| Mathematical conversation is offered in this work as a means to question underlying assumptions and broaden current perceptions of mathematical practice and discourse in mathematics education. Conversation is not defined here as simply a verbal exchange, but involves dwelling with or 'turning' with others to bring forth a reality of mutual significance. An enactivist view of conversation, drawn primarily from the work of Maturana, Gadamer and Bakhtin, orients attention to mathematical activity as it occurs in interaction and through relationships.;Mathematical conversation cannot be described under a metaphor of progress towards a pre-specified goal as is often expected in mathematical activity; instead, conversation appears to resemble continuous play. Through play the participants stretch the boundaries of immediate problems to incorporate new interests and to address concerns raised on a moment-to-moment basis. The emotional commitment to mathematics shifts from the satisfaction of completing problems and subsequently terminating relationships to the maintenance of relationships through an ongoing search for meaning and coherence within the mathematical experiences the participants share with one another. The necessary incompleteness of expression, explanation and understanding becomes apparent in conversation which suggests the need for and provides alternative images of mathematical practice, intelligent action and acceptable mathematical explanations. This dissertation is an exploration of mathematical conversation and its place in mathematical practice. The features of conversation drawn from the illustrative narratives provided are summarized under three orienting gestural domains: (1) Addressivity towards the other highlights the ethical responsiveness and responsibility of persons in conversation and their mutual acceptance of one another to support ongoing interactions. (2) Addressivity towards otherness emphasizes the reciprocal relationship between the conversants and the mathematics brought forth. That is, participants not only respond to the mathematics, but the mathematical field of play responds to the conversants' interactions. (3) The lived curriculum is the path of mathematical activity laid as the addressivity interactions unfold. The lived curriculum in conversation is not directed from within any individual nor is it simply random. The path is oriented by the mutual concerns raised within the ongoing interactions and is bounded by the possibilities for mathematical action and by the participants' experiential histories drawn into the moment. These three domains provide a basis from which to speculate on the potential for and pedagogical consequences of mathematical conversation in classrooms in relation to present conceptions of mathematical practice and discourse. |
