Almost all this thread confuses two things: 1/ being competent on one field 2/ being able and willing to share knowledge especially with newcomers.

To be a good teacher requires both abilities plus the skill to assess the knowledge and interests of ones audience so as to start from the know to lead to knowledge of the unknown. Having both abilities is more the exception than the rule.

I studied maths with teachers who were imposed the Bourbaki style (formal maths). They were good teachers but they were required to teach stuff that they did not really understand. Too bad.

Later, in engineering school (not a math-only cursus) I had as teacher who was a moron who will stay unnamed. He was certainly mastering topology but was unwilling to give examples because it was betraying the generality of the theory. Most people learn from concrete examples and later see in what generalization differs from these examples.

Also, the art to hook an audience without getting off topic is a rare ability. See Conway and Dominus.

-- stefp -- check out TeXmacs wiki