Unfolding of Diagramming and Gesturing between Mathematics Graduate Student and Supervisor during Research Meetings by Petra Margarete Menz B.Ed., University of British Columbia, 1995 M.Sc., University of British Columbia, 1994 B.Sc., University of Toronto, 1992 Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in the Department of Mathematics Faculty of Science Petra Margarete Menz 2015 SIMON FRASER UNIVERSITY Summer 2015 Approval Name: Petra Margarete Menz Degree: Doctor of Philosophy (Mathematics) Title: Unfolding of Diagramming and Gesturing between Mathematics Graduate Student and Supervisor during Research Meetings Examining Committee: Chair: Ralf Wittenberg Associate Professor Jonathan Jedwab Senior Supervisor Professor Nathalie Sinclair Co-Supervisor Professor David Pimm Internal/External Examiner Adjunct Professor Faculty of Education Ricardo Nemirovsky External Examiner Professor Mathematics and Statistics San Diego State University Date Defended/Approved: July 27, 2015 ii Ethics Statement iii Abstract This study has two main purposes. The first is to confirm and advance Gilles Châtelet’s account of the role of diagramming in mathematics invention by extending his results, which were based solely on historical, mathematical manuscripts, to the context of live mathematical activity. The second purpose is to elucidate the enculturation process of a graduate student into mathematical research, which has hitherto received limited research attention. These two purposes are related through the virtual and physical gestures that the graduate student engages in during diagramming thereby providing insights into mathematical invention and the enculturation process. This study adopts a qualitative methodology based on field notes, video-recordings and digital images from nine research meetings, which were held weekly over a period of three months. Current research theorises the role of mathematical diagrams in diametrically opposed ways: diagrams are either a visual representation of already existing mathematical objects and relations, or they are the means through which mathematical objects and relations emerge. The latter is due to Châtelet, who regards the diagram as a material site of engaging with and mobilizing the mathematics. His approach is employed in this thesis to create a window into the realm of mathematical thinking and invention by examining how a graduate student (as the less-expert mathematician) and his supervisor and two research colleagues (as the expert mathematicians) interact with diagrams. An embodied lens, based on the work of de Freitas, Roth, Rotman, Sinclair and Streeck, exposes the similarities and differences in the way that each class of mathematician gestures and diagrams. The analysis in this thesis reveals that gesturing and diagramming support and advance mathematical communication throughout the graduate student’s enculturation process. Furthermore, a collective study of the abundant diagrams produced during research meetings leads to a life-cycle of diagrams, whose phases disclose a variety of distinct relationships between mathematician and diagram. Lastly, a detailed examination of the evolution of a particular diagram uncovers how mathematical invention emerges through gesturing and diagramming. These findings have implications for the teaching and learning of mathematics at all levels. Keywords: diagram, diagramming, embodiment, enculturation, gestures, graduate student, materialism, mathematical research, mathematizing, virtuality. iv Dedication For my family, because they are my roots that give me strength and let me blossom. “But without my family, nothing – not even the beauty of mathematics – would have any meaning at all.” (Zeitz, 1998, p.xii) v Acknowledgements I want to take this opportunity to voice my gratefulness and appreciation to the people who supported and guided my doctoral studies. Foremost, I want to thank my son, Eli, who only had a partial mother during this personal journey. May he find his own journey in life, equally enriching and empowering. I thank and acknowledge my husband, Dr. John Stockie, for supporting my research endeavours, for squeezing my edits into his prolific research life, and for being willing to take a dive into the world of research in mathematics education. Meine Schwester Manuela und ihre Familie, meine Mutti und mein Papa haben mich die Jahre hindurch mit ihrem Verständnis und Gesprächen von nah und fern mit Liebe unterstützt. Dafür bin ich von Herzen dankbar. I thank my senior supervisor, Dr. Jonathan Jedwab, and co-supervisor, Dr. Nathalie Sinclair for agreeing to oversee my joint doctoral thesis in Mathematics and Mathematics Education. I am indebted to both of you for your encouragements and unwavering support, your understanding in light of my illness, your professional expertise and guidance, and your prompt and insightful feedback that helped me to become a more effective writer. I acknowledge the members of Dr. Sinclair’s seminar group, particularly research fellow Oi-Lam Ng, whose thoughtful feedback and edits have challenged me to be critical of my work and to analyse my results in more depth. I thoroughly appreciated all of our meetings, presentations and discussions. Cindy and Mary, thank you for the smiles, hugs and encouragements, and the many dinners to nourish not only my body but also my soul. I express gratitude to the mathematikoi – the participants in my study – particularly the graduate student who answered so many of my questions regarding his research, provided his research description, and shared with me his diagrams in their nearly published form. Lastly, I am grateful for the yoga practices lead by Alix and Delia, who restored my spirit and my body on countless occasion with their technical expertise and wisdom. Lokāḥ samastāḥ sukhino bhavanthu – may we all be happy, healthy and at peace. vi Table of Contents Approval .............................................................................................................................ii Ethics Statement ............................................................................................................... iii Abstract .............................................................................................................................iv Dedication ......................................................................................................................... v Acknowledgements ...........................................................................................................vi Table of Contents ............................................................................................................. vii List of Tables ..................................................................................................................... x List of Figures....................................................................................................................xi Transcription Glossary .....................................................................................................xv Gesturing and Diagramming ........................................................................................... xvi Chapter 1. Introduction ............................................................................................... 1 1.1. Math and Me ............................................................................................................ 1 1.2. Pivotal Moment ........................................................................................................ 6 1.3. Thesis Outline .......................................................................................................... 8 Chapter 2. Three Key Elements of a Mathematical Research Meeting ................. 10 2.1. Introduction ............................................................................................................ 10 2.2. Mathematization ..................................................................................................... 10 2.3. Graduate Student and Supervisor: The Research Mathematician-to-be and the Research Mathematician .................................................................................. 20 2.3.1. Becoming a Researcher ........................................................................... 21 2.3.2. Becoming a Mathematician ....................................................................... 28 Chapter 3. Gestures and Diagrams .......................................................................... 36 3.1. Introduction ............................................................................................................ 36 3.2. Gesturing ................................................................................................................ 37 3.3. Diagramming .......................................................................................................... 45 3.3.1. The Diagram as a Visual Product ............................................................. 47 3.3.2. The Diagram as the Virtual Being Actualized ........................................... 52 3.4. Research Questions ............................................................................................... 64 Chapter 4. Methodology ............................................................................................ 68 4.1. Participants ...........................................................................................................