This summer I've been helping out in a summer-school Algebra I claaroom. I enjoy being with the students and helping them with their assignments. But there is one problem.
I keep wanting to teach them algebra.
For example, one problem required them to handle negative exponents. The basic rule for them is that the negative sign in the exponent turns the fraction upside down. (Using more mathematical jargon, an additive inverse in the exponent implies the multiplicative inverse.) When -- I'm sure they were asking -- will I ever need to know how that negative exponents turn franctions upside down? Probably never, except for the remaining required match classes. If knowing that rule were the only reason for teaching the topic, this would ba a pretty worthless education indeed.
What would make Algebra I more valuable to their future? Learning an algebraic manner of thinking. So, I keep wanting to explain how mathematicians decided that we should use negative exponents to represent division. I want to expound on the beautiful consistency of that convention. Even more, I want to teach that people looked at exponents, and thought about them, wondered what would happen if the exponent became negative, and then decided what that should mean. I want to teach an algebraic way of thinking about problems and creating solutions that make sense and are useful. It wasn't just textbooks all the way back; there are people much like us you came up with these ideas, and -- the most valuable lesson -- we can copy some of the ways they solved their problems when we solve our own problems.
It is interesting when I revisit high school math classes to notice how little of the material I really learned myself. What I learned was how to figure out a way to solve a problem. I learned more about what was really going on in the equations and less of the specific rules for passing the next test.
I have the same urge with geometry. Much of the geometry courses are about performing specific tasks (often using algebraic methods) and too little is about thinking geometrically. I want students to learn both geometry and algebra. I want them to discover that there are multiple approaches to solving problems, even to solving the same problem.
Some students do learn this lesson, but the ones who learn it on their own are less likely to be seeking help from volunteer tutors like myself. Some students will never learn this lesson (at least in the context of mathematics); those are the ones for whom learning practical rules are the only way to get through the required courses. I have to wonder whether this second group might not be better educated with fewer math courses and more wourse with some other kind of problem solving.
For those students who are between these extremes, the goal of teaching algebra (and geometry and especially calculus!) is to wake in their consciousness an awareness of some of the power of human minds -- of their own minds -- to solve whatever problem they might find in front of them.
That's what excited me about math and that is the aspect of mathematics that I most want to pass on to others.