[derefr]

I've thought about this a bit as a game designer. Basically, teachers have to force (induce extrinsic motivation) students to learn things, because the student has no greater goal (intrinsic motivation) which would flag the knowledge to them as being immediately relevant. If, on the other hand, the student needed to be able to know the sums of large numbers quickly while they had their hands full (for, say, estimating a fair tip as a server in a restaurant), they'd quickly absorb a mental addition algorithm--because they would have created a mental gap, a place where their plans say "and then if only I could--".

The things that should be taught, I think, correlate highly to the types of things students will later wish they knew. These are mostly low-level rules and processes for things which are so "obvious" to everyone else that they're hard to communicate unless you have a background in education. To flip that around, the kinds of things we call "trivia"--things equally simple or obvious, but as easily learned as said (e.g. a state capitol) don't really need school to teach them--as, if someone later feels the need to know, they can just ask anyone, and pick it up on their own. School is mostly a place to impart knowledge that's hard to teach (including to self-teach) if you don't know the specific skills of teaching.