In this article we consider data arising from student-teacher-researcher interactions taking place in the context of an experimental teaching program making use of multiple modes of communication and representation to explore three-dimensional (3D) shape. As teachers/researchers attempted to support student use of a logo-like formal language for constructing 3D trajectories and figures in a computer microworld, a system of gestures emerged. Observations of multimodal classroom communication suggest that teachers/researchers and students used similar words and gestures to represent different types of movement. We discuss possible sources of these differences, contrasting formal mathematical and everyday systems of representation of 3D space. More generally, we argue that understanding the structures of everyday discourse and their relationships to the structures of specialized mathematical discourse can provide insight into student interactions. (Contains 7 figures, 5 tables and 9 footnotes.)