Part 12 (1/2)

ARITHMETIC

The arithmeticon-How applied-Numeration-Addition-Subtraction- Multiplication-Division-Fraction-Ariths-Observations

”In arithmetic, as in every other branch of education, the principal object should be to preserve the understanding froorate its powers, and to induce the laudable aeworth

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 ofthe rudi 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 saive children a clear insight into the primary principles of number For this purpose we take care to shew them, by visible objects, that all nues of nu 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 co them to the abstract consideration of number; or what may be terenerally 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 in 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 nu arithinning with a description of the arithreat 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 ti 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 uish the co 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 sreater This object could only be attained by inventing a board to slide in and hide the greater nu their undivided attention to the balls we thought necessary to move out Ti 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, thus937, 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 ht 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 eleht as follows:-

Numeration-Take one ball from the loire, 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 frohth, and say tens of ht; 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

The tablet beneath the balls has six spaces for the insertion of brass letters and figures, a box of which accoure inserted is the 7 in the second space from the top: noere the children asked what it was, they would all say, without instruction, ”It is one” If, however, you tell them that an object of such a forether on a wire, they will at once see the use and power of the number Place a 3 next the seven, merely ask what it is, and they will reply, ”We don't know;” but if you put out three balls on a wire, they will say instantly, ”O it is three ones, or three;” and that they may have the proper naure 7 and figure 3 Put a 9 to these figures, and their attention will be arrested: say, Do you think you can tell , ently out, and, as soon as they see them, they will immediately cry out ”Nine;” and in this way they ures separately Then you may proceed thus: Units 7, tens 3; place three balls on the top wire and seven on the second, and say, Thirty-seven, as you point to the figures, and thirty-seven as you point to the balls Then go on, units 7, tens, 3, hundreds 9, place nine balls on the top wire, three on the second, and seven on the third, and say, pointing to each, Nine hundred and thirty-seven And so onwards

To assist the understanding and exercise the judgure 8 Q What is this? A No 8 Q If No 1 be put on the left side of the 8, ill it be? A 81 Q If the 1 be put on the right side, then ill it be? A 18 Q If the figure 4 be put before the 1, then ill the nuure 4, and put it on the left side of the 8, then ask the children to tell the nu and shi+fting as he pleases, according to the capacity of his pupils, taking care to explain as he goes on, and to satisfy himself that his little flock perfectly understand hiures 5476953821 are in the fra, units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of ht side, and they will say, Five thousand four hundred and seventy-six ht hundred and twenty-one If the children are practised in this way, they will soon learn nu before its application to others was perceived; but at length I found we ht proceed to

Addition-We proceed as follows:-1 and 2 are 3, and 3 are 6, and 4 are 10, and 5 are 15, and 6 are 21, and 7 are 28, and 8 are 36, and 9 are 45, and 10 are 55, and 11 are 66, and 12 are 78

Then the , 12 and 11 are 23, and 10 are 33, and 9 are 42, and 8 are 50, and 7 are 57, and 6 are 63, and 5 are 68, and 4 are 72, and 3 are 75, and 2 are 77, and 1 is 78, and so on in great variety

Again: place seven balls on one wire, and two on the next, and ask them how many 7 and 2 are; to this they will soon answer, Nine: then put the brass figure 9 on the tablet beneath, and they will see how the aht balls and three, when they will see that eight and three are eleven Explain to theure ones which mean 11, but they must put 1 under the 8, and carry 1 to the 4, when youthe, How much are five and nine? put out the proper number of balls, and they will say, Five and nine are fourteen Put a four underneath, and tell theure to put the 1 under, it must be placed next to it: hence they see that 937 added to 482, ht in asre the first ball at the same time to the other end of the frame Then remove one from the second wire, and say, take one from 2, the children will instantly perceive that only 1 remains; then 1 from 3, and 2 remain; 1 from 4, 3 remain; 1 from 5, 4 remain; 1 from 6, 5 remain; 1 from 7, 6 remain; 1 from 8, 7 remain; 1 from 9, 8 remain; 1 from 10, 9 remain; 1 from 11, 10 remain; 1 from 12, 11 reinning at the wire containing 12 balls, saying, take 2 from 12, 10 remain; 2 from 11, 9 remain; 2 from 10, 8 remain; 2 from 9, 7 remain; 2 from 8, 6 remain; 2 from 7, 5 remain; 2 from 6, 4 remain; 2 from 5, 3 remain; 2 froure should be used for the remainder in each case Say, then, can you take 8 froures, and they will say ”Yes;” but skew them 3 balls on a wire and ask them to deduct 8 from them, when they will perceive their error Explain that in such a case they12 balls on the top wire, borrow one froht and they will see the reh the sum, and others of the same kind

In Multiplication, the lessons are performed as follows The teacher moves the first ball, and i the at the same time, twice one are thich the children will readily perceive We next remove the two balls on the second wire for a multiplier, and then re them exactly under the first thich forms a square, and then say twice two are four, which every child will discern for himself, as he plainly perceives there are no more We then move three on the third wire, and place three fro, twice three are six Remove the four on the fourth wire, and four on the fifth, place theht Remove five from the fifth wire, and five fro twice five are ten Remove six from the sixth wire, and six from the seventh wire underneath them and say, twice six are twelve Rehth wire underneath thehth wire, and eight froht are sixteen Remove nine on the ninth wire, and nine on the tenth wire, saying twice nine are eighteen Remove ten on the tenth wire, and ten on the eleventh underneath the, twice ten are twenty Remove eleven on the eleventh wire, and eleven on the twelfth, saying, twice eleven are twenty-two Remove one from the tenth wire to add to the eleven on the eleventh wire, afterwards the re, twice twelve are twenty-four

Next proceed backwards, saying, 12 times 2 are 24, 11 times 2 are 22, 10 times 2 are 20, &c

For Division, suppose you take froether at one end, one fro a perpendicular row of ones: then make four perpendicular rows of three each and the children will see there are 4 3's in 12 Divide the 12 into six parcels, and they will see there are 6 2's in 12 Leave only two out, and they will see, at your direction, that 2 is the sixth part of 12 Take away one of these and they will see one is the twelfth part of 12, and that 12 1's are twelve

To explain the state of the frame as it appears in the cut, we must first suppose that the twenty-four balls which appear in four lots, are gathered together at the figured side: when the children will see there are three perpendicular 8's, and as easily that there are 8 horizontal 3's If then the teacher wishes them to tell how many 6's there are in twenty-four, he moves them out as they appear in the cut, and they see there are four; and the sa branch of nue, which consists in an ability to comprehend the powers of nuns-is imparted as follows:

Addition

One of the children is placed before the gallery, and repeats aloud, in a kind of chaunt, the whole of the school repeating after him; One and one are to and one are three; three and one are four, &c up to twelve