Examples of Oral Exercise. 301 Not till six or seven such sums had been given did I notice that the first two numbers in each group amounted to 100, and the next two also; and that all the questions were framed on the same pattern. Many of the audience did not detect this, but of course the children were in the secret, and were, in fact, confederates with the teacher, in an imposture. It is because so much of what is called mental arithmetic consists of mere tricks of this kind, that the subject has been somewhat justly discredited by good teachers. But the mental Arithmetic which is of real service Its uses, does not consist in exercise in a few special rules, but in rapid, varied, and irregular problems in all the forms which computation may take. It differs mainly from written Arithmetic, in that it uses small numbers instead of large ones. Before attempting to work exercises in writing in any rule, a good oral exercise should be given to familiarize the pupils with the nature of the operation. I will give a few examples to illustrate my meaning: (1) Addition and Subtraction. Take the number 3, add it to 1 Examples and successively to the sums, up to 50. I, 4, 7, 10, 13, 16, 19, 22, 25, 28, 31, &c. &c. You will observe as you do this that there are certain combinations less easy than others. He whose turn it is to say 21 after 18; or to take away 3 from 32 will halt a moment longer than the You observe this, and make up a series of questions in which these two particular numbers shall be brought into relation: 28 and 3, 48 and 3, 19 and 3, 3 from 42, 3 from 21, &c. rest. There are but nine digits, and if in succession you give nine short brisk lessons,‚—one on each,—-requiring the number to be added and subtracted rapidly, you will come in succession upon every possible combination of these digits. You will bear in mind that when you yourself make an error in adding up a line of figures, of oral exercise. rest. 9 you can trace it to some particular pair of units, say the 8 and the 7, or the 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 5d., into 8d. and 4d., into 31d. and 83d., into 41d. and 7ąd., &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, 8 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. (b) of 6d., § of 54, & of 21, 1 of 40, § of 16. Τ is; Of (c) What number is that of which 5 is §; Of which which to is; Of which 2s. is; Of which 1s. 6d. is &? (d) Find other fractions equal to }, to %, to, to %, &c. (e) of a foot, g of 1 lb., of a week, 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 of 60, 1, 1, 1%, 11, 8, 18, 17, &c. (e) Give the components of 60 pence. Of 60 shillings. Of 60 farthings. Of 60 ounces. Of 60 hours. Of 60 yards, &c. (f) Find in how many ways 6o hurdles might be arranged so as to enclose a space, or in how many forms a payment of £60 might be made. (6) Proportion. (a) Name other figures representing the same ratio as 5 7. As 3: 8. As 15: 21, &c. &c. (6) Find a fourth proportional to 2 : 3 :: 4. To 5 : 6 :: 10. To 7: 12 :: 6. 25. 2s. 6d. :: 45. £3 : £1 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 × 24=8×15, 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 not 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. The use 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 measure ment. 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. and measures. As to weights and measures, they are as we all know, Weights a great stumbling-block. The books give us a formidable 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. lessons in sums. It seems hardly necessary to refer to the efforts some Moral 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 F. L. 20 |