A good teacher should know his/her audience : what do they know? what do they expect? what is the goal?
A same lecture will be one's joy and another's torture.
You definitely won't treat the same subject in the same way in front of would-be mathematicians, would-be engineers, would-be biologists, would-be economists, etc! [and that's just mentioning students, not pupils...]
There are audiences in front of which you won't give any definition, any lemma, any theorem. You'll just take examples, first very down to earth to show how to compute/visualize something, then a little more general, but still not abstract, to show off the magic.
And even in front of mathematicians, you should beware : are you talking to a pack of specialists of the subject or in some sort of colloquium? On that matter, I can link you to William Thurston's "On proof and progress in mathematics" (which I recently read because someone pointed it out in an unrelated question).
90% of teaching isn't about the core of the subject, but being receptive to the people you're dealing with.