Abstract:

Non scholae sed vitae discimus (we learn for the life, not for the school) is a guideline that is very often forgotten both by teachers and by pupils. But are these the only stakeholders of teaching? Even if nowadays social nets allow connexions much stronger than in the past, the main points remain the same: in the school and in the university there is an organization that more or less states what must be taught and in most cases establishes also what must be learnt and checked. On the other side young pupils have parents that expect some specific form of teaching, and older students are subject to a continuos exchange of evaluation with other students. Therefore the content of teaching is conditioned by tradition, that in the case of mathematics has a very long history starting with Euclide’s Elements. Is this the best way to ensure lifelong persistence of the knowledge?

Four main points are discussed.
1. Should mathematicians teach only the “kernel” of their subject or should they deal also with the process of mathematization of the real world? Does real world help learning or is it better to use toy examples?
2. Is short term mastery suitable in view of lifelong mastery? What can be recalled when most of the school formation is lost? A well structured knowledge can make a possible recovery easier, but what is its price in time and effort for the teacher and for the student? Are all teachers able to reach this goal?
3. Is the exterior world (pupils, parents, board of education, newspapers…) ready to accept a teacher of mathematics that does not bound himself to pure technicalities of mathematics but speaks also of its occurences in life?
4. Is personalization of teaching through computer expert systems a tool useful in mathematics as it is for example in the learning of foreign languages? Sending mathematical manuscripts, pictures and drawing through photos allows students to communicate at distance in real time, thus improving their capability of autonomous work. Is it a real advantage in comparison with face to face meetings?

Many experiments of connexion of mathematics with real life have been performed, but the problem, at least in our Italian schools, is that pupils have no real knowledge of physics and of economy, what should be very stimulating for learning mathematics. Also spatial perception is very uncertain so that three dimensional geometry is found very difficult and must be reduced to some schemes of linear algebra.

Unlike what could be expected, not always university students of engineering or other scientific faculties are the most successful, since fundamental lifelong subjects happen very often to be better dominated by the students of teaching sciences. This confirms psychological theories that consider mathematics as a discipline that requires both left and right side of our brain.

Keywords:

Life-long learning, teaching of mathematics, lifelong persistence of the knowledge.