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Herbst P. G.
(1999)
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Abstract: |
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The current work of teachers is deployed within an ideological environment that at the same time promotes mathematics classrooms to be "communities of discourse" and emphasizes the "dialogical nature of mathematics" in particular as related to the construction and validation of mathematical objects (conceptions, propositions, etc.). The ideology tends to identify a linguistic notion of discourse (say, conversation) with an epistemological notion of discourse (as in Foucault 1 , 1972). A rhetorical effect is achieved: The aspects of doing mathematics seen as a historic-social practice of a professional community are thus able to be exported to the context of the class. Such an identification is ideological: It serves the reformers to act on teachers (the material agents of the reform) by making them believe that both things go hand in hand and by establishing a believable cause-effect mechanism. The identification of a historical and disciplinary dialogue with a contextualised conversation is helpful to make believe that if dialogue happens in the class, the dialogic nature of mathematics will also happen. I will not contest this relatively unoriginal method of administrative persuasion at the level of action, but address the theoretical problems that such mechanism presents for the conditions of existence and work of the mathematics teacher.2 The mathematical register (as the kind of language used in the professional practices&emdash;oral and written&emdash;of practicing mathematicians; Halliday, 1978) is a functional variation of natural language. Beside having its own words for objects explicitly constructed in the doing of mathematics (e.g., ellipse, line integral, cardinal number, etc.), the mathematical register uses elements of natural language in a specialized way&emdash;yet, a way not explicitly spelled out&emdash;(e.g., and, need, clearly, proof, know, can, and so on, claim, true, etc.). In the class, where at least the student is not a competent speaker of the mathematical register, his or her competence in natural language enables him or her to live the fiction of "speaking mathematics" at the same time that he or she becomes competent in this new way of speaking: Those non-technical but specialized elements of speech are used in non-specialized ways so that mathematical discussions can happen, but, as a result, the meaning that they acquire in the actual use becomes ambiguous&emdash;they neither have the precise meanings that pertain to the mathematical register nor all the possible meanings that pertain to ordinary language, but a relative specialization of the latter whose disambiguation depends on the specific work being done. "Proof" is needed to construct the objects3 of mathematical knowledge. Yet, for this construction to be meaningful, proof must relate to students' experiences in arguing. Those ordinary experiences are different from the mathematical experience in at least two aspects: (1) The nature of the objects of discourse and (2) the criteria for the organization of those objects . But again: The use of everyday argumentation in the construction of the mathematical objects permits this construction to happen (in the sense that it permits the "concepts" to come to a functional existence) and at the same time endows those objects with a fundamental ambiguity regarding their nature (See also Legrand, 1988). The assumptions made in the previous paragraphs help describe the possible "mathematical" work of the student both in a productive and in a restrictive aspect: Ordinary language and ordinary argumentation are conditions of possibility for the student to speak mathematically and about mathematics, and they are also conditions for ambiguity and error. In other words, whereas these features (everyday argumentation, ordinary language) enable a mathematical game of the student, they do not guarantee the mathematical nature of the student's game. To the extent that the classroom is not only responsive to the ideological conditions spelled out in the first paragraph but that it must also respond in continuity with its own past&emdash;namely, the mathematics to be taught and learn is "basically" the same&emdash;one can safely say that the average teacher is not just an agent of the ideology but also its judge. Or, that the teacher cannot refuse its duty of epistemological vigilance (at least in conformity with what was normative in the past&emdash;however limited this can be). For very interesting examples of how these tensions play out in the lives of teachers&emdash;in the form of dilemmas&emdash;see Ball, 1993, and Chazan & Ball, 1995. Therefore, except for those few that will believe in the scientific ineluctability of the "discourse-implies-mathematics"6 method and so be willing to silently accept ordinary talk as mathematical argumentation, the teachers that hang on to their role will not just afford opportunities for argumentation to happen and hope that it becomes mathematical argumentation: Teachers will act on the system student-milieu to maximize the chances for this argumentation to be "mathematical". Of course, like those teachers in Diénes time, those actions may or may not be meaning-preserving. (cf. the Jourdain effect, Brousseau, 1997, chapter 1) Based on the points mentioned, I would like to argue that the conceptualization of the student's game is not enough to describe the possibility and the functioning of mathematical argumentation in the class. There is a theoretical necessity to conceptualize a game for the teacher&emdash;a game that is mathematical in a different sense than the game of the student. This game must confront the teacher with the student's activity (i.e., with the system of interactions student-milieu) and must seek the dual stake of enabling the student to participate in mathematical argumentation and ensuring the mathematical nature of this argumentation. Part of my task today is to analyze some alternative examples of "actual" strategies (or "methods") that illustrate the complexities of this theoretical game of the teacher: To do so amounts to argue for the plausibility of the proposition above. To describe a game will mean to describe the possible moves of the teacher, the possible retroactions from its antagonist system, the possible rules for interaction, etc. As it is apparent, at the exact moment when the game of the teacher takes place, the game of the student (with a related but different goal, and a related but different antagonist) is also taking place. The complexities of integrating these two games exceed what we can discuss in this symposium: To discuss the role of the teacher thus means a radical simplification of the modeling task, namely that of taking the structural perspective of the teacher&emdash;and making some assumptions on the actual game of the student, assumptions that forgo the engagement in the actual problem at hand and the process of negotiation of a contract that makes both games materially possible at the same time7. In our discussion the work of the student is pertinent as an "average milieu" for the work of the teacher. The problem of the role of the teacher seen in this way belongs to a discussion of the custom of a class (Balacheff, 1988): What are the tools and norms that the teacher can naturally and legitimately use in order to claim the dual stake8 of his or her game? What are the structural and functional characteristics of a custom of argumentation that can be deemed viable from the perspective of the teacher, given the structural conditions and constraints annotated before? Part of the task is to discuss if and how such a custom can be viable. Proof in the
history of geometry teaching in the U.S.A. The two-column
proof format. Could it be otherwise? Argumentation and
the custom of a mathematics class. Research into reform-minded mathematics classes, using tools from interactionism, phenomenology, and discourse analysis, has succeeded in showing a-posteriori how argumentation is accomplished. Constructs like formats of argumentation (Krummheuer, 1995, 1998), sociomathematical norms (Voigt, 1995; Yackel & Cobb, 1996), or conversational implicatures (Forman & Larreamendy, 1998) are useful to describe the contingencies of lessons (what eventually happened in those lessons). The value of those descriptions is still to be seen: If they are to inform the theoretical21 or practical22 problems related to the role of the teacher in fostering argumentation, they should also show success beyond the contingency (namely, they should also be useful to distinguish what necessarily happens and what necessarily does not happen, from what "just" happened). In particular, discourse analysis alone cannot account for the "legality" of the teacher's moves and as a result does not distinguish between meanings that are mathematical and meanings that are didactical. A great example of such ambiguity is found in the dialogue between Mr. K and Donna in Yackel and Cobb (1996)23. As the mathematical rationality remains identified with the teacher's rationality, it remains confounded whether the student's opportunity to learn is an opportunity to learn to argue mathematically or to learn to agree with the teacher. We know that the teacher must act and that indeed his or her actions may respond to the responsibility to "represent the discipline of mathematics" (Voigt, 1995, p. 197), yet we also know that this action can be more or less adapted to the construction of a mathematical rationality. The teacher may push for the mathematics but the action of pushing may make the whole situation mean something different from the perspective of that who is not mathematically educated (i.e. the student). The combination of analyses of discourse and analysis of interaction show evidence of two fundamental forms of "participation" of the teacher: On the one hand the teacher is a conversational partner for the student, and on the other hand the teacher is the organizer of a conversation24 in the class25. The analysis of discourse shows that teachers use of selective hearing, return a question to the fore, repeat a statement, revoice, etc.&emdash;strategies that are rarely used by students when responding to the teacher or to other students. Those conversational strategies are linguistic tools that belong in the milieu of the teacher and help establish a game for the teacher. But as those strategies are used within the flow of the "mathematical" conversation with the student (i.e., they constitute part of the milieu of the student), they are also constitutive of the meaning of the student's game. In an extreme, the situation could become a Socratic dialogue: A fluid conversation where the pertinent knowledge does come from the voice of the student, whose activity is however reduced to the detection of verbal cues in the turns of the teacher26. Does "argumentation" offer epistemological tools that can help the teacher control that tension between maintaining a conversation about the student's mathematics and orienting the conversation toward the mathematics to be learned? The two column proof format as an
example: The best thing we can do is, first of all, we've got to keep in mind where we are headed. Okay. First thing to do is to create a plan. Now there was a stroke of genius here, and the stroke of genius was putting this line BD there. Okay? That is not something that is super obvious to do. Okay? I think that's kind of why they [the textbook authors] gave it to you. Of course, the former example does not mean advocacy for the two-column proof format, which has many other epistemological problems already discussed. The former example means to show how an implicit division of the teacher's labor serves to furnish an epistemological control on his or her discursive actions in relation to the proving activity: From outside the actual proving task, the teacher "can know as a teacher" while when proving he or she "can only know" as a student would. Such a division of labor is likely related to the divorce between proving and knowing in the work of the student, so it is not in that sense that it illustrates how to think of this epistemological problem: The two-column proof format is an example insofar as it shows a case of an objective27 tool for the epistemological control of the teacher's discourse. In "argumentation," the epistemological controls on the participation of the teacher seem to depend completely on the idiosyncracy of the teacher. Few teachers identify those controls as critical sites for the conflict of the mathematics of the teacher and of the student, and strive to cope with the dilemmas that such a conflict entails into "what to say," "what to do," and "what is Plan B." (See Ball, 1993; Chazan & Ball, 1995; and Chazan's presentation in this symposium). To identify a dilemma and to observe how they solve it is of interest insofar as those observations can enable one to spot at generic elements of the argumentation game and the role(s) of the teacher. Thus far we have two rather extreme customs of mathematical argumentation. The custom defined around the two-column proof format is established on a compartmentalization of knowledge practices&emdash;a compartmentalization that makes it difficult for "mathematical proof" to be the discursive strategy that work-as-proof (in the class's discourse). The custom defined around "everyday argumentation" is established on an aggregation of discursive practices&emdash;an aggregation that makes it difficult for whatever strategies which work as proof in the class's discourse to have a mathematical meaning. The first custom rests on a quasi-algorithmic role of the teacher, the second custom in a quasi-idiosyncratic role of the teacher. The second is more ideologically viable at the level of the voluntarism of reformers, whereas the first at least satisfies the (culturally reproductive) mathematical values of some teachers. What are the elements available to the teacher that permit him or her to capitalize on the student's abilities in everyday argumentation and at the same time ensure the mathematical character28 of the argumentation accomplished? The custom proposed by Legrand (1988) and others, the "class as a scientific community," relies on some explicit and overarching rules such as "it is okay to hold the opinion of a minority" or "one counterexample is sufficient to falsify a general statement." As Balacheff (1988) indicated, the existence of those overarching rules can foster the dialectic of proofs and refutations, but also stop it: A student can legally refuse being convinced or it may be deemed illegal to adapt a general statement so that it can survive a counterexample. Whereas those rules permit the establishment of a game for the student, the student's construction of a new rationality depends on the negotiation of conditions under which a break in that game can be accommodated. The teacher faces a paradoxical situation: If she always abides by the same rules that the students do she may be renouncing to her responsibility to perturb the old rationality, but if she explicitly breaks those rules using her prerogatives as a teacher she risks to tint the new rationality of an "adapting to the teacher"-meaning. It seems to me that the paradox previously sketched is not a necessary characteristic of the role of the teacher, unless one insists on the assumption that the tools available to the teacher were to be of a general order and a permanent status (such as the rules in Legrand's method). Again there are things that we can learn from the history of the study of plane geometry in America: In the beginning of the century, students were encouraged to use the analytic method 29 to find solutions to construction problems&emdash;it was one of the tricks of the trade that was available for teacher and student in some contexts. In the daily works of professional mathematicians (working in rather specialized areas) there are canonical tools that come to their mind when they are attempting a new problem&emdash;a professional memory of information and tools that pertains to their specific area, a memory that participants develop by working in that particular area. Brousseau and Centeno (1991) have suggested the idea of a didactical memory of the teacher and provided a provisional definition as follows: The (didactical) memory of the teacher is that which orients the modification of his or her decisions in relation to the past school experiences shared with his or her students, however without changing his or her system of decision. (Brousseau & Centeno, p. 172, my translation). To be clear, the notion of the didactical memory does not just refer to what the teacher can personally remember, but to what reasonably belongs to the shared past experiences of the whole class that she can publicly remember (in fact claiming that anybody could remember) so as to use it in the present30. I would like to suggest that the didactical memory plays an analogous role as that which the tricks of the trade (in general, but I count on the fact that they can be spelled out in some detail) play in a specific area of mathematical research for its practitioners. The didactical memory is not just a memory of the institutionalized knowledge (such as 'what is in the previous chapters of the textbook') but can include items that have various statuses (e.g., mathematics knowings, implicit models of action, anecdotal experience of activities). I claim that a custom of argumentation (one that can hope to progress toward mathematical argumentation with the hope of not conflating it with the imposition of the teacher's preferred form of argumentation) would be based on a carefully developed and organized didactical memory. The custom based on the two-column proof format shows us a system with very little memory: It contains, of course, the format itself, plus the institutionalized notations, theorems, and axioms, but for example, the answers to proof-exercises posed to students are not regular elements of this memory 31. It is interesting to ask why that could be the case, because the participants are likely to recall more than what they would normally recall publicly&emdash;what functions does such a little memory serve in the system of geometric proof production and in the geometry course in general?32 The teacher's didactic memory is a tool that the teacher can use to develop a mathematical rationality out of an everyday rationality. I will show a vignette that illustrate this. Gene's algebra
class and the equation of the horizontal
line.33
It could be argued that through the whole event Gene and the students have been talking about different things: Gene has been using a particular line as a generic example to find the equation for any horizontal line, but the students have been finding the equation for that particular horizontal line. In the end, however, it seems to be evident to Gene that the necessary fact that the slope is zero is seen at least by some students as a mere accident of the general formula. In other words, for some students the slope is zero because it happens to come out that way when you do the calculations with selected points on that particular line.34 From the perspective of the mathematically educated teacher, it is clear that y = 0.x + 3 and y = 3 are equivalent: Gene could say that by allowing the proposed response, one is not doing anything wrong, just easing the students' way.35 From an observer's perspective the negotiation of the validity of the student's production has explicitly avoided addressing the meaning of the mathematical objects the class was talking about and has resorted to negotiating the conditions for the teacher to accept a convenient answer.36 The question that one should ask here is whether it could have been otherwise? Would have it been possible to enable the student to work without eventually having to accept the ambiguity of the object one is dealing with and without making a tour de force to resolve the ambiguity37 ? What are the elements of the didactical memory that are drawn upon to make that interaction possible? An evident one38 is the equation of the line through two points&emdash;understood as a joint labor that assumes the teacher to give two ordered pairs and the student to use them to produce the slope and the y-intercept of the line through those two points. Moreover, in that form of labor, a line "exists" because there are two points that determine it. The problem of finding the equation of the horizontal line is uninterpretable in that context: Gene's move of fixing a line and picking two points seems a necessary move to enable the student's game. The possibility for Gene to bring up the necessity of the slope of a constant line to be 0 (as opposed to the result of a calculation that yields slope 0 for "this line parallel to X through 3") is contingent on the possibility to keep in memory the transition between the original problem and Gene's reformulation of it. But whereas the distinction between these two may be clear to Gene personally, they are for all practical purposes identical in the custom of that class. Let's imagine that before the event reported and after learning the equation of the line through two points, the class had worked on the following activity: Pick any point that is not on the axes and trace four lines that pass through that point. Obtain their equations. Could you trace a line through the same point so that its slope is smaller than the ones you traced before? …one whose slope was negative? …one whose slope was 0? …one whose slope was 5? A priori such an activity would aim at making available to memory as an implicit model of action the relation between order among slopes and betweenness among lines (use the knowledge of one to act on the other&emdash;even though some actual affirmative answers may be difficult to get) and as anecdote the independence of that model of action from the point chosen to begin with. On the assumption that this activity has been done39, with varying degrees of success, I ask the same question again: Would have it been possible for Gene to enable the student to work in the question (What is the equation of a horizontal line?) without eventually having to accept the ambiguity of the object one is dealing with and without making a tour de force to resolve the ambiguity? It seems to me that the teacher could refer to the previous activity and ask "what is the equation of a horizontal line in your figure?" and draw on the fact that everybody's answer consists of a 0-slope equation to make the point that the slope is independent of the points picked. The same activity could be used to ease the transition between the conventions y = 0x + b and y = b. (For example by asking students to determine whose horizontal line is farther apart from the x-axis without showing their graphs&emdash;the important thing would be to have them say how they know that). The existence of that previous activity in the didactic memory can help the teacher avoid the ambiguity of which object one is talking about (any horizontal line or this line which happens to be horizontal) therefore rendering the 0-slope as a property of the line rather than as an outcome of the calculation with the coordinates of the points picked. But more importantly, because the activity is in the didactic memory, a sentence like "whenever the line is horizontal you don't need to calculate the slope" may be seen more as a conclusion available to anybody than as an arbitrary direction from the teacher bearing on what to do when.40 The first part of this paper intended to show a way to conceptualize (model) the role of the teacher in fostering mathematical argumentation in the class. In the last paragraphs I elaborated on and illustrated how the didactical memory of the teacher can be an epistemological tool for the teacher to increase the probability that the argumentation game of the student can be mathematical. There may be better illustrations of this and there may be other tools that can help the teacher accomplish that job, but the important point is to that the viability of a custom of mathematical argumentation in a mathematics class relies on the existence and availability for the teacher of tools of that kind: Tools that can be used by the teacher to enable the student's arguing and to ensure the mathematical nature of those arguments. Acknowledgement : Thanks to Humberto Alagia for his comments on a previous version. Notes1. "The term discourse
can be defined as the group of statements that belong to a
single system of formation" (Foucault, 1972, p. 107).
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