Proceedings of the Sixth International Research Forum on Statistical Reasoning, Thinking, and Literacy (SRTL-6), The University of Queensland, Australia, July 10-16, 2009.

I locate the drama of mathematical learning in students’ creative attempts to coordinate two different mental constructions of a situation designed to embody a targeted mathematical notion: A phenomenologically immediate construction of the source situation, and a construction that is semiotically mediated by a mathematical model of the same situation. To successfully synthesize these constructions, students must assimilate mathematical units of analysis into their tacit schematic structures. One strategy for accomplishing this assimilation is to invent realistic, hypothetical, or fantastic images that lend plausible coherence to the blending of these tacit and formal orientations. We focus on Razi, an 11.5 year-old student learning the binomial distribution. She conjures an imaginary mechanical system to concretize her intuitive sense of distribution.