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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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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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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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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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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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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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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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.
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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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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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.
First, besides the book, you need paper and a pen. Second, you must do the exercises of the book. Third, you must do the exercises of the book. Fourth, you must do the exercises of the book. Fifth..., did I mention you must do the exercises of the book? -Do them!