In this chapter, we examine the relevance of and interest in including some instruction in logic in order to foster competence with proof in the mathematics classroom. In several countries, educators have questioned of whether to include explicit instruction in the principles of logical reasoning as part of mathematics courses since about the 1980s. Some of that discussion was motivated by psychological studies that seemed to show that "formal logic … is not a model for how people make inferences" (Johnson-Laird 1975 ) . At the same time, university and college faculty commonly complain that many tertiary students lack the logical competence to learn advanced mathematics, especially proof and other mathematical activities that require deductive reasoning. This complaint contradicts the view that simply doing mathematics at the secondary level in itself suffi ces to develop logical abilities.

This paper discusses the use of the Theory of Didactic Situations (TDS) at university level, paying special attention to the constraints and specificities of its use at this level. We begin by presenting the origins and main tenets of this approach, and discuss how these tenets are used towards the design of Didactical Engineering (DE), particularly adapted at the tertiary level. We then illustrate the potency of the TDS-DE approach in three university level Research Cases, two related to Calculus, and one related to proof. These studies deploy constructs such as didactic contract, milieu, didactic variables, and epistemological analyses, among others, to design Situations at university level. We conclude with a few thoughts on how the TDS-DE approach relates to other approaches, most notably the Anthropological Theory of the Didactic.

It is widely attested that university students face considerable difficulties with reasoning in analysis, especially when dealing with statements involving two different quantifiers. We focus in this paper on a specific mistake which appears in proofs where one applies twice or more a statement of the kind "for all X, there exists Y such that R(X, Y)", and forgets that in that case, a priori, "Y depends on X". We analyse this mistake from both a logical and mathematical point of view, and study it through two inquiries, an historical one and a didactic one. We show that mathematics teachers emphasise the importance of the dependence rule in order to avoid this kind of mistake, while natural deduction in predicate calculus provides a logical framework to analyse and control the use of quantifiers. We show that the relevance of this dependence rule depends heavily on the context: nearly without interest in geometry, but fundamental in analysis or linear algebra. As a consequence, mathematical knowledge is a key to correct reasoning, so that there is a large distance between beginners' and experts' abilities regarding control of validity, that, to be shortened, probably requires more than a syntactic rule or informal advice.RÉSUMÉ. Les difficultés de manipulation, par lesétudiants, desénoncés contenant deux quantificateurs différents, rencontrés dans de nombreux raisonnements en analyse, sont bien attestées. Nous nous intéressons plus spécialement dans cet articleà une erreur qui apparaît dans certaines preuves lorsque l'on applique deux fois ou plus unénoncé de la forme "pour tout X, il existe Y tel que R(X,Y)" et que l'on oublie que dans un tel cas, a priori, "Y dépend de X". Nous analysons cette erreur d'un point de vue logique et d'un point de vue mathématique, puis nous l'étudionsà travers deux enquêtes, l'une historique et l'autre didactique. Nous montrons que les professeurs de mathématiques soulignent l'importance de la règle de dépendance pouréviter ce type d'erreur, tandis que la déduction naturelle dans le calcul des prédicats fournit un cadre de référence logique pour analyser et contrôler l'usage des quantificateurs. Nous montrons que la pertinence de la règle de dépendance dépend fortement du contexte: pratiquement sans intérêt en géométrie, elle est toutà fait fondamentale en analyse et en algèbre linéaire. De ce fait, les connaissances mathématiques sont la clé d'un raisonnement correct, si bien qu'il y a une grande distance entre le débutant et l'expert concernant le contrôle de la validité, que quelques règles syntaxiques ou quelques conseils informels ne permettent vraisemblablement pas de réduire.

In mathematics education, it is often said that mathematical statements are necessarily either true or false. It is also well known that this idea presents a great deal of difficulty for many students. Many authors as well as researchers in psychology and mathematics education emphasize the difference between common sense and mathematical logic. In this paper, we provide both epistemological and didactic arguments to reconsider this point of view, taking into account the distinction made in logic between truth and validity on one hand, and syntax and semantics on the other. In the first part, we provide epistemological arguments showing that a central concern for logicians working with a semantic approach has been finding an appropriate distance between common sense and their formal systems. In the second part, we turn from these epistemological considerations to a didactic analysis. Supported by empirical results, we argue for the relevance of the distinction and the relationship between truth and validity in mathematical proof for mathematics education.