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(Just focus on how to learn and master the stuff pretty well, not involve the aspect of courses or exam)

Because recently I always feel that the time and energy are pretty limited, I want to try another strategy to read some perfect books at middle-difficulty level or written by famous masters in this area , instead of from the introductory big books like traditional thousand-page books.

For instance, the Discrete Maths. Traditionally, the people would firstly read the Rosen's or Epp's thousand-page textbook, following the courses, which almost all important aspects are involved, but not sophisticated, and then continue the middle level books and so forth.

But I want to try the strategy that to read the books seperately on the perfect or written by famous masters in this area. For example, Hardy's Number Theory, Halmo's Naive Set Theory, Jech Set Theory, Logic in UTM series, Chung Kai Lai's First Course in Probability in UTM series, Chartrand's graph theory, and something like that. Instead of reading Rosen's Discrete Maths. Is it okay ?

Because recently I talked with my friends who's majoring in physics, he said it's okay for physics, such as using best and most sophisticated books seperately for mechanics, E&M, waves and so forth, especially some great books written by Nobel winner. Although this way is pretty slow, and you need to focus and read really carefully. Although it's impossible for the course-to-exam procedure, but possible for learning. Therefore, I'm not sure whether it's also okay for maths ?

Because of the limited time and energy, and there're tons of books in one certain topic, it's not possible to read too many books on the same topic. But like Abel said that Read the masters, not the pupils. And famous mathmatician in differential geometry Shiing-Shen Chern said that it's better to learn from masters, because they understand the subject pretty deeply. So, that's where and why my thought of reading books come from.

Or I think about another method, it's to read the so-called introductory thousand-page books thoroghly, and pretty fast to grab the big picture, and then immediately turn to the perfect books mentioned above, to start reading carefully and re-read lots time. Because after put it in experience, I realize that it's somehow impossible to re-read intro thousand-page books for lots of times and then turn to perfect books which still be read lots of times, because the time is not allowed. I have to choose one of them. Hence, I think maybe it's better to put majority on the famous and perfect books.

Summary of strategies:

(1) Read the traditional and popular intro-thousand-page books for lots time(say 50% - 70%), and then to use rest time to focus on perfect and famous books especially written by famous masters in the area(say 30% - 50%)

(2) Putting time and energy averagely, like 50% to 50%. (although I think it's worst idea, because reading two kind of books very fast, the result is I don't get anything, just some memorization of knowledge, not the skills, ideas or thoughts)

(3) Read the traditional and popular intro-thousand-page books for less time, maybe once only, or just as a tool to go through the course-exam procedure.(say 20%-40%) And then, put all rest time and energy on the perfect and famous books especially written by famous masters in the area(say 60% - 80%)

(4) Skip the traditional and popular intro-thousand-page books, focus all just on perfect and famous books especially written by famous masters in the area(say 100%)

The (4) is mostly what I'm talking about in the paragraphs above, and the (1) is what most people doing, the (2) is what I dislike. And the (3) is balancing among them.

Edit not by OP: It may help to consult Math Major: How to read textbooks in better style or method ? And how to select best books?.

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Xingdong
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    Do you have a goal? It's hard to give advice when it's not clear what your aim is. If your aim is only to read, then read what you like. – Ryan Budney Apr 27 '11 at 23:53
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    Work by a "master" won't be of much use if you can't parse it. If you can't parse something by a "master", read something less illustrious but probably more explanatory first, and then go back to the "master" work with fresh eyes. – J. M. ain't a mathematician Apr 27 '11 at 23:55
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    It seems to me that (4) is what you want to do. I think that you should follow your instinct. Different people learn differently, so if studying Steenrod is what inspires you... go for it! There are also great books written by less famous mathematicians -- for instance, I think Spivak's books are fantastic. If you don't just want to learn math, but also need to pass exams, also do some of (3). Ages ago, I made the mistake in an undergrad PDE class of paying all of my attention to existence/uniqueness questions and not learning how to compute solutions quickly. The exam killed me. – Sam Lisi Apr 28 '11 at 00:05
  • @Ryan Budney : I want to do the research in maths in the future, though now it's pretty hard to specify which branch I love the most. In other words, the goal to learn the maths for me is maybe to train more instinction or insight of the concepts and to understand it deeply, not to apply it in engineering projects. – Xingdong Apr 28 '11 at 00:40
  • @J. M. : is it okay that when it comes to a new subject to learn, I firstly start with a "master" work, when somewhere I can't parse , I look it up to the more intro level to get the details of understanding ? Or grab the most important and fundamental concepts on intro level, but not every detail, say more than half, and then put on the "master" one with the intro one as just reference to look up something I can't go through in it? – Xingdong Apr 28 '11 at 00:46
  • @Xingdong: I would say you'll have to experiment; "trial and error", like what I commented in your other question. There are times your first approach is better, and there are times your second approach is better. "Find your groove", so to speak... – J. M. ain't a mathematician Apr 28 '11 at 00:52
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    @Sam Lisi : Thank you for your advice. Because in my experience, if I read in a "smooth road", I realize that I'm always less patient(yeah, although it's a really bad habit). But when I try something pretty difficult, or higher level, although even every step is somehow rough, but it really attracts me to be concentrated mostly. But I'm not sure whether this habit would be hurting for the future learning. – Xingdong Apr 28 '11 at 00:54
  • @Xingdong: something that helps with all reading of mathematics, but especially of books that seem too "smooth" is to approach it as a solutions manual: first try to develop the theory on your own and compare the results. If you have a definition, see what examples satisfy the definition and which ones fail. When you have a theorem, see what happens if you drop some of the hypotheses. If your book presents an algorithm for solving a problem, see why it works, and see if you can reinterpret it so it seems natural. If you do this, you will learn a lot from any book. – Sam Lisi Apr 28 '11 at 10:51
  • Also, a brief warning: some very deep books look smooth, but really aren't if you go slowly and think about them instead of rushing through too fast. This is true for some of the great books by Milnor (e.g. Characteristic Classes or Morse Theory). – Sam Lisi Apr 28 '11 at 10:53
  • I have never heard of "the traditional and popular intro-thousand-page books". I don't see the deciding factor in "famous masters", either. I strongly suggest that you freely browse in your library and that you ask for specific advice on each topic (eg ask us for advice which number theory book to read this summer given that you loved books X, Y and Z in other topics).

    Often, I find it helpful to start out with less comprehensive books, but this is not a function of page numbers. It is more a case of "if I properly understand limits on $\mathbb R$, Banach spaces will be immediate, anyway.

     – Phira Apr 29 '11 at 07:40
  • When (or if) you find a perfect book, let me know; I'd like to read it. Seriously: Some top experts have written wonderful books; some have written lousy books; some people whose research is mediocre have written wonderful books. Judge the book not by its author but by the suitability (for you!) of its content. In most cases, after reading a few pages of a book, you'll instinctively know whether you really want to read the rest of the book. Also, don't forget survey articles; in some cases they're every bit as good as books. – Andreas Blass Aug 07 '13 at 04:40
  • Does this answer your question? Math Major: How to read textbooks in better style or method ? And how to select best books? –  May 28 '23 at 01:02

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You are right. Our time and energy are limited and it is logical to use them in the best manner possible. But reading the "Masters" isn't always the right way to learn mathematics. They may have been masters in their field but that doesn't mean that their books are the best possible books for you.

For example, some years ago I tried to read Hardy's classic book on Number Theory, but unfortunately at that time I felt that the book to be a very dull one, either because I didn't understand it or because it really was dull! However, having read some other "Intro" books on Number Theory, now I don't find that book so much dull as before.

However, there are some things that may be learnt directly from Masters. For example, I found it especially helpful to read Landau's classical text Foundations of Analysis for a rigorous understanding of Analysis. Russell's Books also helped me to clarify my concepts about the logical foundations of mathematical principles.

One word of warning (perhaps out of the place, but still important), DON'T TRY TO IMITATE THE MASTERS. It's okay if you can't learn from Masters but it will be a mess if you ruin your natural genius trying to imitate them.