The Theory of Didactical Situations has had a central position in French mathematics education research since the early seventies. A major component of this theory is the didactical contract, a completely implicit but highly powerful aspect of the relationship between teacher and student. In this article we relate the series of tutorial sessions which provoked the original formulation of that theory, and in which the theory was validated by its first application. Gaël was an intelligent child who was failing exclusively in mathematics. He was one of nine cases studied between 1980 and 1985 (at the Bordeaux COREM ). After observing him in class and offering him various learning situations, both didactical and adidactical, we arrived at the hypothesis that Gaël was implementing an strategy of avoidance of the "conflict of knowing" which we characterized as "hysteroid type avoidance", whereas the others exhibited "obsessional type avoidance" (note that these behaviors should not be confused with the psychiatric categories of the same name, which are serious personality disorders.) It was possible to offer psychological explanations for this behavior, but they did not provide the means for correcting the avoidance, and they focused the interest of the researchers on a characteristic of the child or on his competencies, rather than remaining at the level of his behavior and the conditions which provoked it or which might modify it. This behavior demonstrated the refusal, conscious or not, of the child to accept his share of the decision-making responsibilities in a didactical situation and hence to learn while working with an adult. It made it possible for the experimenters to explore and understand the constraints of the didactical situation, interpreted as a "didactical contract". It is the simulacrum of a contract, an illusion, intangible and necessarily broken, but a fiction which is necessary in order for the two protagonists, the teacher and the learner, to engage in and carry out the didactical dialectic. The didactical means to get a student to enter into such a contract is devolution. It is not a pedagogical device, because it depends in an essential way on the content. It consists of putting the student into a relationship with a milieu from which the teacher is able to exclude herself, at least partially (adidactical situation). The mechanism implemented was devised to engage Gaël progressively but explicitly in a challenge in which the teacher could be "on the student's side." This situation subsequently revealed itself to be one of the fundamental situations of subtraction.