How to learn math properly?

Question

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.

Answer 1

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!