AERA Symposium

19 April 1999, Montréal

Fostering argumentation in
the mathematics classroom:

The role of the teacher

Organizer: Patricio Herbst, Michigan State University

Chair: Jeremy Kilpatrick, University of Georgia

Summary

(Background:) Argumentation and Proof in the NCTM Standards

The NCTM Curriculum Standards (National Council of Teachers of Mathematics, 1989) call for decreased attention to two-column proofs and increased attention to alternative expressions of mathematical argument, such as "deductive arguments expressed orally and in sentence or paragraph form" (p. 126). Teachers are to promote or increase students' opportunities "to make and provide arguments for conjectures" (p. 125), "formulate counterexamples; follow logical arguments; judge the validity of arguments; [and] construct simple valid arguments" (p. 143). The Standards suggest that mathematical argumentation be contrasted with other forms of argumentation (such as political or commercial advertisements), in some of which logical errors can be detected.
   Similarly, the Professional Standards for Teaching Mathematics (NCTM, 1991) indicate the teacher's responsibility to choose tasks that "require students to speculate,… to face decisions about whether or not their approaches are valid" (p. 26). Teachers are encouraged to play a role of orchestrators of classroom discourse in the direction of mathematical reasoning, asking students to explain and justify their ideas (pp. 35-36). The Professional Standards expect that as a result of those efforts from the teacher, students will become competent in the use of mathematical argument to support the validity of their conjectures (p. 45).

(Scientific Importance:) Problematic Aspects of Argumentation and Proof

The vision of the Standards includes enhancing the students' opportunities to participate in practices that are similar to those of working mathematicians. As a consequence, at least implicitly, a contrast emerges between proof and alternative forms of mathematical argumentation, and an opposition between the teaching of a skill (writing two-column proofs, localized in certain curricular areas) and the dissemination of argumentation across the curriculum (with deductive proofs as only one form).
   Some critics have questioned the Standards' presumed challenge to the importance of mathematical proof (Wu, 1997). Voices from mathematicians and mathematics educators have insisted on the centrality of proof for the advancement of mathematical knowledge (Hanna, 1995; Thurston, 1995), and representatives from the community of mathematicians have recommended "the appropriate inclusion of proofs and mathematical reasoning" as "extremely important" for the forthcoming revision of the Standards (Ross, 1998).
   Mathematics educators have investigated argumentation practices in the classroom. Lampert (1990), studying her own teaching, demonstrated that classroom discourse can be orchestrated so as to engage students in forms of argumentation similar to those of working mathematicians. Still, she raises the pertinent question of what kind of knowledge is acquired by students as a result of their participation in such practices. Also examining their own teaching, Chazan & Ball (1995) illustrate the dilemmas that teachers come across when they, on the one hand, try to foster students' involvement in argumentation and, on the other hand, need to ensure the mathematical character of such argumentation practices.

(Objectives:) Questions That Deserve Consideration

The vision of the Standards and the various perspectives on argumentation and proof mentioned above raise some questions:

  • To what extent it is possible for argumentation in the classroom to be mathematical, that is, to maintain the ties between the mathematical practices of the classroom and the way mathematicians advance human understanding?
  • To what extent is argumentation in the classroom necessary in order for students to construct meaning for mathematical proof (rather than just to learn a particular kind of mathematical writing skill)?
  • What is the role of the teacher in fostering genuine mathematical argumentation in the class? How is the role of the teacher in fostering mathematical argumentation different from that in teaching two-column proofs?

 
Organization of the Symposium
 

A group of five mathematics educators will discuss the role of the teacher in fostering mathematical argumentation with respect to the epistemological opposition between argumentation and proof and the educational opposition between mathematical reasoning and two-column proofs. (The discussion will focus on the American curriculum but will be enhanced by our own experiences, as we come from four different countries.) We aim to provide a multidimensional conceptualization that identifies the conditions and constraints of the teacher's work and to illustrate the dilemmas that the teacher may encounter.
   The symposium will have five sections. Each of us will approach the opposition between argumentation and proof from a different angle. We will conclude with a panel discussion in which we will raise the questions posed above, allow the five participants to react to the other presentations, and open the floor for comments and questions from the audience. The following sections describe the parts into which the symposium will be divided.

(First Section:) Argumentation and Proof in Educational Discourse

Presenter: Jeremy Kilpatrick, University of Georgia

Why is argumentation vs. proof an acceptable opposition in the discourse of mathematics education? This section aims at developing an account of the dispersion (Foucault, 1972) of objects of discourse related to argumentation and proof in selected historical and current educational documents. In particular, we will survey materials intended for the preparation of mathematics teachers from various moments in this century (including materials from the new math and Standards eras) and from various educational orientations (such as mental discipline and behavioral objectives). Objects such as proof, argumentation, validity, reasoning, truth, and logic will be traced. In accounting for the emergence of those objects of discourse, the presenter proposes to describe, from the perspective of the argument given to the teacher, the grounds for presuming proof and argumentation to be equivalent so that an opposition can be posed, and the ideological source of their distinction so that argumentation emerges as the viable choice between them.
   For example, Lakatos's (1976) argument that mathematics is quasi-empirical (so there are no such things as formal proofs; see Thurston, 1995) is usually used as a justification by those who argue against two-column proofs (and who seem to equate these curricular objects with formal proofs). Some associations between philosophy and pedagogy (Ernest, 1991, pp.138-139) seem to identify the teaching of proof with the promotion of an authoritarian mathematics (Hanna, 1995). In contrast, arguments that go back to the use of mathematics to teach logical thinking as a skill needed for ordinary life are used to support the explicit teaching of propositional logic and its application in writing proofs (Ross, 1998). By drawing on educational materials from various eras, the presenter will consider the question, What sorts of rhetoric make argumentation and proof comparable, and what sorts of rhetoric make argumentation preferrable to proof from the perspective of the discourse addressed to the teacher?
   By the time of the symposium, the draft version of the revised NCTM Standards (Standards 2000) will be available for a discussion of the various influences regarding the changing and stable features of the "mathematical reasoning" standard.

(Second Section:) An Epistemological Perspective on Argumentation and Proof

Presenter: Nicolas Balacheff, Laboratoire Leibniz, Grenoble, France

When teaching mathematical proof, one cannot ignore that outside the classroom students have already experienced the problem of establishing the truth of a statement either among their classmates, parents, other children, or other adults. The rules for arguing in such situations are rather different from the ones at work in mathematics. Argumentation and mathematical proof differ in several aspects: the value of empirical evidence, the status of counterexamples, the rules allowing one to choose an argument, and so forth. Also, the theoretical nature of mathematical proof requires a theoretical construction of the knowledge that mathematical proof deals with. As a consequence, the teacher has to manage two simultaneous developments: the construction of a new rationality and a new construction of mathematical knowledge. These simultaneous constructions cannot be obtained simply by decree, as the history of the teaching of mathematical proof shows. Rather, one must accept that the learning of mathematical proof goes with the learning of mathematical content, while this content is being constructed. Such a constraint at the level of students' conceptions entails that the mathematics class has to make room for "proofs" that do not necessarily fit the mathematical norm. The conflict between mathematical proof and argumentation is then unavoidable. This conflict is at the core of a basic teaching-learning difficulty. The presenter of this part will characterize argumentation as an epistemological obstacle (Brousseau, 1997, pp. 83-90) to the learning of mathematical proof.

(Third Section:) How Well Suited Is Ordinary Language to
Make Mathematical Argumentation Emerge From Ordinary Argumentation?

Presenter: David Pimm, Michigan State University

Mathematical argumentation develops from within ordinary language and converges to a specialized mathematical register. When the class argues about new mathematical objects, the specialization of a register is not just the shaping of a medium to communicate reasons, but also the construction of the tools with which those reasons can be developed. The presenter of this section proposes to discuss what is the use of modal verbs (would, should, could, must, can't, etc.) in relation to the various forms of mathematical argument (e.g., in historical or current mathematical texts and in classroom discourse or mathematics textbooks). How does that presence compare to their occurrence in other settings such as threats or fights?
   The previous section has made the point that arguments that do not fit the mathematical norm may be necessary for the construction of mathematical knowledge, and that their status of not conforming to the norm may be impossible to disclose (as opposed to Ross's, 1998, injunction that the difference between plausibility and necessity should always be made explicit). If that is the case, what sort of ambiguities of ordinary language contribute to reinforcing ordinary argumentation as an obstacle to mathematical proof? One may ask, What is the role of questions such as "can we say" or "do we know" in the teacher's management of an argument or surveillance of a proof? The presenter will address that role using elements from social semiotics.

(Fourth Section:) Fostering Argumentation: The Teachers' Dilemma

Presenter: Daniel Chazan, Michigan State University

Drawing on data from teachers in a Midwestern high school, the presenter would like to explore the question of fostering mathematical argumentation from the perspective of the difficulties that such "fostering" creates for teachers. On the one hand, teachers are responsible to the school and to parents for helping students learn the mathematics (and the standards of mathematical argumentation) currently accepted in society. Students are aware that teachers "know" this mathematics. Yet, paradoxically, helping students learn, in some views, is contingent on focusing on students' ideas (and indigenous ways of demanding and giving reason), rather than on accepted mathematics, on getting students to take their own thinking seriously, even when their ideas or the arguments they engage in seem to conflict with accepted mathematics. What do teachers do? What can they do? Such dilemmas will be explored by the examination of classroom episodes in which students' ideas conflict with those of accepted mathematics as represented in textbook materials.

(Fifth Section:) What Do the Practices Associated with Two-Column Proofs Say
About the Possibilities of Argumentation?

Presenter: Patricio Herbst, Michigan State University

This section presents an analysis (in terms of the norms that matter in the negotiation of a didactical contract) of two-column proofs as a class practice. The enduring presence of two-column proofs in the curriculum can be explained on the basis of its functionality in the work of the teacher. On the one hand, two-column proofs serve to operationalize the work of the student in ways that make producing a proof like doing an exercise. On the other hand, two-column proofs respond to societal constraints on the reproduction of mathematical knowledge and on efficient management and control of the learning. Thus, two-column proofs maintain the customary relations between teacher, student, and mathematical knowledge, and therefore the practices associated with two-column proofs help constitute the knowledge available to the class.
   The presenter will argue that the production of two-column proofs opposes on the one hand, the final product of a seemingly acceptable mathematical (hypothetico-deductive) argument and, on the other hand, a practice where one can see the division of labor between the teacher's careful manufacturing of an argument and the students' listing and sorting categorical statements. This transactional role will be illustrated with excerpts from a lesson on a proof about angles.
   If argumentation substitutes for proof the class will need to develop efficient procedures to handle that. The teacher is under conflicting injunctions to foster mathematical argumentation: These emphasize either the form of the human communication or the distinction of status between different kinds of argument. Among the resources that the teacher can use, natural language is one of the most important. The customary practices associated with two-column proofs will be used to suggest what a custom of argumentation might look like and to illustrate its features using classroom excerpts.

Conclusion

In the concluding section the five participants will focus on the position of the teacher at the junction of possibly contradicting discourses from outside the class regarding mathematical argumentation. They will draw on the content from the five presentations to illustrate the complexities associated with the role of the teacher in satisfying external constraints and at the same time maintaining a classroom mathematics that students can work in and work with.

References

Brousseau, G. (1997). Theory of didactical situations in mathematics: Didactique des mathématiques 1970-1990 (N. Balacheff, M. Cooper, R. Sutherland, & V. Warfield, Eds. & Trans.). Dordrecht, The Netherlands: Kluwer.

Chazan, D., & Ball, D. (1995). Beyond exhortations not to tell: The teacher's role in discussion-intensive mathematics classes (Craft Paper 95-2 ). East Lansing, MI: National Center for Research on Teacher Learning.

Ernest, P. (1991). The philosophy of mathematics education. London: Falmer Press.

Foucault, M. (1972). The archaeology of knowledge and the discourse on language (A. M. Sheridan Smith, Trans.). New York: Pantheon.

Hanna, G. (1995). Challenges to the importance of proof. For the Learning of Mathematics, 15(3), 42-49.

Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery (J. Worrall & E. Zahar, Eds.). Cambridge: Cambridge University Press.

Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27, 29-63.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: NCTM.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: NCTM.

Ross, K. (1998). Doing and proving: The place of algorithms and proofs in school mathematics. American Mathematical Monthly, 252-255.

Thurston, W. (1995). On proof and progress in mathematics. For the Learning of Mathematics, 15(1), 29-37.

Wu, H. (1997). The mathematics education reform: Why you should be concerned and what you can do. American Mathematical Monthly, 946-954.