The Parents' ReviewA Monthly Magazine of Home-Training and Culture"Education is an atmosphere, a discipline, a life." ______________________________________ The Educational Use of Dynamics.by P. G. O'Connell. It is a strange thing that men who will spend hundreds of pounds a year on the education of their children by others, will yet grudge a few shillings for mathematical instruments, paint boxes, and other things necessary for home education. It is still stranger that they should not realize the pleasure to be obtained by encouraging a child to educate himself. It may be due to excessive modesty, but excess in modesty is a vice, like any other excess. Like all other vices it is severely punished by nature. Disobedient sons and rebellious daughters; these are the more severe punishments; the milder forms are merely deprivations of some of the joys of fatherhood. It is true that this is sometimes a necessary state of affairs. The father who is serving his country in the tropics cannot expose his children to a dangerous climate for the sake of the joys of fatherhood. Yet inexorable nature punishes even him. Only at rare intervals can he enjoy the pleasure of teaching his children to ride or run, to skate or row. Such pleasures are enjoyed by many fathers, and I think the only reason why they do not go a step further and enjoy the pleasure of educating their children's minds, is that they are over-modest. It has been impressed upon most of them in their school days that they were dunces, and, though they may since have become successful in life, they still believe that the art of educating the mind--for they do not consider it a science--is beyond them. Yet a sympathetic study of even one childish mind would teach them more than is known to most pedagogues, and the father has the almost inestimable advantage over the pedagogue that he can leave off "playing at school" the instant he notices that the child is tired of the game, whereas the pedagogue is expected to keep the child glued to his task. Indeed, only a very independent teacher would dare to do otherwise. Most fathers could spare five or ten minutes a day, and any father could do wonders in that time. By an occasional question, he could make sure that his child never uses the name of any number without having a clear mental image of that number of articles. By a little encouragement and praise, judiciously bestowed, with an occasional leading question, he could make the study of Geometry most interesting. On these subjects I have written a few notes already. I propose to-day to show how reckoning may be made more interesting by pressing into the service some of the simpler truths of Dynamics. The commonest method by which the human mind has increased its field of view and conquered further knowledge consists in a preliminary consideration of a number of particular instances, followed by the enunciation of a generalization, which has itself been sometimes, but not always, followed by a formal proof. There is nothing strange in the result usually produced by those teachers who reverse the natural process, presenting first the generalization and the formal proof and then a few examples; for, just as, if we look through a telescope from the wrong end, the better the telescope the further it removes us from the object looked at, so, if we reverse a mind-developing process, the better the process the more it tends to dwarf the intellect. If a man walk twelve miles in three hours, it is obvious enough to the untaught child that his average speed is four miles an hour, yet the question may well present a difficulty to a boy who has been worried with definitions and taught by bitter experience that there is usually a "catch" in the questions his master asks. I can imagine that a child born absolutely incapable of sensation would have no notion of speed, but I cannot imagine that a definition, if it could be supernaturally conveyed to him, would give him even a faint idea of speed. Definitions are mere luxuries of the learned; not aids to education, but rather the reverse. Discarding definitions, let us consider a sufficient number of examples in which the distance travelled in a given time is given; and we shall soon arrive at the generalization that the average speed is the distance divided by the time, the units being the same. Sufficient time should be allowed for this generalization to be thoroughly assimilated. Again, if a man walk at the rate of two miles an hour for one hour, at the rate of four miles an hour for a second hour, and at the rate of six miles an hour for a third hour, it is obvious that the whole distance walked is twelve miles, and therefore the average speed four miles an hour. A sufficient number of examples of this type will lead to the generalization that when the speed is gradually increased, or decreased, the average speed is exactly half way between the initial speed and the final speed, and is therefore equal to half their sum. These examples will also give a clear idea of the meaning of the English word "gradually," which I take to mean by "degrees," or "equal steps." The word thus includes, as a particular case, the meaning of the technical term "uniformly," which supposes the degree to be infinitesimal as well as equal. It is not necessary to introduce this technical term at first, and it is always advisable to put off, as long as possible, the introduction of technical terms. If a man, starting at two miles an hour and gradually increasing his speed, walk altogether twelve miles in three hours, what is his final speed? To a mind that has assimilated the two generalizations already mentioned, there is no difficulty in this question. The average speed is obviously four miles an hour, the initial speed is two miles an hour, therefore the final speed is six miles an hour. If a pupil have a difficulty with such a question it is probable that he has not thoroughly assimilated the first two generalizations that I have mentioned. The examples of this type should be particularly numerous, because the generalization, that the sum of the initial and final speeds is twice the distance travelled divided by the time, is particularly important. In the last example the man's speed is obviously increasing at the rate of two miles an hour every hour. Every example of this type may now be used over again, the pupil being asked at what rate the speed is increasing. Thus the idea of acceleration will be gained, and the word may be introduced. If the long word prove a stumbling-block, the use of it can easily put off for a little longer. If a stone start from rest and, moving with gradually increasing speed, fall four hundred feet in five seconds, what is the final speed and what is the acceleration? That the speed of a falling stone increases gradually, at the rate of about thirty two feet per second every second, is not a necessary fact but a physical fact. It is possible to conceive it untrue, therefore the mind cannot assimilate it unaided by the senses. It is true that it may be taken on trust and rendered familiar by mental contemplation, but a truth that is taken on trust is but feebly held and is easily detached by the first observation that appears to contradict it. Not only is the particular truth detached, but also every other truth learnt on the same authority. Suppose you had learnt from some book which contained a misprint, that water froze at a temperature of thirty-two degrees Fahrenheit and boiled at a temperature of one hundred and twelve, and that you experimentally discovered the untruth of the latter statement, would still believe in the truth of the former statement? I think you would require an experimental proof. Again, suppose you had learnt the facts correctly from a book in your youth, and some years later bought a hundred thermometers in order to commence teaching by experiment; suppose further that every one of these thermometers registered thirty two degrees as the freezing point of water--a thing which might easily occur with instruments manufactured in large numbers--would not your faith in the authority of the old text-book be shaken? Is it said that Newton learnt much from the contemplation of a falling apple; it is certain that we can all learn something from the contemplation of any falling object. We cannot help noticing that it falls in a straight line and that its speed increases as it falls, but the eye can hardly notice that the increase in speed is uniform, though I have come across observers who thought that they could notice this. For accurate determination, an expensive machine is necessary, but the metronome will enable us to perform fairly accurate experiments, and a metronome can always be borrowed or hired. If the instrument be set at sixty, it will tick at intervals of a second; if at eighty, at intervals of three-quarters of a second; if at one hundred and twenty, at intervals of half a second. By letting objects fall through measured distances, with the metronome ticking away all the time, we may experimentally discover how far an object falls from rest during various intervals of time. Moreover, the metronome will register a large number of different intervals of time. If it be set at any given figure, the interval between the ticks will always be that fraction of a second which may be expressed as sixty divided by the given figure. Every observation will necessarily be inaccurate, but, if every observation be repeated a sufficient number of times and the average result recorded, it will be found that Newton's law is probably true. And that is all that can be said of any physical law. We believe these laws to be unchanging, because we have never known them to change or read of their changing--except in the pages of Baron Münchhausen--but they differ as widely from necessary facts as from purely accidental facts. Even Baron Münchhausen would not have suggested that a necessary fact might be untrue; even Hamlet, when feigning madness, does not ask Ophelia to doubt the truth of any necessary fact; but we continually meet people who doubt the truth of physical facts. I have in my mind an old farmer who manages to make farming pay, though nobody can convince him that the earth moves. "Do 'ee mean to tell me," he would say, "that that house isn't in the same place to-day as it was yesterday?" I have never tried to convince him, for I reverence his position as that of a man who will not accept any non-experimental proof of a physical fact. Indeed he is a wiser man than ninety-nine per cent. of the scholars I have met. Assuming that we have convinced ourselves by experiment of the probable truth of Newton's law, and are therefore willing to accept it, provisionally, as true, we are in a position to answer such questions as:-- If a stone be projected vertically upwards with an initial speed of four hundred feet per second, how long will it continue to rise, what will be its average speed during the ascent, and how high will it rise? Moreover, we can answer without using any formula. I mention this because it is usual to teach pupils to solve such problems by applying a formula which has been committed to memory. And this in an intellect-dwarfing process of the first water. As a short method of expressing a fact that has been assimilated, a formula is available; as something that has been committed to memory, it is pernicious, and no formula can be thoroughly known unless the methods by which it has been arrived at can be rapidly passed in mental review. Examples in which the space passed over, the acceleration, and either the initial or the final speed, but not both, are given, and the question is to find the time, are usually worked by a method which involves the solution of a quadratic equation, which is unnecessary. Assuming that the mind has realized that the sum of the two speeds is twice the distance divided by the time, and that the difference of the two speeds is the acceleration multiplied by the time, it is easy to realize that the difference of the squares of the two speeds is twice the distance multiplied by the acceleration. Let us now consider a simple example. If a stone be thrown vertically upwards with a speed of twenty feet per second, how long will it take to rise six feet? Here the attraction of the earth is decreasing the speed at the rate of thirty-two feet per second every second. The difference in the squares of the velocities is therefore twice thirty-two times six, that is three hundred and eighty-four feet per second. The square of one is four hundred, therefore the square of the other is sixteen. This speed is therefore four feet per second, so the speed has been decreased by sixteen feet per second, which, at the rate of thirty-two feet per second every second, takes just half a second. If we realize that the difference between a speed of four upwards and a speed of four downwards is eight, we see that another quarter of a second passes before the stone is at the same distance from the ground on its return journey. This method involves the finding of a square root; in all but the simplest cases the alternative method involves the finding of the same square root. In each case we have to tackle the problem of how best to teach a young mind to find the square root of a given number. I once looked on at a typical scene. A typical British schoolmaster had neatly written various figures on a blackboard. Then, with a most engaging air, he asked:-- "What do you do next?" There was a long pause, and then a timid voice suggested, "You bring down--" "You don't bring down anything!" replied the master, with a trace of haste in his voice. The answer he expected was, as far as I remember:-- "You draw a line." Can anything be more brain-killing than this process which is usually defended on the ground that you cannot teach a boy, who has not studied algebra, how to find a square root otherwise than by rule? Let us see what we can do with mental images, commencing with very simple examples and gradually tackling harder ones. Here are sixteen articles, arranged as in nature:-- ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ The puzzle is how to arrange them in square formation--surely not very difficult. Obviously we must take some of the articles from the right and place them behind the others on the left, thus:-- ¤ ¤ ¤ ¤ ¤ ¤ and then repeat the process till we get a square. Consider now the method of arranging forty-nine articles in the form of a square, thus:-- ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ To arrange them thus, we first of all place twenty-five articles in the left-hand top corner, then we find out what is the largest oblong, of length ten, which we can form out of the twenty-four articles we still have; this oblong we divide into two equal parts, which we place as shown in the diagram; and the remaining four articles fit into the right hand bottom corner. Now let us consider the square root of five hundred and twenty-nine. In the left hand top corner place the largest possible square of a multiple of ten--obviously the square of twenty, then see what is the largest oblong, having one side forty (or twice twenty), which you can form with the remaining one hundred and twenty-nine articles, then divide this oblong into two equal portions and place one in the right-hand top corner and the other in the left-hand bottom corner, the remaining nine articles fit nicely into the right-hand bottom corner. The articles are now arranged thus:-- ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¤ In finding the square root of such numbers as eighty-one, three hundred and sixty-one, and nine thousand and twenty-five, we shall find that the greatest possible oblong does not leave enough articles for the right-hand bottom quarter. In such cases we must obviously use the oblong whose breadth is just one less. Finally, the process may be stated symbolically thus:-- 529 / 23 400 60 60 9 ~ * ~ * ~ * ~ * ~ * ~ * ~ * ~ * After a sufficient number of examples have been worked out, but not before, the symbolical process may be reduced to this:-- 529 / 23 4 12 9 ~ * ~ * ~ * ~ * ~ * ~ * ~ * ~ * This symbolical process can easily be understood, and will lead to the welcoming of the use of pencil and paper as a device for saving the memory unnecessary trouble. The ordinary process may now be described as still further saving the memory trouble. The child's mind will probably now be able to understand it--assuming, of course, that his teacher understands it and does not look upon it as a mysterious magic process which performs what the human mind could not perform unaided. Moreover, with large numbers, the process I have suggested asks rather much of the memory, thought it is easier to understand than the ordinary method. As an instance:-- 483025 / 695 36 1080 81 690 25 ~ * ~ * ~ * ~ * ~ * ~ * ~ * ~ * Each line symbolizes a concrete thing, the 36 a square, whose side is composed of 600 articles; the 1080 an oblong, 1200 by 90; the 81 a square, 90 each way; the 690 an oblong, whose sides are respectively 138 (twice 69) and 5; the 25 a square, whose side is 5. If we now imagine each oblong cut in half, it is easy to imagine the seven blocks fitted together to form a square whose side is 695. In the ordinary method the 1080 and the 81 would be added together before being written down, and also the last two lines, but there are no figures that do not occur in the ordinary method; I have merely omitted those figures and lines which tend rather to confuse than to explain. If a long-division sum by thus treated, it will be rendered much easier to understand, but I do not know that we have yet aroused a desire to learn long division. Before we have aroused the desire, it is folly to attempt to satisfy it. The desire will come soon enough, for there is in every child a fund of natural inquisitiveness which will render the educator's task easy and pleasant, if he will but study it. Unless the educator will allow nature to lay the foundations, his labour is but lost in building thereon. Proofread by LNL, May 2020 |
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