This is my first time on the wikipedia web site. I was poking around and saw this page. I thought my contribution could be sharing an observation as a jazz educator.
Over the years I have tried many different approaches to teaching kids to improvise. Blues scales, modes pentatonic scales etc. I have found that the most beneficial bit of information is basic chord structure. By teaching students that chords are built in thirds and by practicing a universal triad (13th chord) in different key signatures, they understand note choices better. Then they plug in the formulas for Major 7th, Dominant 7th, minor 7th etc. This then leaves choosing rhythms and styles.
As usual, music and teaching are always a work in progress. What works with one group of kids may need to be modified for the next group.
Please share your thoughts.
The most complicated part of improvising is moving outside of the chord and/or key. Students must poke around and learn what works and what doesn't. You cannot simply tell someone what they should play.
If the musician is doing improv as a solo during a song, they should examine the melody and notice what notes of the melody are outside the chord, and try and incorporate those notes into the solo also.