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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.

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    $\begingroup$ For students to think we need to require thinking at every level of their education. They should expect we expect them to know prequisites. For the problem at hand, surely additional intuition about bijections would increase the chances of them finding the proof themselves. The idea that bijections are useful for counting problems is something that could have been more emphasized in previous courses or perhaps the current course. Still, some of these basic proof ideas are only simple because we already know them and we deceive ourselves into thinking students in the wild will do the same... $\endgroup$ – James S. Cook Nov 30 '19 at 3:28
  • $\begingroup$ You write "increase the chances" - surely it's not good enough to just increase chances from 25% to 35%. Even your garden variety math researcher fills gaps that are an order of magnitude bigger with just as little experience of previous solutions (because in their case there are none) regularly. I feel like, as an educational community, we are failing in most cases to pass this skill to our students and neither acknowledging nor trying (except at the margins) to fix it. $\endgroup$ – Alexander Woo Dec 1 '19 at 2:29
  • $\begingroup$ I could answer what I do, but I could not give an "ideal answer...[with] references...," which you and this site would prefer. I could elaborate how I came to do it, some grounding for it if not forgotten, my experience with it (over 30 years), the pitfalls students will have to climb out of, and the teacher's role in helping the students. I feel the goal "to pass this skill to our students" misses the mark. Problem-solving probably is not such a skill.... $\endgroup$ – user1527 Dec 1 '19 at 15:18
  • $\begingroup$ @user1527: Of course we teach these component skills, and our students learn them to some extent. But more than half of our students still cannot find for themselves the proof that subgroups of index 2 are normal. $\endgroup$ – Alexander Woo Dec 1 '19 at 21:33
  • $\begingroup$ @AlexanderWoo I would agree we fail to encourage creativity in proofs. Much of this is based on supposedly good pedagogy which is baked into the metrics on which we are judged as professors. For example, "did the professor use the textbook" or "were the tests representative of the material covered ?" These sort of evaluation questions and/or bad leadership may drive us towards merely regurgitating the book's proof. Instead, it is so much better to start with a sketch and fill in details organically as part of a conversation with students. That is dangerous and requires a lot more prep... $\endgroup$ – James S. Cook Dec 2 '19 at 1:02
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I believe an appropriate strategy that will help students achieve the goals you state is inquiry based learning (IBL) (in math, and ideally, other subjects, as well).

Outcomes were evaluated in this study/research on IBL in mathematics at the college level, which provides background describing IBL in mathematics, as well as outcomes from a two-year experiment, from student and faculty surveys, as well as students' performance outcomes.

You can find many more resources on IBL by searching inquiry based learning in mathematics; college.

Additionally, at the college level, I have found that incorporating, at the Freshman level, having students acquire and read How to Solve It, by George Pólya has been very helpful, along with supplementing an initial proof based course with How to Prove It by Daniel Velleman (pdf).

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    $\begingroup$ I'm running an IBL style course now, and I find that it is not enough. The better students in the course get better. The lower half to two-thirds never manage to do anything without constant hints and don't seem to gain anything. It could be that I don't know how to run the course properly. $\endgroup$ – Alexander Woo Nov 30 '19 at 2:29
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    $\begingroup$ What ages, or level of students, are you running an IBL style course with? I do suspect that the earlier students encounter this approach (elementary?, high school?) the better it is conducive to students thinking and problem-solving independently. I wouldn't rule it out at the college level, until an undergrad math department is invested in freshman to senior IBL, but it can be difficult to get everyone on board with that! $\endgroup$ – Namaste Nov 30 '19 at 2:38
  • $\begingroup$ It is the sophomore/junior level introduction to higher mathematics course that I'm running IBL style. I agree it might be too late. Unfortunately, given the abilities of the students I have who go on to become high school teachers, I wouldn't trust most of them to run an IBL style course, and budgetary constraints means that we are teaching first/second year university courses either to 200 (with 2 TAs) or 55 (with no TAs), neither of which really support IBL style courses. $\endgroup$ – Alexander Woo Dec 1 '19 at 2:16
  • $\begingroup$ Alexander: Have you ever heard of The Moore Method?, typically more appropriate for upper-level undergrads and early grad students. $\endgroup$ – Namaste Dec 1 '19 at 16:40
  • $\begingroup$ Yes I have heard of the Moore Method. I know precisely what would happen if I tried to run a fairly pure Moore class at my university (even in an upper level class), which is that 75% of the students would be observers completely unable to contribute (despite a reasonable amount of effort). $\endgroup$ – Alexander Woo Dec 1 '19 at 21:22

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