The Hardest "R"
Here is a math teacher's joke from around 1975, when the original "new math" movement was in full flood.
A schools inspector is walking round an elementary-school math classroom observing the children engaged in various tasks. One little girl is carefully placing square blocks in a tray to make an array four blocks by six. When she has finished, the inspector says: "Nice job! Now, can you tell me — what is four times six?" The child raises her eyes to the ceiling, furrows her brow and chants: "One six is six, two sixes are twelve, three sixes are eighteen, four sixes are twenty-four. So it's twenty-four." The inspector, somewhat disconcerted, says: "Well, that is the right answer. But why were you stacking those blocks in that tray?" Child: "Because the teacher told me to!"
New math is now a very old story, the opening campaign in a long drawn-out war between theory and reality, in which the advantage tipped sometimes this way, sometimes that. In April this year the National Council of Teachers of Mathematics issued their Principles and Standards for K-12 Math Education. This document has been widely perceived as a retreat on the part of the NCTM, a shifting of the balance back towards reality — "a gesture to the critics," according to the New York Times. For example, it reinstates such practices as the rote learning of multiplication tables, so that children educated under these new standards will have the same fall-back option as the little girl in the 1975 story.
That option has not been available to math students for several years. The previous standard for math teaching, also promulgated by the NCTM, was the so-called "constructivist" approach, defined in a 1989 report and subsequently adopted by most states. The central idea behind constructivism was that students should not merely know the correct answer to a problem, but should understand why it is correct. As reasonable as this sounds, it amounted in practice to a campaign of obloquy against rote learning (for example, of multiplication tables) and even more against "computational algorithms" like long division and the rules for manipulating fractions. With that flair for overstatement so characteristic of the revolutionary Left, the teaching of these algorithms to high school students was declared by Steven Leinwand, a leading proponent of the 1989 standards, and a board member of the National Science Foundation math program, to be "not only unnecessary, but counterproductive and downright dangerous." Dangerous? What was the actual nature of the danger? Such instruction, wrote Mr. Leinwand in 1994, sorts people out, "anointing the few" who master these procedures and "casting out the many." In other words, long division is elitist. So, apparently, are precise answers to math problems; under the 1989 standards you could get full marks for estimating an answer, instead of having to compute it precisely. You better hope that bridge you are driving over wasn't designed by an engineer trained in "constructivist" math.
Constructivism was itself an attempt to impose some order on the chaotic state of math teaching in the 1980s. There was the deceptively-named "basic skills" math: Use your calculator to find out 4 times 6. Then there was "rainforest math": 4 loggers each cut down 6 trees — how did the birds and squirrels feel? To capture the enthusiasm of black children there was "African Math," which proceeded mainly from the pretense that the ancient Egyptians were black Africans, and tried to teach arithmetic via their very clumsy and half-baked system of notation. The 1989 standards at least promised stability. However, when parents discovered that their children were no longer being taught to multiply two-digit numbers, there was a peasant uprising. California was one of the first states to adopt the 1989 standards in its schools (in 1992), and it was in California that parental opposition first made itself heard, leading to the California Math Wars of the mid-1990s. This counter-revolution soon spread, becoming a national movement, and enlisting college teachers disturbed at finding that ever-larger numbers of their freshman intake were in need of remedial math classes. When, last October, the Department of Education endorsed 10 constructivist math programs as "exemplary" and "promising," they were promptly refuted by an open letter from 200 university mathematicians and scholars, who argued that the programs concerned would leave students unprepared for college work.
The difference between math education now and forty years ago is not merely one of changed pedagogic technique. For the kind of person who is good enough at math to teach it, the job market of today is an Aladdin's cave of opportunities. Venture capitalist Jim Woodhill has remarked that: "Whenever I meet a teacher with more than 1300 SAT points, I drag him/her off to one of my high tech startups." So who remains in the classroom?
A recent article by a career high-school teacher in the London Spectator set out the eight types of people to be found on a typical high-school teachers' roster (in general — not specifically math teachers). To abbreviate drastically, they are:
- Evangelist: wants to shape youth.
- Exhibitionist: classroom as theater.
- Bureaucrat: teaching as gateway to a career in administration.
- Scholar: loves his subject but can't get a research post.
- Jock: poor man's way of being a sports pro.
- Fascist: loves power and discipline.
- Pedophile: loves kids … the wrong way.
- Cynic: short hours, long vacations, powerful union — hey.
(The writer says that category 8 is "by far the largest" in England.) Some readers may think this analysis itself too cynical. Certainly there are gifted teachers — "evangelists" and "scholars" — who can work wonders. But how many are there? And what teaching methods are suitable for non-gifted teachers — surely the great majority?
Lurking beneath all these considerations is the fundamental, perhaps intractable problem of math education: math is hard. Probably no large number of people will ever be much good at it, or like it much. Mathematical thinking is, in a sense, deeply unnatural. Mathematical truths are revealed to the human race very, very slowly, after unimaginable intellectual effort and lifetimes of frustration, against all the grain of ordinary human thought and language processes. In the preface to Principia Mathematica Bertrand Russell noted that an everyday object like a whale is immensely more complex, on any scale of complexity that makes sense, than a number like "five"; yet the whale is much easier for the mind to name, encompass and manipulate than the number. Any group of humans that was in regular contact with whales would certainly have a word for them; yet there are plausibly said to be primitive peoples that have no word for any number larger than three, though five-ness is right there, literally, at their fingertips. Given this profoundly unnatural quality of mathematics, it is a wonder we get any of it into children's heads at all. At the Academy of Lagado in Gulliver's Travels, mathematics was taught by writing the propositions on thin wafers which the students then swallowed. So far as I know, this method has not yet been recommended by any of the myriad councils, boards and committees pronouncing on math education. Perhaps it's time to give it a try.