In my opinion, the answer is a resounding yes!!
First of all, what is improvisation? Well, it's composition, but without writing it down. Generally, most people would agree that composition can be taught, so if composition can be taught, why not improv?
Assuming it can be taught, what does it mean to teach improvisation. Well, like anything else in music, there are a million different things to look at. There's form & harmony & theory. Then there's styles, and then there's story-telling. On top of that, you can look at things melodically, harmonically, orchestration-ally.
For me, I'm interested in styles mostly. So how do you understand one? Well, first you have to look at the scales used in the styles. Take the classical style for example, generally it uses either the Major scale or the Minor scale. So first there's this world of scales ... then from there, you have your world of harmony and triads based upon those scales. Almost every style has some kind of tension & resolution, so you can look for that as well.
Today, working with Hillary, we did a little bit with folk music (she asked if we could work on styles and I was a little thrown off... so when in doubt, I thought the pentatonic scale would work - after all, it's applicable cross-style, and the blues scale is a modified pentatonic, so that covers blues & rock as well as folk music). First we went over the basic scale, the notes, what degrees of a classical scale they would be (In this case, EGABDE - 134571). From there we went into story-telling I started the story with a one liner about an Indian (native american) chief, followed by a musical line. She then followed and we went on, basically ad libbing for a few more turns.
I hadn't really though of the story telling approach, but with purely instrumental music, it makes a ton of sense (I think) & it's basically what we do when playing classical music (part of what draws me so much to Steven Isserlis and his playing is his focus on the composer and the story they tell through the piece).
Any thoughts? anyone worked on improv. here, either on your own, or with teacher? If so, what did you do and how effective did you think it was?