It is helpful at this point clearly to distinguish between essential and accidental attributes. The orange may have been kept in the open air when the temperature is low. To the hand it feels cold, and this quality enters into the idea of the first orange which the child has. As other oranges which have been in a warmer atmosphere are brought to the child, the attribute cold is seen to be accidental,—that is, it is not a necessary quality of oranges in general. On the other hand, the qualities which are found in every orange—many of them hard to describe in words—become fixed in the mind as essential attributes of the orange. In course of time many objects of the same kind are presented to the senses, cognized by comparison so as to retain the essential attributes and to omit the accidentals. By this process the general notion or concept is formed.

It is self-evident that the mind’s comparisons and conclusions are unreliable in so far as the gate-ways of knowledge are defective. Few persons have perfect ears; many can never become expert tuners of pianos or reliable critics of musical performances. The man who is color-blind is not accepted in the railway service or as an officer in the navy. The man who is totally blind is never selected as a guide in daylight. On the other hand, the blind girl spoken of by Bulwer could find her way better in the darkness of the last days of Pompeii than other people, because she was accustomed to rely upon the data furnished by the other senses in making her way through the city, and had improved these as gate-ways of knowledge beyond the needs of those gifted with sight.

In building concepts of objects in nature it would be a great mistake to begin with the word instead of the thing. Just as little as a blind man can conceive the qualities color, light, darkness, through mere words, so little can children conceive classes of objects which have never addressed the senses. Hence great stress has been laid by educational reformers upon the cultivation of habits of observation, upon the supreme necessity of teaching by the use of objects, or so-called object-lessons. First, things, then words, or signs for things, was at one time a favorite maxim in treatises on teaching. Consistent application of the maxim would have banished the dictionary from the school-room, or at least its use as a means for ascertaining the meaning of words. In consulting the dictionary for the meaning of a word, we pass not from the thing to its sign, but in the opposite direction,—that is, from the sign to the thing signified, from the symbol to the idea for which the symbol stands. The main essential in good instruction is that the words be made significant. In primary instruction this is best accomplished by passing from the idea to the word; but in advanced instruction it is of less importance whether we pass from the word to the idea or from the idea to the word. The meaning of very many words is acquired from the connection in which they are used. For the meaning of the larger number of words in our vocabulary we never consult a dictionary. The finer shades of meaning we get not from definitions, but from quotations taken from standard authors. This fact should never tempt the teacher to trust to words, definitions, and descriptions in the formation of basal concepts. He should seek to give unto himself a clear and full account of the things or ideas which cannot spring from mere words, however skilfully arranged in sentences. The music-teacher who complained of the public schools because a seven-year-old child did not grasp his meaning when he spoke of half-notes, quarter-notes, eighth-notes, sixteenth-notes, should have known that many children of that age have never been taught fractions, and that the idea of a fraction is obtained not from sounds (who distinguishes between half a noise and a whole noise?), but from objects which address the eye. Instead of complaining about the school which the pupil attended, a teacher acquainted with the mysteries of his art would have started with the comparison of things visible; and after having developed the idea of halves, quarters, eighths, sixteenths, by the division of visible objects into equal parts, he would have applied the idea to musical sounds.

In seeking to build in the mind of the learner the concepts which lie at the basis of a new branch of study, it is a legitimate question to ask by which of the gate-ways of knowledge the materials or elements for the new idea can best be made to enter the mind. At the basis of arithmetic lies the idea of number,—an idea that is evoked by the question of how many applied to a collection of two or more units. Taste and smell must be ruled out from the list of senses which can be utilized to advantage. Three taps on the desk are as easily recognized as three marks or strokes on the black-board. The sense of touch is helpful in passing from concrete to abstract numbers. To think a number when the corresponding collection of objects is not visible, but is suggested by tactile impressions, helps to emancipate the thinking process from the domination of the eye; in other words, it helps to sunder the thinking of number from a specific sense, and thus aids in the evolution of the idea of number apart from concrete objects.

As already indicated, there are some basal concepts, like that of a fraction, in the development of which only one sense can be utilized to advantage. Whilst imparting the idea of a whole number, the appeal may be to the eye, the ear, and the sense of touch; the instruction designed to impart the idea of fractions to the normal child is limited to visible objects. In the instruction of the blind the other senses are addressed from necessity. The extent to which touch can supply the function of sight is full of hints to teachers in charge of pupils possessing all the gate-ways of knowledge.

Moreover, not all units are equally adapted for imparting the first ideas of a fraction. Half of a stick is still a stick to the child, just as half of a stone is still called a stone in common parlance. The half should be radically different from the unit; hence an object resembling a sphere or a circle is best adapted for the first lessons in fractions. In teaching decimals the square or rectangle is better than the circle. It is difficult to divide a circumference into ten equal parts. On the contrary, the square is easily divided into tenths by vertical lines, and then into hundredths by horizontal lines, thus furnishing also a convenient device for the first lessons in percentage.

It is one of the aims of the training-class and the normal school to point out the best methods of developing the different basal concepts which lie at the foundation of the branches to be taught. Many of these are complex, and require great skill on the part of the teacher. The difficulty is well stated in John Fiske’s discussion of Symbolic Conceptions. He says, “Of any simple object which can be grasped in a single act of perception, such as a knife or a book, an egg or an orange, a circle or a triangle, you can frame a conception which almost, or quite exactly, represents the object. The picture, or visual image, in your mind when the orange is present to the senses is almost exactly reproduced when it is absent. The distinction between the two lies chiefly in the relative faintness of the latter. But as the objects of thought increase in size and in complexity of detail, the case soon comes to be very different. You cannot frame a truly representative conception of the town in which you live, however familiar you may be with its streets and houses, its parks and trees, and the looks and demeanor of the townsmen; it is impossible to embrace so many details in a single mental picture. The mind must range to and fro among the phenomena, in order to represent the town in a series of conceptions. But practically, what you have in mind when you speak of the town is a fragmentary conception in which some portion of the object is represented, while you are well aware that with sufficient pains a series of mental pictures could be formed which would approximately correspond to the object. To some extent the conception is representative, but to a great degree it is symbolic. With a further increase in the size and complexity of the objects of thought, our conceptions gradually lose their representative character, and at length become purely symbolic. No one can form a mental picture that answers even approximately to the earth. Even a homogeneous ball eight thousand miles in diameter is too vast an object to be conceived otherwise than symbolically, and much more is this true of the ball upon which we live, with all its endless multiformity of detail. We imagine a globe, and clothe it with a few terrestrial attributes, and in our minds this fragmentary notion does duty as a symbol of the earth.

“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.

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]

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.