## Seminar: 4/6 - Dor Abrahamson, UC Berkeley

**11:00 to 12:30 PM at: 5101 Tolman Hall**

**Discovery Reconceived: Guided Mediated Mathematical Insight**

My approach hinges on decoupling the solution process from its resultant product. In general, students participating in discovery-based learning activities inquire into a realistic source phenomenon, engage in an analytic process of generating a mathematical product that models the phenomenon, and then infer from this model information about the phenomenon. Whereas constructivist theories of learning often focus on process as the site of discovery, I propose from a complementary sociocultural perspective to focus instead on product. Specifically, I view student discovery of mathematical concepts as their guided aligning of two resources: (a) products resulting from guided formal analysis of a source phenomenon; and (b) informal inferences from naively judging perceptual qualities of the same phenomenon. I refer to this design genre as "perception-based," because it steers students to ground mathematical forms in naive gestalt sensations of intensive quantities (see Rochel Gelman's work on "enabling constraints").

Yet in more recent work, we have been investigating "action-based" design. In this genre, we first train students via embodied-interaction remote-control technology to develop new perceptuomotor schemas. Next, we introduce into the interaction space mathematical artifacts, such as a Cartesian grid. Students engage affordances of these artifacts as means of better enacting, explaining, or evaluating their interaction strategy. In so doing, they surreptitiously bootstrap mathematical forms that are the activities' instructional goals.

I demonstrate both design genres with qualitative analyses of videographed vignettes from implementing activities for proportion and probability. I argue that both perception-based and action-based designs enable students to re-invent mathematical ways of seeing, thinking, and speaking via coordinating personal and cultural resources. This theoretically balanced view on discovery learning may give reform-oriented educators license and methodology to implement effective instruction that builds on the best of both worlds.