Laurie O. Cavey

How do Mathematicians Make Sense of Definitions?

Laurie O. Cavey — Thu, 04 Aug 2011 15:12:53 PDT

The purpose of this paper is to share preliminary results from a pilot study on mathematical definitions. Interviews with university mathematicians were designed to gain insight into mathematicians' processes for developing understanding of new definitions. We asked the participants to talk about what helps them understand a new definition and how they support students’ understanding of definitions. We also observed them while they engaged in a definition task. Analysis revealed a noticeable difference in the emphasis on examples between what the participants described that they do and what they actually did while working on the definition task. We hypothesize that mathematicians’ processes for making sense of a definition necessarily involve considering the definition’s usefulness within a particular mathematical setting. Furthermore, these data indicate that mathematicians see examples as a multi-faceted, but not comprehensive, tool for understanding definitions.

Seeing the Potential in Students' Questions

Laurie O. Cavey et al. — Mon, 23 May 2011 10:43:07 PDT

Rich mathematical questions emerge from online dialogues between high school algebra students and prospective teachers.

How Do Mathematicians Make Sense of Definitions?

Laurie O. Cavey et al. — Wed, 23 Feb 2011 13:16:22 PST

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.

Developing Students’ Understanding of Mathematical Definitions: Why Bother?

Laurie O. Cavey — Wed, 09 Dec 2009 13:14:20 PST

Definitions are a fundamental part of doing mathematics, yet studies indicate that many students struggle to learn and apply definitions. In fact, many instructors wonder (myself included) how students can misapply definitions that are so clearly stated. Part of the issue is that a student’s previous mathematical experiences influence how she thinks, even when encountering a new idea that is seemingly unrelated. Not knowing what these experiences might entail, it can be difficult to know how to help students develop a better understanding of a particular definition. So, why bother? I will provide a brief overview of the research in this area including an instructional strategy (student generated examples) that may influence the way we think about developing students’ understanding of definitions.

Proportional Reasoning 101

Laurie O. Cavey — Sun, 30 Aug 2009 19:52:49 PDT

Investigating Teachers' Mathematics Teaching Understanding: A Case for Coordinating Perspectives

Laurie O. Cavey — Sun, 30 Aug 2009 19:48:55 PDT

This paper provides a microanalysis of one Algebra I teacher's instruction to explore the advantages that are afforded us by coordinating two perspectives to document and account for the teacher's mathematical understandings. We use constructs associated with Stein, Grover and Henningsen's domain of mathematical didactics and Realistic Mathematics Education's instructional design theory to infer what the teacher might understand to effectively implement her instructional goals and, more importantly, support student learning. By coordinating these perspectives, we developed a working framework for analyzing the teacher's classroom practice retrospectively. For example, we illustrate how the mathematical possibilities related to one student's question might inform the teacher's decisions as she initiates shifts in students' self-generated models. Additionally, we illustrate how the teacher's decision to capitalize on particular students' models contributes in part to the kinds of mathematical ideas that can be explored and the connections students can make among those ideas. More generally, we explore the utility of coordinating these two perspectives to understand the landscape of ideas that teachers might traverse to align their practices with reform recommendations in the United States.

The Instructor's Important Role In Supporting Mathematical Arguments in a K-5 Mathematics Specialist Program

Joy Whitenack et al. — Sun, 16 Aug 2009 16:33:01 PDT

In this presentation, we use examples from a one lesson taken from an algebra course for K-5 mathematics specialists to illustrate the instructor’s role in supporting argumentation. Krummheuer’s (1995) theory of ethnography was a particularly useful methodological tool for tracing the argument that unfolded during the discussion. As part of this interpretive model, we assume that normative ways of making mathematical arguments are socially accomplished, although individuals contribute to and participate in these arguments in different ways. Preliminary findings suggest that as the instructor highlighted teachers’ explanations he and the teachers established mathematical arguments. In some cases the instructor provided warrants or backings that remained implicit or omitted (cf. Yackel, 2002). In other instances the instructor coordinated different explanations to substantiate or validate the arguments that emerged. By doing so, he made it possible for teachers to engage in making more formal arguments.