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I am an undergraduate really passionate about the mathematics and microbiology. I have few big problems in learning which I would like to seek your advice.

Whenever I study mathematical books (Rudin, Hoffman/Kunze, etc.), I always try to prove every theorem, lemma, corollary, and their relationships in the book. Unfortunately, that determination has been demanding huge time consumption; sometimes, it takes me days to fully understand and able to prove materials in the few pages of book. I am willing to devote my time to understand the topics, but I also wanted to devote time to my undergraduate research projects and other courses. Recently, I started to depend a lot more to the proofs presented in books and websites (like MSE), which has been causing a huge guilt and fear that I am not making the knowledge into my own.

Despite my effort to prove/solve every problem per chapter, I found myself to skip some of the problems and move on to the next chapter, which resulted huge fear as that means I did not fully understand the materials..

  • How do you read the mathematics books and make knowledge on your own?
  • Is it absolutely recommended to prove everything and solve every problems in the book?
  • Also is it recommended to devote more time to the problems than exposition preceding the problems? I found myself devoting a lot time to the actual expositions in the book as I like to play around with definitions and theorems, try to come up with my own ideas, and formulate my own problems (I actually found that making my own problems is much more fun than problems presented in the book).
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closed as primarily opinion-based by user223391, Zev Chonoles, JonMark Perry, user299912, Asaf Karagila Jun 27 '16 at 7:43

Many good questions generate some degree of opinion based on expert experience, but answers to this question will tend to be almost entirely based on opinions, rather than facts, references, or specific expertise. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ For a book like Rudin, a student would typically spend a whole year to cover it. So you can compute how many days per page you should average to do it. $\endgroup$ – GEdgar Jun 26 '16 at 18:00
  • $\begingroup$ More likely you'd do Ch. 1-7 or 8 in a semester, then switch to another book like Spivak's Calculus on Manifolds or Munkres' Analysis on Manifolds for the remaineder. But you do spend a lot of time on these topics. $\endgroup$ – Batman Jun 26 '16 at 18:29
  • $\begingroup$ After Chap. 8 of Rudin, I am planning to move to Munkres or Hubbard/Hubbard. $\endgroup$ – MathWanderer Jun 26 '16 at 18:48
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    $\begingroup$ It sounds like you have impressive discipline. I would think that it is obsessive-compulsive to do everything in a book, although obviously it could never hurt. If an exercise strikes you as routine then it can probably be skipped. I don't think that any general advice can be given beyond -- keep up the good work. $\endgroup$ – John Coleman Jun 26 '16 at 19:09
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    $\begingroup$ This is not primarily opinion based. This is entirely opinion based. $\endgroup$ – Asaf Karagila Jun 27 '16 at 7:44
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Overall answer: no.

I struggled with the same problem as you for quite a long time and, in hindsight, I think I could have spent my time more wisely. Here are my current general guidelines at the time of this post. They may or may not work for you. The overall philosophy I employ is that exercises are usually there to get you comfortable with the material: you are probably usually expected to remember the theorems and (mostly) forget the exercises once you have done them. I expound on this below.

How do you read the mathematics books and make knowledge on your own?

Nowadays I spend more time thinking about the material in the text than doing exercises. I treat the theorems as problems and try to see how far I can get in proving them before reading the proof. Importantly, I don't spend all day doing this! A good strategy might be to set yourself a goal, such as "I want to get to such-and-such page by the end of the week", and pace yourself accordingly; if you spend thirty to forty-five minutes thinking hard about a theorem and having no ideas, perhaps take a peek at the proof and continue from there (alternatively, skip that proof and come back to it later).

If the book has examples in the text, read and understand as many as you can. Otherwise, be sure to allocate generous amounts of time towards coming up with your own examples. Note that this is not necessarily easy, and is arguably the most important stage.

Finally, if the book has exercises, be sure to do at least some of them and try to understand the general idea of the rest. Don't be afraid to ask and try to answer your own questions which might be inspired by some of the exercises.

One downside to this method is that it doesn't work for all books and all subject matters. Some authors place key theorems in the exercises and proceed to use them later, expecting you to have proven them yourself. The hope is that for each topic there exists a book for which this method is not useless.

Another downside is that, in rare cases, too many of the exercises are interesting (this appears to be the case with Rudin, among other books)! In this case it's definitely up to you how much time you spend on the exercises. Allocate time according to how interesting you find the exercises and how comfortable you are with the material in the text.

Is it absolutely recommended to prove everything and solve every problems in the book?

If you can do this and still lead a comfortable life, then by all means do so (if it doesn't hurt, then it can only help, right?). Unfortunately, attempting to do this will probably make life less than comfortable for you, so I would advice against being this extreme.

However, the point of this "advice" is that you should get as comfortable with the theorems and the examples as much as you possibly can, and this is good advice. I would advise just thinking about stuff as much as possible. Thinking about maths in the shower, on the way to the shops, while cooking dinner, etc. will get you used to thinking about the topics that interest you.

Also is it recommended to devote more time to the problems than exposition preceding the problems?

This heavily depends on both the book and the reader. However, if you prefer to come up with your own examples and fiddle around with the theorems, and if the book you are working from is not expecting you to prove key theorems in the exercises, then I would say this approach is a good substitute for doing exercises.

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  • $\begingroup$ As a caveat to this answer, there are definitely cases where I would explicitly not recommend the approach described here. For example, although I have not read these specific books, Atiyah and Macdonald's Commutative Algebra and Hartshorne's Algebraic Geometry are probably two examples where it might be best to just work as hard as you can. But then presumably you are reading a book like Hartshorne because someday you want to do research in algebraic geometry, whence it would help to know the main (?) reference/textbook inside-out and back-to-front. $\endgroup$ – Will R Jun 26 '16 at 23:45
  • $\begingroup$ Solving every problem. I would go further than you. I think this is wholly wrong. Many people use (easy) exercises as a "comfort blanket". You have to spend time (preferably lots of time) solving problems you can only just solve. A good rule for researchers is never to read the proof of a result (especially in a journal) without trying to prove it yourself first. $\endgroup$ – almagest Jun 27 '16 at 8:39
  • $\begingroup$ @almagest: In my experience, I can only just prove the theorems in the book. Presumably, this is unsurprising: if the proofs were easy, many authors would not bother to write down the proof. Is this not the same as being "only just" able to solve a problem? What's the difference? At some point one must acquire the skill of learning mathematics without the aid of pre-made exercises; then it will be useful to have experience proving the theorems by oneself. Ultimately, the question is "why do exercises?" and my answer is "to get comfortable with the material." Proving theorems does the same. $\endgroup$ – Will R Jun 27 '16 at 9:20
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    $\begingroup$ Yes proving the theorems in the book is great! :) If you could prove them all that would be brilliant. I am sad that problem books have not worked out better. The paradigm was Halmos' Hilbert Space. But more recent ones have often dumbed down the problems, sometimes to an absurd degree. However, this approach is really only for those who will excel at research. It is wildly unrealistic for many maths undergraduates. $\endgroup$ – almagest Jun 27 '16 at 9:28
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Mathematics is kind of a subject that is fun as well as scary. I always know the concept and do one or two problems per concept. As solving each & every problem is time consuming and is of no use. My advice is practice all the concepts and theorems, and do two or 3 different types of problems which based on same concept. It helps us to know which concept to apply for certain problem. If you got the concept perfectly, it's application is not difficult, so have no fear and move forward. This subject is very easy when the concepts are understand well.

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    $\begingroup$ Thank you very much for your advice! I actually spend a lot more time on the concepts and expositions rather the problems as I like to spend my time toying around with the theorems, adjusting their conditions, and try to come up with my own proofs and ideas. I actually like to make my own problems, which sometimes surprisingly similar to the problems at the end of chapter. $\endgroup$ – MathWanderer Jun 26 '16 at 19:04
  • $\begingroup$ Lot of concepts have a chance to be related and their analysis may differ. Same concept with different applications is also possible, give more time to these kind of concepts and theorems. $\endgroup$ – Bhavani Chandra Jun 26 '16 at 19:15
  • $\begingroup$ Also, I happen to spend more time on expositions (concepts) presented in the book. I like to play around with those definitions and theorems, deduce my own corollaries and trying to prove them, try to come up with counterexamples, try to generalize the specific conditions, impose new spaces, try to apply different branches of math, and formulate my own questions (which sometimes surprisingly similar to chapter problems). However, due to that result, I could not spend more time to the actual problems at the end of chapter. $\endgroup$ – MathWanderer Jun 26 '16 at 19:21
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I would suggest you read every problem, and in your head if you can see the direction pretty clearly then no need doing that, generally big texts do have repetition, but concise books meant for only problem solving without any theory do try to make sure each problem is unique.
As far as second part of your research goes, I believe your approach gives you more fundamental clarity but definitely not an approach to get 'grades', if you're planning on doing research you should carry on this attitude but if not I would suggest some problem solving.

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  • $\begingroup$ Thank you very much for tour advice. I always do problems that introduce new concepts or ones that seem to be challenging and interesting to me. The problem is that I fear that every problem is designed for increasing my knowledge, but I have to skip quite few of them for time management. I assume you mean Rudin's PMA is one that you described as "concise and problem-solving". Since I want to continue my research, will it be a good idea to spend a lot of time analyzing and playing with expositions in the books? $\endgroup$ – MathWanderer Jun 26 '16 at 18:45
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please be aware that this answers stems from personal experience and is meant more to apply when referring to books such as "calculus year one" or other more generic books that are intended to be large pools of practice problems in early college/high school caclulus/precalculus textbooks. Obviously there are books that are intended to be thouroughly read (but they are probably much shorter/intended to be read over a longer period of time)

Should you solve every single answer in the entire book?

If you actually can in a modern math textbook, then you are certainly more skilled than most A-level students! Either that or you have never had required homework and so you push yourself far too hard... If you notice carefully, each section in modern textbooks maybe have a quarter of the problems required for solving. I have actually calculated problems in books over the years. One book had 30,000 problems in it! That is literally a book of problems! Of course, that is extreme. Most modern (2-semester) books probably have 2,000-5,000. if you can do all of the problems in the entire book, while enjoying them, within the recommended study period (3-4 hours per hour of class time), then definitely do it. Wanna know why? Well, if you are that intellelectually capable, then you shouldn't let your brain get lax. Think of it this way: once you do the required homework you usually are not studying to learn the material. You are studying to excercise your brain.

However, do not force yourself to do all that extra work out of a feeling of obligation or a need to learn. There is a reason why professors dont make students do every problem: it is unfeasible without getting burnt out.

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    $\begingroup$ (-1). Your tone is rude and dismissive, and you're taking the most extreme cases and applying it to the general case. Also, the fact that an instructor may only require a quarter of the textbook problems done is completely irrelevant. $\endgroup$ – T. Bongers Jun 27 '16 at 2:22
  • $\begingroup$ @T.Bongers I am not being rude. I am telling him that it isn't neccessary which is what he asked. I also point out that is, in fact, a bad idea to attempt to do so. $\endgroup$ – The Great Duck Jun 27 '16 at 2:22
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    $\begingroup$ "Are you insane, or are you a genius?" I find that rather rude. And there certainly are contexts where solving (or at least being familiar with) every problem in the book - e.g. classical references like Rudin, Hartshorne, and a few others. The other answers that had a degree of nuance didn't miss this. $\endgroup$ – T. Bongers Jun 27 '16 at 2:25
  • $\begingroup$ @T.Bongers But not a modern college textbook. $\endgroup$ – The Great Duck Jun 27 '16 at 2:26
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    $\begingroup$ The issue is not modern versus original. There are about a half dozen commonly graduate textbooks that I can name off the top of my head where I would say that doing every problem is useful in understanding the material, feasible for a student at that level, and should be done for students working in that area. Since your answer is based on personal (rather than educational) experience, it's going to miss that point. (Regardless, I'd prefer to not get into a much longer discussion about this. Suffice to say that I think this answer misses the point and leaves a lot to be desired). $\endgroup$ – T. Bongers Jun 27 '16 at 2:53

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