Developing a Feel for the Game: Treating Diagrams as Representations of Idealized Mathematical Objects

• In Event: Promising Scholarship in Education: Dissertation Fellows and Their Research

Abstract

My dissertation targets young students’ developing participation in and understanding of a fundamental practice in academic mathematics. In this practice, diagrams are used to represent idealized mathematical objects whose properties are established by definition (as in geometry, where dots are used to represent zero-dimensional points and drawn lines are taken to represent one-dimensional lines with infinite extent). Initiation into this ‘definitional practice’ is critical to students' mathematical development. However, the practice is understudied in educational research. It also presents a source of confusion and miscommunication for students. Instead of using definitions, students may rely on the appearances of the diagram and their knowledge of the physical world—a ‘material’ rather than definitional practice. I have designed two empirical studies to investigate students’ initiation into the definitional practice in the context of points and lines in Euclidean geometry. The first study uses an experimental design to determine whether there are age-related changes in how access to stipulated definitions influences students’ idealization of diagrams. I have collected data for this study and preliminary analyses reveal that with age, children do shift towards relying on definitions rather than the appearances of the diagrams and knowledge of material objects. The second study builds on the first using structured one-on-one interviews, exploring students’ sense-making of definitions and their flexibility in shifting between material and definitional practices. The insights generated by these two studies will contribute to mathematics education research and may inform instructional approaches to address the likelihood of unequal access to developing this foundational practice in mathematics.

Author

• Kenton Dobrin de Kirby, University of California - Berkeley