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A little information about me:

I really like learning math,physics,despite the fact that sometimes i rage when i don't find the pattern of a certain problem in less than 30 minutes (but i guess that you gotta relieve the anger somehow,I'm trying to deal with this).

Soon (I hope) I'll be attending a university to become an IT specialist,luckily for me to get major you need to study both Calculus 1 and Calculus 2 + Statistics but,unfortunately,no discrete mathematics. So eventually I'll be learning discrete mathematics on my own just because I'm both interested in it and due to the fact that IMO it is one of the greatest branches of mathematics for anyone who learns about digital electronics/technologies,programming languages (+ it is always nice to know that it can be used in sandbox video-games which have such simple elements of digital electronics such as logic gates to make sophisticated circuits and logical contraptions).

As you might have guessed I've only studied general algebra,geometry,precalculus.Nothing extraordinary.

I have a few questions regarding ways to study math for years to come without guilt,knowing that I'm doing it properly.

First of all,what really bothers me are times when I'm stuck on problems,I know very well that this will ALWAYS happen to me,it is impossible to avoid these situations,what I want to know is:

In order to improve mathematical skills,strengthen your abilities to solve problems,are you allowed to look at solutions,search for hints IF you set yourself a certain limit,after how much time of "no luck" you are allowed to do this?

Like look at the solution only after 30-35 minutes of absolute zero progress.

Is this the right way? Because I've seen VERY negative reaction of some mathematicians on different forums towards this idea,stating that "there won't be any solutions when you reach graduate math" and "it's better to ponder for hours over a problem than looking in the solution".

Note that I will be abiding by this very method (that i'm still searching for) for years to come,for learning all kinds of math.

Another question is about proof learning:

Is it efficient to learn proofs by:

1)reading author's proof,2)doing best to understand every single step of it,3)trying to write down the very same proof pretending to be an author proceeding from the same logic that author has used,4)repeating 3) until proof is firmly stuck in your head?

Thanks.

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closed as primarily opinion-based by Matthew Conroy, Will Jagy, rschwieb, Namaste, Daniel W. Farlow Jun 22 '17 at 16:49

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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    $\begingroup$ About the frustration: there are aspects of mathematics that are unresolved. There are aspects of mathematics that some of the best minds have tried to solve but were unable. There will be questions you won't know how to answer, this is normal. Step away from the problem and return at another time. About learning proofs: mathematics is an art, learn from other artists but do not replicate it in exactly the same way. Develop your own artistic signature. $\endgroup$ – Tucker Jun 22 '17 at 1:54
  • $\begingroup$ This reminds me of the "experience of being a frustrated artist" portion of A Mathematician's Lament. $\endgroup$ – Wildcard Jun 22 '17 at 2:07
  • $\begingroup$ I've deleted my answer and simply flagged the poor-quality question. Next time I'll know better than to try and help. $\endgroup$ – SZN Jun 22 '17 at 2:20
  • $\begingroup$ @SZN mmm.... waffles $\endgroup$ – Omnomnomnom Jun 22 '17 at 3:07
  • $\begingroup$ "reading author's proof,2)doing best to understand every single step of it,3)trying to write down the very same proof pretending to be an author proceeding from the same logic that author has used,4)repeating 3) until proof is firmly stuck in your head?" That's what I do, but I'm a vagabond, not a mathematician. $\endgroup$ – Steven Thomas Hatton Jun 22 '17 at 4:43
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I think it is fair to look at the problem's solution after you spent a certain time with it where you literally made no progress. Your time is limited, and certainly at university level, you don't have much time to make lots of exercises.

A couple of hints when you are making exercises:

  • You must know your theory before you attempt a problem. Proofs in the theory can give you ideas how to tackle new problems. If you don't know the relevant theorems, you won't get far either.
  • Write out all definitions that you have applied to the information you are given. Try to write down as much as you can and try to get as far as possible. If you get stuck, start working backwards: 'How can I reach this conclusion? Well, if I can prove that statement, I am done.' Chances are that you can prove that statement with the information you have and you can connect the two parts of your proof in the middle.
  • Really don't spend too long on one exercise. When you get stuck, move on. Return later. Maybe take a walk and think about it. I have solved many problems where I didn't even have the problem statement at hand.

Also, try not to get frustrated. This leads nowhere.

Good luck!

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  • $\begingroup$ I think spending long time on an exercise is super useful. Any wrong solutions you tried, you know it leads nowhere. If I can say solve exercise in less that 2 min I consider I learned (almost) nothing. And if by chance you find it, this is very rewarding. Of course most of the time you didn't find it but at least you are sure that you tried everything you could try from the theory, etc $\endgroup$ – user171326 Jun 22 '17 at 11:01
  • $\begingroup$ Yes, it is certainly useful. But if you are a university student (thus speaking for what the OP is becoming), you don't have time to make lots of exercises. Mastering the theory takes a lot of time, so not much time left for exercises. So if you spend many time on 1 exercise, at the end of the day you won't have enough time. $\endgroup$ – user370967 Jun 22 '17 at 11:05
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    $\begingroup$ I'm also a university student, lot of my friends got time for gym, etc ... and could still finish their degree. I don't think it's hard to find say 30 minutes a day and spend time on one problem, trying to solve it, even if you know you have good chance to fail. If you just do some easy exercises, don't spend the time to try hardest one, then little to little you forget everything. The only subject from my undergrad I remember very well were the subjects I was doing more complicated exercises. Of course it's not a generality and everybody has his own vision of how to learn math. $\endgroup$ – user171326 Jun 22 '17 at 11:12

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