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Summary |
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(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. |
(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). |
(Objectives:) Questions That Deserve Consideration |
The vision of the Standards and the various perspectives on argumentation and proof mentioned above raise some questions:
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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. |
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(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. |
(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 |
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? |
(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 |
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. |
ConclusionIn 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. ReferencesBrousseau, 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. |