Strategy for reading math books, is it better to prove the theorems yourself or just read them?

Context: I'm self-studying some mid to upper level undergraduate math subjects. For example, right now I'm reading Munkres' Topology book.

Usually, the approach I use is to go through the book in order, looking at each theorem and trying to prove it without looking at the proof provided in the text. If I get stuck, I look at the beginning of the proof to get a hint and then try again. If I'm still stuck, I repeat this until I have either proved it or read the entire proof provided (even if I am able to prove the theorem myself I still read proof afterwords). I am able to prove most of the theorems without looking at the proof but this approach is very slow. I usually do about 2-3 of the exercises at the end of each chapter and then move on.

An approach where I just read the proofs instead of proving them myself would be much faster and allow me to spend more time on other things. I have two specific questions regarding this.

(1) Given that I have a limited amount of time to allocate to studying math, my current approach means that I spend a lot of time going through the theorems of the main text and not so much time on the exercises at the end of each chapter. Is this good or bad? The theorems of the main text are generally different from those of the exercises in a number of ways, most obviously that the theorems in the main text are usually more important. Given this, I'd like to know how to allocate time between, on the one hand, proving (and understanding the proofs of) the main theorems and, on the other hand, applying/extending/tweaking the main theorems, which is what the exercises usually make you do.

(2) I've heard some people (on SE and elsewhere) say that its best to take several passes through the book with increasing attention to detail with each pass. Given that my current approach is effective for me (I am definitely learning the material) but slow (compared to other students I've talked to), does it seem that I am reading too closely the first time around?

Update: After years of experience, I believe this approach was bad. I think it is better to read most of the proofs and then see if one can prove them a couple days later relying partially on memory but also working sort of from first principles. And then doing the problems which are typically designed to be a good level and to help you learn to apply the theorems which is the important thing since the theorems have already been proved.

Solution 1:

In my experience, it is better to be able to apply the theorems than to prove them (unless you need proofs for an exam). I would personally weight towards spending time on the exercises. Read through the proofs, certainly, to get a feel for which methods are used to prove things in this area, but the main theorems are "main" for a reason: their applications are very useful or important. Get to know their applications by doing the exercises, and spend the remaining time on the theorems.

Otherwise, if the time you're taking isn't causing you problems, then I think you're doing it the right way. A way which isn't much worse but takes substantially less time is to skim each proof as you meet the theorem, and then begin again in your usual way (proving it without any more information). That lets you get a sense of the direction you need to go for the proof, and it will tell you whether a theorem has a horribly long proof you'll take forever on.

If you need to save further time, it can be worth learning to identify which theorems are really important and which are not. Sometimes the author will have written "this is a cute little side-result" or similar. Just read and work through the proofs (or, for even faster reading, omit the proofs) of the results which aren't very central.

If you have to cram for some reason, I find it most effective just to rote-learn proofs, but of course this is awful practice and should only be used as a last-resort before exams or whatever.

Solution 2:

I have studied calculus, real analysis, linear algebra on my own following various books. Currently I am studying multivariable calculus. At first, in Calculus, I tried to prove all theorems myself and tried to do all exercises at the end of the chapter. I was too slow, I couldn't make any progress. Because the definitions and concepts like $\varepsilon-\delta$ were new to me. I couldn't understand what theorems say let alone doing their proofs on my own. So I looked to the proofs and tried to understand them. But I realy tried to understand every step and every proof without skipping. Sometimes I filled the blanks between the steps of the proof. I did examples in the book but skipped the exercises in the end. Since I was not following a course for a term, there were lots of things to learn in a limited time. I asked questions in this site when I struggle.

After a while I have become familiar with math subjects and proofs. Now I can prove theorems in a book without looking to the book and now I am way more faster when learning. I have to admit that I still struggle for hours or days for a few pages of a book when I learn a completely new thing, but it happens less and less as you progress. So it is just a matter of time to get faster. Therefore, you should be very patient. Self-study can be very hard. But no matter how hard it is you should never give up and move on. For the exercises at the end, it is your choice to do or not to do them. I personally skip them if I believe that I completely understand the chapter. I feel more comfartable that way because I don't have to be worried about limited time. But if you have time they can definitely be useful.

Solution 3:

I'd suggest that you don't try to proof all the theorems in the book on your own and instead concentrate on the exercises and maybe on proving the corollarys yourself. Especially in the late chapters of a book the proofs will become more and more complicated and you will waste more and more time only finding a specific kind of idea.

When reading a book I think the following things are more important than devising a proof on your own:

• Understand the idea of the proof. Try to find an intuitive reason as to why certain constructions were made.
• Understand what the theorem conveys. What is the intuitive interpretation of the statement of the theorem?
• Understand why certain assumptions were made. Can you find counter-examples if you drop them? Can the assumptions be weakend in any way? Can you achieve any meaningful stronger assertion if you strengthen the assumptions?
• Try to find an intuitive example for the theorem. Especially in (elementary) topology you can often find a nice example in $\mathbb{R}^2$, $\mathbb{R}^3$ or $C([0, 1])$.

To answer your second question: Personally, I can't read a book about mathematics from front to back. Typically, when I start reading and see a definition or theorem, I can't remember it after reading ten more definitions/ theorems. But these definitions and theorems become important in the later parts, so I usually read until I have a rough understanding of why a concept is useful and then stop reading when I start to have too many gaps to understand proofs. Then I go back a bunch of pages and start re-reading the pages I didn't fully understand.

Solution 4:

Self study can be a nightmare. When you attend a university course, you don't worry about how much time the course takes: the instructor must worry about it. When you study on your own, it is rather difficult to organize your time.

As a pure mathematician I must say that proofs are often (often, not always) more important than statements. The proof of the mean value theorem (Lagrange theorem) is more illuminating than its statement. Physicists and mathematical physicists sometimes recommend to skip proofs because they are too technical.

Many years ago I decided to work in the field of mathematical analysis, but I studied topology, too. In my memory proofs are even more important in topology, since you can learn ideas and techniques that you would not expect after reading a statement. A good book should be able to point out important results and technical lemmas, so my suggestion is that you should follow closely your textbook. I do not have Munkres' book at hand, but I guess that skipping proofs would be a pity.