Cobb, Stephan, McClain, & Gravemeijer (2001)

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Cobb, Stephan, McClain, & Gravemeijer (2001)

Participating in Classroom Mathematical Practices


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.

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.

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:

  1. Students were obliged to explain and justify their reasoning.
  2. Students were obliged to listen to and attempt to understand others' explanations.
  3. Students were obliged to indicate nonunderstanding and, if possible, to ask the explainer clarifying questions.
  4. 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. This second practice, measuring by footstrip, allowed students to place the footstrip and count by 5 steps, recognizing 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

For describing the emergence of the third practice, Cobb et al. focus on two students: Nancy, who understood measurement as an accumulated distance, and Megan, who saw measurement as a sequence of steps. A 10-class session sequence using a narrative about Smurfs measuring objects with food cans led students to measure in a similar manner with Unifix cubes in class Sessions 11 and 12. All students connected the Unifix cubes in a rod that represented the length of an object, then counted the cubes one by one. In Session 13, the teacher said the Smurfs had problems carrying cans around to measure objects, and the solution that emerged in the classroom discussion was to measure using a 10-cube bar that students called a Smurf bar. Students quickly developed a taken-as-shared method of measuring with the Smurf bar while counting by tens.

In Session 14, Nancy measured a table with a Smurf bar iterated the bar twice while counting "10, 20." When the researcher asked Nancy to explain what she meant by "20," Nancy explained, "Twenty is how many little cubes we've done so far" (p. 139). When it was Megan's turn to measure, she used the Smurf bar to measure the height of an animal tank in the classroom. She iterated the bar twice while counting "10, 20" then extended the bar beyond the top of the tank while counting "30." When she tried counting the part of the bar needed to reach the top of the tank, she started counting individual cubes at 31, not 21. In this way, Megan was still seeing measurement as a counting exercise with a focus on a 10-cube bar, and not a length with its own spatial quantity. Nancy measured the tank and after 20 moved the bar and counted by ones: "21, 22, 23." Megan measured similarly but counted "31, 32, 33." Nancy explained her thinking to Megan by measuring again but only counting by ones, starting at 1 and counting cube by cube up to 23, and further explaining that a measurement of 30 would be above the tank, and 33 would be above that. Even though Megan accepted Nancy's explanation, Megan continued to measure as she'd done before.

In the whole-class discussion, the teacher asked Nancy and Megan to show how they would measure the length of the white board at the front of the classroom. After counting partway across the board to 20, a researcher asked "What does 20 mean?" Megan replied it was 20 food cans. The researcher asked if another student, Mitch, could go to the board and show how much space 20 food cans would take, and Mitch only indicated the space between 10 and 20, and Nancy corrected him. Nancy's intervention signaled continued development of the sociomathematical norm of what counted as an acceptable explanation, a practice that was supported by teacher and researcher questioning and intervention in small-group work. The disagreement between Mitch and Nancy, however, showed that measuring as the accumulation of distance was not yet taken-as-shared by the class. Cobb et al. point out that this is not about seeing a difference in two students' interpretations; rather, it's a social perspective that highlights how measuring with a Smurf bar entered the classroom discussion, with an attention to when certain reasoning becomes so widely shared that it needs no justification. Nancy's explanation was treated as legitimate, so she contributed to the third mathematical practice, measuring by iterating the Smurf bar. By changing his mind, Mitch also contributed to the practice. By correctly counting the cubes as the bar extended beyond the end of the board, Megan also contributed to the practice.

The day following the whole-class discussion, Session 16, involved the Smurfs cutting lumber of certain lengths. Using adding machine tape as lumber, Megan and Nancy measured with the Smurf bar and appeared to both think of measurement as an accumulation of distance. In the following whole-class discussion, one student seemed to alternate between measurement as the accumulation of distance and counting within each group of 10, as Megan had done previously with the animal tank. Megan helped the student by measuring while the teacher marked each iteration at her request. A researcher asked Megan to describe how many cans would fit in the first two iterations, and Megan answered 20 cans in the whole space, with 10 in each iteration. For a board 23 cans long, Megan counted three cubes beyond the second iteration, contributing again to the emergence of the third mathematical practice.

It was clear that Megan had reorganized her reasoning around measurement, and Cobb et al. changed their conjecture about Smurf bar use to speculate that Megan's use of the bar only resulted in the measuring-as-accumulation-of-distance when she made or accessed symbolic records of her activity. In an interview after Session 16, Megan reverted to the counting-within-tens strategy, became confused, the worked out the difficulty by labeling her measurement iterations with numerals. As the researchers saw the third mathematical practice emerge, they recognized how the first two practices supported the quick taken-as-shared use of the Smurf bar, and how the interventions by the researchers supported conceptual discussions. The content of these discussions was influenced by the teacher's use of records of students' prior activity, in which students could be prompted to reflect on prior measurement strategies. All together, Cobb et al. described the reflexive nature of their framework in the context of Megan's activity:

These claims about the various ways in which Megan's learning was situated indicate why we find it useful to view her act of reorganizing her reasoning (psychological perspective) as simultaneously an act of participation in the emergence of a new mathematical practice (social perspective). We can further clarify this point by noting that the teacher and researcher both assumed that Megan had already developed aspects of the type of reasoning that they wanted to engender. For example, the possibility that Megan may have been merely moving the Smurf bar end to end while reciting the number word sequence "10, 20," and so forth was not considered by the researcher. Instead, his intervention was premised on the assumption that she and Nancy were structuring space. As a consequence of this intervention, Megan became a participant in a conversation in which the taken-as-shared purpose for measuring may have differed from her initial intentions. What Megan appeared to learn in the course of this exchange was how to create records of her measuring activity so that she could reason in a way compatible with explanations that were treated as legitimate in public classroom discourse. Her learning was therefore supported by her participation in the emergence of the very practice to which she contributed by learning. It is in this sense that we elevate neither individual students' learning nor the emergence of communal classroom practices above the other but instead see them as reflexively related. (pp. 144-145)

Cobb et al. conclude their discussion of the third mathematical practice by discussing implications the practices have for further design and how these practices supported further practices, such as a fourth involving paper strips 100 cans long, and a fifth practice involving mental computation with numbers up to 100 (cf. McClain, Cobb, & Gravemeijer, 2000). The examination of these practices also has implications for the teacher's role in coordinating conceptual discourse in classroom activity (cf. Bowers & Nickerson, 1998; Thompson & Thompson, 1996).

Methodological Reflections

One methodological issue faced by Cobb et al. was how to determine the unit of analysis — in this case, an episode where a single mathematical theme is the focus of classroom activity and discourse. These episodes may or may not be the same as a student solving a problem, or correspond to a classroom activity. The critical episodes in the analysis were those that either supported or refuted a conjecture. These episodes might not seem important by themselves, but become critical when seen chronologically ordered with other episodes.

A second methodological issue for Cobb et al. was their identification of mathematical practices. The analysis describes how distinctions were made and guided by an instructional agenda. Cobb et al. summarized their three practices this way:

The hallmark of the first practice was that it was taken as shared that measuring by pacing structured the physical extension of an object into a chain of single paces. As the second practice of measuring with the footstrip emerged, it became taken as shared that the structured space created by measuring was no longer tied to the physical measuring activity but was instead treated as a property of the object being measured. In the case of the third practice, it became taken as shared that measuring with a Smurf bar involved the accumulation of distance. (p. 147)

Cobb et al.'s third methodological issue was using the analysis to clarify the social perspective. The classroom community is more difficult to distinguish than a physical being in that community, and a classroom practice cannot be observed directly any more than an individual's thought can be observed directly. The alternative for Cobb et al. was to form and test conjectures about communal practices (social perspective) and student reasoning (psychological perspective) in their analyses. Further clarifying, Cobb et al. state:

The distinction between the two interpretative perspectives resides in what may be termed the grain size with reference to which we characterize what they are doing. In the case of the psychological perspective, we view the teacher and students as a group of individuals who engage in acts of reasoning as they interpret and respond to each other's actions. In contrast, when we take the social perspective, we view the teacher and students as members of a local community who jointly establish communal norms and practices. (p. 148)

A fourth methodological issue related to the use of tools. Students used tools for their activities, and each tool did not necessary result in a new mathematical practice. Had students take-as-shared that measurement was the accumulation of distance when they used the footstrip, the use of the Smurf bar might have just supported that practice and not become the activity in which a third practice emerged. Different tools may have afforded different opportunities to elicit the mathematical practices, but this was a result of instructional design and not some simplistic correspondence of tool and practice. Such a correspondence assigns some intrinsic characteristic to the tool, which Meira (1998) and Roschelle (1990) referred to as the epistemic fidelity view. Assuming such a strong relation between a tool and a mathematical idea ignores students' prior activities with the tool. The footstrip, for example, would not have afforded the same opportunity for students to structure space had students not measured with their own steps first. The concept of affordances can be useful, but it's best when the data is analyzed from a psychological perspective to reveal how teachers and students adjust their interpretations and actions during tool use.

Trustworthiness, Replicability, and Commensurability

Typical interpretive analyses highlight critical episodes in the data to clarify assertions (Atkinson, Delamont, & Hammersley, 1988; Taylor & Boydan, 1984), but this presents episodes in isolation from the rest of the data. Conjectures and assertions made in relation to one episode are typically the result of supporting observations from prior episodes. A range of plausible analyses could come from the data, and therefore the trustworthiness of the analysis becomes an issue. Cobb et al. claim a study is trustworthy to the extent that the analysis is "both systematic and thorough" and that the "hallmark of an an analytical approach that satisfies this criterion is that inferences are treated as provisional conjectures that are continually open to refutation" (p. 152). This requires careful documentation of the analyses and testing of conjectures, and not just using the data to illustrate claims.

For replicability, one has to assume that the same mathematical practices would emerge using the same research design in a different setting. Cobb et al. acknowledge that this has been a very difficult task in education research, and mathematics education research is no exception. Cobb et al. assert:

In our view, a primary source of difficulty is that the independent variables of traditional experimental research are often relatively superficial and have little to do with either context or meaning. The conceptualization of the classroom as a matrix of variables is at odds with the approach we have taken in which the classroom microculture is viewed as a semiotic ecology that involves meaning making in which one thing is taken as a sign for another. (p. 153)

Cobb et al. believe that the answer to irreconcilability of previous studies is not a rigid, prescriptive approach to instruction. They state that "The conception of teachers as professionals who continually adjust their plans on the basis of ongoing assessments of their students' reasoning would in fact suggest that complete replicability is neither desirable nor, perhaps, possible (Ball, 1993; Carpenter & Franke, 1998; Gravemeijer, 1994)" (p. 153). The analysis therefore must attend to the enactment of an instructional sequence in a particular context against a backdrop of classroom and sociomathematical norms. Cobb et al. "claim that an analytical approach of this type can lead to greater precision and control by facilitating disciplined, systematic inquiry into instructional innovation and change that embraces the messiness and complexity of the classroom" (p. 154).

Usefulness

Cobb et al. make three points regarding the usefulness of this study. First is the documentation of the learning trajectory of the classroom community. Second is the way in which students' mathematical activity and learning is situated in the classroom community. Elaborating, Cobb et al. state, "For our purposes as instructional designers, the situated nature of this analytical approach is a strength when compared with alternative approaches that aim to produce context free descriptions of cognitive development that apparently unfold independently of history, situation, and purpose" (p. 155). The third point about usefulness relates to implications this analytical approach has for working with teachers and their development (cf. Ball & Cohen, 1996; Hiebert & Wearne, 1992). Cobb et al. suggest that

If the [instructional] sequences were justified solely with traditional experimental data, teachers would know only that the sequences had proved effective elsewhere but would not have an understanding of the underlying rationale that would enable them to adapt the sequences to their own instructional settings. In contrast, the type of justification that we favor offers the possibility that teachers will be able to adapt, test, and modify the sequences in their classrooms. In doing so, they can contribute to both the improvement of the sequences and the development of local instructional theories, rather than merely being the passive consumers of instructional innovations developed by others. (pp. 155-156)

Limitations

Cobb et al. acknowledge that the research was conducted in a reform-compatible classroom where explanation and justification of conceptual understandings was encouraged. In a more traditional classroom, collecting comparable data would likely have come from extensive student interviews, which would have been done out of context and difficult to perform in line with the chronology of the instructional sequence. Another limitation is the exclusion of the influences and contexts that extend beyond the classroom, such as race, gender, and class, all of which influence schooling (Apple, 1995; Zevenbergen, 1996). Citing Lave (1996), Cobb et al. agree that "school as a social institution involves an inherent contradiction between the functions of a universal socialization on the one hand and those of the unequal distribution of particular ways of knowing a cultural capital on the other hand" (p. 156). Alleviating this limitation would require a sociocultural perspective that considers broader contexts (Cobb & Yackel, 1996). Lastly, this study is limited in that it cannot be reduced to a set of procedures. Too much depends on knowing the mathematics deeply yet being able to take a social perspective that is unfamiliar to some researchers.

Conclusion

Cobb et al. remind readers that their goal was to become more effective in designing instruction that helps students learn. They maintain their claim that the social perspective was taken for non-ideological reasons; rather, a perspective that describes activity at the classroom level is needed for design research, and becomes clarified with individual student observation. Cobb et al. hoped their methods provided a path beyond the problems typical of experimental research designs, with added benefits for collaborating teachers.

References

Corrolary

APA
Cobb, P., Stephan, M., McClain, K., & Gravemeijer, K. (2001). Participating in classroom mathematical practices. Journal of the Learning Sciences, 10(1/2), 113–163. doi:10.1207/S15327809JLS10-1-2_6
BibTeX
@article{Cobb2001,
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}
}