Teaching Mathematical Knowledge for Teaching

306 Pages·2013·3.18 MB·English

Teaching Mathematical Knowledge for Teaching:
Curriculum and Challenges

by

Yeon Kim

A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
(Education)
in The University of Michigan
2013

Doctoral Committee:

Professor Deborah Loewenberg Ball, Chair
Professor Hyman Bass
Professor Edward A. Silver
Research Scientist Mark Hoover Thames
© Yeon Kim
2013
ACKNOWLEDGEMENTS
I enjoyed every step of the way, but the work was not easy. Without amazing
guidance and encouragement, I could not have finished it at all.
First and foremost, I would like to express my gratitude to my committee
members. My advisor, Deborah Loewenberg Ball, has given me guidance, support, and
motivation throughout my doctoral studies and this dissertation. Without her
encouragement, it would not have been possible. I thank Mark Thames for his detailed
feedback and helpful suggestions at each stage of my research and on all my drafts. The
quality of my research has greatly improved through his valuable and endless comments
and guidance. I would also like to thank Ed Silver for his insightful questions and
comments. His guidance was influential in clarifying the process of my research and
helping me always keep in mind the contribution of my work to the field of mathematics
education. I also thank Hy Bass for his fundamental perspective of both mathematics and
teaching practice. He has shaped my thinking with his comments. I also have had the
opportunities to work with him and learned a lot his care in thinking as a researcher. All
of their advice and comments were invaluable to the success of this dissertation.
I gratefully acknowledge the kind and warm support from courses and projects in
which I participated at the University of Michigan. Thank you to Vilma Mesa for her
warm and kind concerns for me and to Kara Suzuka for her IRB help and for beginning
this research through the mod4 project. I also thank everyone in suite 1600, which was
always full of kind people.
I was fortunate to have the financial support of several organizations: the mod4
project, the Mathematics Teaching and Learning to Teach project, Rackham Graduate
School, and the School of Education. I am grateful to have had this support. In addition,
I would like to thank the nine participants in this research. Without them, this research
could not have been conducted.
I would never have survived graduate school and finalized my writing without the
support of my colleagues and friends, particularly, Yvonne, Shweta, Yaa, Jenny, Hyun-
Ju, Shanta, Cathy, Lok-Sze, and Anita. They have been a crucial part of my life at Ann
Arbor. Many thanks to all of them for unforgettable years.
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Finally, I would like to express my loving appreciation to my husband, Changsun
Ahn, my parents, and my younger brother. Their sustained love and support made this
work possible. I have just turned one corner of my life; the next has already begun.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS ................................................................................................ ii
LIST OF TABLES ........................................................................................................... viii
LIST OF FIGURES ............................................................................................................ x
LIST OF APPENDICES .................................................................................................. xiii
ABSTRACT ..................................................................................................................... xiv
CHAPTER 1 THE RESEARCH PROBLEM ..................................................................... 1
1.1 Introduction .............................................................................................................. 1
1.1.1 Purpose of the study ........................................................................................ 3
1.2 Overview of Study ................................................................................................... 5
1.2.1 Premise of the study ........................................................................................ 5
1.2.2 Research questions .......................................................................................... 8
1.2.3 MKT in teacher education ............................................................................. 11
1.2.4 Contribution to the field ................................................................................ 12
1.3 Organization of the Dissertation ............................................................................ 13
CHAPTER 2 FOUNDATIONS IN LITERATURE ......................................................... 14
2.1 Introduction ............................................................................................................ 14
2.2 Dynamics in Teaching Phenomena and Attention ................................................. 15
2.2.1 What instructors attend to in the classroom .................................................. 15
2.2.2 Instruction and its components ..................................................................... 17
2.2.3 Dynamics of teaching in mathematics teacher education ............................. 19
2.3 Curriculum in Mathematics Teacher Education .................................................... 25
2.3.1 Content for mathematics teacher education .................................................. 26
2.3.2 Mathematical knowledge for teaching (MKT) ............................................. 28
2.4 Attributes of Mathematical Definitions ................................................................. 32
2.4.1 Mathematical definitions in teaching practice .............................................. 32
2.4.2 Attributes of mathematical definitions in disciplinary mathematics ............ 34
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2.5 Mathematical Work in Teaching ............................................................................ 42
2.5.1 Work of teaching in educational research ..................................................... 43
2.5.2 Work of teaching related to mathematical definitions .................................. 46
CHAPTER 3 METHODS OF DATA COLLECTION AND ANALYSIS ...................... 48
3.1 Introduction ............................................................................................................ 48
3.2 Data Collection....................................................................................................... 49
3.2.1 Curriculum materials ..................................................................................... 49
3.2.2 Video recordings ........................................................................................... 53
3.3 Data Analysis ......................................................................................................... 60
3.3.1 Investigating research question: Developing a framework for
mathematical work of teaching ..................................................................... 60
3.3.2 Investigating research question: Developing a framework for
knowledge about mathematics ...................................................................... 69
3.3.3 Investigating research question: Conceptualizing what challenges
teacher educators face and what teacher educators need to pay
attention to in teaching MKT ........................................................................ 70
3.4 Limitations of Study ............................................................................................... 71
CHAPTER 4 THE COMPLEXITY OF TEACHING MATHEMATICAL
KNOWLEDGE FOR TEACHING .................................................................. 74
4.1 Introduction ............................................................................................................ 74
4.2 Lesson from Curriculum Materials on Hearing Definitions in Children’s
Talk ......................................................................................................................... 75
4.3 Matthew’s Lesson on Evaluating Definitions ........................................................ 94
4.4 Emily’s Lesson on Explaining Why the Units Digit Rule Works ....................... 114
CHAPTER 5 CHALLENGES OF TEACHING MATHEMATICAL
KNOWLEDGE FOR TEACHING ................................................................ 127
5.1 Introduction .......................................................................................................... 127
5.2 Establishing the Focus of an Activity .................................................................. 130
5.2.1 Giving and specifying an activity ............................................................... 131
5.2.2 Helping teachers understand an intended activity ....................................... 135
5.2.3 Identifying purposes of an activity .............................................................. 139
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5.3 Recognizing Mathematical Issues ........................................................................ 140
5.3.1 Identifying and acknowledging teachers’ ideas .......................................... 141
5.3.2 Responding to help teachers have appropriate ideas................................... 142
5.3.3 Identifying mathematical features of teachers’ statements ......................... 143
5.4 Focusing on Mathematical Articulation and Interpretation ................................. 144
5.4.1 Specifying mathematical terms, facts, and relations ................................... 145
5.4.2 Identifying mathematical objects ................................................................ 146
5.4.3 Identifying mathematically appropriate ideas and interpretation................ 147
5.4.4 Using examples for mathematical investigation ......................................... 148
5.5 Emphasizing Ways of Mathematical Thinking .................................................... 150
5.5.1 Asking key questions repeatedly and intentionally ..................................... 151
5.5.2 Explaining mathematical practice ............................................................... 152
5.5.3 Giving comments for teaching practice ...................................................... 153
5.6 Managing Mathematical Ideas ............................................................................. 154
5.6.1 Situating teachers on or away from the context of K-9 classrooms ............ 155
5.6.2 Staying on a mathematical topic ................................................................. 156
5.6.3 Giving mathematical assistance .................................................................. 158
5.6.4 Engineering engagement in mathematical practice..................................... 159
5.6.5 Having mathematical consistency ............................................................... 161
5.6.6 Doing manipulative preparation .................................................................. 162
5.7 Conclusion ........................................................................................................... 164
CHAPTER 6 FRAMEWORK FOR CURRICULUM TO TEACH
MATHEMATICAL KNOWLEDGE FOR TEACHING .............................. 165
6.1 Introduction .......................................................................................................... 165
6.2 Conceptualization of Mathematical Work of Teaching as Content in
Mathematics Teacher Education .......................................................................... 168
6.2.1 Zoomed-out mathematical work of teaching .............................................. 170
6.2.2 Zoomed-in mathematical work of teaching ................................................ 188
6.3 Conceptualization of Knowledge about Mathematics as Content in
Mathematics Teacher Education .......................................................................... 230
6.3.1 Disciplinary facts and structures ................................................................. 231
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6.3.2 Mathematical awareness ............................................................................. 234
6.3.3 Mathematical value ..................................................................................... 239
6.4 Conclusion ........................................................................................................... 242
6.4.1 Potential contributions and uses .................................................................. 244
6.4.2 Limitations .................................................................................................. 246
CHAPTER 7 MATHEMATICAL KNOWLEDGE FOR TEACHING IN
TEACHER EDUCATION: CONCLUSIONS AND THE NEXT
STEP .............................................................................................................. 248
7.1 Summary of Dissertation ..................................................................................... 248
7.2 Potential Contributions to Research in Education ............................................... 252
7.2.1 Talking about MKT in terms of mathematics teacher education ................ 252
7.2.2 Managing MKT in mathematics teacher education .................................... 253
7.2.3 Having a foundation to set up a curriculum to teach MKT......................... 254
7.3 Next Steps: Beginning this Line of Work ............................................................ 255
7.3.1 Using the framework to analyze the curriculum materials and
teacher educators’ lessons to teach MKT .................................................... 255
7.3.2 Using the framework to analyze different curriculum materials and
diverse teacher educators’ lessons to teach MKT ....................................... 256
7.3.3 Studying both MKT in terms of teacher education and MKT in the
practice of teaching ..................................................................................... 257
7.3.4 Developing a tool for teacher educators’ reflection .................................... 258
BIBLIOGRAPHY ........................................................................................................... 273

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LIST OF TABLES
Table 3.1 Lessons in the Curriculum Material, Using Definitions in Learning
and Teaching Mathematics ............................................................................ 51
Table 3.2 Teacher Educators’ Major and Teaching Experience ...................................... 54
Table 3.3 Teacher Educators’ Classes ............................................................................. 54
Table 3.4 Teacher Educators’ Experience of Teaching MKT .......................................... 55
Table 3.5 Participants and Lessons that were Observed for This Research .................... 56
Table 3.6 Participants and Video Recordings .................................................................. 58
Table 5.1 Summary of the Establishing of the Focus of an Activity ............................... 140
Table 5.2 Summary of the Recognizing Mathematical Issues......................................... 144
Table 5.3 Summary of the Focusing on Mathematical Articulation and
Interpretation ............................................................................................... 150
Table 5.4 Summary of the Emphasizing Ways of Mathematical Thinking ..................... 154
Table 5.5 Summary of the Managing Mathematical Ideas ............................................. 163
Table 6.1 The Basic Structure of Zoomed-Out Mathematical Work of Teaching .......... 171
Table 6.2 Specification of Making Decisions regarding One Lesson ............................. 174
Table 6.3 Specification of Making Decisions regarding a Long Period ........................ 178
Table 6.4 Specification of Unpacking Mathematics ideas .............................................. 181
Table 6.5 Specification of Situating in Mathematics ...................................................... 184
Table 6.6 Specification of Using Language Mathematically .......................................... 187
Table 6.7 The Basic Structure of Zoomed-In Mathematical Work of Teaching ............ 189
Table 6.8 Specification of Recognizing and Articulating Mathematical Objects
in Teaching and Its Examples ...................................................................... 196
Table 6.9 Specification of Probing, Interpreting and Comparing Mathematical
Objects in Teaching and Its Examples ......................................................... 205
Table 6.10 Specification of Evaluating and Judging Mathematical Objects in
teaching ........................................................................................................ 214
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Table 6.11 Specification of Selecting and Modifying Mathematical Objects in
teaching ........................................................................................................ 220
Table 6.12 Specification of Constructing Mathematical Objects in Teaching ............... 227

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