I am looking for long-term (over the course of many semesters) strategies, including specific types of in-class activities, for developing the abilities of students to come up with intermediate steps in solving multi-step problems. Ideal answers to this questions will have references that propose strategies and activities as well as study (either qualitatively or quantitatively) their effectiveness.
Let me describe the problem I would like to address. Students in a junior/senior level abstract algebra course have just learned the proof of Lagrange's Theorem (that the cosets of a subgroup $H$ each have the size of the subgroup and partition the group $G$) and the definition of a normal subgroup (in terms of equality of left and right cosets). They are now asked to prove that, if the order of a subgroup is half the order of the group, the subgroup must be normal. It turns out that most students, even only counting students that without a doubt understand the proof of Lagrange's Theorem and the definition of a normal subgroup, are simply unable to do it by themselves. Almost all students need to be given the hint that, given $a$ not in the subgroup, they should prove that both $aH$ and $Ha$ are equal to $G\setminus H$. Once given the hint, most students are able to both see why the hint is true and why it solves the problem for them.
I want to emphasize that the difficulty is not in lack of knowledge or understanding of the concepts involved, but rather in not having the ability to come up with connecting facts.
This is not an issue that really can be solved within the context of the course, because, in the context of the course, the natural thing is simply to tell the students the hint and have them memorize it. The issue is to develop the students' ability to come up with this hint by themselves.
Presumably, this ability should be developed over the entire course of the students' intellectual (and not only mathematical) experiences starting with elementary school or earlier. However, as a college professor, I am particularly interested in knowing about strategies and activities for earlier college courses, starting with calculus, but especially in elementary discrete mathematics, linear algebra, and introduction to higher mathematics courses.
I am most interested in strategies and activities that help almost all students, not strategies that are particularly effective for the best students.