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The advantage of a knowledge of arithmetic has never been disputed. Its universal application to the business of life renders it an important acquisition to all ranks and conditions of men. The practicability of imparting the rudiments of arithmetic to very young children has been satisfactorily shewn by the Infant-school System; and it has been found, likewise, that it is the readiest and surest way of developing the thinking faculties of the infant mind. Since the most complicated and difficult questions of arithmetic, as well as the most simple, are all solvable by the same rules, and on the same principles, it is of the utmost importance to give children a clear insight into the primary principles of number. For this purpose we take care to shew them, by visible objects, that all numbers are combinations of unity; and that all changes of number must arise either from adding to or taking from a certain stated number. After this, or rather, perhaps I should say, in conjunction with this instruction, we exhibit to the children the signs of number, and make them acquainted with their various combinations; and lastly, we bring them to the abstract consideration of number; or what may be termed mental arithmetic. If you reverse this, which has generally been the system of instruction pursued—if you set a child to learn its multiplication, pence, and other tables, before you have shewn it by realities, the combinations of unity which these tables express in words—you are rendering the whole abstruse, difficult, and uninteresting; and, in short, are giving it knowledge which it is unable to apply.
As far as regards the general principles of numerical tuition, it may be sufficient to state, that we should begin with unity, and proceed very gradually, by slow and sure steps, through the simplest forms of combinations to the more comprehensive. Trace and retrace your first steps—the children can never be too thoroughly familiar with the first principles or facts of number.
We have various ways of teaching arithmetic, in use in the schools; I shall speak of them all, beginning with a description of the arithmeticon, which is of great utility.
[Illustration]
I have thought it necessary in this edition to give the original woodcut of the arithmeticon, which it will be seen contains twelve wires, with one ball on the first wire, two on the second, and so progressing up to twelve. The improvement is, that each wire should contain twelve balls, so that the whole of the multiplication table may be done by it, up to 12 times 12 are 144. The next step was having the balls painted black and white alternately, to assist the sense of seeing, it being certain that an uneducated eye cannot distinguish the combinations of colour, any more than an uneducated ear can distinguish the combinations of sounds. So far the thing succeeded with respect to the sense of seeing; but there was yet another thing to be legislated for, and that was to prevent the children's attention being drawn off from the objects to which it was to be directed, viz. the smaller number of balls as separated from the greater. This object could only be attained by inventing a board to slide in and hide the greater number from their view, and so far we succeeded in gaining their undivided attention to the balls we thought necessary to move out. Time and experience only could shew that there was another thing wanting, and that was a tablet, as represented in the second woodcut, which had a tendency to teach the children the difference between real numbers and representative characters, therefore the necessity of brass figures, as represented on the tablet; hence the children would call figure seven No. 1, it being but one object, and each figure they would only count as one, thus making 937, which are the representative characters, only three, which is the real fact, there being only three objects. It was therefore found necessary to teach the children that the figure seven would represent 7 ones, 7 tens, 7 hundreds, 7 thousands, or 7 millions, according to where it might be placed in connection with the other figures; and as this has already been described, I feel it unnecessary to enlarge upon the subject.
[Illustration]
THE ARITHMETICON.
It will be seen that on the twelve parallel wires there are 144 balls, alternately black and white. By these the elements of arithmetic may be taught as follows:—
Numeration.—Take one ball from the lowest wire, and say units, one, two from the next, and say tens, two; three from the third, and say hundreds, three; four from the fourth, and say thousands, four; five from the fifth, and say tens of thousands, five; six from the sixth, and say hundreds of thousands, six; seven from the seventh, and say millions, seven; eight from the eighth, and say tens of millions, eight; nine from the ninth, and say hundreds of millions, nine; ten from the tenth, and say thousands of millions, ten; eleven from the eleventh, and say tens of thousands of millions, eleven; twelve from the twelfth, and say hundreds of thousands of millions, twelve.