I totally understand the place of the algorithm in mathematics. But the argument about the use of algorithms reminds me of a deeper issue, how math education is currently trying to figure out how to adapt to major technological advances in computing that allow us to have computers perform these algorithms for us. I’ve seen a lot of arguments that the math curriculum needs to change drastically to take advantage of new tools like graphing calculators and Wolfram|Alpha (which, of course, both rock). Basically, that any sort of direct computation needs to be phased out of a 21st century curriculum – that we should teach students how to problem solve using these technological tools. Others argue that deep understanding is enhanced by knowing how these processes work. To which others counter that people drive cars all the time to get from place to place without ever having built a car or really knowing how it fully works.

Somehow I was reminded of this when I saw the family dog, Whiskey (who is absolutely hilarious), performing some of his tricks when I was visiting home this summer. Our family’s favorite is the “Bang, you’re dead” trick.

First, my mom puts her gun out, to which Whiskey responds by sticking his paws in the air innocently. Then, my mom yells “bang!” and Whiskey awkwardly flops to the floor, flips over and plays dead. As you can probably tell, it’s a pretty amusing trick, and pretty complicated for such a puny little brain. But… here’s the whole video from which I got these screen shots (no idea why it ended up so stretched out):

Notice that he tries just about everything before he gets it right. He has sort of a general idea of what he is doing, but he has no idea why he is doing each of the steps. When he accidentally does one too quickly, or jumps up instead of putting his paws up, he can’t diagnose his misconception thoughtfully and fix his mistake. He just blunders through trial and error until he figures out something.

The sad part was that this totally reminded me of a few students whom I taught last year (seniors taking Calculus) who had no idea how to do basic algebra because they had no deeper understanding of what was going on and somehow had no idea how to check their answers to see if they was on the right track. It felt like their previous math teachers had taught them how to do tricks, and perhaps I wasn’t doing any better. If they managed to plow through the right steps and stumble on the right order, I would reward them with the treat of a good grade, and that reward exists in both traditional grading methods AND in Standards Based Grading. But the problem for me wasn’t in the algorithms. It was that they had no deeper understanding of mathematics to accompany those algorithms. Teaching just computation, and not teaching it well.

So for me, when math computation technology proponents argue that you can drive plenty of places without every knowing how a car works, I always think about the one time when the car breaks down. What do you do then? You are stuck. You have to call someone for help. You can’t thoughtfully work out the problem on your own. I agree with the hopefully general obvious opinion that learning how to do computation is not the goal of a mathematics education. And I agree about the danger of accidentally teaching meaningless algorithms, which can easily happen unless you conscientiously dig deeper in checking for understanding. But the idea of throwing out the deep conceptual understanding of mathematical structure that goes along with learning some of these processes in favor of using computers for computing just doesn’t really sit well with me. The important, deep conceptual understanding of mathematics certainly doesn’t come from just learning algorithms, but it also not helped by never learning why algorithms work. Technology, though a great tool, is not a replacement for the human mind.

Why I am thinking about things like this and not the hectic as-of-yet-unplanned first week of school is beyond me…

UPDATE: If you haven’t checked out Matthew Brenner’s “The Four Pillars Upon Which the Failure of Math Education Rests” go read it – reading the whole thing is on my to-do list but everything I have read from it so far is wonderful. It was pointed out to me after I wrote this that he wrote something very similar (though about ten times more eloquent) on page 55. Agree to agree I guess! Now I must read the whole thing.