Proceedings of the Annual Meeting of the Cognitive Science Society

We report analyses of the construction and interpretation of mathematical symbols that refer to quantitative properties and relations of a physical system. Middle-school students solved problems that involved contracting tables, equations and graphs to represent linear functions of a device where blocks are moved varying distances by turning a handle that winds string around spools of different sizes. Previous research analyzed activities of reasoning about quantities of this system as attunement to constraints and affordances, a characterization of students' imphcit understanding of concepts of variable and linear functions. This report concerns activities of representing quantitative properties and relations using mathematical notations. W e are developing analyses of constructing and interpreting tables, equations, and graphs in terms of attunement to constraints and affordances of the represented system, the system of notations, and relations between the constraints of the notations and the represented domain. W e present examples that illustrate concepts of semantic clumps, groups, and morphisms; descriptive and demonstrative representations; multiple referent domains; and constructions of meaning in contributions to conversational discourse.