La lettre de la Preuve

ISSN 1292-8763

Introduction to the study of the teaching of reasoning and proof: paradoxes

by
Guy Brousseau

The teaching of reasoning, and in particular of logical and mathematical reasoning, poses for the didactician, as well as for the teacher, several well known paradoxical problems. Essentially, the difficulties ensue from three orders of considerations, one of mathematical origin, one of psychological and sociological order, and one of didactical order.

i. The "orthodox" presentation of mathematical texts gives the impression that formal logic (modus ponens, with perhaps a few other tools of logic) is the fundamental and necessary instrument of mathematics, and that the aim of mathematics is to demonstrate that its author has not produced any contradictions (with himself or with known mathematics). Many teachers tend to deduce from this that since mathematical reasoning is the sole means of establishing publicly that a mathematical statement is true, this reasoning must also necessarily describe (or serve as a model for describing) the thinking that correctly constructs mathematics, hence that describes the thinking of mathematicians and of students. As a result, they try to teach directly how to think, and then how to reason as one does in making a proof. They thus confuse the activity and mathematical reasoning of the students with their cultural product: the standard method of communication.
If, on the other hand, one assumes that the natural functioning of thought produces exact knowledge by some processes (rhetorical, heuristic, psychological…) which cannot be reduced to the presentations and notations that are most convenient (for mathematicians and their research), then what are those processes, and how can they be realized? or how can someone be led to realize them?

This paradoxis akin to the one encountered by beginners in logic: they wish to construct formal logic formally, in other words, to have an autogenesis on which all science can then be based. They must swiftly be disillusioned and learn first to distinguish the logic of the builder from the logic he is building in order to enter conveniently into the study of mathematical logic.

ii. Humanists must postulate that "reason is the best shared thing in the world" in order to establish the universality of their philosophy. Their idea of a human leads them to consider that everyone must have available a personal system with the help of which, as a last resort, he interprets and evaluates what is said to him, and that everyone must recognize in others this capacity and this right. Any influence (coercion seduction, deception, etc.) which cannot be controlled by the person being subjected to it diminishes the person exercising it as much as the person submitting to it. The means of validating influences is clearly "reason", the reason the two protagonists have in common, and the individual reason of each of them. The legitimate means of influencing another person is thus to convince him, and to convince him using his own criteria and knowledge. Respecting someone else doesn't mean accepting his beliefs without examining them, but rather debating them, if necessary, under certain rules.
But then, is being endowed with reason a condition and prerequisite not only for all teaching of mathematics, but also for all legitimate influence and in particular for education?

In these first two questions the problem is comparable: how to teach a student "how to reason" if he doesn't already know it, when reason is exactly what he needs to have available in order to understand and learn what it is to reason? How to teach him to accept conclusions only by exercising his own judgment and without submitting to outside influence when he has not yet acquired this judgment…?

iii The paradoxes of "constructivist" learning (obstacles) and those of the didactical contract (the teacher must not unveil what he wants the student to do by himself) have been advanced by the Theory of Situations in Mathematical Didactique. They are particularly thorny in the case of the teaching of reasoning.
Examples:
· How to make it so that the student feels "responsible" for his answer to a problem when the problem has been posed to him by his teacher precisely for the reason that the teacher thinks he may not be able to do it? What businessman would accept a contract he didn't know how to satisfy?
· Can the teacher use teaching techniques – that is, manipulate the reasons for knowing that the student should ultimately use? Otherwise stated, what are the limits on the teacher's manipulation of the student? Etc.
More concretely, the didactical problem differs according to whether one considers that the teacher "teaches a piece of knowledge" or that he causes this knowledge to be "understood" and "causes knowledge and practices to be appropriated."
In the former case, it suffices to expose the student to the piece of knowledge in an order that economizes maximally on time spent teaching and is self-consistent: systematic, rational, logical, or better yet axiomatic orders realize these conditions. The teacher leaves the student in charge of "understanding", learning and using the knowledge that has been thus passed on to him.
In the second case, with the obligation to have the student "understand", "learn", and "use" mixed into the responsibilities of the teacher, the didactical contract seems more appropriate to the humanistic project. But, especially if one interprets these three requirements as being equivalent to "make the student produce the knowledge himself" and assigns to the teacher the responsibility for achieving this result, the challenge runs into the paradoxes described above.
Because if the teacher cannot modify the orthodox presentation of mathematics, nothing is left for him but to try to correct and complete the first solution by additives:
· by redundancies, repetitions, comparisons and other analogies assumed to produce memorization of the texts
· by "explanations", representations, illustrations, commentaries and other circumlocutions by which he attempts to attach the students' knowledge to that of his axiomatic mathematical exposition
· by more or less "open" exercises and problems which he hopes will stimulate well enough a mathematical activity comparable enough to his model: the activity of mathematicians
· and/or especially by formal (non-didactical) pedagogical arrangements designed to combat one of his numerous so-called "bad tendencies":

o talking, which prevents the student from doing it himself
o having the student talk instead of having him act
o teaching instead of letting the child develop in a way that is consistent with his spontaneous development
o constructing knowledge according to the culture instead of letting the child construct "his" knowledge with all due creativity, novelty and inventiveness
o concentrating the student's attention on himself rather than giving him over to the beneficial influence of small groups working freely
o offering scholarly subjects rather than using the ones from the rich terrain of technical and social activities in the student's "natural" surroundings
o preferring boring scholarly subjects to games which are attractive and consequently instructive
o choosing subjects that are theoretical and consequently useless rather than subjects that are practical and hence intelligible and useful
o establishing a relationship between the work accomplished by each student and the instructional project he has been assigned and by doing so discouraging the students from figuring out the differences between them
o …

Some of these exhortations have an annoying propensity for dissuading the teacher from pursuing his profession. Nonetheless, they are a collection – albeit a "naive" one from the didactical point of view – of respectable "observations" coming from other disciplines, and the dilemma thus remains. Their principal defect is wanting to impose themselves or justify themselves without taking account of the didactical dimension and circumstances. None of these prescriptions is valid absolutely and in all circumstances, none is applicable to all knowledge to be taught without a didactical conversion that takes into account the nature and specific characteristics of the knowledge being taught.
All children try to influence their environment and, under the obligation of following adults' decisions, very early develop rhetorical strategies for satisfying their desires. It is during this period that they begin to develop simultaneously
· an entire hierarchy of means of influence, permitted or forbidden, explicit or inexpressible, in a variety of types of interaction
· and consciousness of their status as a human being with the ability to exercise their own will.
It follows that one should undertake very early to teach children the practice of "situations of rational validation." These are the situations where two players cooperate dialectically with the goal of establishing or rejecting the truth of an assertion. They cooperate, but without concessions, the one proposing, the other opposing him whenever he sees the need, until he arrives at the point of sincerely accepting the evidence.
But what is the type of situation that can require and permit the development of different axioms and theorems of logic and make the student conscious of them?

Guy Brousseau