Four Lenses for Examining Individual- and Collective-Level Mathematical Progress

• In Event: Connecting the Moving Dots: Comparing Approaches to Coordinating Temporal Analyses of Groups and Individuals

Abstract

A pressing challenge in mathematics education is to integrate different theoretical orientations in order to develop a more comprehensive account of teaching and learning (Cobb, 2008; Sfard, 1998). Our approach to this challenge expands Cobb and Yackel’s (1996) interpretive framework for coordinating social and individual aspects of learning and thus contributes to theory building regarding how to conceptualize and coordinate the collective and the individual.

In particular, analysis of individual students may focus on knowledge acquisition or on student participation. Sfard (1998) calls these two perspectives on individual learning the acquisition metaphor and the participation metaphor. A typical approach to individual analysis within the interpretive framework is to focus on individual participation in classroom mathematical practices. For example, Stephan, Cobb, and Gravemeijer (2003) “cast each instance of learning as an act of participation in the mathematical practice that was either emerging or was established.” We show that an equally compelling analysis is one that posits cognitive structures that are manifested as students participate in classroom life.

A collective analysis, on the other hand, may focus on classroom mathematical practices (Stephan & Rasmussen, 2002), or on the how discipline specific practices of proving, defining, algorithmatizing, and symbolizing (Rasmussen, Zandieh, King, & Teppo, 2005) are enacted within the classroom. The difference between the two foci lays in what is foregrounded and backgrounded. In analyzing classroom mathematical practices, the mathematical development of the classroom is brought to the fore, while the relationship of the emergent practices to the broader disciplinary practices is backgrounded. A focus on disciplinary practices foregrounds the ways in which the broader practices of proving, defining, etc. frame and constitute classroom mathematical practices.

Our expansion of the interpretive framework is illustrated in Figure 1. We reframe the bottom row, originally cast in terms of “classroom mathematical practices” and “individual conceptions and activity,” as learning in terms of collective disciplinary practices, classroom mathematical practices, individual participation, and individual acquisition.

We illustrate this expansion of the interpretive framework by analyzing classroom videorecordings and individual interviews from a semester-long classroom teaching experiment (Cobb, 2000) in introductory linear algebra. Data analysis made use of Toulmin’s (1969) argumentation scheme for both individual and collective analyses (Rasmussen & Stephan, 2008; Wawro, 2011). In the full paper we delineate and coordinate (a) a classroom mathematics practice related to reasoning about span and linear independence; (b) the discipline-specific practice of proving; (c) the participation of one student, Abraham, within both of these practices; and (d) Abraham’s cognitive structures regarding his ways of reasoning about span and linear independence. For example, results showed that collective argumentation of the disciplinary practice of proving was characterized by more complex argumentation structures than that of the basic Toulmin scheme. Within these more complex structures the class developed correct ways of reasoning about span, for example, that are not necessarily seen in the broader discipline. The individual case study revealed how Abraham’s reasoning was similar to and different from the larger classroom mathematics practice and specific conceptions enabled and constrained his reasoning capabilities.

Authors

• Chris L. Rasmussen, San Diego State University

• Megan Wawro, Virginia Polytechnic Institute and State University

• Michelle J. Zandieh, Arizona State University