  The learning of mathematics starts early but remains far from any theoretical
considerations: pupils' mathematical knowledge is first rooted in pragmatic
evidence or conforms to procedures taught. However, learners develop a
knowledge which they can apply in significant problem situations, and which is
amenable to falsification and argumentation. They can validate what they claim
to be true but using means generally not conforming to mathematical standards.
Here, I analyze how this situation underlies the epistemological and didactical
complexities of teaching mathematical proof. I show that the evolution of the
learners' understanding of what counts as proof in mathematics implies an
evolution of their knowing of mathematical concepts. The key didactical point
is not to persuade learners to accept a new formalism but to have them
understand how mathematical proof and statements are tightly related within a
common framework; that is, a mathematical theory. I address this aim by
modeling the learners' way of knowing in terms of a dynamic, homeostatic
system. I discuss the roles of different semiotic systems, of the types of
actions the learners perform and of the controls they implement in constructing
or validating knowledge. Particularly with modern technological aids, this
model provides a basis designing didactical situations to help learners bridge
the gap between pragmatics and theory.
