IV
BASAL CONCEPTS AS THOUGHT-MATERIAL

The head may be likened unto a walled city, with comparatively few building materials on the inside, and with a limited number of gate-ways through which all other materials for building purposes must pass. The walls are not made of brick or stone, but of bone; the gate-ways are the different senses through which knowledge enters the mind. The building materials on the inside are intuitive ideas which take shape in conjunction with the entrance of materials from without. The structures which are built up out of the ideas within and the sense-impressions from without are individual and general concepts. Take an orange. Its shape, color, parts, are known through the eye. Its flavor, as sweet or sour, is ascertained through taste; its odor through smell; its temperature, shape, and some other qualities through touch. These various sense-impressions, giving the mind a knowledge of essential and accidental qualities and attributes, are combined in the idea of a particular orange. If the object were a bell, its sound, parts, uses, and qualities would make impressions through different gate-ways of knowledge; the builder inside would combine them into the more or less complete idea of the object presented to the senses. From each sense-impression the mind may get a percept; the synthesis of these percepts produces the individual concept or notion.

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.

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:

Professor Young uses a very suggestive device in his astronomy for showing the comparative sizes and distances of heavenly bodies:

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 its different aspects and relations. In this way the most abstruse subjects are shorn of their difficulties, the most complex problems are solved and elucidated.

The bearing of all this upon the art of teaching is easily shown. A teacher of geometry, whose mind was quite logical, failed, through lack of power, to make things plain. If the class did not grasp the demonstration of a theorem, he invariably started at the beginning, tried to throw light upon every link in the chain of proof, and by the time he reached the point of difficulty the members of the class were thinking of something else. A younger colleague pursued a different plan. Starting some pupil upon the demonstration, he detected the difficulty, and by a few words of explanation, or by a well-framed question, he focussed attention upon the simple elements, into which he resolved the difficulty, and frequently surprised the class by showing the simplicity of what had puzzled their minds. Under the clarifying light of analysis half the difficulties and half the sophistries of human thinking vanish like dew and mist before the morning sun.

For the purpose of making an impression upon the moral nature word-painting is sometimes very helpful. All the text-books on physiology and hygiene intended for use in the public schools seek to teach the evils of strong drink by showing the effect of alcoholic stimulants upon different parts of the human system. Yet the most exhaustive lessons on how whiskey is made, and what are its exhilarating and its pernicious effects, cannot equal the effects of the word painting of Robert Ingersoll and the paraphrase by Dr. Buckley. In making a gift to a friend the former penned the following eulogy on whiskey:

This was Dr. Buckley’s statement of the other side:

There comes a stage of development of the learner at which the word itself becomes the object of thought. Words are then classified as parts of speech, and their function in sentences is studied. Their properties and endings must be learned and compared. There is abundant room for thought in the eleven hundred variations of the Greek verb. The variations of words by declension and conjugation can be made the material for thought, and as these are always at hand in the text-book, no excursions to the field being needed to secure specimens, and no preparation of difficult experiments being required on the part of the teacher, the ancient languages have held their own in the schools with most wonderful tenacity. The study of language has not merely the advantage of supplying material for thought in the words, grammatical forms, and sentences which are always at hand in the text, but through the classics it brings the learner into intellectual contact with the best thoughts of the best men in ancient and modern times. To translate an author like Virgil or Demosthenes is to think the thoughts of a master mind, to weigh words as in a most nicely adjusted balance, and finally to arrange them in sentences that shall adequately convey the meaning of the original text.

Science is, of course, a product of the human mind, quite as much as the so-called humanities, and answers the same purpose when studied as literature; but then it ceases to have the value of training the intellect in the rigid methods of original research and scientific investigation. Whilst it is the function of the laboratory to initiate the student into the mysteries of the methods by which new discoveries are made and verified, and thus to enable him to avail himself of the labors of others through their publications, it does not bring the student into living contact with human hopes, emotions, and aspirations as do the poems of Goethe, Schiller, and Shakespeare.

History deals with what man has achieved. The materials for thought which it furnishes are mostly in the shape of the testimony of eye-witnesses and other original sources of information. The incidents, the achievements, the struggles, the victories and the defeats, the thoughts, feelings, and experiences of historic personages, are an inexhaustible supply of material from which authors, editors, and orators draw illustrations, figures of speech, and other matter for their thinking. Here is a field which must not be neglected by those who would influence their fellows or figure as leaders of men.

Some minds are slow at gathering materials; yet they think vigorously. They look at facts and ideas from every possible point of view, explore their nature and relations, their content and extent, and point out their bearing upon other things by the conclusions they reach. Sometimes they go astray because they do not have sufficient data to warrant a conclusion. Their condition resembles that of the King of Siam, who did not believe that water could become solid because he had been in the nine points of his kingdom and had not seen ice.

Other men are intellectual gluttons. They keep pouring into themselves knowledge from every quarter, carry it in their minds as the overloaded stomach carries food, and end in mental dyspepsia. Better the man with few ideas, who can apply these in practical life, than the man of erudition who cannot apply his knowledge.

Too little food produces inanition and starvation; too much food brings on dyspepsia and a host of other ills and distempers. The haphazard selection of studies by inexperienced youth from the large list of electives offered by a great university is apt to result either in mental overfeeding or in intellectual starvation. The mind can be rightly formed only when it is rightly informed. To expect satisfactory thought-products when the mind lacks proper materials to act upon would be as irrational as to expect good grist from a flour-mill whose supply of grain is deficient in quality and quantity. In the process of making flour very much depends upon the instruments employed. The rude implements of antiquity, the buhr-stones of our fathers, and the improved machinery of the roller process make a difference in the product, even though the same quality of grain is used. In the elaboration of the thought-material the well-educated man uses instruments which may be likened to our modern inventions for saving labor in the domain of the mechanic arts. These instruments of thought will next claim our attention.