Thinking and learning to think

by Nathan Christ Schaeffer

Arabic notation.

The school furnishes numerous examples to illustrate this point. When the teachers of a well-known city began the use of objects to impart the ideas of number and of the fundamental rules in arithmetic, the interest of the pupils and their facility in calculation grew wonderfully. The teaching was in accordance with the laws of mental growth. For fear the pupils would manipulate the Arabic figures without corresponding ideas, collections and equal parts of objects were drawn upon the slate to illustrate addition and subtraction of integers and fractions. The plan was followed for years and carried upward through the grades. Finally the pupils were examined for admission into the high school. A problem involving the four fundamental rules in combinations which could not be illustrated by pictures of objects, or the objects themselves, was set for solution. Out of fifty-nine applicants, only ten succeeded in giving the correct answer. The same kind of problem was given three times by three different persons, and with practically the same outcome. The teachers realized that they had kept up for too long a time the thinking in things, instead of drilling the pupils upon the process of thinking in the symbols of the Arabic notation. It is, of course, possible to think number without using the Arabic digits. The Romans did so by means of their counting-boards, and the Chinese do so by devices of their own. The characters which were brought into Western Europe through Arabic influences are derived, according to Max Mueller, from the first letters of the Sanskrit words for the first ten numerals. Their use facilitated calculation to such an extent that arithmetic gradually ceased to be the prerogative of slaves and ecclesiastics; its operations began to be understood by freemen and by the nobility. If children are denied the use of objects in their early lessons in number, they resort to counting on their fingers. If they are not led from this thinking on their fingers to thinking in figures, they will never become expert in arithmetic. Sometimes the fingers no longer move, but the mind conceives pictures of the hand, and the mind’s eye runs along the fingers of hands not visible to the corporeal eye. It is equally bad if the pupils never think number except by mental pictures of blocks, sticks, balls, and the like. When the pupil sees 7 × 9, he should not conceive seven heaps of nine shoe-pegs each, and then a rearrangement into six groups of ten shoe-pegs, and three stray ones alongside of these groups; but instantaneously the symbols 7 × 9 should suggest, with unerring accuracy, the result,—63.

Fractions.

In the schools of another district the principal proposed concrete work in fractions. The teachers and pupils began to divide things into halves, and thirds, and fourths, and sixths. They added and subtracted by subdividing these into fractions that denoted equal parts of a unit. Whilst the charm of novelty still clung to the process, a stranger who visited the schools asked one of the teachers how the pupils and parents liked the change. “Everybody is delighted,” was the exclamation. A year later the same teacher was asked by the visitor, “How are you succeeding with your concrete work in fractions?” With a dejected air she replied, “We are disappointed with the results.” “Just as I expected,” exclaimed the visitor; “for you were making the children think on the level of barbarism, instead of teaching them to use the tools and labor-saving machinery of modern civilization.”

Reckoning interest.

Still another incident, taken from actual life, will serve to throw light upon the subject under discussion. In the booming days of the iron industry a laborer had saved and put out at interest twelve hundred dollars. The rate was six per cent., and no interest had been paid for one year and four months. Unable to reckon interest with figures, the toiler asked the principal of the schools to tell him the amount of interest due. Next day he greeted the principal by asking, “Did you not make a mistake in your calculation?” The reply was, “In my hurry to avoid being late at school I may have made a mistake.” He found that the man was right, and curiosity led him to ask how the error had been detected. “I reckoned it,” said the man. This aroused still greater curiosity; for the principal knew that, beyond the ability to count, the man had no knowledge of arithmetic. By agreement they met on Saturday afternoon, so that the man might show his method of reckoning interest. At the appointed hour the man laid six pennies on the floor to denote a year’s interest on one dollar, and then laid two pennies alongside of these as the additional interest on a dollar for four months. The supply of pennies being exhausted, he made strokes with chalk, and proceeded to do this twelve hundred times, and then to count them for the purpose of ascertaining the interest. It was thinking in things with a vengeance. And yet the making of strokes with chalk was a step in symbolic representation, and shows the innate tendency of the human mind to use symbols in thinking.

Words.

Dialects.

Even the words used in counting are symbols. In fact, every word that signifies anything is a symbol used by the mind to indicate an idea more or less complex, as well as the thing or things or relation of things in the external world which corresponds to the idea. In advanced thinking the words denote ideas more and more complex as the problems grow in difficulty or involve more of the abstract and general concepts under which the mind classifies the objects of which it takes cognizance. This is more largely true of the words in a developed language than it is of a dialect with little or no literature. A reference to the writer’s early home will be pardoned in this connection. His father, a plain farmer in Eastern Pennsylvania, sent four sons through college and gave each of them a professional or university education. When they gather under the parental roof they use the dialect of their early days in discussing life on the farm and in rehearsing the funny experiences of their boyhood; but when they discuss a question in science or mathematics, in law, medicine, or theology, they drop the dialect of their boyhood and use the instruments of thought furnished by languages having a literature. Some one has facetiously said of one town in the Lehigh Valley that the people pray in seven languages and swear in eight. It is a witty statement of an actual fact. The Welshman can pray as well as swear in his native tongue. The Pennsylvania German can vent his feelings fully in his own dialect when he grows profane. As soon as he says his prayers he reverts to the language of the pulpit and of Luther’s Bible because he there finds the words which express the deepest wants and emotions of the human soul.

Melanchthon.