The example that really stuck with me was synthetic division, a more concise but less intuitive way of performing polynomial long division. Both of these topics are pretty much useless in daily life but polynomial long division does, at least, give the student some insight into the relationship between polynomials and familiar base-ten numbers. Synthetic division has no such value; it's just a faster but less interesting way of doing something you'll never have to do.

I started asking hardcore math people -- mathematicians, statisticians, physicists, rocket scientists*-- if they'd ever used synthetic division. By an overwhelming margin, the answer I got was "what's synthetic division?" Not only did they not need it; it made so little impression that they forgot ever learning it.

Which bring us to this passage from a recent Dana Goldstein post (discussed earlier):

The problem, according to [David] Coleman, is that American curriculum standards have traditionally been written by committees whose members advocate for their pet pedagogical theories, such as traditional vs. reform math.Except, of course, that's not what happened here. As was the case with so many topics in mathematics, synthetic division remained in the curriculum because no one who knew what was going on had bothered to look that closely. Coleman has a clever narrative, but it doesn't fit the facts all that well.

Now I have a request for all the math geeks in the audience (and given that you're reading a blog called Observational Epidemiology...). Since we need to pare down the curriculum, what you choose to cut? Specifically, what mathematical topics that you learned in school can future generations do without?

* Literal rocket scientists -- JPL's just down the road.

Also posted in a slightly different form at Education and Statistics.

I was discussing this with a math ed student the other day: how to find a square root by hand. Drop that in favor of spending more times with logs and exponential functions.

ReplyDeleteI always thought it would be cool to teach students to do this and similar calculations with slide rules but I never got around to it.

ReplyDeleteMy opinion is that the biggest offender is plane geometry. While geometric intuition is useful, many of the theorems students memorize (in order to invoke their usage in proofs) occur in very specific situations: triangles that happen to the similar; lines that happen to be parallel; shapes that happen to be perfect circles, square, etc. The plane geometry curriculum used today is more-or-less the one set down by Euclid in his Elements, but vectors and trigonometry are more general tools for solving the same types of problem. The main purpose geometry serves in the modern curriculum is to teach logic and how to prove things rigorously. However, I believe the vast majority of high school teachers do not realize this because so often they are concerned about getting the theorems out that they forget to teach the metacognitive issues of why the material is important or how to use the material. Indeed, my friend is currently training to be a high school math/science teacher, and next semester he will be assisting a geometry teacher. The geometry class he describes as "geometry appreciation." Students will learn about how to find geometry in nature, geometry in art, and experiment with various geometry computer programs; proofs feature very little in this curriculum.

ReplyDeleteI believe this class is an elective, not a replacement for the required geometry credit, and this underscores an important point: geometry intuition is useful, and it shows students that math is not just a series of equations. However, many of Euclid's theorems are not useful in themselves. I could imagine them being excised from the curriculum, geometric intuition being folded with subjects like vectors, linear algebra, and trigonometry, and mathematical proofs and rigor being emphasized in many other parts of the curriculum. The underclassmen engineering students I teach at Berkeley flinch when we say the word "proof," probably because they relate it to archaic geometry classes, and many don't know how to do a proof by contradiction/contrapositive. However, I hear no complaints about boolean logic, which they learn in their computer science classes. The conclusion I draw from this is that logic and proofs can be taught, but geometry is not the way to do it, yes every student spends a whole year devoted to it.

James,

ReplyDeleteI have mixed feeling about this. Right now, plane geometry is the only real exposure most students get to proofs, axiomatic systems, spatial reasoning, etc. There are certainly better ways to introduce these concepts, but my concern is that, given much of what we've seen from the education reform movement, we'll end up replacing plane geometry with more rote work, cookbook algorithms and glorified test prep.

Of the six trigonometric functions, only three are necessary: sine, arcsine, and tangent. Cotangent, Secant, and Cosecant are useless. ("But wait!" the conservatives will cry. "Without secant, how will we differentiate tangent?" Easy, the derivative of tangent is 1/cos^2.)

ReplyDeleteErm, above, I meant to say "cosine", not "arcsine". I feel sheepish.

ReplyDeleteMark,

ReplyDeleteI hear what you say. However, the fact remains that many teachers, while they work hard at teaching plane geometry, don't understand the purpose it serves within the curriculum, and hence it is wasting both their time and the students' time because, even if it is where students are supposed to learning proofs and axiomatic systems, the students aren't getting it based off what I see when they arrive in college. Furthermore, the fact that the teachers don't realize the purpose of geometry means that who ever designed the curriculum failed to communicate this purpose to them. I can only hope that education reformers realize this and will address this, but I too suspect that they are far more interested in increasing test scores. However, while they usually ask "why aren't test scores improving?," I believe they should first ask "is this the correct test to use?" and say as much here: http://www.boardgamegeek.com/thread/714290/teaching-to-the-test

In short, I believe the problem is this: secondary school math and science curriculum need a serious overhaul. Politicians don't understand the problem, just numbers, specifically test scores and budgets. Someone who actually does understand needs to make sure reform happens in the right direction.

I wrote up a defense of synthetic division, although I agree it shouldn't be taught in high school: http://www.isi.edu/people/lorin/blog/defense_synthetic_division

ReplyDelete