Have you read
Lockhart's Lament yet? If you haven't, you should. It perfectly describes the state of modern mathematics. Let's face it, we're not actually learning MATH in high school. We're learning how to follow processes and solve trivial equations. We're learning to memorize. Ugh, of all subjects, why did they have to ruin math with such a tedious process?
Even as I sit here tapping away at my keyboard, making clickity click noises, my Calculus homework stares me in the face saying, "Go to sleep. Go to sleep." I've repeated the process over and over, up to 11 times now. It's no longer a thinking method; it's a memorized motion. I miss the days of 4x, when Mr. Kilkelly would hand out worksheets with tricky substitutions or elegant proofs. Those worksheets were fun, and they taught us problem solving methods. I remember Mr. Kilkelly saying, "Sometimes, we need to look for where we want to go and let that guide us. It's a powerful method, yet students often overlook it." (Okay, not direct quote, but hey, I can't remember EVERYTHING from 4x. :) Just the math right?) I also remember learning Viete's equations just because they were cool, and then taking tests with fun bonus questions that challenged our understanding and control of the subject.
In Calculus, it's the complete opposite. We spend days, weeks on applications of established ideas when, honestly, it would only take a day or two to learn such narrow subjects. For example, after the chapter where we learned all the tools for differentiation (Note: Tools are useful. Learning how to apply them through problems should be figured out by students in my mind. I mean, Euclid only started out with 5 tools for establishing his geometry.), we had an entire chapter about "related rates" and what not. Learning that lesson took me about one day's class. I never did any homework for the rest of the chapter. Afterwards, we learned integrals, and how to anti-differentiate using methods like U-substitution or Integration by parts. That's useful. Learning how to calculate the volume of a solid of revolution? Not useful.
With that, we come to today. Where we learned L'hopital's rule. Useful? Definitely. The concept of it is immensely powerful. But do we honestly need to study it through endless amounts of trivial problems and tests? No, not at all. I say, give the concept, show how it's used once or twice, practice once or twice, and move on. Instead of listening to endless explanations on homework in Calc (Disclaimer, not the teacher's fault. Skerbitz is awesome. People just have too many questions where they don't try it themselves. Still respect Skerbitz), I find trying to prove a conjecture of mine much more interesting.
But going on a tangent, because tangents are mathematical, why don't we ever teach competition style mathematics? Gosh, that would be so much more fun. See, AMC-style or competition mathematics would teach much more than a shallow process. It would stimulate student's minds and challenge them to look at problems from different perspectives. If it's too hard, let the student ponder. I find that learning through struggling is much more effective than learning through rote memorization. Besides, politicians always champion "critical thinking" and "analysis." Well, competition style math develops that much more than following a process. It also gives students a wider breadth of problem-solving skills than learning strictly one subject, like Calculus. I mean, sure, I know how to solve a basic, basic, basic differential equation, but give me a tough AMC word problem, and I'll still be stumped. Why? I haven't developed good enough problem solving skills.
Hmm, after writing this rant, I might think about changing my Comp proposal topic. After all, the idea of introducing classes like Mathematics for Competition or Combinatorics appeals to me. I am in need of some 1 term classes too, and these would definitely be more interesting than... Human Anatomy... yuck. Dx Bio. Well, time to get back to slave labor! :D
