# 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 Taylor & Francis Online at http://www.tandfonline.com/doi/abs/10.1207/S15327809JLS10-1-2_6 and 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:

- It should enable us to document the collective mathematical development of the classroom community over the extended periods of time covered by instructional sequences.
- It should enable us to document the developing mathematical reasoning of individual students as they participate in the practices of the classroom community.
- 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.

### Methodological Considerations

Cobb et al. focused their analysis on the classroom mathematical practices, as they believed this to be the least developed aspect of their interpretive framework. The data used for analysis included video recordings of all classes during teaching teaching experiments, all students' written work, and video recordings of student interviews collected before and after the teaching experiment. Cobb et al. took Glazer and Strauss's (1967) constant comparison method because it had been adapted for design research (Cobb & Whitenack, 1996). Acting as participant observers, Cobb et al. documented classroom activity in ways that exposed patterns or themes which could then be tested with each cycle of new data. With this method, Cobb et al. treated mathematical learning "as a process of coming to use conventional tools and symbols in socially accepted ways" (p. 127).

Data analysis was approached chronologically, and with each mathematical concept the researchers conjectured about normative reasoning and communication, as well as individual students' reasoning. Separate records were kept for the social and psychological perspectives to make them distinct and to highlight the reflexive relationships between them. Conjectures about mathematical norms took three forms: (a) a taken-as-shared purpose, (b) taken-as-shared ways of reasoning with tools and symbols, and (c) taken-as-shared forms of mathematical argumentation (p. 129). For this paper, these conjectures were made in the context of a measurement activity, where students used tools to assign numerical values to objects and were required to defend both their process and results.

### Measurement Practices

This paper was based on a 14-week teaching experiment with 16 first-grade students, with the first half of the experiment focused on linear measuring and the second half focused on mental computation with numbers up to 100.

#### Background to the Teaching Experiment

The classroom teacher was a member of the research team and allowed the researchers to intervene during classroom work. Two cameras were used to record every lesson while three researchers took field notes. Copies of work was collected from all students and interview data was collected from students before and after the experiment. The research team also recorded their weekly meetings. Five students were interviewed midway through the experiment, selected due to the diversity in their reasoning shown in the initial set of interviews. These five students were watched more closely during the experiment and researchers made detailed notes of the students' reasoning in their field notes after each class.

Initially, Cobb et al. envisioned that by iterating a tool along an object, students would give meaning to the accumulation of distance (cf. Piaget, Inhelder, & Szeminska, 1960; Thompson & Thompson, 1996). This way, students should understand that if measuring by pacing from heel to toe, for example, the number word said with each step represented the accumulated distance and not a label for that particular step. This understanding could be extended to other measuring tools representing multiple units, like a strip of paper that was five steps long. Relating it to their interpretive framework, Cobb et al. claimed "our instructional intent was therefore that a taken-as-shared spatial environment would become established in which distances are quantities of length whose numerical measures can be specified by actually measuring (Greeno, 1991)" (p. 132).

#### The Classroom Microculture

For this experiment, the teacher typically embedded a need to measure into a story that was read at the beginning of each class. Students would then work by themselves or in pairs to solve the measurement problem before discussing their solutions as a class. Stephan (1998) found that in addition to the teacher encouraging students to try to understand and stay involved at all times, the following were social norms for whole-class discussions:

- Students were obliged to explain and justify their reasoning.
- Students were obliged to listen to and attempt to understand others' explanations.
- Students were obliged to indicate nonunderstanding and, if possible, to ask the explainer clarifying questions.
- Students were obliged to indicate when they considered solutions invalid, and to explain the reasons for their judgment. (p. 133)

As for sociomathematical norms, Cobb et al. wanted discussions to focus on the conceptual, not the calculational (Cobb, 1998; Lampert & Cobb, 1998; Thompson, Philipp, Thompson, & Boyd, 1994). Explanations that simply describe using a measurement tool to get a numerical result would not be enough; for discussions to reflect the conceptual, the discourse would have to include justifications of how the measuring procedure structures the physical object into quantities of length.

#### The Emergence of the First Two Mathematical Practices

Cobb et al. describe the first two practices to create a backdrop for the emergence of the third practice, which they describe in greater detail. In Sessions 1 through 3 of the research, students participated in a narrative where a king measured his kingdom by stepping heel to toe. Some students began by placing their heel at the edge of a rug and only counting "one" when they took a second step, while others counted their first foot as "one" and their next step as "two." In the class discussion, some students correctly reasoned that each count should correspond to a physical part of the rug, and the first step should not be "missing." In subsequent sessions, students debated over different measurements due to different foot lengths and what to do when only part of a step was needed to reach the end of a measurement. This led to the first mathematical practice, *measuring by pacing*.

Over the next seven sessions engaged students in the next phase of the story, where the king was too busy to be in all parts of his kingdom to measure things with his feet. The solution that students had was to create a "footstrip," a piece of paper representing five of the king's steps. Now students could place the footstrip and count by 5 steps, and students recognized that this "big step" was more efficient for longer distances. More discussion occurred when only part of the footstrip was needed to complete a measurement, and a student proposed cutting the footstrip. This led researchers to observe an advance in measuring practice as students began to represent measurement as a property of the object being measured, not a physical act of placing the strip. Still, for some students "measuring with the footstrip structured the physical extension of an object into a sequence or chain of individual paces much as had measuring by pacing," while other students saw measurement as "the physical extension of the object being measured constituted into what they called a 'whole space' that was partitioned into paces" (p. 138). Therefore, for a distance of 12, students in the first group were more likely to look at the 12th pace in a sequence of steps, while the second group saw 12 as the distance from the first step through the 12th step.

#### The Emergence of the Third Mathematical Practice

### Methodological Reflections

### Trustworthiness, Replicability, and Commensurability

### Usefulness

### Limitations

### Conclusion

## About

### Mendeley

http://www.mendeley.com/catalog/participating-classroom-mathematical-practices/

### 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. doi:10.1207/S15327809JLS10-1-2_6

### BibTeX

@article{Cobb2001, 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.}, author = {Cobb, Paul and Stephan, Michelle and McClain, Kay and Gravemeijer, Koeno}, doi = {10.1207/S15327809JLS10-1-2\_6}, journal = {The Journal of the Learning Sciences}, number = {1/2}, pages = {113--163}, title = {{Participating in classroom mathematical practices}}, url = {http://www.tandfonline.com/doi/abs/10.1207/S15327809JLS10-1-2\_6}, volume = {10}, year = {2001} }