Imágenes de páginas
PDF
EPUB

by a very simple mechanical contrivance, be rendered as obvious as any preceding part. A piece of pasteboard or wood, with lines ruled from the top to the bottom, and thus divided into six columns at most, with a number of counters not less than a hundred, would serve the purpose very well. But a toy similar to the abacus, which is sold in the shops for the purpose of teaching the multiplicationtable, would be preferable, if it were longer in proportion to its breadth, and had more balls on each wire. The one which we should recommend would be such as is represented in the accompanying diagram.

A wooden frame is traversed by six wires, on each of which are a number of sliding balls. The length should be at least three times the breadth, and the balls, when placed close together, should not occupy a third of the length. There should be at least thirty on each wire. The reader will easily guess that we mean each ball on the first right-hand wire to stand for one, each on the second for ten, on the third for one hundred, and so on. The question now is to convey the same idea to the child. Perhaps it would be advantageous to increase the size of the balls a little in going to the left; at least those on the various wires should be differently coloured : thus the units might be white, the tens red, &c. The instrument being laid on the table, with all

[graphic]

.

* These instruments are made, according to the directions here given, by Messrs. Watkins and Hill, Charing Cross,

the balls at the further end of the wires, the instructor should bring down one unit to the nearer end of the right-hand wire, then another, and so on, causing the pupil to name each number as it is formed. When ten

have thus been brought down, they should be removed back again, and one of the tens brought down, and afterwards one more unit. The pupil should then be asked what number is there, and if he answers two, as is very likely, from his seeing two balls side by side, the ten should be removed back again, and the former ten units brought down again on the right-hand wire. He is then instructed to count the number, and finds eleven. The ten units are then removed again, and the single ten on the second wire substituted in their place. If there is still any hesitation as to the meaning, let the process be entirely re-commenced, and this until the pupil has had occasion to observe repeatedly, that a ball on the second wire is never touched, until there are ten on the first wire. It will be better to avoid verbal explanations at first; the object is to enable the child to lay down on the wires any number with which he is acquainted, and he will do this sooner by actual practice than by any general conception which can be given of the local value of the balls. The step from 10 to 11 once made, no further difficulty will arise before 21 at least; if this happen, the process should be again repeated. No further step should be made until the pupil can readily express any number under 100. Numbers also should be given to him not expressed in their simplest form, but having more than ten on the unit's line; for example, three tens and twenty-two units, which he should be shown how to reduce by taking away collections of ten from the units' wire, and

marking them on the tens. The same number should be varied in different ways, on this principle; and to make it more practicable, we should have recommended a longer instrument, with more balls on each wire, had we not thought it might have been objected to as cumbrous and expensive. The pupil should then be directed to form two different numbers on different parts of the abacus, whose sum is under 100; these he should then add together in his head, as he has already been used to do. The balls of the two numbers should then be placed close together, so as to form one; and the reduction of the units into tens should be made. The first examples, however, should be those in which no such process is necessary, such as the addition of 23 to 55. Examples of subtraction should follow, in which the same rule is observed: for instance, 31 from 59. The number to be taken away should be formed on the lower part of the instrument, and the number which is to be decreased, on the higher. The pupil will immediately be able to bring down from the higher number a similar number of balls to those which compose the lower. At last, an instance should be given in which the borrowing of a ten becomes necessary: for example, the subtraction of 26 from 81. These numbers having been formed, the pupil is directed to take the less from the greater, as he has done before. This he immediately finds to be impossible, on which the teacher removes one of the tens from the higher number, and brings down ten units in its place. The pupil, as has been observed, must be made familiar with this process before he begins this operation. Before proceeding any further, a great number of examples should be given, on practical questions, which can be readily

solved on the abacus; and in no case should the child be allowed to proceed to 100, or beyond, until he is perfectly master of the two left-hand wires.

We have no occasion to enter minutely into the method of proceeding with the other wires; we will, therefore, only observe that they should be added to the instrument, so to speak, one at a time, and whenever a new wire is introduced, all the exercises abovementioned should be carefully repeated. One thing at a time is amply sufficient for the beginner, even when he has already been used to similar things, and it is possible that any inattention or slurring of the process, even at the sixth wire, might introduce confusion among the ideas he has acquired at the previous ones. He may now add three numbers together, for which there are balls sufficient, and may perform combinations of an addition and a subtraction, or of two subtractions.

Multiplication may be performed by repeated additions, which is the only way of introducing it that can be satisfactory. Thus if 117 is to be taken five times, it must be brought down twice, and after the reduction of the tens, again a third time, and so on. If the learner has been previously sufficiently exercised to recollect the multiplication-table as far as ten times ten, he may go through a process more nearly corresponding to that in the books of arithmetic. Previously to this, he must be instructed how to multiply by 10, 100, &c., as follows: the teacher places a simple number on the abacus such as 17, and the same number, in appearance, on the second and third wires, which will stand thus:

[blocks in formation]

The pupil, having named these two numbers, has it pointed out to him that each part of the second is ten times the corresponding part of the first, which may be illustrated by actually forming ten sevens and ten tens in succession. When this has been well understood and practised, 17 may be multiplied by 16, by taking it ten times and six times.

Division can only be conveniently done by continual subtraction of the divisor from the dividend; after which the process may be shortened by subtracting it ten times, or one hundred times at once, if possible. So comparatively complex a process may be deferred to the future and more complete course.

If the child show a capacity a little above the common, it may be useful to try him with other systems of notation. For example, a ball on the second wire may stand for five on the first; one on the third for five on the second, and so on. Of these, the binary scale, being the most simple, should be most particularly attended to, as illustrating the principle of local value in a remarkable degree, by the number and rapidity of the changes. This, however, should be done with very great caution, as it may confuse the pupil's notions of the decimal or common system, and should never be attempted until he is very well grounded in the latter. At the same time, the abacus may be made useful in the explanation of the common system of weights, measures, and money; for example, the first line on the left hand may represent farthings, the second pence, and so on. It will be found that, by this method, the relations of our weights and measures will be much more quickly learnt than by the common usage of committing tables to memory. Practice will suggest many

« AnteriorContinuar »