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Balacheff
N. (1991)
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PresentationWhen teaching mathematical proof is a goal for a curricula, it is sometimes explicitly stated as a content to be taught, sometimes it is indicated as a general aim of teaching. In this last case, the aim is the training in the construction and the formulation of deductive reasoning, "mathematical proof" not being named as such. This objective is very difficult to fulfil as teachers witness it and as it is established by several pieces of research. Usually two reasons are offered by mathematics teachers as to explain these difficulties, one is the lack of students' awareness of the necessity to give any proof, the other is their lack of logical maturity. But, these explanations may appear a bit short, insofar as outside the mathematics classroom, there is some evidence that the same students have quite a different behavior. In order to understand what happens we have to enter the students world, we have to accept to consider the coherency of their own rationale for asserting the truth of a mathematical statement or a result. In other words, we must make our Copernican revolution. So to say, students have some awareness of the necessity to prove and some logic (Balacheff 1988). Perhaps these are not identical to what we expect them to be. The related open question is "why?". One may suggest that the answer to this question cannot be found just by considering the student in isolation. On the contrary we have to consider the context in which he or she behaves: Students' cognitive behaviors are meaningful insofar as they are answers to the situation as these students perceive it. Several research have evidenced that the teaching situation in which students construct the solution to a given problem and in which they formulate it, can lead them to radically transform their initial solution, and the means used to validate it, into something they find more suitable to the teacher expectations as they understand them. We must realize that in such a situation teachers have no way to think anything else than that such students behave at a very low level of logic. This phenomenon is now well known to many researchers, and there is a consensus to consider that "students' apparently bizarre [mathematical] behaviors frequently cannot be accounted for solely in terms of conceptual limitations. What most of the researchers discover, when moving beyond 'purely cognitive', is the crucial role played by social issues. These characteristics of school mathematics are of a social nature, and there is a quite large consensus of the mathematics educators' community about the fact that issues related to the social dimension must be considered more carefully. Evidence of the fruitfulness of using social settings to facilitate mathematics learning consist mainly of stories about the success of such attempts. But, as we know such settings do not work with every teacher or in every classroom. Research is needed to strengthen the foundation of such approaches. Part of the research presented here has been to explore the necessary conditions for the success of such settings and thus to explore both benefits and limits of the use of social interactions in the mathematics classroom. Paper's table of contentThe meaning of mathematical proof : an outcome of school practive Conclusion outlineEfficiency versus rigor Even if we are able to set up a situation whose characteristics promote content specific students interaction, we cannot take for granted that they will engage a "mathematical debate", and finally that they will produce a mathematical proof. We have to realize that most of the time students do not act as a theoreticians but as practical persons. Their task is to give a solution to the problem the teacher has given to them, a solution that will be acceptable with respect to the classroom situation. In such a context the most important thing is to be effective. The problem of the practical person is to be efficient not to be rigorous. It is to produce a solution not to produce knowledge. Thus the problem solver does not feel the need to call for more logic than is necessary for practice. That means that beyond the social characteristics of the teaching situation, we must analyse the nature of the target it aims at. If students see the target as "doing", more than "knowing", then their debate will focus more on efficiency and reliability, than on rigor and certainty. Thus again argumentative behaviors could be viewed as being more "economic" than proving mathematically, while providing students with a feeling good enought about the fact that they have completed the task. Social interaction revisited Social interaction, while solving a problem, can favour the appearance of students' proving processes. Insofar as students are committed in finding a common solution to a given problem, they have to come to an agreement on the acceptable ways to justify and to explain their choices. But what we have shown is that proving processes are not the only processes likely to appear in such social situations, and that in some circumstances they could even be almost completly replaced by other types of interactional behaviors. Our point is that in some cicumstances social interaction might become an obstacle. Some people might suggest that a better didactical engineering could allow to overcome these difficulties; indeed much progress can be made in this direction and more research is needed. But we would like to suggest that "argumentative behaviors" (i) are always potentially present in human interaction, (ii) that they are genuine epistemological obstacles to the learning of mathematical proof. Argumentation and mathematical proof are not of the same nature: The aim of argumentation is to obtain the agreement of the partner in the interaction, but not in the first place to establish the truth of some statement. As a social behavior it is an open process, in other words it allows the use of any kind of means. Whereas, for mathematical proofs, we have to fulfil the requirement for the use of a knowledge taken in a common body of knowledge on which people (mathematicians) agree. Insofar as students are concerned, we have observed that argumentative behaviors play a major role, pushing at the backside other behaviors like the one we were aiming at. Clearly enough, that could be explained by the fact that such behaviors pertain to the genesis of child's development in logic: Very early, children experience the efficiency of argumentation in social interactions with other children, or with adults (in particular with their parents). Then, it is quite natural that these behaviors appear first when what is in debate is the validity of some production, even a mathematical one. What should be questioned is not so much the students' rationality as a whole, but the relationships between the rational of their behaviors and the characteristics of the situation in which they are involved. In order to teach mathematical proof successfully, the major problem appears to be to negotiate the acceptance by the students of new rules, but not necessarily to obtain that they reject argumentation insofar as it is perhaps well adapted to other contexts. Mathematical proof should be learned "against" argumentation, bringing students to the awareness of the specificity of mathematical proof and of its efficiency to solve the kind of problem we have to solve in mathematics. Negotiation is here the key process, for the following reasons:
The solution is somewhere else, in the better understanding of the phenomena related to the didactical contract, the condition of its negotiation, which is almost essentially implicit, and the nature of its outcomes: The devolution of the learning responsability to the students. |