Douek N. (2000) Comparing argumentation and proof in a mathematics education perspective. Contribution to: Paolo Boero, G. Harel, C. Maher, M. Miyazaki (organisers) Proof and Proving in Mathematics Education. ICME9 TSG 12. Tokyo/Makuhari, Japan.

© Nadia Douek

I. Introduction There is a "culture of mathematical proof", which should be developed in teacher training and for direct educational implications. I think that one part of the culture of mathematical proof can be built through and with a "culture of argumentation". I shall try to show how proving and arguing have many common aspects from the cognitive and epistemological points of view, even though significant differences exist between them, particularly as socially situated products.   To begin with, we must distinguish two main aspects attached to argumentations and proofs. One is the process of proving or of arguing when one gathers ideas, tries some calculations etc. The other aspect is the static finished product of an argumentation or a proof. The teaching of proof should not only focus on the product (to conform to the socially accepted products) but also (and even much more) on the learning of the process, keeping into account the fact that the process of arguing or proving is not reduced to the writing of the text of an argumentation or a proof.   That process is complex and difficult enough so that we can agree on the fact that it is the responsability of the teacher to guide the students into this difficult and long term learning, without expecting that they find their way alone towards the expected final product.   Teaching of text production has its specificities too and would be an over charge if it comes along with all the other aspects of learning to prove. In this contribution we shall not analyse the phase of writing such texts.We will analyse and compare argumentation and proof as concerns the production phase and as products. II. Argumentation and proof What is argumentation? Referring to Webster's dictionary, "argumentation" will indicate both the process which produces a logically connected (but not necessarily deductive) discourse about a given subject and the text produced by that process. It may include verbal arguments, numerical data, drawings, etc. An "argumentation" consists of logically connected "arguments". The discoursive nature of argumentation does not exclude the reference to non-discoursive (as visual or gestural) arguments. How to compare argumentation and proof? Keeping into account Duval's "cognitive analysis of deductive organisation versus argumentative organisation of reasoning"(Duval, 1991), we will consider the logical organisation of the products (enchaining steps), the role of the semantic aspects of the steps, the problem of epistemic value (defined as the "degree of certainty or convinction attributed to a proposition").   We will consider the backings that sustain argumentations and proofs.   We will also compare important skills taking part in both processes of production.   Argumentation is often considered very distant from proof, in an educational perspective, because of the formal aspects of proof (as product). To which extent is mathematical proof a formal proof? We consider "formal proof" as proof reduced to a logical calculation.   When considering proof's features like common function (i.e. validation of a statement), reference to an established knowledge (see definition of "theorem" as "statement, proof and reference theory" in Bartolini et al., 1997) and some common requirements (like enchaining propositions), we share Thurston's position (Thurston, 1994): "We should recognize that the humanly understandable and humanly checkable proofs that we actually do are what is most important to us, and that they are quite different from formal proof. For the present, formal proofs are out of reach and mostly irrelevant: we have good human processes for checking mathematical validity". III. Analysis and comparison of argumentation and proof as products About the reference corpus The expression "reference corpus" for argumentation will include reference statements, visual, experimental evidence, physical constraints, etc. assumed to be unquestionable. Argumentation (individual or between two or more protagonists) would be impossible in everyday life if there were no reference corpus to support the steps of reasoning. The reference corpus for everyday argumentation is socially and historically determined, and is largely implicit. Mathematical proof also needs a "reference corpus". Social and historical determination of the "reference corpus" for proof. The reference corpus used in mathematics depends strongly on the users and their listeners/readers, and on the way they are called in.   For example, in secondary school some detailed references can be expected to support a proof, but in communication between higher level mathematicians those may be considered evident and as such disregarded. On the other hand, some statements accepted as references in secondary school are questioned and problematised at higher levels; questions of "decidability" may surface, quoting Thurston (1994): "On the most fundamental level, the foundations of mathematics are much shakier than the mathematics that we do. Most mathematicians adhere to foundational principles that are known to be polite fictions".   Thus for almost all the users of mathematics in a given social context (high school, university, etc.) the problem of epistemic value does not seem to exist. But on a longer time scale things can be different: a mathematician dealing with an unsolved problem may believe he should try to write things this way, describe others in that theory, etc.; these "intuitive" choices at that moment have no epistemic values as true or false statements, they may succeed nevertheless and even for some time become a shared method of work in research and then one day be justified by a new theory, and become statements with true (or false) epistemic value (like in the case of Leibniz' calculations on If we consider the "references" that can back an argumentation for validating a statement in primary school, in an approach to mathematical work, these can include experimental facts. And we cannot deny their "grounding" function for mathematics (see Lakoff and Nunez, 1997), both for the long term construction of mathematical concepts and for establishing some requirements of validation which prepare proving (e.g. making reference to acknowledged facts, deriving consequences from them, etc.). For instance, in primary school geometry we may consider the superposition of figures for validating the equality of segments or angles. Later on in secondary school, this reference no longer has value in proving; it is replaced by definitions or theorems (see Balacheff, 1988). Concerning visual evidence, we may note that, in the history of mathematics, it supported many steps of reasoning in Euclid's "Elements". This evidence was replaced by theoretical constructions (axioms, definitions and theorems) in later theories. In some parts of mathematical analysis, visual evidence plays till now important role in some commonly accepted proofs.   Implicit and explicit references The reference corpus is generally larger than the set of explicit references. In mathematics, as in other areas, the knowledge used as reference is not always recognised explicitly (and thus appears in no statement): some references can be used and might be discovered, constructed, or reconstructed, and stated afterwards. See Euler's theorem discussed by Lakatos (1985). The same occurs in various argumentations. In general, we cannot exchange ideas, whatever area we are interested in, without exploiting implicit shared knowledge.   How to dispel doubts about a statement and the form of reasoning Formal proof "produces" (according to Duval's analysis) the reliability of a statement (attributing to it the epistemic value of "truth") . But as Thurston argues (considering Wiles's proof of Fermat's Last Theorem's example), the "reliability does not primarily come from mathematicians formally checking formal arguments" and the requirements of formal proof represent only guidelines for writing a proof, once its validity has been checked according to "substantial" and not "formal" arguments. The preceding analysis shows many points of contact between mathematical proof and argumentation in non-mathematical fields. But significant differences exist too: - as concerns the form of reasoning visible in the final product, argumentation presents a wider range of possibilities than mathematical proof: not only deduction, but also analogy, metaphor, etc. - argumentation can exploit arguments taken from different reference corpuses which may belong to different theories with no explicit, common frame ensuring coherence. Whereas mathematical proof refers to one or more reference theories explicitely related to a coherent system of axiomatics.   IV. THE PROCESSES OF ARGUMENTATION AND CONSTRUCTION OF PROOF   Experimental evidence shows that "proving" a conjecture often entails establishing a functional link with the argumentative activity needed to understand (or produce) the statement and recognizing its plausibility (see Bartolini et al., 1997). Proving itself needs an intensive argumentative activity, based on "transformations" of the situation represented by the statement. Experimental evidence shows also the importance of "transformational reasoning" in proving (see Arzarello et al., 1998; Boero et al., 1996; Simon, 1996; Harel and Sowder, 1998). Simon defines "transformational reasoning" as "the physical or mental enactment of an operation or set of operations on an object or set of objects that allows one to envision the transformations that these objects undergo and the set of results of these operations. Central to transformational reasoning is the ability to consider, not a static state, but a dynamic process by which a new state or a continuum of states are generated". Metaphors are particular outcomes of transformational reasoning. The process of proving often needs metaphors with physical or even bodily referents. Lakoff and Nunez (1997) suggest that these metaphors can have a crucial role in the historical and personal development of mathematical knowledge ("grounding metaphors"). This shows the "semantic" complexity of the process of proving, and the importance of transformational reasoning as a free activity (in particular, free from usual boundaries of knowledge).. Induction also in general is relevant; for example, the need to produce a deductive chain can guide the inductive search for arguments to "enchain" when coming to the writing process (see Boero et al., 1996).   CONCLUSION We should distinguish situations where knowledge, language, and questions are mastered by the involved people from the situation where such important aspects can be very new. In the second case people generally need semantical backing to sustain their "research" work, to interpret the moves towards the problem solving. On the other hand, we should consider the graduations (continuity) of possibilities between the three "languages" (Grize, 1996, p.47) : natural language, scientific and technical language and formal language. The last one represents a necessary and efficient ideal, but as we argued, generally not possible to work with. We can make use of this continuity to offer students the means of arguing and proving with a certain freedom from the formal language constraints and in the same movement to exhibit the real mathematical problem of rigour and the permanent need of clarification trough formalisation, but also through interpretation.   References Arzarello, F.; Micheletti, C.; Olivero, F. and Robutti, O.: 1998, 'A model for analyzing the transition to formal proof in geometry' , Proceedings of PME-XXII, Stellenbosch, vol. 2, pp. 24-31 Balacheff, N.:1988, Une étude des processus de preuve en mathématiques, thèse d'état, Grenoble Bartolini Bussi, M.; Boero,P.; Ferri, F.; Garuti, R. and Mariotti, M.A.: 1997, 'Approaching geometry theorems in contexts', Proceedings of PME-XXI, Lahti, vol.1, pp. 180-195 Boero, P.; Garuti, R. and Mariotti, M.A.: 1996, 'Some dynamic mental processes underlying producing and proving conjectures', Proceedings of PME-XX, Valencia, vol. 2, pp. 121-128 Duval, R.: 1991, 'Structure du raisonnement déductif et apprentissage de la démonstration', Educational Studies in Mathematics, 22, 233-261 Granger, G. 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