> >>When I'm not sure, I go back to my book and look it up (Happened >>recently, when I wasn't sure what the exact definition of a refinement >>is (in the context of paracompact top. spaces)). You forget, of >>course, unless you solve problems or apply in a regular way. > >Yes, we all forget things over time. But if you are taking a course >in topology that covers paracompactness, I would certainly hope >you have the definition of 'refinement' memorized (by whatever >method) come time for the exam. Or even time to read the next theorem. >If you don't, the definition of paracompactness will be very hard to >understand.
Yes, I didn't understand paracompactness, so I had to read the textbook, looked through the section on partition of unity and so on. I wouldn't have time in an exam, so I would have missed the points. In an examination situation the material would have been a lot closer, timewise, and I could have reconstructed the definition by reflecting on the problems that I solved before. Like: "What has to be subset of what?... Ah, of course!"
Ok, this may not be practical sound advice. If the exam is just designed for the students to dump factual knowledge in contrast to solving problems you'll have to memorize because of the sheer quantity. You have to become a "definition robot". Like - (maybe) medical doctors, who'll have to recite all the bones in the human body in alphabetical order.
> >>>The student should be able to give a rendition that is at least >>>equivalent to the one given in the book and that uses precise language. > >>Absolutely right. But it should go above Parroting, that's what I >>meant. > >Of course it should. But I am lucky to get anything close to an >equivalent statement of the definition from my students. Simply >getting them to understand the difference between 'if...then' >and 'and' has been a struggle.
Hmm, maybe the student should have a minimum amount of talent, maybe it is as simple as that?