Paper presented at the annual meeting of the American Educational Research Association (SIG Research in Mathematics Education), Vancouver, April 13-17.

22 Grade 4-6 students participated in a design study investigating the conjecture that the concept of proportionality can be grounded in perceptuomotor schemas. Students engaged in a non-numerical embodied-interaction problem-solving task, in which they bi-manually manipulated two virtual generic objects in parallel along a vertical axis on a computer screen, in an attempt to elicit a designated feedback. The solution involved moving the objects at different rates. Once they discovered simple solution strategies, the tutor introduced onto the screen a set of symbolic artifacts—a Cartesian grid, and then numerals. Students appropriated these frames of reference, yet in so doing they implicitly reconfigured their strategies so as to incorporate explicit attention to quantitative properties, resulting in strategies that we identified as either additive or multiplicative. We then asked the students to reflect on relations among these various strategies. The paper offers micro-ethnographies of three selected episodes, in which some students expressed logico-mathematical inferences in support of their claim that their additive and multiplicative strategies could be derived one from the other. Students’ naïve deductions hinged on heuristic schematic links that enabled them to ground multiplicative phenomena in additive primitives. Formal coordination of additive and multiplicative conceptualizations of proportion, via elaborating the distributive property of multiplication over addition, may come only through algebraic and number-theory reasoning. We speculate on perceptuomotor intuitions and socio-epistemic norms underlying patterns in students’ informal inferences.