Some Thoughts on Teaching Mathematics

This morning, I was reading the NCTM blog, and the subject was on students struggling with systems of linear inequalities.

First, as background: I don’t have any difficulty with systems of linear inequalities, and I don’t remember ever being taught such things (although I may have, I just don’t remember).

But then I get two questions in my head:

1. Why do students have to learn such things? Why is this specific concept considered “mathematics” so much that we have to fret over its mastery?

2. How is it that this is such an isolated thing that students struggle with it? Where have we gone wrong with the basics?

These are questions I think about often, about a variety of topics. Imagine an art class where a teacher complains that their students are struggling with painting skies. Sure, there are some specific things to do with painting realistic skies that are somewhat unique to the process, but is it something that art teachers isolate to the point of fretting about?

In my opinion, we focus far too much on specific topics and knowledge and making sure students get it, and nowhere near enough on general approaches. I’d gotten the impression that, once upon a time, the point of specific topics and knowledge was so that we’d have something on which to hang the general approaches.

Consider English class. There’s nothing so particularly unique about “To Kill a Mockingbird” that that specific book must be taught. It is selected because it’s considered a modern classic of American literature, but students can be taught the same skills and concepts with a different book.

For “systems of linear inequalities” to be a needful thing, there must be situations in which “the set of data points that are less than (e.g.) two linear functions” is a meaningful enough set. Otherwise, we’re just hanging a useful general concept (the solution set to a group of functions can have multiple elements) on a specific case (systems of linear inequalities).

There are of course real world situations, many of them, where we would want to know if a data point falls within a region defined by functional inequalities. What is the range of prices that will generate profit for me? Is this the right part of the country to grow azaleas? How many pizzas should I order for the company picnic?

But I’m not convinced that the specific topic of “functions of linear inequalities” is so important that non-mastery should be fretted as a gap in the students’ knowledge. It should, instead, be fretted as a gap in the students’ skill-set, and that can potentially be rectified by teaching a different, related skill that can then be transferred.

Ah… transfer… there’s another boogeyman, a much larger specter always lingering at the edge of the honest math teacher’s reality.

The teaching of very specific skills, in my mind, is our act of conceding to the Specter of Transfer. We think that students will never realize that the conjugate of a complex number and the conjugate of a + sqrt(b) are the SAME THING, so we teach them as separate things. We think that students will never realize that the equation of a circle, the distance formula, and the Pythagorean Theorem are the SAME THING, so we teach them as separate things. We actively compartmentalize for the student because we don’t trust students to transfer.

… and, the more we do this, the harder it is for students to transfer. But transfer they must.

Teaching to standardized test is absolutely a fool’s errand. “We’ve been studying the opposition,” says the football coach, “and they always pass, never rush. So we’ll only work on interceptions, not on blocking.” And then on Sunday, the opposition shows up and only runs rushing plays.

Except it’s worse than that: Every year, the SAT and ACT adds to their archive of Things They’ve Tested, and every year “teach to the test” administrators and curriculum developers pour through the new archives to find patterns about what to teach. Oh my God! There’s a secant question! They haven’t asked a secant question in 7 years! This is a new trend! Add two weeks on secants to the Geometry class!

Teach transfer, and you don’t need to play this game. Teach basic skills DEEPLY, and you don’t need to play this game. Because the SAT and ACT provide formula sheets and, on specific question, appropriate formulas, there’s relatively little specific knowledge you need to succeed: What you need, for more, is the ability to tear apart a new question quickly and figure out what it’s really asking.

I think what’s consistently lacking in mathematics education is WHY? And I don’t mean that we’re not asking the students why, although that’s an issue, too. What I mean is that we’re not asking ourselves: WHY are we teaching this specific thing? Wrong answers include “because it’s in the book” and “because it’s on the SAT/ACT.”

In Algebra II, I’m wrapping up exponential functions. I remember as a student teacher, we spent several weeks on Pe^rt. The book is absolutely filled with “A bank offers continuously compounded interest….”

Google it. Find all these mythical banks that offer continuously compounded interest. I found one. There are probably a few others. One of the conclusions from Euler’s natterings was that you don’t really gain very much between “compounded daily” and “compounded continuously”, so you might as well not compound more often than daily. Rather than teaching THAT, though, rather than teaching the beauty of “what if?” in mathematics and how it leads to useful conclusions… rather than even teaching that Pe^rt allows to set an upper bound so we, the consumer, can more quickly check interest ranges… we instead create mythical banks that award interest five seconds after you’ve created an account.

WHY?

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