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

Kenton de Kirby

“Developing a Feel for the Game: Treating Diagrams as Representations of Idealized Mathematical Objects,” 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.

About Kenton de Kirby

Kenton is a PhD Candidate at UC Berkeley in the Graduate School of Education. His interests include culture-cognition relations, social interaction, special education, and linking sociocultural processes with neuropsychological constructs like executive functioning. He is a fellow in the IES-funded “Research in Cognition and Mathematics Education” (RCME) fellowship program. At Berkeley, he has worked with Geoffrey Saxe on children’s developing mathematical understandings with particular regard for social and cultural processes. In the summer of 2015, Kenton conducted a month of fieldwork in a remote mountainous region of Papua New Guinea with Professor Saxe, investigating elementary mathematics education. Reflecting his interest in both typical and atypical development, Kenton also works with Professor Laura Sterponi on language and autism. Kenton earned his BA from UC Berkeley in linguistics, where he studied with noted linguist and cognitive scientist George Lakoff. Following his undergraduate studies, Kenton worked for firms that provide consultation to social change organizations, where he applied insights from cognitive linguistics to analyze the narrative landscape surrounding social issues. Before entering the Graduate School of Education, Kenton co-authored a book on the implications of fundamental findings in neuropsychology for classroom teaching.