The Position of the Student in Geometry Instruction: A Study from Three Perspectives by Wendy Rose Aaron A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Education) in The University of Michigan 2011 Doctoral Committee: Associate Professor Patricio G. Herbst, Chair Professor Magdalene Lampert Professor Karen E. Smith Assistant Professor Elisabeth A M De Groot © Wendy Rose Aaron 2011 Acknowledgements I would like to express my gratitude to all the individuals who have helped me in this endeavor. First, I’d like to acknowledge the honesty and generosity of the teachers and students who participated in these studies. They each shared significant amounts of their time and openly shared their views and feelings about mathematics teaching and learning. They welcomed me into their classrooms and showed me a glimpse of their lives. My dissertation committee, Pat Herbst, Magdalene Lampert, Liz De Groot, and Karen Smith, guided this work along with my academic growth over the past several years. Each member of the committee has contributed differently to the dissertation and to my edification. From Pat I have learned the power of combining rigorous theoretical grounding with a playful imagination and a deep respect for the profession of teaching. Magdalene has guided me in understanding and appreciating the richness and complexity of teaching. Liz has been my guide through the world of educational psychology and aided me in thinking about the art of interviewing. Karen has pushed me to see educational research through a mathematician’s eyes and carefully consider the claims that I am making. I am confident that my time working with each of these scholars will affect my work for the rest of my career, for that I am extremely privileged. ii This dissertation was written within the GRIP research project, alongside many talented colleagues. I am fortunate to have worked with Pat, Talli, Takeshi, Chieu, Jeonglim, Karl, Emina, Michael, Gloriana, Chialing, Angela, Justin, Ander, Annick, Manu, Katie, Kayin, Chrissy, Gina, Jack, Travis, and Mindy. I am proud of the scholar that I am today and I have these colleagues to thank. I am especially fortunate to have formed friendships within GRIP. Gloriana, Chialing, and Angela have become dear friends. They have mentored me and provided invaluable support, both intellectually and emotionally. The school of education is full of wonderful people and I am lucky to count some of them as friends. Laurie, Hala, Sarah, Amy, Emily, Debi, Melissa, Terri, Francesca, and Meghan have been by my side through dark Michigan winters and joyful springs. I would not have wanted to do this without them. My parents and brother are an unending source of love and support. It is because of them that I am Alive, Alert, Healthy, Creative, and Vibrant. My husband, Kayin, is my hero and friend and is the reason that I was able to complete this degree. I am thrilled to have him on my team and I am equally trilled to be on his. iii Table of Contents Acknowledgements ........................................................................................................... ii List of Figures.................................................................................................................. vii List of Tables .................................................................................................................... ix Abstract............................................................................................................................. xi Chapter 1: Introduction .................................................................................................. 1 The three perspectives........................................................................................................................4 The participants...................................................................................................................................6 The data collection tool ......................................................................................................................7 Description of animated scenarios...............................................................................................................7 Description of The Square...........................................................................................................................11 Features of The Square.................................................................................................................................19 Chapter 2: Mathematical Arguments in a Virtual High School Geometry Classroom ................................................................................................................... 24 Relevant literature ........................................................................................................................... 29 Literature on proof in classrooms .............................................................................................................30 The generation of mathematical knowledge: A review of relevant literature...........................46 Definition of proving.....................................................................................................................................55 Data ..................................................................................................................................................... 56 Description of data.........................................................................................................................................56 Map of the argument in the animated scenario.....................................................................................60 Possible mathematical work done in response to the angle bisectors problem .........................65 Available resources........................................................................................................................................68 Methods .............................................................................................................................................. 69 Analysis of mathematical arguments.......................................................................................................71 Earlier uses of Toulmin’s argument model............................................................................................73 Coding of the arguments..............................................................................................................................73 Analysis of mathematical territory ...........................................................................................................78 Results................................................................................................................................................. 79 Maps of arguments.........................................................................................................................................79 Map of content.............................................................................................................................................. 122 Discussion.........................................................................................................................................126 Disciplinary agency of the two-column proof ................................................................................... 127 Arguments that contain chains of implications ................................................................................. 128 Arguments that do not contain rebuttals or backings ...................................................................... 129 Arguments that are compound ................................................................................................................ 129 Arguments that contain congruent triangles....................................................................................... 130 Arguments that cover mathematical territory .................................................................................... 130 Different modes of argumentation......................................................................................................... 131 Conclusion........................................................................................................................................132 References ........................................................................................................................................136 iv Chapter 3: Teachers' Perceptions of Geometry Students........................................ 140 Past research on teachers’ perceptions of students ................................................................143 Researcher generated categories ............................................................................................................ 144 Teacher generated categories .................................................................................................................. 148 Motivation for study ......................................................................................................................154 Theoretical framework .................................................................................................................158 Model of classroom interaction .............................................................................................................