Balacheff N., Gras standard (1991) Treatment of refutations : aspects of the complexity of a constructivist approach of mathematics learning. In : Von Glasersfeld E. (ed.) Radical constructivism in Mathematics Education (pp.89-110) Dordrecht : Kluwer Academic Publisher.
© Kluwer Academic Publishers

Presentation One of the basic hypotheses of research in mathematics education is the constructivist hypothesis which asserts that the student himself participates actively in the construction of his own mathematical knowledge. The starting point for the developmental process according to this view, is the experience of a contradiction which is likely to provoke a cognitive disequilibrium: It is the overcoming of such a contradiction which results in new constructions. I would like to emphasize here the consistency of this Piagetian developmental model with the model proposed by Lakatos (1976) to describe the growth of mathematical knowledge through the dialectic of proofs and refutations. But what Lakatos work shows is the complexity of the overcoming of a contradiction in mathematics due to the diversity of the possible ways of dealing with a refutation. The historical study he presented shows the importance of the social dimension of this dialectic. This dimension also plays an essential role in the learning process taking place in the mathematics classroom. It appears at two levels: Students have to learn mathematics as social knowledge. They are not free to choose the meanings they construct. These meanings must not only be efficient in solving problems, but they must also be coherent with those socially recognized. This condition is necessary for the future participation of students as adults in social activities. After the first few steps, mathematics can no longer be learned by means of interactions with a physical environment, but requires the confrontation of the student's cognitive model with that of other students or of the teacher, in the context of a given mathematical activity. Especially in dealing with refutations, the relevance of overcoming is what is at stake in the confrontation of two students' understandings of a problem and its mathematical content. In this paper, we study the complexity of students' treatment of a refutation by means of an experimental approach within the context of solving a given mathematical problem. The analysis of the data collected takes into account both the cognitive and social dimensions of the phenomena observed. Paper's table of content The problem of contradiction Conditions for the awareness of a contradiction Treatment of refutations Students' treatment of a counter-example The experimental setting: A social interaction about the number of diagonals of a polygon Students' solutions and their foundation The treatment of counterexamples Strategies in the treatment of a counterexample: Stability and dispersion Conclusion outline Three factors appears to determine students' choice in their treatment of a refutation: The analysis with reference to the problem itself. It gives a central place to the discussion on the nature of the objects concerned, and thus on their definition. This analysis potentially leads to any type of treatment which could be considered. The choice the students make can be understood only in the light of a local analysis or of the specific situation and of the knowledge of the students. The type of treatment can change in the course of the problem-solving process: Modification of the definition followed by the introduction of a condition or a modification of the initial conjecture when the conceptions have been stabilized. The origins of the choice between the introduction of a condition and the search for a specific solution, and the modification of the conjecture, cannot be traced in the data gathered. What could be conjectured is that when the refutation brushes aside a wide range of polygons (with respect to the students' conceptions), then a modification of the conjecture is decided which consists in a specific formula. The analysis with reference to a global conception of what mathematics consists of. This could be a serious obstacle to some of the treatments of a refutation: Refusal to treat the counterexample as an exception, refusal of a solution which cannot be expressed by a unique formula, etc. . The analysis with reference to the situation. What is in question is mainly the didactical contract which leads students to favoure some treatments of the counterexample (definition game, riddle game in which they abandon their solution quite easily) or constitutes an obstacle to others (refusal to introduce a condition because it has not been stated in the problem statement). Whatever is the case, the role played by the experimental contract should not be over-estimated. In particular, it should be noticed that the fact that treating the counterexample as an exception or its rejection, during phase II of the experiment, confirms the result already known about the fact that one counterexample is generally not sufficient for students to call a conjecture into question (Burke 1984, Galbraith 1981). The fact that this treatment has been observered in other, and quite different, experimental settings, suggests that it is not specific to the experimental contract in this situation as it might be conjectured at first. One of the questions we have examined is that of a possible influence of the type of conjecture on the choice of a treatment of a refutation. One hypothesis is that if the conjecture is false, then its rejection or its modification or a revision of the definition should be dominant, but that, if the conjecture is correct, then the rejection of the counterexample should be dominant. Actually, we have observed that conjectures like f(n)=n or f(n)=2n are abandoned after their refutation. Such conjectures are very fragile because they are verified by only one polygon. It is quite different when a conjecture like f(n)=n/2 when it is supported (explicitly or not) by the conception of a polygon as a regular polygon and the diagonal as a diameter. In this case it is verified by a large set of polygons. We have checked the type of treatment of the counterexample against the type of foundation of the conjecture: it appears that no treatment of the counterexample is privileged, whether the foundation of the conjecture is naive empiricism or a thought experiment. We have also examined the case of the correct conjectures. It should be noticed that they implicitly refer to a correct conception of polygon and diagonal. These conjectures are the result of a process which has occupied all the first phase of the experiment, whether they have been constructed deductively or are the result of a dialectic between successive attempts and their refutations. The treatment appears to be far less varied than in the case of the conjecture f(n)=n/2. A first type of treatment which is dominant is the rejection of the counterexample after its analysis relative to the students' conceptions, a second type of treatment is to consider the counterexample as an exception or to introduce a condition (actually this last treatment could be a way for some students to escape particular cases). Whatever is the level of proving involved in the problem-solving process, it appears that the robustness of the students' conceptions and the existence of a large domain of validity of their conjecture lead to privilege the treatments which consist of keeping aside counterexamples. On the other hand, we have found that in the case of the conjecture f(n)=n/2, the limitation to even polygons is associated with an important uncertainty about the definition, which is often reconsidered after the refutation. Mathematicians that Lakatos considers, share almost the same rational background. In the case of the students the background is not the same: Naive empiricism, or pragmatic validations, can be the basis of their proof and can constitute the roots of their belief in the truth of a statement. If this is so, the discussion of the criticism of a counterexample or of the modification of the conjecture could appear less relevant to the teacher than the discussion of its background. But, how can we escape the fact that faced with a counterexample produced by the teacher, students claim that it is a particular case when in fact, what should be questioned is the naive empiricism on which their conjecture is based? At a higher level of schooling this problem can still appear: A student may be discussing the legitimacy of a counterexample, when it is his or her understanding of the related mathematical knowledge that should be questioned. To base the learning of mathematics on the students' becoming aware of a contradiction requires that we take into account the uncertainty about the ways they might find to overcome the contradiction. If, as I believe, we cannot assume that there is a strict determinism in cognitive development, what could be the role played by a particular situation? Rather the teacher's interventions will be fundamental. The way he or she manages the teaching situation may bring the students to see that their knowledge and the rationality of their conjectures must be questioned and perhaps modified, because no ad hoc adaptation of a particular solution, or its radical rejection, can by itself lead to a conceptual advance.