The Pi-eyed Pipers
HERE'S an entertaining prospect: Think of all the little children in school. They are sitting around a table, or maybe broken up into small groups--often broken up, those little children--clutching in their little hands some little rulers and tape-measures. Their beaming teacher passes sweetly among them, giving, to this one, the lid of a mayonnaise jar, to that one the mayonnaise jar itself, to another the hula hoop that has been waiting all these years in the Hands-on Educational Supply closet, to another a paper picnic plate, and so on to all. Round and round she goes, grandly bestowing circular objects of every sort, till every child has, in his very hands, at last, after all these ages of ignorance and uncertainty, the golden key to mathematical understanding, the great round lens of Euclid, through which he looked on beauty bare.
This, please be assured, is not the New Math. No, no. This time--this time for sure--the educationists know what to do. Pfui on the New Math; they were deceived in those bad old days. Now they have discovered the New New Math, and everything will be different from now on. Just you wait and see.
Learning by experience. That's the trick of it. Those little kiddies will take their tapes and their rulers in hand, and they will measure. With the tiniest tips of their little tongues protruding, and their little eyes all squinched up, they will decide how far it is from one edge of the hula hoop to the other, and then, with equal meticulousness, how far a little lady-bug would have to toddle to make it all the way around. And then, oh wow, they will divide!
In the long run, after oodles of hands-on measuring, and reams of scribbled division, followed by vigorous bouts of averaging, they will discover, say the preachers of the New New Math, and utterly by their own sociable enterprise, that pi is somewhere more or less in the approximate neighborhood of 3.14.
Now won't that be wonderful, and far better than being told some value of pi by some authority figure? Well, sure. And with all that hands-on experience behind them, won't those children be able to come up with other marvels, the more or less approximate value of the square root of two, for instance? And all by themselves? You better believe it.
We heard about the New New Math just a week or so after we were invited, along with everybody else on the face of the earth, to explore a different sort of mathematical consideration by William Raspberry, a columnist who is sometimes taken in by the educationists, but not always. In this case, he was chewing over a special case of the general proposition of educationism, which holds that whatever it is that the schools obviously don't know how to do should probably not be done in any case.
The special case was this: Look, how much math does the ordinary person do? A little adding and subtracting, maybe, and even a spot of multiplication now and then. If you have to tile the floor, the salesman will help you; he'll likely have a chart. But, be honest, how long has it been since you've found a square root, or solved an equation with more than one unknown? So, come on. Let's not bother the kids with anything more than the little smattering of math with which all the rest of us are getting by very well indeed. We'd save lots of time and money, and we could still offer more advanced math for those few kids who, for some reason or other, seem to like it.
Raspberry was puzzled. He clearly did not like that proposition, but neither could he see any way to reject it. It seems to make sense--common sense--which is the faculty by which we can all know that the earth is flat. So he left himself in puzzlement, and able only to wonder if someone there might be who would take up one cudgel or the other. It was a decent request, nothing less, we hope, than, Come, let us reason together. So we will not be displeased should this piece fall into the hands of William Raspberry, but we will not put it there. And here's why: Any human person can figure it out alone, provided, of course, that he has studied enough math, and language, out of which combined studies we most easily take the propensity--and the skill--for a certain way of thought.
If we study mathematics so that we may do mathematics, the proposition is excellent. We should reject it only if we can find some other purpose for that study, which may provide not what only a few may need, but which many must need. And, if Raspberry, like us, is in the habit of reading the comics section of the New York Times, he may by now have found out what that might be.
It was in that paper, of course, that we heard all about the children who are going to find the value of pi. That, too, was puzzling. To us, as to almost every other living person, the value of pi is--well, not very interesting. We certainly don't use it, and anyone who does can easily look it up, or, to be more accurate, can look up today's value of pi. They keep adding more decimal places; it's sort of like the stock market, up a bit, down a bit. Who knows where it will all end? And if mathematics is something that children study so that they may know the value of pi, then it certainly ought to be stamped out straightaway.
But the New New Math seems to be something even worse, something that children study not in order to know the value of pi, but in order to estimate it, and badly, at that, with the crudest of tools and the grossest of measurements. And whatever for?
Alas, we think we know what for. It is so that the children will come to imagine a vain thing, and to believe what the educationists already believe, to wit, that they can learn from experience.
No one learns from experience. Experience does not teach; it trains and conditions. If experience were truly a teacher, then all who live would be wise, for there is no deficit of experience in any life. What teaches is reflection, the mind's poetry, experience recollected in tranquillity. A man who has had nothing but the experience of sitting on a hot frying pan will not sit back down on that hot frying pan again, just now, but he will see no reason not to sit on it tomorrow, or on a hot waffle iron, for that matter, unless he has done something in his mind.
Although Franklin really called experience not the best teacher, but the hard school and last resort of fools, our educationists, who imagine that training and conditioning are education, are unable to see the difference between "learning" how to build a table, where training and conditioning are useful, and learning mathematics, where they are the barest beginnings.
Measuring hoops and jar lids is surely an experience. So what do the New New Mathers imagine that children will "learn" from it? What can possibly be learned from it? Not even an educationist can suppose that there is some "real" value of pi, just waiting to be discovered with a cracked wooden ruler and a grubby piece of string, and that some lucky child will come up with it, thus proving that we can compete with the Japanese, who are still floundering around in the umpty-umpth decimal. Is there some covert "democratic" goal in this weird exercise, some suggestion that such things as the value of pi ought to be established by consensus and compromise? And if, by some very small but happy chance, some child should happen to reflect on the experience of measuring hoops and lids, would he not conclude that his schooling had provided him with vain and empty busywork whose product, while easily to be had without any measuring, was of no conceivable use to him? Will the New New Math not flourish all the better in the absence of reflection, where it was obviously conceived?
There is, of course, a good reason for revealing the mystery of pi to children, and an even better reason for letting the children discover that mystery for themselves. When they have made a hundred--or a million--measurements, and found them all different, they will be standing at a fork in the path. In the New New Math, apparently, they will be told that they have "discovered" that pi is "about 3.14," and praised, no doubt, for having "learned" something all by themselves. And something will have been settled.
In some other way of studying math, they would be led to notice, not only that they still do not know the value of pi, but that it may be unknowable, and that one who knows that there is something that cannot be known, can nevertheless know something else of great value. It is simply this: that the mind alone can make knowledge that no amount of experience will ever bring.
These school people keep making us think of that strange notion of Aristotle, that a student of astronomy should be careful not to look at the stars. It is not all that absurd. The children would do better to put away their little tools and not even to look at circles, but simply to think. For it is by thinking, and only by thinking, that we can know that a circle and its diameter are what the geometers call incommensurate line segments. That means that there is no unit of measurement, however small, with which both lines can be numbered. That is why pi is called an irrational number; it can not be expressed in any fraction, or as a ratio of one whole number over another, however large those numbers might be. This is strange knowledge.
(You can prove it for yourself, by the way, even if you have forgotten all your geometry, or never even studied it. It's only incidentally math; essentially, it's just mind work. If you need some hints, read the Meno, in which Socrates leads a little boy into the mystery of another pair of incommensurable line segments, the side and the diagonal of the square.)
That knowledge, that the mind can know what experience can never show, is probably the most important event in the history of a mind. It brings countless benefits. It provides a nice distinction between information and knowledge, for such things as the capital of Texas and the betting average of Pete Rose, of which the wisest mind in Greece could have "known" nothing, are now revealed as something not exactly the same as knowledge, but dependent entirely upon experience, and never to be discovered by the mind alone. Not the greatest of minds, by taking thought, can work out the names of the Academy Award winners of 1949, although any clod can look it up. But anyone at all, even the greatest clod, if properly led, can learn to spot a non sequitur.
Between those things that we can learn in ourselves with our minds, and those things that we can learn only out of the experiences of our bodies, because we have eyes and ears, there are interesting differences both of quality and quantity. The latter are countless and cheap, like the vain, innumerable numbers that the children will derive. The former, while few, cost more; far too much to be handed out in the schools.
The Slough of Disparity
The following five areas of disparity in education will be examined:
WELL, honi soit and all that, but we still find that stuff a little bit smutty. So what could all that mean--"based on physical touching," and, even worse, "based on probing students"? Come to think of it, we can't even figure out what "based on" could possibly mean in that passage. Try it for yourself.
And that's not all. Later on in the same document we hear that teachers will observe one another and "code areas of student-teacher interaction." So how would you like it if we coded some of your areas of interaction, based on a little probing, eh? And then there's a bit about some "more satisfying relationships with students." Hmm.
But no, wait. This is not part of the evidence from one of those group child molestation cases out in California. It's strictly on the up and up, and absolutely legit. After all, it comes from nothing less respectable than the Douglas Education Service District, out in Roseburg, Or., where they'd never put up with what used to be called, delicately but quite distinctively, interfering with children. In fact, if you will consider once more what is proposed above, you will see that no one intends to interfere with any children, only with teachers.
You probably don't know what an Education Service District is. Here's how it goes. Education starts, and ends, with two people. One of them generates some sort of current in the other, or maybe not. If not, that's the end of the whole business. Wait till later. Or just forget it. But school is different. It still starts, and ends, with two people, but others get into the act. Because there is a teacher in a classroom, there has to be a crew of people who don't teach but who supervise and coordinate and facilitate one who does. And the members of that crew stand in need of the same services. Thus it is that, in every state, you can find buildings full of really super educators who have never actually seen one of those children whom they educate. An Education Service District is what they call such a building in Oregon.
The Douglas Education Service District is one of twenty-nine such in the state. They are separate from the herds of educators who hang out in the Superintendent of Schools Building; in fact, they educate those people as well as the children. The Douglas District has about 130 position holders, almost all of them people who were trained as what we now call, with a delicacy no less elegant than that out of which our elders named interference with children, special education teachers.
One hundred and thirty times twenty-nine is three thousand, seven hundred and seventy. That is a big bunch of educators. And are they ever busy. Consider the project described in the document we have quoted above. It is nothing less than a full-blown Staff Development Program sponsored by the Curriculum and Career/Vocational Education Departments. Yes, both. And it is called by an appropriately evocative name: Excellence through Equity, which must begin, of course, with an examination of areas--areas of disparity.
Ah, the educators of educators. Notice that the numbered areas are first called "five areas," to guard against the not inconsiderable chance, these days, of innumeracy in schoolteachers. And then the areas are named, perhaps so that we can code them. Instructional contact is an area. Grouping organization, another. The classroom control/discipline area and the enhancing self-esteem area do not exactly share a boundary, but they're both fine areas, and worthy of examination based on something. Last, whether least or not, there is the currently popular evaluation and student performance area, which, a few years ago, was being called the accountability area. But for one reason or another, the name never took.
Maybe it was because an accountability area can't possibly be based on such neat stuff as that higher level questioning and analytical feedback.
Now, we have told you that all this nonsense is named Excellence through Equity, and so it is. That is the name that appears at the top of the page. Farther down the page, however, we find the dotted line along which a school teacher eager to examine areas can tear, so as to send in a little scrap of paper that will take the place of a note asking for further details. In that place, we can read the putative words of the applicant, who is forced by the text to express interest not in Excellence through Equity, but in Equity through Excellence. It is not surprising, of course, that such a mistake should slip through in a document that comes from Sydney Poole, who is, after all, a Community Mentors Partnership Coordinator; only a reader would have noticed it. In all these years of studying the education business, we have yet to encounter even one Community Mentors Partnership Coordinator who can actually read what he himself has written. That takes a bit more reflection than a busy educator has time for. And we do suspect that, even if he had noticed that strange inversion, he would probably have said, So, big deal; they'll know what I mean.
And that's the weirdest part of this weird business. He'd be right. "They" would know what he "means," which is, of course, nothing in particular, which doesn't bother "them" at all. Those are just some words, trendy and attractive.
Try for yourself to attach a clear, concrete meaning to either version of the title. Let's see. Equity through Excellence? If we make everybody excellent they'll be equal? Or, maybe they'll all be equitable? Just what sort of excellence could cause that? Excellence in grades? In self-esteem? How would we know? Or what about Excellence through Equity? Does that mean that everything will be hunky-dory when everybody is equal, or equitable, or both, or what? Or does it mean--which sounds like something an educationist might believe--that if everybody treats everybody equitably, everybody will be excellent? Does either one mean all of those things, or none of those things, or anything at all?
But with such vexing speculation we will be led astray. It is not to the point. The phrase isn't really intended to take its power or worth from its meaning. What counts is how it sounds, and how it will look on the certificate that will be awarded, along with two graduate credits, to any school teacher who will come to six monthly workshops. When the "graduates" apply for new jobs or promotions, the non-teaching educationists who do all the really important stuff in schools will not be distracted by reflection. They do not read; they just look. They will look at Excellence through Equity and say, Oh, good. Or if it comes to that, they will look at Equity through Excellence and say, Oh, good.
In any case, nothing is written in this document about either equity or excellence, or just how the one might produce the other, or the other, the one. The stated goals, in fact, "collegiality" and "relationships" and such, all suggest that school teachers can put off worrying about equity and excellence for quite a while, and sign themselves up for a short course in character building.
Our school teachers must all be sick, or depraved, and much in need of therapy. They are always going to remedial workshops, refreshers, pep rallies, and psych sessions. Physicists and surgeons do go to natter with other physicists and surgeons about their work, but when teachers gather to consider their work, they do not go to listen to other teachers. They go to facilitators and change agents, to functionaries who have never been teachers, or who stopped being teachers as quickly as possible. And always they go not with their minds in mind, but with their sentiments, to get in touch with feelings, to get the feel of touchings. And to learn, of all things, how to be decent and kind, and to pay a little respectful attention to other people, for that, after all, is what all those "areas of disparity" imply. It is as though they became teachers only because they were all unconscionable, inconsiderate egotists and boors, raised in an uninhabited area of disparity by animals, and now must be taught, by their betters, the simplest and most obvious lessons of civil human intercourse.
If that's true, we should give thanks to the functionaries. If not, the teachers ought to give them something else.
A Word from