If we in Higher Ed want to promote abstract representation in learning and if I'm right in my previous post that students learn to do this (or not) well before they go to college, then the utilitarian thing to be doing would be to help the schools encourage this type of change in students' perspective. I am struck, however, that much of what the current intervention does, which happens via student teachers in the schools, achieves quite the opposite.
Lessons are planned, with the "scaffolding" carefully constructed. There is a lot of structure put into place. Students are encouraged to conform to that structure. There may be many benefits to the approach. But I don't believe it encourages learning as abstract representation. For that there needs to be open ended discussion. And there needs to be work on "hard" problems where the solution is "found" most likely through introspection. (Here I'm thinking about math where there are the solutions. There has been a lot written of late about teaching students to deal with messy problems that don't have a "right answer" but only different ways to approach the problem. That is fine for higher ed, but in K-12 before one gets to that point, it would be good for students to spend some significant time on framing where there is a confirming method of success --- the solution has been attained and that is testable. )
The difficulty with hard problems is that people turn away from them, because they don't have a sense that they can make progress --- getting stuck is not fun and from there one often gives up. So the other thing that clearly needs to be learned is persistence -- staying with the idea till it bears fruit or until it is clear it is a dead end.
How do we go about doing this? My broad strokes, unscaffolded idea is that our students in higher ed, those majoring in things other than education, become mentors for students in the schools. Some of that bonding has to occur face to face, most likely during the summer, but with neighborhood schools that would be possible during the regular academic term. And then the rest of the time this would happen online in some sort of asynchronous discussion area. Our students, in turn, would be mentored by faculty, not on the method of teaching, but rather on the problems being discussed and on the approach at framing them. Somehow the teachers in the schools would have to partner so these discussions tied into what was being covered in the more formal curriculum.
Obviously this needs a lot of fleshing out, but the key point is that it is our students and not our faculty who are primary. That is the only way to make the interventions scale, and have some utilitarian impact.
Math might be a good area to start in because of the shortage of teachers, so the interventions might be welcome. There then might be other materials to accommodate so as to give the discussions focus.