I see people like Terry Tao and others take extensive notes. But is this really necessary? When I do this I feel like I am rewriting a textbook.
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closed as primarily opinion-based by Jonas Meyer, Claude Leibovici, Arctic Char, Yiorgos S. Smyrlis, user26857 Apr 25 '15 at 8:34
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About notes and rewriting: if the topic is important and non-trivial, it is all too easy to delude oneself, optimistically, into believing that one grasps all the points... if one is at all passive. To avoid this (as well as to have a perhaps more convenient personal archive), I think there is no substitute for an intensely (self-) critical rewrite.
Yes, this is "expensive" in time and effort, so should NOT be allocated to trivial matters, to "exercises", or to things one doesn't care much about.
Personally, if I've not rewritten something, I don't feel I understand it, except perhaps distantly, passively, as with gossip or hearsay. By the end, the ideal is to perceive the thing that cost effort as being, in fact, as trivial as possible, once one has adjusted one's viewpoint. It is important to remind oneself that this is not an indicator of wasted time or of foolishness, but of success in rewriting/rethinking. Then the significance of things that don't quite become trivial is vastly clearer.
Everyone learns a bit differently.
Some people learn better by listening; others by writing. I find that the best way for me to understand someone else's proof is to write it out in careful detail, rather than simply reading it. For me, taking notes is a way to help me keep up with the material while digesting and internalising it.
Other people don't learn that way. George Bergman (my advisor), for example, throughly dislikes lecturing (in the traditional way) because he says that it is a method for transferring information from the lecturer's notes to the student's notebook without passing through the brain of either (an opinion he has held since he took his first graduate course; so clearly he derived little benefit from taking notes the way you describe).
If all you are doing is writing and this process does not help you think about what you are writing, then it is probably not in your best interest to write notes. Better to keep the pad for any sudden ideas or insights you might want to come back later, and trust the textbook to provide the information you are interested in.
So... whether it is "necessary" is not an absolute, but rather a function of your own personal preference. In short: "whatever floats your boat".
When I go to a lecture I try to discern the general idea and condense the talk into a small summary -- sometimes half a page, sometimes just a couple of lines. Sometimes I get technical details out of a talk, sometimes it's more of a big picture. Sometimes I miss the point. Sometimes I latch onto an idea I don't understand and spend the entire talk trying to work that out, or I try to prove the speaker wrong. But as Arturo mentions, different people have different styles.