Such as it is, I have prepared a reflective (use caution in direct sunlight) teaching statement. I'm neither a deep thinker, nor a proficient writer, so I can't convince myself that the result is any good. Either way, here's what I came up with:
I have no idea why I bother teaching.
I can convince myself that many of my students will need to know mathematics in their future careers. My science students clearly need mathematics, but so do the business students. The fine arts are a little harder to justify, so perhaps pawn it off on "general education." Numeracy, people claim, is practically as important as literacy, even if it doesn't get the same attention (in fact, some people pride themselves on their innumeracy). If nothing else, mathematics courses are a requirement at the University, and I have enough of an understanding of the subject to try to get others through it.
So my issue with teaching isn't that the content isn't important.
My issue with teaching is that, in this connected age, there are effectively unlimited sources for the knowledge that I am supposed to cover in a calculus class. Certainly there has always been the textbook, and I love for my students to read it. But now, online, there are more places to learn calculus than one could possibly use. Entire lectures and individual problems and short snippets get posted as videos to YouTube. Wikipedia contains more information than any one person can know (and it's constantly growing!). I also must compete against the Massachusetts Institute of Technology, who not long ago started posting all course materials online (for free, accessible by anybody).
So why would a student bother coming to my class? Anything I'll be covering in class could be found online, from the relative comfort of a dorm room. It's not my job to tell students when to think about calculus, so I have no attendance policy. Who am I to tell them when they should be learning? If they are motivated enough to learn the material on their own, I don't want to frustrate their ambitions by making them come to class.
It is this sense of competition that drives my lectures. When planning lectures, I make sure to use examples that aren't covered in the book. I also try to find as many "fun" examples and extra problems as possible. Students don't necessarily want to look at other resources to learn the material I will be teaching. Even motivated students might not take that step. Students might not know where to look on Wikipedia for interesting mathematics that is, at the same time, easy enough to digest without having taken lots of math classes. It's my job to show these things to my students.
It's lucky, then, that mathematics is so fascinating. Surprising amounts can be boiled down and understood without having spent years taking math classes and reading math textbooks. This is my job. Anybody can present material, and cover examples. I think it is important to go further, to find more interesting examples, and historical tidbits. I always hope to show my students some of the beautiful things in mathematics.
During the times I am doing fun examples and presenting material that isn't strictly necessary for the class it is easy to be an excited speaker. However, there is something interesting to be found in nearly every example, and I try to keep my enthusiasm high throughout every lecture. Judging from course feedback, my students feed off this energy, and are more motivated to learn the material. If I can create motivated learners, who start seeing interesting things in the content I teach, then all of those external resources are more likely to be accessed and dug through. All the material in world is worth nothing if students aren't interested in looking at it. It is my hope that I can encourage some to embrace this interest.