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.
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