On trying to achieve control and speed of a mathematical topic

Question

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.

Answer 1

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.