La lettre de la Preuve


ISSN 1292-8763

Novembre/Décembre 1999

Questioning argumentation

Raymond Duval
IUFM de Lille


For purposes of initiating middle school students into proof in mathematics, teaching has naturally favored mathematical proof, with all of the constraints of rigor it imposes. But for the past ten years more attention has been paid to argumentation as a means of convincing oneself or others, which is obviously a necessary condition in order for a proof to function as a proof. This note does not propose to search for the reasons for this transfer of interest. Some are clear: there is the accent on the work of research for which the mathematical proof appears as the outcome, and there is also the incomprehensible, for many students, character of the requirements of mathematical proof, and the results of that. We will rather consider what it is that argumentation covers and the questions which its study resolves. In this perspective we will approach successively the emergence of a problematique for argumentation and the two notions fundamental to being able to analyze the process of argumentation and we will indicate some entry points for studying the place of argumentation in the learning of mathematics.

I. The emergence of a problematique of argumentation

The interest in argumentation appeared as an interest in forms of reasoning which escape from norms and logical outlines, and which arise spontaneously as soon as there is an argument with someone. This emergence can be seen as much outside of mathematics as in the teaching of mathematics. What are its principal characteristics?

1. Outside of mathematics

First there was the rediscovery of the irreducible and irreplaceable character of natural language rather than formal languages to fulfill, economically, the function of communication between individuals. This began with Wittgenstein who, beginning in 1930, reacted against all of the philosophy dominating the Principia Mathematica of Russell and Whitehead (1910). And everyone knows the whole pragmatic approach to discourse in natural language which subsequently resulted (Searle 1969, Ducrot 1972). Following that, interest was taken in all situations where it was no longer an issue of communicating only, but of convincing and justifying. Here, the works of Perelman (1958) and Toulmin (1958) were a point of departure. This led, among other things, to a study of the forms of contradiction (Grize 1983) which can be put into effect in arguments, and to an emphasis on the dialogic character of reasoning carried out to convince (Grize 1996).

2. In the teaching of mathematics

Piaget's model of the development of reasoning in the child and the adolescent (1957) have long been the reference for analyzing problems of learning at the middle school level, at least until the mid-seventies. It accorded a central place to implication (the "if…then…") and relativized the role of language in the development of propositional reasoning (the "formal operations"). But such a model swiftly proved unsuitable. It did not permit the analysis of the difficulties encountered by the students when it came to mathematical proof. Furthermore it didn't permit the taking into account of the possibilities opened up by working in groups: investigations became possible as a mode of teaching mathematics (Glaeser 1973) and the interactions among the students could be taken as one of the factors of teaching. The work of Nicolas Balacheff on proof and mathematical proof in middle school (1982) was the first to take this new situation into account. He proposed a more complete approach to the initiation into proof, based on investigation of a problem. It is in this new perspective that there began to be an interest in students' forms of argumentation which appear in the course of resolving a problem. And that led to the question whether that might be a route to discovering mathematical proof.
   Let us bear in mind from this flash-back that the problematique of argumentation is situated at the convergence of a double recognition: that of the important role of communication and social interaction in the acquisition of knowledge &emdash; which leads ipso facto to the recognition of the importance of natural language &emdash; and that of the close bond between proof and conviction&emdash; which likewise leads to favoring communication in order to encourage the confrontation of points of view.

II. Two essential notions for analyzing the processes of argumentation: argument and discursiveness.

1. A first notion is that of argument.

The title of Toulmin's work gives an excellent characterization of argumentation: The uses of Argument.

The notion of argument seems clear. Nonetheless, it is worth pausing over. An argument is considered to be anything which is advanced or used to justify or refute a proposition. This can be the statement of a fact, the result of an experiment, or even simply an example, a definition, the recall of a rule, a mutually held belief or else the presentation of a contradiction. They take the value of a justification when someone uses them to say "why" he accepts or rejects a proposition. An argument is the answer to the question why "do you say that?…do you believe that?…"
   As can be seen, the notion of argument is a purely functional notion. But, contrary to the thought of Toulmin (1958, pp. 99-105) who classified argument as a model of modus ponens (no deduction) while equipping it with "qualifiers" and possibilities of restriction, this notion is structurally indeterminate and a priori indeterminable. Because what can take the value and force of an argument depends not only on the domain of knowledge (mathematics, law, history, politics,…) but also on the particular context which motivates the recourse to arguments. For example, in the course of seeking the solution of a problem a simple question can have the value or force of an argument for dislodging an idea. This point is important. To see it, it suffices to ask oneself whether a theorem can be considered to be an argument. The answer is a good deal less clear than one would believe. If the use of theorems is as central to the resolution of problems as it is to proofs, their use is not that of an argument but that of a "tool". One can only present a theorem as an argument on condition of wanting to justify a proposition as a necessary conclusion from the hypotheses. And experience shows that for the majority of students this use of theorems gives rise to serious difficulties. In fact, a theorem is structurally very determined, although it only has a functional meaning reduced just to the organization of valid deductions or the development of calculations.
   The notion of argument is more global than that of theorem, and involves taking into account two dimensions. To speak of argument is first to refer to the choice of a subject to achieve a determined goal. Next it is to refer to the context for the production of the argument. A production context is determined according to two points. On the one hand there is whatever motivated the recourse to arguments: a weight on the sense of a decision to be taken, the resolution of a conflict of interest, the resolution of a problem presenting technical or logical constraints. On the other hand there is the objective: convince someone else or, on the other hand, diminish the risk of error or uncertainty in the choice of a process. Away from the context of its production an argument frequently loses its "force". And in any case the force of an argument is variable. Also one may need recourse to several arguments to produce conviction.
   In mathematics, or in the sciences, the context of production is radically different from what it is in the other sectors of social activity where one is led to argue. In mathematics the motive and the objective of the argumentation are specific to the problem to be solved. Paradoxically, one could say that these constraints constitute an invariant in communication. Because it is the constraints of the problem which determine the choice of arguments and not first the beliefs of the person to whom the argument is directed. The force of an argument depends primarily on how appropriate it is to the situation and not on its resonance in the universe of the person being addressed: the question is whether the solution "works" or might "work". For that we will speak of heuristic argumentation. But when it is a question of convincing someone about a decision to be made, or resolving a conflict of interest or getting consensus on a question, there is an inversion of priority: one takes into account first the convictions of the person being addressed. In that case we will speak of rhetorical argumentation.

2.The second fundamental notion is that of discursiveness.

Argumentation cannot actually be reduced to the use of a single argument. It requires that one be able to evaluate an argument and oppose it to other arguments. This corresponds to the dynamic of any situation of research or debate. The arguments always take a place in a discourse, in the very wide sense of the term, that is, in a sequence of successive operations mobilizing a semiotic system. Moreover the arguments which might convince someone of the justice of a proposition do not always arise from reasoning. They can consist of a clarification, that is, describing how a system functions and showing the place in it of what the proposition being justified states. Thus the production of arguments, in heuristic argumentation, takes place initially at the level of work on particular cases or examples. Because in particular cases one can examine how things function.
   Let us take the relations stated in the theorem of Pythagoras. To convince someone of the truth of the proposition one can proceed to varied numerical applications and make the person observe that the relationship is always validated no matter what the lengths of the sides of a right triangle. More interestingly, one can make any of the numerous possible reconfigurations of the squares constructed on each of the sides of a right triangle (Padilla 1992, pp. 33-38, 197-218). These numerical verifications or geometrical reconfigurations do not constitute, in the strict sense, a mathematical proof, but they are arguments which will produce conviction of the truth of Pythagoras' proposition. And if the subject is led to change the register of representation, she can justify Pythagoras' proposition by describing, with expressions from ordinary language, what she has observed about the figural transformations between squares and triangles.
   Being able to mobilize multiple forms of discourse, and not only that of reasoning, argumentation always involves mobilizing natural language, even when the arguments used arise from another register of representation. Because then one must clarify why the figural transformations or the calculations can be considered as responses to a problem posed. We rediscover here something that was a strong intuition of those from Wittgenstein to J.B. Grize who tried to understand the mechanisms of argumentation in relationship to the two poles of conviction of a subject and communication between subjects. But nonetheless insisting on natural language is not enough. The decisive point is elsewhere: there are two large mechanisms for the development of a discourse in natural language, while formal languages permit only one. One can get an idea considering these distinctions:

Relationships between a given proposition and another proposition

Relationships of justification
(a component of an argument)
the first proposition is
given as "THESIS"

reasons relative to
the person spoken to


rethorical argumentation

reasons relatives to the constraints of the situation or of tha problem

argumentation heuristique

Relationships of derivation
(constitutif d'un pas de déduction)
the first proposition is given as

instantiation, semantic inference

logic of a language

by énoncé-tiers
theorem, definition

mathematical proof

These distinctions cover very different cognitive functionings. That is why they become essential in the study, from the perspective of learning, of all the questions relative to the relationship between argumentation and mathematical proof.

III. What are the entry points for a systematic study of heuristic argumentation?

Clearly we make no pretence of being exhaustive. We will indicate four with the aim of emphasizing the complexity of the phenomena relative to a problematique of argumentation in the framework of the teaching or learning of mathematics.

1. The context of the production of arguments

There are different factors which determine the context of the production of an argument: the position of the person being spoken to relative to the arguer (cooperation, conflict,…) the motivation of the argumentation (make a decision, find the solution of a problem,…) and its objective (change someone's point of view, diminish the risk of errors or dead-ends in a choice,…). In the case of argumentation in mathematics, the context of the production is determined by the mathematical problem to be solved. That is most certainly one of the most solidly agree-upon points among researchers in didactique. It suffices to look at the frequency of the phrase "solution of a problem" in different communications of work in didactique. Nonetheless it seems to us that the notion of problem remains too general a notion and that the choice of a precise problem for observing students frequently remains too contingent. Between the extreme generality of the notion of problem and the character of the problems posed which, no matter what one says, always remains particular, there does not exist an intermediate level of analysis. To clarify: the analysis of the problem chosen is done downstream, that is, with respect to its solution or solutions and not upstream, that is with respect to possible variations in givens and the variations of distance between the statement and the initialization of the first relevant mathematical treatment which result from them. More radically, there is no elementary classification of the problems available which makes it possible to compare among them the purely mathematical problems or the problems of application of mathematics from the point of view of the process of heuristic argumentation. And the comparison could also need to be made by varying the mathematical domains: geometry, arithmetic, probability, algebra.

2.The modes of expression: verbal or written

The capacities of apprehension and the level of comprehension accessible on a question (topic) are not at all the same in the positions of alternately speaking and listening and those of writing and rereading (one does not read oneself, one rereads.) Until the last few years very little attention was given to the importance of these differences, which were erased in speaking of "language" and "linguistic practices". Nevertheless the passage from an oral mode of expression to a written mode of expression is complex and presents serious difficulties even at the middle school level. Indeed, this passage requires a "reorganization" or a "restructuring" of expression, as Vygotski explained (1985, pp. 360-368, 376).
   This is not without its consequences in the study of argumentation. Rhetorical argumentation is developed largely in the oral mode of expression. The problem which presents itself is knowing whether heuristic argumentation is linked in a favored way to one of these two modes. Which sends us back to the question of knowing whether the practice of mathematics today can be purely oral. But, often for pedagogic or didactical reasons, one favors situations where students cooperate and discuss to solve a problem. Which obviously comes down to favoring a verbal mode of expression. What could then be the function and contribution of a passage to writing? Fulfilling a function of communication and institutionalization, which remains in the prolongation of an oral mode of expression, or on the other hand functions of treatment and control, including for written proofs, which would involve a rupture with the oral mode of expression? As can be seen, behind this question is all of the problem of interferences between the context of a rhetorical argumentation and that of heuristic argumentation. Perhaps one of the benefits of a computer environment is that it makes possible a complete disassociation of these two types of contexts.

3.Discursive arguments mobilized

We insisted on the fundamental character of the notion of discursivity. It necessarily involves the mobilization of a natural or formal "language". Does there exist a mathematical language, as is so often said? This question does not seem to us to be well posed. The problem is not that of the language used but that of the discursive operations which one can carry out with a language. All discursive operations can be grouped around four major discursive functions: designating objects, saying something about those objects which takes on ipso facto an epistemological value (stating a proposition), generating other propositions from a given proposition and finally integrating into the stated proposition its value of epistemological taking-charge by the person who made the statement. Now what is remarkable is the tendency, when one speaks of language in mathematics, to consider only certain of the different discursive operations. The pages which Freudenthal (1978) devoted to the distinction of three levels of mathematical language (the level of display, the functional level, and the level of symbolic conventions making it possible to take variables into account) seems to us to reveal an attitude that is still very widespread: the reduction of language to just the discursive function of designating objects.

4.Argumentation versus mathematical proof

Here the question is that of the homogeneity of processes lasting through the complete development of a mathematical activity: from the first phases of research to the establishment of a mathematical validation of the solution found, that is, until its mathematical proof or its "formal validation", to employ a term whose use frequently has negative connotations. One can look at this question from a strict mathematical point of view and postulate homogeneity of the processes: in this case one could affirm a cognitive continuity between argumentation, explanation and mathematical proof. But if one looks at the question from a cognitive point of view the response is very different. And the cognitive point of view cannot be altogether neglected when one is looking at the learning of mathematics by young children, in whom the different registers of representation mobilized by the practice of mathematics are barely, or not at all, coordinated.
   And that leads us to raise two questions, for which we do not yet have available enough really usable observation results.

- With reference to the work of a mathematician, a lot of emphasis is put on the moment of developing a conjecture. But, at least for the students, do the arguments which lead to the formulation of a conjecture also make it possible to find the means to prove it?
- Are the capacities that a student might have for verification of the pertinence of the arguments produced while he is trying to prove a formulated and maintained conjecture considerably developed when he has understood the differences in discursive functioning between "formal validation" and the rhetorical arguments which are themselves more familiar or more spontaneous?

One can thus see the complexity of the problems attached to the study of the process of argumentation. We would almost be tempted to say that it is easier to give the students access to mathematical proof than to a certain level of mastery of argumentation, at least of rhetorical argumentation. But let us finish by calling attention to a paradoxical situation in the teaching of mathematics. The recognized importance of communication and of social and didactical interactions necessarily leads to giving a priority to natural language. At at the same time, one wishes only to retain the cognitive models of learning in which the role of language, at least of natural language, is given second ranking. One of the tasks of a problematique of argumentation is to bring this paradoxical situation to light.


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Reactions? Remarks?

The reactions to the contribution of Paolo Boero will be
published in the January/February 2000 Proof Newsletter

© R. Duval 1999

Free translation Virginia Warfield


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