I have found that song-based improvisation isn't always the best way to start students improvising. With students who have some classical experience, the stumbling block is often not their understanding of chords or their technique, but rather their own inhibitions.
Some of my students respond to modal improvisation. I always start with a minor mode, because they are easier to get interesting sounds out of, and . I have them play a simple ostinato in the left hand, maybe just the root, or octaves. I might point out where the "juicy" notes are in the scale, and I might introduce the idea of a rhythmic motif, or maybe sequencing. Depending on the student, I might use aeolean, phrygian, or dorian. Mixolydian can be good, too. Sometimes i use harmonic minor or the 5th mode thereof.
Another thing i do is to use a minor i iv V i progression, one chord per measure, to introduce "playing the changes". We begin with the harmonic minor scale in the right hand (G harmonic minor has a nice shape). We divide the progression into two phrases; each phrase is 4 quarter notes followed by a whole. The first phrase begins on beat one of the i chord (on a chord tone), and ends on beat one of the iv chord (also on a chord tone). The second phrase similarly begins on the V chord and ends on the i chord. Then we use different rhythms, including faster notes. After a week or two, I have them use natural minor on the i and iv chords, and harmonic minor only on the V chord.
Meanwhile, they've been using a lead sheet to play a minor key tune like "Beautiful Love", "Black Orpheus", "You Don't Know What Love is", or something else that doesn't stray far from the key. We note that most of the chords are contained in the natural minor, and the few that arent (like the V chord) probably have one note that needs to be altered. Start with 2-bar phrases. Move on to 4-bar phrases.
I haven't totally worked out this method, but it seems to be working pretty well.
Topic: Teaching improvisation