Chapter 10: More than Mere Handwaving: Gesture and Embodiment in Expert Mathematical Proof

For the budding mathematician, the ability to comprehend and produce rigorous proofs marks the transition from a novice student or consumer of mathematics to a more expert practitioner. By the time a student begins graduate school in mathematics, he or she is capable of both understanding and creating formal proofs of known mathematical results and is expected to generate new proofs and theorems. Often the focus is on the proofs themselves, the stable, shared outcomes of a process of proving.

In this chapter, by contrast, the focus is on proof as an active practice carried out by knowledgeable mathematicians, looking specifically at the relationship among gesture, speech, thought, and technical definitions. This case study is part of a larger ongoing research project on expert proof, in which we focus on how advanced students of mathematics conceptualize proof, how they collaborate to produce a proof, and how they coordinate multiple modalities during their practice, modalities that include speech, gesture, and inscriptions, both symbolic and graphic.¹ Our general thesis is that these resources are deployed in a complex but coherent manner to construe and scaffold the building of the desired intellectual product, the mathematical proof.