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July 14th, 2012

The naive way of learning to do something is just to attempt it repeatedly. This works remarkably well for people given how hard it is to teach computers to do the same tasks, but it's not how we usually teach people to do things. Instead we've found you learn faster and at lower cost if you do exercises.

With the naive approach, if we wanted children to be good at handling checkout at stores we would just have them do it. They could pretty quickly learn, under supervision, where to put money that someone gave them, but giving correct change would be hard. The supervisor could simply stand there, pointing out every time they gave incorrect change, but that would lead to slow learning and a lot of supervisor time. It might be better to use exercises: spend some time subtracting typical prices from typical combinations of offered bills. Even this might be too much for ideal practice: what change to give for a $20 when someone gives you $3.52 is kind of hard. So the supervisor recognizes that the semi-abstract skill of giving change has a more abstract subskill of simple subtraction and has the child practice that. You can go another step: 20.00 - 3.52 may even be too hard, so we do exercises in subtracting low whole numbers 10-3, 5-2, 6-1, and build from there.

The risk of an exercise, however, is that it may not match very well to the final goal. I remember two strategies in physics that were very helpful in solving problems:

  1. What have we been recently learning in class? How might that apply here?
  2. Have I used all the numbers they gave me?
These turn out to be great for a lot of physics problems, but they stop helping once you try to use physics to answer real questions.

In the same class, I started using a strategy of assigning variables to all the given numbers, and solving to problem with variables, substituting numbers for the variables only at the end once I'd done all of the algebraic manipulation. I found this helped some in reducing errors and was less writing, so I liked it, but no one else was interested and the teacher thought it was strange. Which is unfortunate: this approach is good practice for automation, a big part of the final goal for anyone actually using physics.

Designing good exercises means understanding what will be useful for larger scale exercises and skills. This is hard, and is why there are so often exercises that don't really teach the skills they're intended to.

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