"Oh, you're studying mathematics? Well, at least there's only one right answer, right?"
I don't know how many times I've been told that. The short answer:"No, there isn't only one right answer. You've just been taught wrong." The long answer? Well, that's what I hope to address a bit on this blog.
First, a little background on me:
Though I have an undergraduate degree in mathematics, my real interests were in philosophy and, especially literature. I even took the GRE topic exam in literature way back in 2004. (More recently, my interests have shifted more toward music rather than literature.) But in 2005, a local private school needed me to teach a couple of more advanced mathematics courses--presumably only for a year before I left for graduate school. But, to my surprise, I found teaching really interesting, and decided to stay on. Four years later, I decided that I wanted to be a teacher, but at a more advanced level--and so I returned to school to pursue a Masters in Mathematics, hoping to get a job at a Community College after I graduated.
During my master's program, I was introduced to the discipline of mathematics education, and several professors encouraged me not to stop with a masters, but to pursue a Ph.D. I applied for Ph.D. programs in mathematics and mathematics education, eventually setteling on mathematics education. At the time, I thought collegiate mathematics education research sounded really interesting, but research into lower levels of mathematics education sounded boring.
That was before I got to graduate school. After I arrived, I fell in love with some of the aspects of elementary mathematics education, particularlly preservice teacher education. Most likely, I'll do my dissertation work on collegiate education--I'm currently working on a NSF Calculus grant--but I definitely have interests in lower level mathematics.
Returning to the mathematics education. Presumably, when we teach "mathematics", we should teach people to be good at mathematics--to begin to do the thing mathematicians do. (Or perhaps to use mathematics as engineers and scientists do.)
But what is mathematics? It's the study of numbers, memorization of long bizarre, nearly nonsensical algorithms, and confusing symbols? It's not exactly supposed to make sense, what we're really after is the right answer. Get that, you succeed, don't get that, you fail. Right?
Wrong. (Though I should confess, that describes my classes far more than I would like it to.) What is mathematics? That's actually a surprisingly difficult question to answer. It's the studey of numbers? But what's a number? And what about branches of mathematics, like Topology, and Category Theory, that hardly use numbers at all?
Hopefully in the following posts we'll begin to investigate mathematics, and make some guesses about its nature. But for now, the important point is: That's not what it is. Mathematicians almost never have the right answer, or an already existing formula, or anything like that. The subject you learned in school probably doesn't have terribly much to do with what mathematicians do.