Generally speaking my best teaching has not resulted from a kind of effortful deliberateness. It has come when I have allowed modes of attentiveness and intuitiveness to come together in a kind of play that goes in its own direction; a direction that, when viewed in retrospect, turned out to be propitious. This is when I was the most effective. This is also when I was the least focussed on predetermined pedagogical paths and on self-conscious carefulness. Good jazz players and good teachers make mistakes, and have bad days, of course. This does not alter my firm belief that we are at our best when we are improvisers, not when we are corporate linear planners.Jim Neyland wrote this toward the end of his delightful article Playing outside: An introduction to the jazz metaphor in mathematics education.
To me, good teaching is like conducting an orchestra of learning. The teacher and students are responding to one another in such an organic way that it's hard to imagine how the teacher makes her decisions.
If you think math is the one subject that just can't work this way, perhaps Neyland's opening will change your mind a bit:
The trouble with mathematics is that it looks a little more logical and consistent than it is. Mathematics has a universally recognised exactitude. It also has an inexactitude that tends to remain concealed. Mathematics contains what we might call ‘illogical truths’; that is, truths whose apprehension requires, not logical reasoning, but insight and imagination. These ‘illogical truths’ are experienced as surprises; as malformations. ... The person who studies mathematics is more than a logician. The study of mathematics has an inescapable element of unpredictability. And, mathematics endlessly escapes being captured within the logical frames of any mathematical structures.[Thanks to Malke Rosenfeld of Math in Your Feet for pointing me to this inspiring article.]