Cognitive and Situated Learning Perspectives in

Cognitive and Situated Learning Perspectives in Theory and Practice PAUL COBB

JANET BOWERS

the traditional concerns of psychological learning theories In their recent exchange, Anderson, Reder, and Simon (1996, and delineate central differences between the two perspec1997) and Greeno (1997) frame the conflicts between cognitive theory and situated learning theory in terms of issues that aretives pri-by considering their underlying metaphors. Our purpose in doing so is to clarify the different meanings that key marily of interest to educational psychologists. We attempt to terms such as social practice, context, and situation have in the broaden the debate by approaching this discussion of perspectives two theoretical orientations. We then demonstrate that, conagainst the background of our concerns as educatorswhoengage tra Anderson et al. (1997), the contrast between the two perin classroom-based research and instructional design in collabospectives cannot be reduced to that of choosing between the ration with teachers. We first delineate the underlying differences and the social collective as the primary unit of between the two perspectives by distinguishing their central individual oranalysis. Against this background, we then develop the raganizing metaphors. We then argue that the contrast between the tionale two perspectives cannot be reduced to that of choosing between the for the particular situated viewpoint that informs our work in classrooms by describing the concerns that moindividual and the social collective as the primary unit of analytivated its development. The issues addressed when consis. Against this background, we compare the situated viewpoint trasting this viewpoint with the cognitive approach advowe find useful in our work with the cognitive approach advocated cated by Anderson et al. by focusing on their treatments of meaning andby Anderson et al. include the differing treatments of meaning, the alternative ways in which instructional goals instructional goals. Finally, we consider the potential contribuare cast, and the conflicting views of the relationship betions of the two perspectives to instructional practice by contween theory and classroom practice. In making these comtrasting their differing formulations of the relationship between parisons, we do not pretend to offer a neutral appraisal but theory and practice. instead assess the perspectives with respect to our interest classroom-based research and instructional design. In in Educational Researcher, Vol. 28, No. 2, pp. 4-15 this regard, the analysis we offer complements those developed by Anderson et al. and Greeno by bringing a different set of questions and concerns to the fore.

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he recent exchange between Anderson, Reder, and Simon (1996, 1997) and Greeno (1997) indicates the depth of the division between adherents to cognitive and situated learning perspectives. Our purpose in this article is to broaden the ongoing debate by assuming a somewhat different vantage point. Anderson et al. and Greeno both frame the central issues in terms of their concerns as educational psychologists. In contrast, our interests are those of mathematics educators who conduct classroombased research and instructional design in collaboration with teachers. Elsewhere, we have argued that our research practice, which involves conducting longitudinal classroom teaching experiments in the course of which we develop sequences of instructional activities, is generally compatible with the activity of reflective teachers as described by Ball (1993), Lamport (1990), and Simon (1995). Thus, the viewpoint from which we approach the issues of learning and transfer is that of educators who view the classroom as a primary site for research and instructional development. As will become clear, the theoretical perspective that we find useful for our purposes is a version of constructivism that sees considerable merit in situated accounts of learning. As Greeno (1997) observes, the cognitive and situated learning perspectives each encompass a panoply of positions. In the first part of the article, we restrict our focus to

Organizing Metaphors In their recent review of research on mathematics learning and teaching, DeCorte, Greer, and Verschaffel (1996) distinguish between what they term the first and second waves of the cognitive revolution.1 The position that Anderson et al. (1997) outline is prototypical of the first wave in that it locates the internal cognitive processes that are the focus of investigation within a stimulus-mediation-response scheme. This basic scheme—which is apparent whether the constructs used to account for mental processes are production systems, schemes, rules, or principles—is illustrated by the studies that Anderson et al. (1997) cite as successful infor-

is a professor of mathematics education in the Department of Teaching and Learning at Peabody College at Vanderbilt University, Nashville, TN 37203. His specialty is mathematics education. PAUL COBB

an assistant professor of mathematics education at the Center for Research in Mathematics and Science Education at San Diego State University, 6475 Alvarado Road, Suite 206, San Diego, CA 92120. She specializes in mathematics education.

JANET BOWERS IS

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mation processing applications to mathematical reasoning.2 An overriding concern of such work is to formulate internal information processing mechanisms that account for observed relations between the external stimulus environment and response behaviors. In this article, we refer to this general theoretical orientation when we speak of the cognitive perspective. It encompasses much of mainstream American information processing psychology but not, for example, Piagetian psychology given Piaget's (1970) emphasis on conceptual operations that are rooted in sensory-motor activity. DeCorte et al. (1996) argue that the second wave of the cognitive revolution evolved as a reaction to the limited emphasis on affect, context, culture, and history in first-wave research. In their view, it involved a shift from a concentration on the individual to a concern for social and cultural factors; from "cold" to "hot" cognition; from the laboratory to the classroom as the arena for research; and from technically to humanistically grounded methodologies and interpretive approaches, (p. 491) Greeno's (1997) situative approach falls squarely within the second wave given its emphasis on interactive systems that are composed of groups of individuals together with the material and representational resources they use. DeCorte et al.'s historical review indicates that the differences between this and first-wave cognitive research are relatively profound. Their analysis therefore leads us to question Anderson et al.'s (1997) contention that the differences between their own and Greeno's position are primarily matters of terminology. As a first step in considering the underlying assumptions of the two perspectives, we follow Sfard (1998) by focusing on their core metaphors. In the case of the cognitive perspective, a central organizing metaphor is that of knowledge as an entity that is acquired in one task setting and conveyed to other task settings. In contrast, a primary metaphor of the situated learning perspective is that of knowing as an activity that is situated with regard to an individual's position in the world of social affairs. In the following paragraphs, we attempt to cast the differing assumptions of the two perspectives in broad relief by considering the entailments of these organizing metaphors first for situated learning theory and then for cognitive theory. Underlying Assumptions of Situated and Cognitive Perspectives Situation and context, as they are characterized in situated learning theory, can be traced to the notion of position as physical location. In everyday discourse, we frequently elaborate this notion metaphorically when we characterize ourselves and others as being positioned with regard to circumstances in the world of social affairs. This metaphor is apparent in expressions such as "My situation at work is pretty good at the moment." In this and similar examples, the world of social affairs (e.g., work) in which the individual is considered to be situated is the metaphorical correlate of the physical space in which material objects are situated in relation to each other. Situated learning theorists such as Lave (1988) and Rogoff (1995) elaborate this notion theoretically by introducing the concept of participation in social practice (see Figure 1). As Greeno (1997) illustrates, this core construct of participation is not restricted to face-to-face interactions with others. Instead, all individual actions are

viewed as elements or aspects of an encompassing system of social practices and individuals are viewed as participating in social practices even when they act in physical isolation from others (Axel, 1992; Forman, 1996; Saxe, 1991; Scribner, 1990). Consequently, when Lave (1988) and Greeno (1997) speak of context, they are referring to a social context that is defined in terms of participation in a social practice. This view of context is apparent in investigations in which situated learning theorists compare mathematical activity in school with activity in various out-of-school settings. The intent in these studies is to compare the forms of mathematical reasoning that arise in the context of different practices that involve the use of different artifacts and are organized by different overall motives (e.g., learning mathematics as an end in itself in school versus doing arithmetical calculations while selling candies on the street in order to survive economically). Thus far in outlining the central metaphor of situated learning theory, we have focused on practices located beyond the immediate social group. However, this same metaphor can be elaborated at the more local level of a particular classroom community. A situated perspective on the mathematics classroom sees individual students as participating in and contributing to the development of the mathematical practices established by the classroom community (cf. Cobb & Yackel, 1996). From this point of view, participation in these communal practices constitutes the immediate social context of the students' mathematical development. As we will clarify, analyses of this type do attend to qualitative differences in individual students' reasoning. In addition, they parallel analyses of broader systems of practices such as those that constitute schooling by locating learning and what is learned in evolving participation frameworks (Hanks, 1991). The central metaphor of the cognitive perspective also involves the notion of position. However, whereas the organizing metaphor of situated learning theory is position with regard to social circumstances, the organizing metaphor of transfer theory is the transportation of an item from one physical location to another (see Figure 1). Contrasts between the entailments of these two metaphors become apparent when we consider how cognitive theorists use context. Anderson et al.'s (1996) discussion of their theoretical perspective is paradigmatic in this regard. Although they do not explicitly state what they mean by "context," the examples they give indicate that it consists of the task presented to students together with the relevant features of the experimental setting. In this regard, they emphasize the importance of analyzing the structure of tasks in terms of com-

Theoretical

Cognitive Theory k

Everyday Discourse

Situated Learning Theory

i

1

Convey from one physical location to another

i

Position with regard to circumstances in the world of social affairs

[Po ition as physical location FIGURE 1. Metaphorical underpinnings of cognitive theory and situated learning theory.

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ponents that are considered to exist independently of situation and purpose. The remaining aspects of a context as they characterize it are those features of experimental or instructional settings that influence the cognitive processes evoked and thus the amount and type of transfer that occurs (Anderson et al., 1996, pp. 7-8). In this theoretical scheme, context appears to be the direct metaphorical correlate of physical location. It is, for example, under the researcher's control just as is a subject's physical location and can therefore be systematically varied in experiments. Given their conception of context, Anderson et al. (1996) can reasonably conclude that "it is not the case that learning is wholly tied to a specific context" (p. 6). However, as Greeno (1997) makes clear in his response, this statement is open to dispute if it is interpreted in terms of situated learning theory where all activity is viewed as occurring in the context of social practice. In a similar manner, Anderson et al.'s contention that "while cognition is partly context-dependent, it is also partly context-independent" (p. 10) is open to an alternative interpretation in the situated paradigm. Differences in basic metaphors also play out in the contrasting characterizations of social processes within the two paradigms. Anderson et al. (1996) cite Lave and Wenger's (1991) contention that "learning is an integral part of generative social practice in the lived-in world" (p. 35). In making this statement, Lave and Wenger argue that participation in social practices does not merely influence or shape otherwise autonomous psychological processes. Instead, in their view, learning is synonymous with changes in the ways that an individual participates in social practices. However, the statement has a different meaning when it is interpreted in terms of cognitive theory. Thus, Anderson et al. (1996) respond to Lave and Wenger by arguing that "while one must learn to deal with the social aspects of jobs, this is no reason why all skills required for these jobs should be trained in a social context" (p. 9). Social context in this and other examples given by Anderson et al. appears to be defined in terms of the interactional demands of the knowledge and skills3 that are being taught. For example, they suggest that learning occurs in a social context when a trainee accountant learns how to deal with clients or when a group of people undergo team training. However, they argue that learning does not occur in a social context when the trainee accountant learns the tax code without simultaneously interacting with a client or when a single member of a team is trained on a new piece of equipment. For their part, situated learning theorists such as Greeno view each of these cases as an instance of learning in social context. For example, the trainee accountant can be viewed as a legitimate peripheral participant who is participating in the practices of the accounting profession even when he or she studies the tax code at home in the evening. Cognitive theorists might, in their turn, legitimately challenge either the explanatory power or the practical relevance of this latter interpretive stance. Our purpose in devoting a considerable portion of this article to the different ways in which key terms such as social context are used in the two paradigms is to help develop a basis for communication between the two perspectives. In our view, a continuing intellectual exchange of the type envisioned by Greeno is virtually impossible unless proponents of each perspective come to understand the basic tenets of the other viewpoint. Although it is not clear

whether it was a conscious rhetorical strategy, the manner in which Anderson et al. (1996, 1997) assimilated central constructs of the situated perspective directly to their own paradigm is particularly striking. We would stress that direct appeals to the empirical findings generated in one paradigm or the other do not allow us to sidestep differences in meanings. As the exchange between Anderson et al. and Greeno repeatedly illustrates, differences in the use of basic terms finds expression in the different conclusions reached when interpreting the same findings. If we fail to undertake the hard work involved in developing a viable basis for communication, we set up ideal conditions for the type of paradigm war with which we are all familiar in which more heat than light is generated. Methodological Differences Between the Two Perspectives In their response to Greeno (1997), Anderson et al. (1997) cast the differences between the situated and cognitive perspectives in terms of a choice between taking the social collective and the individual as the principal unit of theoretical focus. We believe that this rigid demarcation of legitimate units of analysis obscures more significant differences between the perspectives. However, in Anderson et al.'s defense, we note that some situated learning theorists have also framed the debate in these terms. For example, in her critique of the cognitive perspective, Lave (1988) /pits the characterization of isolated individuals drawing on their internal cognitive resources against that of people viewed as social actors drawing on resources available in the settings in which they are acting. In addressing the issue of units of analysis, we first note that the focus of first-wave cognitive research is primarily individualistic even though it acknowledges the importance of social interaction. For example, Anderson et al. (1997) illustrate their reductionist strategy of decomposability by saying that they would "analyze the complex social situation into relations among a number of individuals and study the mind of each individual and how it contributes to the interaction" (p. 11). In this approach, the group or local community is not itself taken as a unit of analysis. Instead, the focus is on the internal cognitive activity of individuals within the group. As Anderson et al. (1997) note, their perspective "does not ignore the social, but it does try to understand the social through its residence in the mind of the individual" (p. 11). In general, first-wave approaches that focus on mental processes that intervene between the observed stimulus environment and subsequent behavior delimit the possibility of taking either local communities or broader systems of practice as units of analysis. By way of contrast, the situated perspective admits a range of units of analysis, the choice in any particular case being a pragmatic one that depends on the purposes at hand (cf. Cobb & Yackel, 1996). To illustrate this point, we briefly summarize analyses reported by Beach (1995) and Bowers (1996). Beach focuses on the ways in which two groups of individuals with different cultural histories participate in broad systems of practices, whereas Bowers focuses on a single group of students who participated in two different classroom microcultures. Both analyses are relevant to the current debate in that they explain instances of transfer in situated terms. To avoid confusion, we differentiate between transfer as an explanatory theoretical con-

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struct within the cognitive perspective and what—for want of more appropriate terminology—we refer to as the phenomenon called transfer. By this, we mean specific individual actions that cognitive theorists would interpret within their perspective as indications that transfer has occurred. Arithmetical Reasoning While Participating in School and in Work

As Greeno (1997) notes in his brief synopsis of Beach's (1995) study, the investigation focused on transitions between work and school and was conducted in a Nepali village where formal schooling had been introduced during the past 20 years. Beach compared changes in the arithmetical reasoning of 13 high school students who were apprenticed to a shopkeeper with changes in the reasoning of 13 shopkeepers who were attending adult education classes. He found that the shopkeepers' arithmetical reasoning was more closely related in the work and school situations than was that of the students. In terms of cognitive theory, one can therefore say that the phenomenon called transfer occurred to a greater extent in the arithmetical reasoning of the shopkeepers than in that of the students. Beach accounts for these results by noting that the students making the school-to-work transition initially defined themselves as students but subsequently defined themselves as shopkeepers when they worked in the shop. In contrast, the shopkeepers continued to identify themselves as shopkeepers even when they participated in the adult education classes. Their goal in the classes was to develop arithmetical competencies that would enable them to generate increased profits in their shops. Beach contends that it is this relatively strong relationship between the shopkeepers' participation in the practices of schooling and shopkeeping that accounts for the close relation between their arithmetical reasoning in the two situations. In contrast, the students' participation in the practices of schooling and shopkeeping involved distinct goals—those of learning arithmetic as an end in itself and of generating profit as a shopkeeper, respectively. This relatively weak relation between studying arithmetic in school and shopkeeping "allowed arithmetic originating with students and school-based activity to achieve a status disconnected from and beyond that originating with shopkeeping and shopkeepers" (p. 301). In reflecting on his analysis, Beach notes that it "offers an alternative to the [theoretical] construct of learning transfer in its attempt to understand change in personal knowledge across situations" (p. 301). As Hanks (1991) puts it, "if both learning and the subject learned are embedded in participation frameworks, then the portability of learned skills must rely on the commensurability of certain forms of participation" (pp. 19-20). This, in succinct form, is the central claim of Beach's analysis. He contends that the phenomenon that Anderson et al. (1996) would call transfer occurred to a greater extent with the shopkeepers than with the students because participating in the practices of shopkeeping and in schooling were experienced as more commensurable by the shopkeepers than by the students. Arithmetic Reasoning While Participating in Two Classroom Microcnltures

The second sample analysis was reported by Bowers (1996) and focuses on the arithmetical reasoning of 23 third-grade

students as they participated in two differing classroom microcultures. One of these microcultures was established during a 9-week teaching experiment in which a member of the research team served as the teacher. The second microculture was established when, later each day, the students' regular classroom teacher also conducted mathematics lessons with them. At the surface level, activity in both classroom settings involved adding and subtracting three-digit numbers. However, there were major differences both in the instructional goals the two teachers sought to achieve and in the general classroom social norms, standards of mathematical argumentation, and specific mathematical practices established in the two settings. With regard to instructional goals, for example, the classroom teacher directly taught the steps of standard paper-and-pencil algorithms from the outset. In contrast, the project teacher attempted to support the students' construction of increasingly efficient but not necessarily standard algorithms that reflected their developing understanding of place-value numeration. To this end, the students first pretended to be workers in a candy factory where 10 candies were packed in a roll and 10 rolls in a box. In subsequent instructional activities, the students developed a variety of ways of symbolizing the process of combining and separating quantities of candies. These included drawing pictures, making tally marks, and writing numerals. j^n issue of particular interest concerns the extent to which the students' participation in each of these classroom microculrures influenced their participation in the other. Bowers' analysis indicates that 12 of the students developed numerical meaning for the standard computational algorithms taught by the regular classroom teacher as a consequence of their participation in the mathematical practices established in the teaching experiment classroom. A further 5 students could interpret standard algorithms in this way when the teaching experiment began, whereas the remaining 6 students did not develop such meanings during the teaching experiment but instead continued to view the standard algorithms as procedures for manipulating digits. Bowers accounts for these findings by focusing on the qualitatively different ways in which the students participated in the mathematical practices established in the teaching experiment classroom. In particular, her analysis indicates that the 12 students who related their arithmetical activity in the two situations participated in the final practices established in the last few days of the teaching experiment by interpreting three-digit numerals as signifying numerical quantities that remained invariant under regrouping transformations. It was therefore possible for them to interpret conventional notation in these terms when various notational schemes were compared in student-initiated discussions in the teaching experiment classroom. As a consequence, they could experience calculating as commensurable in the two classrooms. In contrast, the group of 6 students who continued to view the standard algorithms solely as procedures for manipulating digits participated in these same classroom mathematical practices by making less sophisticated numerical interpretations. As a consequence, they seemed to experience pivotal whole-class discussions that occurred during the last few days of the teaching experiment differently than their classmates and did not come to view calculating in the two classrooms as commensurable.


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It is important to note that, in this explanation, we have not accounted for the phenomenon of transfer by suggesting that the 12 students abstracted the structure of candy factory tasks presented in the teaching experiment classroom and transferred it to the work sheet tasks presented by their regular classroom teacher. Instead, we have treated the various ways in which the students interpreted and understood tasks as aspects of their participation in classroom practices. Thus, whereas cognitive theorists view tasks as part of a stimulus environment that can be analyzed independently of individual and collective activity, we have focused on tasks as they were constituted in activity in the classroom. Bowers's analysis, in fact, indicates that the meaning and significance of what was, on the surface, the same task changed as the mathematical practices and students' ways of participating in them evolved in the course of the teaching experiment.

be seen to complement each other in that they together portray individual students' reasoning as acts of participating in the practices of the local classroom community and in the practices of broader communities.4 The approach of coordinating different units of analysis is also apparent in several other proposals made by situated learning theorists. These include Hatano's (1993) call to synthesize constructivist and Vygotskian perspectives, Saxe's (1991) discussion of the intertwining of cultural forms of cognitive functions, and Rogoff's (1995) distinction between three planes of analysis that correspond to personal, interpersonal, and broad community processes. Each of these theorists explicitly legitimizes a focus on the individual, thereby calling into question Anderson et al.'s (1997) contention that the dispute between cognitive and situated learning perspectives is about the appropriateness of the individual or the social collective as a unit of analysis.

Reflections

Developmental Research

The two sample analyses, when combined with Greeno's (1997) presentation of his situative view of activity, illustrate a range of positions that fall under the rubric of the situated learning perspective. The common thread that runs through Greeno's, Beach's, and Bowers's work is that of viewing individual activity as an act of participation in a system of practices that are themselves evolving. As we have seen, this basic theoretical commitment is inherent in the central organizing metaphor of the situated perspective and serves to differentiate it from the cognitive perspective as exemplified by Anderson et al. (1996). An important difference that comes to the fore when we compare Beach's and Bowers's analyses concerns the units of analysis that they take as primary. Beach's focused on the contrasting forms of arithmetical reasoning developed by two groups with different histories rather than on differences in individuals' reasoning within each group. As a consequence, he interpreted individuals' reasoning as acts of participating in the practices of shopkeeping or schooling and explained the commensurability experienced by the shopkeepers in terms of the way that the goals of the two sets of practices were related in their lives. In doing so, he took participation in two broad forms of collective activity as his primary unit of analysis. Bowers focused on the contrasting forms of arithmetical reasoning developed by a single group of students as they participated in the same practices. Given this purpose, she interpreted the third-graders' reasoning in terms of participation in the local mathematical practices established in two specific instructional settings rather than in terms of participation in broader systems of practices such as those that constitute schooling. Further, she accounted for the commensurability experienced by some students but not others in terms of the qualitatively distinct ways in which the students participated in the local classroom practices. She therefore took individual students' reasoning as a unit of analysis while simultaneously viewing that reasoning as an act of participation in evolving communal practices. Taken together, the two sample analyses illustrate that the situated perspective admits a range of units of analysis that includes an explicit focus on individual meaning. We would argue that the units Bowers and Beach took are each appropriate given the questions they addressed. Sociocultural analyses of the type conducted by Beach and social constructivist analyses as illustrated by Bowers can, in fact,

In comparing the cognitive and situated perspectives thus far, we have limited our focus to priorities and issues that are traditionally of concern to educational psychologists. However, as we noted at the beginning of this article, our interests are those of mathematics educators who conduct classroom-based research and instructional design in collaboration with teachers. Gravemeijer (1994) calls work of /this type developmental research to emphasize that instructional development and classroom-based research proceed hand in hand. Developmental research, as Gravemeijer defines it, is therefore not synonymous with either child development research or psychological research into the development of particular conceptions. Instead, it involves a coordinated program in which instructional design and classroom research each informs the other. In the following paragraphs, we develop the rationale for the particular situated viewpoint we take in our work by describing the pragmatic issues that motivated its development. Although our concern is with the teaching and learning of mathematics, the general notion of developmental research is broad enough to be of interest to researchers who work in other subject matter areas. The basic developmental research cycle is shown in Figure 2. It involves both a domain-specific instructional design theory and an interpretive framework within which to analyze classroom events in a way that feeds back to inform ongoing instructional design efforts. Our overall purpose

Instructional Development

Classroom-based Analyses

(guided by domain-specific instructional design theory)

(guided by interpretive framework)

FIGURE 2. The developmental or transformational research cycle.


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when engaging in developmental research of this type is to investigate ways of proactively supporting and organizing students' mathematical learning. The instructional design aspect of the cycle involves conducting an anticipatory thought experiment in the course of which we formulate hypothetical trajectories for students' learning (Simon, 1995). In doing so, we draw on the design theory of realistic mathematics education (RME) developed at the Freudenthal Institute in the Netherlands (Gravemeijer, Cobb, Bowers, & Whitenack, in press; Streefland, 1991; Treffers, 1987). The resulting trajectories are envisioned within the setting of an assumed classroom microculture and involve conjectures about both the course of students' mathematical development and the specific means of supporting it.5 These conjectures then provide an initial orientation when we work in classrooms and are continually tested and revised on a daily basis during teaching experiments. It is here that the second aspect of the developmental research cycle, that of analyzing classroom events, gains its pragmatic force. In outlining the interpretive framework we currently use, it is worth noting that our theoretical orientation when we first began working intensively in classrooms 12 years ago was primary individualistic. We have modified our position significantly over the past several years in response to problems and issues encountered while attempting to support students' mathematical learning. In a recent article in this journal, Hiebert et al. (1996) summarize what has become a widely accepted position in a number of fields, including mathematics education, when they discuss the critical role attributed to the classroom participation structure (cf. Cobb, Yackel, & Wood, 1989; Erickson, 1986; Lampert, 1990; Schoenfeld, 1987). Beyond these general classroom norms, we have also found it useful to focus on normative aspects of students' activity that are specific to mathematics. Examples of these so-called sociomathematical norms include what counts as a different mathematical solution, an efficient mathematical solution, a sophisticated mathematical solution, and an acceptable mathematical explanation (Simon & Blume, 1996; Voigt, 1995; Yackel & Cobb, 1996). Empirical analyses indicate that these norms can differ radically from one classroom to another, thereby influencing the learning opportunities that arise for students and, indeed, for teachers (Cobb, Gravemeijer, Yackel, McClain, & Whitenack, 1997; McClain, 1996). In addition, we conjecture that students reorganize the mathematical beliefs and values that constitute their mathematical dispositions as they participate in the renegotiation of sociomathematical norms (Yackel & Cobb, 1996). In recent analyses, we have complemented our focus on the classroom participation structure and sociomathematical norms by describing what is traditionally called mathematical content in terms of collective as well as individual activity. Our motivation for doing so stems directly from the approach we take to instructional design. In particular, the conjectures inherent in learning trajectories cannot be about the anticipated mathematical learning of each and every student in a class because there are significant qualitative differences in their mathematical interpretations at any point in time. In our view, descriptions of planned instructional approaches written so as to imply that all students will reorganize their mathematical activity in particular ways at particular points in an instructional sequence are, at best, a questionable idealization. It is, however, fea-

sible to view a hypothetical learning trajectory as consisting of conjectures about the collective mathematical development of the classroom community. Conceptualized in this way, the learning trajectory consists of an envisioned sequence of classroom mathematical practices together with conjectures about the specific means of supporting their evolution from prior practices. Elsewhere, we have illustrated that the actual learning trajectory realized in the classroom over, say, a 6-month period can also be analyzed in terms of a sequence of evolving mathematical practices (Bowers, Cobb, & McClain, in press; Cobb, 1996; Cobb, in press; Cobb et al., 1997; Gravemeijer et al., in press). Such analyses simultaneously document both the instructional sequence as it was realized in interaction in the classroom and the evolving social situation of the students' mathematical development. This is encouraging in that it suggests the possibility of developing a common language in which to frame both instructional conjectures and analyses of classroom events during mathematics instruction.

In a very real sense, students who cannot participate in [classroom ,, mathematical] practices are no longer members of the classroom community...

Regarding units of analysis, the approach we take admits an explicit focus on individual students' reasoning. However, rather than treating that reasoning as solely an internal mental phenomenon, we view it as an act of participating in communal mathematical practices. Further, we view learning as a process in which students actively reorganize their ways of participating in classroom practices. Cast in this way, the relation between individual students' reasoning and communal practices is viewed as reflexive in that students contribute to the evolution of the classroom practices that constitute the immediate social situation of their mathematical development as they learn. Although it is not fully apparent from the brief summary we have given of Bowers's (1996) investigation, analyses conducted in this manner typically bring to the fore the qualitatively distinct ways in which individual students participate in particular practices. We find it important to attend to this diversity in students' reasoning when making instructional decisions during teaching experiments and, in fact, view it as a resource on which the teacher can capitalize. In addition, we also find it informative when conducting retrospective analyses of teaching experiments to trace individual students' actual learning trajectories as they participated in the evolving practices. It is important to note that the viewpoint we have outlined has two major ethical implications. The first is that all students must have a way to participate in the mathematical practices of the classroom community. In a very real sense, students who cannot participate in these practices

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are no longer members of the classroom community from a mathematical point of view. This situation is highly detrimental given that to learn is to participate in and contribute to the evolution of communal practices. Students who are excluded are deprived not merely of learning opportunities, but of the very possibility of growing mathematically. One of our primary concerns when conducting a teaching experiment is therefore to ensure that all students are "in the game." To this end, we attempt to adjust the classroom participation structure, classroom discourse, and instructional activities on the basis of ongoing observations of individual students' activity. In doing so, we once again find ourselves attending to qualitative differences in individual students' reasoning and to the communal mathematical practices in which they are participating. The second ethical implication is closely related to the first and concerns the view one takes of students whose ways of participating in particular classroom practices are less sophisticated than those of other students. In the theoretical orientation we have outlined, these differing interpretations are not viewed as cognitive characteristics of the individual students, but as characteristics of their ways of participating in communal mathematical practices. In the case of the teaching experiment analyzed by Bowers (1996), we therefore did not take a cognitive deficit view of the six students who continued to understand the standard algorithms as procedures for manipulating digits. Instead, our reflections on the teaching experiment have focused on the nature of their participation in the evolving mathematical practices that constituted the immediate social situation of their mathematical development and have led to a number of specific suggestions about how the instructional sequence might be revised (Bowers, Cobb, & McClain, in press). In interpreting the diversity in students' reasoning in this way, we have treated academic success and failure in the classroom as neither an exclusive property of individual students nor of the instruction they receive. Instead, we have cast it as a relation between individual students and the practices that they and the teacher co-construct in the course of their ongoing interactions. In the last analysis, the ethical dimension of this perspective on success and failure in school is perhaps the most important reason for taking seriously a viewpoint that brings the diversity of students' reasoning to the fore while simultaneously seeing that diversity as socially situated. In light of Anderson et al.'s (1996) comments about evidence, we close this discussion of the particular situated viewpoint that we find useful in our work by clarifying what gives rise to conjectures that can be investigated empirically. For example, analyses of sociomathematical norms have led to empirically verifiable conjectures about the types of interactions and discourse that will support students' development of particular types of mathematical dispositions (cf. Bowers & Nickerson, 1998). Similar comments can be made about students' development of specific mathematical understandings and about their general beliefs about their own and others' roles, particularly that of the teacher.6 Further, this approach of attending to both the diversity of individual students' mathematical interpretations and to the communal activities in which they participate is central to the moment-by-moment pedagogical judgments that both we and the teachers with whom we collaborate make during instructional sessions. In this re-

gard, coordinating a focus on both individual and collective activity is an integral part of our classroom-based practice in that it has allowed us to be more effective when supporting students' mathematical development over extended periods of time. On the basis of our experience of working in classrooms, we therefore contend that locating students' mathematical learning in the context of socially organized activity is not a vacuous theoretical commitment as Anderson et al. (1997) suggest. It has pragmatic and ethical consequences that bear directly on the instructional decisions made in the classroom. Experience and Meaning In presenting our viewpoint thus far, we have emphasized the importance of analyzing both communal practices and individual students' diverse ways of participating in them. We turn now to further clarify the contrast between this viewpoint and the cognitive perspective by comparing the alternative approaches taken when analyzing individual students' reasoning. In doing so, we follow DeCorte et al. (1996) in noting that whereas first-wave cognitive analyses attempt to simulate mental processes that mediate between the stimulus environment and observed responses, a situated analysis of the type we favor attempts to account for individuals' inferred, socially situated experience. In this regard, we have already seen that a primary criterion used when assessing cognitive models developed ijl first-wave research involves the input-output match with students' observed activity. In particular, the model should generate the same responses as students in given environmental settings. As Greeno (1997) observes, the models developed in the research program advocated by Anderson et al. (1996) consist of rules for transforming symbolically coded input information. Referring to his production system models, Anderson (1983) clarifies that this approach might legitimately be characterized as "cognitive S-R theories" (p. 6). It is important to appreciate that it is the researcher who imbues the symbols in these models with a representational function. Dreyfus and Dreyfus (1986), in fact, suggest that the symbols might better be called "squiggles" to avoid confusion over who or what is doing the representing. As a matter of principle, models formulated in terms of internal cognitive behaviors cannot, by definition, account for the meanings of the students whose activity is being modeled. This is a significant limitation for us given that our primary concern when conducting a classroom teaching experiment is to support students' development of increasingly sophisticated mathematical understandings. The contrast between Anderson et al.'s (1996) theoretical perspective and the approach we take when analyzing individual students' reasoning is captured by a distinction that MacKay (1969) makes between what he terms the observer's and the actor's viewpoints. Anderson et al. necessarily adopt the observer's perspective when they analyze the observed stimulus environment and posit intervening cognitive processes that generate observed responses. In contrast, we adopt the actor's perspective when we conduct a situated analysis in that our purpose is to understand the meaning and significance that the students' activity has for them. As a consequence, whereas cognitive analyses of the type described by Anderson et al. employ information and internal representations as basic con-

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structs, analyses of the type we conduct admit experience and meaning as a legitimate focus of inquiry. Further, in conducting such analyses, we view knowledge as embodied, or as located in activity, where activity is broadly construed and includes perceiving, reasoning, and talking (cf. Johnson, 1987). An account of individual students' learning developed from this point of view then documents how they reorganize their mathematical activity or, in other words, their ways of participating in communal mathematical practices. This analytic approach therefore stresses intentionality and takes activity and meaning rather than cognitive behavior as its basic currency (Searle, 1983; Taylor, 1985). We can illustrate the implications of this shift from the observer's to the actor's viewpoint by considering the issue of instructional goals. Instructional Goals and the Environmental Metaphor When we conduct classroom teaching experiments, we typically formulate the overall intent of the sequences of instructional activities that we develop in terms of Greeno's (1991) environmental metaphor. This metaphor involves the actor's viewpoint and contrasts with Anderson et al.'s approach of specifying goals in terms of desired cognitive behaviors. As an example, we return to the teaching experiment analyzed by Bowers (1996). This case is appropriate to the discussion in that it addresses what are traditionally described as basic addition and subtraction skills. The global intent of the instructional sequence as we formulated it was that students would eventually come to act in a mathematical environment in which quantities are invariant under certain arithmetical transformations (e.g., regrouping 100 as 10 tens). In such an environment, 254 is the same quantity whether it is structured as 2 hundreds, 5 tens, and 4 ones or as 1 hundred, 12 tens, and 34 ones. The svmbol "254" then signifies the same quantity as "f « 4" and as "2 $ 4"- Observationally, the inference that students were acting in a mathematical environment of this type would be indicated by, among other things, their flexible use of efficient computational algorithms to solve a wide range of arithmetical tasks. In describing our instructional intent in this way, we were concerned with the meaning that students' computational actions might have for them. In particular, we wanted the numerical relationships implicit in their computational methods to be ready to hand for them. They would then not have to consciously recall the steps of a calculational process but would instead have the experience of directly perceiving numerical relationships as they calculated. This goal is much more demanding than those typically set for third graders and obviously requires systematic support. As this example addresses what might, in cognitive terms, be described as the automation of skills, it should be clear that we consider so-called basic skills to be an important aspect of mathematical activity. However, our framing of the instructional intent in terms of Greeno's (1991) environmental metaphor rather than in terms of cognitive behaviors gives rise to a different view of skillful mathematical activity. For example, it orients us to relate the development of arithmetical skills to the development of what is colloquially called number sense. In addition, it suggests the importance of making students' interpretations of tasks topics of conversation in the classroom so that discussions are not limited to calculational processes but also attend to

the underlying reasons for calculating in particular ways (Thompson, Philipp, Thompson, & Boyd, 1994). This shift in emphasis complements Greeno's (1997) analysis by extending his focus on conjecture and argumentation to issues of mathematical content. It is important to note that both the environmental metaphor and the situated approach we follow are nondualist in that they transcend the dichotomy between the individual and the environment. Whereas Anderson et al. (1996) stress the importance of analyzing the environment as it appears to the observing researcher (i.e., independent of individual and collective activity), our concern is to understand the world of signification and meaning in which students act. From this latter viewpoint, the use of tools is viewed as integral to mathematical activity rather than an external aid to internal cognitive processes located in the head (cf. Hutchins, 1995; Meira, 1998; Pea, 1993). From this perspective, it therefore makes sense to speak of students reasoning with physical materials, pictures, diagrams, and computer graphics as well as with conventional written symbols. This nondualist focus on tool use is, in fact, central to the RME design theory that guides our development of instructional sequences including that which focused on computational algorithms (cf. Gravemeijer, 1994; van Oers, 1996). In this regard, we accept sociocultural theorists' claim that the tools acted with profoundly influence both the mathematical understandings that students develop and the process by which they develop them (cf. Vygotsky, 1987; Wertsch, 1995). Relations Between Theory and Practice In describing our situated viewpoint, we indicated that our theoretical commitments have evolved as we worked in classrooms. This notion of theory evolving in response to instructional practice contrasts sharply with Anderson et al.'s (1996,1997) assumptions about the relation between theory and practice. Viewed in broader context, these contrasting assumptions transcend the two specific theoretical positions and affect the extent to which empirical findings and instructional recommendations might, in fact, be relevant to educational practice. In considering differing views of the relation between theory and practice, we first note a point of agreement with Anderson et al. (1996). We concur with their criticisms of several instructional recommendations that are said to stem from situated theories of learning. These include claims that instructional tasks should always be grounded in authentic, real-world settings and that students should always work in groups. In our view, instructional recommendations of this type involve a category error wherein central tenets of situated learning as a theoretical perspective are translated directly into instructional prescriptions. As Anderson et al. (1996) correctly observe, the resulting claims are, for the most part, entirely unsubstantiated. We note, however, that the situated perspective does not have a monopoly on flawed interpretations of this type. In the case of constructivism, for example, the theoretical assumption that learning is a constructive process often leads to the slogan that "telling is bad" because it deprives students of the opportunity to construct understandings for themselves (cf. Smith, 1996). Similar comments can be made regarding social constructivism (as learning involves the negotiation of meaning, students should continually discuss their differ-


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ing interpretations) and distributed intelligence (as intelligence is distributed over individuals, tools, and social contexts, students should always work with computers and other tools). In each of these cases, the difficulty is not necessarily with the background theory, but with the relation that is assumed to hold between theory and instructional practice. As a consequence, although we support Anderson et al.'s (1996) criticisms of specific instructional recommendations, we question their implication that these erroneous claims necessarily reflect theoretical inadequacies. It is instead the practice of translating theoretical tenets directly into instructional prescriptions that needs to be challenged. As will become clear, the relation between theory and practice that we favor rejects both the notion of theory abstracted from instructional practice and the opposite view of instructional practice rarely rising above the concrete particulars of specific cases. Anderson et al. (1997) clarify their view of the relation between theory and practice when they argue that the contributions of the cognitive perspective stem from the analytic power it provides for extracting principles that can generalize from one setting to another. They go on to suggest that teachers should base their instruction on these general principles while relying on common sense and professional experience when issues arise in the classroom that cognitive psychology cannot yet answer. In making this recommendation, Anderson et al. characterize the relation between theory and practice in terms of what Schon (1983) calls the positivist epistemology of practice. Cognitive psychology is seen to stand apart from and above instructional practice. Its role is to generate an empirically substantiated body of principles that can serve as the primary resource to which teachers should turn for answers. In this portrayal, the so-called hard human sciences such as cognitive psychology produce theoretical knowledge that is then applied in fields such as mathematics education that are closer to instructional practice. As DeCorte et al. (1996) observe, this assumed one-way relationship between cognitive theory and educational practice has long been the norm: "Historically, psychologists who have paid attention to the learning of mathematics [have] often exploited selected aspects of mathematics for their own ends" (p. 492). DeCorte et al. continue by discussing the contrasting interests of educational psychologists and mathematics educators and note that the second cognitive revolution involves a more symbiotic relationship between the two communities. As part of this realignment, there is a greater awareness that cognitive research and teaching are different forms of activity motivated by different concerns and interests. Viewed in this way, general principles that Anderson et al. (1997) imply are context free can, in fact, be seen to be situated within the academic practices of their research community. It is by no means self-evident that the principles produced in one form of practice transfer unproblematically to other forms of practice (such as teaching) structured by different purposes, beliefs, suppositions, and assumptions. Thus, we once again find ourselves addressing the phenomenon called transfer. Kessels and Korthagen (1996) discuss this issue when they distinguish between reasoning based on general principles of the type valued by Anderson et al. from the concerns of instructional practice. In doing so, they oppose the view that "there is a fixed solution [to a practical problem] to be found by subsuming the example under a scientific

theory of effective teacher behavior" (p. 18) with practical wisdom that is "not concerned with scientific theories, but with the understanding of specific concrete cases and complex or ambiguous situations" (p. 19). The overall intent of their analysis is to illustrate that pedagogical reasoning is more akin to practical wisdom than to reasoning based on what they term scientific theory. In this regard, Kessels and Korthagen's arguments can be seen to challenge Anderson et al.'s (1997) assumptions about the relationship between theory and practice. However, their conclusion that teaching proceeds largely independently of theory is, in our view, as problematic as Anderson et al.'s (1997) assumption that teachers can readily apply general principles to their instructional practice. The difficulty in both cases can be traced to a single source. In opposing the general with the particular, Kessels and Korthagen follow Anderson et al. in offering a positivist description of science. Their claim that teaching is largely uninformed by theory holds only if one assumes that theories consist of universal, context-free propositions. Kessels and Korthagen's acceptance of this view of theory leads them to develop a parody of the situated perspective in which pedagogical reasoning fails to move beyond the concrete particulars of specific cases. We arrive at a different view of instructional practice when we adopt the more conservative position that theory is shaped by the interests and purposes for which it is produced. ,, We can best illustrate this alternative approach by teasing /out the relation between theory and instructional practice implicit in our classroom-based research activity. As we have noted, one of the central problems with which we have struggled is that of accounting for students' mathematical development as it occurs in the social situation of the classroom in a manner that is specifically tailored to the demands of instructional design. The interpretive framework that we outlined when discussing general classroom norms, sociomathematical norms, and classroom mathematical practices illustrates how we currently address this issue. It is important to stress that this framework emerged over a period of several years as we attempted to understand specific events in the classrooms in which we have worked. The relationship between theory and practice implicit in the framework is therefore reflexive. On one hand, theory as exemplified by the interpretive framework grew out of our efforts to support students' mathematical learning. On the other hand, interpretations of classroom events organized in terms of the emerging framework fed back to inform the ongoing instructional development effort. The key point to emphasize in this process is that theoretical constructs evolve in response to problems and issues encountered in the classroom. As a consequence, theoretical constructs developed in this way do not stand apart from instructional practice, but instead remain grounded in it. A concern central to both Anderson et al.'s (1997) and Kessels and Korthagen's (1996) analyses, that of bridging the gulf between theory and practice, simply fails to materialize. Similar remarks can be made about the process of developing the instructional theory that is central to the first aspect of the developmental research cycle (see Figure 2). The design theory that we draw on, RME, has emerged from, and yet remains grounded in, the activities of designing and experimenting in classrooms over a 20-year period (Gravemeijer, in press). Given Anderson et al.'s (1997) comment that the cognitive and situated perspectives should be


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judged by their abilities to improve education, it is worth noting that the theory of realistic mathematics education has been used with considerable success to guide the development of complete sets of innovative curriculum materials in a number of countries, including the United States. As Gravemeijer (1994) makes clear, the view of mathematical activity central to this design theory is at odds with that implicit in Anderson et al.'s strong cognitive position. It is, however, compatible with the situated view of individual and collective activity that guides our work. As a point of clarification, we should stress that the interpretive framework and the instructional design theory have not emerged from classrooms per se, but instead from the activity of experimenting in classrooms. These approaches therefore reflect the concerns of participants in the learning-teaching process rather than those of a spectator to classroom events. The interpretive framework, for example, explicates how we attempt to make sense of specific classroom episodes while planning for subsequent classroom sessions. Its key constructs therefore summarize and organize our interpretive activity. Similarly, the design theory captures regularities in the process of designing a variety of specific instructional sequences. As a consequence, the design theory and the interpretive framework both describe in general terms a way of coming to grips with and making judgments in concrete cases. In contrast, principles of the type valued by Anderson et al. (1997) describe instructional actions that should be taken irrespective of the concrete specifics of particular cases. Thus, whereas the general remains grounded in the particular in a situated approach that involves a reflexive relation between theory and practice, cognitive research of the type advocated by Anderson et al. has as its focus the production of principles that are independent of the contingencies of particular cases. As we have observed, these supposedly context-free principles are grounded in the academic practices that constitute research in the cognitive paradigm. We find it significant that Greeno (1997) does not attempt to derive instructional principles from his situated position, but instead illustrates how it can inform discussions of educational practice. In doing so, he also seems to be of the opinion that the most important contribution that theory can make to educational practice is to inform the process of making pedagogical decisions and judgments in particular cases. Anderson et al. (1997) interpret Greeno's refusal to divorce his analysis from the reality of educational practice as indicating that the situated perspective lacks abstraction and analytic power. We, in contrast, have attempted to demonstrate that although the extraction of general principles might be valued in certain academic circles, it is not clear that it contributes to the improvement of educational practice. Conclusion We have used the distinction that DeCorte et al. (1996) draw between the first and second waves of the cognitive revolution to organize our discussion of the cognitive and situated perspectives. Our purpose in doing so was to clarify differences in the use of key terms by relating them to differences in the underlying metaphors of the two theoretical perspectives. In the course of the discussion, we also developed our own position and differentiated it from some of the claims made in the name of situated learning theory. For example, we concurred with Anderson et al.'s (1996) argument that

many of the instructional recommendations made in the name of situated learning theory are unsubstantiated. However, we questioned Anderson et al.'s assumption that these erroneous recommendations can be traced back to basic theoretical commitments. We instead suggested that the practice of translating basic theoretical tenets directly into instructional prescriptions involves a category error. The deeper issue is the relationship between theory and practice, and here we also critiqued the positivist stance taken by Anderson et al. (1997). Regarding the alternative view we developed, we would acknowledge that the reflexive relation between theory and practice implicit in our classroom-based research activity is itself situated. In this regard, we agree with Nuthall's (1996) contention that differences between theories of classroom learning . . . reflect differences in the paradigm that researchers have in mind as they think about how their research relates to classroom practice and theory. The evolution of research and theory is itself a sociocultural activity shaped by the working relationships that researchers establish with those whose lives and experiences they study and whose lives and experiences they hope to influence, (p. 6) A second point on which we differ with some of the more uncompromising versions of situated learning theory follows from our discussion of units of analysis. We described the theoretical approach that has emerged in the course of our work in which we view students' reasoning as necessarily situated while simultaneously attending to qualitative differences in the ways that they participate in communal practices. We therefore have difficulty with epistemological behaviorist positions that identify meaning exclusively with use and limit their focus to observed social activity. By the same token, however, we questioned the treatment of meaning in Anderson et al.'s (1996) cognitive perspective. For our purposes, it is essential to go beyond cognitive behavior by analyzing the quality of students' inferred, socially situated mathematical experience. As we noted, the distinction between these two perspectives' treatments of mathematical meaning corresponds to that which MacKay (1969) makes between the observer's and the actor's viewpoints. Our strong preference for the actor's viewpoint is rooted in our activity as researchers who co-participate in the learning-teaching process with teachers and their students. As Rommetveit (1992) and Schutz (1962) both emphasize, to co-participate is to engage in communicative in teractions that involve a reciprocity of perspectives characteristic of the actor's viewpoint. In closing, we stress that our critique of Anderson et al.'s position does not demonstrate that it is wrong in the sense that it somehow fails to capture the essence of individual and collective human activity. Instead, our central claim has been that the characterizations of meaning and learning that this perspective offers do not address our concerns as classroom-based researchers and instructional designers. We readily acknowledge that this theoretical orientation might well be appropriate for other purposes such as designing expert systems or supporting the development of competent performance on a clearly delineated range of tasks. Further, in light of the numerous references that Anderson et al. (1996) make to empirical findings, we accept that the cognitive perspective might be judged to be pro-


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gressive when assessed in terms of its own internal criteria. We do, however, question whether it has made significant progress in recent years on issues that are relevant to those of us who conduct classroom-based research and instructional design. In general, the conclusions we have reached are less generous than those of Greeno (1997), who speaks of exchanges between cognitive and situated learning theorists spurring the development of more comprehensive theories of learning. The contrast in Greeno's and our views on the potential contributions of the cognitive perspective appears to reflect our concerns as members of different research communities. Notes The analysis reported in this article was supported by the National Science Foundation under Grant No. RED 9353587 and by the Office of Educational Research and Improvement under Grant No. R305A60007. The opinions expressed do not necessarily reflect the views of either the Foundation or of OERI. We wish to thank Anna Sfard and an anonymous reviewer for helpful comments on a previous draft of this article. 'Sfard (1994) makes a similar distinction in her analysis of the development of the theoretical notion of concept in mathematics education. 2 We cjuestion the extent to which the work of Case and Griffin (1990) and Palincsar and Brown (1984) instantiates first-wave cognitive research as Anderson et al. (1997) suggest. For example, Wertsch, Tulviste, and Hagstrom (1993) argue that the instructional approach developed by Palincsar and Brown is consistent with their sociocultural perspective, in which agency is considered to be distributed across groups of individuals and the tools they use. -"Anderson et al. (1997) argue that casting their perspective in terms of skills and training involves a rhetorical language game that portrays the cognitive vision as dehumanizing. We wish to avoid this connotation while remaining faithful to the terminology that Anderson et al. (1996) use when describing their position. 4 This complementary relationship reflects the commonality of activity as a central notion in both sociocultural and social constructivist analyses. However, as information and representation rather than activity are basic in the cognitive perspective, attempts to coordinate cognitive analyses with, say, a sociocultural approach are highly problematic at best. 'Although this approach might not seem controversial, most instructional design efforts in mathematics do not involve the formulation of hypothetical learning trajectories that include testable conjectures about the means of supporting students' development of specific understandings. This comment applies particularly to design efforts that are based primarily on psychological analyses (cf. Freudenthal, 1983; Gravemeijer, 1994). 6 Given Anderson et al.'s (1997) remark that they do not really know what Greeno (1997) means by a student's identity as a learner, we note in passing that it is closely related to the student's developing beliefs about the self as learner and doer of mathematics. In the scheme we have outlined, this self is seen to be socially situated and to be constructed as the student participates in the negotiation of classroom norms and practices.

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Revision received March 30, 1998 Accepted April 27, 1998

AERA Annual Meeting Program Available Online The AERA home office and the ERIC Clearinghouse on Assessment and Evaluation are pleased to announce that the 1999 Annual Meeting Program is available online as a searchable database. You can search by words that appear in the session descriptions, including author's name, author's affiliation, session title, paper title, session number, and time.

Go to the AERA web page (http:// aera.net). Thanks to Lawrence M. Rudner, Catholic University, and Gene V. Glass, Arizona State University, for making this service possible.

MARCH 1999 UNIVERSITY 15 Downloaded from http://er.aera.net at VANDERBILT LIBRARY on January 7, 2015