It can be really hard. I'd suggest: 1) Don't take on something too ambitious, it will demoralise you, 2) do _all_ of the exercises, even (especially) the ones that look easy, 3) Learn the basics of LaTeX so that you can ask coherent questions on stackexchange mathematics, 4) don't ask questions on mathoverflow until you actually reached research level.
that is a tall order. Not doing any exercises is the same as not studying at
all, I concede, but more bang for buck can be gotten not getting bogged down
by exhaustive problem solving.
I stand by it, if you're not prepared for exhaustive (and exhausting) problem-solving then don't start down this road, try something easier like Philosophy or bear-wrestling.
Have you written a maths textbook? Do you know how arbitrary some of those
exercises can be? Most textbook writers cobble together sundry, some of which have only
the slightest relevance to the chapter at hand. Maths generally is a
leisurely pursuit, particularly when "self-studying", and will not require
slavish turning of all stones.
I agree with you. Doing everything is not always the best option for many people. If you can do it, great. But sometimes it's better to just do the most important exercises that are the most mentally stimulating and interesting, rather than invest time in the less interesting ones.
If you have a problem and you are reading math textbook to learn methods to do the problem, it would be a lot easier to study textbook. The only problem is to find such a problem, so the most of the textbook would be relevant to it somehow. Or at least it should seem to be relevant long enough while you are studying. Though you could try to find several problems so they cover all the textbook.
It is a funny way human mind works (or at least mine does), it needs goals. When it have goals, it starts to generate guesses how to reach goals. It tries to check these guesses, some of them it fails to check, so I start reading textbook over and over, trying find answers. I begin to see where I lack proper understanding, and I can turn this lack of understanding into questions. Lack of understanding is vague and I do not know what to do with it, I do not know where to begin, while question is something specific and I know how to search for an answer.
Solve problems!
Caveat: Many books seem to have bad problems. If the majority of the problems seem to be the same problem with different parameters, that might raise a warning.
I read and attempt all sample problems before looking at their solutions. I do most of the problems/exercises, and compare my solutions to those in the back of the book. I might search YouTube for some additional help on some topics. The hard part is when I just don't grasp something. For that, I used to go to my neighbor who was a retired physics professor. Nowadays I'd look for a forum, probably Reddit, to get specific help.
Some people say to do all the problems, while others say to do a partial number. I would say just do as many as you can. Don't set a limit to the number of exercises you do, just do enough to the point you feel content. If you feel as if the number of exercises you did is good enough, then move onto the next section. If you ever feel like going back and completing some more exercises, then that's also fine as well.
However, I do think that mathematics builds on itself. I think you need to do a certain number of exercises that allows you to be "fluent" in the particular mathematical subject. Skipping many exercises is terrible for building your skills in maths.
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[ 4.5 ms ] story [ 32.4 ms ] threadGood luck!
that is a tall order. Not doing any exercises is the same as not studying at all, I concede, but more bang for buck can be gotten not getting bogged down by exhaustive problem solving.
It is a funny way human mind works (or at least mine does), it needs goals. When it have goals, it starts to generate guesses how to reach goals. It tries to check these guesses, some of them it fails to check, so I start reading textbook over and over, trying find answers. I begin to see where I lack proper understanding, and I can turn this lack of understanding into questions. Lack of understanding is vague and I do not know what to do with it, I do not know where to begin, while question is something specific and I know how to search for an answer.