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I'm a bachelor student of Math major. The question is about the better way to select and read maths text- or non-text- books, without considering the course and exam. (Because we follow the lecture notes during the course, and the exam is totally covered in lecture notes, so it's not an issue, but just about how to really learn)

Question:

(1) Reading the book one by one, or just focus and re-read lots of times on one certain book, and then continue with little others ? Because recently I've tried the first way, for example, to learn Calculus, I took one book, the read it from first page to the last and finish some exercises, and then immediately read another calculus book, doing the same thing, and then the third one, and so forth. After this, I realize it is really bad result, it seems that I don't master the calculus pretty well, just memorize lots of things. So, I'm thinking about whether the second way is better. Because some friends told me before, said that there're lots of good maths textbooks, I should read as much as possible.

(2): If the second way is better, could I just use the UTM and GTM series(published by Springer, Undergraduate Texts in Mathematics) for all the maths courses ? Because there're too many, tons of books in the market, I'm not sure how to select the perfect books. But lots of people said that the UTM & GTM are two perfect series in Maths. And my teacher told me that it is better to spend more time on the books written by the famous mathematicians who are the masters in their areas. Less time on the so-called popular textbooks, like thousand-page widely-used textbooks. He's not saying that the latter one is bad, it just means that the latter one is too specified for the general course in university, passing exam , or something like that. Although it's organized well, ton's of exercises, examples and so on, the former one is better to grab the deep insight of the field and focus more on thinking, not just knowledge. So, briefly, is it good to just use to the whole UTM series for every course in math bachelor program?

Martin Sleziak
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Xingdong
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    "And how to select best books?" - trial and error. Sometimes it does take a while to find a book you're comfy with... of course, you should have already tried to put in effort with working out included exercises in the book. – J. M. ain't a mathematician Apr 26 '11 at 00:17
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    I think I've done best by asking friends, graduate students, professors and the people on math.se about best books to learn _____ as a beginning student with a certain background. – Michael Chen Apr 26 '11 at 00:39
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    Here's a fun list of "standard" textbooks in their respective areas. This was created by some students in the honors program at the University of Chicago, including one of the math.SE's (former?) members, Pete L. Clark: http://www.ocf.berkeley.edu/~abhishek/chicmath.htm. As a cultural side note, it sounds like your teacher was giving you Abel's advice to "Study the masters and not the pupils" – Tyler Apr 26 '11 at 01:25

2 Answers2

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In my opinion it's much better to pick some book and read it in depth, solving many of the exercises while reading. If you read a 100 books without actually concentrating on what you're doing, it will be of no help. Of course, reading a book thoroughly and attentively is slow, so don't expect it to be quick.

You seem to be somewhat biased by your experience so far. Apart from the first few courses, there are no expensive, shiny, popular textbooks - only the dense, terse, substantial ones. If you really want to understand math (rather than to be just able to apply it), then the shiny books don't help you at all.

Finally, I have no idea whether UTM/GTM books are "enough", but if for some reason they're all that you have access to, you're probably fine. But why limit yourself? It might happen that in a particular subject, none of the "standard" books are UTM/GTM, e.g. Kunen's Set Theory is Elsevier and Jech's is Springer but neither UTM/GTM. Instead of committing to UTM/GTM, just pick whatever friends or professors recommend and is actually available.

Yuval Filmus
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    This is the second time that I felt frustrated that my reading is far too slow and I read your answer and then didn't feel so bad anymore. Wonderful. – Rudy the Reindeer May 27 '14 at 05:34
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For a list of great books, see The Mathematics Autodidact’s Aid.

lhf
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