The Myth that Math is ‘Solving for x'
I read an article in the Atlantic today, titled The Myth of ‘I’m Bad at Math’. It made some very good points, of course the main one being that all scientific evidence points to the belief that skill in mathematics (skill in anything, really) has very little to do with genetics and almost everything to do with practice. This is an important point to make and one that I wish everyone would absorb, because the popular opinion in the United States is quite to the contrary. Most of us believe that you are born with some sort of “math gene” that enables you to solve algebra problems with dazzling ease.
Of course this is ridiculous, but there’s a much deeper problem here. The authors (Miles Kimball and Noah Smith, two economists) perpetuate an even more pernicious myth about mathematics, one that lies dormant in the minds of a startling majority of education policy makers, mathematics teachers, and students.
The myth is that being good at high school mathematics means you have “mathematical ability.”
Kimball and Smith state this very explicitly. They imply that the end-goal of mathematical problem solving is to “solve for x,” and more specifically to manipulate “equations and mathematical symbols.” But every mathematician (the definition of someone with mathematical skills!) knows that the heart of mathematics is not in the ability to manipulate equations just as the heart of cooking is not in learning to chop carrots. Of course there may be a correlation, but you need not master the high school mathematics cirriculum to have a strong mathematical mind, and you can master high school mathematics without having any real mathematical knowhow.
Kimball and Smith go on to claim that many lucrative career opportunities require such abilities (their only example of which is, unsurprisingly, an economist and a software engineer). I’m not saying that lucrative careers don’t benefit from a mathematical mind, because they do, but the truth is that nobody in software solves for x. The only people manipulating equations about computer programs are the ones on the cutting edge of research in computer science, or in particularly mathematical parts of software like graphics (even there, much of the hard mathematics is hidden away in libraries). The vast majority of mathematics used in software is arithmetic and the most basic forms of algebra.
In fact, I claim that for every career, what is meant by “mathematical abilities” is far from the ability to “solving for x.” The truly central mathematical skill is in learning to solve problems, and this usually has nothing to do with symbolic manipulation. I’m surprised that two notable economists didn’t make this point as well, because economics provides one of the easiest examples of the distinction.
In particular, when you’re an economist your job is to design models of economic systems that mirror reality as closely as possible. That is, you need to lay down assumptions about how people behave. You need to translate from real-world fuzzy notions like “supply and demand” to something that can be quantified. You have to evaluate your model against observed data (I wish all economists would do this), and tweak your model to fit unexpected phenomena. You need to revise in the face of criticism, specialize your model or generalize to fit others’ needs, and connect your work to a larger body of economics literature. To sum it up a bit facetiously, the real job about an economist is figuring out how the hell to reason about markets without going completely nuts.
The important fact is: most of the “mathematics” happening here does not “look” like mathematics at all. The mathematics comes in the form of ideas and reasoning. It just so happens that many economists prefer to express their models as succinctly as possible, i.e. in mathematical notation.
And while you may be solving for x in some of the steps of model design, you can always avoid mistakes by: asking a computer to solve for x for you (this is the safest thing to do), checking your work with a colleague, or borrowing the techniques of other economists on similar models. Of course you need to be proficient at algebra to get things done, but the point is that solving equations is not the primary skill, in fact it’s far from it. Again, the primary skill is in reasoning about complex problems in a principled way.
So when Kimball and Smith lambast Andrew Hacker for proposing to remove algebra from the high school cirriculum, I instead wonder why. Could it be that Hacker also agrees that algebra is not the be-all end-all of mathematical reasoning? Could he believe that perhaps true mathematical skills are better taught in subjects like statistics, geometry, graph theory, or basic logic and proofs? I certainly do. I believe students would get a lot more out of rigorously solving puzzles using logic than from memorizing the general equation of an ellipse.
And once the true mathematical problem solving skills are developed, and once one finds that algebra is necessary to one’s studies, then the process of learning to solve for x will be, as mathematicians say, trivial.
So to Kimball and Smith, I applaud you for your part in debunking the myth of the math gene; I just wish that you understood more deeply (or expressed in this article) what mathematical skills really are.
For more of my ideas on how mathematics education should work in high schools, see my post describing my experiences teaching high school students.