Here is a great piece (pdf):
Why don’t we want our children to learn to do mathematics? Is it that we don’t trust them, that we think it’s too hard? We seem to feel that they are capable of making arguments and coming to their own conclusions about Napoleon, why not about triangles? I think it’s simply that we as a culture don’t know what mathematics is. The impression we are given is of something very cold and highly technical, that no one could possibly understand— a self fulfilling prophesy if there ever was one.
I never took a math course in college, which is extremely uncommon for people in my line of work. I had to forget how math was taught to me (and how stupid it made me feel) before I had the confidence to take actuarial exams.
The trouble is that math, like painting or poetry, is hard creative work. That makes it very difficult to teach. Mathematics is a slow, contemplative process. It takes time to produce a work of art, and it takes a skilled teacher to recognize one.
I say I never took a math course, but I did take a statistics course for business majors. I don’t think of this as a math course because, though I feel like I learned some formulas and some concepts, I never really understood them. The best I can say is that the course got me familiar enough with some jargon that when I had to later learn it for real (quite recently, in fact), it took a bit less time. Somewhat valuable, but not math.
The curriculum is obsessed with jargon and nomenclature, seemingly for no other purpose than to provide teachers with something to test the students on. No mathematician in the world would bother making these senseless distinctions: 2 1/2 is a “mixed number,” while 5/2 is an “improper fraction.” They’re equal for crying out loud. They are the same exact numbers, and have the same exact properties. Who uses such words outside of fourth grade?
You start learning math concepts for real when you feel, deep in your mind, the frustration of the problem you can’t solve without them. When I started working full-time on reinsurance problems that required some serious math I picked up much of what I needed on Wikipedia (etc). When I finally started taking the exams I had a framework in my mind for how to use these very abstract tools. Without that, I’d never have passed anything.
In place of discovery and exploration, we have rules and regulations. We never hear a student saying, “I wanted to see if it could make any sense to raise a number to a negative power, and I found that you get a really neat pattern if you choose it to mean the reciprocal.” Instead we have teachers and textbooks presenting the “negative exponent rule” as a fait d’accompli with no mention of the aesthetics behind this choice, or even that it is a choice.
Here the author and I part ways a bit. Math to me is more a tool than a pure art. My art is finding a way of expressing economic/social phenomena (eg. insurance) in mathematical models. I am constantly faced with situations like: this thing happened once before and these other circumstances were present. How are the variables related and how do I think about the risk of this thing happening again?
I get the beauty of math, of course. I get that it’s pretty cool that the normal distribution is a generalization of simple coin-flipping. But in my world any useful application of the normal distribution (or any other in its family) has nothing to do with truth, in the mathematical sense. Business math steals a hodgepodge of tricks from real math when the properties of those tricks are (for no reason anyone understands) convenient. The business mathematician needs to understand the difference between convenience and truth. Many who forget this lose lots and lots and lots of money.
So pure mathematical exploration for its own sake doesn’t turn me on. That’s why I’m not a math geek. I’m a business geek. Those are the problems that fascinate me.
So it falls on the teacher to find problems that students care enough about to get them to learn the material. Sometimes this is hard, particularly for more advanced topics. Here I feel like most math instruction would do well to illustrate the history and historical context of the concept or formula or idea. When was it first discovered? By whom? What motivated the investigation? What was considered the state of the art in other, related, areas of math? What refinements came later? Usually each of these questions is answered in a way that will help a student learn the math.
It takes longer to teach it this way, of course. But for god’s sake let math be slow without making kids feel stupid.