integrative exercises, Aha! exercises, real-world exercises, et cetera

[JohnDenker]

Background:

1. We were discussing equation hunting. We agree this is a Bad Thing. In the real world it is horrifically laborious, if it works at all. There are just too many equations to permit efficient hunting.
2. We face a nasty pedagogical problem: Conventional "end of chapter" exercises invite and reward equation hunting. Exercises keyed to a particular section within the chapter are even worse.
3. We face an obvious, firm constraint: We cannot possibly take away these keyed-to-chapter and keyed-to-section exercises, because teachers depend on them. It's how they teach.
4. As we discussed, such exercises should be considered the first rung on a ladder.

That suggests a path toward alleviating the problem: -- We don't want to take away the first rung of the ladder. ++ Instead we need to add the higher rungs ... and then encourage people to climb.

This leads to some specific action items and opportunities for improvement.

Suggestions:

• For starters, I call attention to the rating system described on page 2 of book II, namely the blue dots and the CR label. It would be nice to have a richer set of labels or badges, perhaps something like this:
  • 1 or more hourglass ⌛ badges, to denote how laborious the exercise is. Even a straightforward exercise can take time.
  • 0 or more lightbulb 💡 badges, to indicate "Aha!" exercises that require insight, creativity, and outside-the-box thinking.
  • 0 or more summation Σ badges, to indicate exercises that require integrating ideas from previous chapters.
  • 0 or more whole-earth ⨁ badges, to indicate exercises that require integrating ideas from outside the book. These may be unsuitable for closed-book exams, depending on how obscure the needed information is.
  • 0 or more expansion ↔ badges, to indicate exercises that teach something new (as opposed to merely reviewing and consolidating stuff that has already been mentioned). -- et cetera

Rationale: The idea of a one-dimensional measure of "difficulty" (e.g. blue dots) is a misconception unto itself. Instead, one should envision multiple types of difficulties, spanning a multidimensional space. There is a theorem that says you cannot change dimensionality in a way that is one-to-one and continuous (Brouwer, 1911) ... so any one-dimensional measure is guaranteed to misrepresent reality.

For example, consider exercise 4-91 on page 67 of book II. I would rate it as ⌛⌛⌛↔. It earns a ↔ because it teaches something about the value of staging (in rocketry). It does not earn any 💡 or ⨁ or Σ badges, because the book spells out the method of solution in nanny-nanny detail, and it involves little more than repeated application of one basic principle.

• Also, as always, teachers are part of the team. We need to serve their needs, not just the students' needs. Therefore in the teacher's guide and/or (better!) in an online search engine, there should be more detailed information, spelling out what topics are covered by the integrative "Σ" exercises and perhaps the underspecified "⨁" exercises. We should make some attempt to keep these details away from the students, so as to discourage equation-hunting, and so that the underspecified questions remain underspecified.

• As you teach the class this year, every time you assign an exercise from the book, ask students to rate it using the multi-dimensional badge system. Collect the answers. They may surprise you.

  Opportunity for improvement: Based on a limited amount of skimming through book II, it seems that there are rather few questions in the 💡 or ⨁ or Σ categories. For example, exercise 24-108 on page 446 of book II is messy and laborious, but does not require much reasoning or insight, so I would rate it ⌛⌛⌛. I reckon it is more like running laps on a well-lit track, not like running a maze. However, students (not experts) should be doing the rating; experts are almost guaranteed to underestimate the difficulties.

  In any case, the point remains that even though running laps is good for you in some ways, a major goal of the course is to teach people how to run a maze.

• Here's something that "should" be easy, but at the moment I'm not sure how to implement it. The wise teacher will always be sprinkling in review questions. For example, when teaching chapter 20, it is super-easy to take a question from chapter 4, strip off the question-number, and then assign it.

  Stripping off the number makes equation-hunting somewhat harder (albeit not impossible). The problem is, a lot of teachers don't realize they should be doing this. They assign questions by number, which takes away all the mystery and invites equation-hunting. It's not immediately obvious how best to improve teaching practice.

  One obvious but possibly wasteful approach is to mix some out-of-context questions among the in-context questions at the end of each chapter. The questions can be verbatim or almost-verbatim copies of ones that appeared in some earlier chapter. For a hardcopy book this seems like a waste of paper. For an e-book it might be OK.

  In any case, this is a poor substitute for truly cumulative, integrative, real-world, reasoning-intensive exercises.

[JohnDenker]

• High up, on the Nth rung of the ladder, we find moderately-large realistic projects, as one sees in a project-oriented course. (At even higher rungs, there are term projects and PhD thesis projects.) Projects are surely good things, but still we need exercises to cover intermediate rungs of the ladder, i.e. more integrative than an isolated building-block, but not quite as complex as a large edifice.