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I know there have been many popular threads about how to learn math, and that it is universally agreed that solving problems is a key part of the learning process.

However, I would like to hear what is specifically wrong or inefficient or suboptimal about the following approach to learning math, which emphasizes understanding proofs but not solving problems at the end of the chapter:

1) When you read a math book, before reading a proof of a theorem, first try to prove the theorem yourself. (It's hard to say how much time should be spent on this step.)

2) When you read the proof, try to understand it very clearly. This includes trying to understand the intuition behind the result, and understand how someone might have thought of the proof (even if you were unable to prove it yourself). Possibly consult other books for more enlightening explanations.

3) Check that you understand the theorems and proofs well enough that you could state the theorems and write down proofs yourself. Write down the proofs on paper or using latex, trying not to look at the book for help. Ideally, you might often find that your explanation is a little more clear, in your own opinion, than the explanation given in the book.

Throughout this process, no time is spent solving end of chapter problems.

Please tell me what you think is suboptimal or wrong or misguided about this approach to learning math.

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  • $\begingroup$ It is not meaningful to discuss optimality without stating an objective. $\endgroup$ – mathreadler May 8 '17 at 22:46
  • $\begingroup$ Well, let's say that the objective is to master / understand deeply a particular math subject in a reasonable amount of time. $\endgroup$ – eternalGoldenBraid May 8 '17 at 22:48
  • $\begingroup$ Maybe in that case you are right. It could simply be that not everyone agree on that objective. $\endgroup$ – mathreadler May 8 '17 at 22:57
  • $\begingroup$ @mathreadler Out of curiosity, what other objectives do you have in mind? Developing problem solving ability? $\endgroup$ – eternalGoldenBraid May 9 '17 at 7:05
  • $\begingroup$ There can probably be many different reasons/objectives. Just wanted to give you the idea that differing objective functions could be the case. $\endgroup$ – mathreadler May 9 '17 at 8:41
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Learning is a highly personal experience; what is optimal for you might not be optimal for others.

But while some people prefer reading maths and other people rather have it explained to them, there's a general consensus that 'getting your hands dirty' is essential to really mastering, as you call it, a topic, rather than just being able to recite it.

That is what exercises should be for, although this is not always the case in practice. Sometimes exercises might be solely computational in nature, in which case they help you understand certain algorithms, but not necessarily why they work. A good mathematics books, though, provides exercises which are like theorems, in that they require creative thinking and applying the theory in new and varied ways. In a good book, this is probably a more optimal way to study, since lemma's and theorems might either be too hard to come by yourself, or so technical that they do not necessarily provide deeper insight either. Exercises can be chosen precisely to sharpen your intuition and skills further.

To sum up, try different methods of studying (maybe different books if the exercises in your book are lacking) and in the end, pick whatever suits you. But as quasi said, you have to be honest with yourself about your understanding, and creative exercises leave you little room for pretending.

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There's nothing wrong with treating lemmas and theorems as exercises; in fact, in my opinion, if there's no time pressure, that's a great way to achieve a deep understanding of the subject.

But that should be complemented with actual exercises, else you could be fooling yourself about your level of understanding. If you can't apply the lemmas and theorems to concrete scenarios, what is the value of the learned material? How do you know you have truly mastered the material (as opposed to just knowing how the proofs progress, line by line)? The exercises confirm your knowledge, and amplify your understanding as you work through the issues that arise in diverse examples.

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