Autour de l'algèbre: les entiers relatifs et la modélisation algébrico-fonctionnelle

We present two studies that, even if they are located “before” and “after” the teaching of algebra, they aim at filling up two important gaps left by the didactic transposition process of elementary algebra. The first one proposes a motivated and functional construction of integer numbers in an algebraic context ; the second one approaches the problem of the transition from elementary algebra to calculus driven by what we call algebraic-functional modelling. Both studies lead to the experimentation of research and study courses for French and Spanish secondary schools and they identify some constraints and conditions of different level of generality affecting their didactic ecology.