Cobb, Stephan, McClain, & Gravemeijer (2001)

The article Participating in Classroom Mathematical Practices was written by Paul Cobb, Michelle Stephan, Kay McClain, and Koeno Gravemeijer and published in The Journal of the Learning Sciences in 2001. The article is available from JSTOR at http://www.jstor.org/stable/1466831.

Abstract

In this article, we describe a methodology for analyzing the collective learning of the classroom community in terms of the evolution of classroom mathematical practices. To develop the rationale for this approach, we first ground the discussion in our work as mathematics educators who conduct classroom-based design research. We then present a sample analysis taken from a 1st-grade classroom teaching experiment that focused on linear measurement to illustrate how we coordinate a social perspective on communal practices with a psychological perspective on individual students' diverse ways of reasoning as they participate in those practices. In the concluding sections of the article, we frame the sample analysis as a paradigm case in which to clarify aspects of the methodology and consider its usefulness for design research.

Detailed Summary of Participating in Classroom Mathematical Practices

In this article, Cobb et al. describe the type of design research they conducted with 1st grade students to study the evolution of classroom mathematical practices. Using a design research approach and teaching experiments, the authors developed both sequences of instructional activities as well as a theory and process for interpreting classroom events from both a social and a psychological perspective.

Design Research

Rooting themselves in more than a decade of research involving students' mathematical learning in the social context of classrooms, Cobb et al. use a design research approach (which both Cobb and Gravemeijer had previously called developmental research) to develop instructional activities and study student learning. Design research is based on iterative cycles of design and analysis, with Cobb et al. basing their instructional design on the theories of Realistic Mathematics Education (RME). The theories of RME inform the design of activities along with the researcher's predictions about how the activity will further student learning. These predictions are tested upon enactment of the activity, which generates data to inform future development and refinement of instructional activities. Cobb et al. list three criteria for their analytical approach:

1. It should enable us to document the collective mathematical development of the classroom community over the extended periods of time covered by instructional sequences.
2. It should enable us to document the developing mathematical reasoning of individual students as they participate in the practices of the classroom community.
3. It should result in analyses that feed back to inform the improvement of our instructional designs. (p. 116)

Cobb et al. carefully worded the first of the criteria to focus on the classroom community, as predictions and instructional design cannot possibly account for each and every individual student's learning at a given point in time. The second criteria, however, takes advantage of individual student thinking not in the planning of activities, but in the enactment, where observed differences in student reasoning help to highlight key ideas in the mathematics. The third criteria is perhaps the most obvious in design research, where data collected in each cycle is used to inform the design of the next.

Interpretative Framework

Cobb et al. take two theoretical viewpoints in this paper: a psychological perspective for thinking about and describing individual student learning, and a social perspective for thinking about and describing the classroom microculture.

Social and Psychological Perspectives

Cobb et al. describe the social perspective as "concerned with ways of acting, reasoning, and arguing that are normative in a classroom community" (p. 118). This means that individual student reasoning is described in the ways they participate in normative classroom activities. In contrast, the psychological perspective "focuses squarely on the nature of individual students' reasoning or, in other words, on his or her particular ways of participating in communal activities" (p. 119). The key difference is that the social perspective focuses on normative activities, and the psychological perspective highlights differences in individual student participation. Cobb et al. build their social perspective from sociocultural theory (e.g., Cole, 1996; Lave, 1988; Rogoff, 1997) and ethnomethodology and symbolic interactionism (Blumer, 1969), and base their psychological perspective on constructivism (Piaget, 1970; Steffe & Kieren, 1994; Thompson, 1991) and distributed accounts of intelligence (e.g., Hutchins, 1995; Pea, 1993). Concerning their use of a psychological perspective, Cobb et al. clarify:

The goal of analyses conducted from this psychological perspective is therefore not to specify cognitive mechanisms inside students' heads. Instead, it is to infer the quality of individual students' reasoning in, with, and about the world, and to account for developments in their reasoning in terms of the reorganization of activity and the world acted in. (p. 121)

Together, Cobb et al. did not seek to describe the social and psychological perspectives together as some grand theory; rather, each perspective compliments the other, just as individual student activity shapes classroom activity and vice versa.

Aspects of the Classroom Microculture and Individual Students' Reasoning

Cobb et al. use the following interpretive framework for studying both classroom and individual mathematical activity and learning (p. 119):

Social Perspective	Psychological Perspective
Classroom social norms	Beliefs about own role, others' roles, and the general nature of mathematical activity in school
Sociomathematical norms	Mathematical beliefs and values
Classroom mathematical practices	Mathematical interpretations and reasoning

Social norms are features of the classroom participation structure (Erickson, 1986; Lampert, 1990) that include things like how a class explains and justifies solutions, makes sense of explanations, and questions alternative interpretations. For the research, Cobb et al. looked for occasions when norms might be violated to inform their understanding of the classroom's social norms. Because this is a social perspective, this view of social norms avoids singling out student behavior and instead views participation in norms as a joint activity. For the corresponding psychological perspective, Cobb et al. sought to understand how students saw their role and the role of others in mathematical activity. The social and psychological are related as social norms affect how individuals see their own roles and vice versa, with neither more important than the other.

Because social norms are not unique to mathematics classrooms, Cobb et al. looked for math-specific norms (Lampert, 1990; Simon & Blume, 1996; Voigt, 1995; Yackel & Cobb, 1996) they referred to as sociomathematical norms, including things such as agreement on what counts as a mathematical solution, a sophisticated solution, an elegant solution, and a mathematically acceptable explanation. For students to engage in these sociomathematical norms, Cobb et al. believed students must co-develop beliefs about mathematics that encourage them to act autonomously. Cobb et al. described this development of autonomy as moving from relatively peripheral participation towards more substantial participation, instead of an individual act (cf. Forman, 1996; Lave & Wegner, 1991).

As instructional designers, Cobb et al. also wished to define classroom mathematical practices, arguing that "it is feasible to view a conjectured learning trajectory as consisting of an envisioned sequence of classroom mathematical practices together with conjectures about the means of supporting their evolution from prior practices" (p. 125). Unlike sociomathematical norms, which are only specific to a mathematical classroom community, classroom mathematical practices are analyzed in relation to specific mathematical tasks or ideas. Cobb et al. see these practices from a social perspective, but reflexively relate them to the mathematical interpretations and reasoning of individual students.

About

APA

Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. The Journal of the Learning Sciences, 10(1/2), 113–163.

BibTeX

@article{Cobb2001,
author = {Cobb, Paul and Stephan, Michelle and McClain, Kay and Gravemeijer, Koeno},
journal = {The Journal of the Learning Sciences},
number = {1/2},
pages = {113--163},
title = {{Participating in classroom mathematical practices}},
volume = {10},
year = {2001}
}