“The case becomes still more striking when we have to deal with conceptions of the universe, of cosmic forces such as light and heat, or of the stupendous secular changes which modern science calls us to contemplate. Here our conceptions cannot even pretend to represent the objects; they are as purely symbolic as the algebraic equations whereby the geometer expresses the shapes of curves. Yet so long as there are means of verification at our command we can reason as safely with these symbolic conceptions as if they were truly representative. The geometer can at any moment translate his equation into an actual curve, and thereby test the results of his reasoning; and the case is similar with the undulatory theory of light, the chemist’s conception of atomicity, and other vast stretches of thought which in recent times have revolutionized our knowledge of nature. The danger in the use of symbolic conceptions is the danger of framing illegitimate symbols that answer to nothing in heaven or earth, as has happened first and last with so many short-lived theories in science and in metaphysics.”
The word conception as used in this quotation is synonymous with concept, but elsewhere it is also used in two other senses,—namely, to signify the mind’s power to conceive objects, their relations and classes, and to name the activity by which the concept is produced. Hence the term concept is preferred in this discussion.
Concepts of distance.
Large cities.
To give a full account of the development of the basal concepts in the different branches of study would require a treatise on the methods of teaching these branches. All that can be attempted is to draw attention to some of the typical methods and devices adopted by eminent teachers in the development of the concepts which Mr. Fiske calls symbolic conceptions. Distance is one of the concepts at the basis of geography and astronomy. To say that the circumference of the earth is twenty-five thousand miles, that the distance of the moon from the earth is two hundred and forty thousand miles, and that the distance of the sun is ninety-two and one-half millions of miles may mean very little to the human mind, especially to the mind of a child. Supposing, however, that a boy finds a mile by actual measurement, and that he finds he can walk four miles an hour, he can gradually rise to the thought of walking forty miles in a day of ten hours, or two hundred and forty miles in the six working days of a week. In one hundred and four weeks, or two years, he could walk around the globe. To walk to the moon would require a thousand weeks, or about twenty years. It is by the method of gradual approach that concepts of great distance, of immense magnitudes, of the infinitely large and the infinitely small, must be developed. To this category belong large cities like New York and London, quantities denoting the size of the earth and its distance from the sun and the fixed stars, the fraction of a second in which a snap-shot is taken, or an electric flash is photographed; such quantities are apt to remain as mere figures or symbols in the mind of the learner unless the method of gradual approach is adopted. Starting with a town or a ward with which the pupil is familiar, several may be joined in idea until the concept of a city of fifty or sixty thousand population is reached. It takes about twenty of these to make a city like Philadelphia, and five cities like Philadelphia to make a city like London. A lesson on how London is fed will add much to the formation of an adequate idea of such a large city.[5]
Shape of the earth.
An adequate idea of the shape of the earth can be formed only by gradual development. The three kinds of roundness (dollar, pillar, ball) must be taught; then the various easily intelligible reasons for believing it to be round like a ball may follow in the elementary grade. As the pupil advances he may be told of the dispute between Newton and the French, the former affirming it to be round like an orange,—that is, flattened at the poles,—the latter asserting that it resembled a lemon with the polar axis longer than the equatorial diameter; and how, by measuring degrees of latitude and finding that their length increases as we approach the poles, the French mathematicians, in spite of their wishes to the contrary, proved Newton’s view to be correct. The same lesson might be taught by starting with the rotation of the earth, showing by experiment the tendency of revolving bodies to bulge out at the equator, and then drawing the inference that the degrees of latitude are shortest where the curvature is greatest, and that they are longest where the curvature is least. Either method is strictly logical; but the method which follows the order of discovery, whenever it is feasible, is calculated to arouse the greater interest in minds of average capacity. The teacher who is a master of his art will supplement the historical lesson by a lesson passing from cause to consequence, so as to fix and clarify the concept formed by passing from the ground of knowledge to the necessary inference. Finally, by drawing attention to the fact that the equatorial diameters are not all of the same length, he will build up in the pupil’s mind a concept of the real shape of the earth,—a shape unlike any mathematical figure treated of in the text-books on geometry. The attempt to give a complete idea of the shape of the earth in the first lessons on geography would have ended in confusion of thought; the wise teacher develops complex concepts gradually and not more rapidly than the learner is able to advance. This process may be called enriching the concept. The successive concepts, although only partial representations of what is to be known, are adequate for the thinking required at a given stage of development; the number of complete or exhaustive concepts in any department of knowledge is small indeed.
The order of discovery and of instruction.
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]