Studying mathematics as an Undergraduate This is not a question related to a certain topic in mathematics, but more a question about studying mathematics generally. (please forgive me if my vocabulary isn't appropiate, I'm not a native speaker since I grew up and live in Germany)
Well, where to begin with? I started studying mathematics in October 2015, and I already passed my exams in calculus 1 and linear algebra 1, both with good grades. (and the exams in calculus 2 and linear algebra 2 are standing right in front of my door step) Before I started studying mathematics, I was a horrible math student, but when a good friend of mine introduced me to the world of mathematics on an academic level, I knew that this was actually what I wanted to do for the next few years or even the rest of my life.
Despite the typical problems that an average undergraduate math student has to face, I really like the subject itself, and I can clearly see that I get better and better at solving problems and understanding mathematics in general.
But there is one thing that bothers me: I feel like my skills don't develop fast enough, and therefore I don't know in which direction I'm heading right now. I don't know if studying mathematics is really worth it when I have the constant feeling of not being or not getting good enough at what I'm doing. I always think that there is something wrong with my way of approaching problems. (although there are a lot of people around me that tell me they wouldn't be able to solve those kind of problems that I solve, but I guess that's just a result of them not even trying to do so)
I wonder if my thoughts are normal and that everything gets much better when I  give my skills plenty more time to unfold themselves - or if I should seriously consider to adjust myself in a completely new way. Do you recognize yourself when you reflect upon your time as an undergraduate student? 
I would be glad if you could advise me on this topic. 
 A: Your thoughts are certainly normal. In fact, one could generalise and say that no mathematician is ever a success in his own eyes. How can he be, given that so much of maths involves working for weeks on a problem which, once you have solved it, has a solution that can be understood in minutes?
I strongly recommend that you read section VIII.6, "Advice to a Young Mathematician", of The Princeton Companion to Mathematics. It has four top mathematicians as co-authors and I think you will find it reassuring and encouraging. (The whole book is of course a pleasure, being mostly written by Europeans).
Beyond that, only the people who know you can really answer your question. But I really do suspect that you are having too much self-doubt too early.
You will derive great benefit from the maths you are doing even if you do not go on to be a professional mathematician. You will have developed and exercised your mind. So the time you spend on maths now will never turn out to have been wasted.
It is good that you are facing these questions now, because genuine research mathematics involves long periods of dryness and not being sure if what you are doing makes sense. The sooner you get used to dealing with those feelings, the better. Even Richard Feynman decided he was useless at one point, but his mentors at the time simply told him to relax and not worry about it, and within a couple of weeks he spotted a new problem (when you throw a plate down on the table in the cafeteria so that it spins and wobbles before coming to rest, what is its motion?) and before he knew where he was, he was going at full speed once more.
A: 
I knew that this was actually what I wanted to do for the next few years or even the rest of my life.

That's great!

I feel like my skills don't develop fast enough, and therefore I don't know in which direction I'm heading right now.

It's okay. Just enjoy the journey.

I always think that there is something wrong with my way of approaching problems.

We always improve, the more problems we solve. Everyone has different ways of approaching problems anyway, and all that matters is that you're open to exploring new techniques or learning from others.

I wonder if my thoughts are normal and that everything gets much better when I give my skills plenty more time to unfold themselves - or if I should seriously consider to adjust myself in a completely new way.

They are normal, and yes you should just find a way of learning mathematics that suits you and press on. If you're interested in some particular topics, especially, then it is completely worth the time and energy that you put in. A lot of results come out of hard work. Talent or flashes of insight may be helpful at times, but without hard work they are still of no use.
