindered Discourses related to mathematics learning and teaching. This conclusion is not
an indictment of Kia, but a conclusion drawn based upon data collected and analysis of
classroom interactions related to Discourse or community. Specific interaction episodes
were described through classroom snapshots selected as exemplars that illustrated
Discourse or community in Kia’s classroom interactions. There were examples of teacher
behaviors (e.g., informer and expert), student behaviors (e.g., thinking and ideas and
Discourse challenging), Discourse influencers (e.g., answers only and exercise), and
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Discourse types (e.g., authoritative and contributive). The literature offers much with
respect to aspects of learning and teaching that enhance and hinder Discourses (e.g.,
Megan Loef Franke & Kazemi, 2001; Megan L. Franke et al., 2007; Kazemi, 1998;
Sherin, 2002). The theoretical model that guides this investigation included two other key
constructs, classroom culture and relationships that support learning opportunities, the
findings related to both are discussed next.
Classroom Culture and Relationship/Opportunities
The analysis of Kia’s classroom interactions yielded frequently referenced
classroom culture categories depicted in Table 4.21 including collaborative sense making
(26%), teacher explaining (32%), and negative social norms (42%). Similarly, frequently
referenced relationships that support learning opportunities categories included
traditional/autocratic (43%), fixed intelligence (28%), and math as procedural (29%).
In the previous section, a frequently observed Discourse challenging student
behavior was described in Classroom Snapshot 4.20 and Classroom Snapshot 4.21
(Interactions 2.7 and 3.3, respectively summarized in Table 4.22) as talking at the same
time. As articulated previously, this simultaneous talking could sometimes be referred to
as a negative social norm (42%), a frequently coded category of classroom culture. There
were several other observed behaviors coded as negative social norm including students
consistently not responding to polling type questions27, writing and passing notes28
27 For more information, see Appendix K interaction summaries for 3.6; 3.10; and 4.4.
, and
only one or two students answering most questions. An emblematic example of limited
participation occurred during interaction 4.2 coded interaction type teacher-class and
28 For more information, see Appendix K interaction summaries for 2.3; 3.13; and 4.1.
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interaction focus interpreting meaning. The interaction was about understanding the
information given on a worksheet that was comprised of step-by-step instructions for
creating a histogram to represent the ages of award winners at a fair, a worksheet
comprised of exercises.
Classroom Snapshot 4.22: Limited Participation Line 1: Kia: What does frequency mean? Line 2: Student A: It's a fraction?. Line 3: Kia: What does frequency mean? Line 4: Student A: Ah, like out of like a survey or something? Line 5: Kia: What does frequency mean? Line 6: Student A: Frequent guess? [puts hands out, palms up] Line 7: Student K: Oh, I forgot... Line 8: Kia: [~ 2 second pause] Hhhooow mmmaany or how much. [speech is
drawn out and Student K said the word ‘many’ in unison] Yeah, I know, that was years ago, Ms. Kia, back in September when you taught us that. Frequency is how many or how much. What's the next column?
Line 9: Student K: 16 through 25. Line 10: Kia: You know it isn't gonna work if we're not gonna be part of the lesson.
Where it says relative frequency they give you a fraction, 6/20. What does that mean? 6 out of?
Line 11: Student K: [shrugs shoulders] Line 12: Student A: 6, 6 out of 20 Line 13: Kia: Were between the ages of Line 14: Student A: 6 and 15? Line 15: Kia: Right. So, we were making a fraction. What are the total amount of
numbers up there? What are the total amount of numbers up there? Line 16: Student A: 20 numbers. Line 17: Kia: 20 becuase of that denominator
(Interaction 4.2)
During this portion of the interaction, the negative social norm of very few
students participating was exemplified. Only 2 of the 18 students present in the classroom
participated. The other students appeared to be disinterested and engaged in Discourse
challenging student behaviors, such as passing things around; snacking; or sitting
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sideways, not facing front, and appeared to do nothing. The majority of the class
appeared to have been choosing not to participate. This type of Discourse led to the
emergence of the identity perception category chosen non-participation, which was
densely coded in Kia’s classroom.
In the middle of the dialog, Kia attempted to address the non-participating student
behavior when she said, “You know it isn't gonna work if we're not gonna be part of the
lesson” (Line 10). However, she waited less than 1 second for students to respond to her
words. Instead, she was showing students where she wanted them to focus on the
worksheet; nothing appeared to changed. Kia turned slightly so her body faced the two
participating students and continued the lesson.
Earlier during the lesson, Kia offered an extrinsic reward for student cooperation;
she told the students her goal was to quickly complete this activity so they could return to
their groups to complete work started a day or two earlier and they would get to present
to the class. This did not work to transition students away from chosen non-participation
identity perception, a densely coded category for Kia’s classroom. The implications of
student behavior chosen non-participation is documented in the literature, and typically
hinders Discourses related to learning and teaching (Haberman, 2002; C. West, 1993).
This episode, Classroom Snapshot 4.22 suggests much about perceptions related
to the nature of mathematics and intelligence. The day before, students had created a
histogram to represent restaurant ratings in collaboration with the MCP coach, and then
they independently, with limited assistance from Kia or the coach, created a histogram to
represent test scores. In this day’s activities, for the first histogram the coach provided
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students with step-by-step guidance, solicited ideas from students, and problem
parameters governing the creation of the histogram (e.g., the number of intervals to use
and the range for the ratings scale). The second problem started out with teacher/coach
guidance that lessened once students got started and the problem was structured
identically to the first problem. Even so, students had decisions to make related to
histogram creation.
Conversely, Classroom Snapshot 4.22 (Interaction 4.2) followed the creation of
histograms for restaurant ratings and test scores. This day’s task was for students to
create a histogram about state fair winners’ ages using a step-by-step guided worksheet,
with lots of blanks to fill in, an activity with arguably lower cognitive demand than the
day before. The guided task left nothing for students to think about beyond knowledge,.
The opportunities for students to come up with authentic strategies for creating a
histogram to represent the data were nonexistent, all of the decisions were already made,
and there was little left beyond counting, interpreting mathematical vocabulary, and
symbolic number representations (e.g., tally, frequency, and relative frequency). The
guided step-by-step worksheet treated mathematics as a series of steps, very procedural;
coded as math as procedural, a frequently coded category. That is, doing mathematics is
about knowing the steps or algorithms that lead you to answers (Baroody et al., 2007).
Additionally, task such as the guided step-by-step worksheet suggests a
perception of students with fixed intelligence, effort does not positively influence
learning (Dweck, 2006). Students with fixed intelligence are often perceived as only able
to do things after they have been explicitly taught or shown (Blackwell, Trzesniewski, &
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Dweck, 2007); the worksheet showed students explicitly how to create a histogram. The
students were deprived of the more cognitively demanding elements of histogram
creation such as, thinking about the number of intervals to use and the values at the
interval boundaries needed to adequately represent the data. They did not have the
opportunity to make or view different choices and then compare which choices yielded
the best representations. Neither the guided worksheet nor the teacher/coach guided
histogram tasks led to these types of cognitively demanding opportunities.
Traditional (43%) authority type was a frequently coded category for Kia’s
classroom and shown in Table 4.21. The activities described thus far within this case
were all examples of traditional authority for several reasons. The questioning was
primarily knowledge level, such as: a) “Now look up here [pointing at the word wall],
what do we call range?” (Line 6, Classroom Snapshot 4.19); b) “So, it’s from umm to
umm?” (Line 8, Classroom Snapshot 4.19); or c) in reference to the fraction 6/20, “6 out
of” (Line 10, Classroom Snapshot 4.22). Each of these questions would yield very
predictable students’ responses. The first question shows students the answer, the second
is a fill in the blank, and the third is a low-level fact question for most grade six students.
Traditional authority is as much about managing student behavior (control) as it is
about disseminating information; however, if generative mathematical Discourses were
the goal, learning would be less predictable and less controlled because everyone would
be engaged in thinking, contributing, and learning (Apple, 2004; Fine, 2006; Pace &
Hemmings, 2006; Resnick, 1988; Sztajn, 2003). For generative mathematical Discourses
to emerge, the opposite of traditional authority type is required.
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Consider another interaction (5.5) with traditional authority type, a more
cognitively demanding task, and a more effective mathematical Discourse This
interaction was coded interaction type teacher-class and interaction focus teacher
explaining math idea more students participated. In this interaction episode, the class was
estimating fractions to the nearest benchmark fraction (0, ½, and 1) using a pizza
analogy.
Classroom Snapshot 4.23: Teacher explaining how to compare estimates Line 1: Kia: 8 out of 11. Line 2: Student A: That would be considered almost a whole. Line 3: Kia: OK. Did you hear Student A? Why do you think that is about a whole? Line 4: Student K: Because it's only 3 pieces down. Line 5: Kia: Its only about 3 pieces away; and its out of 11. Now, what could be
another option for an answer? Line 6: Student G: For 1/2 Line 7: Kia: Now I'm gonna go over both of them real quick . . . What is 1/2 of 11? Line 8: Student J: Ah, 5 and 1/2 Line 9: Kia: Isn't that 5 and 1/2 out of 11? and What is 1 whole if its 11? Line 10: Students: [multiple students talking] 11 Line 11: Kia: Its 11. So, we are going to say that 1 whole is 11 out of 11. So let's
look at how close we are. How much is 8 away from 5 and 1/2? Line 12: Students: [multiple students talking] 3... um... 3 and 1/2... Line 13: Kia: about 3 but actually its 2 and 1/2. [in a playful voice] Don't give me
that look. and How much is 11 away from 8? Line 14: Students: [multiple students talking] 3 Line 15: Kia: 3. So really either answer is OK. You would say that 8 is half of 16.
So, it's kind of like right smack in the middle. So sometimes as long as you have a reasonable explanation, your answer is fine.
(Interaction 5.5)
This interaction was also a traditional authority type, but this time more students
participated. Kia was explaining how to compare estimates in this interaction episode, but
she engaged the students to support her effort. Teacher explaining (32%) was a frequent
coded category as shown in Table 4.21, but teacher explaining can be accomplished via
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contributive Discourse type as this interaction exemplifies or using an authoritative
Discourse type. This dialogue was an exemplar of collaborative sense making (26%),
another frequently coded category for Kia’s classroom. Again, collaborative sense
making can be achieved through respect and listening as was exemplified in this
interaction or assumed because of proximity of language as was described in Classroom
Snapshot 4.20 when students said, “That ain’t got nothin’ to do wit it…that’s
right…no…” (Line 6).
Kia, in this interaction episode (Classroom Snapshot 4.23), asked higher-level
questions (analysis or evaluation), more thinking questions, than before, such as: “Why
do you think that is about a whole?” (Line 3) and “Now, what could be another option for
an answer?” (Line 5). Arguably, the task and the types of questions used during this
snapshot presented sufficient cognitive demand that encouraged more students to
participate. Additionally, strict control appeared to be lessened (Lines 3, 5, 15) and
students input appeared to be valued (Lines 3, 13), each likely contributed to student
participation and the more effective Discourse in this interaction episode (Pace &
Hemmings, 2006; Yackel & Cobb, 1996).
Summarizing Matrix for Kia’s Classroom
The nature of classroom culture, Discourse or community, and relationships that
support learning opportunities have been articulated through the lens of the theoretical
model and supported with the data and analysis. The primary research question: How do
classroom interactions influence the Discourse related to mathematics learning and
teaching in Kia’s classroom? and key elements from the preceding discussion are
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depicted in Table 4.23. This table represents a summarized response to the primary
research question for Kia’s classroom interactions.
Table 4.23 was organized to illustrate the case study through the lens of the
theoretical model. The matrix columns represent the three theoretical constructs: a)
classroom culture; b) Discourse or community; and c) relationships that support learning
opportunity. The rows of the matrices depict the four concepts that define the construct
group. The theoretical construct classroom culture the four concepts include: a) cultural
influencer; b) mathematical practices; c) sociomathematical norms; and d) social norms.
Classroom culture Discourse/community Relationship/opportunities
Cultural influencers included collaborative sense making and fact or procedural reproduction
Discourse influencers included frequent use of exercises and focus on answers
Authority type (and leadership style) was overwhelmingly traditional (autocratic)
Mathematical practices often included communication
Discourse types varied and were split almost evenly between contributive and authoritative
Identity perceptions were split between chosen non-participation and PAID a(positive and negative)
Sociomathematical norms included teacher explaining and low-level questioning
Student behaviors were split between Discourse challenging and thinking and ideas
Learning influencers included teacher action
Social norms did not enhance the learning environment
Teacher behaviors included informer, expert, and catalyst
Teaching influencers included perceptions of mathematics as procedural and fixed intelligence
a
PAID – participation and agency identity
Table 4.23: A summary of the analysis of Kia’s case study through the lens of the theoretical model.
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The theoretical construct Discourse or community the four concepts include: a) Discourse
influencer; b) Discourse types; c) student behaviors; and d) teacher behaviors. The
theoretical construct relationships that support learning opportunity the four concepts
include: a) authority type or leadership style; b) identity perceptions; c) learning
influencer; and d) teaching influencer. Specific elements of this table are discussed in
detail within the cross case analysis in the next chapter.
Results & Findings Summary
In this chapter, the participating district and schools were introduced and then the
case studies were presented. Each of the three cases were presented using the same
format and structure. The cases were ordered alphabetically using the teachers’
pseudonym; the case studies for the classrooms were presented Ada, Eva, and Kia.
The cases were structured as follows. First the teacher’s perceptions about the
class and interactions were presented. Then the case data was summarized, including:
student and teacher survey results; interaction generalizations by type and focus; analysis
coding was summarized; and interaction summaries were presented for the interactions
used during the case study. Next, the findings were presented, organized by the three
theoretical constructs. Finally, the case study was ended by presenting a summarizing
matrix, which was the response to the research questions. The next chapter presents the
cross case analysis, which fully elaborates the responses to the research questions.
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CHAPTER 5: DISCUSSION
Overview
This chapter interprets the findings from this investigation. The discussion begins
with a presentation of data used to inform the cross case analysis of the three cases
described in the previous chapter. The cross case analysis serves to summarize, compare
and contrast, and draw conclusions from the findings articulated within the case studies.
The implications and recommendations that emerged from this investigation about
developing mathematics Discourses related to learning and teaching are articulated. Then,
the chapter is concluded and the dissertation report culminates with reflective comments
focused on next steps related to both methodology and future research.
Cross Case Analysis: Summarizing the Data and Analyses
In this section, I address the primary research question: How do classroom
interactions influence Discourses related to mathematics learning and teaching in MCP
supported classrooms? The cross case analysis was organized around the theoretical
model by using the summarizing matrices at the end of each case study. Each of the
individual matrices was compiled into one comparative matrix by construct. However,
first the pertinent data are presented that support the cross case analysis.
Teacher Surveys: Reform Orientation and Teaching Efficacy
Two teacher instruments were used, the Teachers’ Commitment to Mathematics
Education Reform (TCMER, Ross et al., 2003) and the Teacher Self-Efficacy Scale
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(TSES, Tschannen-Moran & Woolfolk Hoy, 2001). The teachers’ responses were
analyzed and summarized and the data are compared in Table 5.1.
Classroom Classroom Culture Discourse or community
Relationship opportunity
Ada 40% 35% 25%
Eva 42% 26% 32%
Kia 26.5% 47% 26.5%
Table 5.1: Percent favorable responses to teacher surveys (TCMER and TSES) for each teacher.
These data suggest that Ada (40%) and Eva (42%) perceived themselves as being most
reform-oriented and efficacious with respect to classroom culture, while Kia (47%)
perceived her strengths in these areas with respect to Discourse or community.
Student Surveys: Attitude and Identity Perceptions
The students in each classroom were given a student survey that measured
mathematics related attitudes and identity perceptions. During analysis, I created
meaningful categorical names for each of the six questions scored using a Lickert scale.
The attitude questions were categorized as confidence, efficacy, agency, and persistence.
The identity questions were categorized as mathematics identity and fixed intelligence.
Examination of the data from the student survey across classrooms and by categorical
coding is illustrated in Table 5.2. The data suggests that the differences among students’
attitude or identity perceptions were not substantial.
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Attitude or Identity Ada (n=11) 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚, �̅�𝑥 (𝑠𝑠𝑠𝑠𝑚𝑚 ,𝜎𝜎)
Eva (n=22) 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚, �̅�𝑥 (𝑠𝑠𝑠𝑠𝑚𝑚 ,𝜎𝜎)
Kia (n=15) 𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚, �̅�𝑥 (𝑠𝑠𝑠𝑠𝑚𝑚 ,𝜎𝜎)
Mathematics Confidence 3.3 (0.9) 3.7 (0.5) 3.3 (0.8)
Mathematics Efficacy 3.8 (0.4) 4.0 (0.2) 3.7 (0.5)
Mathematics Agency 3.5 (1.0) 3.5 (0.6) 3.4 (0.9)
Mathematics Identity 2.7 (0.7) 2.8 (0.9) 2.5 (1.1)
Fixed Intelligence 2.8 (1.3) b
2.3 (1.3) 1.9 (1.2)
Persistence 1.3 (0.7) b
1.4 (0.6) 1.3 (0.8) a sd – standard deviation b
Low scores are most favorable for these questions Table 5.2: Students’ survey responses about attitude and identity perceptions by class.
The largest differences (∆�̅�𝑥 ≥ 0.4), calculated by subtracting the next closest
mean between classrooms, occurred for mathematics confidence (∆�̅�𝑥 = 0.4) and fixed
intelligence (∆�̅�𝑥 ≥ 0.4). The mean (�̅�𝑥) representing mathematics confidence as perceived
by students was greatest in Eva’s (3.7) classroom when compared to those for Ada’s and
Kia’s (3.3) classrooms. However, there was limited consensus among students in either
classroom as evidenced by the large standard deviations, for Ada’s (0.9) and Kia’s 0.8)
classrooms.
A high score for fixed intelligence suggests that students’ perceive they cannot
improve mathematics intelligence by trying hard. The mean (�̅�𝑥) representing fixed
intelligence as perceived by students was greatest in Ada’s (2.8) classroom and least in
Kia’s (1.9) classroom; Eva’s (2.3) students’ perceptions were in the middle. However,
consensus among students in all classrooms was limited given the large standard
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deviations for each. Therefore, students in Kia’s classroom, on average, perceived they
had growth intelligence and they could improve their mathematics intelligence by trying
hard.
Analysis Summary of Coding
The analysis of classroom interactions yielded coding densities of theoretical
construct group categories by classroom. Qualitative analysis software, NVivo 8 (QSR
International, 2008) was used to count the number of times data sources (e.g., video
segments or transcribed data) were coded to categories which were separated into
construct groups. Specifically, several categories were related to classroom culture,
others were related to Discourse or community, and the remaining categories were related
Classroom (N) Classroom culture Discourse or
Community Relationship Opportunity
Ada (742) 34% 33% 33%
Eva (308) 39% 34% 27%
Kia (579) 29% 41% 30%
Table 5.3: Summary of coding density of theoretical construct group categories as a percent by classroom.
across all construct groups and converting the quotients into percents. As stated earlier, to
relationship opportunity. The results depicted in Table 5.3 were derived by dividing the
construct group category coding counts by the total number (N) of categories coded
Ada’s and Eva’s classroom interactions were most densely coded using classroom culture
categories and Kia’s were most densely coded using Discourse or community categories.
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Summary of Classroom Interactions by Type and Focus
The interaction type and interaction focus categories were emergent codes that
were defined during the analysis phase to describe the interactions from a more general
perspective. The interaction type categories have literal meanings29
Table 5.4
. For example,
interaction type of student-student was coded for interactions when the primary
interaction involved students; and teacher-group was coded for interactions when the
primary interaction involved the teacher and a small group of students (i.e., not the whole
class). The interaction type categories are summarized by classroom in .
Interaction Type Ada Eva Kia
Student-student 33% 18% 9%
Teacher-class 29% 69% 87%
Teacher-group 2% 5% 2%
Teacher-student 13% 8% 2%
Table 5.4: Summary of interaction type categories by classroom.
Both Eva’s and Kia’s classrooms had substantially more teacher-class interactions coded
(analyzed) than all other interaction types. Ada’s classroom had more student-student
interaction type coded (analyzed) than all other interactions. Note well, that a classroom
with a large number of interactions coded as one type or another does not suggest that
that classroom was primarily comprised of those interaction types, but rather that there
29 The code book with descriptions for all codes is in Appendix C.
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were classroom interactions of a specific type that presented good opportunities for
analysis governed by the adopted theoretical model. Therefore, in the case of Ada’s
classroom where there was a large number of student-student interaction type coded, this
does not mean that this classroom was very student-centered.
Interaction Focus Descriptions Ada (67)
Eva (49)
Kia (72)
General explaining Explaining that is not mathematical 7% 4% 3%
Interpreting meaning Developing clarity or shared understanding 1% 20% 25%
Math problem solving Mathematical problem solving 18% 12% 17%
Reflecting or evaluating Contemplation about prior event, activity, or action 19% 16% 19%
Social problem solving Resolving sociocultural situations 19% 12% 12%
Student explaining math idea
Students revealing mathematical thinking 15% 22% 7%
Teacher explaining math idea
Teacher revealing mathematical thinking 19% 12% 17%
Table 5.5: Summary of the interaction focus categories by classroom.
The interaction focus category was coded to capture the essence or interpreted
purpose of the interaction. The categories shown in Table 5.5 were used to categorize all
interactions analyzed during this investigation and every interaction analyzed was
assigned one or two interaction focus categories.
The interaction focus categories in Table 5.5 were distributed differently between
classrooms. Several of the comparisons made within the cross case analysis were chosen
based in part because of this table. Interestingly, all classrooms were coded less densely
for the general explaining category. This can be explained because the analysis was
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guided by the research questions and the theoretical framework, which are on the subject
of classroom interactions related to Discourses related to mathematics learning and
teaching. Therefore, in most instances, classroom interactions that would be interpreted
as being focused on general explaining would not warrant analysis. Nonetheless, there
were several interactions coded as interaction type general explaining.
Cross Case Analysis of Three Classrooms
In this section, cross case analyses are used to summarize the three cases studies
presented in the previous chapter by comparing and contrasting them. Additional
interpreted meanings emerged that further informed the descriptive responses to the
research questions. Finally, conclusions were drawn that increased understandings and
expounded on the research questions responses based upon the findings from this
investigation. The cross case analyses and presentation of interpretations were organized
by the three theoretical constructs: a) classroom culture; b) Discourse or community; and
c) relationships that support learning opportunity. Key interpretations from each of these
sections are presented before moving to the next section.
Cross Case Analysis for Classroom Culture
In Chapter 4, each case study was concluded with a summarizing matrix. Those
matrices were summarized and separated by construct and then all classrooms combined
to create three new matrices used to guide the cross case analysis. These matrix
summaries compare the three classrooms, one matrix for each of the three theoretical
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Classroom Culture Concepts Ada Eva Kia
Cultural influencers
° Fact or procedural reproduction
° Powerful other
° Collaborative sense making
° Teacher expectations
° Collaborative sense making
° Fact or procedural reproduction
Mathematical practices ° Communication ° Communication
° Math connections ° Communication
Sociomathematical norms
° Questioning ° Teacher explaining ° Student explaining ° Questioning
° Teacher explaining
Social norms
° Low risk environment
° Negative social norms
° Listening and respect
° Negative social norms
Table 5.6: Classroom culture summarizing matrix for each classroom.
constructs. The rows of the matrices depict the four concepts that define the construct
group. Specifically, for the theoretical construct classroom culture the four concepts
include: a) cultural influencer; b) mathematical practices; c) sociomathematical norms;
and d) social norms. The combined summary matrix comparing the three classrooms’
theoretical construct classroom culture is presented in Table 5.6.
Collaborative Sense Making, the Same, but Different
All classrooms evidenced collaborative sense making as a cultural influencer;
however, Eva’s and Kia’s classrooms were densely coded for this category. Examining
each of the classrooms revealed that the ways these collaborations manifested for
mathematics learning and teaching was different, specifically with respect to talk time
and the tasks. To illustrate these differences, consider classroom interaction episodes
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from Kia’s classroom in Classroom Snapshot 5.6 and from Ada’s classroom in
Classroom Snapshot 5.5.
Kia’s Classroom – Collaborative sense making. The interaction episode shown
in Classroom Snapshot 5.1 was first introduced in Chapter 4 (Classroom Snapshot 4.23).
In this interaction episode, the class was estimating fractions to the nearest benchmark
fraction (0, ½, or 1) using a pizza analogy..
Classroom Snapshot 5.1: Teacher explaining how to compare estimates Line 1: Kia: 8 out of 11. Line 2: Student A: That would be considered almost a whole. Line 3: Kia: OK. Did you hear Student A? Why do you think that is about a whole? Line 4: Student K: Because it's only 3 pieces down. Line 5: Kia: Its only about 3 pieces away; and its out of 11. Now, what could be
another option for an answer? Line 6: Student G: For 1/2 Line 7: Kia: Now I'm gonna go over both of them real quick . . . What is 1/2 of 11? Line 8: Student J: Ah, 5 and 1/2 Line 9: Kia: Isn't that 5 and 1/2 out of 11? and What is 1 whole if its 11? Line 10: Students: [multiple students talking] 11 Line 11: Kia: Its 11. So, we are going to say that 1 whole is 11 out of 11. So let's
look at how close we are. How much is 8 away from 5 and 1/2? Line 12: Students: [multiple students talking] 3... um... 3 and 1/2... Line 13: Kia: about 3 but actually its 2 and 1/2. [in a playful voice] Don't give me
that look. and How much is 11 away from 8? Line 14: Students: [multiple students talking] 3 Line 15: Kia: 3. So really either answer is OK. You would say that 8 is half of 16.
So, it's kind of like right smack in the middle. So sometimes as long as you have a reasonable explanation, your answer is fine.
(Interaction 5.5)
In this example of collaborative sense making from Kia’s classroom, she
explicitly asked if students had heard what was in reference to a student’s idea (Line 2),
and then requested an explanation for the idea (Line 3). This appeared to set an
expectation for the Discourse, that is, students should listen to what others say. There is
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more to say about this classroom interaction, but not until after introducing Ada’s
classroom interaction example.
Ada’s Classroom – Collaborative sense making. The interaction episode shown
in Classroom Snapshot 5.2 was first introduced in Chapter 4 (Classroom Snapshot 4.5).
In this interaction (5.11), students were volunteering to come to the overhead to explain
achievement test items they had been using for practice during class.
Classroom Snapshot 5.2: Student at overhead during problem solving Line 1: Student D: [Reading] Tomorrow is the fourth of June. What day will it be
three days from today? Line 2: Ada: OK. How would you do that? Line 3: Student D: [~ 2 second pause] Line 4: Ada: [interrupting] Remember, you can ask… Line 5: Student D: [end pause] 4, 4th
Line 6: Ada: [Interrupting, over talking] You know you can ask people?
Line 7: Student K: [Immediately raises hand, makes a sound, Student D is looking down] Bing!
Line 8: Student D: [abandoning his thinking and idea] Student K. Line 9: Student K: Four plus three equals seven, so it'd be of June. So, it would be
the 7th
Line 10: Ada: OK. Then, can you write it up there? Like put 1... up higher…very good.
of June.
Line 11: Student D: [Writing per Ada's directions] Line 12: [Students chatter softly] Line 13: Student B: [speaking above the other students, chatter stops] I don't get it
because if it’s the 3rd and tomorrow is the 4th, and you add three days, wouldn't it be the 6th
Line 14: Student K: Nooo! ?
Line 15: Ada: [Thinking aloud] ...because if tomorrow is the 4th
Line 16: Student B: If it's the third...
of June, what day would it be? 4..., you're right! Student B, can you explain what you did?
Line 17: Student D: [interrupting] I'm wrong? Line 18: Student B: [Continuing his thinking and explanation] If it’s the third of
June, you add three more days it’s going to be the 6th
Line 19: Ada: So, it’s not the 4 of June.
th. It says if tomorrow is the 4th. So, it's actually the 3rd
Line 20: Students: Nice job, Student B! . Did you see that word there?
Line 21: Ada: Very good Student B! [More accolades from other students]
(Interaction 5.11)
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This interaction snapshot showed multiple students contributing to the final
solution for a problem (Lines 5, 9, 13, 14, 17, and 18). While this was not the norm for
mathematical Discourses in this classroom, it illustrated collaborative sense making.
Comparing and contrasting collaborative sense making between Kia’s and
Ada’s classrooms. Examination of the two classroom interactions, at first glance they
appear to be very similar; however, they are quite different. The two classroom
Discourses were compared in Table 5.7 using three elements related to mathematical
Discourse including: a) talk time; b) questioning; and c) the task. When the amount of
teacher talk was compared to that of student talk, Ada’s (47%) talk time was less than the
students’ (53%), and Kia’s (87%) talk time far exceeded the students’ (13%).
The questioning was very similar, with respect to the number of questions asked
and of those, the number of higher-level questions. However, the outcome after the
higher-level questions were posed was different. For example, in Kia’s classroom, after
posing each of the high-level questions (Lines 3 and 5), students offered ideas in response
(Lines 4 and 6). Conversely, in Ada’s classroom, after posing high-level questions (Lines
2 and 15), responses ensued, but with them were social problems30
Finally, in
(Lines 4, 6, and 17),
which perhaps needed solving. These sociocultural elements of the classroom appeared to
cause a slight divergence from the mathematical Discourse.
Table 5.7 were the mathematical tasks; Ada’s classroom was finding
the date for a given start date and number of days displacement, and Kia’s classroom was
30 Social problem solving was an emergent category defined during the analysis phase of this investigation and defined as resolving sociocultural situations in Table 5.5.
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estimating fractions to the nearest half and explaining. According to the state standards
(Ohio Department of Education, 2001), Ada’s task was below grade level and Kia’s task,
while at grade level focused only on the skill of rounding a fraction to the nearest half;
hence, suggesting that both tasks offered low cognitive demand for grade six students.
Comparing the Discourses Ada Kia
Teacher vs. student talk
47% words vs. 53% words 87% words vs. 13% words
Total # of questions asked; question levela
9 questions; 2 higher level questions (Lines 2 and 15)
8 questions; 2 higher level questions (Lines 3 and 5)
Mathematical Task What day will it be 3 days from today?
What is a valid estimate for 811
of 0, ½, or 1? Explain.
a
Level of questioning is at application or higher, according to Bloom’s taxonomy.
Table 5.7: Comparing Ada’s and Kia’s classroom Discourses related to cultural influencer, collaborative sense making.
Also of import was the affect of each Discourse with respect to mathematical
understanding. According to Hiebert and colleagues (1997), “ we understand something
if we see how it is related or connected to other things we know” (p. 4). In the classroom
interactions illustrated, Kia’s Classroom Snapshot 5.1 and Ada’s Classroom Snapshot
5.2, there was no evidence of students or teachers relating or connecting what they were
learning to what students’ know. There was no evidence that suggests students
understood what they were learning; therefore, perhaps the collaborative sense making
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did not affect the mathematics Discourse related to learning and teaching to the extent of
understanding.
Mathematical Practice – Communication and the variability among classrooms
All of the classrooms evidenced mathematical communications; however, the
mathematical content, communication affect, and communication substance varied
among the three classrooms. Subsequently, mathematical Discourses related to learning
and teaching were sometimes enhanced and other times hindered in relation to the
effectiveness of communications in the classrooms.
Eva’s Classroom – Communications. Consider the interaction depicted in Classroom
Snapshot 5.3 from Eva’s classroom, which was initially introduced in Chapter 4
(Classroom Snapshot 4.13). The class reflected on the mathematical efficiency of their
approach, counting multiples for each denominator when seeking least common
multiples. The idea for counting multiples had come from a student, but midway through
the counting approach, Eva challenged them to find a more efficient approach.
Classroom Snapshot 5.3: Reflecting and Evaluating a Mathematical Procedure Line 1: Eva: By 4's Line 2: Students: 4, 8, 12, … [Counting by 4's up to 48]. Line 3: [Students are unenthusiastic, and lose synch. Eva writes on the overhead
sighs, and then runs out of space] Line 4: Eva: Do we have to count like this? Seriously guys, what would be the
easiest way to do this? We'll be counting forever. What's an easier way to do this Student A?
Line 5: Student A: You know how we put the numbers at the bottom and circle them? Instead of going through the whole thing.
Line 6: Eva: OK. OK. Student G? Line 7: Student G: Factor tree Line 8: Eva: Guys? Factor tree. Awesome. I would say factor tree. You're probably
gonna spend less time than if you do it the other way. Let's try it? Let's try factor tree.
(Interaction 2.5)
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In this interaction, Eva used body language, expressions, and props to elaborate
the need for a better way to find LCMs (Lines 3 and 4). Her body language and
expressions included looks of great fatigue and occasional sighs, even writing on the
overhead appeared to be taxing. The combined affect appeared to be motivational for the
students, as evidenced by raised hands waiting to be called upon to offer ideas.
Additionally, in this interaction episode (2.5), students were collaborating to make
sense of the proposed approach (Lines 5 and 7). Eva, in Line 6, momentarily retreated
from the Discourse by saying nothing to clarify the initial idea, but called on another
student for clarification. The class appeared to share some sense of shared meaning for
finding LCMs using factor tree (Line 7), as evidenced by hands being lowered at its
mention. In fact, the lowering of the hands appeared to indicate a kind of class agreement
or consensus related to the idea of factor tree.
Comparing and contrasting among Eva’s, Kia’s, and Ada’s classrooms. Table 5.8
was created by expanding Table 5.7; a column for Eva’s classroom and row 2 were
added. Eva’s and Kia’s classrooms were similar in that they both had more teacher talk
compared to student talk (row 1). All three classrooms were similar with respect to the
number and types of questions asked (row 3). As described earlier, the tasks in both
Ada’s and Kia’s classrooms were cognitively limited. Eva’s task for reflecting on an
approach for finding an LCM and selecting a more efficient approach; however, was
grade appropriate and included benchmarks from the Mathematical Processes Standard
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for grades 5 – 7 (Ohio Department of Education, 2001), which suggests more cognitive
demand, a more rigorous task than the others.
Comparing the Discourses Ada Eva Kia
Teacher vs. student talk
47% words vs. 53% words
66% words vs. 34% words
87% words vs. 13% words
Teacher vs. student (T : S) explanations
2 : 3 1 : 1 3 : 1
Total # of questions asked; question levela
9 questions; 2 higher level questions (Lines 2 and 15)
6 questions; 2 higher level questions (Lines 4 and 6b
8 questions; 2 higher level questions (Lines 3 and 5) )
Mathematical Task
What day will it be 3 days from today?
Is there a better way to find the LCMc
What is a valid estimate for 8
11 of 0,
½, or 1? Explain. ?
a Level of questioning is at application or higher, according to Bloom’s taxonomy. b The question Student G was answering was implied, but initially asked in Line 4. c
LCM – least common multiple
Table 5.8: Comparing mathematical Discourses among the three classrooms related to mathematical practice, communication.
Examining the three classroom interaction episodes Kia’s Classroom Snapshot
5.1, Ada’s Classroom Snapshot 5.2, and Eva’s Classroom Snapshot 5.3 and focusing on
communications, there were two concepts that appeared to alter the tenor of the
Discourses including: a) who explained the mathematics, teachers or students, and b)
teacher’s expectations for students.
Who explained the mathematics, teacher or students? – As illustrated in Table
5.8, Eva’s interaction (Classroom Snapshot 5.3) was short, but there was one student
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explanation (Line 5) and one teacher explanation (Line 8). Kia’s interaction (Classroom
Snapshot 5.1) was lengthier, and there was one student explanation (Line 4) and three
teacher explanations (Lines 9, 11, and 15). Ada’s interaction (Classroom Snapshot 5.2)
was the longest of the three, and there were three student explanations (Lines 9, 13, and
18) and two teacher explanations (Lines 15 and 19). If we were to rank classrooms by
ratio of student explanations to all explanations, the top rank equaling the classroom with
the greatest ratio, the classrooms would be ranked with Ada’s at the top, followed by
Eva’s, and then Kia’s classroom. This ranking is the same had we compared the
classrooms by teacher to student talk time.
Teacher’s expectations for students – Ada in Classroom Snapshot 5.2 did not
expect all students to participate as evidenced when she encouraged Student D to
relinquish his task of sharing his solution with the class (Lines 4 and 6). He had
volunteered to share and he appeared to have an idea to share (Line 5), but it was left
unacknowledged and unexplained during the Discourse. Kia in Classroom Snapshot 5.1
in Line 7 made it clear that she was going to explain, and there were no more instances of
students explaining after Line 7, only teacher explaining. Eva in Classroom Snapshot 5.3
made no explicit utterances as to her expectations, but she did commend students for their
thinking (Line 8). Similarly, Ada followed her students’ lead and commended Student B
for work well done (Lines 20 and 21).
The Significance of Social Norms for Discourses
All of the classrooms had an established set of social norms; some norms were
shared by all class members, others were accepted or practiced by only a few class
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members, but in both instances, social norms influenced mathematical Discourses related
to learning and teaching. Both Ada’s and Kia’s classrooms evidenced social norms,
coded as negative social norms because they hindered mathematical Discourses, but were
manifested in ways that impacted each classroom differently. Eva’s classroom social
norms were not typically coded as negative social norms and they were very different
from the other classrooms; they often enhanced or supported mathematical Discourses.
To illustrate this, a series of excerpts from classroom interactions were included in Table
5.9 for Ada, Table 5.11 for Eva, and Table 5.10 for Kia; the selected excerpts captured in
these tables illustrate the essence of social norms observed.
Comparing social norms across the three classrooms – In Ada’s classrooms, in
each of the excerpts in Table 5.9 students’ ideas were not clarified. In Interaction 1.4,
instead of asking a student to clarify what they meant by “I hate it” or “I don’t like it,”
which are feelings and not reasons, Ada offered a reason, “. . . it’s like you really didn’t
get an answer.” Perhaps, that was the reason the students felt as they did, but without
asking clarifying questions or creating an opportunity for students to articulate a reason,
one cannot know if that was the students’ reason. In Interaction 4.2, a student indicated
that he or she had a different way to multiply mixed numbers, but in this situation, his
idea was not allowed for consideration because Ada declared and ‘proved’ that the way
she had just modeled was the only way if they wanted to get the correct answer. This was
referred to in Table 5.6 as fact or procedural reproduction.
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Ada’s Classroom
Interaction 1.4 Interaction 4.2 Interaction 5.11 Ada: Student A and I just had this conversation, that's an OK answer; but, it's kind of like a ‘because I said so answer,’ and how do you feel when mom tells you, ‘because I said so?’ Students: [students talking at once] I hate it...I don't like it... Ada: OK. So it's like you really didn't get an answer.
Students: [lots of mubling by students, but one says] I know another way to do that. Ada: So, that is the way you do that. And if you notice up here, I have to go through all those steps. If I just would have multiplied. [multiplied numerators and denominators of the mixed numbers as proof] So you see, if I would have just done it my own way it just does not work. You have to follow the steps.
Ada: OK. How would you do that? Student D: [~ 2 second pause] Ada: [interrupting] Remember, you can ask… Student D: [end pause] 4, 4Ada: [Interrupting, over talking] You know you can ask people?
th
Interpretive Notes by Interaction
• Students’ ideas not clarified
• Teacher’s idea replaced students’ idea
• Student’s idea not attended
• Teacher “proves” that her way is the only way
• Student not afforded think time
• Student encouraged to seek help, perhaps prematurely
Table 5.9: Excerpts that depict typical social norms within Ada’s classroom.
In Interaction 5.11, the student was encouraged to ask another student for help
before he was able to offer his idea. He offered an idea, “4, 4th,” but he opted to take the
teacher’s ‘advice’ to seek help from others and did not share his thinking further. The
affect of what occurred was that Student B’s ideas was never acknowledged or clarified.
The social norms in this classroom included: a) not to ask clarifying questions about
students’ ideas; b) not to allow approaches outside of those modeled by the teacher; c)
not to afford students time (< 1 or 2 seconds) for thinking once called upon for ideas; and
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d) for the teacher to share her thinking and ideas in the absence of perceived viable
student ideas.
Similarly, in Kia’s classroom, the social norms included: a) not to ask students’
clarifying questions about their ideas; b) not to afford students time for thinking once
called upon; c) for the teacher to share her thinking and ideas in the absence of perceived
viable student ideas; and d) diminishing rigor to motivate participation. Illustrated in
Interaction 2.1, was Kia’s idea replacing the student’s idea; the student’s idea was
inaudible because she was interrupted and over talked. Kia recognized the interruption,
apologized, and perhaps gave voice to the student’s idea. The specifics of the student’s
comments were inaudible on the videotaped data, so her reference to the nature of what
was said could not be confirmed. Nonetheless, Kia appeared to be redirecting instruction,
but at the expense of not listening to the student’s idea.
Related to students not participating in polling type questions was many students
appeared engaged in what was coded as chosen non-participation. This negative social
norm was depicted in at least one and perhaps two of the excerpts in Table 5.10,
Interaction 1.6 and 4.2. In Interaction 4.2 was exemplified an instance when most
students chose non-participation because only two students were responding to questions
posed, Student A and Student K, and in Kia’s final comment in the excerpt, she addressed
the situation explicitly and appeared to lower the cognitive demand of what was being
asked. When she asked students for the meaning of relative frequency, 620
; the anticipated
response should be interpreting meaning in context, not restating the fractional amount.
This action appeared to diminish the cognitive demand. Even so, there was no evidence
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Kia’s Classroom
Interaction 1.6 Interaction 2.1 Interaction 4.2 Student K: adding? Kia: By whatever you look at, how many of you agree? Class: [no one raises their hand] Kia: They’re just not very supportive of you. What do you think it is Student D?
Student J: So like . . [Kia spoke over the student making her comment inaudible]. Kia: What is the 4? I'm sorry sweetheart. I don't want to dismiss you, but I'm gonna show you what we're gonna relate it to. She's absolutely correct that we did this, didn't we? We did this all the way back in the fall, but I want to continue it.
Kia: What does frequency mean? [asked 3 times] Student A: [gave 3 answers] a) It's a fraction?; b) Ah, like out of like a survey or something?; and c) Frequent guess? [puts hands out, palms up] Student K: Oh, I forgot... Kia: [~ 2 second pause] Hhhooow mmmaany or how much. [speech is drawn out; Student K said ‘many’ in unison with Kia] . . . What's the next column? Student K: 16 through 25. Kia: You know it isn't gonna work if we're not gonna be part of the lesson. Where it says relative frequency they give you a fraction, 6/20. What does that mean? 6 out of? Student K: [shrugs shoulders] Student A: 6, 6 out of 20
Interpretive Notes by Interaction
• Students did not respond to polling questions
• Kia appeared to accept students’ nonsupport of peers
• Student’s idea was over talked
• Teacher’s idea replaced student’s idea
• Repeating question was perceived by student as wrong answer
• Chosen non-participation by majority of class
• Diminished rigor used to motivate participation
Table 5.10: Excerpts that depict typical social norms within Kia's classroom.
that either strategy, speaking explicitly about non-participation or diminishing
cognitive demand, were effective because following her interventions, Student A and
Student K continued to be the only students participating.
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On the other hand, in the other interaction (1.6), which may not be an example of
chosen non-participation, illustrated the typical observed response to polling questions.
The class was polled to see who agreed with Student K’s idea. No students raised their
hands; perhaps no one participated because they disagreed with Student K. However, in
this interaction and the reason for its inclusion in Table 5.10, Kia positioned the students’
non-response as an act of non-support for Student K, rather than non-support of the
student’s idea; an important distinction for students to understand when engaging in
effective Discourses. Supporting students’ opposition to one another is not productive for
building a community of learners (McClain & Cobb, 2001), and hence, mathematical
Discourse.
Conversely, in Eva’s classroom, her social norms typically did not include any of
the negative social norms described. Several social norms that enhanced Discourses in
her classroom and depicted in Table 5.11 included: a) creating space for students’ ideas
and explaining; b) affording students’ time to think and explain their ideas; and c) making
comments and listening to students in ways that supported students’ developing positive
mathematics attitude and identity perceptions.
In Interaction 3.1, Eva created space for Student E; he interrupted Eva’s talk to
ask a question relevant for his understanding and Eva attended to him without hesitation.
She responded to his question and then elaborated further by making a connection to a
prior mathematical topic they had learned. This was one of many instances when the
Discourse direction was determined by students’ ideas, thinking, and/or learning needs.
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Eva’s Classroom
Interaction 1.2 Interaction 2.5 Interaction 3.1 Student B: There would be 5 groups of 3 with 1 left over. Eva: So, what are we going to do with you, Student B? Class: Throw Student B out! [The class erupted with laughter, including Eva and Student B] [Within seconds they were refocused selecting groups; Student B, selected the first and only group made up of four students and included his friend Student A]
Eva: I ran out of room. [B has his hand up] Um. Student B. Student B: Well this is how I did it. Eva: I liked how you did it! Student B: I said that well 4 can't go into. . . [spoke for ~ 60 seconds without interruption]
Eva: Where's the largest group of 2’s grouped together? Student B? Student B: 6 Eva: 6 is. So [Student interrupts] Student E: Wait, Ms. Eva, wouldn't 6 be 3 too? [Referring to one of the factors of 6] Eva: Yeah. You could choose either one. [Eva explained further making a mathematical connection to prior learning]
Interpretive Notes by Interaction
• Students and Eva have fun, but return to work quickly
• Eva created a mathematical problem to structure class and used students’ solution
• Students offer unsolicited ideas
• Eva creates time and space for students to express their ideas
• Eva encourages individual thinking
• Students interrupt instruction to get clarity
• Eva attends to student’s learning needs
• Eva points out mathematical connection to students’ prior learning
Table 5.11: Excerpts that depict typical social norms within Eva's classroom.
In Interaction 2.5, affording students’ time for explaining was exemplified.
Student B explained how he had found a least common multiple in a way that differed
from counting multiples. He spoke for about one minute without being interrupted by
anyone in the classroom. Kia’s comments following his explanation suggested that she
did not know how he would represent his thinking in writing, but she commended him for
his effort. Additionally, in this interaction and in Interaction 1.2 were examples of Eva
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making comments or taking actions that supported students’ development of positive
mathematics attitude and identity perceptions. In this interaction (2.5), Eva comments, “I
liked how you did it!” in reference to what she had observed while circulating the room
as students worked. Comments such as this and others were commonplace in this
classroom.
In the other interaction (1.2), students are experiencing mathematics and having
fun, both positive perceptions related to mathematics learning and teaching. In this
excerpt students were determining if and how they could create groups of 3 with 16
students. Student B solved the problem, and Eva asked a question that resulted in another
student making a joke at Student B’s expense. In some circumstances, making a joke
about another student would be inappropriate, however, in this situation, with these
students, and their established relationships, everyone laughed including Eva and Student
B and the incident infused fun and enjoyment into a mathematics classroom that was
otherwise focused on mathematics learning and teaching.
Summary: Classroom Culture Influenced Mathematical Discourses
There were several instances when classroom culture enhanced Discourses and
other instances when it served to hinder them. Using findings from this investigation,
elements of classroom culture that hindered Discourses included: a) fact or procedural
reproduction; b) low-level questioning; and c) negative social norms. Conversely,
elements of classroom culture that enhanced Discourses included: a) mathematical
connections; b) student explaining; and c) listening and respect. On the other hand,
elements that sometimes enhanced, but could hinder Discourses and were dependent
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upon other circumstances included: a) collaborative sense making; b) communications;
and c) teacher explaining.
Cross Case Analysis for Discourse or Community
In Chapter 4, each case study was concluded with a summarizing matrix. Those
matrices were summarized and separated by construct and then all classrooms combined
to create three new matrices used to guide the cross case analysis. These matrix
summaries compare the three classrooms, one matrix for each of the three theoretical
constructs. The rows of the matrices depict the four concepts that define the construct
Discourse or Community Concepts
Ada Eva Kia
Discourse influencer
° Answers only ° Exercise tasks ° Static mathematics
° Connected mathematics
° Answers only ° Exercise tasks
Discourse types ° Authoritative ° Contributive ° Authoritative ° Contributive
Student behaviors
° Help and support ° Math efficacy ° Thinking and ideas
° Math efficacy ° Persistence ° Self-regulating ° Thinking and ideas
° Discourse challenging
° Thinking and ideas
Teacher behaviors
° Expert ° Informer ° Validator
° Connector ° Reflective
° Catalyst ° Expert ° Informer
Table 5.12: Discourse or community summarizing matrix for each classroom.
group. Specifically, for the theoretical construct Discourse or community, the four
concepts are listed in the first column and include: a) Discourse influencer; b) Discourse
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types; c) student behaviors; and d) teacher behaviors. The combined summary matrix
comparing the three classrooms’ theoretical construct Discourse or community is
presented in Table 5.12.
Answers in Relation to Mathematical Discourses
Answers were important in all of the classrooms; however, the ways they
influenced the mathematical Discourses in the classrooms varied. In each of the three
classrooms, answers were shared and validated by the teacher, and in most situations they
emerged early during the review process. For both Ada’s and Kia’s classrooms, the
answers only category was densely coded. This category was used when it appeared that
answers were the primary focus for an interaction. There were several instances in Kia’s
classroom, after answers were shared, a large number of students retreated from
participation until the next answer was given and they appeared to write them down. For
a comparative examination of the classroom interactions, consider the following episodes
from Ada’s and Eva’s classrooms.
Ada’s Classroom – Answers in relation to mathematical Discourses.
Typically, finding answers was the point for doing most mathematics and once answers
were known, mathematical Discourses rarely developed. This interaction occurred after
students had been learning steps for adding and subtracting fractions with unlike
denominators by hand.
Classroom Snapshot 5.4: Adding fractions with calculators. Line 1: Ada: Now that I have shown you the rough way [adding/subtracting
fractions w/ unlike denominators by hand]. What you need to know is you need to understand. It's vitally important that you know what your calculator is doing here. . .
Line 2: Ada: [Sets up the overhead calculator as students chat]
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Line 3: Ada: [Telling and showing students which buttons to push; students follow using their calculators]
Line 4: [The calculator displays the number sentence: 14
+ 23
= 1112
]
Line 5: Students: [Several speaking at once] You get 1112
Line 6: Ada: Why is our bottom number 4, then 3, then 12? Student A? Line 7: Student A: Becasue it's the least common denominator. Line 8: Ada: It's the least common mmultiple, OK. So, your calculator has done the
least common multiple for you, it has found equivalent fractions, and it has given you a final answer. Now you know what went into finding that answer in 2 seconds. OK, clear it out, and on your paper you can just write down the answer for number 3, which was 11
12.
(Interaction 3.7)
After the Ada and the students arrived at an answer on the calculator, she asked a
question (Line 6) and then she summarized what the calculator had done (Line 8) using
the same language she had used when teaching the steps for adding fractions with unlike
denominators. It appeared that she believed telling students once would be sufficient for
understanding because she did not ask that they write anything except the answer or that
they explain what had happened in their own words. Her instructional goal and her belief
about meeting that goal were stated, “. . . you need to understand.” (Line 1) and “Now
you know. . . “ (Line 8), respectively. If you only considered what students’ wrote or
explained during this interaction episode, it follows that what was most important was the
answer only.
Conversely, in Eva’s classroom, typically after answers were known, Discourses
ensued about different strategies for finding answers and for clarifying understanding.
Some of these mathematical Discourses lasted for 30 minutes or more, often included two
or more students, and the majority of students appeared to be listening when they were
not speaking.
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Eva’s Classroom – Answers in relation to mathematical Discourses. This
instructional segment started with Eva asking students how to show 34 using a pictorial
representation of a fraction bar; she appeared confident that they could do this
independently. Several students contributed and described what to do as she drew. Then,
Eva asked, “So, I've got 710
but I have 100 squares. How would I divide this up?”
Classroom Snapshot 5.5: Developing fraction representations. Line 1: Eva: [waits ~10 seconds, repeats the question several times as she waits,
and more students raise their hands] Student A1? Line 2: A1: You can make boxes of ten. Line 3: Student: no [shouting out] Line 4: Eva: So, I'd have to make boxes of ten.You're right. What do I know about
the boxes of ten Student A1? Line 5: A1: Um Line 6: Eva: They need to all look how? Line 7: A1: The same. Line 8: Eva: They all need to look the same. A1, So, could I go like 1, 2, 3, 4, 5 and
could I make my boxes of ten like this? [drawing 5 X 2 rectangular arrays on the overhead displayed grid paper]
Line 9: A1: Yeah Line 10: Eva: You bet I could. How else could I make them? Student A2? Line 11: A2: You could take one like, one set of ten, like a row [gesturing with her
hands as she speaks] and color it in. (Interaction 3.3.)
In this interaction (3.3), Eva appeared to be a member of the learning group, in
that she appeared to be an integral part. Her roles during the Discourse included revoicing
(Lines 4 and 8), asking questions (Lines 1, 4, 6, 8, and 10), validating input (Line 4), and
encouraging students’ contributions (Lines 1 and 10). The result of the class’ Discourse
was two different approaches for dividing the 10 X 10 grid so that 710
could be
represented, shown pictorially on the overhead, and articulated verbally.
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The Discourse continued, using the same pictorial representation, Eva asked
students for an equivalent fraction and they responded with 70100
. Without prompting from
Eva, Student A asked, “What if you took one bar [10 of the 100-square grid] and colored
in 7?” Additional Discourse ensued about what the fractional value would be for the new
pictorial representation. This added Discourse potentially preempted misconceptions
about the importance of the whole when representing fractions because they considered
the difference between 710
when the whole was 10 versus when the whole was 100. This
was one of many mathematical Discourses in this classroom that exemplified how the
focus for mathematics learning and teaching went beyond answers only, the focus
appeared to be mathematical understanding for teacher and students.
Discourse Types and Students’ Thinking and Ideas
Students’ in each of the three classrooms offered thinking and ideas. Some of the
students’ ideas appeared to be well thought out and articulated while others appeared to
be less so. Specifically, when teachers used authoritative Discourse type (teacher tells and
shows with little input from students), students’ thinking and ideas were short and often
not well articulated. Conversely, when teachers used contributive Discourse type
(includes ideas from students), students’ thinking and ideas appeared to be more
substantive and well articulated, comparatively speaking. To illustrate, consider
interactions from Kia’s and Eva’s classrooms.
Kia’s Classroom – Discourse types related to students’ thinking and ideas. In
this interaction, Students were working to make sense of a worksheet with Kia’s
authentic graphical representation of subtraction of two decimal numbers using two 10 X
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10 grids, a circle between the grids, some squares colored (marked with X’s), and a
portion of both grids circled with an arrow pointing out is shown in Figure 5.1. In the
interaction (1.6), Kia asked the students, “What kind of problem do you think number 1
is?” Two students responded to the question that led to clarification by Kia and then
mathematical Discourse ensued as captured in the following classroom interaction
episode.
Figure 5.1: Kia’s Classroom – a graphical representation for adding or subtracting decimal numbers
In this interaction, the Discourse was more contributive than authoritative and Kia
appeared to have been a part of the Discourse, not as a member of the learning group, but
more as a guide or manager. Her contributions to the Discourse included supporting
students (Lines 4 and 8) and asking questions (Lines 2, 4, and 12). Kia’s questions
appeared to enhance students’ articulation of their ideas.
Classroom Snapshot 5.6: Kia's classroom engaged in contributive Discourse.
Line 1: A: Subtraction Line 2: Kia: Why? Line 3: A: Because, I . . .don't know.
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Line 4: Kia: You don't have to know why. Does anybody have a reason why its either subtraction or addition, either one?
Line 5: C: I think it's subtraction Line 6: A: oh! [raising her hand] Line 7: C: Come on now! [reprimands A] Line 8: Kia: Let him finish [joins in reprimand] Line 9: C: I think its subtraction because when you have like the circle covering that
box is saying that that box is subtracting that amount right there [pointing at Figure 5.1 on the overhead]
Line 10: Kia: OK. Student K. Line 11: K: Subtracting, [inaudible]... the colors and it looks like it took it away Line 12: Kia: What tells you it looks like its taking it away? Line 13: K: ‘Cause it's colored in differently? Line 14: [Student J contributes, no follow up questions asked] Line 15: Student D: I think that subtraction because that arrow is saying like
subtraction [he gestures, his right arm showing a sweeping away from his body ]
Line 16: Kia: OK. He's saying that arrow has something to do with it. (Interaction 1.6)
The affect of Kia’s and the students’ Discourse was several students shared ideas
(Lines 1, 3, 5, 7, 9, 11, 13, and 15) about mathematical meanings for the authentic
representation. The student contributions were mostly ideas, but in Line 7, Student C
contributed a comment directed at another student that appeared to enhance the
sociocultural environment for the remainder of the interaction. The other student
contributions served as inputs for developing the mathematical task of making sense of
an authentic representation. The Discourse in this interaction was not typical for this
classroom, but was an exemplar for illustrating a contributive mathematical Discourse
that resulted in increased student thinking with more substantively articulated ideas. A
second example of a contributive Discourse occurred in Ada’s classroom in Classroom
Snapshot 5.2, specifically see Lines 13 – 21.
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Eva’s Classroom – Discourse types related to students’ thinking and ideas.
Refer back to Classroom Snapshot 5.5, when Eva’s class was developing fraction
representations. Her Discourse was more authoritative than contributive in that
interaction episode and the students’ thinking and ideas appeared shorter and less
substantive. Eva was engaged in the Discourse as a manager and guide. Her contributions
included Discourse guiding questions (Lines 1, 4, and 10), but she was also directive
(Lines 6 and 8) and provided validation (Line 4 and 10). The Discourse appeared to begin
to transition toward a contributive Discourse type after Eva’s final question, “How else
could I make them?” (Line 10). Most of the students’ contributions were comprised of
one or two words (Lines 3, 5, 7, and 9), which appeared to be emblematic of authoritative
Discourse types as observed in this investigation. However, most of Eva’s classroom
interactions appeared to be contributive and students’ thinking and ideas were substantive
and in several instances unsolicited by her.
Expert/Informer Teacher Behaviors and Discourses
Teaching behavior presented to students as expert or informer appeared to
decrease students’ participation in mathematical Discourses. Expert teaching behavior
occurs when the teacher is the only recognized source for right or wrong. Informer
teaching behavior occurs when the primary teaching approach is telling and/or showing.
In this investigation, each of the teachers presented expert and/or informer teaching
behaviors. In Ada’s and Kia’s classrooms, the effect of this behavior was for students’
ideas to be invalid until after the teacher validated it. This behavior and others appeared
to have decreased the number of unsolicited students’ ideas and perhaps not supported
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positive students’ behaviors, such as, help and support; mathematics efficacy;
persistence; and self-regulation (as depicted in Table 5.12). To illustrate these ideas,
classroom interaction episodes for Kia’s and Ada’s classrooms are examined.
Kia’s Classroom – Teacher behaviors and students’ behaviors. There was
evidence of expert informer teacher behavior in Interaction 1.6 and it followed what was
presented in Classroom Snapshot 5.6.
Classroom Snapshot 5.7: Expert informer teacher behavior Line 1: Kia: So, obviously we have a little indecisiveness. We're not quite
sure. How many of you are just unsure and you're just waiting for Ms.Kia to tell you what it is?
Line 2: [No one acknowledged or responded to her polling question.] Line 3: Kia: That's fine.
(Interaction 1.6)
The last line of this interaction episode encapsulated Kia’s perception of the
outcome of the Discourse, in spite of the more positive aspects described previously in
reference to the earlier dialogue. Kia appeared to believe that the students lacked
understanding of the authentic representation of subtracting decimal numbers and she
needed to tell them. This stance exemplified the expert informer teacher behavior.
Reflecting back on Classroom Snapshot 5.6, students offered several viable ideas
(Lines 9, 13, and 15) from which to develop a mathematical Discourse about the
authentic representation for subtracting decimals; however, the expert informing teacher
behavior led to a starting the Discourse from the teacher’s prescribed script. This
behavior does not promote positive student behaviors because their ideas appear to be
excluded from the Discourse.
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Ada’s Classroom – Teacher behaviors and students’ behaviors. In Ada’s
classroom, one student appeared to challenge her expert informer behavior, and perhaps
that explained, in part, why he and Eva appeared to be engaged in an adversarial
relationship, which did not enhance Discourse. To illustrate this point, in interaction
(3.1), a glimpse of the expert informer behavior and this Discourse challenging
relationship can be seen.
Classroom Snapshot 5.8: Student challenges expert informer teacher behavior Line 1: Ada:[Pauses during description of the day's objectives] Line 2: Student K: Didn't we already do this? Line 3: Ada: Student K, I want you to remember to raise your hand and wait to be
called on Line 4: Student K: [Interrupting] Didn't we already do this when... Line 5: Ada: [Interrupting] Student K, you have to raise your hand and wait until I
say, "Yes, student K" 'cause if I'm in the middle of a sentence, I'm going to wait until I'm done before I address you. And then the second part we're doing . . . [Resuming with the day's objectives]
Line 6: Ada: . . . so, a little motivation to stay focused to do what you are supposed to do. Please, do not go ahead. Do not go ahead. I want you to stay with me where I'm at unil I tell you that I know you know what you are doing and you can go. Student K, you're talking while I'm talking.
(Interaction 3.1)
In this interaction, Student K’s idea (Lines 2 and 4) was neither acknowledged nor
discussed, and this was the norm for his unsolicited ideas. Ada often selected him to
respond to specific questions, but in some instances, unsolicited ideas may have
potentially developed into substantive mathematical Discourses. Therefore, the practice
of not recognizing Student K’s unsolicited ideas could hinder mathematical Discourse.
Additionally, Student K was usually reprimanded for attempting to offer
unsolicited ideas, which likely did not improve his student behavior (i.e., mathematics
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efficacy or persistence). On the other hand, he did not appear to require Ada’s support for
his behavior because he continued to participate regularly and to challenge her expert
informer behavior. Even so, developing mathematical Discourse communities within
classrooms is sufficiently challenging (McClain & Cobb, 2001; Sherin, 2002; Sherin et
al., 2004) and limiting learners who appear to have positive mathematical attitude and
identity perceptions likely complicates the challenge.
Summary: Discourse or Community Influenced Mathematical Discourses
There were several instances when Discourse or community enhanced Discourses
and other instances when it served to hinder them. Using findings from this investigation,
elements of Discourse or community that hindered Discourses included: a) answers only;
b) expert informer teacher behaviors; and c) Discourse challenging student behaviors.
Conversely, elements of Discourse or community that enhanced Discourses included: a)
contributive Discourse type; b) positive student behaviors, such as, help and support,
mathematics efficacy, persistence, and self-regulation; and c) reflective teacher behavior.
On the other hand, elements that sometimes enhanced, but could hinder Discourses and
were dependent upon other circumstances included: a) students’ thinking and ideas; and
b) authoritative Discourse type.
Cross Case Analysis for Relationship Opportunity
In Chapter 4, each case study was concluded with a summarizing matrix. Those
matrices were summarized and separated by construct and then all classrooms combined
to create three new matrices used to guide the cross case analysis. These matrix
summaries compare the three classrooms, one matrix for each of the three theoretical
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constructs. The rows of the matrices depict the four concepts that define the construct
group. Specifically, for the theoretical construct relationships that support learning
opportunity the four concepts include: a) authority type or leadership style; b) identity
perceptions; c) learning influencer; and d) teaching influencer. The combined summary
matrix comparing the three classrooms for the theoretical construct Discourse or
community is presented in Table 5.13.
Relationship Opportunity Concepts
Ada Eva Kia
Authority type or Leadership style
° Charismatic ° Traditional or
autocratic
° Charismatic ° Distributed
° Traditional or autocratic
Identity perceptions ° Positive PAID ° positive PAID
(participation and agency identity)
° Chosen non-participation
° Positive PAID ° Negative PAID
Learning influencer ° Student action ° Teacher actions ° Teacher actions ° Teacher actions
Teaching influencer ° Mathematics as
procedural ° Fixed intelligence
° Mathematics as science
° Mathematics as procedural
° Fixed intelligence
Table 5.13: Relationships that support learning opportunity summarizing matrix for each classroom.
Authority Type Appears Connected to Participation
Traditional authority was regularly used within the three classrooms. For Ada and
Kia, traditional was their primary authority type; Eva used it in some instances, but she
often used distributed authority. Traditional authority is characterized by instruction that
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is teacher-centered, limited instructional flexibility, and the Discourse type is mostly
authoritative. Distributed authority is characterized by instruction that is more student-
centered, student-inspired instructional flexibility, and the Discourse type is more
contributive. Additionally, within each classroom, students exhibited identity perceptions
categorized as positive participation and agency identity (positive PAID); however,
positive PAID appeared to manifest in different ways dependent upon the teacher’s
authority type. Distributed teacher authority appeared to foster student positive PAID in
ways that enhanced mathematical Discourses; traditional authority appeared to foster
student positive PAID that were accompanied by elements that hindered mathematical
Discourses. Classroom interaction episodes from each classroom follow to elucidate the
relationships.
Ada’s classroom – Authority type and students’ identity perceptions. In this
interaction, Ada models and shows students how to multiply fractions, and the students’
role was to provide computation facts. Ada’s authority type was traditional and the
students exhibited identity perceptions of positive PAID; they readily responded to
questions and appeared compliant but not engaged.
Classroom Snapshot 5.9: Defining procedure for multiplying mixed numbers Line 1: Ada: Take the mixed number and the first step is change to improper
fraction. Line 2: [Writes out step 1; a student writes then starts beating softly on the desk] Line 3: Student A: Do we need to write what you're writing? Line 4: Ada: No, I'm just putting it up here for a reminder for you. OK. Step 2 is
going to be to simplify. Line 5: [Writes out step 2] Line 6: Ada: You're gonna simplify in an 'X' ... Line 7: [Describes and shows reducing by factors the numerators and denominators
of two fractions separated by a multiplication symbol] Line 8: Ada: Is there anything that will make 4 and 8 smaller by dividing?
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Line 9: Students: [multiple students responding] yes...2... 2 Line 10: Ada: Are there any bigger numbers? Line 11: Students: [multiple students responding] 4... 24... 4 Line 12: Ada: 4, alright. So, you take 4 and you go 4 divided by 4 is [pause] Line 13: Students: [multiple students responding] 1 Line 14: Ada: [Writing] so you cross it out and put a 1. So all these steps I want you
doing on your paper. and then you take the 8 and you divide it by 4 again and you get [pause]
Line 15: [Simplifying continues, using the same teaching strategy] Line 16: Ada: The second step is to simplify. The third step is gonna be to multiply
and I put it like that because you're gonna multiply your numerators and you’re gonna multiply your denominators. [emphasis on the word multiply]
Line 17: [Ada writes the multiplication, while asking students for computation facts] Line 18: [Several students look bored, begin mumbling, heads on hands, sighing,
yawning, and making other noises] Line 19: Ada: Not done yet. Last step is to simplify again. Line 20: [Converts improper fraction to mixed number, while asking students for
computation facts] Line 21: Ada: ...and your answer is 3 and 1/3. Line 22: Students: [students talk softly, but one says] I know another way to do that. Line 23: Ada: So, that is the way you do that. And if you notice up here, I have to go
through all those steps. If I just would have multiplied...[multiplies across the mixed number numerators and denominators as proof] So you see, if I would have just done it my own way it just does not work. You have to follow the steps.
(Interaction 4.2)
In this interaction, Ada’s instruction was teacher-centered in that she made all
decisions and choices with respect to multiplying fractions, many of which are innately
flexible, including: a) naming and ordering steps (Lines 1, 4, 16, and 19) ; b) articulating
verbally what was done (Lines 6, 7, 14, 16, 19, and 21); and c) recording the mathematics
in writing (Lines 2, 5, 14, 16, 17). Additionally, there appeared to be limited or no
instructional or mathematical flexibility (Line 23) and the Discourse type was
authoritative.
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The students’ were not required or encouraged to take notes about the procedure
being taught (Line 4); it appeared that their only responsibility was to provide
computational facts (Lines 9, 11, and 13). The students appeared to be compliant, in that
they proffered requested facts eagerly; hence, evidence of positive PAID. Even so,
several students appeared to not be fully engaged based upon their actions, including: a)
making noises (Line 2 and 18); b) not writing the mathematics along with Ada (Line 14
and 18); and c) talking softly among themselves and doing other non-participatory things
(Line 18). Thus, in this instance of traditional authority, positive PAID was manifested as
compliance without full engagement. The affect was an inclusive mathematical Discourse
did not emerge in spite of the presence of students’ positive PAID.
Kia’s classroom – Authority type and students’ identity perceptions. In this
interaction, Kia tells students how to interpret mathematical meaning on a worksheet, and
the students’ role was to participate. Kia’s authority type was traditional and 2 of the 18
students present in the classroom exhibited identity perceptions of positive PAID; they
readily responded to questions while the rest of the class chose non-participation.
Classroom Snapshot 5.10: Limited Participation Line 1: Kia: What does frequency mean? Line 2: Student A: It's a fraction?. Line 3: Kia: What does frequency mean? Line 4: Student A: Ah, like out of like a survey or something? Line 5: Kia: What does frequency mean? Line 6: Student A: Frequent guess? [puts hands out, palms up] Line 7: Student K: Oh, I forgot... Line 8: Kia: [~ 2 second pause] Hhhooow mmmaany or how much. [speech is
drawn out and Student K said the word ‘many’ in unison] Yeah, I know, that was years ago, Ms. Kia, back in September when you taught us that. Frequency is how many or how much. What's the next column?
Line 9: Student K: 16 through 25.
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Line 10: Kia: You know it isn't gonna work if we're not gonna be part of the lesson. Where it says relative frequency they give you a fraction, 6/20. What does that mean? 6 out of?
Line 11: Student K: [shrugs shoulders] Line 12: Student A: 6, 6 out of 20 Line 13: Kia: Were between the ages of Line 14: Student A: 6 and 15? Line 15: Kia: Right. So, we were making a fraction. What are the total amount of
numbers up there? What are the total amount of numbers up there? Line 16: Student A: 20 numbers. Line 17: Kia: 20 becuase of that denominator
(Interaction 4.2)
In this interaction, Kia’s discourse was traditional in the sense that it was teacher-
centered and the Discourse type was authoritative; Kia asked and answered the questions
(Lines 1, 3, 5, and 8) or asked skill or knowledge questions (Lines 8, 13, and 15). The
instruction appeared to be inflexible because she continued to pursue the instructional
path even after students’ made it clear they were not going to participate (Lines 10 - 17).
Student A participated by offering several ideas (Lines 2, 4, 6, 12, 14, and 16).
Student K’s participation included one idea (Line 9) and claims or gestures related to not
knowing (Lines 7 and 11). The non-participating students passed things from one person
to another, tapped lightly on the desk, and some appeared to be looking away from their
papers, perhaps daydreaming. Kia addressed the non-participating student behavior
briefly in Line 10, but did not follow up when things did not appear to change. Instead,
she turned her head toward the participating students and continued the instruction.
The two students’ positive PAID in this instance manifested to pacify the
teacher’s expectation that students participate, and indirectly supported the other
students’ chosen non-participation. Specifically, because Kia was able to continue
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instruction with the two students participating, she opted to ignore the non-participating
from the remaining students. In this instance of traditional authority, positive PAID was
manifested as a pacifier for the teacher; her desire for addressing students’ non-
participation was diminished. The affect was an inclusive mathematical Discourse did not
emerge in spite of the presence of students’ positive PAID.
Eva’s classroom – Authority type and students’ identity perceptions. In this
interaction, two students were afforded an opportunity to share how they thought about
solving a problem in a unique and unanticipated way. Eva’s authority type was
distributed and the students exhibited identity perceptions of positive PAID; they each
articulated their conceptual mathematical thinking with significant mathematical
confidence and autonomy.
Classroom Snapshot 5.11: Contributive Authority and Positive PAID Line 1: Eva: Student B and Student J, you didn't work it out, you had good logic
behind it. Alright 3/5, 3/4, and 3/12 and I said that I wanted you to compare them and of course show your work. . . . They didn't show a picture, they didn't find LCM, they just rewrote the fractions. Would you guys tell me, from least to greatest. How would you write it? Student B?
Line 2: Student B: 3/12, 3/5, 3/4 Line 3: Eva: . . . and I walked by in my irritation and I said, 'Gosh, you have to
show your work." and they both basically told me they didn't have to. Who wants to tell me why? Who wants to tell them why you didn't have to show your work. You didn't draw a picture. you didn't multiply anything out. And I was iritated! Alright, Student J, what did you do?
Line 4: Student J: Because um, I looked at the denominator and the biggest number is the fewer pieces and I started with 3/12 and then I looked at 3/5, and 3/4 and I just compared them. and I put 3/5 before 3/4.
Line 5: Eva: Um. Good. Good. Student B, tell me how you explain it. Line 6: Student B: I was like, well if the denominator is greater, then that is like the
opposite of the real numbers like 1,2,3,4 but if the denominator is least its actually like a bigger fraction.
Line 7: Eva: Good. Good. Good. Line 8: Student B: So, I did like 3/12, 3/5, and 3/4
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Line 9: Eva: Good. He said, you know I looked at my denominator and I said, Student I, that these pieces are smaller 12 pieces are smaller than 5 pieces are smaller than 4 pieces I have. . . What do you notice about the numerators? They are all, what?
Line 10: Class: The same Line 11: Eva: Well, the same. What was I thinking? They're all the same. um. So, do
you want 3 of these itty bitty pieces or 3 of these medium sized pieces or 3 of these big pieces? Um so yeah, you could have looked at it certainly that way. and you could have explained it.
(Interaction 2.4)
In this interaction, Eva’s distributed authority type was evident when she solicited
students (Lines 1, 3, and 5) to explain their thinking and her actions during and after their
explanations. During each student’s explanation of their thinking (Lines 4 and 6), no one
interrupted, and afterwards Eva revoiced (Line 9) the students’ explanations for what
appeared to be added clarity and assurance that other students heard the softly spoken
explanations. Additionally, in Line 9, Eva made students aware that she was recounting
the words of their peers when she started with, “He said,” a strategy known as animating
(Empson, 2003). Finally, it appeared from Eva’s comments (Line 1 and 3) that the
students’ approach for ordering the fractions was unexpected; the students’ unpredicted
approach was met with appropriate instructional flexibility and created an opportunity for
learning for the class. This was an exemplar for distributed authority; Eva’s instruction
was student-centered and appropriately flexible, and the Discourse type was contributive.
The students’ positive PAID manifested as mathematical confidence and
autonomy. The two students’ demonstrated mathematical confidence and autonomy when
they dared defy Eva’s request to show their work, and then conceptualized an approach
that Eva had not considered; they were beholden to neither her request to show work nor
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the mathematics she advocated. There were few examples from all of the classrooms of
students evidencing this level of mathematical confidence or autonomy to the extent that
Students B and J had in this interaction. The two students, as Eva explained had not
shown their work, caused her to be irritated, and then told her that “they didn't have to”
show any work (Lines 1 and 3); the boys showed mathematical moxie (i.e., gumption).
The students’ behavior was an exemplar of mathematical autonomy and
confidence. In this interaction (2.4), Eva used distributed authority, the affect was
positive PAID manifested as demonstrated students’ mathematical confidence and
autonomy, which enhanced the mathematical Discourse.
Beliefs about Mathematics Manifest in Teacher Actions
Table 5.14 summarized teaching actions for each classroom using one specific
interaction, which is described in row one; however, the teaching actions captured
represent norms of practice for each during the observation phase, that is, several other
classroom interactions could have been used in lieu of those specified.
The beliefs about mathematics, row two, suggests how each teacher perceived
mathematics based upon this analysis. Mathematics as procedural suggests mathematics
knowledge that is based on sets of steps and limited connectedness; mathematics as
science suggests an integrated conceptual and procedural knowledge that is richly
connected (Baroody et al., 2007). The notion of mathematics as procedural or science is
misleading in that these are not static states, but each exists on a continuum. Therefore,
Ada’s and Kia’s teaching actions appeared to be more closely aligned with giving steps
and/or guidance to get to correct answers. This type of instructional focus appeared to
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hinder mathematical Discourses because teachers often adopted traditional authority
types and Discourses were authoritative, which excluded students from participation in
some ways.
Relationship Opportunity Concepts
Ada Eva Kia
Interaction 4.2 Teacher tells and shows steps for multiplying mixed numbers
Interaction 2.4 Students explain how they compared fractions without showing any work
Teacher interprets mathematical meaning for things on a worksheet
Interaction 4.2
Beliefs about Mathematics Procedural Science Procedural
Teaching actions
• Steps were given • Espoused one way
to find correct answers
• Allotted time for students’ sharing their approaches with class
• Asked/answered questions
• Ignored non-participants
Table 5.14: Comparing beliefs about mathematics and teaching actions by classroom.
Whereas, Eva’s teaching actions appeared to be more aligned with developing
integrated mathematical knowledge. This type of instructional focus appeared to enhance
mathematical Discourses because teachers adopted distributed authority and contributive
Discourses emerged, which enabled authentic ideas to be shared (Classroom Snapshot
5.11, Lines 4 and 6). Additionally, inclusive Discourses appeared to reveal elements of
inquiry or exploration and led to multiple student approaches for finding correct solutions
and mathematical connections (Line 4, 6, and 9); restated, using a traveling metaphor, the
journey is valued as much as the destination.
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Summary: Relationship Opportunity Influenced Mathematical Discourses
There were several instances when relationships that support learning opportunity
enhanced Discourses and other instances when it served to hinder them. Using findings
from this investigation, elements of relationship opportunity that hindered Discourses
included: a) traditional authority type; b) mathematics as procedural; and c) chosen non-
participation. Conversely, elements of relationship opportunity that enhanced Discourses
included: a) mathematics as science; and b) distributed authority type. On the other hand,
elements that sometimes enhanced, but could hinder Discourses and were dependent
upon other circumstances included: a) students’ positive PAID (participation and agency
identity); and b) teacher actions.
Review of the Investigation
This study was about developing understanding about how classroom interactions
in three classrooms influenced Discourses related to mathematics learning and teaching.
To that end, data was collected and analyzed, three case studies were created, and cross
case analysis was used to interpret understandings that emerged from the case studies
related to the research questions. This research process enabled me to offer descriptions
that addressed the following research questions:
1. How do classroom interactions influence the Discourse related to mathematics learning and teaching in MCP supported classroom?
a) What is the nature of classroom culture in each classroom?
b) What is the nature of Discourse and community in each classroom?
c) What is the nature of relationships that support learning opportunity in each classroom?
2. How do classroom interactions that influence Discourses related to mathematics learning and teaching compare among the classrooms?
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My descriptive responses to these questions were extensive and key points are
summarized for clarity.
Summarized Response to the Research Questions
The nature of classroom interactions that enhanced mathematical Discourses in
the three classrooms, included: a) classroom culture that was inclusive, collaborative, and
focused on mathematics; b) Discourses or community that were student-centered,
supported authentic thinking and ideas within a connected mathematics; and c)
relationships that supported learning opportunity treated mathematics as science,
leadership was moving toward being transformational, and students’ attitude and identity
perceptions were nurtured. None of these ideas related to reform were unanticipated
given the review presented in Chapter 2 of the mathematics education reform literature
(e.g., Megan L. Franke et al., 2007).
This investigation was focused on sociocultural elements of mathematics
education, which was proposed as a needed research trajectory in the mathematics
education literature (e.g., Megan L. Franke et al., 2007; Hufferd-Ackles et al., 2004;
McClain & Cobb, 2001). Additionally, two findings from this investigation contribute to
the mathematics education literature.
The first finding is referred to as classroom interactions on the boundary, that is,
classroom interaction elements that either enhance or hinder (sabotage) mathematics
Discourses as determined by extenuating classroom contexts or circumstances. Sfard and
Kieran (2001) suggested the need to understand actions that contribute to or hinder
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mathematics learning and teaching. The other finding from this investigation is the
introduction of a practical framework for developing ROC that is referred to as a
Reflective Triad. This finding contributes to the mathematics community because it is
offered as a tool for teachers (in-service and pre-service) and those supporting them to be
reflective. The mathematics education literature has long articulated the need for helping
teachers to be more reflective as a method for improving practice (e.g., Ball, 1993; Blase
& Blase, 2000; Breyfogle, 2005; Manouchehri, 2001; Senger, 1999). Each of these
contributions is discussed next.
Classroom Interactions on the Boundary – A Result of this Investigation
Table 5.15 depicts the boundary classroom interaction concepts organized by the
three theoretical constructs (i.e., classroom culture, Discourse or community, and
relationship opportunity). These concepts appeared to straddle the boundary for
influencing mathematical Discourses for the classrooms that were the focus in this
investigation; however, that does not suggest that the concepts listed would emerge as
boundary interactions in other classrooms.
Column two of Table 5.15 shows examples of potential enhancers for Discourse,
examples of the types of extenuating circumstances that potentially enhanced Discourse
given classroom interaction concepts listed in column one. For example, teacher
explaining (from column one) likely enhanced mathematical Discourses when students’
sharing choices and ideas (from column two) were an integral element of the classroom
interaction.
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Classroom Interaction Concepts Potential Enhancers Potential Saboteurs
Classroom culture • collaborative sense-
making • communications • teacher explaining
• problem-based tasks that includes students’ options • students’ sharing choices
and ideas • students’ comparing
options and choices
• tasks without choices • no options for students • not creating opportunities
for students to talk about mathematics
Discourse or community • students’ thinking and
ideas • authoritative Discourse
type
• students’ afforded opportunities to think about meaningful things related to mathematics • teachers relinquish
authority and trust students to think of good ideas
• teachers make all of the choices and do most of the thinking • teachers hold all authority
and only relinquish the hold in very small ways (e.g., computation only)
Relationship Opportunity • students’ positive
participation and agency identity
• teacher actions
• create opportunities for substantive participation • effective listening and
respect for all
• limit participation to surface mathematics • accept Discourse
challenges without opposition
Table 5.15: Classroom interaction concepts by theoretical construct on the boundary for either enhancing or hindering mathematical Discourses.
Conversely, column three consists of potential Discourse saboteurs or hindrances.
For example, collaborative sense-making (from column one) likely hindered
mathematical Discourse when students were offered tasks without choices (from column
three) as an integral element of the classroom interaction. Table 5.15 was not presented as
an exhaustive representation, but illustrated several types of extenuating classroom
circumstances that either enhanced or hindered Discourses when boundary concepts
existed in classroom interactions and emerged as a result of this investigation. These
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situations may be valid for any mathematics classroom, but they would need to be
observed and independent conclusions drawn.
Reflective Triad – A Result of this Investigation
The Reflective Triad, shown in Figure 5.2 was conceptualized following this
investigation. The impetus for the conceptualization of this practical framework included:
a) the responses to the research questions for this investigation; b) the mathematics
education researchers’ call for increased teacher reflection; and c) the theoretical model.
The Reflective Triad offers three focal perspectives for reflection, including a)
flexible pedagogical approaches; b) student-teacher cooperative actions; and c)
communal learning perspective. The focal perspectives align with the three theoretical
Figure 5.2: A practical framework for supporting teacher reflection
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model constructs as indicated by the arrows in Figure 5.2. For example, Discourse or
community is related to student-teacher cooperative actions.
This reflective tool was designed for teachers and those who support them to
reflect collaboratively on specific elements of their practice in ways that include
sociocultural elements of classrooms. Consider a teacher reflecting on one of the focal
points (the bulleted items) related to classroom culture, for example, choices and options.
He or she would likely consider questions with sociocultural implications, such as: a)
What options did the task offer students? b) Who participated? or c) What evidence
suggests that students’ perceived they had options? These questions and others conceived
by users of this tool, in most instances, would include elements related to sociocultural
elements in addition to issues of curriculum, pedagogy, and mathematics content.
An underlying assumption related to the Reflective Triad is that the mathematics
underlying instruction should be substantive. That is, the more substantive the
mathematics, the greater reflective opportunity for the teacher. Consider instruction that
offers little mathematical substance, such as 20 exercises multiplying whole numbers
between 1 and 12 in a grade nine mathematics class. The Reflective Triad would not
yield substantive reflection for the teacher until he or she recognized that the mathematics
was insufficiently substantive because few, if any, of the focal points would apply.
Finally, the Reflective Triad was designed for collaborative use. Ideally, the tool
would be used with two or more persons who had experienced the instructional episode
being reflected upon, such as: a) teacher and coach; b) pre-service teacher and mentor
teacher or supervising teacher; or c) lesson study group. However, the tool could still be
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used collaboratively between experienced teachers who were engaged professionally as
critical friends or within a professional learning community. Advantages of reflective
collaboration are more questions and/or ideas and professional integrity. It is much easier
to see through rose-colored glasses when you are the only observer because when two or
more people look at the same things they each see through their own perspective, even if
all perspectives are rose colored. This inherent characteristic of individuality offers much
when collaborating to solve ill-defined problems as those encountered within classrooms
often are.
Limitations of the Investigation
The primary limitation of this investigation is the generalizability of the results.
Case studies do not lend themselves to generalizing because what happened in three
classrooms would never be sufficient evidence for predicting across many (Merriam,
2002b; Stake, 1994; Stark & Torrance, 2005). There are several factors to consider that
may have improved the results for this investigation.
Factors that may have Affected the Results
The research design and procedures enabled the research questions to be
addressed; however, there were factors that may have affected the results. First and most
importantly was the data collection for classroom observations. On occasion, the sound
quality was poor for portions of the video-taped observations, which made portions of the
classroom interactions inaudible. The use of powerful microphones perhaps centrally
located about the classroom and connected to the video cameras may have improved the
sound quality. Initially, attempts were made to capture sound using a digital audio
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recorder, but placement in the classroom was problematic, synchronization of video and
audio was challenging, and in the end, the issue of poor sound quality remained.
Secondly, scheduling classroom observations so that the instructional goals or
curricula were similar across all classrooms may have added another dimension to the
analysis. Specifically, comparisons could have been made about the ways classroom
interactions influenced Discourses when instructional goals were related among
classrooms.
Finally, of interest might be observing classrooms over extended time or tied to
specific outcomes, such as covering a specific mathematics concept or during an entire
semester. Some of the data collected and subsequent analysis may have varied because
teachers’ or students’ preference for or against specific topics taught during the limited
observation phase, which may have influenced the resultant Discourses. Additionally,
given a longer data collection phase, opportunities for interviewing participants or
holding student focus groups may have added new dimensions. Finally, more time may
have made videotaping virtually transparent and afforded use of a handheld device for
capturing less public classroom interactions.
Implications and Recommendations
This investigation made clear that the sociocultural elements of classroom
interactions that influence mathematical Discourses are not set and therefore cannot be
prescribed. Even so, there were several implications that emerged for mathematics
practice and teacher support. The implication for the mathematics practice was effective
mathematical Discourses require classrooms with reform-oriented culture (ROC); and
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developing ROC would occur more readily if there were classroom-embedded support
that focused on student understanding.
Implications for Mathematics Practice
In the past, mathematics education reform has been articulated in terms of content
(NCTM, 2000), curriculum and assessment (NCTM, 1989), and teaching (NCTM, 1991);
and in each of these standards documents and related research were implied or mentioned
recommendations for sociocultural elements for mathematics classrooms. The
implication from this investigation is the need to bring sociocultural elements of
mathematics classrooms from the background to the foreground as a way to improve
mathematics Discourses related to learning and teaching. This is essential for
transitioning from traditional to reform-oriented cultures (ROC) in mathematics
classrooms.
There were glimpses of ROC within the case studies and in the cross case
analysis. ROC emerges within classrooms when students’ perceive they have
opportunities to share their profound ideas, that is, ideas that are contrary to the
instructional flow or accepted ideas. Consider two specific examples, one from Ada’s and
another from Eva’s classroom. In Ada’s classroom when Student B (Line 13) turned the
class away from an accepted solution to a problem to an alternate solution (Interaction
5.11, Classroom Snapshot 5.2). Alternatively, in Eva’s classroom when Students B and J
had each autonomously thought about comparing fractions conceptually instead of using
one of the procedural approaches that had been the focus for instruction (Interaction 2.4,
Classroom Snapshot 5.11).
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These example suggests teachers must intentionally establish pedagogical
approaches that enable students to perceive they possess opportunities to share their
profound mathematical ideas. Inherently to achieve such a lofty goal requires teachers’ to
create opportunities for students to think in ways that their profound ideas emerge. A type
of thinking that comes for students who feel free to think autonomously or differently
from their teacher and their thinking is respected and valued by all in the mathematics
classroom.
Implications for Teacher Support
The second implication of this investigation was the need for classroom-
embedded support. Bringing sociocultural elements of mathematics classrooms from the
background to the foreground, and establishing ROC within classrooms that are often
more traditional than reform oriented, will require support. Workshops and providing
resources do not appreciably improve practice (Loucks-Horsley, Love, Stiles, Mundry, &
Hewson, 2003; L. West, 2007). Support that is classroom embedded and collaborative
would likely result in teacher reflections capable of challenging epistemological
regularities (Spillane & Zeuli, 1999) and reified norms of practice (Patrick & Pintrich,
2001) because perspectives from another will vary from that of an individual and thereby
creating the potential for change.
Recommendations
Clearly, all classrooms are different and their cultures unique, defined by the
contributions from all its members. Therefore, there are no specific recommendations for
developing ROC in all classrooms, but a set of general guidelines emerge from this
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investigation. ROC requires effective listening, if ideas are to be heard, especially those
that are contrary to what is accepted should be vetted in some way, preferably in ways
that are inclusive for all class members. This is a task that must be championed by
teachers. However, listening and vetting implies environments where power and/or
authority are shared. Additionally, ROC requires respect from everyone and their ideas;
all classroom members must perceive that their ideas will be listened to, respected, and
assumed viable and worth consideration.
The classrooms selected for participation were in part selected because the
assigned Mathematics Coaching Program (MCP) instructional coaches were second year
MCP participants when this investigation was underway. The focus of the MCP is to
support the implementation of reform-based instructional approaches and develop
effective mathematical Discourses via classroom-embedded support (Erchick & Brosnan,
2009a, 2009c). Specifically, the goal of MCP is to support teachers’ development of
reform-oriented cultures (ROC) within their classrooms. During this investigation, we
know from both coaches and teachers that the prescribed MCP support was not
implemented in the three classrooms; however, there is reason to believe that the
prescribed MCP support would help teachers to establish ROC and improve
mathematical Discourses in each of the classrooms (Erchick & Brosnan, 2009c).
Suggestions for Future Research
This investigation revealed much about classroom interactions and the ways they
influenced mathematical Discourses in three sixth grade mathematics classrooms.
However, there is so much more to learn about how classroom interactions influence
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Discourses. This investigation left significantly more questions at the end than those
considered at the beginning. For example, what is the nature of Discourses in grade levels
above the sixth grade? How do classroom interactions influence them? What is the nature
of mathematical Discourses in classrooms supported by prescribed MCP support? Which
of the three theoretical constructs contributes most to developing effective mathematical
Discourses and student learning? We know it when we see it, but what exactly is ROC?
How can the Reflective Triad be improved to support ROC development?
Finally, the methodological approach for this investigation was case study and
cross case analysis for interpreting findings. Ethnographic or phenomenological research
methodology approaches would offer more nuanced understandings. Ethnography
research studies are used to study people’s beliefs, values, and attitudes in relation to
community or situations and phenomenology research studies are used to identify the
nature or structure of an experience or phenomenon (Merriam, 2002a). Using either
methodology would result in deeper understandings of the sociocultural contexts within
specific classrooms and perhaps shed light on how beliefs and perceptions influence
mathematical Discourses. This type of study might enable further elaboration on the
theoretical model so it could be made more specific and perhaps better articulated,
thereby opening a doorway for exploring classrooms on a larger scale for experimental
studies that include rigorous quantitative analyses that lead to greater generalizations.
Discussion Summary
This chapter started with an interpretation of understandings by examining across
the case studies presented in chapter 4 using cross case analysis and the research findings
260
were summarized. The investigation was recapped, limitations were articulated, and
implications and recommendations were described. Finally, the chapter was closed by
sharing emergent questions from this investigation and several suggestions were offered
261
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APPENDIX A: VIDEO TAPING PROTOCOL
2 video cameras with tripods
Equipment needs:
2 video tapes for each lesson
Capture video using 2 video cameras, both cameras are stationary
Video Recording Protocol:
Position cameras to capture a full room shot (excluding non-consenting students)
Video and audio files will be coded and selectively transcribed for analysis
Use and Storage of Video and Audio Files:
All files will be stored on an external hard drive within a secured location for the following uses:
o Dissertation analysis and write up o Data source for related future article publication (identities will not be
revealed) o Data source for future teacher education (identities will not be revealed)
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APPENDIX B: INTERVIEW PROTOCOL
Coach Interview Questions
1. Describe consenting teacher(s) practices in terms of the following:
a. Instructional style
b. Classroom environment
c. Classroom interactions
d. Student engagement/participation
2. Tell one memorable story that exemplifies this teacher’s practice.
3. Describe your coaching approach when working with this teacher(s).
Teacher Interview Questions
1. Describe your teaching practice in terms of the following:
a. Instructional style
b. Classroom environment
c. Classroom interactions
d. Student engagement/participation
2. Tell one memorable story that exemplifies your teaching practice.
3. Describe the coaching support you have experienced in terms of:
a. Interactions
b. Influencing your practice
c. Influencing your mathematics content or pedagogical knowledge
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APPENDIX C: CODE BOOK
Classroom Culture (CC) Theoretical Construct Group 1 Cultural Influencers (CI) • Collaborative sense-
making Evidence of discussion and talk involving two or more persons related to mathematical development, solution reasonableness, and alternate arguments are considered reflectively.
• Dyadic talk Evidence of teacher-student dialog with quality student cognition related to collaborative mathematical development
• Fact/process reproduction Evidence of students recreating teacher presented facts or procedures; limited student ‘doing’
• Powerful other Evidence of the learning community looking to text or other expert for validation or correctness
• Ownership for learning Evidence of the classroom community or members owning their mathematical understandings or constructed knowledge
• Teacher expectations Evidence of teacher actions or expectations for students that influence classroom culture
Mathematical Practice (MPr) • Argument & proof Evidence of mathematical arguments and proof
influencing mathematics learning and teaching • Communication Evidence of Discourse and talk influencing mathematics
learning and teaching • Mathematical connections Evidence of mathematical connections influencing
mathematics learning and teaching • Multiple representations Evidence of multiple representations influencing
mathematics learning and teaching • Problem solving Evidence of problem solving or mathematical modeling
influencing mathematics learning and teaching Sociomathematical Norms (SMN)
273
• Questioning Evidence of questioning that contributes to mathematics learning and teaching
• Student conjectures Evidence of students making mathematical guesses using their own mathematical understanding
• Student explaining Evidence of students articulating meaning for mathematical ideas
• Teacher explaining Evidence of teacher articulating meaning for mathematical ideas
Social Norms (SN) • Listening Evidence of teacher and students focusing on hearing the
talk of learners • Low risk environment Evidence of students feeling free to participate in
mathematics learning and teaching • Negative SN Evidence of social norms that hinder mathematics
Discourses related to learning or teaching within the classroom
• Respect Discourse or Community
(DC) Theoretical Construct Group 2
Discourse Influencers (DI) • Answers only Evidence of teacher or students perceiving mathematics
as answers only, the goal is to get the answers, not thinking or process
• Connected math Evidence of teacher or students perceiving mathematics as connected and coherent science
• Exercise tasks Evidence of learning and teaching guided by exercises, e.g., procedural problems solved via steps or simple algorithms
• Problem-based tasks Evidence of learning and teaching being influenced through focus on a problem solving and related processes
• Static math Evidence of teacher or students recognizing mathematics as a collection of never changing facts to be memorized, not necessarily understood
Discourse Types (DT) • Authoritative Evidence of univocal discussion and talk, a one-way
dialog that is likely dominated by the teacher • Contributive Evidence of interactional discussion and talk where
student voice is included, procedural focus • Dialogic Evidence of generative conversation that focuses on
collaborative sense making Student Behaviors (SB)
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• Collaborations Evidence of two or more people collaborating in ways that contribute to learning and teaching; collaborations around problem solving, consensus seeking, genuine communication attempts (beyond surface talk)
• Compliance Evidence of students meeting minimum expectations • Discourse challenging Evidence of student behaviors that hinder Discourses
and/or community development related to learning and teaching. For example, participating in learned helplessness, a failure to cooperate with others, etc.
• Efficacy Evidence of students demonstrating mathematically • Helping & supporting Evidence of students supporting peer learning, explicitly
or implicitly • Persistence Evidence of students showing tenacity • Self-regulating Evidence of students controlling their learning or
behaviors that influence learning, such as listening to peer explanations
• Thinking & ideas Evidence of student thinking manifested through ideas contributed to Discourse and/or community development related to mathematics learning and teaching
Teacher Behaviors (TB) • Catalyst Evidence of teacher using and supporting student ideas
toward developing understanding • Connector Evidence of teacher making connections among math
topics or between math and other things • Enforcer Evidence of imbalanced teacher focus on student
behavior versus effective instruction • Expert Evidence of teacher using his/her mathematical skills or
knowledge in lieu of students’ • Informer Evidence of significant teacher talking; telling and
showing; overly guided teacher dialogues • Reflective Evidence of teacher contemplating on instruction related
to mathematics learning and teaching • Revoicing Evidence of teacher restating students words,
representations, or other communication modes to improve clarity for others
• Validator Evidence of teacher determining correctness and/or validity
Relationships that Support Learning Opportunities (RO) Theoretical Construct Group 3
Authority/Leadership (AL)
275
• Charismatic (distributed) Evidence of authority garnered through personality and charm; leadership is shared
• Distributed (transactional) Evidence of authority is delegated; transactional leadership is fostered through rewards (e.g., grades, prizes, etc.)
• Professional (transformational)
Evidence of authority through mathematics content and pedagogical expertise; transformational leadership transforms followers to leaders via development of mathematical power and understanding
• Traditional (autocratic) Evidence of traditional hierarchical authority via teacher-directed and student compliance; autocratic leadership via status or position
Identity Perceptions (IP) • Chosen non-participation Overt evidence of students choosing to not participate and
no effective teacher intervention • External identity Evidence from student’s actions that infer or give insights
into how students see themselves outside the math classroom related to mathematics learning and teaching
• Math autonomy Evidence of students showing mathematical independence
• Math identity Evidence of students showing themselves as mathematical learners
• Negative PAID Evidence of students participating in ways that negatively contribute to learning and teaching, negative PAID (Participation & Agency Identity)
• Positive PAID Evidence of students participating; the act of participating; making contributions that positively enhance learning and teaching Positive PAID (Participation & Agency Identity)
Learning Influencers (LI) • Teacher action Evidence of specific teacher actions that impact
mathematics learning • Student action Evidence of student’s actions that contribute to
mathematics learning or teaching, e.g., scaffolding, animating, revoicing, etc
• Peer pressure Evidence of peer pressure that influences mathematics learning and teaching
Teaching Influencers (TI) • Fixed intelligence Evidence of students’ effort not improving mathematics
success
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• Growth intelligence Evidence of students’ effort improving mathematics success
• Math as chaos Evidence of mathematics being perceived as an unrelated and disjointed
• Math as procedural Evidence of mathematics being perceived as an ordered set of step and procedures
• Math as science Evidence of mathematics being perceived as a social activity that involves systematic observation, evidence gathering, and experimentation in pursuit of generating rules (e.g., axioms, theorems, etc.) models, and generalizations
Emergent Codes for Categorizing Interaction Episodes Interaction Type • Student-student Two or more students engaged in an interaction • Teacher-class Whole class interaction • Teacher-group Teacher and small group engaged in an interaction • Teacher-student One-on-one teacher and student interaction Interaction Focus • Interpreting meaning Explaining that is not mathematical • General explaining Developing clarity or shared understanding • Reflecting and/or
evaluating Mathematical problem solving
• Mathematical problem solving Contemplation about prior event, activity, or action
• Social problem solving Resolving sociocultural situations • Student explaining
mathematical idea Students revealing mathematical thinking
• Teacher explaining mathematical idea Teacher revealing mathematical thinking
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APPENDIX D: RESEARCH TIMELINE
278
Research Timeline
Month (time estimate)
Research Task
Oct. 2008 Prepare IRB application Proposal approved by dissertation committee (DC) Revise IRB application based on DC recommendations Submit and receive IRB approval
Nov. 2008 *
Acquire letter of support for data sharing from MCP Coordinate informant selection with MCP PIs Identify year 2 MCP coaches (Y2C) interested in participation Describe the study and expectations verbally and in writing to
interested Y2C Y2C informally (verbally) consent to participate*
Request permission from targeted school districts Dec. 2008 District grants permission for research
Send an informational e-mail to principals and teachers to find interested participants to visit
Contact Y2C to support coordination for informational site visit (e.g., logistics such as teacher schedules, etc.)
Form an Expert Review Panel (ERP) to review claims & warrants (C&W); share summary of expectations; get signed confidentiality agreement
Jan./Feb. 2009
40 min. per teacher
Teacher Recruitment * Describe study, expectations, and answer questions for
principals and interested teachers
Ask principals to sign school consent forms and teachers to sign teacher consent forms
Leave copies of signed consent forms for principals and teachers
Collect teacher information forms Schedule site visits for Student informational meeting, one-on-
one interview, video taping sessions, and student pizza party
Feb. 2009
40-50 min. per classroom
Student Informational Meeting*
Verbally describe study and expectations for students
Answer questions related to the research project or video taping Ask students to read and sign the Student Assent forms if they
want to participate in the study Distribute Letters of Information and Parental Consent forms (2
copies per student) for parent signatures (Note: parents retain
279
Month (time estimate)
Research Task
letter of information and 1 copy of the signed parent consent form and return the other to school with their student)
Leave envelops for teachers to collect signed parental consent forms (Note: researcher will consult signed parental consent forms to inform pre-video set up)
Leave a video taping schedule to be posted in the classroom for teacher and student reference
Administer the Student Survey to students and the Standards Implementation Survey and TSES instruments (~ 30 min.) to teachers
Data Collection
30 min per coach
Request coaches sign Coach consent forms Leave copies of signed coach consent forms for coaches
Data Collection Audiotape one-on-one interview with coach
Feb. - Apr.
2009
20-30 min. per teacher for interview
20-30 min. per classroom for
set up
Pre-Video Setup Collect signed parental consent forms Review parental consent forms and set up for video-taping (e.g.,
arrange desks, adjust seating assignments, determine camera placement, etc.)
*
Observe teacher during instruction, if possible, to get a feel for his or her teaching style for video set up (e.g., wanderer, stationary, A/V and technology use patterns, etc.)
Data Collection Separate student surveys by those with or without parental
consent, researcher retains student surveys of those with parental consent. Teachers will be provided copies of consenting student surveys upon request.
Audiotape one-on-one interview with teacher (~ 20 min) prior to video taping. If instructional observation is not possible, be sure to ask about the teacher’s teaching style relative to video capture.
6 instructional episodes per classroom
Data Collection Video tape instructional episodes to capture classroom
interactions during mathematics teaching and learning Use the video taping protocol for video captures.
280
Month (time estimate)
Research Task
Collect Y2C MCP data related to (i.e., LAMP and LMT responses, all versions available, year 1 site evaluations, coaches’ reports)
Collect teacher MCP and school data for (i.e., LAMP responses, all versions available; OAT building scores)
Feb. – Aug.
2009
Using descriptive statistics to situate consenting teachers’ schools among all MCP participating schools
Quantitative Data Analysis
Using MCP and school data, compare and contrast teachers and Y2C with respect to their beliefs about mathematics, teaching, and learning
Create field notes based on analysis Draw preliminary conclusions based on data and reflections Make data collection adjustments based on preliminary findings
Repeat multiple times as data corpus
grows
Import data into NVivo v8 for analysis Qualitative Data Analysis
Listen to Y2C and teacher audio taped interviews and selectively transcribe
Create field notes based on audio and transcriptions Draw preliminary conclusions based upon data, theory, and
reflections Make data collection adjustments based on preliminary findings
20-30 min. consultations
as needed
ERP review C&Ws based on preliminary data and offer feedback
Validity Testing
Member checks with coaches and/or teachers to consider C&W and offer feedback
Import data into NVivo v8 for analysis Data Analysis
Listen to interview audio tapes, selectively transcribe, and code using theoretical framework
Review and code student surveys using theoretical framework Create field notes based on audio and transcriptions Draw preliminary conclusions based upon data, theory,
reflections, and preliminary findings
281
Month (time estimate)
Research Task
Make data collection adjustments based on preliminary findings
ERP – Review C&W related to:
Validity
Quantitative analysis Student surveys Teacher and coach interviews
Import data into NVivo v8 for analysis Data Analysis
Watch video and listen to audio classroom segments, selectively transcribe, and code using theoretical framework and other emergent themes
Review each classroom video segment at least 3 times to focus specifically on each of the primary constructs – Discourse (community), culture, and relationships.
Create field notes based on each video/ audio review and all transcriptions
Draw preliminary conclusions based upon data, theory, reflections, and preliminary findings
Make data collection adjustments based on preliminary findings
Begin preliminary write up of findings using format for
member checks with Y2C and teachers (use format from Demerath’s P&L 967 class)
Preliminary Write Up
Data Collection
Schedule additional data collection cycles as needed to fill data collection/analysis gaps
Aug. 2009
Continue preliminary write up of findings using format for member checks with Y2C and teachers
Final Draft Write Up
Begin final report write up of findings using dissertation format Review earlier chapters of report and adjust as needed
Sept. 2009
Review findings and warrants. Final Data Analysis
282
Month (time estimate)
Research Task
Critically reflect on the strength of claims and warrants. Identify additional warranting data examples as needed to
strengthen analysis.
Member check
Final Draft Validity
Schedule time with teacher (or use e-mail based on informant preference) to share/clarify preliminary findings, as needed
Schedule time with Y2C to share/clarify preliminary findings ERP – Review specific elements of: Final report and the evidentiary support from the data corpus
Sept. 2009
Submit initial dissertation draft to committee for review Final Write Up
Revise according to feedback and resubmit as needed Schedule Dissertation Defense
Nov. 2009
Meet with committee to defend dissertation (1Dissertation Defense
st or 2nd
week of Nov.)
* A timeline influencing task if not completed according to plan
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APPENDIX E: STUDENT SURVEY
284
Student Survey31
Stro
ngly
A
gree
O
vera
ll A
gree
O
vera
ll D
isag
ree
Stro
ngly
D
isag
ree
I feel confident that I can learn most math topics. I can learn anything in math if I put my mind to it. If I find the work hard, I know that if I keep trying I can do it My friends say that I keep trying when math gets hard You are either good at math or not and you cannot get better by trying.
If I can’t do the work in math I give up.
Your friend Loretta is really good at math, but she never tries.
Explain why this could be.
What advice would you give Loretta?
What can Loretta do to help herself?
What can the teacher do to help Loretta?
31 This survey is an adapted form of that used in Sullivan, P., Tobias, S., & McDonough, A. (2006). Perhaps the Decision of Some Students Not to Engage in Learning Mathematics in School Is Deliberate. Educational Studies in Mathematics, 62(1), 81.
285
APPENDIX F: TEACHERS’ SENSE OF EFFICACY SCALE
286
Teachers' Sense of Efficacy Scale
Teacher Beliefs32
Directions: This questionnaire is designed to help us gain a better understanding of the kinds of things that create difficulties for teachers in their school activities. Please indicate your opinion about each of the statements below. Your answers are confidential.
How much can you do?
Not
hing
Very
litt
le
Som
e in
fluen
ce
Quite
a b
it
A g
reat
dea
l
1. How much can you do to motivate students who show low interest in school work?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
2. How much can you do to get students to believe they can do well in school work?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
3. How much can you do to help your students value learning?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
4. To what extent can you craft good questions for your students?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
5. How much can you use a variety of assessment strategies?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
6. To what extent can you provide an alternate explanation or example when students are confused?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
7. How much can you assist families in helping their children do well in school?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
8. How well can you implement alternative strategies in your classroom?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
9. How much can you do to help your students think critically?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
10. How much can you do to get through to the most difficult students?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
11. How much can you gauge student comprehension of what you have taught?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
12. How much can you do to improve the understanding of a student who is failing?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
13. How much can you do to adjust your lessons to the proper level for individual students?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
14. How much can you assist families in helping their children do well in school?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
15. How well can you provide appropriate challenges for very capable students?
(1) (2) (3) (4) (5) (6) (7) (8) (9)
32 This survey is an adapted form of the TSES from Tschannen-Moran, M., & Woolfolk Hoy, A. (2001). Teacher Efficacy: Capturing an Elusive Construct. Teaching and Teacher Education, 17, 783-805.
287
APPENDIX G: TEACHERS’ COMMITMENT TO MATHEMATICS EDUCATION REFORM
288
Teachers' Commitment to Mathematics Education Reform
Name: ______________________________________
Mathematics Education Reform Statements33
Directions: This questionnaire is designed to help us gain a better understanding of commitments to mathematics education reform. Please indicate your level of agreement for each of the statements below. Your answers are confidential.
Level of Agreement
Stro
ngly
ag
ree
Agre
e
Som
ewha
t agr
ee
Som
ewha
t dis
agre
e
Dis
agre
e
Stro
ngly
di
sagr
ee
1. I like to use math problems that can be solved in many different ways. (6) (5) (4) (3) (2) (1) 2. I regularly have my students work through real-life math problems that
are of interest to them. (6) (5) (4) (3) (2) (1)
3. When two students solve the same math problem correctly using two different strategies I have them share the steps they went through with each other.
(6) (5) (4) (3) (2) (1)
4. I tend to integrate multiple strands of mathematics within a single unit. (6) (5) (4) (3) (2) (1) 5. I often learn from my students during math time because my students
come up with ingenious ways of solving problems that I have never thought of.
(6) (5) (4) (3) (2) (1)
6. It is not very productive for students to work together during math time. (6) (5) (4) (3) (2) (1) 7. Every child in my room should feel that mathematics is something
he/she can do. (6) (5) (4) (3) (2) (1)
8. I integrate math assessment into most math activities. (6) (5) (4) (3) (2) (1) 9. In my classes, students learn math best when they can work together
to discover mathematical ideas. (6) (5) (4) (3) (2) (1)
10. When students are working on math problems, I put more emphasis on getting the correct answer than on the process followed. (6) (5) (4) (3) (2) (1)
11. Creating rubrics for math is a worthwhile assessment strategy. (6) (5) (4) (3) (2) (1) 12. In my class it is just as important for students to learn data analysis
and probability as it is to learn basic math facts. (6) (5) (4) (3) (2) (1)
13. I don’t necessarily answer students’ math questions but rather let them puzzle things out for themselves. (6) (5) (4) (3) (2) (1)
14. A lot of things in math must simply be accepted as true and remembered. (6) (5) (4) (3) (2) (1)
15. I like my students to master basic math facts before they tackle complex problems. (6) (5) (4) (3) (2) (1)
16. I teach students how to explain their mathematical ideas. (6) (5) (4) (3) (2) (1) 17. You have to study math for a long time before you see how useful it is. (6) (5) (4) (3) (2) (1)
33 This survey is an adapted form of the TCMER from Ross, J. A., McDougall, D., Hogaboam-Gray, A., & LeSage, A. (2003). A Survey Measuring Elementary Teachers' Implementation of Standards-Based Mathematics Teaching. Journal for Research in Mathematics Education, 34(4), 344-363.
289
APPENDIX H: ANALYSIS GRID USING THEORETICAL CONSTRUCTS
Theoretical Constructs & Implications from the
Literature
Analysis Tasks Relationship to Research Question
Response Discourse/Community
Community characteristics: • Members are active
participants to set purpose, direction, or approach
• Collective contributions to shared purpose (Hodge, 2006)
• Mathematical culture exists (i.e., members share “linguistic practices, beliefs, values, & ways of acting” (Cobb & Hodge 2002, p. 262)
Discourse Types (Lloyd, 2008) • Authoritative • Contributive • Dialogic Tasks/Curriculum implications for Discourses (Ball, 1993): • Answers only • Student thinking & ideas • Student-student collaborative
constructions &/or interactions • Mathematical connections Teacher Roles/Behaviors (Ball, 1993): • Teacher as expert
1. Examine classroom elements: Code using theoretical concepts, such as: • Discourses type • Task/curricular
influences 2. Examine interactions: Code using theoretical concepts, such as: • Teacher roles/behaviors • Student roles/behaviors • Group dynamics
3. Look for patterns in coding and use definitions and theory to identify communities and/or Discourses • Characteristics • Artifacts of existence • Instances of Discourse
related to mathematics teaching or learning
4. Draw inferences and cite specific evidence from data (look for data
1. How do classroom interactions influence the Discourse related to mathematics teaching and learning in four MCP supported classrooms? a) What is the nature of Discourse(s) within the classrooms? 2. How do classroom interactions influence Discourses related to mathematics learning and teaching compare among the classrooms? Focusing on mathematics learning and teaching, as appropriate and warranted by the data: • Describe communities
and Discourses in each classroom
• Describe tasks/curriculum implications for
290
Theoretical Constructs & Implications from the
Literature
Analysis Tasks Relationship to Research Question
Response • Teacher as listener • Teacher attends to students
ideas for developing understanding (Hufferd-Ackles, et al., 2004)
Student Roles/Behaviors: • Helping/Supporting (Lloyd,
2008) • Efficacious (Sullivan et al.,
2006) • Engaging • Self-regulating (Sullivan et al.,
2006) • Negative (e.g., avoiding,
distracting, etc.) Group Dynamics • Similarities between teacher
and student Discourses (Webb, et al., 2006)
• Classify group norms (Webb, 1991)
• Positive/negative for student achievement
• Direct/indirect development of increased Discourse
source triangulation) 5. Generate claims and warrants 6. Consult with expert review panel
communities and Discourses in each classroom
• Describe teacher and student role/behaviors related to communities and Discourses in each classroom
• Describe group dynamics related to communities and Discourses in each classroom
• Compare and contrast the above across all classrooms
For each classroom and then across classrooms, articulate connections between: • tasks/curriculum and
student engagement and participation
• student and teacher roles/behaviors and student engagement and participation
Culture
Mathematics classroom microculture characteristics (Cobb & Hodge, 2002): • Social norms – accepted ways
of doing, knowing, acting, etc. • Sociomathematical norms –
accepted ways of doing mathematics, such as
1. Examine mathematics classroom microculture Code using theoretical concepts, such as: • Classroom microculture
characteristics • Cultural capital
1. How do classroom interactions influence the Discourse related to mathematics teaching and learning in four MCP supported classrooms? b) What is the nature
291
Theoretical Constructs & Implications from the
Literature
Analysis Tasks Relationship to Research Question
Response questioning, establishing validity, sharing ideas, etc.
Mathematical practices– accepted modes of practice, mathematical processes Cultural Capital • Ways of knowing • What counts as knowledge? • Who decides? Community – cultural influencer (Hufferd-Ackles, et al., 2004): • Questioning • Explaining • Authentic sources of
mathematical ideas • Ownership for learning Teacher actions – cultural influencer: • Uses student ideas to develop
mathematical concepts (Hufferd-Ackles, et al., 2004)
• Social scaffolding (Nathan & Knuth, 2003)
• Analytic scaffolding (Nathan & Knuth, 2003)
• Filtering (Sherin, 2002) • Revoicing & animating
(Empson, 2003) Student actions – cultural influencer: • Analytic scaffolding • Revoicing & animating
2. Examine mathematics cultural influencers & contexts Code using theoretical concepts, such as: • Community influencers • Teacher actions • Student actions • Mathematical contexts 3. Look for patterns in coding and use definitions and theory to identify mathematics classroom culture and cultural influencers: • Characteristics • Artifacts of existence • Instances of culture
related to mathematics teaching or learning
4. Draw inferences and cite specific evidence from data (look for data source triangulation) 5. Generate claims and warrants 6. Consult with expert review panel
of cultures within the classrooms? 2. How do classroom interactions influence Discourses related to mathematics learning and teaching compare among the classrooms? Focusing on mathematics learning and teaching , as appropriate and warranted by the data: • Describe classroom
microculture in each classroom
• Describe what counts as cultural capital in each classroom
• Describe community cultural influencers in each classroom
• Describe teacher and student cultural influencers in each classroom
• Compare and contrast the above across all classrooms
For each classroom and then across classrooms, articulate connections between: • Cultural influencers
292
Theoretical Constructs & Implications from the
Literature
Analysis Tasks Relationship to Research Question
Response • Meta-messages related to
participation and engagement (Sfard & Kieran, 2001)
Mathematical Cultural Communication Contexts • Argument & proof
(Stylianides, 2007) • Problem solving & modeling
(Lesh & Jawojewski, 2007; Schroeder & Lester, 1990)
• Discourse & Talk (Zevenbergen, 2000; Wood, et al., 2006)
• Connections • Multiple representations
and student engagement and participation
• Cultural contexts and student engagement and participation
Relationships Opportunity The nature of opportunity related to mathematics learning: • Participation & agency –
learning influencer (Nasir, 2002) • Affords learning • Stifles learning
• Engagement – learning influencer • Thinking (Ball, 1993;
Lampert, 1990/2004) • Struggling to learn
(Smith, 2000; Ball, 1993) • Talk – learning influencer
(Sfard, 2001; Sherin, 2002) • Enhances learning • Hinders learning
Authority (Leadership styles)
1. Examine mathematics classroom opportunity learning influencers: Code using theoretical concepts and states, such as: • Affords/enhances
learning • Hinders/stifles learning • Enables/suppresses
thinking • Allows/prevents student
struggle 2. Examine mathematics classroom authority & social control: Code using theoretical concepts, such as: • Authority types
1. How do classroom interactions influence the Discourse related to mathematics teaching and learning in four MCP supported classrooms? c) What is the nature of relationships within the classrooms? 2. How do classroom interactions influence Discourses related to mathematics learning and teaching compare among the classrooms? Focusing on mathematics learning and teaching, as
293
Theoretical Constructs & Implications from the
Literature
Analysis Tasks Relationship to Research Question
Response Authority types (Pace & Hemmings, 2006) Leadership styles (Bass, 1990): • Traditional (autocratic) • Charismatic (distributed) • Legal-relational – referred to
as distributed (transactional) • Professional (transformational) Social control strategies (Pace & Hemmings, 2006): • Exchange • Influence • Coercion • Structural Structural Control – Who controls what & why: • Flow of talk (Cazden, 1988) • Compliance (Haberman, 2002) • Responsibility for learning
(Sullivan, et al., 2006) Identity Identity perceptions – teaching and learning influencer • Talk reveals evidence of
identity (Cazden, 1988) • Participation identities (Hodge,
2006) • Mathematical identity
• Productive • Less productive (e.g.,
resistance, avoidance, alienation, etc.)
Intelligence perceptions –
• Social control strategies • Structural control 3. Examine mathematics classroom identities: Code using theoretical concepts such as: • Identity perceptions • Intelligence perceptions
4. Look for patterns in coding and use definitions and theory to identify mathematics classroom opportunity, authority, and identity for articulating relationships: • Characteristics • Artifacts of existence • Instances of
relationships related to mathematics teaching or learning
5. Draw inferences and cite specific evidence from data (look for data source triangulation) 6. Generate claims and warrants 7. Consult with expert review panel
appropriate and warranted by the data: • Describe classroom
relationships related to opportunity, authority, and identity in each classroom
• Describe how relationships influence mathematics teaching and learning in each classroom
• Describe how relationship teaching and learning influencers have manifest in each classroom
• Compare and contrast the above across all classrooms
For each classroom and then across classrooms, articulate connections between: • Identity (teaching and
learning influencers) and student engagement and participation
• Authority and student engagement and participation
• Control and student engagement and participation
294
Theoretical Constructs & Implications from the
Literature
Analysis Tasks Relationship to Research Question
Response teaching influencer (Dweck, 2006) • Fixed intelligence • Growth (Incremental)
intelligence Perceptions of mathematics – teaching influencers (Schoenfeld, 1992) • Chaotic – disorganized, not
connected • Procedural – facts, right
answers, fixed • Science – a social activity that
involves systematic observation, evidence gathering, and experimentation in pursuit of generating rules (e.g., axioms, theorems, etc.), models, and generalizations
295
APPENDIX I: ADA’S CLASSROOM INTERACTION SUMMARY
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
02/24/2009 Observation 1: OAT review; drawing representations of 3-D objects; estimating fractions; solving problems using fraction estimation 1.1
Teacher-class Math PS
Practice solving achievement test items together
reading items aloud; thinking and ideas; creating Q's for peers; answering Q's from peers; explaining
asking Q's; helping student with words during reading [@ first sign of trouble]; tasking students to ask Q's of each other; encouraging students to explain their responses; making connections
1.2
Teacher-class General Explaining
Ada explains how students are to respond to OAT constructed response Qs
during teacher talk students do not look at teacher, have bored expressions (e.g., head on hands, looking at ceiling, etc.); after teacher talkwriting students are writing, sharpening pencils (loud noise and continuous)
Explains expectations for written responses on OAT, gives sentence starters for their written responses
1.3 Teacher-student Interpreting meaning
Ada addresses students Q about the task
Asks Ada for task clarification; listens; selects from options; begins writing
Ada repeats sentence starter, "C is correct because..." listens to student; offers student a limited choice about what he can write about and repeats sentence starter again
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Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
1.4 Teacher-class Reflecting and/or evaluating
Ada evaluates written responses
sharing their thinking & ideas; revising written responses based on teacher input; respond to Qs; share thinking & ideas
circulating looking at student work; asking Qs; critiquing written responses; asking Ss to reflect on "because I said so" respnse from their moms'; asking Ss for ideas for improving written responses; asked leading Qs to improve on ideas
1.5 Teacher-class T & S explaining math idea
Ada develops definitions for supplementary and complementary angles
responding to Qs; thinking & ideas; shouting out responses; comment about student ("he's fast" [wrt coming up with answers])
Q'g; selecting Ss to respond; build on from supplementary defn to complementary; acknowledge and agrees w/ students' observation about another student, "yes, he is fast, math wizards"
1.6 Teacher-class Reflecting and/or evaluating
Review of leanring from previous Friday
thinking & ideas; guessing asked Qs seeking facts; recalling process from text
1.7 Teacher-class & student-student Reflecting and/or evaluating
Checking for understanding using manipulatives
following directions; responding to Qs; student K shows class his thinking
giving directions; asking Qs; validating; informing
297
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
1.8 Teacher-class Reflecting and/or evaluating
Ada reflects on instruction aloud
reading directions aloud revising directions based on experience from an earlier period
1.9 Teacher-class T & S explaining math idea
Ada summarizes lesson and connects learning to a purpose
collecting papers; off-task behaviors (e.g., looking around, messing with stuff on their desks, etc.); responding to Qs
Asking Qs about what they just did; summarized why they did it
1.10 Teacher-class T explaining math idea
Ada models acceptable approaches for estimating fractions
reading from text; interrupts T; limited participation; asking Qs; not watching teacher @ board; playing w/ cubes; looking bored (slouched in chairs, playing w/ calc)
shares expectations ("must be able to do each of my ways"); models methods; asks skill/knowledge Qs; T atends to S interruptions & off-task behavior; solves a second problem using the method w/ student input
1.11 teacher-class Math PS and T explaining math idea
Problem solving as teacher defined procedure
reading; sharing; responding to Qs
asking leading Qs; validates, telling
1.12 Teacher-student Social PS and S explaining math idea
Ada reprimands student
corrects teacher; makes flipant remark; explains
gives wrong answer; reprimands student's approach
02/25/2009 Observation 2: Homework review; timed multiplication/division test; designing spaces activity using snap cubes
298
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
2.1 Student-student; Student explaining math idea
Students sharing HW solutions
presenting HW solutions using overhead; responding to teacher Qs; students watching peer
encouraging students to correct HW; selecting student to present; asking clarifying Qs; sharing her strategies for reading fractions for understanding; simplifying cognitive demand with comments; waiting; encouraging
2.2 Teacher-class Reflecting and/or evaluating, T explaining math idea
Intervention informed by observing student work
offering examples; looking around; raising hand; non-participating
reflecting on student work; asking for examples; selecting students; asking clarifying Q; explaining
2.3 Teacher-class Math PS
Soliciting ideas from students about solving a word problem
reading; thinking & ideas; asks for two things that have to be done to solve the problem in words; accepting all ideas; encouraging students; waiting for ideas; telling after getting multiple ideas from students
2.4 Teacher-class Math PS
Students to share solutions
several raised hands (5); selected Ss (2) share what they wrote
connects story problem to Ss lives; selects Ss for sharing; (some Ss are rarely called on); acknowledges Ss contributions w/o validation; expresses not understanding Student M's response, but encourages her to listen and perhaps add on to what she has written
299
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
2.5 Teacher-class Math PS
Determining the better deal (the computation)
bored faces/actions (yawning, slouching, leaning back, etc.); asking Q that reveals misconception
asking leading Qs; telling; waiting; atttends to misconception
2.6 teacher-class: General explaining
Introducing Mad Minute
compliant, looking around; no enthusiasm
talking; directing students how to complete the mad minute (left to right)
2.7 Teacher-class Reflecting and/or evaluating
Articulating what was learned the day before
sharing learning; responding to Qs (verbally and with gestures); increased participation (e.g., more hands raised, gesturing, etc.)
asking Qs (leading); asking about connections to OAT; refining/extending Ss' contributions (not revoicing); limitied or no acknowledgement of Ss' contributions
2.8 teacher-class: General explaining
Communicating directions for an activity
reading; limited or no evidence of following along in text
selecting readers (Students T & D); helping with words; adding comments (not related to comprehension per se); giving specific directions for completing the task
2.9
Student-student Reflecting and/or evaluating
Students compare block designs
Studen K calls to T; T turns around; K shows his blocks; T shows his blocks; T turns around
N/A
300
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
2.10
Teacher-student T & S explaining math ideas
Students articulating 3-D block solutions given orthogonol drawings
Student K (8 blks), other students offer other solutions (4 blks); K disputes Ada's block solution, begins building the solution (validation), and offers justification; Student K, "whatever"; K puts blks down
describing orthogonol drawings; asks class how many blocks; waits as Ss shout out answers; telling/showing block solution; ignores, then dismisses Student K's argument
2.11
Teacher-class General explaining
Giving directions
reading; responding to Qs
Revising directions; asking Qs to check understanding; telling which drawings were easiest in other classes; offering choices
2.12 Teacher-student Math PS
Ada scaffolds student
watching Ada draw on paper; responding to Qs; begins using pencil to draw; drawing using Ada's descriptions
takes pencil; drawing on paper; gives it back; asking Qs; describing how to create drawing
2.13 Student-student Math PS
Students helping one another during drawing time
one S get's anothers attention; asking Qs; looking/pointing @ paper; showing papers
N/A
2.14 Teacher-class Reflecting and/or evaluating
Students scoring student work
participation; responding to Qs; reading aloud as class when Ada stops; most students read along; evaluated the reading wrt ability to recreate block plan; limited participation comparing S work samples
Introduced student work and task; asking Q (skill); elicited Ss' participation (e.g., read aloud when I stop); asked evaluative Qs about reading; repeated for second student work; asked Ss to compare S work samples; Ada did not comment on comparison
301
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
2.15 Teacher-student Reflecting and/or evaluating
Student makes case for using blocks for final assessment
Explaining why blocks are needed for completing the task
Collecting blocks; telling to use the paper, his thinking, and learning from the past 3 days to complete the task
2.16 Student-student S explaining math idea
Partners (T & E) debate issues on final assessment
asking Qs; explaining; showing evidence; debating enthusiastically (i.e., gesturing and body language); writing
N/A
03/02/2009 Observation 3: Adding & subtracting fractions w/ unlike denominators using LCMs and equivalent fractions; reviewing the three different types of OAT Qs; and working w/ 3-D 3.1
Teacher-class Social PS
Ada sets participation expectations for Student K
Asks Q (during Ada's pause); asks Q again; sits quietly
Reprimands S for Q'g @ wrong time (perceived interruption); Sets expectation (Raise hand and wait to be called on); reprimands S again; does not respond to S's Q
3.2 Teacher-class T explaining math idea
Reprimands S for Q'g @ wrong time (perceived interruption); Sets expectation (Raise hand and wait to be called on); reprimands S again; does not respond to S's Q
Responding to leading Qs (quickly); looking bored (yawning, not looking @ Ads's work, leaning head on hands, not taking notes); most students are compliant; some Ss write down the problem;
Display problem on board; draw two circles, one for each fraction (pizza representations); asking leading Qs to solve fraction addition; telling Ss to write down the problem if they have not done so; describes what to write; tells Ss to write down problem 2
302
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
3.3 Teacher-student Social PS
Student K wants to tell the answer to the problem being solved
Raises hand and waits for Ada to call on him; asks if he may give the answer
Calls on K; tells him no; goes on with the process of showing the class how to solve the problem
3.4 teacher-class T explaining math Ideas
Lecture for example 2
listening; limited note taking; some Ss working independently; thinking & ideas
telling; showing; asking Qs (leading & open ended); attending to some ideas; not interrogating most ideas (i.e., multiply by 2)
3.5 teacher-student T explaining math idea
Finding LCM of 3 and 6
thinking & ideas; guessing
Asking Qs (leading, unclear, open-ended); validating some responses; using unclear language; positing that Ss are confusing multiplication and division
3.6 teacher-student T explaining math idea
Finding simplest form of a fraction
Several not actively participating (i.e., heads on hands, heads down, interested in paper ball, ); several actively participating - obedient compliance (i.e., shouting out expected responses, looking @ the board & copying what's being written); thinking & ideas; debating w/ peers
Offering rationale for doing the math procedure; asking Qs (leading & skill/procedural); categorizing responses as right and wrong; telling; asking students to take notes; writing on the board
303
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
3.7 Teacher-class T & S explaining math ideas
Understanding fraction ops w/ unlike denominators using the calculator
Making comments about the overhead calculator; most Ss participating; pushing calc buttons; explaining; telling answer; writing answer on paper
Giving rationale for calculator use (i.e., the easy way); telling which buttons to push; asking Ss to explain what/why for calc results; summarized calculator math experience; instructing Ss to write answer only on paper
3.8 Teacher-class S explaining math idea and T general explaining
Ss sharing written responses to OAT short answer type Qs
reading response; Ss raising hands to volunteer to read their sentences; not listening (i.e., raising hands while Ss talking; looking around; playing w/ pencil and writing)
selecting Ss for sharing; giving feedback (i.e., highlighting strengths, weaknesses, and/or giving advice)
3.9 Teacher-class T & S explaining math ideas
Teacher giving hints
S interrupts Ada; Ss sharing ideas about surface area of 3-D block drawings; thinking & ideas; explaining ideas
Ada reprimands S for interruption; suggesting a situation; asking Qs (leading and explain); interrupting S talk
3.10
Student-student Social PS
Ss select partners
moving w/excitement and anticipation; gesturing across the room; reseating; finding/negotiating locations; rearranging desk; settling down to work
Stopping selection process b/f directions are completed; completing directions; collecting and distributing materials
304
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
3.11
Student-student Math PS
Ss working w/ partners
working (discussing, debating, listening, sharing ideas); laughing; raising hands for help/validation
Circulating the room and helping, supporting, and encouraging
03/09/2009 Observation 4: Multiplying mixed numbers; using estimation to approximate products involving mixed numbers; and practice 4.1 Teacher-
student Math & Social PS
Student K's conjecture
Raises hand; offers conjecture before solution is presented; offers an explanation (cannot be heard)
Recognizes Student K; acknowledging K's intelligence; over talks K's explanation (tells K he is wrong, does not listen to his rationale); posits that K is giving addition answer, not multiplication; requests he wait so she can tell him how to give right answers
4.2 Teacher-class T explaining math ideas
Giving steps for multiplying mixed numbers
Taking notes; making noises (sighs, yawns, pencil tapping); rubbing eyes; asking to copy less than is written; chosen non-participation; no S talking beyond computation/skill utterances
Tells/shows each step; asks skill/fact Qs; ignores most wrong answers, sometimes corrects; repeats Qs until correct answer is given; instructing Ss to copy only problems not written steps; refutes K’s earlier conjecture related to multiplying mixed numbers
4.3 Teacher-student S explaining math idea
Student A offers unsolicited input
Raising hand; clarifying his input as idea for next step and not answer
Recognizing student A; telling A no answer now; interrupting student
305
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
4.4 Student-student Social PS
S's idea not used for class
J offers idea for next step (idea is rejected or not heard); another student repeats idea louder; J claims ownership for idea (via facial expression, side comment, gestures); Ss acknowledge idea as J's; J accepts recognition from peers (via heads nod and smiles exchanged)
N/A
4.5 Teacher-group Reflecting and/or evaluating
Checking for evidence of student understanding - small groups (4 or 5 Ss @ a time) working problems on chalk board
Working independently; showing work on chalk board; Ss have limited or no understand of feedback (i.e., as evidenced by written work, facial expressions, and indecisive/hesitant body language)
Circulating among Ss; erasing S's work; writing changes; giving terse instructions; telling & showing; no Q'ing (i.e., probing, explanation, or check for understanding)
4.6 Student-student Social PS
Student points another student in the right direction
Student B walking aimlessly to put an object away; Student C walks to him and points out where
N/A
4.7 Student-student Social & Math PS
S joins existing group
Asking to join grp; inquiring about work progress; rearranging desk; group catches new member up; asking Qs; thinking & ideas; debating; explaining; showing evidence
N/A
306
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
03/10/2009 Observation 5: Dividing fractions w/ mixed numbers; practice using white boards; worksheet practice (for grading) 5.1 Teacher-
class T & S explaining math idea
Ss sharing math HW answers and explaining
Telling answer; explaining answer
Selecting Ss to share (T, K, J, A, A, M, D, A, J2, A2); extending Ss' explanations and/or revoicing; creating math misconception
5.2 Teacher-class Math PS
Reviewing procedure for multiplying mixed number
Responding to fact/procedural Qs; writing on white boards
Asking fact/recall Qs; validation; not listening to Ss beyond expected response; did not ask Ss to hold up white boards for inspection
5.3 Student-student Math PS
Ss tasked to solve multiplication w/ mixed numbers
begin working; almost immediately Student J requests to solve problem on overhead (OH); solution is on OH; some Ss appear to be checking out solution on OH
Assents to J's request; circulating the room looking at evidence; thanking Ss for writing neatly on white boards
5.4 Teacher-class Math PS and T explaining math idea
Division of fractions using algorithm
Responding to Qs; thinking & ideas; writing on white boards
Writing on OH; asking fact/recall Qs; leading Ss through solution process; validating responses
5.5 Student-student S explaining math idea
Peer tutoring Student J: asking clarifying Q; listening, watching, trying; Student K: helping like Ada (showing, taking pencil, writing, telling steps)
N/A
307
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
5.6 Student-student Reflecting and/or evaluating and S explaining math idea
More peer tutoring
Student J: reflects on problem K did for him on his paper, asking reflective Q; K: helping; J: listening & watching; "aha" verbal, gesture, and facial expression at the end
N/A
5.7 Student-student Reflecting and/or evaluating
Student reflection about division of fractions
Ss are working & talking; One student says, "I don't get this though, how can I divide and the answer is higher?"; announcement refocused Ss’ attention
Announcing remaining time for work
5.8 Student-student Social PS
Student poses Q to peers
A1 taps J’s back for attention; asks Q about calculator; Student A2 teases A1 for wanting help; J is unable to help or chooses to say nothing; A1 teases J for seeking lots more help previously; no one intervenes on behalf of A1 or attends to the negative taunting
N/A
5.9 Student-student Reflecting and/or evaluating and S explaining math idea
Helping & learning
Copying; not much talking; standing around; playing around (timed copying); sitting; Black students working together and not mixing w/ others
N/A
308
Ada’ Classroom Interactions
ID Interaction
Structure & Focus
Interaction Description
Student Actions Teacher Actions
5.10 Student-student Social PS
Peer helping & learning
A1 said, "I don't need help" he rejected several students; Student A3's "help" (allowing A1 to copy) was accepted; polling to see who has answers
N/A
5.11 Student-student Math PS
Student D solving problem @ overhead
Read the problem; D was about to begin, then opted to select someone else to speak; Student K was selected; Student B corrected response supporting his thinking using the wording of the question; peers commended B for his thinking and answer
Just as D was about to start Ada asked: You can ask someone if you like; teacher commended B's thinking and answer
5.12 Student-student S explaining math idea
M facilitating geometry PS w/ class
M: drawing; asking Qs; selecting Ss for input; class: thinking & ideas; describing & explaining
Giving additional information
5.13 Student-student S explaining math idea
K facilitating fraction multiplication problem
K using addition procedure; others chime in; A2 suggests just times it, but before he can explain Ada explains
Interprets A2's input as the steps she taught previously
5.14 Student-student S explaining math idea
D facilitating PS about ratios
Reading aloud; Ss giving input; lack confidence
Telling; explaining; asking extension Q; coming to the front; taking the overhead pen
309
APPENDIX J: EVA’S CLASSROOM INTERACTION SUMMARY
Eva’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
02/24/2009 Observation 1: The focus of instruction is comparing fractions using words, pictures, and symbols 1.1 Teacher-
student; Explaining mathematical idea
Eva question: Which would you rather have 6/8 or 3/8 piece of candy?
Student A answers question and tells why using mathematical language
Affords student think time, encourages her to use calculator for mathematical language support
1.2 Teacher-class; Problem solving
Determining the number of students in each cooperative group (16 kids in groups of 3)
Three or more students propose ideas about the question
Selects a go to kid, then selects another student to build on what was said
1.3 Teacher-class; Interpreting meaning
Eva question: What does it mean to compare 6/10 and 5/6?
Three or more students propose ideas about what it means to compare and ways to show a mathematical comparison
Selects different students to contribute to the conversation; offers a simpler whole number example for students to consider the ways they mathematically compare two numbers
310
Eva’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
1.4 Small group-teacher; Problem solving
The students call Eva to the group for help. They are trying to determine how to compare two fractions 6/10 and 5/6.
The two girls laugh with the teacher and one another, the boy gets the group’s paper and begins working on the calculator, the boy pushes paper to the girls to draw pictorial models
Eva laughs with the girls, suggests they listen to what the boy has to offer, she waits as the boy does something on the calculator, before he is done, she suggests the group draws a pictorial representation
1.5 Student-student small group; Problem solving
The group has taken Eva’s advice and have drawn a pictorial representation of the two fractions, but the two girls do not know what to do next. They ask the boy about multiplication.
The boy tells the girls to multiply the 5 until they get 30 (i.e., find multiples of 5 until you reach 30); One girl (C) says 5 times 6 is 30; the other girl (E) puts something in her calculator and tells A to copy it on the paper.
N/A
1.6 Student-student; Reflecting or evaluating
The students are reflecting on their task and deciding if they are done.
The group reflects on the task: we have to compare the fractions using pictures and words. They add a final written statement to the group’s paper.
N/A
311
Eva’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
1.7 Teacher-students; Explaining mathematical idea
The students are collaboratively explaining through reasoning why 5/6 is larger than 6/10 by considering the number of pieces and the size of each piece.
Compare 1/6 to 1/10 pieces given the same whole; generalize about fraction pieces and denominators - the larger the denominator the smaller the pieces and the smaller the denominator the larger the pieces
Selects students to speak; asks question to extend thinking to making generalizations; checks for understanding with an example (i.e., would you rather share a candy bar equally with two friends or five?)
1.8 Student-teacher; Explaining mathematical idea
Eva: It’s not always reasonable to draw a picture to compare fractions. What if I wanted to compare 4/7 and 12/28. I wouldn’t want to use a ruler to draw 28 equal pieces.
Student B offers an unsolicited explanation of how to compare fractions using least common factor; he attempts to connect prior knowledge to the teacher’s dilemma
Eva listens patiently to B’s explanation; she acknowledges B’s attempt at connecting mathematical concepts; she reminds the class that the things they learn in mathematics are useful and connected; then she picks up where she left off before B’s contribution.
1.9 Teacher lecture; Explaining mathematical idea
Eva explains a new way to compare fractions without using pictorial representations
Writing notes; looking around; sitting quietly; some look bored at times
Explaining using a teacher pre- organized approach; sounding animated; writing on the overhead using colored markers; asking leading questions
312
Eva’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
1.10 teacher-class Reflecting or Evaluating
Reflect on student work and offering suggestions for improvement
reflect on work; offer suggestions
selects papers for reflection; highlights positive aspects of paper; selects students for input; locates work samples from other groups that match student explanations
1.11 teacher-class T Explaining Math Idea
Student seeks validation of work
notices difference in work; get's Eva's attention and shows her
Reassures C; shares work; explains
1.12 teacher-class Reflecting or Evaluating
Students reflect on student work and offer suggestions for improvement
reflect on work presented; offer ideas for improvement; raise hands for selection
present a group of student work that is incomplete; ask for improvement suggestions; select student
02/25/2009 Observation 2: Comparing fractions using mathematical approaches (no pictorial representations) 2.1 Teacher-
student; S explaining math idea
Comparing 6/7 and 3/12 using prime factorization
Corrects teacher error; explains comparing two fractions by prime factorization; uses the recording methods modeled by Eva
Thanks student for pointing out her error; fixes the error; listens, revoices student explanation, records students process, and asks clarifying (and sometimes leading) questions to keep things moving
313
Eva’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
2.2 teacher-class Interpreting meaning
Come up with mathematical terms for 84 (i.e., the new denominator found during Q's fraction comparison using the prime factorization method)
thinking and ideas; raising hands and waiting to be called; taking advice from Eva
Offering advice when stuck; selecting students for input; accepting all ideas offered with positive praise
2.3 Teacher-class Math PS
Students collectively compare two fraactions
Thinking and ideas; raising hands and waiting to be called; wait quietly when peers speak; take advice from Eva; respond to questions from Eva
Selecting students for input; revoicing for clarity for other students; called on a total of 6 different students to contribute during the 2.5 min. interaction
2.4 Student-student Students explaining math idea
Using reasoning to compare three fractions (an approach Eva had not thought about)
explained in their own words and in their own ways; listened quietly to peers
complimented students for their thinking, revoiced student explanations for clarity
2.5 Teacher-class Teacher explaining math idea
Explain a mathematical way to compare three fractions (3/4, 3/5, 3/12)
thinking & ideas; explaining ideas; contributing to math process
Listening; controling Discourse flow; making connections w/ previously studied math topics; giving limited hints to students who need help
314
Eva’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
2.6 teacher-class: General explaining
Explaining how to work collaboratively
listening; responding when asked
explain collaborative behaviors
2.7 Student-student Math PS/Social PS
A partner group figures out why their answers do not match
compare answer; assume mathematical error; figure out a way to communicate mathematically
N/A
2.8 student-student: Math PS/Social PS
Social skills collide with matheamtical PS
express their own thinking; not open to hearing the others point of view; calls Eva to mediate; rethinks math after Eva intervenes
Refocuses boys attention on the math; offer suggestions for thinking about the fractions; encourages students to use mathematical approaches to compare fractions
03/02/2009 Observation 3: Use fractional understanding to create pictorial representations of decimal numbers on grid paper 3.1 teacher-
student Reflecting or evaluating
Student reflects on a mathematical approach for finding LCM and decides he doen’t like it
asking questions about mathematical approach; admitting not liking a mathematical approach
Asking guiding questions to model finding LCM by prime factorization; accepting student’s negative feelings about using prime factorization for finding LCM
3.2 teacher-student Math PS
Student shares a different way for comparing fractions.
Volunteers another way to compare fractions without finding the LCM
Validates answer; thanks student for sharing; compares the new approach to Eva's selected approaches
315
Eva’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
3.3 teacher-class T explaining math ideas
Teacher and students work collaboratively to represent decimals and fractions using grid paper
shared their thinking and ideas; listened to peers; raised hands and waited to be called upon before speaking
asked leading questions; selected a variety of students; made mathematical connections; revoicing and drawing student ideas for clarity
3.4 teacher-class S explaining math Ideas
Students working collaboratively to explain denominator
sharing thinking and ideas; listening to peers
encouraging; asking leading questions; drawing to communicate meaning
3.5 teacher-student S explaining math idea
Student A created a representation of 7/10 on a 10X10-grid that is different
mathematical autonomy; agency
did not attend to student's idea; focused on teaching plan
3.6 student-student general explaining
Student helping another to meet teacher expectations
squabbling; helping and support; listening; complying
N/A
3.7 teacher-class: Reflecting and evaluating
Eva waits (almost 1 min.) before calling on a student
thinking; looking at notes; waiting
asks questions; waits
3.8 teacher-class Math PS and teacher explaining math idea
Eva guides student collaborative PS
thinking and/or recall; raising hands; contributing ideas; listening
calls on 7 different students; asks leading questions
316
Eva’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
3.9 student-student S explaining math idea
One student asks a fact question and another answers
asks Q; answers Q N/A
03/09/2009 Observation 4: Apply understanding of fractions to read and understand ruler markings, including fractional equivalents 4.1 teacher-class
Interpreting meaning
Eva and class look for the most reasonable answer
explaining math ideas; responding to questions; asking questions; using calculators
asking questions; calling on students; responding to questions; revoicing
4.2 teacher-class Interpreting meaning
Teacher questions students about the meaning of 2b
respond to questions; thinking and ideas; note taking; using calculators; reasoning; making conjectures about calulator operations with variable
request students to read 2b without saying, "two bee"; using student work to inform instruction; posing related what if questions; correcting misconceptions related to calculator function
4.3 teacher-class social PS
Students are transitioning to the next part of the lesson
helping and supporting peers to meet Eva's expectations
requests students to help neighbors kindly; encourages student to rely on peers; offer rationale for doing so
4.4 teacher-class Interpreting meaning
Eva evaluates student interpretations of ruler markings as fractional values
interprets ruler mark as a fraction; writes fraction on individual white board; holds white board up for inspection
marks ruler; inspects student responses; reinforces equal spacing within the 1 inch whole
317
Eva’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
4.5 teacher-class T explaining math idea and interpreting meaning
Eva explains how to create equal one-eighth pieces within one inch
watching; answering questions; asking questions; copying drawing from overhead to photcopied ruler image
drawing and coloring on overhead image of a ruler; asking questions; allowing for think time by waiting; revoicing for clarity; providing individual help
4.6 teacher-class Interpreting meaning and reflecting or evaluating
Teacher addresses student misconceptions using iterative formative assessment
incorrectly represent ruler markings using fractions; responds to questions; reflects on fraction understanding; explains why denominator can be set to 8; correctly represents ruler markings using fractions
assesses student understanding; simplifies task; reinforces connection between fractions and ruler markings; guides students with questioning through the process of representing a ruler marking as a fraction; sets denominator to a constant (8); reassesses student understanding; praises student success
4.7 student-student social PS
Student-student help not wanted
Student directing student; back talking; others attempt to intervene; name calling; accompanying Eva to hall; soft wispering during Eva's absence; refocusing attention upon Eva’s return (self-regulation)
invites student to hallway; returning without comment to restart lesson
318
Eva’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
4.8 teacher-class Interpreting meaning and reflecting or evaluating
Eva marks a mixed number on the ruler for students to interpret fractionally and mathematical discussion emerges
fractional representations vary; explain responses; asks mathematical questions related to student solutions
praises students "I like what some of you put;" asks reflective questions about their responses; responds to student questions
4.9 teacher-class Interpreting meaning
Eva draws fourths over eighths on ruler; asks students for another name for 6/8
Some students see 4ths and want to tell before pictorial representation is complete and hands wave with quiet sounds
makes students wait until she finishes making new markings on ruler that show 4ths over the 8ths before responding
319
APPENDIX K: KIA’S CLASSROOM INTERACTION SUMMARY
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
02/24/2009 Observation 1: Using pictorial representations to perform decimal +/- operations 1.1 Teacher-
class; General explaining
Kia explains rules for a game by modeling
listening; aswering Q's; heads resting on hands
Explaining how to play; asking Q's
1.2 Teacher-class Social PS
Students describe rules for behavior during the game
thinking and ideas Asking questions, evaluating responses, adjusting tools based on student ideas (i.e., markers/crayons instead of pencils)
1.3 Teacher-class Math PS
Class plays the decimal number game (object: roll 10-sided number cube, get largest number by filling in one digit per roll)
reads #'s on die, writes #s on their paper, excitedly participate in game
rolls die, selects who reads die, polls to see who has the highest #, validates papers, enforces rules, summarizes strategy for winning
320
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
1.4 Teacher-class Reflecting or evaluating
Determining who won the game
Checking number against teacher voiced criteria; raising hand when number meets criteria; victory shout outs; accusing peers of cheating; talking to neighbors
asking descriptive questions (Who has an 8 in the ten-thousand’s place?), validating papers, collecting winning papers
1.5 Teacher-class Interpreting meaning; Reflecting or evaluating
Introducing the day's lesson & reflecting on the lesson from challenges experienced by first period
looking around (not looking @ Kia); fidgeting; raising hands; shouting out answers
explaining; asking Qs; revoicing; encouraging; reflecting on first period; reflecting on worksheet; pointing out mistakes she made during 1st period
1.6 Teacher-class Interpreting meaning
Students attempt to make sense of the worksheet (representations of +/- of decimal #s using 10X10 grids and no symbols)
making conjectures; raising hands to be called
asking Q's; revoicing; selecting students to speak; encouraging students to listen to peers and not interrupt; managing Discourse flow
321
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
1.7 Teacher-class T explaining math idea
Explains meaning of worksheet & her thinking related to the pictorial representations
raising hands; thinking and ideas; answering Qs
explaining, reflecting on worksheet design; asking leading Q's; explaining subtraction process using the pictorial representation
1.8 Teacher-class T & S Explaining math idea
Subtracting decimals using pictorial representation
thinking and ideas; answering Qs; heads down; yawning
asks leading Qs; selects students; revoicing; answering her own Qs
1.9 Teacher-class S explaining math idea and reflecting or evaluating
Student corrects Kia's computation error
student identifies Kia's mistake; another student defends teacher; student explains mathematical error
stands up for mistake finding student; encourages students to point out mistakes; acknowledges error; revoices correction
1.10 Teacher-class Reflecting or evaluating
Kia reflects aloud about curriculum's approach
fidgeting; head down; bored expressions
Comparing curriculum approach to Kia's past approach for decimal addition; concludes it's better to follow the book b/c of testing
322
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
1.11 Teacher-class Interpreting meaning/ reflecting or evaluating
Kia leads class to interpret subtraction problem then she makes a connection to money
thinking & ideas; respond to Qs; reflect and comment on money connection raised by Kia
showing and drawing on pictorial representation; asking leading Qs
1.12 Student-student Reflecting or evaluating
Students reflect on student's interpretation and collaborate to make sense
student makes a conjecture; reflect on peer's response; evaluate peer's response; offers ideas for sense making
encourages students to reflect on response; asks extending Qs; manages Discourse flow, assesses understanding, revoices
1.13 Teacher-class General explaining
While assigning independent work, Kia struggles with telling
asks clarifying Q; listening; students appear more awake
explaining and clarifying directions; giving hints (e.g., #4 is an addition problem that you have to make up)
1.14 Teacher-class Math PS
Converting improper fraction to mixed number
thinking and ideas; asking Qs; heads down; not paying attention
leads students through process of simplifying improper fraction; validating responses; notices inattention; simplifies problem
1.15 Teacher-group Reflecting or evaluating
Kia explains task, encourages one student to help another
asks Q; listens to hints; wants peer help
clarifies tasks; offers hints; evaluates student work; asks if she wants partner to explain
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Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
03/12/2009 Observation 2: Estimating addition and subtraction of decimal numbers; prepare for state achievement test by solving problems from achievement test practice workbook 2.1 Teacher-
class Social PS
Kia stops student input to focus lesson
asking Q; “we did this” interrupts student; asks redirecting Q
2.2 Teacher-class Teacher & Student explaining math idea
Eva guides students to find prime factorization of 36
thinking and ideas; defining math terms; responding to Qs; eating; revoice; taking notes
Asking leading and fact Qs; writing on the OH; invited student to revoice
2.3 Teacher-class Interpreting meaning
With multiple factor trees written on OH, How can 36 be represented?
responding to Q; looking around for support [lots of math facts & ideas on walls]; waiting; writing/passing notes
Asking probing Qs; encouraging thinking; giving hints; showing validation; asking fact Qs
2.4 Teacher-class Interpreting meaning
Interpreting factor trees as prime factoriaztion
asking Q; responding to Q; thinking & ideas; guessing ; talking & laughing
Asking Qs; giving hints; summarized prime factorization ; correcting off-task behavior
2.5 Teacher-class General explaining
Kia makes an implied math connection - no explicit mention of specific mathematics
guessing to interpret teacher hints; articulating prime factorization using exponents
giving hints; filling in missing operation; making implied connections
324
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
2.6 Teacher-class Interpreting meaning and student explaining math idea
Understanding a story problem by interpreting meaning of range (measure of spread)
thinking and ideas; recalling prior problems; recalling definition and process for finding range; taking notes
telling; refering to info posted on walls about measures of center and spread; validating student responses; asking Qs; offering advice to support thinking to answer Q
2.7 Teacher-class Reflecting or evaluating
selecting the best answer from multiple choices and explaining why.
listening to Kia reading; making a selection; shouting out choice; shouting out reasoning; appearing to not listen to peers - multiple students talking at once
reading each response; asking students to decide if response can be eliminated; telling her reasoning; addressing inapporopriate student behavior; asking Q and not waiting or response
2.8 Teacher-class Student explaining math idea
Explaining how to multiply fractions using Achievement test workbook
reading from notes about multiplying fractions; talking at once; thinking and recollections about multiplying fractions; off-task behavior
revoicing to clarify correct student responses; correcting student behavior; polling students who don't remember procedure; validating student idea; reviewing procedure for multiplying fractions
325
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
2.9 Teacher-class Teacher & Student explaining math idea
Checking Achievement test workbook problems.
talking at once; relfecting and evaluating; heads down; subset of students (6) responding; thinking and ideas; giving answers
validating answers; asking Qs; telling and giving hints; encouraging students to listen to peers; revoicing; making implicit math connections; revoicing
2.10 Teacher-class Teacher & Student explaining math idea and Reflecting or evaluating
Kia corrects teaching error by explaining the directions on the worksheet and discussing rounding
look bored; guessing; talking at once; head down; off-task behavior; telling procedure for rounding; guessing responding to Q
asking students to recall what she told them to do two days ago; waiting; attending to off-task student behavior; asking procedural and fact Qs
2.11 Teacher-class Teacher & Student explaining math idea
Consider solving problems using estimation
thinking and ideas; raising hands before offering ideas; sharing ideas; listening; offering different estimates for the same price; responding to what if Q's;
asking S to estimate prices; selecting students to respond; accepting all estimates; offering her own estimate; asking/aswering Qs; asking what if Qs; pointing out differences between estimating and rounding; sends student out of class
326
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
2.12 Teacher-class reflecting or evaluating
Deciding whether high or low estimates are better for shopping
reflecting and deciding position; reasoning; shouting out reasons
asking Q; polling students; deters students from sharing reasons; giving examples of both sides; asking reflective Qs; settling dispute, telling
2.13 Student-student Math PS
Cooperative group of 3 girls solve an estimation comparison problem from the text
ask for help; read the problem multiple times (individually and in group); thinking and ideas; working to solve indipendently; peer support; discuss answers along the way; compute with decimals; make a comparison
tell them to read the problem
03/16/2009 Observation 3: Problem solving, representing data with a histogram; engage in math talk during PS to improve understanding; and state achievement test preparation and practice 3.1 Teacher-
class Social PS
Kia polls to see who needs pencils before the lesson can begin
talking; raise hands if no pencil; comply with teacher requests to face front and stop talking
Polls students; passes out pencils; asks students to face front in their chairs; attends to student issues
327
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
3.2 Teacher-class Math PS
Coach asks students to read problem that they will solve together using accountable talk.
some reading, others appear off-task and mostly quiet
talking; waiting; read aloud
3.3 Teacher-class Interpreting meaning
Whole class interpreting meaning of problem
identifying key things in the problem (e.g., vocabulary, key words, etc.); raising hands before speaking
asking Qs; recording things students' identified; summarizing student ideas; adding comments (e.g., "hmm, scale has been mentioned twice in this problem")
3.4 Teacher-class Math PS
Creating the histogram and defining intervals
thinking and ideas; playing with rulers; making conjectures about unfamiliar math terms; daydreaming
asking Qs; giving hints about writing on the state test answer form; passing out rulers; defining unfamiliar terms (e.g., interval, histogram, etc.)
3.5 Teacher-class Math PS and interpreting meaning
Continue constructing the histogram and defining label
playing with rulers; thinking and ideas; correcting teacher; making conjectures about what the numbers mean; interrupting peers; talking at once
asking Qs; revoicing; Kia circulating room; accepting correction from student; noticing where students are writing on the answer sheet
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Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
3.6 Teacher-class Math PS and interpreting meaning
Determining what the histogram height means
thinking and ideas; accusing peers of changing their minds; responding to poll; supporting response; coming to consensus on poll
Coach asking Q; Kia clarifying question; Kia telling; Kia Shh students; Kia showing student where to look on paper for answer; polling students; Coach requesting support for response
3.7 Teacher-class Teacher explaining math idea
Kia summarizes the histogram and tells students what the numbers mean prior to asking students
listening; thinking aloud; raising hands to respond;
Kia explaining meaning of histogram; telling students where to look and how to get answers; Coach asking Q; Kia shushing loud thinking; instructing students to raise hands with responses
3.8 Teacher-class Reflecting or evaluating
reflecting on histogram and determining if it meets the criteria specified by the question
reflecting; thinking and ideas (i.e., provide meaningful title);
asking Q; refocusing attention on problem criteria identified at start
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Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
3.9 Teacher-class Teacher explaining math idea
Students are asked to make sense of the state achievement test scoring rubric for the problem they just solved (histogram of restaurant ratings)
reading aloud; side talking; playing with rulers
asking definition Q, asking evauating Q and answering; describing rubric and its use; explaining how the class' work meets 2 pts.; selecting students for participation; comparing rubric scoring criteria; Kia talking to students as others are reading
3.10 Teacher-class Reflecting or evaluating
Student reflect on three student work samples and evaluate each using the scoring rubric
looking at student work; justifying score; thinking and ideas; communicating; comparing student work samples to their own work; tapping rulers on desks; shouting out; abstaining from polling; using justifications not on the rubric (i.e., sloppy)
asking students to score student work samples; giving hints; polling students for scores; encouraging students to listen to peers; explaining scores from state; declaring consensus with limited or no participation; refocusing students to use rubric for justification
3.11 Teacher-class Teacher explaining math idea
Introducing the problem students are to solve, a similar problem
responding to Qs; looking bored; playing with rulers;
attending to inappropriate use of ruler; telling; asking leading Qs
330
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
3.12 Teacher-class Student-student Math PS
Students work on solving the problem, creating a histogram
students working; sneakily working as Kia gives hints; seeking validation; calling Kia to solve problems; helping peer; sitting with nothing to do
waiting; giving hints; reminding students of her thinking; stopping work; asking Qs; encouraging students to create 2 pt. histograms; showing; encouraging; supporting; saying answers aloud; congratulating
3.13 Teacher-class Reflecting or evaluating
Kia asked students to reflecto on their work and evaluate their score according to the scoring rubric
some raised hands in response to polling; most opted not to particpate; continued doing quiet other stuff (e.g., writing notes, etc.)
asked Q; polled student responses for 2pt and 1 pt; kicked student out for talking
3.14 Teacher-class Teacher and student explaining math idea
Teacher reviews problem solution
guessing for fill-in Qs; comparing teachers' graph to their own graph; talking at once; off-task behavior (heads down, talking, back to teacher, playing with rulers, etc.); thinking and ideas; identifyng what's missing for 2 pt. score; asking Qs; defining purpose of histogram
Asking fact Qs (e.g., what are the intervals?) and fill-in; scolding students for inattention; telling; encouraging hand raising
03/17/2009 Observation 4: Review histograms and state achievement test preparation
331
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
4.1 Teacher-class Teacher and student explaining math idea
What is a histogram?
thinking and ideas; explaining; writing & passing notes; describing pieces and parts of histogram; comparing to bar graphs
asking probing Qs; addressing student issue; revoicing; recording students' ideas
4.2 Teacher-class Interpreting meaning
Introducing a worksheet about creating histograms
talking; thinking and ideas; 1 or 2 students responding to Qs;
Describing the data; asking definition Qs; encouraging students to participate (2); revoicing; giving hints for definitions; asking students to complete worksheet
4.3 Teacher-class Interpreting meaning
Using the frequency table and histogram to answer questions about the situation
answering Qs using chart and histogram; participating; talking at once; answering Qs before asked; copying histogram bars; thinking and ideas; reading notes; playing games
askig Qs; revoicing; connecting situation to her life; attending to inappropriate behavior (4); asking fill-in the blank Q; hearing selectively (only right answers); drawing histogram bars; telling students to copy
4.4 Teacher-class Interpreting meaning
Introducing a worksheet about interpreting histograms
helplessness; thinking and ideas for starting
encouraging best effort; explaining what to do for each question; suggestions for starting
332
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
4.5 Teacher-class Interpreting meaning
Going over worksheet
Sharing answers and explaining; collaborative sense making; students abstaining from polling; explaining answers/strategies
asking Qs; revoicing; expanding on student input; solving problems; stating listening equates to understanding; polling students for answers; doing most of the thinking
03/18/2009 Observation 5: Getting caught up (with pacing), including doing an assignment for her class (i.e., accountable talk). Using estimation for problem solving using decimals and fractions. 5.1 Teacher-
class Math and social PS
Using estimation to solve a problem about buying candy. What's the first thing you need to do?
flipping through books; covering face; working on homework for another class; not listening; guessing what peer said; listening
creating problem; writing problem; revoicing; asking for paraphrase; assessing listening; acknowledging no evidence of listening; asking for listening; giving rationale for listening (i.e., "awesome ideas")
333
Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
5.2 Teacher-class Math PS and reflecting/evaluating
Using the best estimates to decide how much candy bars cost
more students participating (4-6 students); reading; face covered; thinking and offering alternate input;
asking for alternate estmates, students pick best; acknowledging all input
5.3 Teacher-class Math PS and reflecting or evaluating
Using reasonable estimates to decide if you have enough money for candy
thinking and ideas; performing arithmetic calculating; working on other things; face covered;
asking questions; acknowledging input; asking for reasonable estimates; writing answer; guiding process; asking and answering Qs; requesting authentic mathematical descriptions
5.4 Teacher-student Reflecting or evaluating
Extreme case of student choosing abstinence from learning
down and writing; looking at teacher with no response; refusing to acknowledge teacher and Qs
assessing student understanding; noticing non-participating student; engaging all students; asking explicit Q to non-participating student; refocusing on class
5.5 Teacher-class Teacher and student explaining math idea
Estimating fractions using a pizza analogy
thinking and offering ideas in context; responding to questions; heads down writing; covering face; explaining
explaining by analogy; asking questions; drawing representations; acknowledging input; attending to inappropriate behavior; soliciting input
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Kia’s Classroom Interactions
ID Interaction
Structure & Purpose
Interaction Description
Student Actions Teacher Actions
5.6 Student-student Social PS
Student copying peer's paper in teacher’s absence
talking; not participating; taking paper from neighbor; neighbor allows paper to be taken; copying paper
N/A
5.7 Teacher-student Social PS
Student perpetrates participation
holding up copied paper; evidencing participation; explaining attentivness; raising hand; sharing answers
asking student to put non-math things away; asking for answers
5.8 Teacher-class Reflecting or evaluating
Polling class for understanding
not responding; not responding again
asking Q; reasking Q; responding to poll Q
5.9 Teacher-class Interpreting meaning and Math PS
reading story problem and identifying important information for solving
thinking and ideas; estimating for easy calculations; explaining; generating number sentence representations; explaining in context of situation
telling; connecting pretzel dough to something students know (i.e., Aunt Annnie's or play dough); giving hints; explaining the mathematical situation