Improvisation on the Piano

I have been working - in my mind, at least - on an outline re: teaching improvisation. Usually my starting point is either a book of some sort, from which I pick up a pedagogical outline, or my own experience. In terms of improv, though, the books I've found seem insufficient. They are either way too theoretical, or way too tune-based. By the latter I am referring to the design where a book throws several tunes and basically says, "Here, play this".

There deosn't seem to be material which walks a student through the process of working on an improvisation, which has started me thinking about the process you go through when improvising. What musical skills and knowledge do you have to have in order to imrpvo successfully? What ear capabilities?

In this, also, my own experience is proving deficient. I don't remember how I started improvising - other than I began by adding in parts to the boring piano accompaniments I was playing for my high school choir.

So, my initial thoughts:

1. Scales and keys play a part. You need to know instinctively what notes fit a particular key.
2. Knowing chords is also a part. You could work off the notes in a particular chord.
3. Some Rhythmic capabilities is necessary - being able to play rhythmic patterns for comping work, for example.
4. A sense of form - you'd want to play differently between the verse and chorus.
5. There's also the non-note-based but oh-so-important performance aspects such as tempo, dynamics, expressions, and articulations. Intonation, too!

So in what order would you teach these concepts? Using what material? At this point I am certain you can't segment the areas listed, but how to mix them together into a cohesive whole is a fairly knotty problem - not to mention the problem of keeping things interesting!

More entries as the thinking warrants.