I recently came across an article by the mathematician Ethan Akin, whose work in topology and dynamical systems I admire greatly, called “In Defense of ‘Mindless Rote’“. In the article he defends the traditional education model of having students memorize mathematical facts and techniques. He begins with the following quote from Alfred North Whitehead’s Introduction to Mathematics.
It is a profoundly erroneous truism repeated by all copybooks, and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of operations which we can perform without thinking about them. Operations of thought are like cavalry charges in battle—they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.
Akin begins by drawing a parallel with physical tasks—playing sports, playing music, riding a horse—and how it is a necessity to practice the rudiments so thoroughly that they become second nature and subconscious.
As a beginner you move slowly, thoughtfully, with conscious attention. In a disciplined way you repeat the same movements again and again. Think of Audrey Hepburn at the cooking school in Sabrina: “one-two-three, crack. New egg. One-two-three, crack. New egg…” … As you practice, you speed up and your movements alter so that they are less in your mind than “in your fingers”. The skill is gradually incorporated into muscle memory.
Then he turns to mathematics and writes about the importance of mastery of the rudiments.
My real defense of all this symbolic manipulation is that it is easy. I hasten to add that when I speak of solving a system of two simultaneous linear equations in two unknowns as easy, I am using the word “easy” as a term of art. None of this stuff is easy when you start learning it. But these routines all have the capacity to become easy given disciplined practice. They are easy after they have become automatic.
He admits that people forget these skill over time, but that it comes back quickly.
Not everything that you learn has to be at your fingertips. However, there are lots of tricks that you have to learn well the first time so that when you need them later you can easily relearn them. The method of Completing the Square is learned and then gratefully forgotten after it has been used to get the Quadratic Formula. However, the return of the repressed occurs first in analytic geometry and later in some integration techniques. You have to be able to say “O yeah. How did that go again?” and then dust it off with a few examples once it is retrieved from storage in the mental attic.
Although he begins the article by insinuating that we should not allow students to think, but only to memorize, by the end he acknowledges his bluff.
What is hard is thinking. Despite the initial quote, neither Whitehead nor I really intend to disparage thinking. The algorithms and algebra routines are resources which can be deployed to help with problems where thought is required.
Moreover, he encourages educators to keep the material fun.
We traditionalists may insist on the value of drill but we don’t have a commitment to making education boring and hateful.