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When I'm taking courses in Calculus I & II, and Linear Algebra, the lecturers are always telling us to do as many exercises as possible. But when it comes to practical situation, I realize it takes so much time on exercises especially calculation parts.

So I'm thinking about whether it's really necessary, because I found that I love proof exercises mostly, with that, I could think more, and make connections among many theorems, to see how they are related and what it really means. But for many 'calculation' exercises(say 400-500 problems for each chapter totally) I realize for most of them are just substituting the data into the formulas, and doing calculations are somehow less helpful in understanding compared with proof problems or just read & re-read text part and thinking.

So in this case, I want to ensure that whether or not what I'm doing is suitable, because lots of people around told me that it's dangerous. Thus, I want to know more advices or explanations. (It's somehow impossible to focus both proof and calculation as much as possible, and I've enjoyed and benefited from the former but not latter one.)

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    $\begingroup$ In the long run the proof problems will do more for your understanding than the computational exercises, especially if you’re a good mathematics student. As a practical matter, though, you need to do enough of the computational exercises to be able to do them reasonably quickly and easily, simply because you’re likely to be tested on computational skills on exams for which you have a limited amount of time. $\endgroup$ Oct 14, 2011 at 21:45
  • $\begingroup$ If after glancing at a problem, you are absolutely certain that you can do it, and quickly at that, it makes sense to go on to the next problem, say 50 percent of the time. $\endgroup$ Oct 14, 2011 at 21:47
  • $\begingroup$ @André Nicolas , so does it mean that I don't need to force myself to really 'do' every problems in the textbook(especially in lots of books they have even&odd numbered problems, I always just finish odd ones and have answers to check, except for proofs, finish all of them) ? Just searching more 'valuable' or 'beneficial' problems to focus on ? $\endgroup$
    – Xingdong
    Oct 14, 2011 at 22:01
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    $\begingroup$ @That is not precisely what I wrote. Do look at every problem, but there is no need to do more than half of the problems that you are certain* that you can do. Even better would be a time machine, to take you back to when you were $13$, and had boundless available time. At $13$, you could do every problem. $\endgroup$ Oct 14, 2011 at 22:07

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I would agree that the proof exercises are more interesting. However I would still recommend that you complete a lot of the calculation exercises. If you are really focused, 500 exercises really shouldn't take that long to complete. Unless, that is, they aren't as easy for you as you believe. Understanding an exercise is not the same as solving it quickly.

You might not think that you need the mechanics but if you practice the mechanics, you have more brain power left to do the interesting parts. Exercises are not just about understanding, they are also about execution. You should be able to fill out the details of a calculation without hesitation or even thinking. That only comes from practice.

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