What is teaching? Something whole, created by man to communicate from one life to another. The way that we embody ourselves, so that others may receive us. A wholly spiritual venture, often mistaken for a quantitative exercise.
To teach is to tell, to put oneself forth almost vulnerably and say, here, here it is what you must know; no--what you must know you must know, trust me; pouring one's heart out by faith in another. It is becoming a parent for however long an apprentice is beside you, and living a mother's life over the course of a week, a semester, or four years. Seeing that person become, or deviate, or rebel.
There are some people who pretend to be teachers, who do things teachers are "supposed" to do: sit behind a desk, stand behind a podium, and most famously, speak at--certainly not to the people. Does a teacher need to do these things? Of course not. What is a teacher if not a poet, musician, or friend? Are dreams teachers? Perhaps it is not about who has taught you, but where you have learned.
Because that is certainly all that teaching is about, not proclamation; that is for the clergy and politicians and ye olde mountebanks; it is the learning that is the stuff. That is the combined element, the end, but also the beginning. Once a thing is truly learned, it cannot be undone, it can only be carried forth, taught, or submersed into the infinite mind.
Once a thing is really learned, what is left? The propensity of something to be taught again.