MATH

One of the most irritating things about being a mathematician in the United States is the tendency of the American educational system to leave its students convinced they hate mathematics, when they haven't actually seen mathematics. The "math" curriculum, up through high school, and all too often through the first years of college (when it is not practical to choose freely among courses), consists largely of rote unexplained drill: first through the four arithmetical operations, and then through a sort of higher arithmetic, ending up with calculus.The worst part of this is that the rules are never explained--or at least the explanations do not get through to the students. Teachers present rules like "Don't divide by zero" as if they were shibboleths instead of communicating their logic: dividing by zero doesn't work, because it reaches conclusions which are so far from being necessary that they are actually false. For example, 1 x 0 = 2 x 0, since both equal 0, but we still can't divide through by zero; 1 is not equal to 2. (This can only happen with 0. If 1a = 2a, then 0 = 2a -1a = (2-1) a = a; or in short, a = 0.)

There are reasons, good and bad, that we do this; I'll get back to them. But let it first be said that this is not mathematics. In the words of Benjamin Peirce, "Mathematics is the science that draws necessary conclusions". Any particular sudoku puzzle is a piece of mathematics: Given the requirements that no number be repeated, and the given digits, there's anecessary solution. Every sudoku is an isolated piece of mathematics, and the methods used are not particularly graceful.

But many people enjoy sudoku, although they never hear it called math. Indeed, it has nothing to do with arithmetic: one could use A through G, or the nine former planets (Poor Pluto, no longer inexorable!), just as well as numbers, for the nine symbols. But it is mathematics: a special case of the theory of Latin squares. More general questions about sudoku (How many sudoku puzzles there can be? What is the fewest number of digits one can posit and have a unique solution?) are quite challenging and many of them remain unsolved.

For more conventional mathematics, we have the proof that there is an infinite number of primes:

Suppose there were a finite number of primes, and no more: [2, 3, 5,...,769,...,P]. We could then multiply them all together, and add one, getting N = [2*3*5...*769*...*P] + 1. N is a very large, but finite, number; like all numbers greater than 1, it has a prime factor. But N is odd, since N-1 is divisible by 2. Likewise, N has a remainder of 1 when divided by 3, since N-1 is divisible by 3. So the prime factor is not 2 or 3, or by the same logic, 5 or 769 or P. So there must be another prime, contrary to the hypothesis.

Again, if there are six people at a party, either there are three of them who all know each other, or there are three of them who all don't know each other:

Suppose I'm at a party that is an exception to the rule. There are five other people. There are two cases; either I'm among friends, and know at least 3 people, A and B and C, or I'm not. If A and B know each other, then I, A, and B are all mutually acquainted; likewise if A and C or B and C know each other. But if none of them know each other, then A, B and C are three mutual non-acquaintances, and again the rule is satisfied.

But if I know less than three people, there are D, E, and F, all of whom I don't know. The same logic works in reverse: If D and E don't know each other, then I, D, and E are all mutually unacquainted; likewise if D and F, or E and F, don't know each other. But, on the other hand, if all of them know each other, then D, E and F are three mutual acquaintances, and still the rule is satisfied.

I find these things both persuasive and beautiful, and the few tries I've made to explain them seem to have been understood. Understanding mathematics is a great pleasure, the mind's pleasure in its consciousness of its own power; being dragged through a bit of drill and hoping your guesses at the answers are right, though, is one of the worst parts of schooling. We do much more of the latter than the former, and it is the latter that happens in compulsory education.

Heraclitus would find this natural; every beast is driven to pasture with blows, and we must make the drill compulsory, or no one would do it. But we don't, any more, teach history by reciting meaningless dates, or reading by driving kids through boring exercises; we try to mingle power with pleasure.

So what has happened; what goes wrong? Part of the problem, of course, is bad teachers, bad students, inadequate resources. But, as usual, the road to here was paved with good intentions. Society teaches arithmetic because it values citizens who can balance their checkbooks; it teaches the further branches of high-school "mathematics" because it values citizens who do surveying and calculate the volume of barrels. All this is rote learning, and efforts to explain interfere with it. I know, because I've been a Teaching Assistant myself.

If you're driving a class of thirty through any drill, it will go easier if the class understands it. But doing more than a cursory explanation has risks: you may get through to one or two students, who don't really need the drill anyway and the other twenty-eight or -nine may understand no more than they did before. This is the worst option--the class still doesn't understand, and the time spent on the explanation hasn't been used for drill either.

The inherent problem of teaching mathematics is that there are various levels of explanation. "Division by zero is forbidden" is a level-zero explanation, in the sense of Ring Lardner: Shut up, he explained. The explanations that introduced this piece may perhaps count as a level-one explanation. Above that there are higher-level explanations, using language of greater generality, or tools of greater power. The assertions used in the level-one explanation are all summed up in a level-two convenience: the numbers are a division-ring which has inverses, is distributive, has no zero divisors... If you know the language, this is as much further clarification as the level-one explanation can offer beyond "Just because". There are higher levels yet, if you want them, up to the giddying heights of category theory.

Once you know the language. That's the problem; it is always tempting to take the class to a level-two explanation, because it seems so much clearer than a level-one. So it is--to the teacher, but not to the students. Furthermore, the level-two explanation requires that the teacher spend even more time defining the terms: what distributivity, associativity, and so forth, mean. This is worth it if you're going to do what a college algebra course does: study systems that have different combinations of these properties (are distributive, but not associative, for example), and see what various combinations of properties imply. But not for a middle-school math class, which is not going to deal with any system more recondite than fractions.

This mistake was actually made. After Sputnik, there was a revision of textbooks, in the interests of a new, "more eddicated and scientific" citizenry, which could keep up with the pesky Commies. The mathematical part of this was the New Math, which included a lot of these second-level explanations; not only were these clearer sub specie aeternitatis, several of the professors who designed the thing felt that if students had this modern formalization available to them when they got to college, much time could be saved not having to explain it to them once they got there.

This might have been the effect if it had worked. I went through the New Math; I had associativity explained to me in fourth grade and in seventh grade, when I didn't care; it stuck when it was explained to me as a Princeton junior, and I was also taught why it mattered. (Making matters worse, this sort of formalization of arithmetic was, in the long history of mathematics, relatively new: a handful of mathematicians, including Peirce, had worked out the language a century or so before. It had been widely known among mathematicians for fifty years, but it had taken longer to work its way down to teacher's colleges. Many teachers had never seen it before and many who had seen it had done so in a brief flutter through chapter one of the books they had learned from, with no more real idea of its importance than I had in fourth grade.)

Some parts of mathematics are also genuinely tough to explain. Unfortunately, the parts of mathematics that people take just before they are permitted to drop out of it are geometry and calculus, which are among the most difficult to really explain. It took two centuries for some of the brightest minds in Europe to work out how to draw necessary conclusions in calculus without the risk of "proving" such things as the equivalence of infinity and negative one.

There still is a course, Mathematics 104, which is calculus done right. But the Math Department still discourages you from taking it unless you've been through calculus done wrong first. --Paul M. Anderson '79