La lettre de la Preuve

       

ISSN 1292-8763

Mars/Avril 2000

 
The articulation and structuring of conceptions
in the mathematics class:
argument and public knowledge

by

Patricio Herbst
The University of Michigan, USA

 

If the mathematics class is considered as a system of public knowledge, what are the possible uses of argument in the articulation and structuring of knowledge within this system. The term public knowledge is used here to indicate two simultaneous distinctions. The mathematics of the class as public knowledge is distinguished (1) from the mathematics (prescribed) to be transmitted and acquired (the official knowledge), and (2) from the mathematics (known or developed) by the individuals that play roles of learners or teachers (their personal knowledge). These distinctions are analytical; they mean not to discredit or neglect official or personal knowledge. Rather, these distinctions serve to identify a challenge to undertake: That of fleshing out a notion of public knowledge which is dialectically tied to the other two notions.
   We need to find ways of talking about the relation between argument and mathematical knowledge within the complexity of the mathematics class. The problems with proof in the mathematics class have often been framed in terms of the logical structure of mathematical discourse or the personal conviction of the participants. A focus on the mathematics of the class aims at developing a way of talking about proof more closely related to the class's construction of mathematical knowledge.
   Recent examples of mathematics classroom research (e.g., Yackel & Cobb, 1996; Ball & Bass, forthcoming) have encouraged thinking about the mathematics class as a place where a specific kind of mathematical knowing exists and functions. Ball's example of Sean's Numbers (numbers that are odd multiples of two--such as 6 or 10) illustrates very clearly that such specificity means not just an indigenous way of doing, but it involves indigenous objects (1) of that doing (Ball, 1993). I suggest that observations like that support the hypothesis that the mathematics class is a cognitive locus: A sort of ecosystem that produces its own objects of knowledge and, one would expect, its own communication codes, substantial forms of argumentation, indigenous theories, and inscription and memory devices. According to the hypothesis that the mathematics class is a cognitive locus, the mathematics (of the) class is enabled (and constrained) as much by the cognition of the individuals involved as by the structure of the knowledge to be taught, but is irreducible to either of those factors. Studying such system of public knowledge may help us understand more systematically what mathematics might be known and done in schools. Whereas knowledge about how individuals develop specific mathematical ideas or about ways to organize the knowledge to be learned provide motivation and warrant for suggesting that changes be made in school mathematics, they are not enough to estimate their cost or even demonstrate that they are possible. We need to know more about the logic that underlies the knowledge practices of school mathematics in order to envision the conditions of possibility for those changes.
   The mainstream perspectives of mathematics education research on proof can be correlated with the ways of conceiving of the mathematics class outlined in the previous paragraph. Some researchers interested in the individual development of notions of proof have defined proof as those strategies by which people convince themselves or persuade others of the validity of their mathematical assertions (see for example Harel & Sowder, 1998). In contrast, scholars interested in determining the import of mathematical proof (as a general practice of the mathematical sciences) to the practice of mathematics education have considered proof as the generic ways of validating and explaining truth in mathematics (see for example Fawcett, 1938; Hanna, 1995). Those perspectives may be useful to study the development of the notion of proof by individuals or to monitor the fit between what is called proof in the mathematics class and in mathematics. However, the conceptualization of the class as a system of public knowledge recommends one to avoid a reduction of proof to conviction or to logic. Instead, the conceptualization of the class as a system of public knowledge recommends to take up other sorts of problems that seem to be critical for an investigation of a classroom-based notion of proof. Those problems concern the possible roles of argument in constituting, articulating, and structuring a system of public knowledge.
   The problems of constitution, articulation, and structuring of public knowledge involve, at least, tensions between knowledge as product (as public wealth) and production of knowledge (as public work), both at a global level (accumulation and growth) and at a local level (creation and trade). At a global level one can envision a tension between the structure of knowledge (as a product) and the structuring of knowledge (as part of its production). These tensions can be located in the history of mathematics as well as in the individual development of mathematical ideas. The following reflection on mathematical practices and their history should help make the point that there is an intimate connection between a substantive, practice-based, notion of argument (or proof) and the problems of public knowledge presented above.
   The creation and trade of knowledge in mathematical practice concerns fundamentally the articulation of definition of objects and proof of theorems (if-then statements about those objects). There is always a tension between the interest to claim a conclusion and the possibility to establish such a claim in continuity with what is known: This tension relates the conclusion (stating a categorical fact) and the proposition (providing a hypothetical statement). Inasmuch as one can load more or less the conditions under the "if," a proof will more or less likely be achieved and, thus, warrant claiming the conclusion. But clearly, those conditions can be so demanding or the proof so trivial that the hypothetical statement may become unimportant. Mathematicians do not just strive to provide the most impeccable proof that a conclusion is warranted, they also search for the (relatively) minimal conditions that enable an acceptable proof to exist&emdash;and this naturally raises the expectation of the ingenuity in the proof itself (2). The work of proof in this local aspect of producing mathematical knowledge cannot be reduced to the logical form of the argument (as it involves substantive knowledge and values about knowledge). The work of proof in this local aspect of producing mathematical knowledge cannot be reduced to the personal knowledge of the participants either (as it involves a synthesis between epistemological positions that usually coexist in a person).
   The accumulation and growth of mathematical knowledge concerns fundamentally the structuring of mathematical theories, memoirs, discourses, or texts (which could be understood as intellectual architectures that integrate a body of objects and proved propositions, and suggest problems for future work). There is always a tension between the capacity of theories to house large number of results within a coherent structure (the accumulation aspect) and the capacity of theories to enable the actual formulation of problems and generation of solutions (the growth aspect). This tension affects the constraints to which a proof submits. If a theory is going to enable the community of researchers to foresee what to do, it has to allow insights that are not formalized or axiomatized, concentrate the rigor where it is needed, avoid handcuffing intuition. Reciprocally, if a theory will give a coherent account of what a community knows, it needs tools to avoid the circularity of arguments or the ambiguity of information. (3) The work of proof in this global aspect of production of mathematical knowledge cannot be reduced to the building of a logical architecture (as it involves not just the existence of a logical structure but also a substantive, prospective, rationale for it). The work of proof in this global aspect of production of mathematical knowledge cannot be reduced to the sharing of the personal knowledge of the participants (as theories involve the reconstruction of fictional but common pasts and enable the envisioning of common futures).
   The search for a classroom-based notion of proof may, therefore, benefit from addressing analogous questions of articulation and structuring of the mathematics (of the class). This proposal thus intends to attack the problem of proof in the classroom by circumventing it. Instead of abstracting a generic notion of proof in mathematics as a discipline or aggregating the ways in which classroom participants are individually convinced, the metaphors of economic production help give a working definition of proof from the perspective of public knowledge. The provisional definition I suggest includes and extends the one proposed by Balacheff (1987, p. 147):

Proof is an explanation [of the truth character of a proposition] accepted by a given community at a given moment of time. The decision to accept it can be the object of a debate whose principal objective is to determine a common system of validation for the speakers.

I suggest that proof can initially be conceived of as a prospective and retrospective tool for the public control of knowledge production: What counts as proof depends on what is being (and to be) done and known at various levels of cognitive activity (problems, propositions, models, theories).
   A focus on the objects of public knowledge dealt with in the class makes one aware of the fact that to recognize a claim being formulated needs an appraisal of the normative ways of representing and enunciating that are specific of the objects dealt with. The recognition of the conditions under which a claim is formulated and of the validity of its "proof" depend on the conceptions being handled and those being kept away. What things count as a claim or as a proof in the mathematics of the class are not understandable just in terms of an indigenous but generic logic or an indigenous but generic language. A refined search for understanding about proof in the mathematics (of the) classroom could use working inside specific conceptual fields rather than in mathematics in general.
   There is an important relationship between what may possibly count as "proof" and the way conceptions are articulated. The various conceptions related to a mathematical notion are not likely to be available at any given time (even if students have used them in the past). This seems to fulfill the purpose of enabling some work to be done as well as placing bounds on what arguments and counterexamples can be offered. What are the arguments by which various conceptions associated with a single mathematical notion are integrated into public knowledge? If various conceptions are reconciled into a unified theory, how are they brought to "argue" against each other, what do they "say," and how do they develop a "normal argument"? If such reconciliation does not happen in fact, what are the ways in which parallel and disconnected conceptions maintain their own senses of "normal argument"? These questions are formulated metaphorically because an important part of the work to do involves developing theory that can formulate them in more precise terms.
   Nicolas Balacheff (largely inspired by Gerard Vergnaud's theory of conceptual fields and by Gaston Bachelard's notion of epistemological obstacle) has provided an abstract characterization of conceptions that permits to describe substantive examples of argument. Mathematical conceptions are defined as coherent domains of practice and modeled as a quadruplet of four interrelated systems, namely:

• the problems that can normally be attacked,
• the operators that normatively handle those problems,
• the systems of representation that permit to express and attack those problems, and
• the structures of control (including the kinds of proof) that permit to determine whether a
  problem has been satisfactorily solved.

I have suggested that the question of what proof might look like is intimately related to the articulation and structuring of student conceptions: What counts as proof and what works as proof depend on what the class knows and could know. Whereas such conceptualization of proof may respond to a vision of school mathematics more closely related to what mathematicians do, it also upgrades the stakes of the teacher. In such a vision of proof in school mathematics, what are the tools available for the teacher to engage students in argument and to ensure that such arguments are mathematical?
   The study of how conceptions are articulated and structured provides a base to understand the rationality behind current uses or lack of use of argument. One will then be in better position to understand more precisely what are the specific challenges that teachers face in trying to make the practices of school mathematics more similar to those of working mathematicians. Further, one may be able to work toward the development of technical support to meet those challenges.

Notes

(1) The fact that Sean's numbers admit of a precise mathematical definition does not make them objects of the mathematics of mathematicians (by themselves they are relatively uninteresting to be created). Insofar as there is no cultural imperative to transmit and learn Sean's numbers, but these numbers nevertheless emerge as mathematical objects in Ball's class, they can be an example of an indigenous object. [Back]
(2) For example, it is relatively easy to prove Euler's theorem (that vertices plus faces minus edges equals 2) for the case of a convex polyhedron, using projective geometry. The observation that this invariant holds for some other (but not all) polyhedra imposed the need to define the largest set of objects for which this theorem was true (hence, the definition of simple polyhedron) and to search for new tools that enabled a proof to exist. The resulting argument was still compelling, yet rather more metaphorical than the projective argument (see Frechet & Fan, Lakatos ). What counted as an acceptable proof was an emergent of the best synthesis between the interest in claiming the conclusion and the importance of the proposition stated. [
Back]
(3) It seems to me that a prime example of this equilibrium is found in Euclid's Elements (exactly with all its merits and faults). [
Back]

References

Balacheff N. (1987). Processus de preuve et situations de validation [Processes of proof and situations of validation]. Educational Studies in Mathematics, 18, 147-176.
Ball D. L. (1993). With an eye on the mathematical horizon: Dilemmas of teaching elementary school mathematics. The Elementary School Journal, 93, 373-397
Ball D., Bass H. (in press). Making believe: The collective construction of public mathematical knowledge in the elementary classroom. In D. Phillips (Ed.), Constructivism in education: Yearbook of the National Society for the Study of Education. Chicago: University of Chicago Press.
Fawcett H. (1938). The nature of proof&emdash;The National Council of Teachers of Mathematics Thirtheenth Yearbook. New York: Bureau of Publications of Teachers College, Columbia University.
Hanna G. (1995). Challenges to the importance of proof. For the Learning of Mathematics, 15(3), 42-49.
Harel G., Sowder L. (1998) Students' proof schemes: Results from exploratory studies. In: A. Schonfeld, J. Kaput J., and E. Dubinsky (eds.) Research in collegiate mathematics education III. (Issues in Mathematics Education, Volume 7, pp. 234-282). American Mathematical Society.
Yackel E., Cobb P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 27, 458-477.

 

 

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