My son is nearing the end of his Art of Problem Solving precalculus class, and it is still going well, as I reported earlier. We are now looking at what to do next. Should he take Calculus BC at his high school in the Fall? Should he take the AoPS Calculus class this fall? Or should he detour into a different branch of math?
The Art of Problem Solving people have written a couple of essays about pre-college math preparation:
- The Calculus Trap. The basic premise of this piece is that the lock-step march through arithmetic, algebra, geometry, algebra, trigonometry, precalculus, and calculus is not the only or best way to study math. Elementary and secondary math education is not a race to see who gets to the “finish line” (calculus) soonest. Indeed, from a mathematician’s viewpoint, calculus is just one of many starting points for interesting math. A lot of what gets tossed aside in a race to calculus is more interesting.
Even more important: “the standard curriculum is not designed for the top students.” They point out that being the top student in your class is not the way to make progress in your learning—you are better off getting to a level of challenge that really makes you exercise your problem-solving skills. Racing through courses full of drill problems is not going to do that the way working on harder problems will. There are plenty of hard problems that do not need a lot of mathematical machinery to solve, so students can start on them before having learned all the machinery.
- Why Discrete Math Is Important. As a computer scientist and computer engineering who taught applied discrete math a few times, and as a bioinformatician who has to teach grad students all over again how to count (that is, how to do simple combinatorics) and how to do simple Bayesian probability, I am certainly in agreement that discrete math is important. In fact, a big part of the reason I ended up in computer science rather than pure math was that I liked discrete math (graph theory and combinatorics) better than real or complex analysis, and I had made the mistake of starting my graduate math education in a department that had no discrete math. Luckily there were four or five great people doing combinatorics, graph theory, and graph algorithms in the computer science department, and I was able to switch departments. That turned out very well for me, so perhaps it was a good thing that I’d chosen the wrong math department.
Most likely we’ll continue the standard progression, doing either Calculus BC or AoPS calculus next year. After that, there will be time for applied discrete math and for probability and statistics before he goes off to college.