The experience of a group of teacher-researchers, starting from the analysis of teaching practice, has allowed to spotlight some difficulties in learning the concept of limit, but also highlight the presence of pre-conceptions in very young pupils. The theoretical reflection which followed focused, on the one hand, on a historical-epistemological analysis; on the other hand, on planning research activities aimed firstly at progressively establishing the concept of limit in students and secondly at outlining the basics of teacher's classroom activities on this subject. Having taken into account the complexity and importance of such a concept, we believe that a gradual approach is appropriate for pupils, trying to develop in them an understanding from the first years at school, by taking the pre-conceptions of primary school children as a starting point. In particular, we wish to emphasize the role and importance of approximation in this itinerary, and, more in general, the need to modify, through training courses, the practice of the teachers of mathematics.

mathematics, limit, approximation, measure, pre-conception.