Idealized Objects and Material Diagrams: a Cultural Practice-Approach to Understanding Definition Use in Early Geometrical Problem Solving

This dissertation targets young students’ developing knowledge relevant to a fundamental practice in academic mathematics: the use of diagrams to represent idealized mathematical objects whose properties are established by definition (as in the use of drawn dots to represent zero-dimensional points and drawn lines 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 significant source of confusion for students. Instead of using definitions, students may rely on the appearances of the diagrams and their knowledge of the physical world—an ‘empirical’ rather than definitional approach./nI have designed three studies to investigate students’ developing understanding of the definitional practice, using points and lines in Euclidean geometry as a potentially fruitful mathematical context. Adopting a design research approach that draws on Vygotsky’s method of double stimulation, these studies employ a pedagogical strategy that is rarely observed in mathematics classrooms: providing definitions and making explicit the distinction between drawn diagrams and the idealized objects they symbolize, pointing out that the defining features of points and lines are not perfectly embodied in their conventional representations. These studies investigate students’ uptake of this support and capture students’ developing understanding of the definitional practice./nThe first study uses an experimental design to determine whether there are age/grade-related changes in students’ uptake of the intervention. Participants include students from a San Francisco Bay Area charter school in fourth grade (n=46), sixth grade (n=53), and eighth grade (n=43). Students are assigned to one of two treatment conditions whose purpose was to manipulate exposure to the definitions of points and lines. In the experimental treatment, students are presented with a sheet that contained mathematical definitions of points and lines, which are unavailable to students in the control condition. Students are then administered a paper-and-pencil assessment consisting of eight multiple choice items, some of which included diagrams of points and lines. Analyses suggest that with age, children shift towards relying on definitions rather than the appearances of the diagrams and knowledge about material objects./nThe second study addresses unanswered questions about the general learning trend identified by the first study. Specifically, it uses structured one-on-one interviews to determine whether (a) students are indeed drawing on provided definitions when selecting idealized (rather than empirical) answers, and (b) whether students are constructing a conceptual differentiation between material diagram and idealized object. Participants include fourth (n=40) and sixth (n=37) grade students./nThe third study explores how students make sense of the definitions of points and lines presented in Study 1 and 2. In particular, this study considers how students may be constructing analogies that draw on ideas related to the material world. Results demonstrate that students’ materially-based analogies incorporate other ideas typically learned in school to make sense of the definitions of points and lines. For example, students frequently refer to scientific ideas related to the microscopic world of atoms and molecules to explain the zero-dimensionality of a mathematical point. Participants were identical to those of the second study./nThe insights generated by these three studies contribute to our understanding of sociocultural processes in cognitive development as well as to mathematics education research. This dissertation also has practical implications for mathematics instruction, suggesting potentially promising pedagogical possibilities for giving students a better ‘feel for the game’—an understanding of what academic mathematics is all about.

Advisor: Saxe, Geoffrey B