TITLE: Gaining Mathematical Maturity
QUESTION [31 upvotes]: 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

REPLY [13 votes]: 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.