I've been purposefully drowning myself in math for quite some time now, after having done no formal math at all for about 18 years. This has given me a rather strange, distanced point of view on how math appears to be presented, or taught.
There are two things that seem to be taught: proofs (sometimes, if you're lucky), and steps for solving a problem. There is an emphasis on the latter, and an expectation that the student will learn those steps and use them. "If you follow 1., 2., 3., you will have the solution." This isn't just how things are being presented in the college calculus classes I'm currently taking, but I also recall it rather poignantly from my grade school and high school experiences years ago (most of which were frightful, despite the fact that I love math and understand it easily, b/c I usually need to invent my own solution steps).
It is curious to me, especially now that I am older and have a clearer sense for what works for me and what does not, that a key concept recognized in other disciplines is not recognized in mathematics--at least it's not being recognized at the academic level I'm currently at. That key concept is this: if a person has deep, real understanding and ownership of an idea, they can use that idea creatively. Conversely, creative application of an idea leads to deeper understanding. One is expected to make creative use of learned concepts in the arts, right? And even the graduate science course I am taking expects (and grades on) creative problem exploration. Why is the expectation in math that one will simply plug in values to a memorized series of steps that were invented by someone else and be happy with that? That that's enough. Why isn't there an emphasis on using concepts creatively to find an individually optimized series of steps for solving a problem, or using concepts creativity to get at the core of what is wrong or right with methods of solution. If a concept is understood enough to be used creatively, then that implies a much deeper understanding of the concept than rote regurgitation.
I wonder if it is because we are taught in grade school that there is always "one right answer" to a math problem that we get bogged down into thinking that mathematics is purely formulaic. The closer I get to more advanced mathematical concepts, the less I find this paradigm useful (not that I've ever found it particularly useful). What would be different if from the start children were taught mathematical concepts instead of someone else's steps, and were told to find optimal methods of solution? What if from the start children were encouraged to play with their math, to discover for themselves how each jeweled facet of it works?
Eh, maybe things have changed in the way math is taught in primary education in the 20-30 years since I was a kid, but still, I've been thinking about this a good deal. I'm not entirely certain where I'm going with it yet, but I thought the ramble had relevance to this community :-)