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.”
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.