Characterizing and Supporting Practices of Defining in a Mathematics Classroom

• In Event: Representing and Supporting the Development of Epistemic Practices in Classroom Communities

Abstract

Recently, educators have advocated rethinking how we treat definitions in mathematics classrooms. Scholars suggest that rather than having students memorize definitions, we more closely align classroom practices to those in the discipline. In mathematics, members participate in the co-construction of definitions; these definitions play a significant role in the development of mathematics as a system of interrelated entities and objects (Lakatos, 1976). Although a few scholars have looked at students’ participation in defining (e.g., Lehrer, Jacobson, Kemeny & Strom, 1998), most of the existing research focuses on supporting conceptual understanding of definitions (e.g., Ambrose & Kenehan, 2009). This paper expands current work by investigating more extensively the nature of defining as a classroom practice.

To attend to our goal, we pursue three questions: 1) How might we characterize definitional practices? 2) How is the development of definitional practices supported? and 3) How do definitional practices participate in the development of a mathematical system? We present data from video records of whole class activity where sixth-grade students created and refined mathematical definitions of geometric objects. Our design for instruction capitalized on students’ everyday experiences and conceptions of space, especially bodily motion (e.g., Abelson & diSessa, 1980), and on everyday forms of argument.

Here, we trace initial explorations that emerged as students pursued the question, “What is a polygon?” We focus on the first six days of instruction because the activity largely involved defining and because we hoped to see how initial forms of practice arose and were supported. We divided the data into definitional episodes – segments of time in which students participated in definitional activity. We then conducted three pieces of analysis. First, through iterative observation of the video, we categorized aspects of definitional practice. Second, we employed a similar method to identify how the teacher supported these practices. Finally, in order to represent the mathematical system developed by the class, we looked across neighboring definitional episodes to identify moments of interrelationships (e.g., defining polygon created the need to establish what a side was). Using this representation, we looked at how practices and supports aided development of the mathematical system (see Figure 1).

We observed five aspects of definitional practice: a) asking definitional questions, b) engaging in definitional argument, c) revising definitions, d) evaluating and creating cases and e) considering definitions in new forms. The teacher supported these aspects by a) modeling definitional practice, b) keeping the goal of defining at the forefront, and c) positioning the practice of defining as an argument. Additionally, the teacher encouraged the development of mathematical system by asking definitional questions that probed into new relations between objects. As students began to appropriate definitional practices, they, in turn, became supporters of their collective enterprise of defining. These characterizations of definitional practices and supports may provide resources for developing similar classroom environments. Moreover, ongoing analyses suggest that defining contributed to a longer sociogenetic trajectory (Saxe, 2002) involving conjecture, theorem and proof, illustrating its potential for supporting other mathematical practices.

Authors

• Marta Anna Kobiela, McGill University

• Richard Lehrer, Vanderbilt University