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For concreteness lets keep our discussion to calculus courses where there is a balance of proof and computations (computing limits but also doing epsilon-delta proofs)

I can understand that in more computational based courses one would like to use flow diagrams such as this:

https://www.khanacademy.org/math/ap-calculus-ab/ab-limits-new/ab-1-7/a/limit-strategies-flow-chart

where a student just wants to obtain a bag of tools. In more proof based courses the argument against using them is that it encourages the bad habit of rote memorization and so when a slightly different situation arises, the student will be stuck.

In discussions with other instructors, they told me there is research showing that using such summarizing diagrams is harmful for learning. So I would appreciate if you can also include some references in your answers.

Here are some arguments I can think of for either side.

Good for learning

  • Let me first take the perspective of a good student who actively does problems. A good student could use such a diagram to obtain a bird eye view of the techniques. Then as new techniques and refinements arrive, he/she will update their mental model.
  • These diagrams are not good or bad per se, but only as good as the instructor using them. So ,for example, the instructor can accompany these diagrams with homeworks that refine them and challenges the students to modify the existing techniques.

Bad for learning

  • From the POV of a bad student this diagram is a god send. So it might discourage them from doing problem solving to help them figure out the skeleton of possible techniques and their refinements. They will rely on these diagrams and give up on a problem that doesn't quite fit them. In other words, they will take the summarizing strategies as absolutes.
  • Students will try to save time and focus on learning the strategies and not the material. Therefore, we are depriving them of learning and discovering the techniques for themselves. So they will soon forget that material because they were not challenged with it and didn't form those precious neural connections.

  • In more proof based courses, the emphasis should be on deriving and thinking, so any such diagram will deviate them from that. The instructor should guide the students to form their own mental models, rather than imposing external ones.

Update: I now realize that this goes under the debate of IBL (inquiry based learning) versus procedural based learning.

Q2: Are there any ways we can use strategy diagrams in an IBL setting?

In IBL we want the students to discover many of the techniques themselves. So we avoid giving these diagrams at the start, that way students will be forced to do a lot of problems themselves.

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  • $\begingroup$ (No references, therefore comment): I think these frameworks are helpful. For the weak student, they need them. For the strong student, it won't hurt them. Look at how people use frameworks in other fields (e.g. business analyses). Doesn't mean your brain turns off. But helpful to the weak and doesn't hurt the strong. $\endgroup$ – guest Oct 14 '18 at 21:18
  • $\begingroup$ I would also add that this impulse to create such frameworks is actually a thematic unifying conception to what otherwise is a bag of tricks. You can look at Stephen Wolfram writing integration methods into a flow chart (as a young student, long before his work with computers). I actually was not that sort. I did SO much calculus (every single problem in the textbook plus an extra book) that I was so honed that I did not need flow charts and had instinctive ability to sort of sense what way to go. But rationally, I should have slowed down and created a flow chart. $\endgroup$ – guest Oct 14 '18 at 21:20
  • $\begingroup$ One question is if it is better to introduce these frameworks before or after learning the components or both. I remember learning detailed NPV analysis and having a flow chart of the different components (leveraged beta and all that). And we got it at the beginning and referred to it as we learned concept after concept. It worked reasonably well. Although I don't think it was a difference maker either way. But still impressive to see how we were plowing through this complex flow chart (of the course) concept by concept and that it all went into one equation. $\endgroup$ – guest Oct 14 '18 at 21:24
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    $\begingroup$ Maybe have students create strategy flowcharts, summarizing the ways a class tackled different problems. If they can reason at this higher-level viewpoint, it would surely be beneficial. $\endgroup$ – Joseph O'Rourke Oct 17 '18 at 23:11

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