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I've spent most of my studies focused on learning proofs of theorems, as that was what took the most time to learn well enough in order to do well at exams. After seeing how some mathematicians I know are able to solve some problems with ease, I've started trying to improve my problem solving ability and general control of a topic.

The first goal I set for myself was to go through Isaac's Finite Group Theory, and do all the exercises. But I'm not sure it's working well enough for my goal - after doing the exercises in a few of the sub-chapters, I still feel like if I were to given a similar type of problems as the ones I did, I would probably not be much faster than I was when I did the first set.

My aim isn't to just somewhat understand the topic, and feel like I can do the exercises - I want it to become something relatively easy, that doesn't take effort, that's almost 'intuitive' in some sense, and that I can do it without much focus.

But I don't know whether it's a good idea to aim for this so directly - maybe it will just (hopefully) come with time as I improve generally and get a slight understanding of some topic... but I have considered that maybe I should just do tons of exercises of a given type, until it becomes easy and fast, and then move on, rather than moving on after I do the exercises at the end of a chapter.

Also, I do ask my own questions concerning the topics, the exercises, and play around with it a little bit, but only as much as I feel like, as I enjoy it more to just move to the next exercise - which might not be enough.

I would appreciate opinions on whether trying something like this (very large amount of exercises of 'similar' difficulty and topic, and possibly even repetitive practice of a certain type exercises) is a good idea - and whether even directly trying to get control of a certain topic is a good goal, etc.

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    $\begingroup$ To be fast the solution is simple PRACTICE, PRACTICE,PRACTICE. This will improve your speed on questions you have seen before, but this will not necessarily improve your speed on questions you have never seen before. It is an unrealistic goal to think you would be able to solve all questions with lightning speed. A more realistic goal would be solve tonnes of questions that require only a few assertions and be very quick with them. Then you can use your understanding to break larger problem into smaller parts, once you have broken the problem down you will be able to solve it quickly. $\endgroup$ – Sonal_sqrt Dec 20 '17 at 5:54
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    $\begingroup$ Depending what you are trying to do, there is very little purpose to being able to solve exercises quickly. If you are doing research, no one is timing you. Also, practicing similar problems repeatedly is only going to help you solve problems that look just like them. What is your goal? I'm not understanding the motivation for your question. $\endgroup$ – Morgan Rodgers Dec 20 '17 at 6:02
  • $\begingroup$ My goal is for certain topics/problems to feel intuitive, and very easy, so that I can use them very efficiently as tools, for solving more complicated problems. My situation seems similar to say.. elementary school - by the time you're learning to multiply large numbers, wouldn't it be a good idea that you can sum them up quickly and easily, without thinking about it? I'm not sure how much of a good analogy it is, but the point should be understandable. $\endgroup$ – John P Dec 20 '17 at 6:12
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    $\begingroup$ I cannot agree enough with @PiyushDivyanakar enough that practice is key. It also may be valuable to study the proofs of standard results from the texts. You shall see that in every field of math there are techniques that come up again and again. (For example, in analysis it is very common to write $\epsilon = \epsilon/2 + \epsilon/4 + \epsilon/8 + \cdots$. One might never think to do this having never seen it before, but after seeing it a bunch, this becomes an almost obvious trick.) It can be helpful when you read someone else's proof to ask yourself: "how could I have come up with myself?" $\endgroup$ – eepperly16 Dec 20 '17 at 6:23
  • $\begingroup$ @john p Different types of problems require different levels of knowledge for them to be intutive. The math in college isn't the sort where you mechanically calculate stuff unlike elementary school. In college it is very analytical. So being able to solve the problems more times than being unable is already a very high bar. Proofs aren't trivial mechanical problems, or we would have computers do it. For you to be extremely fast with proofs you need either an algorithm that can prove anything or a vast wealth of experience and knowledge in addition to creativity. Such algorithm doesn't exist. $\endgroup$ – Sonal_sqrt Dec 20 '17 at 6:34
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Different people learn differently, so take the responses to your question, even this one, in that light.

  • Ramanujan learned math by reading a book that was just a list of Theorems, which he set about proving, and then went on to prove his own.
  • Many people think that solving problems is the best way to learn mathematics.
  • Whether you should try to do every problem in a text book depends on the number and quality of the problems. If there a lot or they seem repetitious, you could leave half or more for future review. Even if there are only a few, the author might sneak in a very hard one that you could skip. Another problem with text book exercises is that they typically focus on what you have just learned, so you are focused on that material.
  • More important are the problems you set for yourself as you are reading and trying to understand what the author is saying and its implications. Making up your own examples is a good exercise too. Some of the best math books are Do It Yourself books; they come with no problems at all.
  • I recommend reading the 10 page essay "How to Learn Math" which is chapter 7 of Ian Stewart's book "letters to a young mathematician."
  • I strongly recommend against spending time doing problems over again, esp. text book exercises. What happens is that you tend to learn the problem, not the math.
  • But the best advice to improve your proficiency, intuition and speed, is as the first commenter said: PRACTICE, PRACTICE, PRACTICE, not just solving problems but talking about the subject, reading other peoples solutions, reading published papers, and books.
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