RUME 14 (Volume 4)
2011
Portland, Oregon
HOW DO MATHMATICIANS MAKE SENSE OF DEFINITIONS?
Page: 41
It seems clear that students' activity while working with definitions differs from that of mathematicians. The constructs of concept definition and concept image have served to support analyses of both mathematicians' and students' work with definitions (c.f. Edwards & Ward, 2004; Tall & Vinner, 1981). As part of an ongoing study, we chose to look closely at how mathematicians make sense of definitions in hopes of informing the ways in which we interpret students' activity and support their understanding of definitions. We conducted interviews with mathematicians in an attempt to reveal their process when making sense of definitions. A striking observation relates to the role of examples. We will share a preliminary analysis of these interviews and engage the audience in reflecting on the ideas. KEY WORDS: mathematical definitions, advanced mathematical thinking, mathematicians' practice, examples How do we come to understand mathematical definitions? Is the process different for students than it is for mathematicians? What can be learned from the practice of mathematicians that could support students' learning? In their chapter on advanced mathematical thinking, Harel, Selden, & Selden (2006) identified mathematical definitions as one area of focus when comparing the activity of students with the practice of mathematicians. The constructs of concept definition and concept image have served to support analyses of both mathematicians' and students' work with definitions (c.f. Edwards & Ward, 2004; Tall & Vinner, 1981). Our current research attempts to bring such ideas together into explanatory models for mathematicians' and students' activity. In this presentation we will focus on mathematicians and how their ability to build adequate concept images might develop. Mathematicians encounter definitions in their work in a variety of ways. There are definitions included in courses they teach, definitions proposed by other mathematicians, and perhaps even new definitions created in the course of their own research work. In instructional settings, mathematicians must decide how to present definitions to students. In the context of current mathematical work, mathematicians must judge the clarity and appropriateness of stated definitions. In preparing to share proposed definitions, mathematicians must also consider presentation, clarity, and usefulness. Each of these settings requires some level of making sense of a given definition within a mathematical setting. We set out to create an interview context in which aspects of this activity were brought out and thus became accessible for analysis. The interviews provided opportunities for the mathematicians to articulate their perspectives on making sense of definitions and to participate in definition-related tasks (Watson & Mason, 2005). In the interviews, participants were first asked to describe how they make sense of new mathematical definitions and to provide a recent example of doing this, if possible. The second interview question asked participants to share how they support students' work with definitions. Participants were then asked to engage in an example-generation activity, and finally were given a formal definition from an unfamiliar context and asked to share how they would go about 441