Tuesday, February 16, 2010

Thinking Inside the Box

Nathan Carter - Bentley University

I love computers and related gadgets, but have been wary about integrating technology into my classroom. Calculators are not allowed on most of my exams, my students and I use plenty of chalk, and PowerPoint rarely shows up. I graph by computer only if a hand sketch would be messy. But as the math world gazes with interest on a shiny, new WolframAlpha, formerly dormant debates over technology begin anew. And they got me thinking.

At first, I leaned right back on my old favorite argument, shared by many, that can be applied to many different pieces of high-tech math tools: “Technology is a black box that can actually get in the way of real learning when pushing buttons replaces a more rooted understanding of what’s going on below the surface. When used as a teaching tool, students may come away able to produce a few impressive answers, but they do so without real comprehension or the ability to apply their knowledge in any context other than the basic setting of the problems they’ve encountered in the assignment.”

There’s a lot of truth to this argument. Sure, some instructors might use it to justify a pre-existing preference—not wanting to rework the whole curriculum in response to a shiny, new WolframAlpha!—but that doesn’t mean the argument isn’t correct.

And I still think it is compelling, but I recently made an important realization. It’s also irrelevant. To see what I mean, let’s apply the same argument to a piece of mathematical technology that’s a little older than WolframAlpha, even older than the calculator—yes, even older than the slide rule. Let’s apply it to…algebra! (I’m talking quadratic formula and completing the square, not groups and rings.)

But is algebra a technology? Merriam-Webster defines technology as “the practical application of knowledge especially in a particular area.” The American Heritage Dictionary is less brief, but allows any “technical means” even if only from “pure science.” No sprockets or circuits are required! Algebra is a technology.

Why compare algebra, which takes so much thinking, to using a calculator or computer, which (often) takes comparatively less thinking? I suppose I could stave off this question by saying that in each case you must learn a technical skill, or you’ll make an error and thus get wrong answers. This is true, but there is a better answer.


Our old, faithful friend algebra has just as much potential to be a “black box” as calculators and computers do. This includes not only the too-common example of using algebra to derive just as ridiculously incorrect an answer as you might with a calculator, but it includes much more sophisticated missteps as well. Consider the mathematician who attacks a problem or a proof with all the metaphorical levers, buttons, and knobs in the algebra arsenal and comes out the other side victorious. Then the referee’s report points out a far-more-elegant, two-sentence argument. In such a case, the referee’s report might very well say, “The author clearly doesn’t understand what is going on in this argument.” Oh, what we miss by fleeing to the trusted algebra crutch too soon!

But isn't algebra useful precisely because it works even at times when we either don't know why or at least don't care to focus on why? Surely not every algebraic argument can be turned into elegant prose—at least not in short order. And more importantly, haven't we as instructors justified students' study of algebra for this exact reason—its utility?

If the power of algebra, when used rightly, to churn out correct answers from correct inputs is the reason that students should become proficient at it, then shouldn’t that same reasoning justify their becoming proficient at even more powerful tools? In fact, if we use that reasoning to justify requiring students to be proficient with algebra, how can we do anything but require them to be proficient with the likes of WolframAlpha? (Software engineers may now cackle and/or cheer.)

So this is how I saw the light. I am a new mathematics professor and I say that a mathematician who wants students to learn algebra should also want them to learn any similarly powerful mathematical invention, even if it has sprockets or circuits! Wait. This means that I have to rework my curriculum, doesn’t it?

About the author: Nathan Carter is Assistant Professor of Mathematics at Bentley University in Massachusettes and author of the acclaimed new book Visual Group Theory, which employs the graphic power of computers to explore abstract algebra.

Aftermath essays are intended to be editorials and do not necessarily reflect the views of the MAA. Contact information is available here.

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