Thinking and learning to think - Nathan C. Schaeffer (romantic story to read .txt) 📗
- Author: Nathan C. Schaeffer
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Instructive as it often is to follow the order of discovery, it must not be inferred that this is invariably the best order of instruction. What teacher of astronomy would be so foolish as to lead a student through the nineteen imaginary paths which Kepler tried before he discovered that an elliptical orbit fitted the recorded observations of Tycho Brahe![6]
Much may be learned from the methods pursued by eminent teachers. It will abundantly pay any teacher of science to study Faraday’s lectures on the chemistry of a candle,—a series which for models of developing the fundamental concepts of chemistry is unsurpassed. The devices used by such teachers are often very suggestive. For instance, in teaching the concept of the new geography that the earth revolves not like a body with a liquid interior, but like a body with an interior as rigid as glass, Lord Kelvin suggests a comparison of the spinning of a hard-boiled egg and of an egg not boiled at all,—an experiment easily made in every school-room.
A few quotations from the astronomer Young will show how concepts of great distances can be developed so as to be more than a numeral with a row of ciphers annexed:
“If one were to try to walk such a distance, supposing that he could walk four miles an hour, and keep it up for ten hours every day, it would take sixty-eight and one-half years to make a single million of miles, and more than sixty-three hundred years to traverse the whole. If some celestial railway could be imagined, the journey to the sun, even if our trains ran sixty miles an hour, day and night, without a stop, would require over one hundred and seventy-five years. To borrow the curious illustration of Professor Mendenhall, if we could imagine an infant’s arm long enough to enable him to touch the sun and burn himself, he would die of old age before the pain could reach him, since, according to the experiments of Helmholtz and others, a nervous shock is communicated only at the rate of one hundred feet per second, or one thousand six hundred and thirty-seven miles a day, and would need more than one hundred and fifty years to make the journey. Sound would do it in about fourteen years if it could be transmitted through celestial space, and a cannon-ball in about nine, if it were to move uniformly with the same speed as when it left the muzzle of the gun. If the earth could be suddenly stopped in her orbit, and allowed to fall unobstructed towards the sun under the accelerating influence of his attraction, she would reach the centre in about two months. I have said if she could be stopped, but such is the compass of her orbit that to make its circuit in a year she has to move nearly nineteen miles a second, or more than fifty times faster than the swiftest rifle-ball; and in moving twenty miles her path deviates from perfect straightness by less than one-eighth of an inch.”[7]
Professor Young uses a very suggestive device in his astronomy for showing the comparative sizes and distances of heavenly bodies:
“Representing the sun by a globe two feet in diameter, the earth would be twenty-two-hundredths of an inch in diameter—the size of a very small pea or a ‘twenty-two caliber round pellet.’ Its distance from the sun on that scale would be just two hundred and twenty feet, and the nearest star (still on the same scale) would be eight thousand miles away at the antipodes.”[8]
Sometimes the employment of a new unit aids in realizing the idea of very great distances. The ordinary astronomical unit is the distance of the sun from the earth; it is not large enough to be convenient in expressing the distances of fixed stars. Hence astronomers have found it more satisfactory to take as a unit the distance light travels in a year, which is about sixty-three thousand times the distance of the sun from the earth. The tables of fixed stars give distances in terms of this unit from 3.5 upward. A glance at these figures fills the mind with an idea of the infinite grandeur of the universe and with feelings of awe and sublimity akin to those which must fill the soul on approaching the throne of Almighty God.
Scientists assert that the infinitely great is more easily conceived than the infinitely small; that quantities represented by billions and trillions are more easily grasped than fractions of a unit with a million in the denominator; that ages of time are more easily comprehended than fractions of a second. In a lecture delivered at the International Electrical Exhibition, Professor Charles F. Himes employed a very ingenious device for giving an idea of how a “snap-shot” may be made, or a photographic impression taken of an electric spark, or a flash of lightning. He exhibited a photograph of the sparks of a Holtz machine, which are of shorter duration than any instantaneous drop or slide could be made to give. “They impressed themselves upon an ordinary collodion plate as they passed. Suppose we assume one-twenty-thousandth of a second as the time, and we will be within bounds. That is a fraction difficult to comprehend. Our mental dividing engine fails as we work towards zero. The twenty-thousandth of a second is so small that it eludes our mental grasp.... Looking at it from another point of view, let us regard the effect as a space-effect instead of a time-effect. Light has a velocity, in round numbers, of one hundred and ninety thousand miles per second. That would be one hundred and ninety miles in one-thousandth of a second, nineteen in one-ten-thousandth, or, say, ten miles in our one-twenty-thousandth of a second. Ten miles of light drive in upon our plate in that time; or, if we held the corpuscular theory of Newton, a chain of these little pellets ten miles long would have delivered themselves upon the sensitive surfaces. Ten miles is comprehensible, one mile is, so that we could easily conceive of an effect in one-tenth of the time allowed to our electric sparks. But let us take another look at it. Light is not corpuscles, but undulations, tiny wavelets, ripplets of ether, eight hundred million million in a second for violet, a number we can easily understand, as Sir William Thomson[9] has told us. That would make eight hundred thousand million in one-thousandth, eight thousand million in one-ten-thousandth, or forty thousand million impulses striking our sensitive molecules in our one-twenty-thousandth of a second. Surely that number should produce an effect. We can readily conceive that one thousand million wavelets would produce an appreciable effect. They would represent one-eight-hundred-thousandth of a second, say one-millionth of a second. That would seem, then, to be ample time to produce a photographic effect.”[10]
Many teachers of science spend all their spare time in reading scientific literature and in posting themselves upon the latest achievements in their specialty. It might be to them a less delightful occupation if they traversed fields of investigation already well explored for the purpose of seeing how the student can be led over these most expeditiously and with minimum expenditure of time and effort. Thought bestowed upon the best way of imparting the elements of science would have a most beneficial effect upon their methods of instruction, and would greatly increase their skill in teaching. Many of the most abstruse and complex ideas can be resolved by analysis into their elements, and thereby be made intelligible to people of ordinary training. An eminent teacher of theology felt called upon to impart to a promiscuous audience an idea of the doctrine of total depravity as taught by the Church. He started by referring first to the popular mistake that the doctrine teaches the utter depravity of the human race, then to the ancient heresy that the depravity of human nature resides in the body, and not in the soul, and, finally, to the meaning of total as signifying not that man is as bad as he can become, but that he is depraved, or has a tendency towards sin not merely in his physical body, but in the totality of his being. Analysis prepared them to see that by total depravity is not meant that men are as bad as they can be, nor that they do not have in their natural condition certain amiable qualities or certain laudable virtues; that the doctrine means that depravity, or the sinful condition of man, infects the whole man,—intellect, feeling, heart, and will,—and that in each unrenewed person some lower affection, and not the love of God, is supreme. Such analysis of a complex concept into its elements, the explicit setting forth what it is and what it is not, followed by the synthesis of the parts into a thought-unit, is the plan pursued by the best teachers in teaching difficult subjects. By analysis we resolve complex concepts into their elements, which may be simple percepts or their relations. Things are separated in thought which go together in time, space, motion, force, or substance. Every essential attribute or constituent can then be viewed by itself until the mind has gone around it with the bounding line of thought, grasped its nature and essence, and explored it in
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