ABSTRACT
We argue that research beyond that reported in this volume is urgently needed. This research must abandon basic assumptions: (a) that the future curriculum will be organized in ways that are only local modifications of the current curriculum; (b) that the technology used in research should parallel that available to schools; and, most importantly, (c) that the appropriate representations be restricted to the traditional “big three,” namely, symbols, numerical tables, and coordinate graphs. We explore alternative perspectives to all these assumptions and offer particular examples to illustrate the third, which then force rethinking of the first two. These examples suggest ways of using the new representational capacity of computers to bridge the highly persistent gap between the rich experience of living and the sterile formalisms that have dominated our approaches to mathematical learning and knowing. The research questions raised by these alternatives are myriad and deep, and are not yet being explored.
