The contents of this page were taken from a question on Math Stack Exchange.
How is it that you read a mathematics book? Do you keep a notebook of definitions? What about theorems? Do you do all the exercises? Focus on or ignore the proofs?
I have been reading Munkres, Artin, Halmos, etc. but I get a bit lost usually around the middle. Also, about how fast should you be reading it? Any advice is wanted, I just reached the upper division level.
This method has worked well for me (but what works well for one person won't necessarily work well for everyone). I take it in several passes:
Read 0: Don't read the book, read the Wikipedia article or ask a friend what the subject is about. Learn about the big questions asked in the subject, and the basics of the theorems that answer them. Often the most important ideas are those that can be stated concisely, so you should be able to remember them once you are engaging the book.
Read 1: Let your eyes jump from definition to lemma to theorem without reading the proofs in between unless something grabs your attention or bothers you. If the book has exercises, see if you can do the first one of each chapter or section as you go.
Read 2: Read the book but this time read the proofs. But don't worry if you don't get all the details. If some logical jump doesn't make complete sense, feel free to ignore it at your discretion as long as you understand the overall flow of reasoning.
Read 3: Read through the lens of a skeptic. Work through all of the proofs with a fine toothed comb, and ask yourself every question you think of. You should never have to ask yourself "why" you are proving what you are proving at this point, but you have a chance to get the details down.
This approach is well suited to many math textbooks, which seem to be written to read well to people who already understand the subject. Most of the "classic" textbooks are labeled as such because they are comprehensive or well organized, not because they present challenging abstract ideas well to the uninitiated.
(Steps 1-3 are based on a three step heuristic method for writing proofs: convince yourself, convince a friend, convince a skeptic)
From Saharon Shelah, "Classification Theory and the Number of Non-Isomorphic Models"; quoted in Just and Weese, "Discovering Modern Set Theory I":
So we shall now explain how to read the book. The right way is to put it on your desk in the day, below your pillow at night, devoting yourself to the reading, and solving the exercises till you know it by heart. Unfortunately, I suspect the reader is looking for advice on how not to read, i.e. what to skip, and even better, how to read only some isolated highlights.
Sorry... I just love that quote.
By accident I came to this question-discussion only today.
The theme of several answers and comments, that many readings in different styles is best, I'd second, at least up to a point.
I would disagree with all advice to refuse to move forward without "mastery of all details prior"... certainly for nearly all textbooks, and even many higher-level monographs. The reasons is that textbooks currently seem to have the style of belaboring every possible detail, in the name of "rigor", as well as being rather sub-verbal about it. That is, the relative significance of different details/lemmas/whatever is not at all delineated. Since at least 90 percent of details are not at all "dangerous", and not even terribly surprising or illuminating, this results in gross inefficiency. Textbooks are 10 times longer than they need to be, and the critical points are lost in a 10-times-larger mess of fussy details. Terrible.
The only serious approach to avoiding drowning in the faux-rigor fussy details is to make at least one pass through material to see the big points, the higher-level plot-arcs. This lends coherence to the lower-level details. "Hindsight" of a sort.
In particular, "exercises" are an extremely volatile issue. Contemporary textbooks "must" include lots-and-lots of exercises to please publishers and meet other expectations. Thus, one has scant idea of the nature of a given one! Also, one can observe the schism in many texts between the "theoretical" nature of the chapter, and "problem-solving" nature of the exercises, with dearth of prototypes in the chapter itself, to maintain a sort of misguided "purity".
So: distinguishing the relative significance of details, and seeing the larger story-arc, are the most important things to cultivate. Some acquaintance with lower-level details is obviously useful, but the purported "ultimate" significance of low-level details is mostly an artifact of the way mathematics is taught in school.
It really depends on the book. There will be certain books that you won't like or wont be able to get on with. Other books you will sail through and enjoy immediately.
I have many books but tend to find that the majority of maths books are written to be concise rather than interesting (by this I mean the readers have to find the interesting bits themselves rather than the author transferring his/her interest). I am not a fan of this style but you should note that you can nearly always find some supplementary materials to fill in those gaps.
Sometimes it takes a mix of resources to develop sound knowledge in a specific area.
It cannot be too strongly emphasized that a long mathematical argument can be fully understood on the first reading only when it is very elementary indeed, relative to the reader's mathematical knowledge. If one wants only the gist of it, he may read such material once only; but otherwise he must expect to read it at least once again. Serious reading of mathematics is best done sitting bolt upright on a hard chair at a desk. Pencil and paper are nearly indispensable; for there are always figures to be sketched and steps in the argument to be verified by calculation.
L. J. Savage
My approach is to skim every chapter first so I'm aware what I'm going to do. I belive this is psychologically beneficial and it keeps me motivated and hungry to get the next "block of knowledge" - fatigue is bad and counterproductive. Then I read every section slowly and twice, so that I get the confidence to attack exercises. I try to give serious attemp for every problem, but I don't spend days with the problem set (max. 1-2 days, depending on book).
A good way to maintain the important positive attitude for learning is to borne in mind: "This material may seem difficult, but if I read it slowly, then I will understand it all."
When I have finished the book, then I will quickly read it again to refresh my memory. Then I should be fine with the material.
When you read a book,
(Lovely book!)
Read the first part and see how the layout looks.
If some sections are elective
Then don't read them, be selective
With your books books books books books books books
Be selective and you'll sail on through your books.
When you come to exercises
In your book
Keep a notepad and a pencil by your book.
Do each interesting problem,
All the easy, and some hard ones
In your books books books books books books books
Don't forgo the exercises in your book.
I will just add by saying reading a mathematics text can be very daunting initially (especially for degree level mathematics). You can't expect to understand everything you read the first time. This will also depend on your level of mathematical understanding and how advanced the text is. But before reading, you need to establish what you intend to get out of the reading. Are you reading to pass an exam or for personal fulfilment?
Hence, I tend to read initially to get a scope of the subject and then my subsequent readings will be to understand the finer details. Its good to understand proofs and making notes might be helpful too. Some proofs may be too complex to understand fully at one reading. You only truly understand mathematics if you can solve problems in the relevant areas-reading mathematics can be falsely re-assuring without putting pen on paper.
I came across some useful advice on reading mathematics which I will share here (https://www.people.vcu.edu/~dcranston/490/handouts/math-read.html):
I would recommend a book by Lara Alcock which I found really helpful-https://www.amazon.co.uk/Study-Mathematics-Degree-Lara-Alcock/dp/0199661324. The whole of chapter 7 is dedicated to reading mathematics.
She also wrote about "How to think about analysis" which was even more helpful-https://www.amazon.co.uk/Think-About-Analysis-Lara-Alcock/dp/0198723539/ref=pd_lpo_sbs_14_t_2?_encoding=UTF8&psc=1&refRID=0WK87FS6FG62H50Z29RD
This is a seven-year post but I am putting here my experience learning mathematics as an undergraduate.
First, I would explain a bit what's usually inside a mathematics book. For every chapter or sections, there are usually two parts, the contents and the exercises. The purpose of contents is to let you expose yourself to the knowledge while the exercises are to let you improve until you really understand the material well. So, first and foremost, do not feel disappointed if you cannot solve some of the exercises as you have met the exercise at your level! If you can solve all exercises easily or it's so hard that you can't solve any of them, that book is not for you. For me, a book suitable for you would be the one you can solve at least 60% of the questions most of the time. Also, do not try to solve every single exercise in a book as it leads to diminishing returns. There are a lot of fun subjects that you should go for it. This paragraph is to let you have a correct attitude towards exercises in mathematics. Now I will share my way to study a mathematics book.
The first level - Read the chapters briefly. If you are following a course, lecture's class would be at this level. If you are self-studying, read a chapter that you can understand at least half of the content including proofs. If at this point, it takes so long for you to understand (even the first chapter), consider changing to an easier book.
The second level - Read the chapters in detail. Jot down the concept or part of proofs that you can't understand. Copy down all the theorem statements and definitions on a piece of paper. This is extremely helpful when you attempt the exercises.
The third level - Do the exercises in each chapter. After struggling, do not feel embarrassed to look the answer to the questions you cannot solve. Make sure to understand how they solve it.
The fourth level - After you have completed half of the book. It's time for revision! Try to prove everything in the text once again. Revise the exercise that you had done. If you can't prove some of them, reread again and try to memorise the key parts of the proofs. Then reprove them after a few days.
The fifth level - First you do the same as in the fourth level to the last half of the book. Then, write a revision sheet for all the chapters. This revision sheet is for your future reference, so it should be short and concise.
Usually, you can read 4 mathematics book in 3 months by following this process provided the level of the book is suitable for you. After this, you may study another subject of your interest or further sharpen your skills by reading a more advanced text of the same subject. For me, it's the fastest way. Of course, to do this, you need to invest at least 8 hours per day on it.