Figure 1. A part-to-whole fraction as a conceptual composite. MATHEMATICAL REPRESENTATIONS AS CONCEPTUAL COMPOSITES: IMPLICATIONS FOR DESIGN

Positing that mathematical representations are covert conceptual composites, i.e., they implicitly enfold coordination of two or more ideas, I propose a design framework for fostering deep conceptual understanding of standard mathematical representations. Working with bridging tools, students engage in situated problem-solving activities to recruit and insightfully recompose familiar representations into the standard representation. I demonstrate this framework through designs created for studies in three mathematical domains. This design-theory paper presents a framework that spells out intuitive aspects of the craft of design for mathematics education so as to formulate these aspects, giving designers tools for progressing from domain analysis and diagnosis of learning problems toward design, implementation, and data analysis. The proposed framework focuses on mathematical representations and attempts to provide specificity, a “design template, ” for implementing radical-constructivist philosophy of didactics in terms of actual objects, activities, and facilitation guidelines for mathematics learning environments. The paper emanates from reflection on a decade of design-based research my collaborators and I have conducted on students’ mathematical learning, in three separate projects with designs for the content domains o