Collective Mathematical Understanding as Improvisation

This article explores the phenomenon of mathematical understanding, and offers a response to the question raised by Martin (2001) at the Annual Meeting of the Psychology of Mathematics Education Group (North American Chapter) about the possibility for and nature of collective mathematical understanding. In referring to collective mathematical understanding, we point to the kinds of learning and understanding we may see occurring when a group of learners work together on a piece of mathematics. We characterize the growth of collective mathematical understanding as a creative and emergent improvisational process and illustrate how this can be observed in action. In doing this, we demonstrate how a collective perspective on mathematical understanding can more fully explain its growth. We also discuss how considering the growth of mathematical understanding as a collective process has implications for classroom practice and in particular for the setting of mathematical tasks.