Many of you have sent very kind messages. I haven’t sent out individual acknowledgments because of my rather packed schedule at the moment. But of course I’ve been very grateful. However, there was one point that was brought up about the weight of proofs in the exam that some felt was more than was expected. So I thought I’d comment on it very briefly. That is, I was quoted as writing:
“I wouldn’t quite divide the material up in this way. But the portion of the exam where you are expected to prove an assertion coming from a standard theorem is worth around 10~30 points, depending on which problems you select.”
I hoped it would be clear that the point I wished to get across was *memorizing proofs is not enough*. That’s why I prefaced that remark with the observation about difficulty in dividing up material, and referred only to `proofs of standard theorems.’ I have made such points many times quite emphatically. In fact, in the same post (3 March, 2008), one could read on:
“This does not mean you can avoid *understanding* many proofs. More on this below.
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4) could you please list the theorems you expect us to reproduce. i know i need to cover all, but some are required to do calculation while others are to be reproduced. i understand them but reproducing somehow is difficult for me. if i have a list, i can revise them in a way, keeping in mind that i need to reproduce them; such as if a is congruent to a’ and b is congruent to b’ (mod m) then a+a’= b+b’ and aa’=bb’
Reply:
Many students have expressed concerns about `reproducing’ the proof of a theorem. I would rather say that you need to come to a confident understanding of the proofs. This means you should feel comfortable with the main points rather than being able to read (or even memorize) line by line. In practice, this also means connecting the different parts of a section, chapter, or the whole course. With any given proof, ask yourself what the key idea is. If you’ve done this through sufficiently many readings, then memorization becomes unnecessary. I will try to illustrate this a bit with another of your questions below. But rest assured that if you’ve understood the main point of a theorem, a small defect in memory will not cause a problem in the exam.”
So it’s somewhat hard to admit that I’ve misled people in some serious way. As for the rough estimate I gave, choosing 1,3,4,5, will minimize the `standard proof’ portion to the Chinese remainder theorem, worth 10 points. Of course the word `proof’ occurs in 5(a), but I hope you’ll agree that this question (with a two line answer) could hardly be described as requesting the proof of a standard theorem. If anything, it merely tests your knowledge of the definition of `positive definite.’ 4(f) could perhaps also be construed as a `proof question,’ since it asks for justification. But again, knowing that the occurrence of -1’s or 0’s in the canonical form contradicts positive definiteness is much more of a definition (of the real canonical form and positive definiteness) question than a proof question. You’ll notice in the online notes that it’s a brief `remark.’ That is to say, for both these questions, whatever is required to be justified looks entirely consistent with my reply to (4) above. As for comparison with previous years I certainly don’t see a smaller proportion of proof questions. In fact, in 2006, every problem but one contained one or two `standard proof’ questions.
To summarize, the misapprehension I wished strongly to avoid was that much mileage could be gained by memorizing a bunch of proofs.
Once again, I sympathize with the generally difficult experience, and I apologize for this one attempt at a point-wise rebuttal, a rather undignified exercise that could add insult to injury.
