Delphi complete works of.., p.348
Delphi Complete Works of Stephen Leacock, page 348
I am of course stepping out here on ground where wiser feet might hesitate to tread. But I think that for most of us something goes wrong, very early in school, with our mathematical sense, our mathematical conceptions — or rather with the conceptions that we fail to get. We get lost in the symbols of mathematics and can’t visualize the realities — visualize or dramatize, or whatever you do with them. Mathematics is always, for most of us, a sort of mystery which we don’t even expect to understand. Let me illustrate the attitude by recalling a joke of a stage “review” of a few years ago. Some boys are seen coming out of school, comically overgrown and comically under-dressed, grown too long and dressed too short, so as to make them look funny.
“Well, my little man,” says a stock stage gentleman, in the stock voice of a stage question, “and what are you learning at school?”
“Reading and writing,” says one of the comedian boys, his immobile face a marvel of wooden imbecility, blank as the alphabet.
“Reading and writing,” repeats the stock gentleman, so as to let the audience get it, “and anything else?”
The “boy” answers, with no facial movement, “We learn gazinta.”
“You learn what?”
“Gazinta.”
“But what is gazinta?”
“Why,” explains the boy, “like ‘two gazinta four’ and ‘five gazinta ten.’”
The roar of the audience’s laughter ends the mathematics. They laugh because in the contrast between the clarity of reading and writing and the mystery of “gazinta” they see their own experience. For them all mathematics is, and always will be, “gazinta.”
Here is a particular example, familiar to all school and college people, of what I mean by our failure to get a proper grasp of mathematical thought. We all learn that the attraction of gravity exercised on or by a body varies in direct proportion to its mass, and inversely as the square of its distance. The square? That’s the sticker for most of us. What’s the square got to do with it? We understand, or we think we do, that of course the more “mass” a thing has the more it pulls. In reality this is the real philosophical difficulty, since mass means power to “pull,” and “pull” means having mass. But we don’t look into it so far as that; the bigger the mass, the bigger the pull, all right. But the square of the distance we accept, learn it by heart, use it, multiply it — in short, it becomes “gazinta.” It seems an odd thing. Why the square? Why not the cube, or the anything else? We don’t see, till we learn to get it straight, that the thing is self-evident.
The pull varies with the amount of surface, a thing of two dimensions, broad and long. A tower at a certain distance (don’t call it x or we’ll get mixed) looks a certain height and looks a certain breadth. A tower twice as far away would have to be twice as high to look level with it and twice as broad to look of the same breadth. So the far-away tower at twice the distance of the near one, in order to look the same size, would have to be twice as high and twice as broad and would present to the eye four square feet to one, in order to present an apparently equal surface. The attraction is in proportion to the surface and gets less and less for any given size of surface as you go further away. And it doesn’t matter if the surface is square or round or triangular, or any other shape, since they are all proportional. Here I believe is where pi comes in — but don’t let us go too far with it.
There are ever so many of these mathematical conceptions that turned into mystification because we never got them right at the start. The trigonometrical ratios — sine, cosine, etc. — seemed just an arbitrary iniquity. If we had thought of them as moving arms, like traffic signs, we would have felt them to be the natural and inevitable way of measuring an angle.
It seems to me therefore that something might be done, at the very opening of education, to strengthen our grip on the mathematical idea. This would bring us back, I presume, to those mathematical judgments synthetically a priori with which I started. The question involved is the nature of number and magnitude, and why does one and one make two? and the consideration whether a statement of that sort is just a fact or an inference from one judgment to another. I imagine that if we could see into one another’s minds we should find a great difference in our grasp on the sequence of numbers. A hen, it is understood, can distinguish two from one but is lost at three. Primitive languages count a little way and then say “a whole lot.” Here figures end and lies begin. Even the Greeks used to say “a myriad” to mean not an exact number but ever so many.
We have fallen heirs to the wonderful ingenuity of what we call Arabic notation. In reality the Hindus started it, but the Arabs made it plainer still by writing into it a “cipher” or “zero” to mark a blank place. We learn it so early in life and so artificially that we don’t appreciate it. We think of ten as an arbitrary point, whereas the shift to a new “place” could have been set anywhere, and would be better if set at something more divisible than ten. If the people on Mars have brains as much better than ours as their planet is older, they may use a set of numbers that would go thousands at a jump and write the population of the United States in three figures. We couldn’t of course do that. The multiplication table used for it would be beyond our learning. But I am sure that we, the non-mathematical people among whom I belong, would get a better grip on mathematics if we had a better conception of the relationship of numbers and symbols.
I am aware of course that there are many recent books that attempt to shed new light on mathematics. But the light seems dim. One or two well-known “series” contain what are really admirable presentations of the philosophy of mathematics. But, for the ordinary person, to mix philosophy with mathematics only makes it worse. Other popular works undertake to bring mathematics to the intelligence of the millions; it would be invidious to name the books, but, apart from their optimistic titles, I cannot see much success in them.
I am aware also that various new methods of teaching mathematics are adopted, especially in teaching mathematics to beginners. But in any that I have seen there is little else than one more example of the present tendency to turn children’s education into fun. Kindergarten children waving little flags, forming themselves into squares and cubes and separating themselves into fractions, may look very pretty, but they are no nearer to the mysteries of number. Singing the multiplication table doesn’t make it less relentless.
Here on my desk, for instance, is a widely known pretentious book of “new method.” It undertakes to “individualize arithmetic” by teaching the children what the author calls “number facts” by the use of numbered cards. “Cards,” says this authority, “are invaluable for learning number facts.” Many of us found this out long ago. The children “individualize” their arithmetic by sitting in a ring, dealing out cards with numbers and pictures on them, and then seeing whose “number facts” win out against their opponents. The children might learn poker from this but not mathematics. What they are doing sounds like a “show-down” of “cold hands,” a process as old as California.
The basic idea of my discussion is that somehow we don’t get our minds mathematically adjusted as they might be. I am aware that there are great differences of natural aptitude. We are told that Isaac Newton when he was a boy took a look through Euclid’s Elements and said it seemed a “trifling book.” That meant that, when Euclid said, “the three angles of a triangle are together equal to two right angles,” little Newton said, “Why, of course, obviously so.” Probably the Pythagorean theorem about the squares on the sides of a right-angled triangle only held him back a minute or two. These things took the rest of us a year of school. But, all said and done, I think that it is not only a matter of aptitude but of approach. We don’t “go at it” right.
With that I leave the subject, with the hopes that at least it may be stimulating to professors of mathematics. A little stimulant won’t hurt them.
CHAPTER V. PARLEZ-VOUS FRANÇAIS? OR, WHY CAN’T WE LEARN MODERN LANGUAGES?
SIT DOWN, GENTLEMEN — The annual mass attack on French and its repulse — How to learn French: forget English — Never learn a rule: learn to think subjunctively — Swallow the phrase whole — Read for reading’s sake — When you get it, it’s like swimming
I remember that when I was a student taking German at college, a criticism reached the ears of our good old professor to the effect that the students never spoke German and never heard it. He was hurt at this, and so at the end of that term he put on what was announced as an oral examination. We were directed, four at a time, into a little room, where the professor and two “outside examiners” were sitting in state as a board of examination.
I went in with three other students and we lined up across the room.
“Setzen Sie sich, meine Herren,” said the professor, very impressively. This means, “Sit down, gentlemen.”
We stood right there.
“Meine Herren,” repeated the professor, as casually as he could, “nehmen Sie, bitte, Platz.”
No, sir; not us; we never budged.
“Sit down, gentlemen,” said the professor curtly.
Down we sat, all together.
After that year there weren’t any more oral examinations.
The same thing could have happened in any North American university, and could have been carried out in any of the modern languages — except in French in French Canada. Yet any of us, with the instruction we received, could have translated ordinary German or French into English, and even put English sentences into French or German by a process like working with a hammer and saw.
I have selected French as the main object of discussion in this chapter because I have had over sixty years of dealing with it and enjoy the advantages for observation that go with residence in a bilingual province. But all that is said here about learning French could be said about learning German, Spanish, Russian or Norwegian.
The fault with our teaching of modern languages is not so much that we teach them wrongly as that we don’t succeed in teaching them at all. Ask anyone who “took” Freshman French at college, or learnt French in high school. Only don’t ask him in French.
Every year in English-speaking North America a vast phalanx of high school and college students, millions of them, gather for a mass attack on French. They come on against a heavy barrage of declensions, conjugations and exceptions; they are beaten back, gather again and re-form each year till their school-days end in defeat — as glorious and as hopeless as Pickett’s charge at Gettysburg. Twenty-five years later, when the pupils and students have grown up into adult life, there will be practically nothing left of their French except a few fragments and a little wistful regret and wonder. Ask your friend, the father of a family, what French he knows, and he will say that he knows such things as, “Donnez-moi un bock, s’il vous plaît” and “Garçon, encore un bock.” But he learned those on a trip to Paris, in the proper way, by eating and drinking them in.
Let me speak here on my own experience, not from vanity over it or egotism in telling it, but because I think it is typical of that of thousands of others. I learned, or mislearned, my French in the English-speaking Province of Ontario; but what I say of Ontario, for which I have nothing but affection, is not directed against it singly. I am certain that its faults are shared by practically all our English-speaking continent. Of England I am not speaking for the moment; over there the proximity of France and the fact that languages were learned for generations before the schools spoiled the process make things a little different. But not altogether so.
Let me then explain about my experience in Ontario. I am not offering here any criticism against the efficiency and the industry of the many hundred people who teach French in the schools of Ontario. They do what they are compelled to do to meet the strange and disastrous kind of test applied to their pupils. They have to prepare their pupils to pass the matriculation examination of the universities; and they do so. Some of their pupils even pass with distinction; others carry away what is called honours, and are so badly damaged thereby for learning French that a residence of ten years in Paris would hardly effect a complete recovery of their native faculties.
And the amazing thing about the situation is that if Anatole France or Victor Hugo had been sent up to write on an Ontario matriculation examination in French there is not the slightest chance that either of them would head the list; they would be beaten right and left by girls from Seaford High School who never saw the red wings of the Moulin Rouge, and by boys from the Hamilton Collegiate Institute who wouldn’t know enough real French to buy a boiled egg in the Café de la Paix. Indeed it is doubtful whether Anatole France and Victor Hugo would have passed at all. The whole examination being a test in English, they would probably have been ploughed and have had to be put under the care of an Ontario special teacher for six months to enable them to get through.
The point that I am endeavoring to make and reinforce with all the emphasis of which I am capable is this: the ability to translate French into English in writing is not a knowledge of French. More than this, it is the very opposite of it. It involves, if exercised persistently and industriously, a complete inability ever to have a knowledge of French. The English gets in the way. The French words are forever prevented from acquiring a real meaning in connection with the objects and actions indicated, because the mind has been trained always and for ever and hopelessly to associate them with English words instead of with things. The process is fatal. The whole system is not only worthless but it is a fraud and an imposition practised upon all those who learn French in such a school method; and the schools are driven to use the method because the colleges impose a written examination of translation and grammar as the criterion of a knowledge of French. For the proof of it I appeal to the candid confession of all those who were trained in this machine. I appeal to such people for corroboration of what I say. All that they learned was directed toward nailing the English word so tight to the French one that nothing can ever prize them apart.
I, myself, speak of what I know. When I was a little boy in England I learned to use a few small phrases in French, such as “Bonjour, Monsieur” and “Au revoir,” in the proper and real way; not connecting them with any link to English words but letting them spring out of the occasion. Anybody who understands the matter will understand what I mean. Later on I learnt French in Ontario and entered, traversed, and left the Provincial University with all sorts of distinction in it. Part of the teaching, like part of the curate’s egg at the Bishop’s table, was excellent no doubt, but the base of it was worthless; it had all been undermined and spoiled and forever rendered futile by the unspeakable matriculation examination which preceded it and which was a necessary preliminary to entrance to the French classes.
I mean it literally and absolutely when I say that I knew more French in the real sense of knowing it when I was a child of six years in England than when I was given first-class honours at graduation by the University. In the first case I knew a little; in the second case I knew not a single word that was not damaged by false association and contact. All the energy and industry and determination that I had put into my college work, all the interest and fascination that I felt for the language, all the pride that I could have felt in really knowing and using it — was dashed to pieces against the stone wall of the barrier erected in my path.
When I graduated I could not use a single word of French without thinking of English. I had to begin painfully and wearily all over again at the very bottom. Somehow I had stumbled upon the secret of a true beginning, and I began to try to collate in my mind the French words and the objects and ideas and to exclude the English. But it was hard work. The college had left its fatal mark deep stamped upon my brain. But at last, many years after my graduation, and with advantage of residence in Montreal, the light began to break.
If I live long enough to forget a little more of what I learnt at school I shall soon be able to speak French as well as a Montreal cab-man talks English. More than that I do not ask. But for my academic education I might have spoken French with the easy fluency with which the girls behind the notion counters of the Montreal department stores rip off their alternative languages. For such higher competence I can only have a despairing admiration. It is not for me. Yet let me speak as the cab-man and the car conductor speak, and I am content to depart in peace. For I shall know that if a French angel (such is the kind I should prefer) opens the gate to me and says “D’où venez-vous?” I shall answer “Je viens de Montréal,” without first framing the thought in English.
Let us consider a little further the matter under discussion. The whole of the teaching of school French is directed toward passing the matriculation examination of the colleges. This examination is conducted on paper in English. It has therefore absolutely no connection with the use of the ear as a means of hearing language. In fact, the language is regarded as a thing seen but not heard. I am told that people thus taught, when they land at Calais or Dieppe, are often seen to grasp their ears at the first tingling of the new sensation of hearing a language spoken. Moreover, the examination in question consists, entirely, or almost so, of writing out English translations of French words and of translating written English words into French ones.
The typical form of a French examination test is to hand out to the candidate a rapid-fire series of silly-looking little grammatical difficulties involving a queer sequence of pronouns or something of the sort. Some such exercise as this is given:
