Inscribing the Winch: Mechanisms by Which Students Develop Knowledge Structures for Representing the Physical World With Algebra

Abstract

I propose and test an account of mechanisms by which students develop knowledge structures for modeling the physical world with algebra. The account begins to bridge the gap between current mathematics curricula, in which modeling activities play an important role, and theoretical accounts of how students learn to model, which lag behind. After describing the larger study, in which I observed 12 pairs of 8th-grade students introduce and refine algebraic representations of a physical device called a winch, I then focus on 1 pair that generated an unconventional yet sound equation. Because the prevailing genetic accounts of knowledge structures in mathematics education, cognitive science, and information-processing psychology do not explain key characteristics of the data, I begin to construct a new developmental account that does. To do so, I use forms, a class of schemata that combine patterns of algebra symbols with patterns of experience in the physical world, and 2 mechanisms, notation variation and mapping variation. I then use forms and the 2 mechanisms to analyze how the selected pair of students introduced and refined initial, faulty algebraic representations of the winch into an unconventional yet sound equation.