15

Supporting the Improvement of Learning and Teaching in Social and Institutional Context

Paul Cobb

Vanderbilt University

 

The charge the editors gave me for this chapter was to “push the envelope” by raising methodological issues and by indicating phenomena that might be included within the scope of future cognitive science research. In seeking to fulfil this charge in what I hope will prove to be a constructive manner, I consider two points of methodology in the first part of the chapter. The first of these concerns the relation between the development of cognitive or learning theory on the one hand and the improvement of instructional practice on the other. The second methodological point brings to the fore the need for interpretive frameworks that enable us to analyze students' learning as it occurs in the social context of the classroom. This discussion then provides a backdrop for the second part of the chapter in which I focus on two general phenomena that typically fall beyond the purview of cognitive research but that are significant in the current era of educational reform. These phenomena concern the institutional context of the school and the cultural diversity of the students whose learning we seek to support.

COGNITIVE THEORY AND INSTRUCTIONAL PRACTICE

It is helpful if I clarify at the outset that I am neither a psychologist nor a cognitive scientist. Instead, my professional identity is that of a mathematics educator. As a consequence, my colleagues and I spend a considerable amount of our time in classrooms attempting to understand and improve the process of learning and teaching mathematics in specific content domains. As the work of several contributors to this volume indicates, this focus on classroom instructional processes and students' learning as they participate in them also motivates an important strand of cognitive science research (e.g., Griffin & Case, 1997; Lehrer & Romberg, 1996; Lehrer, Schauble, Carpenter, & Penner, in press; Moss & Case, 1999). In our own work, the basic methodology that we have sought to refine over the past 13 years is that of the classroom teaching or design experiment (Brown, 1992; Cobb, in press; Confrey & Lachance, in press; Simon, in press). Our goal in these experiments, which can last up to a year, is both to develop sequences of instructional activities and associated tools, and to conduct analyses of the process of the students' learning and the means by which that learning is supported and organized. Research of this type falls under the general heading of design research in that it involves both instructional design and classroom-based research.

Gravemeijer (1994) wrote extensively about the first aspect of the design research cycle shown in Fig. 15.1, instructional design, and clarifies that the research team conducts an anticipatory thought experiment when preparing for the design experiment. In doing so, the team formulates a hypothetical learning trajectory that involves conjectures about both a possible learning route or trajectory that aims at significant mathematical ideas, and the specific means that might be used to support and organize learning along the envisioned trajectory. It is important to emphasize that these two aspects of a hypothetical trajectory are interrelated in that the realization of the conjectured learning route is seen to depend on the use of the proposed means of support. Thus, the proposed learning route might well depart quite radically from accounts of cognitive development in the domain of interest that are based on either naturalistic observations or on analyses of the learning of students who have received traditional forms of instruction. The envisioned trajectory is therefore hypothetical in the sense that it embodies hypotheses about what might be possible for students' mathematical learning in a particular domain.

The speculative nature of this trajectory does not of course imply that it is fabricated in an unrestrained ad hoc manner. Instead, its formulation is subject to two types of constraints. The first type stems from the designers' general theoretical perspective on learning in instructional situations. Carver (chap. 12) illustrates constraints of this type when she discusses what she terms metaprinciples of domain-independent cognitive theory, such as building on prior knowledge. In the case of myself and my colleagues, the background theory that orients our design work is a version of social constructivism.¹ The second type of constraint on the design process derives from the syntheses that we develop of prior research on learning in the mathematical domain of interest. Significantly, we have found that exclusively cognitive studies that merely report misconceptions or sequences of levels in students' reasoning have been, at best, of limited value to us. In contrast, studies that attempt to document the process of students' learning in innovative instructional situations have proven to be of great value. Unfortunately, the number of studies of the first type heavily outweighs the number of studies of the second type in most mathematical domains.

Turning now to consider the proposed means of support inherent in a hypothetical learning trajectory, it is important to clarify that they are construed broadly and include:

• Resources typically considered by materials developers (e.g., instructional activities together with the notational schemata and the physical and computer-based tools that students might use).
• Classroom social context (e.g., the general classroom participation structure and the nature of specifically mathematical discourse).
• The teacher's proactive role in supporting the emergence of increasingly sophisticated ways of reasoning.

Although instructional planning at this level of detail is unusual in the United States, there are several notable exceptions (e.g., Confrey & Smith, 1995; Lehrer et al., in press; Simon, 1995). In addition, an encompassing approach of this type is the norm in Japan where members of professional teaching communities often spend several years teaching and revising the hypothesized learning trajectories that underpin a sequence of mathematics lessons (Stigler & Hiebert, 1999).

It is important to stress that the conjectures inherent in a hypothetical learning trajectory are just that—they are tentative, provisional, eminently revisable conjectures that are tested and revised on a daily basis once the experiment begins. Our goal when experimenting in a classroom is therefore not to try to demonstrate that the instructional design formulated at the outset works. Instead, it is to improve the design by testing and modifying conjectures as informed by ongoing analyses of both students' reasoning and the classroom learning environment. As a consequence, although we formulate a hypothetical learning trajectory in advance and also outline possible types of instructional activities, we develop the specific instructional activities used in the classroom only a day or 2 before they are needed. I mention this to clarify that the methodology is relatively labor-intensive. It is essential, for example, that the senior researchers be present in the classroom each day and that they lead debriefing meetings of the research team after every classroom session. Formal design experiments should therefore not be confused with informal explorations in which research assistants are delegated to work in a classroom in a less principled way.

My immediate purpose in outlining the methodology that my colleagues and I use is to highlight what I take to be a defining characteristic of design research, the tightly integrated cycles of design and analysis. Gravemeijer (1998) differentiated what he refers to as daily minicycles from macrocy-cles that span an entire teaching experiment. This latter, longer term cycle involves a retrospective analysis that is conducted once the design experiment is completed and that can feed forward to guide the formulation of a revised learning trajectory for follow-up teaching experiments.

My larger purpose in discussing design experiments is to describe a methodological approach in which instructional design serves as a primary setting for the development of theory. This characteristic of design experiments becomes apparent once I clarify that the questions and concerns that arise while an experiment is in progress are typically pragmatic and relate directly to the goal of supporting the participating students' learning. In contrast, the intent when conducting a retrospective analysis of an experiment is to contribute to the development of a domain-specific instructional theory (see Fig. 15.2). This theory emerges over the course of several macrocycles and consists of a demonstrated learning route that culminates with the emergence of significant mathematical ideas, and substantiated means of supporting and organizing learning along that trajectory. As Steffe and Thompson (in press) clarified, it is this theory that makes the results of a series of design experiments potentially generaliz-able, even though they are empirically grounded in analyses of only a small number of classrooms. In their terms, this is generalization by

means of an explanatory framework rather than by means of a representative sample in that the insights and understandings developed and tested during a series of experiments can inform the interpretation of events and thus pedagogical planning and decision making in other classrooms.

Design experiments in which the development of theory and the improvement of an instructional design coemerge can be contrasted with an alternative approach to design that involves a sequence of steps:

1. The development of cognitive theory.
2. The derivation of principles for design from the cognitive theory.
3. The translation of the principles into concrete designs.
4. The assessment of the designs to test whether they work as anticipated.

It is, of course, debatable whether this putative sequence is ever strictly adhered to in practice. However, research reports are often written so as to imply that the development of the design involved a one-way chain of reasoning from cognitive theory to instructional practice. The general approach is therefore alive in research discourse, where it is frequently treated as the ideal. It is this ideal that I want to scrutinize from my perspective as a mathematics educator.

As an initial observation, I should clarify that assessments conducted as part of this design approach can involve the generation of qualitative and process data as well as quantitative outcome measures. Thus, my primary concern is not with the nature of the assessment data, but with the intent that underlies their generation. The key point to emphasize is that, in this orientation to design, the anticipated or targeted ways of reasoning (often those of experts) are typically used as normative standards against which students' reasoning is compared. As a consequence, the purpose of the assessment is usually to evaluate rather than to understand students' reasoning in that the analysis focuses on the extent to which their reasoning deviates from the standard rather than on understanding how they are actually reasoning and on explaining why they came to reason in these ways. As someone who sees considerable merit in Toulmin's (1963) argument that explanation and understanding constitute the driving forces of scientific inquiry, I find the priority given to evaluation troubling. In addition, I am also concerned about the relatively weak feedback loop from the final assessment phase to the design principles and cognitive theory. It is not clear, for example, that these principles and the underlying theory are open to the results of an unfavorable assessment of a specific design. This is disconcerting if one takes the view that educational reform should be an ongoing, continual process of iterative improvement characterized by what Lampert (1990) termed a “zigzag between conjectures and refutations.”

Beyond these general observations, I also wonder about the practical feasibility of this idealized approach to design in specific mathematical domains. For example, my colleagues and I have recently completed a sequence of two classroom design experiments that focused on statistical data analysis at the middle school level. In preparing for these experiments, we conducted a reasonably extensive review of the literature but found only a small number of studies that could guide our formulation of a hypothetical learning trajectory.² There did not even appear to be a consensus on the overarching statistical ideas that should constitute the potential endpoint of a learning trajectory. However, rather than waiting for development of cognitive theory, we drew on the available literature to formulate initial conjectures about major shifts in students' statistical reasoning and the means by which these shifts might be supported. Not surprisingly, a number of these conjectures proved to be unviable when we began experimenting in the classroom. We therefore began the process of revising conjectures online and eventually formulated a new learning trajectory that was empirically grounded in our work in the two classrooms. We contend that what we learned about the learning and teaching of statistical data analysis by proceeding in this way can contribute to an emerging instructional theory that aims at a significant statistical idea, that of distribution.

It is important to note that the lack of an adequate research base is not unique to statistical data analysis. The same observation applies to areas such as algebra and geometry that are also undergoing relatively profound disciplinary changes (Cobb, 1997; Kaput, 1994; Lehrer, Jacobson, Kemney, & Strom, 1999). In domains such as these, the idealized top-down approach to instructional design in which cognitive theory trickles down to instructional practice appears to be untenable. Instead, the bootstrapping approach integral to design research in which theory and instructional designs evolve together appears to be more feasible. The contributions to this volume by Case and colleagues and by Lehrer and Schauble serve to substantiate this claim. Some years ago, the Dutch mathematician and mathematics educator Hans Freudenthal (1973) argued that psychology should follow instructional design rather than the reverse. Although this might be too strong, it does seem reasonable to think in terms of a symbiotic relationship between cognitive theory and instructional practice, and between the discipline of cognitive science and so-called applied fields such as mathematics education.

In concluding this discussion, it is worth clarifying that the arguments I have advanced about the relative merits of design research have not been cast in absolute terms. Instead, they reflect the view that methodologies are conceptual tools that are more appropriate for some purposes than for others. It would, therefore, be naive to reject treatment-control experiments of the type frequently conducted by cognitive scientists out of hand. I would, however, contend that such experiments are relatively blunt instruments that are not well-suited to the demands of instructional design. A fundamental difficulty is that these experiments do not produce the detailed kinds of data that are needed to guide the often subtle refinements made when improving an instructional design. In contrast, treatment-control experiments are highly appropriate for another purpose for which design research is less well-suited, that of generating the types of data that we need in order to participate in public policy debates about education. Relevant assessment studies of this type include those that compare matched groups of students in experimental and traditional classrooms on a range of measures such as basic computational skills, mathematical understanding, motivations for engaging in mathematical activity, and beliefs about the general nature of mathematics in school (e.g., Cobb et al., 1991; Wood & Sellers, 1996). We have found the results of studies of this type to be invaluable in persuading school district administrators and other policymakers of the potential value of the instructional sequences developed during design experiments. In addition to conducting further studies of this type as part of our ongoing work in statistics, we also plan to compare the capabilities of middle school students in experimental classrooms with university undergraduate and graduate students. These studies should enable us to both triangulate the findings of a series of design experiments and to frame our claims in the linguistic currency of public policy discourse, that of inferential statistics.

LEARNING AND TEACHING IN SOCIAL CONTEXT

The second methodological issue I raise builds on the first by focusing on the need, within design research, for interpretive frameworks that enable us to analyze students' learning as it occurs in the social context of the classroom. As we know all too well, classrooms are messy, complex, and sometimes confusing places. One of the concerns that my colleagues and I have struggled with as we have worked in classrooms is that of developing an analytic framework that enables us to come to terms with this complexity so that we can begin to see some pattern and order in what appear at first glance to be ill-structured events.

I would argue that an interpretive framework that is appropriate for the purposes of instructional design should satisfy the following criteria:

1. It should result in analyses that feed back to inform the improvement of instructional designs.
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 enable us to document the collective mathematical learning of the classroom community over the extended periods of time spanned by design experiments.

The rationale for the first criterion follows directly from the tightly integrated cycles of design and analysis that I discussed at both the micro and macrolevel (see Figs. 15.1 and 15.2). The second criterion, which emphasizes the importance of focusing on the nature and quality of individual students' reasoning, probably needs little justification for most cognitive scientists. Nonetheless, I give a brief rationale to clarify the pragmatic origins of this criterion. Briefly, the classroom sessions that we conduct during a design experiment are frequently organized so that students initially work either individually or in small groups before convening for a whole-class discussion of their interpretations and solutions. A pedagogical strategy that we have found productive involves the teacher and one or more members of the project staff circulating around the classroom during individual or small group work to gain a sense of the diverse ways in which students are interpreting and solving instructional activities. Toward the end of this phase of the lesson, the teacher and project staff members confer to prepare for the whole-class discussion. In doing so, they routinely focus on the qualitative differences in students' reasoning in order to develop conjectures about mathematically significant issues that might, with the teacher's proactive guidance, emerge as topics of conversation. Their intent in doing so is to capitalize on the diversity in the students' reasoning by identifying interpretations and solutions that, when compared and contrasted, might lead to substantive mathematical discussions that advance their current pedagogical agenda. It is because of this crucial role that a focus on individual students' reasoning plays in our instructional decision making that we require an analytic approach that takes account of the diverse ways in which students participate in communal classroom practices.

The third criterion, which introduces a focus on the mathematical learning of the classroom community, is probably more controversial. Its rationale stems from the observation that the conjectures the designer develops when preparing for an experiment cannot be about the trajectory of each and every student's learning for the straightforward reason that there are significant qualitative differences in their reasoning at any point in time. In my view, descriptions of instructional approaches written so as to imply that all students will reorganize their thinking in particular ways at particular points in an instructional sequence involve, at best, questionable idealizations. An issue that arises is therefore that of clarifying the focus of the envisioned learning trajectories that are central to our (and others') work. The resolution that we propose involves viewing a hypothetical learning trajectory as consisting of conjectures about the collective mathematical learning of the classroom community. This proposal in turn indicates the need for an interpretive framework that enables us to analyze collective mathematical learning as well as the learning of the individual students as they participate in this communal process.

The interpretive framework that we currently use to organize our analyses of classroom data is shown in Fig. 15.3. As can be seen, it addresses the last two criteria by coordinating a psychological perspective on individual students' reasoning with a social perspective on communal classroom processes (cf. Cobb & Yackel, 1996). The three entries in the column headed “Social Perspective” indicate three aspects of the classroom microculture that we have found it useful to differentiate. The first of these, classroom social norms, provides a way of documenting what both Erickson (1986) and Lampert (1990) termed the “classroom participation structure.” Examples of social norms for whole class discussions include that students are obliged to explain and justify solutions, to attempt to make sense of explanations given by others, to indicate understanding

or nonunderstanding, and to ask clarifying questions or challenge alternatives when differences in interpretations have become apparent.

It is readily apparent that these social norms are not specific to mathematics, but instead apply to any subject matter area. For example, one might hope that students would explain their reasoning in science or history classes as well as in mathematics. In contrast, sociomathematical norms focus on regularities in classroom actions and interactions that are specific to mathematics (cf. Simon & Blume 1996; Voigt, 1995; Yackel & Cobb, 1996). Examples of sociomathematical norms include the criteria that are established in a particular classroom for what counts as a different mathematical solution, a sophisticated mathematical solution, and an efficient mathematical solution, as well as for what counts as an acceptable mathematical explanation.

If sociomathematical norms are specific to mathematics, then classroom mathematical practices are specific to particular mathematical ideas and are thus concerned with the emergence of what is traditionally called mathematical content. A hypothetical learning trajectory can in fact be viewed as consisting of an envisioned sequence of mathematical practices together with the means of supporting and organizing the emergence of each practice from prior practices. To give one brief example that is taken from a second-grade design experiment, the students typically solved arithmetical tasks by counting by ones at the beginning of the school year. However, some of the students also developed solutions that involved conceptualizing numbers as composed of units of 10s and 1s. When they did so, they were obliged to explain their numerical reasoning. This contrasted with observations we made at the midpoint of the school year in that students who conceptualized numbers in this way were no longer obliged to give justifications. Instead, it appeared to be taken for granted, at least in public classroom discourse, that numbers were composed of units of 10s and 1s. This way of construing numbers was now treated as a self-evident mathematical fact. It was at this juncture that we inferred that this way of talking and reasoning had been established as a classroom mathematical practice.

It is important to clarify that an assertion that a mathematical practice has been established does not enable us to make claims about the reasoning of any particular student. As a consequence, we find it essential to complement an analysis of the mathematical development of the classroom community with analyses of individual students' mathematical activity. These psychological analyses invariably reveal that there are qualitative differences in students' reasoning even as they participate in the same practice. In general, a practice can be thought of as constituting the immediate social context in which students' learning occurs. We take the relationship between practices and the reasoning of the participating students to be reflexive (cf. Cobb et al., in press). Although relationships of this type are nondeterministic, they are nonetheless relatively strong and imply that individual students' reasoning does not exist apart from communal practices and vice versa. On the one hand, we view the practices being continually regenerated by the teacher and students as they mutually adapt to each other's actions in the classroom. On the other hand, we view individual students' reasoning as acts of participation in communal practices and contend that the types of reasoning we observe would not have emerged but for the students' history of participation in specific classroom mathematical practices. This does not, of course, imply that students can only reason in particular ways if they are actually engaging in collaborative activity with others in the classroom. The eventual goal is that students will be able to participate in communal practices on their own outside the classroom.³ The processes involved in this development are discussed in some detail by Beach (1999).

It is apparent from this brief overview that analyses developed by following the general approach of coordinating social and psychological perspectives result in situated accounts of learning. I can further clarify this point by drawing on an analogy that Herbert Simon introduced in his presentation at the symposium that led to this book. He proposed that the relation between mind and the environment is analogous to that between jello and the mold. In developing the entailments of the analogy, he argued that just as we should study the mold if we want to understand the shape of the jello, so we should analyze the environment if we want to understand the mind. Extending this analogy to the analytic approach I have presented, individual students' reasoning corresponds to the jello, and communal practices correspond to the mold. The analogy helps to make explicit a contrast with the cognitive paradigm discussed by Simon in that the mold (i.e., communal practices) is not viewed as pregiven with respect to the mind. Instead, the mold is viewed as a collective accomplishment of the community of which the students are members. Further, the reflexive relation between the psychological and social perspectives implies that students actively contribute to the continual regeneration of the mold that enables and constrains the development of their reasoning.

Thus far, in outlining the interpretive framework that my colleagues and I currently use, I have said little about the first of the three criteria that I stated earlier, namely that analyses should feed back to inform the ongoing instructional design effort. In this regard, I contend that the situated nature of our analyses is a good thing given the concerns and interests that motivate our work in classrooms. It is a strength rather than a weakness in that it results in accounts of students' learning that are tied to analyses of the actual environment in which that learning occurs (i.e., the classroom microculture). As a consequence, we can disentangle aspects of this environment that served to support and organize the development of the students' reasoning. This, in turn, makes it possible for us to develop testable conjectures about ways in which we might be able to improve those means of support and thus our instructional design.

It is important to note that this argument in support of a situated approach is not ideological, at least in the pejorative sense of the term. I have not, for example, presented a barrage of citations to claim that cognition is situated and that anyone who thinks otherwise has got the mind wrong. Instead, the gist of my argument is that it is useful for the purposes of design research to view students' reasoning as situated with respect to communal classroom processes. The argument is, therefore, pragmatic and reflects the view that the interpretive framework, as an aspect of our methodology, is a conceptual tool that is appropriate for certain purposes but not others.

BROADENING THE CONTEXT

Thus far, in raising methodological issues, I have limited my focus to the social context of the classroom. I now discuss two broader phenomena that are pragmatically significant in the current era of reform, the institutional context of the school, and the cultural diversity of the students whose learning we attempt to support. Although my intent is to bring to the fore two complex sets of issues that are not usually considered in cognitive science research, I should acknowledge that my own research has also failed to take explicit account of these phenomena to this point. Consequently, in proposing issues that might be included in future cognitive science research, I also indicate directions in which I believe my own and my colleagues' work should move. As our attempts to develop an initial theoretical orientation are still ongoing, my primary concern is to justify why cognitive scientists might want to give the two phenomena serious attention.

The Institutional Context of the School

To frame issues relating to school context, I return briefly to the design experiment methodology. An important aspect of our work concerns the negotiations that occur as we seek entry into a school. In these negotiations, we request that we be given responsibility for a group of students' mathematics instruction for an agreed-upon period of time. Further, as part of the negotiation process, we seek to insulate the design experiment classroom from many of the institutional constraints of the school (e.g., use of adopted textbooks, content coverage prescriptions, norms for the structure of lessons and students' engagement in them). Negotiations of this type are, of course, not unique to design research but are instead undertaken routinely in a range of school-based research endeavors. Typically, there are good reasons for attempting to insulate the school-based research site in this way. In the case of design research, for example, I would argue that this insulation is justifiable when the purpose is to develop a local instructional theory by investigating what might be possible in students' mathematics education. However, it can become problematic in this and other types of research if the researchers overlook the effort they expend to create and maintain temporary lacunae within the institutional structure of the school. Documenting these efforts is particularly important if the eventual goal is to use the products and findings of the research as a basis for the development of a professional teaching community to be established in collaboration with a group of teachers.

A primary difficulty that arises concerns the potential tensions and conflicts between the forms of instruction institutionalized in the school (or school district) and those that emerge both within the capsule of the research site and within the professional teaching community. Frequently, these two forms of instruction are organized by contrasting overall intentions or motives. The motive of school instruction might, for example, be competent performance on a relatively limited range of tasks as assessed by teachermade, textbook, and state-mandated tests of skills. However, the overall motive of the forms of instruction nurtured in the professional teaching community might be mathematical understanding as assessed by the teachers' observation and documentation of their students' reasoning.

The need to make this potential conflict a focus of inquiry becomes particularly pressing if one is concerned about the sustainability of the instructional approaches the teachers create in the course of the collaboration. In this regard, Cole (1996) noted that although it might be possible to nurture novel forms of instruction in almost any institutional setting by supplying the material and human resources necessary to create appropriate conditions, it is often not possible to sustain those conditions once the resources have been withdrawn. This observation obviously indicates the importance of involving administrators, parents, and other stakeholders in the activities of the professional teaching community (cf. Lehrer & Shumow, 1997; Price & Ball, 1997). The intent in doing so is to bring the instructional goals of the professional teaching community and of the school into closer alignment, for the teachers' emerging instructional approaches must come to fit with institutional goals if they are eventually to be sustained by local resources.

This discussion of sustainability illustrates the complexity of teacher professional development in that the goal is to support both the profes-sionalization of teaching and the emergence of institutional conditions that enable teachers to act as professionals. The realization that professional-ization is distributed in turn gives rise to a theoretical challenge, that of coming to understand the relations between teachers' participation in multiple communities. The design experiment methodology provides a point of reference in that the focus is on students' learning as they participate in the practices of a single community, that of the classroom. In contrast, analyses of teachers' learning that are adequate for our purposes necessitate the development of interpretive frameworks that make it possible for us to understand how teachers' participation in two distinct communities, the professional teaching community and the school community, influence their pedagogical activity in the classroom. To be sure, an appropriate framework should enable us to analyze individual teachers' planning and decision making. However, in addition to enabling us to locate that reasoning in the immediate social context of their interactions with students, it should also enable us to see their reasoning as situated within the broader institutional context of their schools.

The extent of this theoretical challenge is indicated by a dichotomy that is apparent in the literature on teacher change (cf. Engestrom, 1998). One body of scholarship focuses on the role of professional development in supporting teachers' views of their instruction and themselves as learners. A second body of scholarship is concerned with the structural or organizational features of schools and with how changes in these conditions can lead to changes in classroom instruction. The task, as I see it, is to transcend the dichotomy between these two largely independent lines of work, one oriented toward teachers' learning as they participate in professional teaching communities and the other toward broader policy considerations. Although this challenge is central to our own and others' current work (e.g., Cobb & McClain, 1999; Spillane, 1999; Stein, Silver, & Smith, 1998). I contend that it is also of critical importance to cognitive scientists who want to leverage their research by collaborating with teachers and administrators to influence public school instruction.

Cultural Diversity

As was the case with the social context of the school, the design experiment methodology provides a useful starting point from which to approach issues of cultural diversity. I have already noted that the design experiment classroom is insulated from the institutional constraints of the school to a considerable degree. A second kind of insulation is inherent in the analytic approach that my colleagues and I follow in that we characterize students' reasoning solely in terms of their participation in the practices of the classroom community. In adopting this analytic stance, we fail to take account of students' participation in the practices of either their local home communities or of the cultural groups within the wider society of which they are members. This second form of insulation is, I believe, less justifiable than the first, particularly as our espoused goal is to support all students' development of what current reform documents label as mathematical power (NCTM, 1989).

The consequences of this limitation become apparent when we consider the various ways in which students' participation in out-of-school practices can influence their activity in the classroom. There is evidence, for example, that students make a range of suppositions and assumptions when interpreting mathematical problems set in real-world scenarios that reflect the practices of their communities. As an illustration, Ladson-Billings (1995) presented the following task to suburban students and to their inner city counterparts:

It costs $1.50 to travel each way on the city bus. A transit system fast pass costs $65.00 a month. Which is the more economical way to get to work, the daily fare or the fast pass? (p.123)

Ladson-Billings reported that the suburban students assumed that the scenario was about a person who commuted to work in the city. They therefore calculated the cost of the daily fare as $3.00 per day for 5 days each week. In contrast, the inner city students asked a number of questions: How many jobs are we talking about? Are these part-time or full-time jobs? As Ladson-Billings noted, these differing responses reflect differences in the economic practices of the two groups of students' home communities. Although there has been some acknowledgment in the cognitive science literature that the process of interpreting story problems involves implicit suppositions and assumptions of this type (e.g., De Corte, Verschaffel, & De Win, 1984), they are typically characterized in purely individualistic psychological terms. In exploring links with the practices of students' home communities, Ladson-Billings oriented us toward an explanation that, in my view, makes such issues more tractable for teachers.

In addition to influencing how students interpret instructional activities in the classroom, students' home communities can involve differing norms of participation, language, and communication, some of which might actually be in conflict with those that the teacher seeks to establish in the classroom (Cazden, 1988; Delpit, 1988; Fine, 1987; Philips, 1972). For example, Cazden (1988) identified two distinct types of narrative style, topic-centered and episodic, that differ in thematic structure and organization when she analyzed a group of first-grade students' activity during sharing time at school. As she clarified, topic-centered narratives focus on one topic whereas episodic narratives focus on a number of events with shifting scenes. Cazden reported that most of the students who gave topic-centered narratives were White whereas most who gave episodic narratives were African American. This finding takes on added significance in light of a follow-up study in which Cazden investigated adults' reactions to videorecordings of children giving topic-centered and episodic narratives. She discovered that the White adults found the episodic narratives hard to follow and typically judged the child to be of low ability. In contrast, the African American adults were more likely to value both narrative styles even though they were aware of differences between them. These results are important given that schools, in general, and mathematics classrooms, in particular, privilege topic-centered narratives. In particular, the results illustrate how differences in the discourse patterns of students' home communities may influence how they are perceived in the classroom and how their contributions are treated by teachers and other students. These reactions can, in turn, influence whether students perceive themselves as supported or as marginalized and silenced in the classroom, thus leading to inequities in both opportunities and motivations to learn.

It should be clear from this discussion of Cazden's work that the implications of potential conflicts between home and school norms are not limited to the establishment of general classroom social norms and the classroom participation structure but also relate specifically to the teaching and learning of mathematics. For example, the notion of mathematical proof is central to any coherent instructional program. Within the discipline, proof serves both as a means of inquiry for individual mathematicians and as a means by which they convince others of the validity of their reasoning (Hanna, 1989; Schoenfeld, 1983). It is part of the linguistic currency of mathematics and can be seen to originate in discussions conducted in the early grades in which students are expected to justify their reasoning by developing and refining arguments. In this case of proof, potential conflicts between the forms of discourse targeted in the mathematics classroom and the norms of communication in students' home communities therefore penetrate to the core of teachers' instructional agendas. This observation is not, however, unique to proof but also holds for other central aspects of mathematical activity such as modeling, symbolizing, inference, and indeed the very process of mathematizing (cf. O'Conner, 1998).

From the point of view of instructional design, the issues I have raised emphasize the importance of attending explicitly to the diversity in the practices of students' home communities when formulating the starting points of a hypothetical learning trajectory. In addition, they indicate the need to be cognizant of these out-of-school practices when analyzing the ways in which individual students participate in the practices of the classroom community. This broadening of the analytical approach is crucial given that analyses of classroom events inform pedagogical judgments and thus impact the quality of students' mathematics education. At a minimum, a broader perspective would enable cognitive scientists to guard against the danger of attributing inherent cognitive deficiencies to students from home communities whose practices conflict with those that are privileged both in school and in psychological research. In addition, it would enable cognitive scientists interested in influencing school instruction to view the cultural diversity of students' discourse and reasoning as an instructional resource to be capitalized on rather than as an obstacle to be overcome. In science education, design experiments conducted by Warren and Rosebery (1995) that focused on scientific argumentation indicate the viability of such an approach.

In stepping back still further to consider the practices of cultural groups within wider society, I draw primarily on the work of my colleague Lynn Hodge (1999). She developed an initial orientation by introducing the notions of dominant cultures and cultural capital (Bourdieu, 1977). As she clarified, cultural capital refers to the ways of speaking, writing, dressing, and so forth that are, in a sense, possessed by members of identifiable cultural groups and that are passed from one generation to the next (see also Zevenbergen, 1996). The idea of dominant cultures indicates that although all cultural groups within society possess cultural capital, these diverse forms of capital are treated differently by a range of societal institutions, including schools. As a consequence, schools can be seen to play a central role in maintaining the dominance of certain cultural groups. Students who talk and act in ways consistent with a dominant culture are, for example, usually at an advantage in gaining access to educational, economic, and professional opportunities.

For mathematics educators and, I suggest, cognitive scientists, the notions of dominant cultures and cultural capital indicate the importance of scrutinizing the instructional goals that constitute the endpoints of learning trajectories. The purpose in doing so is to examine whether the goals can be justified in terms of participation in a democratic society or whether they are tied to taken-for-granted policies and practices that serve to perpetuate the regeneration of inequities. In the two design experiments I referred to earlier that focused on statistical data analysis in the middle grades, we did in fact conduct such an examination by clarifying for ourselves why statistics should be taught in school. Although we identified several rationales in the literature, the argument that we found most compelling took account of the increasing premium being placed on statistical reasoning as a consequence of the increasing use of computers, not just in the discipline but in society more generally (cf. Cobb, 1997). We noted, for example, that there is much talk of preparing students for “the information age,” but without fully appreciating that much of this information is statistical in nature. This shift has significant implications for the discourse of public policy and thus for democratic participation and power. It is already apparent that debates about public policy issues tend to involve reasoning with statistical data and that this discourse is increasingly becoming the language of power in the public policy arena. These observations led us to conjecture that the inability to participate in this discourse results in de facto disenfranchisement that can spawn alienation from and cynicism about the political process. From this, we concluded that statistical literacy that involves reasoning with data in relatively sophisticated ways relates directly to both equity and participatory democracy.

Given this image of students, as increasingly substantial participants in public policy discourse, the competencies that we considered critical were those of developing and critiquing data-based arguments. As we have noted elsewhere (Cobb, 1999), in supporting students' development of these competencies in the design experiments, we were simultaneously supporting their eventual participation in the practices of dominant groups. As a consequence, the approach we took to statistics is compatible with Delpit's (1988) admonition that instruction should aim to support students' appropriation of what she called the culture of power. I would argue that learning trajectories that involve the type of analysis I have illustrated, therefore, have the potential to contribute to the reconstruction of schools as institutions that redress rather than regenerate inequities. In my view, such an approach is worthy of consideration by cognitive scientists, given that their work has sometimes been criticized for unquestion-ingly accepting traditional instructional goals and thus, for tending to support the status quo.

As a further observation, it should also be clear from the illustrations I have presented that issues of diversity, like those relating to the institutional context of the school, give rise to the theoretical challenge of accounting for participation in multiple communities. Hodge's (1999) analysis is again helpful in that she proposed that cultural diversity can be treated productively as a relational notion. The approach she developed grounds students' activity in the practices of the three types of communities I have discussed; the classroom community, the local home community, and the identifiable cultural groups within wider society. She argued that issues that fall under the general rubric of diversity can then be seen to emerge from students' participation in the practices of these different communities. Further, in this scheme, the mathematics classroom is viewed as the immediate arena in which issues of diversity play out at the microlevel of face-to-face interaction. As a consequence, diversity is not equated with students' culturally situated ways of reasoning and talking per se, but is instead framed in terms of relations between those ways of acting and the norms and practices established in the classroom (Crawford, 1996; Moll, 1997; Warren, Rosebery, & Conant, 1994). The arguments I have presented to support a focus on diversity when formulating the starting points and ending points of learning trajectories can, in fact, be viewed as illustrations of the relational view. Although both the specific remarks I have made and the more general relational view are preliminary, I nonetheless hope that they might in some way sustain the emergence of diversity as a shared interest of cognitive scientists and of subject matter specialists such as mathematics educators. The immediate challenge, then, is to develop analytical approaches that treat students' reasoning as situated with respect to their history of participation in the practices of particular communities.

CONCLUSION

In the first part of this chapter, I raised two methodological issues and, in doing so, developed a pragmatic justification for analyzing students' reasoning in situated terms. In the second part of the chapter, I critiqued the situated perspective inherent in the design research methodology as not going far enough. My concern was not whether this situated perspective remained faithful to the basic tenets of what might be termed a strong situated cognition paradigm. Instead, my concerns were again pragmatic and stemmed from the need to integrate a focus on both the institutional context of the school and the student's cultural diversity into our work. This led me to argue that it is insufficient to see teachers' and students' activity as situated merely with respect to the norms and practices of the classroom community. The overriding theoretical issue that then emerged involves coming to understand how teachers' and students' participation in multiple communities plays out in the arena of the classroom. It is this issue that I propose as a prime candidate for inclusion in future cognitive science research.

ACKNOWLEDGMENTS

The analysis reported in this chapter was supported by the National Science Foundation under Grant No. REC9814898 and by the Office of Educational Research and Improvement under Grant No. R305A60007. The opinions expressed do not necessarily reflect the views of the Foundation or of OERI. I am grateful to Sharon Carver for her constructive comments on a previous draft of this chapter.

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¹ A discussion of our general theoretical orientation can be found in Cobb, Stephan, McClain, and Gravemeijer (in press). It is important to stress that this background theory has itself evolved as we have worked in classrooms in that we initially took a relatively individualistic psychological con-structivist perspective.

² McGatha (2000) described the year-long process of preparing for the first of these experiments in considerable detail. As she documented, it involved reviewing the research literature, the available research-based curriculum materials, and the available data analysis tools. In addition, we analyzed videorecordings of individual student interviews and whole class performance assessments that we conducted as part of our pilot work.

³ Stated in this way, it might appear that we frame our overall instructional goals in purely functional terms. However, in preparing for a design experiment, we attempt to explicate the overall goals in terms of Greeno's (1991) metaphor of knowing in a particular domain as acting in and reasoning with the resources of a particular environment. One of the attractive features of this metaphor is that it enables us to transcend the dichotomy between subject and object by simultaneously specifying both the taken-as-shared mathematical environment in which we hope that students will come to act and the taken-as-shared ways of reasoning and communicating mathematically that we hope will emerge in the course of a design experiment.