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RobinZ comments on Understanding your understanding - Less Wrong
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This would suggest a typology of understanding: type N* understanding knows the passwords, type I understanding knows the equations (edit: or can otherwise find the answer - thanks, SilasBarta), type II understanding knows the connections to other fields of knowledge, and type III understanding generates type I and II knowledge within the domain and connections to other fields of knowledge (type II knowledge in other domains).
* Zero in roman numerals. Which is appropriate, of course, because the passwords are just names and can be wrong in a number of ways.
Very good, but I just want to emphasize that Level 1, as I've defined it, doesn't necessary involve equations; it just means that you get the right answers somehow, as long as it's not cheating. (So I think the "calculate" part in its name is a bit misleading and I should probably pick a different one.)
To put it another way: for purposes of determining whether you have type I understanding, ad hoc is okay, but post hoc is not.
I see - yes, that's a good point. I'll edit in a note.
(By the way, I switched from Arabic to Roman numerals to distinguish the typology from the hierarchy - it's level 1 and type I because the two are related, not perfectly identical.)
Oops, I had thought to correct my first reference (which should have been "Level 1"), but only corrected it halfway! Fixed now.