Now that 2.2 metric Ages of Internet Time have passed since Andrew Hacker’s ill-advised “math is hard!!” ramble, I figure it’s a good day to propose my own way of improving high-school mathematics education. Be advised: this is a suggestion about the curriculum, not about how to train teachers, buy books and all that un-TED-friendly stuff which reformers happily gloss over. And I’ll be talking about changes late in the game, which won’t address problems at the “why can’t Johnny add?” level.
When I was in high school—at a pretty well-supported public school, out in the ‘burbs at the comparatively unimpoverished end of town—I took a “precalculus” class my eleventh-grade year. Most of the advanced-track students I knew did the same thing. (If you’d gotten yourself on the even-more-advanced track back in eigth grade, you took precalculus in tenth.) This was supposed to prepare us for taking the AP Calculus class our senior year, which would allow us to get college credit. Instead, it was a thoroughgoing waste of time. The content was a repeat of Algebra II/Trigonometry, which we’d taken the year before, with two exceptions thrown in. The first, probability, was a topic our teacher didn’t know how to teach. In fact, she admitted as much: “I don’t know how to teach probability, so you’re all going to read the book today.” The second, limits, served no purpose. I’ll explain why in a moment.
I suggest the following: scrap “precalculus” and replace it with a year-long statistics course. This plan has several advantages:
It takes care of the legitimate point raised by Hacker’s op-ed, what he called “citizen statistics.” If we need students to grow up being well-informed, numerate contributors to civilisation, let’s have a class on exactly that. Required reading: Darrell Huff’s How To Lie With Statistics (1954, but we could update it easily).
It’s easier to implement than demolishing all of post-arithmetic mathematics education and trying to invent a replacement. Just swap out one class! We could develop the necessary material comparatively quickly and try it out with a straightforward pilot program.
As an ancillary benefit, the course materials could be put online as OpenCourseWare for the general welfare.
Would replacing precalculus with statistics impair the students who go on to take AP Calculus? I doubt it. In fact, by teaching as much statistics as it’s possible to do without calculus, we’d prepare the way for improved calculus lessons. For example, when it comes time to show what you can actually use integration for, we could teach about probability distributions. Statistics class would, in part, be a precalculus course itself—just one with an actual raison d’être.
And as for the limits lesson, what serious harm could there be in missing one topic which had been shoved disjointedly into a clone of another class, and which everyone sees again the next year anyway? Personally, I’m of the mind that limits make more sense if you see them after the basic ideas of differential calculus. That way, you know what you’re aiming to do and why it matters. (This alternative also tracks better with the subject’s history: Newton and Leibniz came long before Weierstrass.) Otherwise, as a maths professor friend of mine told me, they look like a solution dying to meet a problem.
How did that precalculus class of mine teach us limits? Well, we had to do a lot of worksheets. Bear in mind, I’d already read about the subject in my father’s old calculus textbook (a doorstopper whose size and yellowing reminded me of a dictionary—far thicker than necessary for first learning the subject!). The book started with a healthy dose of examples, comparing a question one could answer with algebra or geometry and a corresponding question you’d need calculus for. Calculus, it said, was algebra and geometry “with the addition of the limit process.” What did “taking a limit” mean? If you had a function $f(x)$, then writing
$$ \lim_{x\rightarrow a} f(x) = L $$
means that you could make the value of $f(x)$ as close to $L$ as you wanted, by choosing $x$ to be sufficiently close to $a$.
Now, in my high-school precalculus class, this translated into plodding through several pages of Xeroxed problems. Almost all of them belonged to one of two types. Either you could get the answer by just plugging in the number $a$, so you weren’t learning anything new and couldn’t see any reason to care, or you had to find the limit as $x$ approached 0 of some hairy thing which looked like
$$ \frac{\sin(2y + x) – \sin(2y)}{x}. $$
If you already knew a little calculus, you could see that all the hairy problems were just taking the derivatives of functions. And knowing a bit about derivatives—I mean, knowing as much as a physics class teaches in half an hour—you could get the right answer every time in two lines of work. Those who didn’t have this pipeline of wisdom floundered for half a page of trigonometry, rewriting things back and forth and ending up with the wrong answer. Of course, for knowing what was going on and getting the right answer every time, we got the red pen: “That’s not how we learned to do this problem in class.”
Come to think of it, most of my fond memories of those years are about skipping class.
What would the curriculum for this proposed statistics class look like? We could do worse, I think, than cover the topics here: data description and summarization, means and standard deviations, measures of correlation, hypothesis testing… Flesh it out with examples of the “citizen statistics” variety—why was the SEC’s pilot study on repealing the “uptick rule” flawed?—and you’d have a full year’s worth. Alternatively, a one-semester introductory course with a strong “citizen statistics” flavour could be a warm-up for AP Statistics the following term. (In my experience, AP classes were so focused on taking the official exam at the end of the year that there was no time to spread out and look at anything which wasn’t easily testable in multiple-choice form.) The key point, assuming most school administrators are like the ones I had to deal with, would be revising the official tangled web of prerequisites, so that students could take AP Calculus without having taken precalculus first.
Incidentally, the professor friend I mentioned earlier is going to be teaching precalculus to college kids this fall, and is hoping to make it a better experience than the class I shuffled through in eleventh grade.