La lettre de la Preuve 

ISSN 12928763 

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 argumentationThe 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 mathematicsFirst 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 mathematicsPiaget'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 midseventies. 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. 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?…" 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.
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 argumentsThere 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 deadends 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 agreeupon 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 writtenThe 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. 360368, 376). 3.Discursive arguments mobilizedWe 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 takingcharge 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 proofHere 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.  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? 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. ReferencesBalacheff N. (1982) Preuve et démonstration
en mathématiques au collège. Recherches en
didactique des mathématiques. 3(3) 262306
The reactions to the contribution of Paolo Boero will
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