Question: Stack-Exchange,
I am a freshman in Mathematics Department.

I always learn mathematics the hard way. Everytime I learn mathematics on my own by reading from Mathematics textbooks, I would read the definition, try to figure out what it really means, stick new definitions which concepts that I already know, prove theorem without ever reading given proof.

But recently, I find this method cumbersome. In order to get a deep level of understanding those concept and solve a good amount of problems, I trade it off with a lot of my time, and my energy.

My seniors tell me that I should learn from different source of material to get more insight view, and separate the processes of learning theory and problem solving. I should read all the theory before trying to apply them to save times. But I'm used to the way of Pólya's mouse, so I am worried that I won't be able to catch up with all my classes like this. I want to try the way my seniors told me; that might be faster. But I want to know if the method would come with a loss of depth in understanding.

So I am asking for help. Also, please give me some tips to boost up the performance and the quality of learning mathematics all by myself. Thank you all very much.

Answer: 
I will try to give an idea of what I did when I was in your situation (It worked for me, and I hope it will work for you too).

I had a teacher who used to say self learning is best learning. I was one of the strong believer of his philosophy. He once said, being a teacher I can only do one things for you which is Introducing the beauty of subject to you. The main thing in learning is the efforts you give.

Now since you are studying such a great subject on your own, You should have two things.

$1$. **Interest in the subject**

$2$.**Patience**.

Mathematics is a language that attract at first, then repel and then stick to you forever. You just have to bear the period of repulsion.

The period of repulsion comes when you can't solve problems, the formulas go over your head and at that time you just start reading the solutions of problems and start convincing yourself that "Yes I am learning (Though you are not)".

The basic idea is to stick to a topic until you have a complete belief that you can do any question of that particular topic. The mathematics means solving Problems and if you can't solve problems then you can't do mathematics.

I agree with your seniors but to a small extent. You should do problems from different books but avoid use of too many books as that leads to confusion.

> 
> So, I will sum up all the discussion regarding self learning in points as:
> 
> 
> $1$. Bring your interest in mathematics.
> 
> 
> $2$. Be patient and don't lose confidence.
> 
> 
> $3$. Do problems as much as possible.
> 
> 
> $4$. Avoid use of so many books until you have completed one.
> 
> 
> 

Many other people can disagree with my opinions but that doesn't matter. I respect opinion of everyone that is why if you think I m wrong somewhere leave comment.Hope it helps.
I wish I had you for a teacher or a student. I am long past the age when I can take classes in school, and everything I do is driven by curiosity. Recently, I became interested in retro-tech and began with the table of four-place logarithms we were given in high school. (Before computers or calculators log tables were the one of the instruments by which human computing power could be amplified.) Wow! What a bunch of errors in that thing, but I am finding ways to weed out the worst mistakes and improve the accuracy. 

I think I have discovered things about log tables that nobody else knows. It is an empowering feeling. And some day, when some conference needs some kind of paper, I will have one pre-generated. Most likely this would be the HHC 2017 conference dealing with HP calculators.

It is a good way to learn. Never quit!