I'm going to share a deep, dark secret with you:
I don't know as much math as you might think I do. And I've always been afraid of being found out.
I've always known where my education was deficient, even when I was receiving that education. In high school I never learned about hyperbolas, or about factoring a cubic polynomial. I'd never heard of the Rational Root Theorem or Descartes's Rule of Signs. In college I earned a degree in applied math, and could calculate my butt off, but so often didn't fully understand what I was doing. I knew what to do, and I could understand why one step logically led to the next, but I didn't have a "big picture" understanding. As an example, in differential equations I could calculate eigenvalues all day long, but to this day I don't know what an eigenvalue is or what it does for me or why I need to calculate it. I've taught myself plenty--sometimes just days before I had to teach it to my students.
This came to a head today when I was talking to one of my students who's heading to Cal Poly. I told him not to make the mistake I made; ask the questions, go for the deeper understanding.
I've never understood the Fundamental Theorem of Calculus. Why, exactly, are an integral and an antiderivative the same thing? I've followed the steps in my calculus books, and understood each step, but never really understood how they all fit together. So today I pulled a different calculus book out of my closet and I started studying. I found one that provided a very user-friendly explanation, which then allowed me to understand the very rigorous (read: dry and difficult) proof in a second text. It took a few minutes to replace decades of deficit.
I learned something today. Tackling the Fundamental Theorem of Algebra, which one book says is "beyond the scope of this textbook", is next.
I remember my senior project at West Point. I was writing a computer program that would aim a gun at an airplane, and an instructor asked me, "Why are you using that algorithm? There are others that will converge much more quickly." I knew that there were others, and I knew what "converge much more quickly" meant, but what I didn't know was what others there were or how they'd converge more quickly. I was able to throw him off, but I remember the fear of being caught that day.
So now I want to learn. I want to understand. I'm looking forward to that masters program I'll be starting in the fall, a Masters in Teaching Math through the University of Idaho's Engineering Outreach Program.
Explaining this to another teacher today, I was told that now I'm experiencing the difference between learning and merely completing a degree.
I'm looking forward to this.