Well, they're judged on output of new stuff. But it's hard to be productive in one area of math without "having a good working knowledge of" the areas on which it is based. And "working knowledge" usually means "able to perform the standard activities by habit".
The example that drove this home for me was watching my brother struggle with calculus. The problem was that he hadn't learned algebra to the point where he could execute it reflexively. During any particular calculus derivation, there would be one or more places where his work was not identically the same as the book's version. If he'd really learned algebra, he would have seen at a glace that the two were equal, but he couldn't think through the algebra at the same time he was thinking through the calculus.
Unfortunately, time pressures are ubiquitous in the real world, and it seems to me that the best course is to help the kid learn how to control her emotional state when under time pressure. (How else will she be able to drive, where every task is real-time?)
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Well, they're judged on output of new stuff. But it's hard to be productive in one area of math without "having a good working knowledge of" the areas on which it is based. And "working knowledge" usually means "able to perform the standard activities by habit".
The example that drove this home for me was watching my brother struggle with calculus. The problem was that he hadn't learned algebra to the point where he could execute it reflexively. During any particular calculus derivation, there would be one or more places where his work was not identically the same as the book's version. If he'd really learned algebra, he would have seen at a glace that the two were equal, but he couldn't think through the algebra at the same time he was thinking through the calculus.
Unfortunately, time pressures are ubiquitous in the real world, and it seems to me that the best course is to help the kid learn how to control her emotional state when under time pressure. (How else will she be able to drive, where every task is real-time?)