For mathematical communication, the great promise of computers is interactivity, the ability to make mathematical objects respond to user actions. Thus an equation transforms at a mouse click into an equivalent form, a geometric configuration retains its integrity when its components are dragged to new positions, or a graph changes shape continuously as the parameters describing it are changed. New modalities are still to come: what sound would a curve make, for example?

I look at questions of *communicating* with interactivity, i.e. the process of designing, refining and confining interactivity appropriately in order to demonstrate a given mathematical idea, or to focus the reader experience in a chosen direction. Numerous examples of gratuitous, ineffective or misleading interactivity can be found on educational CDs, on the web and elsewhere. I'll look at some of them, and attempt to formulate some general design principles of interactivity for effectively communicating mathematics.