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Mathematical Defining As a Practice: Investigations Of MATHEMATICAL DEFINING AS A PRACTICE: INVESTIGATIONS OF CHARACTERIZATION, INVESTIGATION, AND DEVELOPMENT By Marta Kobiela Dissertation Submitted to the Faculty of the Graduate School of Vanderbilt University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Learning, Teaching, and Diversity December, 2012 Nashville, Tennessee Approved: Professor Richard Lehrer Professor Leona Schauble Professor Ilana Horn Professor Philip Crooke To my advisor, Rich Lehrer, for providing infinite support and guidance and To my parents, Bogda and Paul Kobiela, for always believing in me ACKNOWLEDGEMENTS I would first like to thank Dr. Richard Lehrer, the chairman of my committee, for his never-ending support and guidance. As my teacher and mentor, he has taught me more than I could ever give him credit for. I would also like to thank the members of my Dissertation Committee, Dr. Leona Schauble, Dr. Ilana Horn and Dr. Philip Crooke, for their constructive feedback and guidance. I am grateful to all of the graduate students and post docs, too many of whom there are to list here, who have provided me feedback along the way. I would also like to thank friends and family who supported me during this time, including my parents, Paul and Bogda, my brother, Sebastian, and Deb Lucas. iii TABLE OF CONTENTS Page DEDICATION ..................................................................................................................... ii ACKNOWLEDGEMENTS ............................................................................................... iii LIST OF TABLES ............................................................................................................. vii LIST OF FIGURES ......................................................................................................... viii Chapter I. INTRODUCTION ................................................................................................... 1 References ............................................................................................................... 5 II. MATHEMATICAL DEFINING AS A CLASSROOM PRACTICE: A REVIEW .................................................................................................................. 6 Introduction ...................................................................................................... 6 Theoretical perspectives ................................................................................... 9 Disciplinary perspectives on definitions and defining .............................. 9 Supporting classroom disciplinary practice ............................................ 16 Method ............................................................................................................ 20 Inclusion criteria ...................................................................................... 20 Procedure ................................................................................................. 21 Results of Review ........................................................................................... 27 Overview of results ................................................................................. 28 Nature of mathematical defining in classrooms ...................................... 28 Potential of mathematical defining in classrooms .................................. 51 Supporting defining ................................................................................. 57 Discussion ...................................................................................................... 70 References ...................................................................................................... 73 III. ESTABLISHING A MATHEMATICAL PRACTICE IN A MIDDLE SCHOOL CLASSROOM ...................................................................................... 77 Introduction .................................................................................................... 77 Characterizing defining as a practice ............................................................. 78 Disciplinary perspectives on definitions and defining ............................ 78 A framework for analyzing defining in classrooms: Aspects of definitional practice ................................................................................. 79 iv Method ............................................................................................................ 81 Participants, setting and data collection .................................................. 82 Analysis ................................................................................................... 83 Establishing the co-constitution of practice and knowledge .......................... 85 Discussion ...................................................................................................... 87 References ...................................................................................................... 88 IV. CHARACTERIZING AND SUPPORTING PRACTICES OF DEFINING IN A MATHEMATICS CLASSROOM ..................................................................... 90 Introduction .................................................................................................... 90 Characterizing defining as a practice ............................................................. 91 Disciplinary perspectives on definitions and defining ............................ 91 A framework for analyzing defining in classrooms: Aspects of Definitional Practice ................................................................................ 92 Method ............................................................................................................ 95 Participants, setting and data collection .................................................. 95 Analysis ................................................................................................... 97 Establishing the co-constitution of practice and knowledge .......................... 99 Posing question that elaborated system components .............................. 99 Provoking contest through the generation of examples and non-examples ........................................................................................ 103 Keeping definition at the forefront ........................................................ 105 Discussion .................................................................................................... 105 References .................................................................................................... 107 V. INVESTIGATING THE CO-DEVELOPMENT OF MATHEMATICAL KNOWLEDGE AND THE PRACTICE OF DEFINING IN A MIDDLE SCHOOL CLASSROOM .................................................................................... 109 Introduction .................................................................................................. 109 Theoretical Perspectives ............................................................................... 111 Characterizing defining as a practice .................................................... 111 A situational approach to learning ........................................................ 113 Supporting disciplinary practice in classroom discussions ................... 114 Method .......................................................................................................... 119 Instructional context .............................................................................. 119 Data collection procedure ..................................................................... 122 Analysis ................................................................................................. 123 Results .......................................................................................................... 156 Overview of results ...................................................................................... 156 Excerpt 1: Initial forms of practice and knowledge .............................. 159 Excerpt 2: Students take on authority for expanding the mathematical system .................................................................................................... 165 Excerpt 3: Student positioning of definition at the fore ........................ 173 Excerpt 4: Student agents in orchestrating defining ............................. 181 v Other contributions to creating a culture of defining ............................ 191 Discussion .................................................................................................... 196 References .................................................................................................... 201 VI. CONCLUSION ................................................................................................... 204 Appendix A. TRANSCRIPTS ................................................................................................... 209 vi LIST OF TABLES Table Page Chapter 2 1. Overview of selected works for the review ........................................................... 22 2. Aspects of the Practice of Defining ....................................................................... 33 3. Occurrence of Aspects of Definitional Practice among the reviewed works ........ 52 Chapter 5 1. Definitional Episodes from days 1, 4, 6, and 26 .................................................. 128 2. Coding scheme for engaging in Aspects of Definitional Practice ....................... 138 3. Coding
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