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How do you effectively study mathematics? How does one read a maths book instead or just staring at it for hours?

(Apologies in advance if the question is ill-posed or too subjective in its current form to meet the requirements of the FAQ; I'd certainly appreciate any suggestions for its modification if need be.)

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    $\begingroup$ Do the exercises, then come up with your own exercises. The best exercises are the ones which test your understanding of two different books. (These are the ones you have to come up with.) $\endgroup$
    – Qiaochu Yuan
    Feb 20, 2011 at 12:31
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    $\begingroup$ I definitely agree with the advice of "coming up with your own exercises". It is certainly the best way to learn mathematics. Let me also add the advice of "coming up with your own proofs of results in the text". If you find it difficult to come up with your own exercises, then it is also a good idea to search online or look at many different books to find good exercises. $\endgroup$
    – Amitesh Datta
    Jun 25, 2011 at 1:35
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    $\begingroup$ @Quiaochu Yuan: What did you do if you could not solve ''your own exercise''? Normally, I got alot of my own problems when I study a new mathematical theory, and I could not answer them. Then I try to abandon them and come back in a beautiful day :D $\endgroup$
    – Arsenaler
    Mar 29, 2012 at 2:12
  • $\begingroup$ Your question is probably appropriate for the nearly-in-beta-SE area51.stackexchange.com/proposals/64216/…. Check out the proposal and commit to it if you're interested. Then we can get it off the ground and get the site in beta! $\endgroup$
    – Xoque55
    Mar 3, 2014 at 4:58

14 Answers 14

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The only way is to put in a lot of time, to not give up, and to keep studying. As Agusti Roig mentions, the importance of doing exercises cannot be downplayed - working things out for yourself is absolutely necessary when trying understand things more deeply.

For me one of the most important things has been asking myself questions. Lots of questions Why does the theorem have theses hypothesis? Why this definition? What was the key idea in the proof? Can I apply this idea, this method of proof to other questions? For what kinds of questions will this method fail, and why does it fail? Is there another proof? Terence Tao has a good blog post about this titled "Ask yourself dumb questions – and answer them.

The more time you spend thinking about a subject, the better you will understand it.

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    $\begingroup$ I agree with you on asking oneself questions. That really motivates self-learning. $\endgroup$
    – Tim
    Feb 20, 2011 at 1:49
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    $\begingroup$ Great advice, Eric. I forgot how important it to study with a question in mind. $\endgroup$
    – user7273
    Feb 20, 2011 at 1:52
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    $\begingroup$ "I don't believe in "shortcuts."" -- "There's no royal road to geometry". $\endgroup$
    – The_Sympathizer
    Aug 7, 2013 at 0:32
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First, besides the book, you need paper and a pen. Second, you must do the exercices of the book. Third, you must do the exercises of the book. Fourth, you must do the exercices of the book. Fifth..., did I mention you must do the exercises of the book? -Do them!

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    $\begingroup$ I think you should also mention to "Do the exercises of the book". $\endgroup$
    – user17762
    Feb 20, 2011 at 1:36
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    $\begingroup$ Curious, should one do every exercise in the book? $\endgroup$
    – yunone
    Feb 20, 2011 at 1:39
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    $\begingroup$ Sure. I guess this is good advice. I mean if you're trying to actually learn the material or something... $\endgroup$
    – Matt
    Feb 20, 2011 at 1:39
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    $\begingroup$ @yunone Not always, and depends on the book! $\endgroup$
    – Eric Naslund
    Feb 20, 2011 at 1:45
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    $\begingroup$ Hehe... yes. Thanks... I was thinking advice more along the plane of web.stonehill.edu/compsci/History_Math/math-read.htm although even this just seemed like re-phrased common sense. $\endgroup$
    – user7273
    Feb 20, 2011 at 1:51
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Two points :

1) Work the proofs. Work them hard. I said work, and not learn, for a specific reason. Many students just think if you are able to recite the proof, it's ok. What you have to do is (in my opinion) :

  • Understand which part of the proof is a key idea. Not all statements are equally important.

  • Understand why there is this condition, and not another one. Aka find counter-examples. It's one of the first exercice you should do. It can be really hard for some theorem, but it is very instructive. As already stated, ask yourself questions !

2) Do exercices. A lot of them. And by a lot, if you are undergrad, I mean a lot.

  • Never skip a correction even if you have the good results. You should always try to see differences between your answer and the book one.

  • Learn to be good in mental calculus. Yeah, it's annoying to work, but you have to do it.

  • Do not work only short exercises. Working problems helps to get the "big picture", especially when you mix tools like analysis and algebra together.

  • If you are stuck in an exercise after some time, ask help, dont put it away. Not be able to do an exercise on your own is the way to make some progress. Even the best mathematicians ask help to their collaborators.

But the most important point (third one) is to be patient, and to enjoy working. Math is not a quickly rewarding field, but it's worth it.

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    $\begingroup$ +1 for mentioning "Understand which part of the proof is the key idea." $\endgroup$
    – Eric Naslund
    Mar 6, 2011 at 0:20
  • $\begingroup$ +1 for math is not a quick rewarding field but it's worth it. $\endgroup$
    – Arsenaler
    Mar 29, 2012 at 2:22
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    $\begingroup$ Can you explain what you mean by being good in mental calculus, and what are the benefits of this? $\endgroup$
    – littleO
    Nov 23, 2012 at 1:24
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If “staring” means “I have no thoughts”, then you are probably stuck at some tough place. Mathematics was developed for centuries, do not expect of yourself to develop it by your own in minutes. Put in words what you are stuck at, ask people.

I'd found revealing to switch between textbooks on the same subject, as textbooks have different strengths and weaknesses. However, this method has its own disadvantage, because you should integrate knowledge from different books on your own. They may use different notations etc.

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    $\begingroup$ I think different books may have different view point and in my case, if I get troubles with chapters, definitions,... I switch to other books. The advantage is that you can see the bigger picture and improves your synthesis ability :) $\endgroup$
    – Arsenaler
    Mar 29, 2012 at 2:18
  • $\begingroup$ Thinking about how much "mathematics was developed for centuries, do not expect of yourself to develop it by your own in minutes" would have helped me if I had listened in the beginning. $\endgroup$
    – Powder
    Jun 30, 2022 at 2:01
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To add one more idea, which may work for some of you (it works great for me): try invent applications of what have you just learned. Naturally, they can be purely theoretical uses and have nothing to do with applied math.

  • What corollaries follows from the theorem? Does it allow you to gain any deep insights or to work out some intuition?
  • With what other lemmas you can combine it? What do you gain?
  • Can you construct a non-trivial example for it? Is the result meaningful in any way?
  • Sometimes: can you find a non-trivial example in real world (i.e. formulate the theorem using real-world entities, e.g. for a sphere you could use the globe, for some sequences you could use stock market prices, also there are many real-world posets, and all the probability theory fits into real life just perfect)?
  • What are the downsides of the theorem (e.g. is it only existential or maybe computationally impossible)? Are there any nice things that are just beyond the scope of the theorem? What would you need to have to close the gap, is it possible?

I have used that approach for quite some time with success in wide range of domains including not only calculus, functional analysis or topology, but also logic, abstract algebra, combinatorics and category theory. This also works in problem solving -- starting with special cases may help you a lot if you are stuck (this is also one of the advices given by Pólya in his famous How to Solve It).

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A math book usually represents a way of thinking about a topic, a perspective on that topic. Hence, you have to agree with the authors opinion on how to present the topic in order to get a useful learning approach. (Contrary, if you completely disagree with the book's perspective, you may regard it as a challenge.)

In my experience, learning achievements are enhanced if you let the knowledge flow through you own hands. This means you have to put down the content of the book in a way which fits your way of thinking best. The author will probably have a (slightly) different perspective than you have, due to taste and ability.

A good approach towards a book is: "The author is lying." - each line of the book has to be justified. If you can't do so, you do not understand the topic in full.

Furthermore, I have made the experience it is inevitable to spend lots of time with the matter. Difficult and inaccessible proofs may unveil if you read it over and over again (imo, good examples of these are Hörmander's books on linear pde).

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It depends on what level you are at. But Schaum's outlines can introduce you to the basics of topology, group theory, abstract algebra, and several other areas in mathematics. Just do the exercises.

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I like your question. For me, I prefer to get a big picture first. If books that I have do not work for me in this way, perhaps because of my weak math background, I will search online for a comprehensive overview and mostly be led to Wikipedia. More details will be pursued when needed.

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Spending lots of time doing math is necessary but not sufficient. To do math effectively, there must be an intensity to your study. I wear earplugs while I work, and use the StayFocusd extension to Chrome so that when the going gets tough, I have to actively decide to start dicking around by going through the somewhat involved process of disabling that app.

I also have to keep in mind that Wikipedia is not always my friend. It is rarely inaccurate, but the writing style is horrid and the notation is invariably different than that used in your text. Plus, when your text says something inscrutable, it's best to just battle it rather than pray to God that Wikipedia will have that magical statement that makes it facile. Once you're on Wikipedia, you can get hopelessly diverted for hours. Limit your Wikipedia time per day to (say) 15 minutes via the StayFocusd app.

When doing homework, write it out on paper, then TeX it up for submission to your professor. I cannot tell you how many times I've found huge gaps in my proofs while typing it up. Never TeX before writing a proof out completely on paper.

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I agree with tim. If ur a right brian learner, u should first get a general idea of what the topic is trying to do. Ex: calc 1 is all about integration and differentiation, after u get the general idea then work in the details

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Try to learn a subject you find more difficult by reading more then just your notes from class. You can rent math books from your local libary or even at your school. In my case I used to watch math videos about several subjects I had problems with and it helped me alot. A place where you can find a collection of several math video website is: http://www.efficientmath.com/

As last almost all teachers say "practice, practice and practice" and it's true but practice the subjects with a focussed mind. Another extremely helpful website is Wolram Alpha, take some time to learn how to work with this website and it'll be a great help in checking your solved excersises and learning.

Regards

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there is only one way to study maths...LEARN THEM! this is the bit that is always missing in maths. having acquired a certain amount of knowledge and mental dexterity is is necessary to 'learn' how to do a new problem. this ,learning is done by

  1. get an example
  2. go through it and through it ( with pencil in hand) until you understand it and you have clarified what steps to take at each stage.
  3. write the same question out on a rough page
  4. do it without looking at the example
  5. if its right great!...If its wrong look at example and un stick yourself. As you would do if you were learning anything else )
  6. once you understand the basic premise and have learned the basic method do questions.
  7. identify the anomalies that can be involved and again 'learn' how to deal with them. ( this is where you need the mathematical knowledge and mental dexterity) shouldn't be too much of a problem if you have a solid foundation of the basic method...no matter how complex! There is alwaws a basic framework of steps to use as a base for more complex questions on a topic.
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Mathematics nowadays is a very rich field and has many applications. Doing the exercises, fighting with your own questions... is very good advice. I just want to mention another interesting activity, that is finding the connections between various fields of maths, like the connections between commutative algebra and representation theory or commutative algebra and combinatorics... It may not have connections in the whole subject, but it may have in some specific theorems. Finding such connections can help us seeing the bigger picture and may be some astonishing proofs...In my case, it motivated me a lot.

You can also read the paper "Advice to a young mathematician" here.

it contains very useful advice of Micheal Atiyah, Allain Connes...

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I gave my answer to this question here: What is the proper way to study (more advanced) math?

Hope you find it helpful.

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