A shorter version of this paper was presented at
the annual meetings of the American Anthropological Association,
Chicago, November 1999.
This work is part of an ongoing project in search of a better understanding of the institutions and practices of public schooling, and of the way schools transform the kinds of person a child is. This paper examines the way academic learning is culturally situated by focusing specifically on a math lesson in a 5th grade classroom.
Math as Cultural
Both educators and researchers are coming to appreciate that mathematics is a cultural phenomenon. For example, Brian Rotman argues that the formal procedures of math are not self-sufficient, they are sustained by the informal practices of the community of mathematicians. He writes, "Mathematics is before all else self-consciously produced; and it is so according to an agenda formed out of its historically conditioned role: as instrument in relation to the needs of both commerce and technoscience and, with greater autonomy, out of the image of itself as the exercise and play of pure, abstract reason engaged in the production of indubitable truths" (Rotman, 1993, p. 25) (cf. note 1).
And it has become apparent that the axiomatic systems of math rest on postulates that are conventional rather than being logically necessary. The discovery of 'alternative' geometries in the nineteenth century, for example, brought into relief the differences between axioms as "truths" and axioms as descriptive claims that offer a model of viewing physical space. But the notion that axioms somehow represent truth is difficult to abandon, having become, as Putnam (1983) puts it, "built into our notion of rationality" (cf. Rotman, 1993, 59ff, 117ff).
If mathematics is cultural, it follows that learning mathematics is, in a very real way, a matter of socialization--socialization into a community of practice--socialization into what Terezinha Nunes (1999) has called the "ways of knowing" of a community. Learning math is social in at least two respects: first, the mathematical systems to be learned are at root conventional and thus social and, second, learning math is typically something accomplished in interaction with others.
This paper focuses on the later. How does the social interaction in a math classroom, for example, accomplish this socialization? The term "socialization" is in certain respects a little dangerous, if it were to connote an unthinking, even unconscious adherence to collective norms. And of course it would be a mistake to assume too quickly that what happens in school is that children become mathematicians, for there are significant differences between school math and research mathematics (Ernest, 1992). But even if we restrict our focus for the time being to school math, how does classroom instruction "socialize" children into new "ways of knowing?"
To answer this question we've taken a detailed look at classroom discourse, using an interpretive analysis that owes something to ethnomethodology (Garfinkel, 1967; Sacks, 1984) and something to conversation analysis (Nofsinger, 1991; Levinson, 1983), and so attends to the turns and moves of the "language games" that are played. Our interpretive analysis, however, is also informed by an ontological hermeneutics (cf. Heidegger, 1927). As a result, it attends to indexical features of discourse that enable speakers to invoke contextual backgrounds and position both objects and people against these backgrounds.
Such an analysis is necessary if we are to make good on our programmatic claim that learning and development are not solely epistemological matters, but also involve ontology. That is to say, development introduces the learner to new worlds, worlds with new species of object, and at the same time transforms the kind of subject, the kind of person, the learner is.
An Ontology of Mathematics
It might seem odd to say that mathematics has an ontology. Isn't math a species of knowledge, a purely epistemological enterprise? But knowledge must be of something, some kind of entity or another. As Anna Sfard points out, in a paper titled "Symbolizing mathematical reality into being," (1998) "The common referent of the symbols '2/3' and '12/18'is an elusive entity, the ontological status of which has been puzzling philosophers for ages."
David Lachterman gives the issue a historical dimension in his book "The ethics of geometry" (1989). What is meant by the "existence" of a mathematical entity? Lachterman argues that mathematics has an "ethos": "settled or characteristic ways human beings have of acting in the world" (p. xi); mores and styles with which people "comport themselves as mathematicians both toward their students and toward the very nature of those learnable items (ta mathemata) from which their disciplined deeds take their name" (p. xi). Classical Greek and modern Cartesian mathematics have, Lachterman argues, very different ethe: the Greeks considered mathematical entities--numbers, say--to be already existing in a Platonic realm--underlying, preexisting geometrical forms, and math merely a way of finding them. Post Descartes--with his famous X & Y coordinates--math became viewed as active and creative, showing the mind's essential power of making. Mathematical entities became seen as constructed, having no existence prior to their construction.
Sfard, too, considers the construction of objects as central to mathematics, (the "central theme" of her paper is "[t]he process through which the objects 'represented' by the symbols come into being retroactively" p. 15). She argues that the search for the elusive referents of mathematical discourse has led to a reformulation of the problem of the relation of symbol and referent (cf. note 2).
Sfard reminds us of "Foucault's central claim that the objects ‘referred to' by symbols, far from being primary to signs and speech acts, are an added value (or the emergent phenomenon) of the discursive activity." She adds, "This is particularly true for the evanescent objects of mathematics" (p. 14). Like Sfard, we would like to follow Foucault in an analysis of classroom interaction: "[The task] consists of not--or no longer--treating discourses as groups of signs (signifying elements referring to contents or representations) but as practices that systematically form the objects of which they speak" (Foucault, 1969/1992, p. 40, emphasis added).
The Academic Situation
Our analysis of a math lesson in a 5th grade classroom, then, will demonstrate the way in which a group of students is shepherded into a new understanding of what counts as good practice in the language game of school math. An analysis of the social organization of an episode involving fractional equivalents will offer a glimpse of the children's socialization into math practices. We shall show that the academic task in the classroom has embedded within it a cultural task. That's to say, certainly school uses social interaction to achieve academic ends, but at the same time school uses academic tasks to achieve cultural ends, embedding cultural tasks within an academic framework. These children are not just learning math; they are learning to be. Disclosing and doing justice to both these cultural and academic goals is possible with an analysis that attends to ontology.
The class we'll examine was located in an elementary school in the industrial U.S. midwest, in an ethnically mixed district predominantly working class.
First, a quick overview of the task. The students
are given colored segments of a circle: light blue wholes, yellow halves,
dark blue thirds, white quarters, red sixths, orange eighths. The activity
of the lesson is highly structured, yet the organization is not announced
in advance. Nor is it entirely emergent--one imagines the teacher had some
plan ahead of time--but for the children it is something they are led through,
unfolding in time:
Notice the iteration here, and the way it's nested (cf. note 3). The task is iterated through a series of targets: wholes, then halves, then thirds (cf. note 4). For each of these targets, the students are guided through another iteration, as they find equivalents to the target in sequence.
This task structure is a social accomplishment, orchestrated by the teacher but carried and shaped by student participation. Throughout there are subtle but important changes in the manner of the students' involvement. (Describing these in detail is not possible in this short paper.) But, again, what is accomplished, precisely? What is the aim of this activity, its academic goal? We get evidence about the answer to this question by looking at the way the lesson ends.
Closing Accomplishments: Appropriate Actions, Emerging Objects
T: If you put four eighths isn't that the same size as three sixths?
T: Yes. Yes, and so is six twelfths but we don't have twelfths. We don't have the denominator for twelve, we don't have the denominator- what else would it be, sixth twelfths, what else?
T: What else could you go up to?
S: Twelve twenty-fourths.
T: Twelve twenty-fourths.
S: Sixteen thirty-second- twos.
T: Thank you! [Laughs] Alright. Now, I'm gonna- I want you to keep... [they put the pieces away]
"Thank you!" says the teacher, with a laugh. "Alright." And she has them put the plastic pieces away.
In other words, the lesson ends when the students display a capability to "go up to" fractional equivalents that they don't actually "have"-- when they demonstrate an ability to iterate beyond what is immediately given. That's to say, the students can perform a particular kind of action on a particular kind of object. The action is iteration; the objects are fractions. This is one piece of evidence that this lesson accomplishes subtle shifts in the mathematical objects which the children are able to recognize and act upon. That this is a central aspect of the socialization taking place.
Species of Object
What are the various species of mathematical objects, the changing ontology, constructed in this math lesson? We can identify three distinct types. (The detailed basis of this identification would require more extensive consideration.)
The interaction starts with the plastic pieces, indexed with color terms, "the blue ones," "different shapes," "pieces," and the like. Their instantiation is actual and self-evident--we can all see these pieces of plastic. They are acted on by being arranged, physically manipulated, to "make" a given shape (a whole circle, or a segment of a circle).
Second are fractional parts, indexed with formulations such as "one half," "two quarters," "a whole." They can be instantiated as the plastic pieces ("This blue piece is one quarter"), and also, importantly, as written inscriptions: "1/2." They are acted on by being- well, put together: "One third plus one third...." The equivalence relation here is, so to speak, a syntactic one. Things are made with fraction parts.
What things? Well, third and finally we have fractions, indexed thus: "three sixths," "two fourths," "two over two." They can be instantiated as arranged plastic pieces (e.g., three continguous red pieces forming a segment) and also as inscriptions: "3/6." Equivalents are stated ("three sixths"), and they are constructed by iteration: viz "3/6," "6/12," "12/24," "24/48...."
It is important to note here that there is ambiguity to some (but by no means all) of the verbal and written designations with which these different types of object are referred to. "Three sixths," for instance, can refer to an arrangements of fractional parts, or to a fraction. And the same is true for "3/6." The ambiguity can generally be resolved by attention to context, both verbal and non-verbal. (Sometimes there are hints that the teacher utilizes and exploits this ambiguity. Exploring this would require more detailed treatment.)
The Action: Making, Dividing, Making
What's done with these different objects? The lesson starts with students "making" wholes. ("Making" is how the teacher characterises their activity: lines 20, 21, 22, 23, 24, 34). Then they turn, as we've said, to finding what's equal to one whole, then what's equal to one half. But there is a significant difference between the way they deal with wholes and the way they deal with halves. In the first case they are "dividing," in the second case they are "making."
In the first case the teacher and students (but the teacher plays the primary role) are "dividing" (67, 75, 78, 82) the whole into parts. In the second case--finding what's equal to a half--the teacher and students (but this time primarily the students) are "making" (161, 167, 177, 242) the target fraction from its fractional parts. "[T]ake two from the white circle and make a half" (166), the teacher says, for example.
Let's consider the "dividing" in more detail. The teacher consistently characterises the activity of finding what "equals one" as a matter of dividing. At the outset she says "This is a whole, whole, whole. [Pointing to circles of different colors.] Okay. So. Let's divide our fraction circle into halves" (66).
And finding how many thirds make a whole is also done by dividing. Then she says, "Okay we have a whole, one, okay. And then we divided the next one we had yellow, okay, right?" (75). Shortly thereafter the next equivalent, in thirds, is introduced with "So look [at] the next division. What would you go to?" (82).
But while this enactive division is taking place--as the students are dividing the whole into its "pieces," its "fractional parts," into "halves" and "thirds" and "fourths"..., the teacher is putting these pieces together again, reconstructing the whole symbolically, writing on the chalk board. As the fraction circles are "divided," taken apart enactively (finding what is equal to one whole), at the same time they are put together again, ("plus..."), symbolically, inscriptively ("1/2 + 1/2 = 2/2 = 1"):
The teacher says "Okay, so (I'm gonna) do something" (72) and she writes:
1They turn to the whole circle made of two yellow halves. She says "Let me do it like this it'll be a little easier." She writes:
1/2 + 1/2 = 1and says "one half plus one half equals one, okay" (82).
They move on to the whole circle made with blue thirds. The teacher says, "So it's be, one third, plus one third, plus one third, equals-?" (87), while writing:
1/3 + 1/3 + 1/3...She then completes a self-initiated repair. She goes back to modify the formula with halves, saying as she does this "Let me do it like this so you can see the difference." (89) She inserts the symbol 2/2. Thus:
1/2 + 1/2 = 1becomes
1/2 + 1/2 = 2/2 = 1....as she says, "Would be- would equal two over two which would equal one." Then she returns to the case of thirds. "So one third plus one third plus one third would equal what? Three over three, equals one, okay. Get it now?" (90-94)
1/3 + 1/3 + 1/3 = 3/3 = 1What's accomplished here? We propose that the teacher is showing the students how fractional parts not only make a whole circle, they also make a fraction. The "it" that she's made "easier," rewriting so they "can see the difference," is the difference between two species of object: fractional parts, and fractions. This difference surely is the "it" when she asks "Get it now?"
Thereafter, finding the equivalents to a half becomes a matter of making rather than dividing. The fractional parts, now ready-to-hand, can be put together, and the fraction "made," enactively without the support of being inscribed symbolically, and it's striking that the teacher now inscribes only the product, the equivalent, the fraction (e.g. "2/4"; lines 209, 211, 213): the element she added to her inscriptions in the examples just examined. She doesn't bother to inscribe either the target fraction or the parts used to make it.
Math as Mastery of the Imaginary
We have suggested that this task teaches the students math (school math, at least) by socializing them into new forms of activity with new types of object. Over the course of the task they come to both talk about and appropriately manipulate fractions: objects they they "don't have" in the way they have the plastic pieces. At the end of the lesson they demonstrate that they can iterate indefinitely with these new objects.
It is this capacity for infinite iteration--this ad infinitum principle--that Rotman finds at the heart of modern math. Using Rotman's terms, the child, now an indefinite iterator, has become a "mathematical Subject"--"a semiotic agency made available by the [mathematical] Code" (Rotman, 1993, p. 145). Mathematics, Rotman proposes, is "a kind of fantasy action" (1993, p. 139), "its entire discourse consisting of a web of waking dreams, conscious fantasies that operate on a separation between activity performed or peformable by the Subject and that which is dreamed, imagined to be performed by the Agent" (p. 87). The fantasy, as Valerie Walkerdine has pointed out, is "the dream of a possibility of perfect control in a perfectly rational and ordered universe" (1988, p. 187).
The lesson ends with the students iterating again, but this time the iteration is not bounded by the objects, the plastic pieces, the fractional parts, that are immediately present. The students have become able to iterate indefinitely, and this is possible only because the objects now being named in the iteration are no longer colored pieces, but fractions. No longer actual objects, but ideal objects. When the students demonstrate their new ability to iterate indefinitely, when they demonstrate they have "got it," they show they've bought the fantasy of universal mastery at the center of modern math.
The Academic Task as a Cultural Task
At the same time, ironically, they are willingly doing what they're told. When the children come to operate on fractions, to iterate indefinitely on objects they "don't have," they are entering into this fantasy of power and mastery--by following the directions of an authority.
The lesson we've just examined occurred the day following a field trip to the middle school. This had been a visit before, during, and after which the teacher emphasized the new and difficult challenges her students would face next year. From the start of the present year she had presented herself as knowing better than her students what was important for them. Specifically, she knew how they would need to behave if they were to survive in the rough new world of the middle school, which we can attest is a very different institution, large and complex, emphasizing discipline and control.
Jean Lave has pointed out that while street math is typically an end in itself, school math is generally a means to an end, and the end is pleasing the teacher. Lave is surely one of the smartest social scientists around, so we always pay attention when she says or writes something that sounds wrong, in part because it provides an opportunity for a good debate. Lave is correct, we believe, in seeing school tasks as means to an end, but wrong in then criticising this. It is no accident that children in school do math in some sense for, and certainly in relationship with, an adult teacher; indeed, we would argue that this is crucial to the cultural tasks of schooling.
In this case, the children have accepted the legitimacy
of their teacher's authority; they have accepted her message that to thrive
in the middle school they must do what they are told. At the same time
that they have become masters of the universe, they've also become subjected.
This transformation is a central part of what schooling is about.
1 We return to Brian Rotman's work throughout this paper because he is one of the few people to systematically explore this new view of mathematics. Rotman himself notes that "there has been little sustained attempt to develop the philosophical and conceptual consequences of saying what it means for mathematics to be a language or to be practiced as a mode of discourse" (p. 17), especially "the question of the being, or rather the becoming, of basic mathematical objects such as numbers and points" (p. 23). Rotman's work is not very well known by developmental psychologists or educational researchers; better known is the writing of Valerie Walkerdine (1988), who Rotman has influenced.
2 She traces successive formulations from realist to constructivist and then an abandonment of the classic dichotomy with interactionist views of symbols and meaning, semiotics of Saussure and Peirce.
3 Webster's defines "iterate" as: "to say or do again or again and again."
4 It is not insignificant, we'll see later, that the teacher refers to the first of these targets as "our fraction circle" (67), and to each subsequent target as a "fraction" (154, 229, 240). Line 96 is a rare exception to this pattern.
Ernest, P. (1992). The nature of mathematics: Towards a social constructivist account. Science & Education,1, 89-100.
Foucault, M. (1969/1972). The archaeology of knowledge. (A. M. Sheridan Smith, Trans.). New York: Harper.
Garfinkel, H. (1967). Studies in ethnomethodology: Englewood Cliffs, NJ: Prentice-Hall.
Heidegger, M. (1927/1962). Being and time. (J. Macquarrie & E. Robinson, Trans.). New York: Harper & Row.
Lachterman, D. R. (1989). The ethics of geometry: A genealogy of modernity. New York: Routledge.
Levinson, S. C. (1983). Pragmatics. Cambridge: Cambridge University Press.
Nofsinger, R. E. (1991). Everyday conversation. Newbury Park: Sage.
Nunes, T. (1999). Mathematics learning as the socialization of the mind. Mind, Culture, and Activity, 6(1), 33-52.
Rotman, B. (1993). Ad infinitum: The ghost in Turing's machine - taking god out of mathematics and putting the body back in: Stanford University Press.
Sacks, H. (1984). Notes on methodology. In J. M. Atkinson & J. Heritage (Eds.), Structures of social action: Studies in conversation analysis, (pp. 21-27). Cambridge: Cambridge University Press.
Sfard, A. (1998). Symbolizing mathematical reality into being: How mathematical discourse and mathematical objects create each other. In P. Cobb, K. E. Yackel, & K. McClain (Eds.), Symbolizing and communicating: Perspectives on mathematical discourse, tools, and instructional design. Mahwah, N.J.: Erlbaum.
Walkerdine, V. (1988). The mastery of reason: Cognitive development and the production of rationality. London: Routledge.