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I went to a very interesting talk today, extending a modern theory to an ancient example.

In 1907, Francis Galton went to the fair, and watched a weight-guessing contest, in which men put in tickets to guess the weight of meat that an ox would be when it was killed and dressed. The buyers were all sorts and conditions of men,  including professional farmers and butchers, and idle souls who came by and guessed their best. He thought this a fair analogue to democracy, and asked to see the cards afterwards. He found, that although the cards included estimates over a hundred pounds off, the median (1207 lb) was only nine pounds from the actual weight (1198 lb).  (Galton chose the median, as most analogous to the 50% + 1 result of democracy; the mean was 1197 lb, even closer.)

This is not what Galton expected, by the way; he was the snob who invented eugenics. It is to his credit that he published anyway.

This has recently become a hot topic in economics, called the "wisdom of crowds" with many experiments: one, to guess the number of jellybeans in a jar, had the mean estimate of a classful of students closer than all but one of them, IIRC. 

Our lecturer, Prof. Herman, of Hebrew University, had, however, a new example to propound: the Athenian direct democracy was also composed of all sorts of men, and made decisions through the collective decisions of a large random sample: although its decisions were not always perfect, they were good enough - if sometimes just barely good enough - for two centuries. 

He quotes sections from Aristotle, Plato, and Thucydides which argue for the wisdom of the crowd; in the last two, these are from speeches not expressing the author's view. (All of them, like most surviving Greek literature, disliked democracy, or preferred something else. Herman concludes that the Athenian democrats indeed argued for the wisdom of the many.
 
James Surowiecki and Scott Page have written books on the subject, offering Deep, Thoughtful, explanations, involving diversity of knowledge and approach (thirty people know more together than any one expert knows separately, and have more techniques for problem-solving), and information exchange.

What is truly striking, if I read correctly, is that neither of them discusses, even to refute, the obvious default explanation: the central limit theorem, from which follows: one way to get the average weight of a boxful of rocks is to draw out a committee of thirty rocks, weigh all of them, and take the average of the thirty. The weights of the committee may vary widely, but their average weight is likely to be quite close to the average of all the rocks, quite possibly closer than any of the thirty individual weights. 

This involves two assumptions: that the committee is large, and chosen randomly without bias.  If you vibrate the box, and shake the big rocks down to the bottom, the thirty on top will average light; if you pick a committee of one or two, they will be no closer to the grand average than any other rocks.

Similarly, if there is a large, randomly chosen bunch of people, their average opinion on the number of jelly beans in the jar will be close to the average opinion of all possible human beings. Unless there's some reason for people to guess fewer  (some optical illusion, for example) this average will be the number of beans in the jar. The conditions implied are the same the economists come up with: a large number of people, and as diverse as possible.

But then, few economists know much about mathematics: I have been convinced of this since I took Macro and they solved the problem of how to explain calculus right by explaining it wrong.

Understanding mathematics is a great pleasure, the mind's pleasure in its consciousness of its own power; being dragged though a bit of drill, and hoping your guesses at the answers are right, 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 schooling. (See the first part for more.)

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 criving 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; but 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, but the time spent on the explanation hasn't been used for drill either.

There are other problems inherent in mathematics:

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 explanation in my last post may perhaps count as a level-one explanation; but 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 the level two explanation that the numbers are a  division ring, which has inverses, is distributive, has no zero divisors....; and if you know the language, this is as much further clarification as the level 1 explanation has over "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-2 explanation, because it's so much clearer than a level 1. So it is, to the teacher; but not to them. Furthermore, the level two explanation requires 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 which 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 this 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 Benjamin 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; many who had seen it had seen it flutter briefly through chapter I 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 toughest 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:   ∞ =  −1.


There still is a course, Mathematics 104, which is calculus done right; I see they've adapted a standard textbook for the purpose; I had it out of the truly glorious Calculus by Michael Spivak (Long live the Yellow Pig!), which explains all of this, beginning with the rationals and  least upper bounds; but the Math Department still discourages you from taking it unless you've been through calculus done wrong first.