This paper explores the phenomenon of mathematical understanding, and offers a response to the question raised by Martin (2001) at PME-NA 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, of any size, work together on a piece of mathematics. In employing and extending the theoretical work of Becker (2000), Sawyer (1997; 2000; 2001) and Berliner (1994; 1997), we characterise collective mathematical understanding as a creative and emergent improvisational process and illustrate how it can be observed in action. (Contains 3 footnotes and 1 figure.) [For complete proceedings, see ED500858.]