The medium and the curriculum: Reflections on transparent tools and tacit mathematics

This paper explores the phenomenological and curricular dynamics of implicit mathematical structures embodied in "transparent" computer-based tools. Examples from a clinical study of students working with the Tabletop database/data analysis environment illustrate the process by which disruptions of transparency can provoke increasingly reflective use of a tool and bring students into engagement with valuable mathematical ideas. The interaction among learner, medium and curriculum is seen to have important implications for pedagogy, tool design, and evolving conceptions of mathematics.