The construction of a didactic itinerary of calculus starting from the students concept images (age 16-19).

It is well known that calculus is a topic difficult to teach because of educational and epistemological reasons. The Italian programmes enforced by the Ministry of Education for the age range 16-19 years contemplate calculus in many types of schools. These programmes are precise for the content, but do not suggest any methodology; so the freedom left to teachers mainly concerns this last point. We will describe the main features of a didactic itinerary which combines compulsoriness for the contents and freedom for the methodology. The main points of this itinerary are as follows. (i) We approach calculus starting from the ‘pre-knowledge’ of students and proposing problems set inside mathematics and outside mathematics (physics, philosophy). In this way we hope to provide students with motivation and to offer them stimuli for constructing concepts. (ii) Afterwards we attach the traditional programme and construct the concepts using the textbook mediated by worksheets we prepared. In these worksheets there are problems and exercises: problems are aimed to construct concepts, exercises are aimed to fix concepts. The final step of the work consists in giving the students examples of applications. So the complete scheme of work is: Construction→formalization →application.