From Indiscrete Thoughts, by Gian-Carlo Rota:
The best introduction to mathematics is not achieved by rigorous presentation. No one can learn calculus, linear algebra, or group theory by reading an axiomatic presentation. What one wishes is a feeling for a piece of mathematics. Let the student work with unrigorous concepts that lead as quickly as possible to a half-baked understanding of the main results and their applications. Rigorous presentation can occur later, as an afterthought, to be given only to a fraction of the students, those who develop a genuine interest in mathematics.
It is a dreadful mistake to expect that everyone who learns an area of mathematics should be subjected to the learning of the foundations of that particular area. This mistake was made in the sixties, with the introduction of new math, a step backwards in the teaching of mathematics from which we are a long way from recovering. The student needs to develop an understanding, however partial and imperfect, by descriptions rather than definitions, by typical examples rather than grandiose theorems, by working out dozens of menial exercises.
Most mathematicians who teach mathematics fail. They bask in the illusion that the majority of their students should become mathematicians, or their teaching is wasted; or in the illusion of immediate, effortless understanding, the illusion that it suffices to present the “facts” and let students figure out the “sense” of the mathematics, a sense that substantially differs from the “meaning.”
(This post first appeared at my other (now deleted) blog, and was transferred to this blog on 22 January 2021.)