Here is Michael direct modeling a separate problem. Michael would
rather do math problems in his head, and only direct models when he
can't do it any other way (this is the third separate problem I
tried--the first 3 he answered quickly without direct modeling, so I
had to keep experimenting to find something that would prompt him to
use the counters). Michael understands separate problems, but working with larger numbers (15 and 13) is in his zone of proximal development (he can do it with help, but makes mistakes without help). Anyway, he doesn't have a lot of
practice with direct modeling, and I thought you would find it
interesting what mistakes he made, and what questions I asked to get
him to figure it out correctly (the questions are the scaffolding for the problem).
(be patient, it will take a while to load)