Question is: For a first year math PhD student, should the student read their graduate level math textbooks page by page, sentence by sentence?

I often hear professors say that that's a bad idea, and that it's better to look at problems first, and then go back to read as necessary.

From undergrad training though, I like reading math books and working through them methodically, namely, going cover to cover (e.g. for calculus, linear algebra, introductory analysis, probability, etc.)

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    Different approaches work better for different people. Why not try it one way or another for a period of time, and see how it feels to you? Jun 25, 2019 at 19:33
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    I think you'll find that you simply won't have enough time to read through your textbooks sentence by sentence. This means you will have to figure out what you can get away with knowing more superficially. Keep in mind that your eventual goal is to produce a piece of original research, not to know everything that's already known. Jun 25, 2019 at 19:38

7 Answers 7


While it may be good for something, it isn't really the best way to learn anything. To learn, you need to get practice (reinforcement) and feedback. The exercises in the book will do two things for you if the book is a good one. First, they will exercise the more important ideas, giving you the practice, but also giving you an outline of the chapters. Second, the problems will point you, indirectly, to the examples, theorems, and proofs that are the most important to study.

The big idea here is that not every word in a math book has equal weight.

Trying to "memorize" the textbook is also a terrible way to prepare for exams. Find more exercises and solve them.

Your brain isn't like a thumb drive that you can pour information in to an then expect that you have "learned" it. Think of a novel that you read a year ago. How much of it do you really remember? If you read a lot, then it is unlikely that you will have retained very much about that old book. A few people can do that, but they are very rare. Chances are slim that you are one of them.

Finally, if you can't solve an exercise you know what you need to ask the professor about. Hopefully the prof will give you feedback on your work - the other essential to learning. That way you are less likely to develop misunderstandings that need to be corrected later.


The generic answer to this question is: it depends. Textbooks are written in such a way that both instructors and students can use different approaches to teach and learn the material, respectively. There is no one “right” way to use a textbook.

Professors who recommend certain approaches are likely speaking from experience and their own personal preference. It can be a good idea to follow the advice of those who are familiar with the material (because they learned it themselves at one point) and who have been tasked to teach it to students; they will likely have good suggestions.

If you personally feel that you can learn better by using a particular method (even if it is unpopular and not genetically recommended), you should use the book in that way.

The bottom line: if you are actually learning the material and are motivated to continue studying in that way, you should do that.


I would stick with the process that worked well for you before, until, you find it doesn't. As far as glancing at the problems, before. I don't think you have to--think you can use your old practice (doing the problems after). As long as you are reading the text in an engaged manner (working examples, doing skipped steps to the side, etc.) you should be fine. Of course, do the problems after.

One other answer mistakenly thinks you are talking about memorizing the text, but I don't think that was your question, just if you should pre-look at the problems. I don't think it matters that much, especially if the pre-look is reasonably fast. But in any case, there will be some recursion. Because as you work the problems, it will force you to go back and look at the text again.

P.s. Good pedagogical question. Might also be of interest on one of the math forums.

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    Not the question, no, but too many students think that memorization is the key. It is not. Just a warning.
    – Buffy
    Jun 25, 2019 at 20:43

The answer to your question depends not only on your style of learning but also on your professors and the textbooks they choose. Some professors regard the textbook as just a convenient source of homework problems or a reference for details that don't fit into the class schedule. Other professors regard the textbook as the first place you should learn the material; their lectures are not so much systematic presentations of the material as commentaries on the textbook. This difference in professors' attitudes affects not only what they do in their lectures but also which textbooks they choose for their courses. In particular, some but not all of your textbooks are likely to be appropriate for self-study. (I found that some textbooks, from which I was unable to really learn anything as a first-year Ph.D. student, were excellent references later when I needed some more detailed information about those topics.)

So I suggest you see (or ask) how your professors view the textbooks and then use the books accordingly.

Of course, if you have the time, it's never a bad idea to read more, to attack more problems, and generally to do more mathematics than your professors require. But your first priority, especially at the beginning of your Ph.D. studies, should be to learn thoroughly what your professors want you to learn.


I am a second year PhD student in statistics (I know, if mathematics at all it is applied maths). I like to have books around me. I consult them all the time.

Sometimes I realise that for my particular research problem I need to understand some point in complete detail, so complete that I could explain it to a first year undergraduate. In that case, for someone with my learning style, there is no alternative but to slog through every line, in detail, and do all the exercises.

But sometimes all I need to know is that at some point in my thesis I will find myself writing "it can be shown that ..." and just applying the result.

My experience so far in my PhD research is that it is not always easy at first to identify which of these two options is right. But I do know that if I chose option 1 as my default, I would not have got as far as I have with my own research. So my adivce would be: try to understand the main results first until you know which of them you will need to examine in such detail that maybe you might overturn them or add significantly to them.


There is lots of good advice in these answers. I think it is also helpful to try a method that is in between yours and your professors' suggestions. Go ahead and read the textbook line by line, but whenever you get to a theorem or lemma, try to prove it yourself before you read the proof. This can give you a chance to test your understanding in a way where you can get immediate feedback. If you have no idea how to start the proof, read the first couple lines and then try to finish it on your own.

Of course, there will be several times that you can't do the proof on your own, and you shouldn't be discouraged by this! Trying it first will help you better understand the proof when you read it, though.

Along a similar line, whenever you read a definition, try to think of some examples and non-examples that you're familiar with already. For example, if you were reading the definition of a group for the first time, you might think about how the integers under addition are an example, but the natural numbers are a non-example.


Depends on the book. Undergraduate-level textbooks tend to be easier to read cover-to-cover. After a while books end up being less self-contained and are better as references, like encyclopedias. That suits research-level work which revolves around research papers. Encyclopedic references are not meant to be read page by page.