If we’re to believe the press, mathematics education is in the midst of a war. Education saw the reading wars a couple of decades ago, and academia still seems to be in the middle of a science war. At least according to the press.
Before getting into the ideas, let’s just stop for a moment and say something about the metaphor. Real wars are violent, disgusting affairs that result in death, destruction and social instability. Nothing of the sort is going on. The math wars are blood sport only insofar as political careers can be destroyed by standing on the wrong side of an angry press.
In short the problem is simple. Elementary school children are currently being taught a number of pencil and paper methods that are clearly
- Not the same as the methods their parents were taught;
- Not mathematically efficient;
- Reportedly frustrating for students, parents and teachers.
So what’s going on?
Traditionally (for the last century or so, anyways) elementary school mathematics has largely been devoted to developing knowledge about numbers and to developing facility in computation. You begin by learning to count, learning the names of numbers, learning to add, subtract, multiply and divide. Students typically began with whole numbers and after developing knowledge and skill, they extended their work to fractions. Later on (typically Junior High, or Middle School), this arithmetic knowledge and skill expanded further to integers and real numbers. High School typically saw further extension to computation with radicals and polynomials. Depending on the location and political climate, complex numbers were added to the mix. Sure there were other aspects to the study of high school mathematics—a bit of geometry, theory of functions and so on—but school mathematics has traditionally been about taking arithmetic from whole numbers in the early grades to polynomials in high school. The central mathematical and educational idea was that this progression allowed for increasing mathematical generalization and increasing computational sophistication, leading to the development of some rudimentary mathematical understanding. (Before you get too excited about the “depth” of high school trigonometry and calculus, remind yourself that at the university level, first and second year mathematics textbooks invariably contain the words “introduction” or “elementary” in their titles. There’s a reason for this.)
As a general plan, I like the sketch I’ve given. Mathematics is largely about generalizing and abstracting the particular context away from particular problems, calculations, and whatnot. What’s the evidence that the “traditional” system gave students the payoff that was planned? To be sure, most English-speaking countries do not enjoy a population that by and large loves mathematics and has fond recollection of school math studies. Worse yet, the great majority of people do not need more than the most rudimentary mathematics to fulfill their life-plans, including education, employment and personal satisfaction. It would be nice to know if people were somehow better off because of their school mathematical experiences, but alas, we have nothing to indicate that this is so. Maybe we were on the wrong path. Hold that thought. I’ll return to it in a couple of days. There is some uncertainty whether the math education of the past few decades adequately prepared academically competent students for further study, and whether it provided much of value to those students whose futures were not dependent on mathematical achievement.
A second line of thought looks at the basic achievements of students past. Computation is a very valuable thing. You can’t do your taxes without it; you can’t take care of a significant amount of daily business without it. Accurate computation keeps the trains on time, and the books balanced. Nobody doubts this. But the days of accurate manual computation are gone, and there is no sign of them returning. A $5 calculator is better at all basic computation than anyone would ever want or need to be. Better, in fact, than any human ever could be (if you include the capacity to compute trigonometric ratios and logarithms). So why bother teaching children to compute? Why indeed?
There are two schools of thought on this question. One says that we should abandon manual computation because machines are all we need. Perhaps that’s too strong. Computation should be learned, says this camp, only insofar as it can be demonstrated to increase student understanding of mathematical concepts that are worth learning—whatever those turn out to be. Not so fast, says the other school. Computation is valuable in and of itself. It not only is a worthwhile thing to learn, it is the key to unlocking number sense in young minds, and it is the best way to increase student understanding of deeper mathematical concepts.
How to adjudicate? To date there is absolutely no compelling research evidence that is useful in addressing this matter. We have two ideological positions, each grounded in respectable intuitions, but neither having the weight of empirical evidence to support it. This saddens me deeply. With all the educational research published every year, why in the world is no one pursuing this question in a responsible way? I’m sorely tempted to give up my position in bureaucracy to get my feet back into a research institution just to answer this question.
In the meantime, what is happening in schools? Not surprisingly, two things. In some jurisdictions, you’d never notice that the times are changing and students are computing away like it’s 1969. In others, we have (yet another) “new” math. Students are no longer given extensive drill on basic number facts, and are learning a number of pencil and paper algorithms that look very strange to the students’ parents. These algorithms are not the tidy methods that were taught decades ago; they are often long-ago-discarded methods that involve a lot of extra writing and computing, but which are supposed to reveal the deeper structures that hide inside arithmetic. (E.g. Division by repeated subtraction.) Are these students gaining deeper understanding into arithmetic than their peers that learn the same algorithms as their parents? Nobody knows. To the best of my knowledge, there are no reliable evaluations of these new programs.
So here we are, stuck between two ideological positions, with a distressing lack of evidence. We’ll look at a bit more detail in a few days.