How to build "Mental stamina" when it comes to studying mathematics.

Question

I'm currently studying undergraduate mathematics and I found that my interest for high rigour currently goes beyond that of my peers. This means that I often end up doing a lot of self study from books that are more theoretical than my peers are interested in.

This also means however, that I've started to read highly theoretical books without really having the "mental stamina" to read them effectively. What I mean by this is that I often find that I feel unable to digest more than maybe a dozen pages in a session, for maybe 3 or 4 sessions a day at maximum, and even that often feels like glossing over it. In simple terms; studying rigorous texts tires me out very quickly and I would like to build the toolbox I need to tackle such texts more effectively.

My question(s) are about how to build this toolbox. Specifically:

• How would you recommend getting used to high-density rigorous texts?

• How did you get used to learning in higher volumes?

I know that similar questions have been asked before, but I haven't really found anything that goes in to building the stamina to study better as opposed to general studying techniques.

Answer 1

Mathematics is not something a normal person can just read once and then use. You need practice and experience. You can read all the theorems you want, study all the big books and works, and still, you will have forgotten most of it or not gotten the idea.

What helps me the most to understand something, is to translate it into my own language. I have a certain amount of knowledge and a certain way how to see things. You might notice that the very same thing can be seen and discussed in various different ways, e.g. linear algebra could be done in matrix notation or in map/function notation. The first step to understand a new work is to find out how it embeds into what you already know, how it translate into the language you are using.

Another important point are repetitions. You can't just read a mathematical text once and suddenly you know everything in there. But, different from other topics, reading it 100x also doesn't help, learning it by heart you will never fully understand it. Mathematics is very much learning by doing, even though it seems rather theoretical at first sight. Thus, if you really want to understand a topic, use it yourself. There are multiple ways to do it. For example, the authors might sometimes leave out proofs as they are easy or they just say "this can be seen by applying theorem X.Y to the special case of ...". Do not just read over such passages but take them as an exercise: If the author of the book thinks that you, as the reader, should be able to prove what he wrote there, then go on, do it. This not only helps you develop a deeper understanding of the topic and the tools to use it, but also repeats the previous theorems, as you will most likely use them. A textbook that comes with exercises (and maybe even solutions for them) is of great help here when first learning a topic. For example, you might consider looking for higher grade textbooks at your university and try to get your hands on the exercises and solutions from a few years back.

As a third point to help understand and memorize a topic, try (after doing the first two points) to filter out the relevant parts. Assume that you are preparing for an exam on this topic and you are only allowed, say, one page of notes to bring, so you should choose them wisely. After you did the proofs and the exercises in the book, you should be able to see which results are important to use and which ones are technical and are only there to help prove the main theorem - thus will most likely not be important outside of this chapter.

I know I did not answer your question on how to learn more in a setting, but that is because in my opinion, you can't. There is a certain amount you are able to learn, for some it is more, for others it is less, but you can't just go and fully understand the contend of a whole semester on a weekend (you could read through it maybe, but you would still not be able to use it). What I tried to give you are hints on how to properly study on your own, how to make sure that it does not go to waste and that you really memorize what you did, that you will be able to use it even outside of the setting of the book/text. For me, my limit when I really want to learn (as described above) a completely new topic from a high density book is maybe 10-15 pages a day. So 4x12 already sounds quite much, that is the reason why I assume that you are not properly learning it and hence will not be able to use it fully.

However, don't loose your motivation: If you extend your knowledge and remember the first step to translate it into your own language, then you will notice (rarely at first, more often as time goes on) that a topic is not completely new but rather an extension of what you already know; maybe a different point of view on a topic you already studied. This allows you to go through it faster (as you might for example already know some of the tricks used) and on the other hand also allows for a fourth point, the one that is the most fun of them all: Apply the new theory to extend and simplify your knowledge. It is often the case in mathematics, that a fact that was really difficult at first and took the professor hours to prove, becomes easy and obvious once you have developed the right language to properly understand it (often only years later). Revisiting the points that caused you problems, maybe even open questions or little riddles you found for yourself, later on with new knowledge and seeing how some of them just resolve like nothing is simply beautiful.

edit: One of the comments above reminded me of a fact I found out through teaching as a student assistant: I have only fully understood a topic, if I am able to teach it understandable to others and motivate it for them, show them how what they learned in the lecture the day before and that sounded really strange is in fact not that hard but rather nice. If you get the change, go for a part time job as a teaching assistant, it will really help you to review topics you once thought you fully understood and see that there are even more to learn there.

Answer 2

Do as many exercises as possible.

If the source you're reading has exercises, do those. Otherwise, you'll need to invent your own. When I'm reading a dense proof, I make a list of all the steps that I don't understand (often there are very many!) and do them as exercises.

Read about the subject elsewhere.

I'm talking about cursory reading: learn about the history of the subject, read its Wikipedia article, look at old Math Stack Exchange questions. This is good to do when you're feeling mentally depleted, but still want to learn something. It can also give you a more rounded picture of the subject.

Talk to other math people about what you're learning.

Many of us have an image of "mathematician" as someone working alone at a desk, but I think it's nearly as common for mathematicians to talk to and learn from their peers. Learning about a subject is much easier if someone who understands it gives you a high-level overview first.

Shop around before you commit to learning a subject from a specific text.

For example, the "standard" reference to learn scheme theory is Hartshorne's algebraic geometry book. But there are at least eight other texts that would serve the same purpose; see the discussion at Math Overflow. There's no reason to use a text just because it's "standard," or someone recommended it to you.

I often like to use course notes (found online), because they are usually more concise than textbooks.

But once you decide on a source, commit to it.

That is, make sure you don't switch between sources too often. It's useful to stick with one text because that makes measuring your progress easier: look at how many pages you've read.

Work through your source regularly.

Reading five pages a day might seem extremely slow, but if you do this for a month, you'll have read 150 pages, which is actually quite a lot. For example, most books by Serre are shorter than this! (littleO's comment is along these lines.)

More generally, this is a good strategy for any personal project: distribute the work consistently instead of working irregularly.

Constantly ask yourself "Why do I care?"

Why are you learning this subject to begin with? Why would anyone make that definition? Why is it helpful to know that this theorem is true? The goal is to constantly challenge the text, and make sure you have sufficient motivation to progress.

Keep in mind that some of the answers to these questions will only come in time -- you may have to suspend your disbelief until you understand the subject better. But make sure that you ultimately have an answer to these questions.