When introduced at the wrong time or place, good logic may be the worst enemy of good teaching.

Look around when you have got your first mushroom or made your first discovery: they grow in clusters.

Pedantry and mastery are opposite attitudes toward rules. To apply a rule to the letter, rigidly, unquestioningly, in cases where it fits and in cases where it does not fit, is pedantry. [...] To apply a rule with natural ease, with judgment, noticing the cases where it fits, and without ever letting the words of the rule obscure the purpose of the action or the opportunities of the situation, is mastery.

In order to solve this differential equation you look at it until a solution occurs to you.

You should not put too much trust in any unproved conjecture, even if it has been propounded by a great authority, even if it has been propounded by yourself. You should try to prove it or disprove it.

Mathematics is being lazy. Mathematics is letting the principles do the work for you so that you do not have to do the work for yourself

It may be more important in the mathematics class how you teach than what you teach.