ARITHMETIC.

The arithmeticon-How applied-Numeration-Addition-Subtraction- Multiplication-Division-Fraction-Arithmetical tables-Arithmetical Songs-Observations.

"In arithmetic, as in every other branch of education, the princ.i.p.al object should be to preserve the understanding from implicit belief, to invigorate its powers, and to induce the laudable ambition of progressive improvement."-Edgeworth

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.

[Ill.u.s.tration]

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 b.a.l.l.s, 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 b.a.l.l.s painted black and white alternately, to a.s.sist 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 b.a.l.l.s 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 b.a.l.l.s 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 bra.s.s 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.

[Ill.u.s.tration]

THE ARITHMETICON.

It will be seen that on the twelve parallel wires there are 144 b.a.l.l.s, 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.

The tablet beneath the b.a.l.l.s has six s.p.a.ces for the insertion of bra.s.s letters and figures, a box of which accompanies the frame. Suppose then the only figure inserted is the 7 in the second s.p.a.ce from the top: now were 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 form stands instead of seven ones, and place seven b.a.l.l.s together 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 b.a.l.l.s on a wire, they will say instantly, "O it is three ones, or three;" and that they may have the proper name they may be told that they have before them figure 7 and figure 3. Put a 9 to these figures, and their attention will be arrested: say, Do you think you can tell me what this is? and, while you are speaking, move the b.a.l.l.s gently out, and, as soon as they see them, they will immediately cry out "Nine;" and in this way they may acquire a knowledge of all the figures separately. Then you may proceed thus: Units 7, tens 3; place three b.a.l.l.s 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 b.a.l.l.s. Then go on, units 7, tens, 3, hundreds 9, place nine b.a.l.l.s 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 a.s.sist the understanding and exercise the judgment, slide a figure in the frame, and say, Figure 8. Q. What is this? A. No. 8. Q. If No. 1 be put on the left side of the 8, what will it be? A. 81. Q. If the 1 be put on the right side, then what will it be? A. 18. Q. If the figure 4 be put before the 1, then what will the number be? A. 418. Q. Shift the figure 4, and put it on the left side of the 8, then ask the children to tell the number, the answer is 184. The teacher can keep adding and shifting 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 him. Suppose figures 5476953821 are in the frame; then let the children begin at the left hand, saying, units, tens, hundreds, thousands, tens of thousands, hundreds of thousands, millions, tens of millions, hundreds of millions, thousands of millions. After which, begin at the right side, and they will say, Five thousand four hundred and seventy-six million, nine hundred and fifty-three thousand, eight hundred and twenty-one. If the children are practised in this way, they will soon learn numeration.

The frame was employed for this purpose long before its application to others was perceived; but at length I found we might 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 master may exercise them backwards, saying, 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 b.a.l.l.s 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 bra.s.s figure 9 on the tablet beneath, and they will see how the amount is marked: then take eight b.a.l.l.s and three, when they will see that eight and three are eleven. Explain to them that they cannot put underneath two figure ones which mean 11, but they must put 1 under the 8, and carry 1 to the 4, when you must place one ball under the four, and, asking them what that makes, they will say, Five. Proceed by saying, How much are five and nine? put out the proper number of b.a.l.l.s, and they will say, Five and nine are fourteen. Put a four underneath, and tell them, as there is no figure to put the 1 under, it must be placed next to it: hence they see that 937 added to 482, make a total of 1419.

Subtraction may be taught in as many ways by this instrument. Thus: take 1 from 1, nothing remains; moving 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 remain.

Then the b.a.l.l.s may be worked backwards, beginning at the wire containing 12 b.a.l.l.s, 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 from 4, 2 remain; 2 from 3, 1 remains.

The bra.s.s figure should be used for the remainder in each case. Say, then, can you take 8 from 3 as you point to the figures, and they will say "Yes;" but skew them 3 b.a.l.l.s on a wire and ask them to deduct 8 from them, when they will perceive their error. Explain that in such a case they must borrow one; then say take 8 from 13, placing 12 b.a.l.l.s on the top wire, borrow one from the second, and take away eight and they will see the remainder is five; and so on through the sum, and others of the same kind.

In Multiplication, the lessons are performed as follows. The teacher moves the first ball, and immediately after the two b.a.l.l.s on the second wire, placing them underneath the first, saying at the same time, twice one are two, which the children will readily perceive. We next remove the two b.a.l.l.s on the second wire for a multiplier, and then remove two b.a.l.l.s from the third wire, placing them exactly under the first two, which 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 from the fourth wire underneath them saying, twice three are six. Remove the four on the fourth wire, and four on the fifth, place them as before and say, twice four are eight. Remove five from the fifth wire, and five from the sixth wire underneath them, saying twice five are ten. Remove six from the sixth wire, and six from the seventh wire underneath them and say, twice six are twelve. Remove seven from the seventh wire, and seven from the eighth wire underneath them, saying, twice seven are fourteen. Remove eight from the eighth wire, and eight from the ninth, saying, twice eight 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 them, saying, 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 remaining ball on the twelfth wire, saying, 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 from the 144 b.a.l.l.s gathered together at one end, one from each row, and place the 12 at the other end, thus making 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 b.a.l.l.s 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 same principle is acted on throughout.

The only remaining branch of numerical knowledge, which consists in an ability to comprehend the powers of numbers, without either visible objects or signs-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 two; two and one are three; three and one are four, &c. up to twelve.

Two and two are four; four and two are six; six and two are eight, &c. to twenty-four.

Three and three are six; six and three are nine; nine and three are twelve, &c. to thirty-six.

Subtraction.

One from twelve leaves eleven; one from eleven leaves ten, &c.

Two from twenty-four leave twenty-two; two from twenty-two leave twenty, &c.

Multiplication.

Twice one are two; twice two are four, &c. &c. Three times three are nine, three times four are twelve, &c. &c.

Twelve times two are twenty-four; eleven times two are twenty-two, &c. &c.

Twelve times three are thirty-six; eleven times three are thirty-three, &c. &c. until the whole of the multiplication table is gone through.

Division.

There are twelve twos in twenty-four.-There are eleven twos in twenty-two, &c. &c.

There are twelve threes in thirty-six, &c.

There are twelve fours in forty-eight, &c. &c.

Fractions.

Two are the half (1/2) of four. " " " third (1/3) of six. " " " fourth (1/2) of eight. " " " fifth (1/5) of ten. " " " sixth (1/6) of twelve. " " " seventh (1/7) of fourteen. " " " twelfth (1/12) of twenty-four; two are the eleventh (1/11) of twenty-two, &c. &c.

Three are the half (1/2) of six. " " " third (1/3) of nine. " " " fourth (1/4) of twelve.

Three are the twelfth (1/12) of thirty-six; three are the eleventh (1/11) of thirty-three, &c. &c.

Four are the half (1/2) of eight, &c.

In twenty-three are four times five, and three-fifths (3/5) of five; in thirty-five are four times eight, and three-eighths (3/8) of eight.

In twenty-two are seven times three, and one-third (1/3) of three.

In thirty-four are four times eight, and one-fourth (1/4) of eight.

The tables subjoined are repeated by the same method, each section being a distinct lesson. To give an idea to the reader, the boy in the rostrum says ten shillings the half (1/2) of a pound; six shillings and eightpence one-third (1/3) of a pound, &c.