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you can trace it to some particular pair of units, say the 8 and the 7, or the 9 and the 5, which habitually give you more trouble than the
It is only practice which can set you right. So the moment you observe any hitch or difficulty in special combinations or subtractions, it is well to work at them till they become thoroughly familiar, till for example the sight of 8 and 7 together instantly suggests 5 as the unit of the sum, or the taking away of 6 from a number ending in 3 instantly suggests 7.
(2) Money. Little exercises on the arithmetic, first of a shilling, afterwards of half a crown, and afterwards of a sovereign are very interesting, and require no slate or book. The scholars should be practised in rapid adding, and subtracting, in dividing it into parts, in reduction to half-pence and farthings; in telling different ways in which the whole may be made up, e.g. a shilling into 7d. and 31., into 8d. and 4d., into 3}d, and 8}d., into 47d. and 7d., &c., until every form of arithmetical exercise possible with this sum of money shall be anticipated.
(3) Simple Calculations in time, e.g. the time it will be 3 hours hence, hours, 12, 24; the date and day of the week, three days, four weeks, seventeen hours, two months hence; and in like manner easy calculations respecting lengths and weights, may fitly precede all attempts to work sums in compound arithmetic by written exercise
(4) Fractions. The first oral exercises should be founded on familiar sums of money, and on the products already known in the multiplication table and may be graduated in some such way as this:
(a) The third of a shilling, the 8th, the 12th, the 4th, the 6th, &c. The fifth of 30, the ninth of 27, the third of 18, the twelfth of 72, &c.
(6) 3 of 6d., of 54, of 21, io of 40, 5 of 16.
(c) What number is that of which 5 is ç; Of which 4 is į; of which ro is ő ; Of which 25. is $; Of which is. 6d. is ?
(d) Find other fractions equal to š, to $, to 1, to , &c. (e) i of a foot, sof i lb., 4 of a week, iç of an hour.
By selecting your examples from fractions which present no complications or remainders, and by rapidly varying and often repeating them, it is easy to advance a considerable distance in the manipulation of fractions, before talking at all about numerators and . denominators, or giving out what is called a rule.
(5) Exercises on special numbers. (a) Take the number 60. Its half. Its third. Its fourth. Its fifteenth. Its sixteenth, &c. (6) Find two numbers which make 60; 24 and 36, 18 and 42, &c. three numbers
11, 14 and 35; 21, 19 and 20; 7, 35 and 18, &c.
(c) Take from 60 in rapid succession, fours, sevens, elevens, eights, threes, &c.
(d) Find i of 60, j, 10, 15, 11, }, dó, 37, &c.
(e) Give the components of 60 pence. Of 60 shillings. Of 60 farthings. Of 60 ounces. Of 60 hours. Of 60 yards, &c.
V Find in how many ways to hurdles might be arranged so as to enclose a space, or in how many forms a payment of £60 might Le made.
(6) Proportion. (a) Name other figures representing the same ratio as 5 : 7. As 3:8. As 15 : 21, &c. &c.
() Find a fourth proportional to 2 : 3 :: 4. To 5: 6 :: 10. To 7:12 :: 6. 25. : 25. 6d. :: 45. £3:£55. :: 6 oz.
(c) Find two pairs of factors whose products are equal, and arrange the whole four in several ways so that they shall form proportions: e.g. Because 5 x 24 = 8 x 15, Therefore 5 :15 :: 8 : 24 and 24 : 15 :: 8 : 5, &c. &c.
A good teacher will invent hundreds of such exercises for himself, and will not need a text-book. There is nothing unsound or meretricious in mental arithmetic of this kind. On the contrary, it will prove to be one of the most effective instruments in making your scholars good computers. It will give readiness, versatility, and accuracy, and will be found an excellent preliminary training for the working of ordinary sums in writing. Keep in view the general principle that the nature of each process should be made familiar by oral exercise before recourse is had to pen or pencil at all, and that the oral exercises should be of exactly the same kind as written sums, but should differ only in their shortness, and in the fact that each problem requires only one
or at most two efforts of thought, and deals only with figures such as can be held in the mind all at once, without help from the eye. Much activity of mind is needed on the part of a teacher who conducts this exercise ; and it is its least recommendation that when so conducted it challenges the whole thinking faculty of the children, concentrates their attention, and furnishes capital discipline in promptitude and flexibility of thought.
In beginning to give lessons on money, weights, and of near and fami- measures, you may do well to make an occasional use of liar objects actual money, to give a few coins in the hand and to let as units of them be counted. In French and Belgian schools, not
only is a diagram shewing the form and proportion of the legal weights and measures displayed, but a complete set of the weights and measures themselves is deposited in every school : so that the children may be taught to handle and to use them, occasionally to weigh and measure the objects near them, and to set down the results in writing. The dimensions of the school-room and of the principal furniture should be known, and a foot or a yard, or a graduated line of five or ten feet should be marked conspicuously on the wall, as a standard of reference, to be used when lengths are being talked about. The area of the playground; the length and width of the street or road in which the school stands; its distance from the church or some other familiar object, the height of the church spire, should all be distinctly ascertained by the teacher, and frequently referred to in lessons wherein distances have to be estimated. Children should be taught to observe that the halfpenny has a diameter of exactly one inch, and should be made to measure with it the width of a desk or the dimensions of a copy-book. It constantly happens, that if I ask
elder children, who have gone through' as it is called a long course of computation in ‘long measure,' to hold up their two hands a yard apart, or to draw a line three inches long on their slates, or to tell me how far I have walked from the railway station, or to take a book in their hands and tell me how much it weighs, their wild and speculative answers shew me that elementary notions of the units of length and weight have not been, as they ought to be, conveyed before mere "ciphering' was begun.
As to weights and measures, they are as we all know, Wrights a great stumbling-block. The books give us a formidable and list of tables, and children are supposed to learn them by heart. But a little discrimination is wanted here. It is needful to learn by heart the tables of those weights and measures which are in constant use, e.g. avoirdupois weight, long measure, and the number of square yards in an acre; but it is not worth while to learn apothecaries' weight, cloth measure, or ale and beer measure, because in fact, these measures are not in actual or legal use; and because the sums which the books contain are only survivals from an earlier age when the technical terms in these tables, puncheons, kilderkins, scruples, and Flemish ells, had a real meaning, and were in frequent use. Keep these tables in the books by all means, and work some sums by reference to them : they are of course all good exercises in computation ; but here, as elsewhere, abstain from giving to the verbal memory that which has no real value, and is not likely to come into use. It seems hardly necessary to refer to the efforts some Moral
lessons in teachers have made to use Arithmetic as a vehicle for the inculcation of Scriptural or other truths. Such efforts have been commoner in other countries than our own. “How admirably,” says an enthusiastic French writer on
Arithmetic, “ does this science lend itself to moral and religious training !"
Père Girard composed a manual of Arithmetic in which, for the most part, the problems given had a distinctly hortatory character, and were meant to embody economic and moral instruction. Here is an example.
“Un père de famille avait l'habitude d'aller tous les soirs au cabaret et laissait souvent sa famille sans pain à la maison. Pendant quatre ans qu'il a mené cette vie il a depensé la première année 197 fr., la seconde 204 fr., la troisième 212 fr., et la quatrième 129 fr. Combien de francs aurait epargné ce malheureux père s'il n'eut pas eu le gout de la boisson ?”
And another French writer seeks on this wise to give a moral tone to his arithmetical lessons. the Curé to visit the school, and the teacher to say,
What does the number 7 remind you of?
The 7 deadly sins, the 7 sacraments, and the 7 golden candlesticks.
What have you tell me about the number 12?
The number of the Apostles, the number of the minor prophets and of the gates of the Apocalyptic Jerusalem.'
And then he turns to the children, “Mes enfants,” he says, we have thus shewn to our worthy pastor that we establish true relations between the art of computing and the principles of virtue and religion. Who will say after this that Arithmetic is not a moral and edifying study?"Who indeed? Of course sums founded on Bible facts, on the
age of Methusaleh, or the length of Goliath's spear are innocent enough. But I suspect that all attempts of this kind to kill two birds with one stone, so to speak, are very unsatisfactory. Moreover it does not seem quite reverent to use books or names with which some of us have very sacred associations for the sake of manufacturing arithmetical puzzles for school boys. After all, in just the proportion in which children pay attention to the