Teaching functions in a graphic milieu: What forms of knowledge enable students to conjecture and prove?

In many countries, the first concepts of calculus (such as functions)are taught by looking at examples, noticing their properties and generalizing from them in some implicit ways. Students have no means to discuss the general truth of a statement, or to examine the validity of a theorem, relative to the mathematical field. This knowledge is nonetheless required by teachers at the university level. The question is, therefore, if it is possible to organize activities for beginning calculus students, which would nevertheless lead them to working on statements and validity of theorems. This paper presents a teaching approach related to the concept of function, which aimed at leading students working within a graphic milieu to producing, discussing and testing the validity of mathematical statements and theorems. The intention of the approach was to use the procedural aspect of the graphs to provide a favorable milieu for linking the intuitive and the formal knowledge(such as required at the university for establishing proofs). The approach was experimented with a group of students. After the experiment, the students indeed became able to think of functions as objects and to engage with questions of validity of mathematical statements.This revised version was published online in October 2005 with corrections to the Cover Date.