ࡱ> 57234 @ <bjbj00 RbRb&X4X4X484L4HTl5&:"H:H:H:H:H:H:SSSSSSS$>VRXPS;H:H:;;SH:H:S>wDwDwD;tH:H: O~wD;SwDwDrLMH:`5 ̑"X4 >M NTHHTMYB$YMYM H:0x:"wD::H:H:H:SSDX Supporting a Discourse About Incommensurable Theoretical Perspectives in Mathematics Education Paul Cobb Vanderbilt University The issue in which I focus in this article is that of coping with multiple and frequently conflicting theoretical perspectives. This issue relates directly to the question of how is mathematic learned and how should it be taught, and has been the subject of considerable debate in both mathematics education and the broader educational research community. The theoretical perspectives currently on offer include radical constructivism, sociocultural theory, symbolic interactionism, distributed cognition, information-processing psychology, situated cognition, critical theory, critical race theory, and discourse theory. To add to the mix, experimental psychology has emerged with a renewed vigor in the last few years. I approach the thorny issue of coping with multiple perspectives by first questioning the repeated attempts that have been made in mathematics education to derive instructional prescriptions directly from background theoretical perspectives. I argue that it is instead more productive to compare and contrast various perspectives by using as a first criterion the manner in which they orient and constrain the types of questions that are asked about the learning and teaching of mathematics, the nature of the phenomena that are investigated, and the forms of knowledge that are produced. I then go on to propose that mathematics education can be productively construed as a design science, the mission of which involves developing, testing, and revising conjectured designs for supporting envisioned learning processes. This perspective on mathematics education gives rise to a second criterion, namely how various theoretical positions might contribute to this collective enterprise. In the next section of the article, I sharpen the two criteria for comparing theoretical perspectives. The first criterion focuses on how various perspective orient the types of questions asked and the forms of knowledge produced, and is often framed in terms of whether a particular perspective treats activity as being primarily individual or social in character. I argue that this dichotomy is misleading in that it assumes that what is meant by the individual is self-evident and theory neutral. As an alternative, I propose that it is more fruitful to compare and contrast different theoretical positions in terms of how they characterize individuals, be they students, teachers, or administrators. The second criterion concerns the potentially useful work that different theoretical positions might do and here I draw on Deweys sophisticated account of pragmatic justification and his related analysis of verification and truth. I then illustrate the relevance of the two criteria by presenting a brief comparison of three broad perspectives: cognitive psychology, sociocultural theory, and distributed cognition. In doing so, I conclude from the comparison of the three perspectives that each has limitations in terms of the extent to which it can contribute to the enterprise of formulating, testing, and revising designs for supporting learning. This leads me to propose that mathematics educators should view the various theoretical perspectives as sources of ideas to be appropriated and adapted to their purposes. In the final section of the article, I step back to locate the approach I have taken within a philosophical tradition that seeks to transcend the longstanding dichotomy between the quest for a neutral framework for comparing theoretical perspectives on the one hand and the view that we cannot reasonably compare perspectives on the other hand. THE POSITIVIST EPISTEMOLOGY OF PRACTICE Donald Schns ( ADDIN EN.CITE Schon19834931Schön, D. A.1983The reflective practitionerNew YorkBasic Booksteaching1983) book The Reflective Practitioner in one of the most widely cited texts in the filed of teacher education. In this book, Schn explicitly critiques what he terms the positivist epistemology of practice wherein practical reasoning is accounted for in terms of the application of abstract theoretical principles to specific cases. As Schn notes, this epistemology is apparent in attempts to derive pragmatic prescriptions for mathematics teaching and learning directly from background theoretical perspectives. The most prominent case in which such attempts have been made is that of the development of the general pedagogical approach known as constructivist teaching. This pedagogy attempts to translate the theoretical contention that learning is a constructive activity directly into instructional recommendations. Constructivism does not, however, have a monopoly on questionable reasoning of this type. For example, advocates of small group work have sometimes justified this instructional strategy by drawing on sociocultural theory. Similarly, adherents of the distributed view of intelligence have argued on occasion that as cognition is stretched over individuals, tools and social contexts, it is important to ensure that students mathematical activity involves the use of computers and other tools. In each of these cases, the difficulty is not with the background theory per se, but with the relation that is assumed to hold between theory and instructional practice. As I have argued elsewhere ( ADDIN EN.CITE Cobb200210850Cobb, P.2002Theories of knowledge and instructional design: A response to ColliverTeaching and Learning in Medicine1452-55philosophy, instructional design, constructivism, pragmatismCobb, 2002), pedagogical proposals developed in this manner involve a category error wherein the central tenets of a descriptive theoretical perspective are transformed directly into instructional prescriptions. The resulting pedagogies are underspecified and are based on ideology (in the disparaging sense of the term) rather than empirical analyses of the process of students learning and the means of supporting it in specific domains. Work in the philosophy, sociology, and history of science, sparked by the publication of Thomas Kuhns ( ADDIN EN.CITE Kuhn196210861Kuhn, T. S.1962The structure of scientific revolutionsChicagoUniversity of Chicago Press2ndphilosophy of science1962) landmark book, The Structure of Scientific Revolutions, has challenged the view of theoretical reasoning as the application of abstract theoretical principles to specific cases in the natural sciences. Kuhn followed Polanyi ( ADDIN EN.CITE Polanyi195810881Polanyi, M1958Personal knowledgeChicagoUniversity of Chicago Pressphilosophy of science1958) in questioning the assumption that scientists fully explicate the bases of their reasoning by presenting analyses of a number of historical cases. His goal in doing so was to demonstrate that the development and use of theory within an established research tradition necessarily involves tacit suppositions and assumptions that scientists learn in the course of their induction into their chosen specialties. Kuhn extended this argument about the tacit aspects of scientific reasoning when he considered how scientists choose between competing research traditions by arguing that there is no neutral algorithm of theory-choice, no systematic decision procedure which, properly applied, must lead each individual in the group to the same decision (p. 200). In making this contention, Kuhn was not questioning the rationality of scientists. Instead, he was challenging the dominant view that scientific reasoning could be modeled as a process of applying general rules and procedures to specific cases (Bernstein,  ADDIN EN.CITE Bernstein19832261Bernstein, R. J.1983Beyond objectivism and relativism: Science, hermeneutics, and praxisPhiladelphiaUniversity of Pennsylvania Pressphilosophy of science, pragmatism1983). Kuhn ( ADDIN EN.CITE Kuhn197010877Kuhn, T. S.1970Reflections on my criticsLakatos, I.Musgrave, A.Criticism and the growth of knowledgeCambridge, EnglandCambridge University Press231-278philosophy of science1970) subsequently clarified that what I am denying is neither the existence of good reasons [for choosing one theory over another] nor that these reasons are of the sort usually described. I am, however, insisting that such reasons constitute values to be used in making choices rather than rules of choice. Scientists who share them may nevertheless make different choices in the same concrete situation [In concrete cases, scientific values such as] simplicity, scope, fruitfulness, and even accuracy can be judged quite differently (which is not to say that they can be judged arbitrarily) by different people. Again, they may differ in their conclusions without violating any accepted rule. ( ADDIN EN.CITE Kuhn197010877Kuhn, T. S.1970Reflections on my criticsLakatos, I.Musgrave, A.Criticism and the growth of knowledgeCambridge, EnglandCambridge University Press231-278philosophy of sciencep. 262) As Bernstein (1983) observes, Kuhns analysis of rationality in the sciences is highly compatible with Gadamers ( ADDIN EN.CITE Gadamer19753411Gadamer, H-G.1975Truth and methodNew YorkSeabury Pressdiscourse, philosophy1975) philosophy of practical activity. The specific phenomenon that Gadamer analyzed was the process by which people interpret and understand texts, particularly religious texts. In developing his position, Gadamer responded to earlier work that instantiated the positivist epistemology of practice by differentiating between general methods of interpretation and understanding on the one hand, and the process of applying them to specific texts on the other hand. Gadamer rejected this distinction, arguing that every act of understanding involves interpretation, and all interpretation involves application. On this basis, he concluded that the characterization of theoretical reasoning as the application of general, decontextualized methods serves both to mystify science and to degrade practical reasoning to technical control. Kuhns and Gadamers arguments call into question attempts to develop a philosophy of mathematics education that is normative and prescriptive and that seeks to identify foundational principles from which mathematics education researchers and practitioners are expected to derive their practices. My goal in this article is therefore not to formulate a general method or procedure for choosing between different theoretical postions, but to initiate a conversation in which mathematics education researchers can begin to work through the challenges posed by a proliferation of perspectives. As the neo-pragmatist philosopher Rorty ( ADDIN EN.CITE Rorty19794051Rorty, R.1979Philosophy and the mirror of naturePrinceton, NJPrinceton University Pressphilosophy, pragmatism1979) clarified in his much cited book, Philosophy and the Mirror of Nature, philosophy so construed is therapeutic in that it does not presume to tell people how they should act in particular types of situations. Instead, its goal is to enable people themselves to cope with the complexities, tensions, and ambiguities that characterize the settings in which they act and interact. As will become apparent, such an approach seeks to avoid both the unbridled relativism evident in the contention that one cannot sensibly weigh the potential contributions of different theoretical perspectives and the unhealthy fanaticism inherent in the all-too-common claim that one particular perspective gets the world of teaching and learning right. As an initial criterion, I propose comparing and contrasting the different theoretical perspectives in terms of the manner in which they orient and constrain the types of questions that are asked about the learning and teaching of mathematics, and thus the nature of the phenomena that are investigated and the forms of knowledge produced. Adherents to a research tradition view themselves as making progress to the extent that they are able to address questions that they judge to be important. The content of a research tradition, including the types of phenomena that are considered to be significant, therefore represent solutions to previously posed questions. However, as Jardine ( ADDIN EN.CITE Jardine199110921Jardine, N.1991The scenes of inquiry: On the reality of questions in the sciencesOxfordClarendon Pressphilosophy of science1991) demonstrates by means of historical examples, the questions that are posed within one research tradition frequently seem unreasonable and, at times, unintelligible from the perspective of another tradition. In Lakatos ( ADDIN EN.CITE Lakatos197010937Lakatos, I.1970Falsification and the methodology of scientific research programmesLakatos, I.Musgrave, A.Criticism and the growth of knowledgeCambridge, UKCambridge University Press91-195philosophy of science1970) formulation, a research tradition comprises positive and negative heuristics that constrain (but do not predetermine) the types of questions that can be asked and those that cannot, and thus the overall direction of the research agenda. In my view, delineating the types of phenomena that can be investigated within different research traditions is a key step in the process of comparing and contrasting them. As Hacking ( ADDIN EN.CITE Hacking200010351Hacking, I.2000The social construction of what?Cambridge, MAHarvard University Pressphilosophy of science, identity2000) observed, while the specific questions posed and the ways of addressing them are visible to researchers working within a given research tradition, the constraints on what is thinkable and possible are typically invisible. This initial criterion is therapeutic in Rortys ( ADDIN EN.CITE Rorty19794051Rorty, R.1979Philosophy and the mirror of naturePrinceton, NJPrinceton University Pressphilosophy, pragmatism1979) sense in that the process of comparing and contrasting perspectives provides a means both of deepening our understanding of the research traditions in which we work, and of enabling us to de-center and develop a basis for communication with colleagues whose work is grounded in different research traditions. MATHEMATICS EDUCATION AS A DESIGN SCIENCE The second criterion that I propose for comparing different theoretical perspectives focuses on their potential usefulness to the concerns and interests as mathematics educators. My treatment of this criterion is premised on the argument that mathematics education can be productively viewed as a design science, the collective mission of which involves developing, testing, and revising conjectured designs for supporting envisioned learning processes. As an illustration of this collective mission, the first two highly influential Standards documents developed by the National Council of Teachers of Mathematics ( ADDIN EN.CITE National Council of Teachers of Mathematics19897601National Council of Teachers of Mathematics,1989Curriculum and evaluation standards for school mathematicsReston, VAAuthormathematics education1989,  ADDIN EN.CITE National Council of Teachers of Mathematics19917611National Council of Teachers of Mathematics,1991Professional Standards for teaching mathematicsReston, VAAuthormathematics education1991) can be viewed as specifying an initial design for the reform of mathematics teaching and learning in North America. The more recent Principles and Standards for School Mathematics ( ADDIN EN.CITE National Council of Teachers of Mathematics20009251National Council of Teachers of Mathematics,2000Principles and standards for school mathematicsReston, VAAuthorNational Council of Teachers of Mathematics, 2000) propose a revised design that was formulated in response to developments in the field, many of which were related to attempts to realize the initial design. As a second illustration, the functions of leadership in mathematics education at the level of schools and local school systems include mobilizing teachers and other stakeholders to notice, face, and take on the task of improving mathematics instruction ( ADDIN EN.CITE Spillane2000950Spillane, J. P.2000Cognition and policy implementation: District policy-makers and the reform of mathematics educationCognition and Instruction18141-179institutional contextSpillane, 2000). Ideally, this task involves the iterative development and revision of designs for improvement as informed by ongoing documentation of teachers instructional practices and students learning ( ADDIN EN.CITE Fishman20049880Fishman, B.Marx, R. W.Blumenfeld, P.Krajcik, J. S.2004Creating a framework for research on systemic technology innovationsJournal of the Learning Sciences1343-76design researchFishman, Marx, Blumenfeld, & Krajcik, 2004). At the classroom level, the design aspect of teaching is particularly evident in Stigler and Hieberts ( ADDIN EN.CITE Stigler19992631Stigler, J. W.Hiebert, J. I.1999The teaching gapNew YorkFree Pressteaching, PTC1999) description of the process by which Japanese mathematics teachers collaborate to develop and revise the design of lessons, and in empirical analyses of mathematics teaching as an iterative process of testing and revising conjectures ( ADDIN EN.CITE Ball19936150Ball, D. L.1993With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematicsElementary School Journal934373-397teachingLampert200110961Lampert, M.2001Teaching problems and the problems of teachingNew Haven, CTYale University PressteachingSimon19951590Simon, M. A.1995Reconstructing mathematics pedagogy from a constructivist perspectiveJournal for Research in Mathematics Education26114-145teaching, constructivism, design researchBall, 1993; Lampert, 2001; Simon, 1995). Although the three illustrations focus on developing, testing, and revising designs at the level of a national educational system, a school or local school system, and a classroom respectively, the ultimate goal in each case is to support the improvement of students mathematical learning. As a point of clarification, I should stress that the contention that mathematics education can be productively viewed as a design science is not an argument in favor of one particular research methodology such as design experiments. Rather than being narrowly methodological, the claim focuses on mathematics educators collective mission and thus on the concerns and interests inherent in their work. This framing of mathematics education research therefore acknowledges that a wide spectrum of research methods ranging from experimental and quasi-experimental designs to surveys and ethnographies can contribute to this collective enterprise. The fruitfulness of the framing stems from the manner in which it delineates core aspects of mathematics education research as a disciplinary activity. These core aspects include: Specifying and clarifying the prospective endpoints of learning Formulating and testing conjectures about both the nature of learning processes that aim towards those prospective endpoints and the specific means of supporting their realization ( ADDIN EN.CITE Confrey20001237Confrey, J.Lachance, A.2000Transformative teaching experiments through conjecture-driven research designA. E. KellyR. A. LeshHandbook of research design in mathematics and science educationMahwah, NJErlbaum231-266design researchGravemeijer199491Gravemeijer, K.1994Developing realistic mathematics educationUtrecht, The NetherlandsCD-ß Pressinstructional design, RME, Design ResearchSimon19951590Simon, M. A.1995Reconstructing mathematics pedagogy from a constructivist perspectiveJournal for Research in Mathematics Education26114-145teaching, constructivism, design researchConfrey & Lachance, 2000; Gravemeijer, 1994a; Simon, 1995) This formulation is intended to be inclusive in that the learning processes of interest could be those of individual students or teachers, of classroom or professional teaching communities, or indeed of schools or school districts viewed as organizations. The framing of mathematics education as a design science gives rise to a second criterion for comparing and contrasting background theoretical perspectives that concerns their usefulness. Stated as directly as possible, the usefulness criterion focuses on the extent to which different theoretical perspectives might contribute to the collective enterprise of developing, testing, and revising designs for supporting learning. This second criterion reflects the view that the choice of theoretical perspective requires pragmatic justification whereas the first focuses on the questions asked and the phenomena investigated. In the next two sections of this article, I sharpen each criterion in turn. THE NOTION OF THE INDIVIDUAL Historically, mathematics education researchers looked to cognitive psychology as a primary source of theoretical insight. However, during the last fifteen years, a number of theoretical perspectives that treat individual cognition as socially and culturally situated have become increasingly prominent. In their historical overview of the field, De Corte et al. ( ADDIN EN.CITE DeCorte19962587De Corte, E.Greer, B.Verschaffel, L.1996Mathematics learning and teachingBerliner, D.Calfee, R.Handbook of educational psychologyNew YorkMacmillan491-5491996) speak of first wave and second wave theories to distinguish between these two types of theories. The goal of first-wave theories is to model teachers and students individual knowledge and beliefs by positing internal cognitive structures and processes that account for their observed activity. De Corte et al. portray the emergence of second-wave theories as a response to the limited emphasis on affect, context, and culture in first-wave research. Theories of this type typically treat teachers and students cognitions as situated with respect to their participation in particular social and cultural practices. The distinction that De Corte et al. ( ADDIN EN.CITE De Corte19962587De Corte, E.Greer, B.Verschaffel, L.1996Mathematics learning and teachingBerliner, D.Calfee, R.Handbook of educational psychologyNew YorkMacmillan491-5491996) draw between cognitive and situated perspectives serves as the primary way in mathematics education researchers compare and contrast theoretical perspectives. Although these comparisons throw important differences between perspectives into sharp relief, they rest on the misleading assumption that the notion of the individual is theoretically neutral. Adherents to different perspectives conceptualize the individual in fundamentally different ways. Furthermore, these differences are central to the types of questions that adherents to the particular perspectives ask, the nature of the phenomena that they investigate, and the forms of knowledge that they produce. I can best exemplify the difficulties that arise when the notion of the individual is taken as self-evident by focusing on distributed theories of intelligence. The term distributed intelligence is perhaps most closely associated with Pea ( ADDIN EN.CITE Pea1985820Pea, R. D.1985Beyond amplification: Using computers to reorganize human mental functioningEducational Psychologist20167-182distributed intelligence, situated cognitionPea1993837Pea, R. D.1993Practices of distributed intelligence and designs for educationG. SalomonDistributed cognitionsNew YorkCambridge University Press47-87distributed intelligence, instructional design1985; 1993). A central assumption of this theoretical perspective is that intelligence is distributed across minds, persons, and symbolic and physical environments, both natural and artificial ( ADDIN EN.CITE Pea1993837Pea, R. D.1993Practices of distributed intelligence and designs for educationG. SalomonDistributed cognitionsNew YorkCambridge University Press47-87distributed intelligence, instructional designPea, 1993, p. 47). This theoretical orientation is consistent with the basic Vygotskian insight that students use of symbols and other tools profoundly influences both the process of their mathematical development and its products, increasingly sophisticated mathematical ways of knowing ( ADDIN EN.CITE Dörfler200087Dörfler, W.2000Means for meaningCobb, P.Yackel, E.McClain, K.Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional designMahwah, NJErlbaum99-132discourse, mathematical learning, symbolizingMeira1998240Meira, L.1998Making sense of instructional devices: The emergence of transparency in mathematical activityJournal for Research in Mathematics Education29121-142symbolizing; mathematical learningvan Oers20001877van Oers, B.2000The appropriation of mathematical symbols: A psychosemiotic approach to mathematical learningCobb, P.Yackel, E.McClain, K.Symbolizing and communicating in mathematics classrooms: Perspectives on discourse, tools, and instructional designMahwah, NJErlbaum133-176Drfler, 2000; Meira, 1998; van Oers, 2000). Pea ( ADDIN EN.CITE Pea1993837Pea, R. D.1993Practices of distributed intelligence and designs for educationG. SalomonDistributed cognitionsNew YorkCambridge University Press47-87distributed intelligence, instructional design1993) has been outspoken in delegitimizing analyses that take the individual as a unit of analysis, arguing that a functional system comprising the individual, tools, and social contexts is the appropriate unit. Not surprisingly, Peas admonition has been controversial. For example, Solomon ( ADDIN EN.CITE Solomon1993805805Solomon, G.1993No distribution without individuals' cognition: A dynamic interactional viewG. SolomonDistributed cognitionsCambridgeCambridge University Press111-138information processing,psychology,distributed intelligence1993) rejected Peas claim and argued that, in distributed accounts of intelligence are truncated and conceptually unsatisfactory (p. 111). It is worth pausing before adopting one or the other of these opposing positions to clarify what Pea might mean when he speaks of the individual. His arguments indicate that individuals, as he conceptualizes them, do not use tools and do not take account of context as they act and interact. They appear to be very much like individuals as portrayed by mainstream cognitive science who produce observed behaviors by creating and manipulating internal symbolic representations of the external environment. Peas proposal is to equip these encoders and processors of information with cultural tools and place them in social context. This proposition is less contentious than his apparent claim that all approaches that focus on the quality of individuals reasoning should be rejected regardless of how the individual is conceptualized. The relevant issue once we clarify what we are talking about is not whether it is legitimate to focus on individual teachers and students reasoning. Instead, it is how the individual might usefully be conceptualize d given mathematics educators concerns and interests. USEFULNESS AND TRUTH A concern for the usefulness of different theoretical perspectives carries with it the implication that they can be viewed as conceptual tools. This metaphor fits well with the characterization of mathematics education as a design science. However, for this criterion itself to do useful work, we have to clarify what it means for a theoretical perspective to contribute to the collective enterprise of mathematics education. Prawats ( ADDIN EN.CITE Prawat19954530Prawat, R. S.1995Misreading Dewey: Reform, projects, and the language gameEducational Researcher24713-22pragmatism1995) discussion of three types of pragmatic justification identified by Pepper ( ADDIN EN.CITE Pepper194211081Pepper, S. C.1942World hypothesesBerkeley, CAUniversity of California Pressphilosophy1942) is relevant in this regard. The first type of pragmatic justification is purely instrumental in that actions are judged to be true if they enable the achievement of goals. Prawat ( ADDIN EN.CITE Prawat19954530Prawat, R. S.1995Misreading Dewey: Reform, projects, and the language gameEducational Researcher24713-22pragmatism1995) indicates the limitations of this formulation when he observes that a rat navigating a maze has as much claim on truth, according to this approach, as the most clear-headed scientist (p. 19). In the second type of pragmatic justification, it is not a successful act that is true, but the hypothesis that leads to the successful act. When one entertains a hypothesis, Pepper ( ADDIN EN.CITE Pepper194211081Pepper, S. C.1942World hypothesesBerkeley, CAUniversity of California Pressphilosophy1942) points out, it is in anticipation of a specific outcome. When that outcome occurs, the hypothesis is verified or judged true. ( ADDIN EN.CITE Prawat19954530Prawat, R. S.1995Misreading Dewey: Reform, projects, and the language gameEducational Researcher24713-22pragmatismPrawat, 1995, p. 19) As Prawat goes on to note, this is a minimalist view in that hypotheses are treated as tools for the control of nature. This portrayal sits uncomfortably with Tolumins ( ADDIN EN.CITE Toulmin19638201Toulmin, S.1963Foresight and understandingNew YorkHarper Torchbookphilosophy of science1963) demonstration that the primary goal of science is to develop insight and understanding into the phenomena under investigation, and that instrumental control is a by-product. The third type of pragmatic justification, which Pepper ( ADDIN EN.CITE Pepper194211081Pepper, S. C.1942World hypothesesBerkeley, CAUniversity of California Pressphilosophy1942) calls the qualitative confirmation test of truth, brings the development of insight and understanding to the fore. This type of justification builds on Deweys analysis of the function of thought and the process of verification. Dewey viewed ideas as potentially revisable plans for action and argued that their truth is judged in terms of the extent to which they lead to a satisfactory resolution to problematic situations ( ADDIN EN.CITE Sleeper19865241Sleeper, R. W.1986The necessity of pragmatism: John Dewey's conception of philosophyNew HavenYale University PresspragmatismWestbrook19919021Westbrook, R. B.1991John Dewey and American democracyIthaca, NYCornell University Presspragmatism, situated cognitionSleeper, 1986; Westbrook, 1991). Furthermore, he treated foresight and understanding as integral aspects of thought, the primary function of which is to project future possibilities and to prepare us to come to grips with novel, unanticipated occurrences. The anticipatory and adaptive functions of thought are central to Deweys analysis of verification as a process in which the phenomena under investigation talk back, giving rise to surprises and inconsistencies. Crucially, Dewey also viewed verification as a context in which theoretical ideas are elaborated and modified. As Pepper ( ADDIN EN.CITE Pepper194211081Pepper, S. C.1942World hypothesesBerkeley, CAUniversity of California Pressphilosophy1942) put it, ideas judged to be true are those that give insight into what he termed the texture and quality of the phenomena that serve to verify them. In the case of mathematics education, for example, ideas are potentially useful to the extent that they give rise to conjectures about envisioned learning processes and the specific means of supporting them. However, these ideas are not simply either confirmed unchanged or rejected during the process of testing and revising designs. Instead, Dewey drew attention to the process by which ideas that are confirmed evolve as they are verified. It is worth observing in passing that Deweys analysis has far reaching implications in that it challenges a distinction central to the traditional project of epistemology, that between the context of discovery and the context of justification. However, the key implication to note for my purposes is that the truth of fallible, potentially revisable ideas is justified primarily in terms of the insight and understanding they give into learning processes and the means of supporting their realization. This is the criterion that I will use when I focus on the potentially useful work that cognitive psychology, sociocultural theory, and distributed cognition might do in contributing to the enterprise of formulating, testing, and revising conjectured designs for supporting envisioned learning processes. COMPARING THEORETICAL PERSPECTIVES Cognitive Psychology My discussion of cognitive psychology is limited to theoretical orientations that involve what MacKay ( ADDIN EN.CITE MacKay1969201MacKay, D. M.1969Information, mechanism, and meaningCambridgeMIT Pressconstructivism, philosophy1969) termed the actors viewpoint. The goal of psychologies of this type is to account for the process of students mathematical learning by positing changes in internal cognitive structures and processes that account for the development of their inferred interpretations and understandings. Examples of theories of this type include Pirie and Kierens ( ADDIN EN.CITE Pirie19941530Pirie, S.Kieren, T. E.1994Growth in mathematical understanding: How can we characterize it and how can we represent it?Educational Studies in Mathematics2661-86constructivism, mathematical learning1994) recursive theory of mathematical understanding , Dubinskys ( ADDIN EN.CITE Dubinsky19913087Dubinsky, E.1991Reflective abstraction in advanced mathematical thinkingTall, D.Advanced mathematical thinkingDordrecht, The NetherlandsKluwer95-123constructivism, mathematical learning1991) theory of encapsulation, and Vergnauds ( ADDIN EN.CITE Vergnaud19828257Vergnaud, G.1982A classification of cognitive tasks and operations of thought involved in addition and subtractionCarpenter, T. P.Moser, J. M.Romberg, T. A.Addition and subtraction: A cognitive perspectiveHillsdale, NJErlbaummathematical learning1982) analysis of the process by which students gradually explicate their initial theorems-in-action. Each of these theorists considers the primary source of increasingly sophisticated forms of mathematical reasoning to be students activity of interpreting and attempting to complete instructional activities, not the instructional activities themselves. Furthermore, they each characterize mathematical learning as a process in which operational or process conceptions evolve into what Sfard ( ADDIN EN.CITE Sfard1991330Sfard, A.1991On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coinEducational Studies in Mathematics221-36constructivism; mathematical learning1991) terms object-like structural conceptions. Proficiency in a particular mathematical domain is therefore seen to involve the conceptual manipulation of mathematical objects whose reality is taken for granted. Greeno ( ADDIN EN.CITE Greeno19911340Greeno, J. G.1991Number sense as situated knowing in a conceptual domainJournal for Research in Mathematics Education22170-218distributed intelligence, situated cognition1991) captured this aspect of mathematical proficiency when he introduced the metaphor of acting in a mathematical environment in which tools and resources are ready at hand to characterize number sense. For her part, Sfard ( ADDIN EN.CITE Sfard2000347Sfard, A.2000Symbolizing mathematical reality into beingCobb, P.Yackel, E.McClain, K.Symbolizing, communicating, and mathematizing in reform classrooms: Perspectives on discourse, tools, and instructional designMahwah, NJErlbaum37-98discourse; mathematical learning2000) describes mathematical discourse as a virtual reality discourse to highlight the parallels between this discourse and the ways in which we talk about physical reality. It is important to note that each of the theoretical schemes does not focus on the mathematical development of any particular student but are instead concerned with the learning of an idealized student that Thompson and Saldanha ( ADDIN EN.CITE Thompson20008647Thompson, P. W.Saldanha, L. A.2000Epistemological analyses of mathematical ideas: A research methodologyM. FernandezProceedings of the Twenty-Second Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics EducationColumbus, OHERIC Clearinghouse for Science, Mathematics, and Environmental Education2403-407constructivism, mathematical learning2000) refer to as the epistemic individual. Researchers working in this cognitive tradition account for variations in specific students reasoning by using the constructs that comprise their conceptual framework to develop explanatory accounts of each students mathematical activity. This approach enables the researchers to both compare and contrast the quality of specific students reasoning and to consider the possibilities for their mathematical development. The cognitive theorists I have referenced portray both the epistemic individual and specific students portrayed as active constructors of increasingly sophisticated forms of mathematical reasoning. This metaphor of active construction is drawn from Piagets ( ADDIN EN.CITE Piaget19707721Piaget, J.1970Genetic epistemologyNew YorkColumbia University Pressconstructivism1970) epistemological theory of the origins and development of logico-mathematical knowledge. Piaget drew on his early training as a biologist to characterize intellectual development as an adaptive process in the course of which children reorganize their sensory-motor and conceptual activity ( ADDIN EN.CITE Piaget19807751Piaget, J.1980Adaptation and intelligence: Organic selection and phenocopyChicagoUniversity of Chicago PressconstructivismPiaget, 1980). In appropriating Piagets general constructivist orientation to the process of development, cognitive theorists have necessarily had to adapt the theoretical constructs that he proposed given that their concern is to gain insight into the process of students mathematical learning. In doing so, they seek to understand how the epistemic individual successively reorganizes its activity and comes to act in a mathematical environment. Their goal is therefore to delineate how the world of meaning and significance in which the epistemic individual acts changes in the course of development. Proponents of this cognitive tradition frequently assume that the types of explanatory frameworks they produce constitute an adequate basis for both classroom instructional design and pedagogical decision making. However, it is questionable that such frameworks are, by themselves, sufficient. The primary difficulty is precisely that the forms of knowledge produced are cognitive rather than instructional and do not involve positive heuristics for design ( ADDIN EN.CITE Gravemeijer19941300Gravemeijer, K.1994Educational development and developmental researchJournal for Research in Mathematics Education25443-471design research, rme, instructional designGravemeijer, 1994b). In my view, researchers working in this tradition have inherited perspectives and associated methodologies from cognitive psychology but have not always reflected on the relevance of the forms of knowledge produced to the collective enterprise of mathematics education. The issue of how these perspectives and methodologies might be adapted so that they can better contribute to the concerns and interests of mathematics education has rarely arisen because their sufficiency has been assumed almost as an article of faith (see  ADDIN EN.CITE Thompson200211257Thompson, P. W.2002Didactical objects and didactical models in radical constructivismGravemeijer, K.Lehrer, R.van Oers, B.Verschaffel, L.Symbolizing, modeling and tool use in mathematics educationDordrecht, The NetherlandsKluwer197-220instructional design, symbolizingThompson, 2002, for a rare discussion of these issues). Given that cognitive theories of mathematical learning do not, by themselves, constitute a sufficient basis for design, the question of clarifying the contributions that they can make remains. I can identify three potentially important contributions. First, domain-specific theories typically include analyses of the forms of reasoning that we want students to develop, thereby giving an overall orientation to the instructional design effort. Second, domain-specific frameworks can alert the designer to major shifts in students mathematical reasoning that the design should support. Third, the designers or teachers use of a domain-specific framework to gain insight into specific students mathematical reasoning can inform the design of instructional activities intended to support subsequent learning. As I will clarify when I discuss distributed cognition, I consider this emphasis on instructional activities as the primary means of supporting students mathematical learning to be overly restrictive. In summary, general and domain-specific theories of mathematical learning are both concerned with the learning of an idealized student, the epistemic individual (see Table 1). Variations in specific students reasoning are accounted for by using the constructs central to theories of this type to develop explanations of their mathematical activity. Cognitive theories of the type that I have referenced characterize mathematical learning as a constructive process in the course of which students successively reorganize their sensory-motor and conceptual activity. The forms of knowledge produced by cognitive psychology can contribute to the development of classroom instructional designs by delineating central mathematical ideas and by informing the design of instructional tasks. Table 1. Contrasts Between Four Theoretical Perspectives Theoretical PerspectiveCharacterization of the IndividualUsefulnessLimitationsCognitive PsychologyEpistemic individual as reorganizer of activitySpecification of big ideas Design of instructional activitiesMeans of supporting learning limited to instructional tasksSociocultural TheoryIndividual as participant in cultural practicesDesigns that take account of students out-of-school practicesLimited relevance to design at classroom levelDistributed CognitionIndividual element of a reasoning systemDesign of classroom learning environments including norms, discourse, and toolsDelegitimizes cognitive analyses of specific students reasoning Sociocultural Theory In contrast to cognitive theorists focus on the epistemic individual, the forms of knowledge produced by sociocultural theorists concern the process by which people develop particular forms of reasoning as they participate in established cultural practices. This theoretical perspective equates intellectual development with the process of becoming an increasingly substantial participant in various cultural practices. Consequently, sociocultural theorists investigate the activity of the individual-in-cultural-practice. Contemporary sociocultural theory draws directly on the writings of Vygotsky and Leontev. For Vygotsky ( ADDIN EN.CITE Vygotsky19621001Vygotsky, L. S.1962Thought and languageCambridge, MAMIT Presssociocultural theory, discourseVygotsky19781011Vygotsky, L. S.1978Mind and society: The development of higher psychological processesCambridge, MAHarvard University Presssociocultural theory1962; 1978), human history is the history of artifacts such as language, counting systems, and writing that are passed on from one generation to the next. In formulating his theory of intellectual development, Vygotsky developed an analogy between the use of physical tools and the use of intellectual tools such as sign systems ( ADDIN EN.CITE Kozulin1990681Kozulin, A.1990Vygotsky's psychology: A biography of ideasCambridgeHarvard University Presssociocultural theoryvan der Veer1991991van der Veer, R.Valsiner, J.1991Understanding Vygotsky: A quest for synthesisCambridge, MABlackwellsociocultural theoryKozulin, 1990; van der Veer & Valsiner, 1991). His central claim was that just as the use of a physical tool serves to reorganize activity by making the formulation of new goals possible, so the use of sign systems serves to reorganize thought. Consequently, in Vygotskys view, childrens mastery of a cultural artifact such as a counting system does merely enhance or amplify an already existing cognitive capability. He instead argued that childrens ability to reason numerically is created as they appropriate the counting systems of their culture. This example illustrates Vygotskys more general contention that childrens minds are formed as they appropriate sign systems and other artifacts. Although Vygotsky brought sign systems and other cultural tools to the fore, he gave less attention to material reality. In building on Vygotskys ideas, Leontev ( ADDIN EN.CITE Leont'ev1978721Leont'ev, A. N.1978Activity, consciousness, and personalityEnglewood Cliffe, NJPrentice-Hallsociocultural theory1978) argued that material objects as they come to be experienced by the developing child are defined by the cultural practices in which he or she participates (cf.  ADDIN EN.CITE Minick1987797Minick, N.1987The development of Vygotsky's thought: An introductionRieber, R. W.Carton, A. S.The collected works of Vygotsky, L.S.New YorkPlenum117-38sociocultural theoryMinick, 1987). For example, a pen becomes a writing instrument rather than a brute material object for the child as he or she participates in literacy practices. In Leontevs view, the child does not come into contact with material reality directly, but is instead oriented to this reality as he or she participates in cultural practices. He therefore concluded that the meanings that material objects come to have are a product of their inclusion in specific practices. This thesis serves to underscore his argument that the individual-in-cultural-practice constitutes the appropriate analytical unit. I contend that sociocultural theory is of limited utility when actually formulating designs at the classroom level. The contributions of Davydov ( ADDIN EN.CITE Davydov19882940Davydov, V. V.1988Problems of developmental teaching (Part I)Soviet Education3086-97sociocultural theory, instructional design1988), notwithstanding, it is in fact difficult to identify instances of influential designs whose development has been primarily informed by sociocultural theory. This becomes understandable once we note that the notion of cultural practices employed by sociocultural theorists typically refers to ways of talking and reasoning that have emerged during extended periods of human history. This construct makes it possible to characterize mathematics as a complex human activity rather than as disembodied subject matter ( ADDIN EN.CITE van Oers1996377van Oers, B.1996Learning mathematics as meaningful activityNesher, P.Steffe, L. P.Cobb, P.Goldin, G. A.Greer, B.Theories of mathematical learningHillsdale, NJErlbaum91-114sociocultural theory, mathematical learning, symbolizingvan Oers, 1996). The task facing both the teacher and the instructional designer is therefore framed as that of supporting and organizing students induction into practices that have emerged during the disciplines intellectual history. While the importance of the goals inherent in this framing is indisputable, they provide only the most global orientation for design. A key difficulty is that the disciplinary practices that are taken as the primary point of reference exist prior to and independently of the activities of teachers and their students. In contrast, the ways of reasoning and communicating that are actually established in the classroom do not exist independently of the teachers and students activity, but are instead constituted by them in the course of their ongoing interactions ( ADDIN EN.CITE Beach19991100Beach, K.1999Consequential transitions: A sociocultural expedition beyond transfer in educationReview of Research in Education24103-141transfer, sociocultural theory, institutional context, situated cognitionCobb20009457Cobb, P.2000The importance of a situated view of learning to the design of research and instructionJ. BoalerMultiple perspectives on mathematics teaching and learningStamford, CTAblex45-82situated cognition, design research, copBauersfeld19806210Bauersfeld, H.1980Hidden dimensions in the so-called reality of a mathematics classroomEducational Studies in Mathematics1123-41symbolic interactionism, CoPBauersfeld, 1980; Beach, 1999; Cobb, 2000). A central challenge of design is to develop, test, and refine conjectures about both the classroom processes in which students might participate and the nature of their mathematical learning as they do so. Sociocultural theory provides only limited guidance because the classroom processes on which design focuses are emergent phenomena rather than already-established practices into which students are inducted. Extending our purview beyond the classroom, there is one area where, in my judgment, sociocultural theory can make a significant contribution. A number of researchers working within the sociocultural tradition have documented that the out-of-school practices in which students participate can involve differing norms of participation, language, and communication, some of which might be in conflict with those that the teacher seeks to establish in the mathematics classroom ( ADDIN EN.CITE Ladson-Billings19989530Ladson-Billings, G.1998It doesn't add up: African American students' mathematics achievementJournal for Research in Mathematics Education28697-708equityLadson-Billings, 1998). An emerging line of research in mathematics education draws on sociocultural theory to document such conflicts and to understand the tensions that students experience ( ADDIN EN.CITE Boaler20009437Boaler, J.Greeno, J. G.2000Identity, agency, and knowing in mathematical worldsJ. BoalerMultiple perspectives on mathematics teaching and learningStamford, CTAblex45-82identity, equity, situated cognition, COP, structure/agencyMoschkovich20029390Moschkovich, J.2002A situated and sociocultural perspective on bilingual mathematics learnersMathematical Thinking and Learning4189-212equity, discourseGutiérrez20029380Gutiérrez, R.2002Enabling the practice of mathematics teachers in context: Toward a new research agendaMathematical Thinking and Learning4145-189equityBoaler & Greeno, 2000; Gutirrez, 2002; Moschkovich, 2002). The value of work of this type is that it enables us to view students activity in the classroom as situated not merely with respect to the immediate learning environment, but with the respect to their history of participation in the practices of particular out-of-school groups and communities. It therefore has the potential to inform the development of designs in which the diversity in the out-of-school practices in which students participate is treated as an instructional resource rather than an obstacle to be overcome ( ADDIN EN.CITE Gutstein2002, April9683Gutstein, E.2002Roads towards equity in mathematics education: Helping students develop a sense of agencyannual meeting of the American Educational Research AssociationNew OrleansAprilequity, identityCivil200212240Civil, M.2002Culture and mathematics: A community approachJournal of Intercultural Studies23133-148equityCivil, 2002; Gutstein, 2002). In summary, sociocultural theory characterizes the individual as a participant in established, historically evolving cultural practices and accounts for learning in terms of peoples increasingly substantial participation in particular cultural practices (see Table 1). I have questioned the relevance of sociocultural theory to the development of designs at the classroom level but have suggested that analyses that compare the out-of-school practices in which students participate with those established in the mathematics classroom are important from the point of view of equity in students access to significant mathematical ideas. Distributed Cognition In contrast to sociocultural theorists focus on relatively broad cultural practices, distributed cognition theorists typically restrict their focus to the immediate physical, social, and symbolic environment. Empirical studies conducted within the distributed tradition therefore tend to involve detailed analysis of either specific peoples or a small groups activity rather than analyses of peoples participation in established cultural practices. The intent of these analyses is to understand how cognition is distributed across minds, persons, and symbolic and physical environments. As Pea (1985, 1993) and other distributed cognition theorists make clear, this perspective directly challenges a foundational assumption of both mainstream cognitive science and of the cognitive tradition that I have discussed (cf.  ADDIN EN.CITE Brown1989510Brown, J. S.Collins, A.Duguid, P.1989Situated cognition and the culture of learningEducational Researcher1832-42situated cognitionGreeno19971330Greeno, J. G.1997On claims that answer the wrong questionsEducational Researcher2615-17distributed intelligence, situated cognition, information processingBrown, Collins, & Duguid, 1989; Greeno, 1997). This is the assumption that cognition is bounded by the skin and can be adequately accounted for solely in terms of internal processes. Distributed cognition theorists instead see cognition as extending out into the immediate environment such that the environment becomes a resource for reasoning. As a consequence, the individual is characterized in this tradition as an element of a reasoning system. In developing to this position, distributed cognition theorists have been influenced by a number of studies conducted by sociocultural researchers, particularly those that compare peoples reasoning in different settings. One particularly influential line of work demonstrates that people develop significantly different forms of mathematical reasoning as they engage in school-like activities, and in everyday and workplace activities ( ADDIN EN.CITE Lave1988691Lave, J.1988Cognition in practice: Mind, mathematics and culture in everyday lifeNew YorkCambridge University PressCoP; situated cognitionScribner1984897Scribner, S.1984ing working intelligenceRogoff, B.Lave, J.Everyday cognition: Its development in social contextCambridge, MAHarvard University Press9-40situated cognition, mathematical learning, transferNunes1993801Nunes, T.Schliemann, A. D.Carraher, D. W.1993Street mathematics and school mathematicsCambridgeCambridge University Presssituated cognition, mathematical learning, transferLave, 1988; Nunes, Schliemann, & Carraher, 1993; Scribner, 1984). These findings led distributed cognition theorists to question whether traditional school-like tasks that are often used to investigate cognition constitute a viable set of cases from which to develop adequate accounts of cognition. Distributed cognition theorists have also critiqued typical school instruction, in the process drawing attention to the nature of the settings within which the tasks take on meaning and significance for students. In contrast to sociocultural theory, distributed cognition treats classroom processes as emergent phenomena rather than already-established practices into which students are inducted. For example, distributed theorists consider that aspects of the classroom learning environment such as classroom norms, discourse, and ways of using tools are constituted collectively by the teacher and students in the course of their ongoing interactions. In my judgment, the distributed perspective therefore has greater potential than sociocultural theory to inform the formulation of designs at the classroom level. As I noted, designs developed from the cognitive perspective on which I have focused typically emphasize the development of instructional activities. In contrast, distributed cognition theorists characterization of students as elements of reasoning systems orients them to construe the means of supporting students mathematical learning more broadly. The means of support that are incorporated into designs are usually not limited to instructional activities but also encompass classroom norms, the nature of discourse, and the ways in which notations and other types of tools are used. Thus, design from the distributed perspective focuses on the physical, social, and symbolic classroom environment that constitutes the immediate situation of the students mathematical learning. Given my generally positive assessment of the usefulness of the distributed perspective in informing the development, testing, and revision of designs, it is also important to note a potential limitation that concerns the scant attention typically given to issues of equity. This limitation stems from an almost exclusive focus on the classroom as the immediate context of students learning, thereby precluding a consideration of tensions that some students might experience between aspects of this social context and the out-of-school practices in which they participate. Adherents to this perspective could address this limitation by coordinating their viewpoint with a sociocultural perspective that enables them to see students classroom activity as situated not merely with respect to the immediate learning environment, but also with respect to their history of participation in the practices of out-of-school groups and communities. In summary, the distributed perspective characterizes the individual as an element of a reasoning system that also includes aspects of the immediate physical, social, and symbolic environment (see Table 1). In contrast to sociocultural theorists focus on peoples participation in established cultural practices, distributed theorists usually conduct detailed analyses of reasoning processes that are stretched over people and the aspect of their immediate environment that they use as cognitive resources. In my view, the distributed perspective has greater potential than sociocultural theory to contribute to the formulation of designs at the classroom level because it treats classroom processes as emergent phenomena. However, I also noted the limited attention given to issues of equity as well as the disavowal of cognitive analyses of specific students mathematical reasoning. Taken together, these limitations indicate the value of attempting to capitalize on the ways that multiple perspectives can contribution to the enterprise of developing, testing, and revising designs for supporting learning. THEORIZING AS BRICOLAGE Given the limitations that I have discussed of the cognitive, sociocultural, and distributed cognition perspectives with respect to the collective enterprise of developing, testing, and revising designs for supporting learning, the question that arises is not that of how to choose between the various perspectives. Instead, it is how they can be adapted to the concerns and interests of mathematics educators. In addressing this question, I propose to view the three perspectives as sources of ideas that mathematics educators can appropriate and modify for their purposes. This process of developing conceptual tools for mathematics education research parallels that of instructional design as described by Gravemeijer ( ADDIN EN.CITE Gravemeijer19941300Gravemeijer, K.1994Educational development and developmental researchJournal for Research in Mathematics Education25443-471design research, rme, instructional design1994b). [Design] resembles the thinking process that Lawler ( ADDIN EN.CITE Lawler198511451Lawler, R. W.1985Computer experience and cognitive development: A child’s learning in a computer cultureNew YorkWileyconstructivism1985) characterizes by the French word bricolage, a metaphor taken from Claude LeviStrauss. A bricoleur is a handy man who invents pragmatic solutions in practical situations. [T]he bricoleur has become adept at using whatever is available. The bricoleurs tools and materials are very heterogeneous: Some remain from earlier jobs, others have been collected with a certain project in mind. (p. 447) Similarly, I suggest that rather than adhering to one particular theoretical perspective, we act as bricoleurs by adapting ideas from a range of theoretical sources. As an illustration, sociocultural theorists notion of a cultural practice is attractive because it makes it possible to characterize mathematics as a complex human activity rather than as disembodied subject matter. However, the notion of a practice as framed in sociocultural theory is problematic from the point of view of design because practices are typically characterized as existing prior to and independently of the teachers and students activity. It would therefore be useful to modify this notion by explicitly defining the practices established in the classroom as an emergent phenomenon that are established jointly by the teacher and students in the course of their ongoing interactions. This is of course a non-trivial adaptation in that students are then seen to contribute to the development of the classroom norms and practices that constitute the social situation of their mathematical learning. As a second illustration, the notion of learning as a process of reorganizing activity proposed by cognitive theorists also seems potentially valuable. However, it would be important to broaden the characterization of activity so that it is not restricted solely to solo sensory-motor and conceptual activity but instead reaches out into the world and includes the use of tools and symbols. The rationale for this modification is at least in part pragmatic in that the work of instructional design involves developing notation systems, physical tools, and computer-based tools for students to use. The work of distributed cognition theorists is also relevant in this regard provided we modify how they characterize the individual. As I have noted, distributed theorists appear to have accepted the portrayal of the individual offered by mainstream cognitive science and propose equipping this character with cultural tools and locating it in social context. The individual might instead be conceptualized as already acting in its immediate physical, social, and symbolic environment. Once this modification is made, distributed cognition theorists notion of a student-tool system is displaced by the view of an individual student participating in classroom practices that involves reasoning with tools and symbols. With regard to usefulness, this latter viewpoint yields analyses of students mathematical learning that are tied to the classroom social setting in which that learning actually occurs. These analyses, in turn, enable us to tease out aspects of this setting that served to support the development of students reasoning, and thus to develop testable conjectures about ways in which those means of support might be improved. INCOMMENSURABILITY The contrasting ways in which the three different theoretical perspectives that I have discussed characterize the individual indicate that they are incommensurable. The approach I took in comparing and contrasting the perspectives is consistent with Feyerabends ( ADDIN EN.CITE Feyerabend197511491Feyerabend, P.1975Against methodLondonVersophilosophy of science1975) claim that people typically cope with immensurability both in research and in other areas of life by delineating similarities and differences. Feyerabend also argued that there is no single ultimate grid for comparing theoretical perspectives, and demonstrated that they can be compared in multiple ways. The primary challenge posed by incommensurability is therefore to develop a relevant way of comparing and understanding different perspectives. It was for this reason that I justified the two criteria I used in some detail. As Bernstein ( ADDIN EN.CITE Bernstein19832261Bernstein, R. J.1983Beyond objectivism and relativism: Science, hermeneutics, and praxisPhiladelphiaUniversity of Pennsylvania Pressphilosophy of science, pragmatism1983) observed, the process of comparing immensurable perspectives has parallels with anthropology in that the goal is to figure out what the natives think they are doing. As Geertz( ADDIN EN.CITE Geertz19736901Geertz, C.1973The interpretation of culturesNew YorkBasic Booksculture, identity1973) emphasizes, the absence of abstract, cross-cultural universals does not condemn anthropology to absolute relativism. If we want to discover what man [sic] amounts to, we can only find it in what men are, and what men are, above all other things, is various. It is in understanding that variousness its range, its nature, its basis, and its implications that we shall come to construct a concept of human nature that more than a statistical shadow and less than a primitive dream has both substance and truth. 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