This is a chapter in David Kirshner and Tony Whitson, eds, Situated Cognition: Social, Semiotic, and Psychological Perspectives, Erlbaum, 1996.
Please do not quote from this version, which may differ slightly from the version that appears in print.
In the last several years, Jean Lave and Valerie Walkerdine have developed searching analyses and critiques of mathematics education, particularly in their books Cognition in Practice (Lave, 1988) and The Mastery of Reason (Walkerdine, 1988). Despite their many differences, they share a compelling view of mathematics not as an abstract cognitive task but as something deeply bound up in socially organized activities and systems of meaning. The influence of their analyses has been growing steadily as educational researchers have looked for conceptions of cognition and learning that locate knowledge in particular forms of situated activity and not simply in mental contents (Brown, Collins and Duguid, 1989). Their work has also been part of an increasing interest in the ethnographic study of everyday mathematics (Leap, 1988; Saxe, 1991), the shared construction of mathematical knowledge in school (Newman, Griffin and Cole, 1989), and the relationship between school activities and the rest of life (Carraher, Carraher and Schliemann, 1985; Resnick, 1987; Scribner, 1986). Their understanding of mathematics as fundamentally a sociological phenomenon is a striking turn, holding out the hope for an analysis of math -- and perhaps of schooling and learning in general -- that includes homes, workplaces, and aspects of social relations that structure knowledge and learning across the whole of an individual's life.
The promise of Lave and Walkerdine's sociological analyses of mathematics, then, is considerable. But the very sophistication of their theories puts up equally considerable obstacles. Not only are their respective sociological methods unfamiliar to many readers, but the application of these methods to mathematical activities is demanding even as social-scientific research goes. As a result, future research in this area may benefit from a complex understanding of their conceptual frameworks. In this chapter I propose simply to draw some distinctions and pose some questions within the shared horizons of their research. Lave's Cognition in Practice and Walkerdine's The Mastery of Reason are challenging on their own for the scope of their intellectual ambition and for the powerful theoretical frameworks on which they draw in analyzing their materials. And they are particularly challenging to read in conjunction, given that despite their broad commonality of purpose and subject matter, they get ahold of their materials in remarkably different, and indeed often almost incommensurable, ways.
Lave on Mathematical Activity
First, though, let me summarize Lave and Walkerdine's respective theories in just enough detail to motivate the deliberations below. Lave is concerned to refute the view of mathematics as the kind of abstract thinking that a long tradition of research in cognitive psychology has called "problem solving." The salient feature of this process is that it unfolds in three distinct steps: life situations are translated into formal cognitive structures or "problems"; these structures are then manipulated through mental processes whose outcomes are called "solutions"; and these outcomes are finally interpreted within the larger life situation as actions to take or answers to give.
Mathematical calculation is often held to reflect this theory of cognition. In particular, Lave suggests that word problems presuppose certain ideas about mathematics and cognition: that occasions for mathematical reasoning present themselves as clear-cut "problems," that these problems can be solved by extracting mathematical information from a situation and then "setting up" the necessary calculations, that the outcomes of these calculations can be interpreted as "solutions" to these problems, and so on.
Lave takes particular issue with a theory of mathematical learning that she calls "transfer." Educationalists have used this word in a variety of ways, but Lave has something specific in mind. She associates the problem solving theory of cognition with a particular view of knowledge, according to which the contents of knowledge can be specified in abstract, formal terms independent of the larger social organization of the activities within which this knowledge is used. Learning is held to consist in the construction of this kind of knowledge, in large part through the discovery of structural analogies between one situation and another.
In contrast to this theory, Lave argues that mathematical reasoning must be viewed as deeply bound up with the activities within which it takes place. Her argument has several steps, which I can only briefly indicate here. Her central conclusion is that ordinary life activities do not present clear-cut "problems" in the sense required by the problem solving theory of cognition. She argues this point ethnographically, by exhibiting some of the properties of naturally occurring mathematical reasoning in particular settings. For example, she shows people bringing such a complex variety of considerations to bear on their decisions that one cannot find clear boundaries among the different phases of problem solving. In particular, one does find clear "solutions" but simply "resolutions" that keep things moving along, subject to later revision; things are not so much solved as usefully transformed. Even when the people are set to performing school-like calculations, for example in scientific diet programs, the activity quickly evolves and the boundaries between calculation and the rest of the activity steadily blur.
Lave concludes from this that the natural unit of analysis is not abstract knowledge but structured activity. People certainly do know things, but this knowledge is to be understood as an attribute of the varieties of structured activity that the people are capable of engaging in while pursuing particular socially organized ends in organized settings. Most importantly, knowledge is to be understood relationally -- that is, as something located in the evolving relationships between people and the settings in which they conduct their activities.
This is a tremendously difficult idea and it helps to understand the kind of motivation that Lave offers for it. Her project is ethnographic: she and her students hung out with some people, documented their activities, intervened in them to some extent, and searched for language that adequately described the things they saw. The language of problems and transfer, they argue, does not allow us to formulate adequate descriptions of what the people did, and this can be demonstrated by telling an orderly sequence of representative stories about their field informants and pointing to the difficulty or impossibility of describing the action in those stories using the conventional vocabulary. The point is not that the problem solving theory makes the wrong predictions but that the theory cannot even be applied. If the problem solving theory can be applied to laboratory situations or even to some classrooms, she would argue, that is because the social and physical structures of those places have specifically provided for its applicability. It then stands to reason that forms of activity that arise in those settings may have little consequence, whether scientific or educational, for the quite different forms of activity found elsewhere.
Having found the problem solving theory inapplicable to describing the activity she found in ordinary activities, Lave proceeds to develop her own descriptive vocabulary in relational terms. This descriptive vocabulary has two properties: it is dialectical, and it operates on several levels. For example, when a person is engaged in cooking dinner, one would provide a series of descriptions of the relationship between the cook and the kitchen, each of which would be framed in dialectical terms.
For Lave, two entities (e.g., a person engaged in an activity and the setting in which that activity takes place) are related dialectically when three conditions obtain: the entities interact with one another, the entities are changed over time through their interactions, and the cumulative changes are sufficiently extensive or complicated that it is difficult or impossible to give an account of either one except in terms of their unfolding relationship to one another. These changes need not be equal in magnitude -- for example, supermarkets have changed me more than I have ever changed them -- but the reciprocal influences should not be neglected.
The descriptive levels on which activities are described dialectically are:
1. How various activities fit together, for example when someone is simultaneously preparing dinner, putting the groceries away, minding the children, and listening to the news.
2. The interaction between the person engaged in the activity and the setting of that activity; for example, the evolving arrangements of various tools and materials on countertops and stoves and the guidance that these arrangements provide to the cook.
3. The interaction between the person considered as a social agent in a much larger sense and the socially organized "arena" within which the activity takes place; for example, the cook who was brought up to regard food and cooking in a certain way, working in a kitchen that was designed by the efficiency and marketing experts of a particular era.
4. The political economy and the cultural meanings of these things, as historically coevolving systems.
This is a complex story, and Lave would not claim to have proven its necessity or adequacy in any complete way. It does, however, do justice to her central observations about the actual nature of everyday mathematical reasoning.
Walkerdine on Mathematical Activity
Before assessing Lave's story any further, let us turn to a brief account of Walkerdine's theory. Whereas Lave draws on the dialectical tradition of social thought, Walkerdine draws on a more recent tradition of French post-structuralist theory, specifically a synthesis of certain ideas of Michel Foucault (1977; 1979) and Jacques Lacan (1977). Although it is impossible to give a brief account of these thinkers' views, Walkerdine's use of them in The Mastery of Reason can be summarized briefly.
Walkerdine also views mathematical reasoning as deeply bound up with the larger activity within which it takes place; she furthermore understands activity itself in relational terms. Nonetheless, her analysis of these propositions is significantly different. Walkerdine is centrally concerned with processes by which something called "the child" is produced in various social settings, particularly the classroom. This too is a tremendously difficult idea. Its core intuition is that people with authority over children -- teachers, parents, school administrators and others -- orchestrate classroom and home activities in elaborate ways that allow them to "see" something that they can understand as a proper child. In doing so, they are guided by complicated and historically specific discourses about child development. That is, they actually put these discourses into practice, going to tremendous lengths to arrange their relationships with children in ways that make their "truth" evident in the visible details of the various joint activities that make up an average day in a school or other institution.
Walkerdine develops a set of conceptual tools for analyzing the activities within which authoritative adults manage to discover the child they are looking for. She fashions these tools out of materials provided by the tradition of semiotic analysis. The child that is produced in a classroom is specifically a sign. But Walkerdine's notion of a sign is distinctive and requires careful attention. For Saussure (1974), a given language distinguishes a set of meanings, called signifieds, which are organized as a field of differences. Put in another vocabulary, languages divide the world into different categories, perhaps distinguishing different types of cheeses, boats, or sexualities. The signifieds are defined by nothing more than these relationships of being-different-from-one-another. Each sign, then, comprises one of these signifieds and the particular signifier ("cheddar," "yacht," "straight") that names it.
Walkerdine, although retaining the vocabulary of semiotics, has a different understanding of what signifieds are and how they go together with signifiers to make signs. For her, a signified is not an abstract meaning but rather a form of activity, and specifically a form of joint or shared activity among a number of people. To produce the child as a sign, then, is to orchestrate a form of joint activity that can be recognized as exhibiting the signified that goes together with a particular signifier, namely "the child." More generally, this activity will exhibit the whole interconnected system of signs organized by a particular discourse about childhood, development, learning, motivation, and so forth.
Let me remark in passing that these ideas combine themes from Foucault and Lacan in a complex way. Walkerdine's semiotic ideas, and particularly the emphasis on the connections and interrelations among signifiers that will be central to her theory of mathematics, derive from Lacan. Her ideas about historically specific discourses and their role in organizing activities and "producing" particular kinds of people derive from Foucault. Her principal goal is to develop concepts to analyze particular kinds of empirical materials, but it should be noted that she makes little effort to develop the wider consequences for Foucault's and Lacan's larger philosophies of the particular synthesis she has developed. Indeed, Foucault regarded psychoanalysis as one more discourse aiming at producing a historically specific "truth" about people, and Lacan had next to nothing to say about the organization of material activity. Nonetheless, Walkerdine's synthesis is not obviously incoherent, and further conceptual analysis would probably be rewarding.
Be this as it may, Walkerdine's theory of discourses and signs provides an exceptionally sharp instrument for the analysis of everyday mathematics. Numbers for her are signifiers that signify forms of activity. For example, a child's counting forms a series of signs through the verbalization of a sequence of signifiers in synchrony with a sequence of pointing motions. Mathematical activity, then, is activity whose forms are orchestrated so as to exhibit specifically mathematical signs. This is where the distinction between signifiers and signs becomes crucial. Walkerdine demonstrates that a mathematical signifier such as "more" can be embedded in wholly different signs in different settings. In the homes of the people she studied, for example, children and adults employ the signifier "more" in the context of contests over the consumption of food and other things. (As in, "A: Can I have more? B: No, you cannot have more.") This "more" has quite a different discursive logic than the "more" of classroom mathematics assignments, and it is no wonder if children stumble in their acquisition of the pairing of signifiers and signifieds that is specific to mathematical activity.
Mathematical signs, as we have seen, are produced in a series of operations by which objects, images, fingers, heads, and so forth are made visible as participants in mathematical signifieds. What is specifically mathematical about these signs is the ending point of the series: a signifier such as "three" that has become effectively shorn of any empirical signification. Submitted to a mathematical operation such as counting, arrays of real-world things such as three bears, three fingers, three books, and three knocks on a door all yield the same signifier, "three." Having no inherent attachment to particular signifieds, mathematical signifiers become universally applicable. As signifiers go, they are uniquely self-sufficient; whereas most signifiers are more or less inconsequential unless bound up in signs within the complex and contingent world of activity, mathematics offers a closed system whose relations, consequences, outcomes are not contingent at all. And when read back onto material activities to form signs, mathematical signifiers promise to represent the world in a uniquely transparent way. For example, when the "seven" that results from a calculation might then be formed into a sign denoting "seven apples," we feel confident that everything we wish to know about those apples is already present in "seven." Of course, this is not true in reality; the application of numbers to real-life situations is fraught with numerous problems of interpretation. But this is an easy point to miss when the situation is only presented through a fantasy story.
In an altogether remarkable sequence of chapters, Walkerdine argues that the detachability of mathematical signifiers from empirical signification affords a particular kind of emotional investment. The process by which arithmetic yields useful signifiers is -- at least given conventional algorithmic means of teaching arithmetic -- pretty magical to begin with. And people who proceed to treat these signifiers as transparent representations of the world are engaged in a fantasy -- the fantasy in which the differences between signifiers and signifieds no longer matter. Of course, all fantasy suspends the complex link between thought and reality. But mathematical fantasy is supported by an elaborate system of prestigious social institutions. These institutions effectively encourage children in the idea that they can comprehend and control the world by encoding and manipulating it symbolically.
Of course, many children never develop or consolidate this fantasy life or the formal skills that support it. Walkerdine suggests that one reason for this lies in the nature of mathematical story problems themselves. Story problems are frequently unrealistic, for example in the prices they assign to desirable objects. As a result, children regularly become involved in playing out the wrong fantasies. Perhaps, she suggests, the children who find these fantasies appealing are distracted from taking on the deeper and more formidable fantasies of symbolic control. This may help to explain the tendency of mathematical skill to become gendered: inasmuch as fantasy is part of the development of personal identity, the many encouragements and discouragements that shape gender identity will also tend to shape a child's choice of fantasies -- and thus his or her affinity to mathematics.
Views of Activity
These brief accounts of Lave and Walkerdine's positions neglect many points and risk oversimplification of the points that remain, but they are perhaps sufficient to begin exploring the complex relationships between them. To begin with, the tremendous differences in their vocabularies should not blind us to several points of convergence. They are both concerned with math education, and they both wish to describe math education against the broad background of the production of ordinary activities. They both understand activity in relational terms; that is, as something that can only be described through the socially organized relationships into which people enter. They both recognize that mathematical reasoning occurs, but they are also both critical of the abstraction and formality of mathematical reason.
The similarities between Lave and Walkerdine's work extend to the level of institutions as well. Both of them argue that school and nonschool activities organize mathematical reason differently, and their critiques begin with those differences. Although neither presents a detailed analysis of school institutions (see, however, Henriques, Hollway, Urwin, Venn and Walkerdine, 1984; Lave & Wenger, 1991), they both regard school activities as historically specific. School activities interconnect with much else in society, and must be understood in those terms. The organization of school activities is influenced to a substantial degree by complex systems of ideas; these in turn are historically specific and should not be accepted uncritically. To the contrary, the ideas themselves are very much part of the larger phenomenon requiring explanation.
These vaguely expressed commonalities provide starting points for the more difficult analysis of the two authors' differences. To begin with, despite their shared interest in mathematical activity, they understand both mathematics and activity in quite different ways. Specifically, they understand mathematics differently as a vehicle by which forms of reasoning can be abstracted from concrete situations. For Lave, abstraction is prototypically something that happens in school. It is part of the practice of problem solving and proceeds by formalization, calculation, and the formal derivation of conclusions. For Walkerdine, abstraction is a feature of discourse; it is part of the practice of incremental recoding that proceeds from nonmathematical discourses to mathematical discourses, and from signs that have empirical references within concrete activities to signs that do not. Lave believes that mathematics can occur without abstraction, for example in the everyday practice of cooks and grocery shoppers. Walkerdine, on the other hand, believes that mathematical discourse can be found produced in a wide variety of milieux, provided only that the necessary practices of resignification are present. In other words, Walkerdine finds mathematical abstraction in places where Lave does not.
As a result, the authors provide different accounts of school and its effects. Although both recognize that school varies historically, they are both concerned with particular kinds of contemporary schooling. And each of them analyzes school in terms of a set of ideas about children. But whereas for Lave these ideas relate learning to cognition, for Walkerdine they relate learning to emotion. In Walkerdine's view, schools are centrally concerned with arranging for students to develop emotional investments in the forms of activity and of subjectivity within which the normative "child" can be discursively produced. Lave does not analyze this aspect of schooling at all, instead analyzing in greater detail than Walkerdine the whole system of ideas around problems, transfer, and knowledge.
They do converge, however, in accusing school-based activity of being organized in such a way that children are encouraged in an unfortunate kind of relationship to reality. In each case, this occurs through the construction of artificial problems that are amenable to wholly abstract treatment. But they differ markedly in their view of the outcome of this process -- the problem solving model of activity versus fantasies of control -- and especially in how successful they view school as being in bringing these things about. For her part, Lave views schools as failures, inasmuch as the problem solving methods they teach have little application to everyday life. Walkerdine, on the other hand, views schools as only too successful in inculcating a particular kind of fantasy life and a particular connection between these fantasies and the child's gendered identity.
More generally, Lave and Walkerdine differ significantly in their analysis of activity as such. For Lave, activity is dialectical engagement; it is the self-organizing interaction and reciprocal influence of socially constituted persons and socially constituted settings of activity. Activity for Lave can be analyzed on several levels, bringing into focus the sociology and history of particular arenas of activity such as schools and grocery stores. For Walkerdine, activity is produced discursively. Discourse, for Walkerdine, is not simply verbal or simply symbolic. Although books and speeches might arrange clouds of signifiers into complex formal relationships, the signifieds with which they form signs are precisely human interactions; school activity can be interpreted as producing the sign "counting to five" or the sign "a child who is ready to begin work with fractions."
These vocabularies of dialectical activity and discourse are very different in their surface forms and it is not a simple matter to reconcile them. One point of comparison is their respective treatments of educational theory itself. Lave understands the effectivity of the concept of transfer, for example, within the vocabulary of ideology and consciousness. Transfer and its embracing functionalist framework are held to be ideologies in the sense, among other things, that they are wrong. They can be disproven by exhibiting their inapplicability to ethnographically observed activities; they can be demonstrated to be absent in their proponents' data through a reanalysis of their experiments; and they can be viewed as mystifications in relation to some different, deeper, and more accurate story about activity, in which they play a part -- for example, as apology or rationalization -- but that they do not totally constitute.
As such, Lave's theory leads to a complicated story about the understandings of mathematics that people take away from school; these understandings are precisely ideologies. As such, they can be compared and contrasted to the understandings of things that people actually have -- that is, in more precise language, the consciousness of their world with which they live their lives. The ideas of ideology and consciousness both require more analysis than we have any need for here. Suffice it to say that they are not, for example, simply propositional statements or beliefs. Rather, like everything else in Lave's theory, they are aspects of people's relationships to the arenas in which they conduct their activities. She observes, for example, that the cultural prestige of mathematics regularly combines with the frustration of problem solving to produce the harmful experience of oneself as being "bad at math." Lave is willing to argue that this idea is simply mistaken and can be disproven by carefully observing actual arithmetic calculations in supermarkets and kitchens and working up a schoolish score for them.
This dialectical approach is to be contrasted with Walkerdine's Foucauldian approach, which has no concept of ideology at all. In place of ideology, Foucault employs concepts like discourses, disciplines, and the laborious production of both subjects and signs. Again, activities on this view are constituted by discourses; on this view, school itself is one big discourse. What makes school different from any other site of activity is simply which particular discursive formation it is. It is important to understand that a discourse, in Foucault's sense, is not right or wrong; it is simply what the people in a given institution are producing. This is the controversial sense in which Foucault speaks of disciplinary practices as producing "truth," or "truths," with quotation marks. This is troubling to many people, for whom it sounds like relativism. Although it is not entirely unproblematic, it is not any simple kind of relativism. Neither Foucault nor Walkerdine would ever argue that institutions can produce anything at all; their concern, rather, is to describe the historical processes and the practical means by which particular "truths" are produced, in classrooms or anywhere else.
The contrast between Lave's and Walkerdine's theories seems so irreconcilable in part because of the contrasting details of their case studies. Lave is focusing on adults in their largely solitary and self-directed activities; she wants to provide a vocabulary in which we can describe the structure of those activities and the history through which they acquired that kind of structure. In particular, it is we, the investigators and readers, who are finding significance in the individuals' activities, which assume their special forms without our influence. (But this is not quite true, because the fieldworker necessarily shapes, to some degree, the activities in which she participates by observing; Lave remarks on this, but only in passing.)
Walkerdine, for her part, focuses on interactions between adults and children; within these relationships particular signs are produced that are held to be predicative of the children on their own. These signs "are produced," in the passive voice, in that they are not necessarily consciously or deliberately negotiated but rather are manifestations of the anonymous power relations within which the very social beings of the adult and child are organized. Walkerdine's theory, like Foucault's, has a certain ambiguity on this point: to what extent should we view the children as putty that is being passively shaped into the discursive "child," and to what extent should we view the children as active participants in the process? This is, to be sure, a point of instability in many theories of development. Each theory offers an explanation for the negative labels that are applied to children who fail to conform to norms. But what is missing in each case is a substantive account of the children's active noncooperation with the adults' plans for them.
To put the contrast crudely, whereas Lave's is a two-level model describing contradictions within subjectivities and between ideologies and realities, Walkerdine's is a one-level model in which the social order constitutes the activities as such. Put in simpler words, Lave contrasts the falsehood of cognitivist psychology, which she treats as an ideology, to the (provisional, approximate) accuracy of her own description of activity, whereas Walkerdine simply describes a discourse that is part-and-parcel of particular kinds of activity, judgments of truth and falsehood being beside the point. Whereas Lave's negative assessment of cognitivism is both empirical (it fails to explain the phenomena) and ethical (it distracts people from the reality and causes them to discount their own abilities), Walkerdine's negative assessment of school discourse is wholly ethical (it marginalizes whole categories of students and it inculcates regrettable fantasies of control).
I will not attempt to reconcile these differences. Instead, I see them as organizing a horizon of new research topics. The best way to proceed, in my view, is not to generate a list of contrasting predictions by which we might perform some kind of differential diagnosis. Instead, we should take these projects in the spirit in which they were intended, as provisional ethnographic accounts of enormously complicated things. The descriptive vocabularies that each author has proposed will be thoroughly tested, and presumably extended and transformed, as ethnographic research begins to fill out the picture of institutional and social relationships that surround the practices of school and of mathematics.
One crucial direction for further development of Lave and Walkerdine's ideas is in critical studies of educational institutions. Teachers, for example, are not simply passive transmitters of discourses and ideologies passed down from on high. Each theory would also benefit from an encounter with the practices of classroom discipline that are prerequisites of mathematical activities -- and that should probably be viewed as part of those activities.
Given that both authors are concerned with the relationship between school and nonschool activities, both of their theories would benefit from more extensive analyses of the cultural and practical resources that children carry with them in each direction. The processes of discursive recoding that Walkerdine regards as defining mathematics also apply, for example, to the semiotics of the Teenage Mutant Ninja Turtles. These television sign systems, as well as those from shopping, have been omnipresent in my exposure to elementary school mathematics. Such investigations would help us ask more carefully what mathematics and mathematics education should be like.
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