If you could be given a magical amulet that let you teach students better, more quickly, and more permanently than ever before but at the expense of never being able to test them to see exactly what it was that they learned, or you could be given a magical apparatus that let you test them perfectly so that you knew exactly what it was that they learned and did not learn but gave no insights or help into how to teach them those things that they failed at, which would you choose?
Which would your local school administrator choose? Which would your children's teachers choose? Which would the legislator writing education policy choose?
Everything else is post hoc rationalization. Having decided what it is that we want public education to be, we need to have some sort of justification for it even if it doesn't make sense.
Do you know what people who don't show work do when they move on to more difficult problems? They start scribbling it out on paper, without any prompting. The more difficult problems are interesting enough that they want to get them right, and when they notice that basic mistakes are interfering they strive to avoid those.
Or, in some cases, they just don't bother. When you solve the Poincaire Conjecture (spelling? didn't want to cheat and look it up) no one gives a crap whether or not you "showed your work" because most of the other mathematicians can also "just see" the boring details, and are interested primarily in the truly insightful portion of the proof.
I suspect that we're actually selecting for accountants and not math geniuses when we harp on "showing your work". How many Perelmans did we discourage and how many math stooges were praised last year in public schools?
