The Parents' Review
A Monthly Magazine of Home-Training and Culture
"Education is an atmosphere, a discipline, a life."
Children and Arithmetic.
By Rev. R. H. Quick, M.A.
For those who study the development of thought in children, the formation of their ideas of number has a very special interest. In most subjects we cannot be quite sure that the idea in the child's mind is the same as in ours. May-be our complex idea has elements in it which it cannot have in the mind of the child. But with regard to the first notions of number, this common source of uncertainty and error is absent. The ideas are simple, and we have ample means of testing their accuracy.
But my object now is not simply to stimulate the spirit of inquiry, but to point out the great importance of getting children's thoughts about numbers clear and accurate. Many mothers will have been led by Mrs. Hart-Davis's paper in the April number of this Review to form good resolutions on the subject; and if they carry out her suggestions, both their children and their children's future teachers will have cause to be grateful.
We often hear the question asked, What is the use of teaching children things they cannot help learning without teaching, e.g., that the cat has four legs and one tail, or that grass is green and snow is white? The right answer is, no doubt, The less we teach the better. We want our children to use their eyes, not ours, and our object must be not to put in but to bring out. In the matter of numbers it is the teaching or telling method that produces such hopeless confusion in the minds of children; and Mrs. Hart-Davis has shown not how to teach children what they would otherwise learn for themselves, but how children should learn for themselves what nobody else can teach them.
It is found that savages of the lowest type have no word to indicate a definite number four or five. All greater numbers are included in the word for "many." We see, then, that they can think of four or five things, but then get lost. It would indeed be impossible for any one to go through life in a civilised nation without a much wider range of conception than this; but in the case of most children, and of many grown people, the range even with us is more limited than it seems. But among us people without conceptions do not break off with "Many! Many!" like the savages.
Where conception fails, a word
Indeed we all take refuge in symbols when we come to high numbers, and advanced arithmetic is the art of dealing with symbols no less than algebra. But the science on which this art should rest depends on the clearness and accuracy of the conceptions formed about numbers less than 100, and it is, therefore, extremely important that these conceptions should be rightly formed. They must come through the senses, through handling and dealing with a small number of things. The Germans give a whole year to a course of this kind with things up to ten, and experience justifies this slow advance.
Mrs. Hart-Davis's suggestions about dominoes, and about forming patterns with round counters, seem to me excellent. With the counters children may play at a game of seeing "how many?" when a number of counters are uncovered before them and then covered again. Children with practice in this soon get to beat grown people without practice.
It is a great matter to keep to tens. This is the base of our numeration--a fact somewhat obscured by our notation, as we have but nine significant digits. With children I should for a long time give no notation but the Roman, which they will want for the clock, and which is less conventional than the more perfect Arabic system.
The clock reminds me of another difficulty which in the early years of arithmetic may with advantage be avoided. Our numeration, as I said, is by tens. The primitive man counted by his fingers, and when in counting a number of things he had one for each finger, he made a heap of them, and started afresh. The things in his second heap he called "1 and 10, 2 and 10," &c., &c. But when he came to the end of this heap he got muddled in his nomenclature. Sometimes he said, "one and twain-ten," or one and twenty, "two and twenty," &c., but sometimes he named the piles of ten first, and said, "twenty-one," "twenty-two," &c. As he got further on, he adopted more and more the plan of naming the tens first, until the method he set out with was lost entirely. So we have to name the units first, up to twenty; then we have a choice of tens or units for a little while, but this liberty leaves us again about fifty. If I ask the age of a man under thirty, I am told, say, that he is "about three and twenty." "About twenty-three" is allowable, but only just allowable. If the man were ten years older, I should be told, either "about three-and-thirty" or "thirty-three," and here one form would be as natural as the other. If we raise the age another ten years, it would be spoken of as "about forty-three," rarely "three-and-forty": another ten, and "fifty-three" is almost the only form possible; and still more of the higher decades. If we heard any one talking of "three-and-eighty" we should say he could not be an Englishman. It seems to me a great pity that the symmetry of our numeration should be obscured by freaks of language. After ten I should go on "ten-one, ten-two," &c. With young children then I should name the tens first throughout. Instead of "twelve, thirteen, fourteen," I should say, "ten-one, ten-two, ten-three, ten-four;" for "twenty" I should use "two-tens;" for thirty, "three tens."
Rev. R. H. Quick, M.A.
Typed by Niki McAlister, Oct 2015
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