Yesterday one of my students, one who almost never engages me in conversation, asked me a question right before class was over: he wanted to know how "infinity" could have different sizes.
My degree was in applied math, not theory. I did calculus, differential equations, partial differential equations, separable differential equation, numerical solutions to differential equations, some math modeling (probably with differential equations), etc. Math history, number theory, set theory, graph theory--those weren't the classes I took in college.
Interestingly, they're covered in varying degrees in the courses I'm taking for my master's degree. Yes, I could have gone to National University and in 10 months picked up a Master's in Education with an Emphasis on Curriculum and Instruction and gotten a mambo-sized pay raise, but instead I chose to pursue a degree that would make me a better math teacher rather than just a better-paid one. I can't fault people who did go the National (or similar) route, as they just played by the system's rules, I just want more.
And it's working. I'm a much better statistics teacher than I was because now I have both a broader and a deeper understanding of what I teach, I can answer the "why" questions and tempt students with a taste of what university math could have in store for them.
So when this student asked me about infinity, I was able to answer his question somewhat. I told him I'd like to review my notes and to check with me tomorrow, which was today. Last night I consulted my notes and wrote up 2 pages of commentary and examples to show how the size of the infinity that encompasses the set of natural numbers (1, 2, 3, ...) is the same size as that of the integers or even the rational numbers, but the infinity of the set of real numbers (or even just the numbers between 0 and 1) is larger than the infinity of the natural numbers. When my instruction was done today, he and I got together and went through the integers and rational numbers but didn't have time to go through the real numbers. I told him if he couldn't figure out my examples by Monday, we'll meet again then and go through it.
A year ago I wouldn't have been able to answer his question, now I can. That makes me a better math teacher. That's why I'm getting the degree I am.