Article: When Good Teaching Leads to Bad Results

Which tasks best help students meet the learning objectives?

In our very first C&I discussion, we touched upon some of the issues that came up for me as I read the Schoenfeld article. In that discussion, Birthday Boy and I had a fun back and forth about "math as a utilitarian method of solving problems" vs. "math as beauty, joy, and the answer to all that is great and good". Or, if you like, math as method vs. math as art. I'm all method,baby. I don't want some annoying problem that I'm supposed to stare at for fifteen years and then finally approach a solution. What do people think I am, a mathematician? Nix! Am I teaching mathematicians? Negatron! Am I giving my students an "interpretive framework" that "shapes the ways that students see the world and act in it -- in particular, how they see and use their mathematical knowledge" (Schoenberg, in case you missed it) No, no, a thousand times no.

See, my problem is that I agree with all the supposed misconceptions that math teachers are supposed to disprove. Like "math has no relationship to real life" and "it's ridiculous to get fussed about remainder issues when in real life, you'll remember that people can't be split in half"--wait, I talked about that one before. So I'm supposed to get all worked up about the horror, the horror, of the kids fussing about accuracy of construction, when I'm thinking "hey, the only reason they are working on construction in the first place is for the test, so yeah, get it right." I've mentioned this before: my math knowledge, such as it is, is entirely self-taught, and I can know and use formulas long, long before they have any relevance to me. I still think it's dumb that (x-2)² moves the parabola to the right instead of the left--yes, I know why. It's just dumb. Annoying. Counterintuitive. And my feelings on this point, I believe, make me a better math teacher.

No, I don't believe that math is all formula and no concept. Yes, believe it or not, I almost always show my students how a formula is derived, and remind them of it constantly, because derivation helps them remember. There is a middle ground to be found, and as a country we've been struggling to find it for years. We'll keep looking.

As for the lens question on point, I genuinely believe that open-ended tasks are a terrible way to teach most students math. I like repetition, I like increased complexity, moving from discrete chunks of the problem to putting things together to word problems/applications. I think the hardest part of a math problem, as students develop competence, should be the identification process: what tools or process should be used to solve this problem? Like the broomstick probability problem--at its heart, it involved the triangle inequality theorem (or at least, it did for me). How do we get students to look at a problem and understand what to apply? We need to move them towards this, over time, but when, and how do we develop mid-level questions for the students that don't master the recognition process? Getting the bulk of students to that point is a long process, and not everyone gets to the same place. And for the students who aren't there yet, I'm fine with them going the long way around, or not seeing that adding a number three times and dividing by three will get you that number, provided they know how to find the average anyway.

Fundamentally, I believe math instruction is only tangentially about math, and most importantly about giving students a clear construct that they can use to acquire knowledge and solve problems using that knowledge. Then some small percentage of them can go on and become mathematicians.

And that's the real question, isn't it? Who is our audience? Should we be trying to turn most students into mathematicians, or should we be trying to give our students a sense of competence and mastery they can carry with them out into the real world?