The arithmeticon—How applied—Numeration—Addition—Subtraction— Multiplication—Division—Fraction—Arithmetical tables—Arithmetical Songs—Observations.
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"In arithmetic, as in every other branch of education, the principal object should be to preserve the understanding from implicit belief, to invigorate its powers, and to induce the laudable ambition of progressive improvement."—Edgeworth
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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.
The tablet beneath the balls has six spaces for the insertion of brass letters and figures, a box of which accompanies the frame. Suppose then the only figure inserted is the 7 in the second space 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 balls 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 balls 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 balls 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 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 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 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 amount is marked: then take eight balls 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 balls, 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 balls may be worked backwards, beginning 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 from 4, 2 remain; 2 from 3, 1 remains.
The brass 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 balls 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 balls 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 balls 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 balls on the second wire for a multiplier, and then remove two balls 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 balls 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 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 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.
Sixpence the half (1/2) of a shilling, &c. Always remembering, that whatever the boy says in the rostrum, the other children must repeat after him, but not till the monitor has ended his sentence; and before the monitor delivers the second sentence, he waits till the children have concluded the first, they waiting for him, and he for them; this prevents confusion, and is the means of enabling persons to understand perfectly what is going on in the school.
In a book lately published, which is a compilation by two London masters, it is stated, in the preface, that they were at a loss for proper lessons: had they used those in existence I cannot help thinking they were enough for the capacity of children under six years of age.
254 ARITHMETICAL TABLES.
Numeration, Addition, Subtraction, Multiplication, Division, and Pence Tables.
s. d. 10 0 are half 6 8 —- third 5 0 —- fourth 4 0 —- fifth 3 4 —- sixth 2 6 —- eighth 1 8 —- twelfth 1 0 —- twentieth
Of a shilling.
6_d_. are half 4 —- third 3 —- fourth 2 —- sixth 1 —- twelfth
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Time.
60 seconds 1 minute 60 minutes 1 hour 24 hours 1 day 7 days 1 week 4 weeks 1 lunar month 12 cal. mon. 1 year 13 lunar months, 1 day, 6 hours, or 365 days, 6 hours, 1 year.
Thirty days hath September, April, June, and November; All the rest have thirty-one, Save February, which alone Hath twenty-eigth, except Leap year, And twenty-nine is then its share.
36 pounds 1 truss of straw 56 pounds 1 do. of old hay 60 pounds 1 do. of new hey 36 trusses 1 load
MONEY.
Two farthings one halfpenny make, A penny four of such will take; And to allow I am most willing That twelve pence always make a shilling; And that five shillings make a crown, Twenty a sovereign, the same as pound. Some have no cash, some have to spare— Some who have wealth for none will care. Some through misfortune's hand brought low, Their money gone, are filled with woe, But I know better than to grieve; If I have none I will not thieve; I'll be content whate'er's my lot, Nor for misfortunes care a groat. There is a Providence whose care And sovereign love I crave to share; His love is gold without alloy; Those who possess't have endless joy.
TIME OR CHRONOLOGY.
Sixty seconds make a minute; Time enough to tie my shoe Sixty minutes make an hour; Shall it pass and nought to do?
Twenty-four hours will make a day Too much time to spend in sleep, Too much time to spend in play, For seven days will end the week,
Fifty and two such weeks will put Near an end to every year; Days three hundred sixty-five Are the whole that it can share.
Saving leap year, when one day Added is to gain lost time; May it not be spent in play, Nor in any evil crime.
Time is short, we often say; Let us, then, improve it well; That eternally we may Live where happy angels dwell.
AVOIRDUPOISE WEIGHT.
Sixteen drachms are just an ounce, As you'll find at any shop; Sixteen ounces make a pound, Should you want a mutton chop.
Twenty-eight pounds are the fourth Of an hundred weight call'd gross; Four such quarters are the whole Of an hundred weight at most.
Oh! how delightful, Oh! how delightful, Oh! how delightful, To sing this rule.
Twenty hundreds make a ton; By this rule all things are sold That have any waste or dross And are bought so, too, I'm told.
When we buy and when we sell, May we always use just weight; May we justice love so well To do always what is right.
Oh! how delightful, &c., &c., &c.
APOTHECARIES' WEIGHT.
Twenty grains make a scruple,—some scruple to take; Though at times it is needful, just for our health's sake; Three scruples one drachm, eight drachms make one ounce, Twelve ounces one pound, for the pestle to pounce.
By this rule is all medicine mix'd, though I'm told By Avoirdupoise weight 'tis bought and 'tis sold. But the best of all physic, if I may advise, Is temperate living and good exercise.
DRY MEASURE.
Two pints will make one quart Of barley, oats, or rye; Two quarts one pottle are, of wheat Or any thing that's dry.
Two pottles do one gallon make, Two gallons one peck fair, Four pecks one bushel, heap or brim, Eight bushels one quarter are.
If, when you sell, you give Good measure shaken down, Through motives good, you will receive An everlasting crown.
ALE AND BEER MEASURE.
Two pints will make one quart, Four quarts one gallon, strong:— Some drink but little, some too much,— To drink too much is wrong.
Eight gallons one firkin make, Of liquor that's call'd ale Nine gallons one firkin of beer, Whether 'tis mild or stale.
With gallons fifty-four A hogshead I can fill: But hope I never shall drink much, Drink much whoever will.
WINE, OIL, AND SPIRIT MEASURE.
Two pints will make one quart Of any wine, I'm told: Four quarts one gallon are of port Or claret, new or old.
Forty-two gallons will A tierce fill to the bung: And sixty-three's a hogshead full Of brandy, oil, or rum.
Eighty-four gallons make One puncheon fill'd to brim, Two hogsheads make one pipe or butt, Two pipes will make one tun.
A little wine within Oft cheers the mind that's sad; But too much brandy, rum, or gin, No doubt is very bad.
From all excess beware, Which sorrow must attend; Drunkards a life of woe must share,— When time with them shall end.
The arithmeticon, I would just remark, may be applied to geometry. Round, square, oblong, &c. &c., may be easily taught. It may also be used in teaching geography. The shape of the earth may be shewn by a ball, the surface by the outside, its revolution on its axis by turning it round, and the idea of day and night may be given by a ball and a candle in a dark-room.
As the construction and application of this instrument is the result of personal, long-continued, and anxious effort, and as I have rarely seen a pirated one made properly or understood, I may express a hope that whenever it is wanted either for schools or nurseries, application will be made for it to my depot.
I have only to add, that a board is placed at the back to keep the children from seeing the balls, except as they are put out; and that the brass figures at the side are intended to assist the master when he is called away, so that he may see, on returning to the frame, where he left off.
The slightest glance at the wood-cut will shew how unjust the observations of the writer of "Schools for the Industrious Classes, or the Present State of Education amongst the Working People of England," published under the superintendance of the Central Society of Education, are, where he says, "We are willing to assume that Mr. Wilderspin has originated some improvements in the system of Infant School education; but Mr. Wilderspin claims so much that many persons have been led to refuse him that degree of credit to which he is fairly entitled. For example, he claims a beneficial interest in an instrument called the Arithmeticon, of which he says he was the inventor. This instrument was described in a work on arithmetic, published by Mr. Friend forty years ago. The instrument is, however, of much older date; it is the same in principle as the Abacus of the Romans, and in its form resembles as nearly as possible the Swanpan of the Chinese, of which there is a drawing in the Encyclopaedia Brittanica. Mr. Wilderspin merely invented the name." Now, I defy the writer of this to prove that the Arithmeticon existed before I invented it. I claim no more than what is my due. The Abacus of the Romans is entirely different; still more so is the Chinese Swanpan; if any person will take the trouble to look into the Encyclopaedia Britannica, they will see the difference at once, although I never heard of either until they were mentioned in the pamphlet referred to. There are 144 balls on mine, and it is properly simplified for infants with the addition of the tablet, which explains the representative characters as well as the real ones, which are the balls.
I have not yet heard what the Central Society have invented; probably we shall soon hear of the mighty wonders performed by them, from one end of the three kingdoms to the other. Their whole account of the origin of the Infant System is as partial and unjust as it possibly can be. Mr. Simpson, whom they quote, can tell them so, as can also some of the committee of management, whose names I see at the commencement of the work. The Central Society seem to wish to pull me down, as also does the other society to whom reference is made is the same page of which I complain; and I distinctly charge both societies with doing me great injustice; the society complains of my plans without knowing them, the other adopts them without acknowledgment, and both have sprung up fungus-like, after the Infant System had been in existence many years, and I had served three apprenticeships to extend and promote it, without receiving subscriptions or any public aid whatever. It is hard, after a man has expended the essence of his constitution, and spent his children's property for the public good, in inducing people to establish schools in the principal towns in the three kingdoms,—struck at the root of domestic happiness, by personally visiting each town, doing the thing instead of writing about it—that societies of his own countrymen should be so anxious to give the credit to foreigners. Verily it is most true that a Prophet has no honour in his own country. The first public honour I ever received was at Inverness, in the Highlands of Scotland, the last was by the Jews in London, and I think there was a space of about twenty years between each.
