Here it is, though. At least I accept my own submission. Enjoy:
Why Not Teach Math for Math’s Sake?
It is quite conventional wisdom, with lots of supporting evidence, to believe that the way we teach primary and secondary mathematics here in the US is generally failing our young students. This was detailed in the recent article in this forum “How to Fix our Math Education”, by Sol Garfunkel and David Mumford, and I see the general effects of pre-university education in students daily from my perspective as the director of an undergraduate program in mathematics at an American university. I believe the problems discussed in that article are real and demand action, and I applaud the authors for writing the piece. I disagree, however, with the conclusions of Professors Garfunkel and Mumford.
From my perspective, students come to university with a view of mathematics as a giant tool box they carry around with them, the tools being techniques useful to solve many kinds of diverse math-based problems. Pre-university education seems to be filled with disparate situations where a new concept is introduced (abstract or applied) to solve a certain kind of numerical problem, a technique is drawn up and the student receives a worksheet containing 40 or so variations of the same type of problem. Once completed, the class moves on to the next idea. While this assessment of pre-university education is simplistic, the outcome is that students never really learn how to think analytically, reason deductively, understand why these tools exist in the first place, or see just how each idea fits into the whole. Context via applications to real world phenomena (the kernel of the above authors’ proposed solution) may help in this regard, but there is a deeper problem with simply embedding math into applications to prove its usefulness.
The idea that mathematics lives only to serve its applications and functions only as the language of the sciences is absurd on many levels. More like music and poetry than physics or engineering, mathematics is an art, the art of pure reason. One could say that mathematics is the distillation of pure rational thought. When we teach math, we are not teaching how to solve problems. Instead, we are teaching how to think analytically; how to analyze any given complex situation, discover and understand its underlying logical structure, and figure out how to abuse that underlying logical structure to say something useful or conclusive about that situation. Numerical problem solving is solely one manifestation of this process. In the general sense math has very little to do with actual numbers at all. It is just that the use of a number system as one of our basic building blocks allows for a natural logically consistent system. Instead of giving students tools for solving problems, we should be teaching them the very nature of how and why these tools exist and were developed. Questions like why there is a quadratic formula, and why does the sine function exist at all are much more thought provoking and fundamental than how they work. We should be engaging students to actually design and redesign the tools themselves, a process of self-discovery which enables them to own the math they create. We should be giving them the confidence and experience to be able to see a problem as an opportunity for creativity and ingenuity, rather than an obstacle to overcome. And we should be teaching them that mathematical constructions have an innate aesthetic quality. They exist simply for what they are: beautiful constructions, often useful, whose existence lies entirely in the imagination, but whose manifestations in the real world are everywhere.
Math, like music and poetry, has a few constituent parts (notes and keys in music, words with contextual meanings and rhyming schemes in poetry) and a few logical rules which they follow. But with these few rules and parts, no one questions the infinite beauty and variance of musical creations or the fact that a few well-placed and possibly rhyming words can draw such emotion (think Shakespeare). And no one questions the value of teaching primary and secondary students music for music’s sake. Why not teach math for math’s sake? View it as a ground up endeavor where applications can serve as motivations for new mathematical ideas, but where the math lives outside of any application; where the beauty of self-exploration and discovery of fascinating concepts arises simply out of the aesthetic appeal of the constructions; and where the process of developing the skills of analysis and deduction in abstract logical systems becomes the goal of mathematics at the primary and secondary level. The application-based problem-solving skills could come along for the ride, and be reinforced in the other science-based classes. But the math would exist on its own.
Someone once said to me, “When are we going to stop getting students to solve problems and start getting them to POSE problems?” At the research level in math, we design and use the tools we need to pose and solve questions and problems as we need them. Teaching children the rudimentary process of doing this would go a long way to curing our math education woes.
What is that old saying “Teach a student a technique, and she will be able to solve some problems. Teach a student how to develop techniques, and she will be able to solve any problems.” I just made that up. But if we can teach our children to think analytically (read mathematically) before they reach university, imagine what we can do with them in university and beyond.