Long division: when and why?

The long division algorithm is a standard point of tension between “reform” and “traditional” ways of conceiving mathematics education. To reformers it embodies all the evils of old-fashioned rote learning and should be scrapped. To traditionalists it is a matter of pride and principle to keep it.

I say both are wrong. And their errors are symptomatic of core problems with their ideologies.

Consider first the traditionalist case. A famous and typical expression of this point of view is “The Role of Long Division in the K-12 Curriculum” by Klein & Milgram. Here we find the long division algorithm defended on the grounds that it illuminates the nature of the real number system and forms the foundations for certain techniques in higher mathematics such as calculus, differential equations and linear algebra. This is of course true but it is a poor argument for teaching long division in middle school or even high school. It makes no sense to teach millions and millions of children a tedious algorithm simply because a tiny fraction of them will become engineers and need it to do Laplace transforms. The vast majority of children will only “learn” long division through meaningless rote. Very many of them will struggle and mathematics will be for them a cause of anxiety and a drain of enthusiasm. Ironically, these children will not go on to a science career, so they will have struggled for nothing, whereas those who do go on to become engineers and do Laplace transforms are precisely those students who can pick up long division in five minutes once in college, so the middle school drill was meaningless for them as well. This general pattern––specific skills being cram-taught to lots of students at an early age, while the tiny minority who actually need the skills are precisely those who don’t need any cramming––is a major shortcoming of “traditional” education which, to my knowledge, is never addressed by the anti-reform movement. As we say in the biz, don’t teach it “just in case”––teach it “just in time.”

On the other hand the traditionalists are certainly right that “reformers” are by and large ignorant of the full mathematical significance of the topics they are so busy scrapping. The reform movement’s reputation as being, frankly, lacking in intelligence is well-deserved.

In a nutshell, “reform” advocates fail because they do not have enough of a vision of higher mathematics, while establishment mathematicians fail because they have nothing but this vision. In the hands of the former the mathematics curriculum will degenerate into unintelligent fluff and students will run around like headless chickens instead of being led systematically to the greatest pinnacles of human thought. In the hands of the latter the curriculum will degenerate into a mathematician’s top-down, “royal road to me” utopia––the fact that a mathematician can see the value of a topic is all the reason needed to force it upon any number of children at any age.

I advocate therefore a third way, the intellectual mathematics way. Which is this: teach it when you can convey its importance. It is all very well to say that long division is the foundation for understanding real numbers, but to this I reply: then teach it that way. If this is the reason for teaching it, then let this be the guiding principle of how you teach it. This is of course never done. This is the step the traditionalists never take and this where I depart from them.

Traditionalists are right that students deserve better than great mathematics being replaced by playing with colour blocks and plugging fake “real world” problems into a calculator. But I say that students also deserve better than being left in the dark as to the actual reasons for the importance of what they are studying and in fact being taught in a way that is profoundly out of touch with the supposed rationale for it.