Should Students be Asked to Regurgitate Known Proofs?

by Justin Skycak (@justinskycak) on

Imitating without analyzing produces a robot / ape who can't think critically; analyzing without imitating produces a critic who can't act on their own advice.

Cross-posted from here.

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If you’re training musicians, you might have them do exercises in 2 different categories:

  1. improvising from scratch in real time, on the spot, without prior rehearsal
  2. rehearsing and analyzing difficult pieces by master composers

Category 1 is about developing creativity within your repertoire, completely mastering the techniques that you’re already familiar/comfortable with.

Category 2 is about extending your repertoire to include new techniques that are initially very unfamiliar/uncomfortable. Initially, these techniques are advanced relative to your ability, so the only way to practice them properly is by imitating and analyzing them. But over time, as you build a baseline level of competence imitating/analyzing these techniques, they can gradually be moved into category 1 (and replaced with even more advanced techniques in category 2).

Basically every skill domain is the same way, including math.

  • Category 1 contains general techniques like proof by induction.
  • Category 2 contains key theorems like the fundamental theorem of calculus.

On an exam, it would be reasonable to ask students to

  • solve previously-unseen problems in category 1, and
  • imitate and analyze known solutions to previously-seen problems in category 2. ("Analyze" could mean, for example, identifying an error in a given proof.)

Lastly, I should emphasize that moving techniques from category 2 to category 1 requires both imitation and analysis, not just one or the other.

  • If you don't imitate, then you won't be able to execute the mechanics of the technique. One who analyzes but doesn't imitate is called a critic.
  • If you don't analyze, then you won't know how to tweak the mechanics of the technique to new situations, and you also won't know whether it's even appropriate to apply the technique in a new situation unless you're explicitly told to use the technique.

Imitating without analyzing produces a robot / ape who can’t think critically; analyzing without imitating produces a critic who can’t act on their own advice.


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