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.
The Journal of the Learning Sciences, 10(1&2) (2001), 113–163.
Copyright © 2001, Lawrence Erlbaum Associates, Inc.
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Notes
- 1.
- 2.
Following Gravemeijer (1994a), we have previously called this type of research developmental research. Our purpose in relabeling it design research is to reduce the possibility that it may be confused with either child development research or with noninterventionist research into the development of particular mathematical concepts.
- 3.
For a justification of the notion of a classroom microculture, see Cole (1995).
- 4.
We speak of normative activities being taken as shared rather than shared to leave room for the diversity in individual students’ ways of participating in these activities. The assertion that a particular activity is taken as shared makes no deterministic claims about the reasoning of the participating students, least of all that their reasoning is identical.
- 5.
For a number of years, we spoke of the relation between the two perspectives as being dialectical. However, German colleagues pointed out that, in their country, this term was associated with neo-Marxist theories of dialectical materialism. It was to avoid this miscommunication that we began to describe the relation as reflexive.
- 6.
Given our discussion of social and psychological perspectives, we need to say explicitly that we are not attempting to reconcile Vygotskian and Piagetian theory as some have assumed (e.g., Lerman, 1996).
- 7.
In reflecting on his activity as an instructional designer, Gravemeijer (1994b) argued that design resembles the thinking process that Lawler (1985) characterized by the French word bricolage, a metaphor taken from Claude Levi–Strauss. A bricoleur is a handy man who invents pragmatic solutions in practical situations. The bricoleur has become adapt at using whatever is available. The bricoleur’s tools and materials are very heterogeneous: Some remain from earlier jobs; others have been collected with a certain project in mind. Extending this metaphor, we would add the interpretative framework we use can also be viewed as a bricolage. In developing it, we have acted as bricoleurs who have drawn on and adapted ideas from a range of theoretical sources for pragmatic ends. Casting our work in these down-to-earth terms serves to differentiate it from more grandiose efforts that aim to fashion theoretical cosmologies (cf. Shotter, 1995).
- 8.
The characterization of the individual is generally consistent with Roth’s use of the individual’s life world or the individual in the world as his unit of analysis (Roth & McGinn, 1998). In his terms, our concern when we adopt the psychological perspective of the interpretive framework is to delineate the ontology of the world in which an individual student acts.
- 9.
This approach of coordinating psychological and social analyses is closely related to several other proposals. These include Hatano’s (1993) call to synthesize constructivist and Vygotskian perspectives, Saxe’s (1991) discussion of the intertwining of cultural forms and cognitive functions, and Rogoff’s (1995) distinction between three planes of analysis that correspond to personal, interpersonal, and community processes.
- 10.
In psychological terms, this learning can be described as the internalization and interiorization of the activity of creating two coordinated logs. This activity was socially situated in that we were participating in the practices of our research community as we conducted the analyses.
- 11.
These interventions clearly changed the social interactions in which the target students engaged. We justify the interventions on the pragmatic grounds of generating information that we needed to make pedagogical decisions.
- 12.
The emphasis that we give to quantitative reasoning can be contrasted with the almost exclusive emphasis on numerical reasoning both in traditional classrooms and in many classrooms where instruction is compatible with recent reform recommendations (cf. Smith, 1997; Thompson, 1993, 1994; Thompson & Thompson, 1996).
- 13.
This is not to say that all explanations involved a backing. It was not necessary for a student to give a backing when other students indicated that they understood a particular method of measuring structured space.
- 14.
Nancy’s gesturing as she explained, “Twenty is how many little cubes we’ve done so far” illustrates Roth’s (2001, this issue) arguments concerning the integration of word and gesture in establishing the entities that are spoken about. Several other examples of this phenomenon are apparent in our account of the emergence of the first three mathematical practices. In this connection, it is worth noting that we attended to the teacher’s gesturing as part of our instructional design. In particular, we conjectured that it would be important for the teacher to indicate by gesture the entire distance that had been measured when iterating a Smurf bar rather than the successive placements of the bar.
- 15.
Bowers and Nickerson (1998) reported a similar pattern in their analysis of a university mathematics course that focused on quantitative reasoning.
- 16.
See Thompson et al. (1994) for a more extensive discussion of the role of inscriptions in supporting conceptual conversations.
- 17.
- 18.
This semiotic ecology can be made explicit by delineating the chain of signification (Lacan, 1977; Walkerdine, 1988) that is constituted as successive classroom mathematical practice emerge. Stephan (1998) described the chain of signification that was constituted during the measurement teaching experiment. Examples of other analyses of this type can be found in Cobb et al. (1997) and Gravemeijer et al. 2000).
- 19.
Wertsch (1991) made a similar point when he observed that much contemporary research in psychology does not in fact have the practical implications claimed for it: “In my view, a major reason is the tendency of psychological research, especially in the United States, to examine human mental functioning as if it exists in a cultural, institutional, and historical vacuum” (p. 2).
- 20.
It could be argued that the forms of instruction developed in the course of a teaching experiment are unfeasible for any teacher working alone. We would acknowledge, for example, that the entire research team in effect constitutes a collective teacher with some members of the team actually teaching, whereas others observe and analyze classroom events. The demands of this collective activity are, however, balanced by the possibility that the collaborating teachers will be able to capitalize on our learning as represented by instructional sequences and learning trajectories. This conjecture about the proposed role of instructional sequences as a means of supporting the development of professional teaching communities is discussed in some detail by Cobb and McClain (2001). As part of our work with teachers, we are currently developing a series of companion CD-ROMs to support their learning that are based on the video recordings and other data sources. Readers who are interested in the practical implications of our work are referred to a series of articles and book chapters that we have developed for practitioner audiences (e.g., McClain & Cobb, 1999; McClain, Cobb, & Gravemeijer, 1999; McClain, Cobb, Gravemeijer, & Estes, 2000).
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Acknowledgments
The analysis we report was supported by the National Science Foundation under Grant REC 9814898 and by the Office of Educational Research and Improvement under Grant R305A60007.
We are grateful to Sasha Barab, Joanna Kulikowkich, Geoffrey Saxe, and Michael Young for helpful comments on a previous draft of this article.
The opinions expressed do not necessarily reflect the views of either the Foundation or OERI.
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Cobb, P., Stephan, M., McClain, K., Gravemeijer, K. (2010). Participating in Classroom Mathematical Practices. In: Sfard, A., Gravemeijer, K., Yackel, E. (eds) A Journey in Mathematics Education Research. Mathematics Education Library, vol 48. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-9729-3_9
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