My dissertation research, “Developing a Feel for the Game: Treating Diagrams as Representations of Idealized Mathematical Objects,” was supported by a fellowship from the Spencer Foundation and the National Academy of Education.
My dissertation engages a seldom-studied but important idea in mathematics education: students’ developing understanding that geometric diagrams are representations of imaginary objects, which are actually too perfect to draw exactly. (For example, in geometry, dots are used to represent zero-dimensional points and drawn lines are taken to represent one-dimensional lines with infinite length.) As a shorthand, I refer to this way of using diagrams “definitional practice,” since it’s definitions that establish the properties of the mathematical object, not the inevitably imperfect appearances of the drawn diagrams.
This topic interests me because it highlights the fact that mathematics, when we really stop to think about it, is a peculiar kind of “language game” whose rules are often not made fully explicit to students. My dissertation topic reflects my interests in language, semiotics, culture, cognition, and pedagogy—as well as my habit of mind of making the familiar seem strange as a fruitful starting point for inquiry.
Initiation into the “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 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 analyses reveal that with age, children do shift towards relying on definitions rather than the appearances of the diagrams and knowledge of material objects.
To better understand the nature of this shift, I conducted a second study. The purpose of this one-on-one interview study is to investigate how students make sense of the items, the definitions, and relations between the definitions and the diagrams—in other words, how students may be constructing a differentiation between the diagrams and the ideas of points and lines.
The insights generated by my dissertation make an important contribute to mathematics education research and offers suggestions for instructional approaches that can give students better access to this central aspect of doing mathematics.