Gaining Mathematical Maturity I was redirected here by a kind fellow from math.overflow. This is not a typical math question, so I apologize if that is discourteous.
I am currently a sophomore in my undergraduate mathematics program. It has taken me a while to take school seriously; I was one of "those" students who just skated by without studying until Linear Algebra. However, I did not start to take my education seriously until about two months ago. I took the semester off to re-assess what I truly wanted to study and decided to follow my (difficult) passion of mathematics. I am now enrolled in summer semester at my university and am taking ODEs and Real Analysis 1. Like many students, I find analysis to be challenging yet very exciting. Sometimes, however, I catch myself getting frustrated with my own self because I feel I am not making enough progress. Have you ever felt self doubt in your career as a mathematician? How did you overcome those worries? Also, what are some good techniques or resources to advance one's skills as a undergraduate level mathematician? Thank you for taking the time to read this.
Sincerely, Rebecca
 A: Im not studying and Im not going to study mathematics officially (Im a bit "old"), Im an amateur. When started to follow my hobby more seriously sometimes I feel that Im not understanding anything just memorizing things.
But the brain is a giant mystery: one or two years after I started the "hobby", with long periods of time not seeing something about mathematics, I get back and I was VERY surprised that I understood everything very clear and not only as a kind of "memorization", i.e., I understood really where comes all the things, in a human way (why measure is measure, where it comes from our relationship with common experiences in life, etc...)
Another think I learned reading books of math is that a topic can be seen from many points of view, someones are more in harmony with your own vision or pre-understanding of the topic and many others not too much. By example: I read a lot of analysis books of first year of university, I understood (with less or more effort) but I was feeling everytime that I was just "memorizing" things and not understanding (what in the moment was true: I was not understanding, just trying to understand and memorizing), but oh surprise!, now after some time and reading books of other mathematical topics (probability, topology, etc...) and reading different books of analysis from different perspectives (Abbott book or Tao's book) I started to see clearly analysis.
The same happen to me with topology: I didnt understood it when started to see it from the classical approach of metric spaces. Only after reading a book about topology from the point of view of topology (not analysis) I started to understand very clear. In many other topics of mathematics happened the same: some approach is not as understandable as other.
And Im starting to think that mathematics are really easy, the key stones are powerful ideas but very clean and simple. The difficulty comes, to me, from understanding the language of mathematics and having some kind of examples (images) and contexts that help see the questions with clarity.
Another thing that happened to me is that I started to NEED to prove every theorem I see, or least I need to see some proof about it. I started to see proofs as something essential and, moreover, very funny.
I dont know if this answer will be interesting or useful to you, I hope it will be. And talking more in general: in any kind of learning (not only maths) there are time where all is more boring/complicated/frustrating, and times where everything is very interesting and funny.
P.S.: my recommendation for analysis is Understanding analysis from Abbot. For topology my recomendation is Topology without tears from Morris.
A: 
Have you ever felt self doubt in your career as a mathematician? How did you overcome those worries? Also, what are some good techniques or resources to advance one's skills as a undergraduate level mathematician? 

You bet I did. I graduated with a B.S. Mathematics, Statistics emphasis two years ago, and I am now pursuing a M.S. Statistics at a top-20 school part-time while working full-time in a job that I love. 
I remember the first week of my Analysis I course. I was scared to death about the class, because all of the math majors were telling me about how difficult it was, and here I was, this actuarial science major who is taking Analysis I to raise my GPA. (I, quite frankly, didn't do so well in my actuarial science courses because I felt that I was just regurgitating facts, and that the professors didn't teach the results in generality.)
Before the drop-date, I asked my professor to grade my first homework and use that to give me a recommendation on continuing in the class. He told me to keep going.
I got an A in Analysis I. This was quite pleasing, considering that I was an actuarial science major. And then pure math just clicked: I got As in Abstract Algebra I, Analysis II, and Abstract Algebra II. By then, I realized that I had completely diverged from the actuarial science major, and then on my last semester, I declared the new major.
How did I overcome these worries? I didn't look at math in fear, but as a challenge I could solve. You can't study math with the fear of math in advance of studying it. 
Techniques/Resources:


*

*Read many, many math books. I found that spending some time each day - just merely reading the textbook, not necessarily trying to understand the concepts in-depth, but just get a small idea for what's going on - helped immensely. Don't keep yourself to one math book, either, per subject.

*Talk to a lot of people about what you're studying. This helps you communicate mathematics with others, along with helping you learn more about how others view mathematics. I've learned a lot of techniques that I didn't learn in class that I learned from tutoring others.

A: Even some of the most extraordinarily talented mathematicians, scientists, and scholars experience doubt--personal doubt that they've chosen the right career, worked on a solvable or important problem, that they can ever solve the problem (before a competitor) and so forth.  This happens with every challenging discipline.
Judge your progress both on a personal level (are you learning as fast as you think you should) and at a group level (are you excelling in your class).  
Mathematics is hard, and if you want a career that exploits it, you'll be buoyed if you take pleasure in the raw acts of mathematics:  casting problems into mathematical form, performing the calculations, interpreting the results, communicating them to others effectively, and so on.  
Depending upon your deepest interests, though, you might consider focusing on mathematics that will help you in another, mathematically rigorous career.  If you're interested in finance, then study statistics and differential equations; if you're interested in medicine, then study statistics; if you're interested in computer science, then study discrete math,... and so on.
Finally:  talk to teachers and fellow students to express your interests, goals... and yes, doubts.
But good luck to you!
