Mathematics lecturing: Gestural practices in contexts

The purpose of this research is to examine the actions of an instructor in the undergraduate mathematics classroom over a full term of lecturing on abstract algebra. A micro-ethnographic, natural history approach was adopted, guided by the Natural History of an Interview project, and especially influenced by the application of such a methodology by Jürgen Streeck. The analytic framework owes much to George Herbert Mead's focus on the act, on the necessity of social interactions for the emergence of meaning, mind, and the self, and on the critical importance of gestures in meaningful interaction; together with Gregory Bateson's focus on metacommunication and the creation of contexts in interaction; synthesized with Streeck's analysis of gestures as a human praxis engaging directly and actively with a material world. Thirty-five lectures were video-recorded, transcribed and summarized in multiple ways. Contexts, and the gestural practices achieving mathematical ends that occurred within them, were analyzed. Lecturing was found to be segmented, using a constellation of bodily resources, into local contexts: stanzas and lines. Six varieties of gestural practice were found. Manipulating the object: interacting, using the hands, with a physical, textual, or imagined object. Looking at side-by-side: moving back and forth between two pieces of writing, handling each in turn. Regarding as: expressing with the body and hands the manner in which some writing is to be considered. Deducing that: touching multiple pieces of writing to figure out what ought to be written next. Communicating about: stepping back from the writing action, gesturing and speaking about writing to come or that was just finished. Correcting self and others: occasions when the lecturer interrupts themselves, or an interactant fixes an ongoing mathematical action. The structure and function of these gestural practices, alone or in combination, were studied. Three broad mathematical situations were considered: the three lectures on isomorphisms; the appearances of the mathematical object D4, namely the group of symmetries of the square; the appearances of the notion of well-definedness.