can be brought within the limits of their capacities. But these are very different in different individuals; and this must be attended to in teaching a number of them together. The rule we should propose is, make them arithmeticians, rational ones if you can; but, at any rate, make them master the processes of computation. There is no reasoning in arithmetic which does not become extremely simple, if the numbers on which it is em ployed be simple. So much is this the case, that we know they can be comprehended, even by children whose previous education in counting has been very far below the one proposed by us. We should recommend the following detail: the pupils having been formed into classes, and provided with books of arithmetical reasoning, and not merely, as is almost always the case, consisting of nothing but dogmatical rules, the master should explain to them the principle of a rule, as nearly as may be in the words of the book, questioning the pupils as he goes on, to see that they understand every step. When a difficulty arises, the principle on which it depends should, if possible, be separated from the rest, and announced in a distinct form. Copious examples should then be given of it, and on no account should the class be allowed to proceed until it has become familiar to every one. When the demonstration has been thus finished, it should be repeated by one or more of the pupils, with different numbers, first in the words of the book, and then in their own. This is done to help the memory, and the instructor may be nearly sure that no one of his pupils will be able to substitute other data in a process of reasoning, unless he understand it. The rule is then to be reduced to its simplest form, which will usually require one or two additional

observations. We are not against learning these rules by heart, provided they be reduced to the utmost degree of conciseness. One or two simple examples should then be worked, first by the master, and then by the pupils; after which the latter should be dismissed to practise what they have learned. The same examples should not now be given to all; those who have shown the greatest facility of comprehension should have the more difficult ones. As soon as a rule has been thus finished, other questions should be given which combine the previous ones with it. Thus, in multiplication, questions should be given in which certain additions and subtractions are necessary to form the multiplicand and multiplier. The commercial rules should go together with the corresponding rules for abstract numbers, as the second differ in no respect from the first in their principles.

We consider arithmetic as a preparation for algebra, and the higher parts of mathematics. With each rule, therefore, we should introduce the algebraical signs and terms which are connected with it, so that, on beginning algebra, the pupil may be familiar with the signs + x=, &c., and the words second power, third power, &c. The words square and cube are perhaps objectionable, in their arithmetical sense, though we do not see what harm could arise from using them, if it were distinctly explained that they are incorrect terms, sanctioned by common usage. We are advocates for the use of many words which have gradually glided out of our books; such, for example, as minuend, subtrahend, resolvend, &c. We would even propose to coin addend. It must not be imagined that these terms are hard, because they are Latin and polysyllable; the

ideas attached to them are very distinct. Certainly, if simple Saxon equivalent terms could be found, it would be an advantage, with regard to these and many others, particularly numerator and denominator. We would even carry our extensions of arithmetic as far as the gates of algebra; not giving any reasoning with algebraical symbols, but accustoming the pupil to translate literal expressions into their corresponding arithmetical results in particular cases. Thus, instead of telling the learner to add together 18, 19, and 20, we would ask him, after previous explanation, what a+b+c stands for, where a stands for 18, b for 19, and c for 20? This would give the instructor an unlimited command of examples, while it would prepare the pupil to reason upon general symbols, by accustoming him to their sight. We would not, however, introduce him to the use of exponents, but would write aa for a2, aaa for a3, and so on. But here, as everywhere else, the progress should be very gradual from simple to more difficult expressions. It would also tend to interest the student, if problems, of which the algebrai cal solution is given, were presented to him for the application of arithmetic in particular cases. For example: It is found that if one man can finish a job in a days, which another can do in b days, they will do it a b

both together in days. What number of days will a+b

both do it in, which the first would finish in 72, and the second in 36 days?' Such instances as these would give views of the nature of general expressions, and the use of reasoning upon them. It would also teach the pupil to look forward to a higher science, and would relieve what the taste and constitution of most lead them

to call the drudgery of computation, in the application of which term we heartily join them.

Our limits will not allow of any further observations on this branch of the subject. We propose, however, in the next number to follow up our remarks by some others upon the teaching of fractions, and the higher parts of arithmetic, and, if we have room, the principles of algebra. We shall not quit the subject until we have gone through the elementary branches of mathematics. Whatever we may think of the higher parts, we are sure that these may be made accessible to any capacity, if the discipline be begun at an early age. We would caution those who teach, against measuring the progress of a student by the number of results he has learned, even if he is really ready in their application. If any of the processes we have described should seem a waste of time, let them recollect that, as the case actually stands, many years are passed (as far as this subject is concerned) in learning a few rules very badly. We think that much more might be well learned in the same time than is now learned at all; but if not, and if experience should at last oblige us to decide, that six or seven years are necessary to acquire only the facility of computation necessary for common purposes, it would be a great change for the better if a system could be introduced by which the pupil should think as well as work.

Since this article was printed we have learnt that flattened glass beads are made at Birmingham, at a very cheap rate, which might be strung in tens or hundreds, and supply the place of the boxes in page 5.

VOL. II.

H

74

ON THE METHOD OF TEACHING
FRACTIONAL ARITHMETIC.

By A. DE MORGAN,

(From the Quarterly Journal of Education, No. X.)

We

IN the preceding article we developed a method of giving the first notions of whole numbers to children. We now proceed to treat the fractional part of arithmetic in the same manner; premising, however, that on no account should this ground be entered until the pupil has the clearest notions, not only of the method of numeration, but of the first four rules in whole numbers. do not mean that he should be ready at the solution of questions which involve high numbers, that is, at the mechanical part of the subject; but he should at least be competent to perform any addition or subtraction of not more than four figures, any multiplication of two figures by two figures, and division of three figures by two others.

In treating of whole numbers, where it was sufficient that each one should be like the others, we used marbles or counters: these should now be entirely rejected; the child will be confused by any attempt to divide them into parts, as the whole and its parts will not then be entirely of the same character. So long as nothing more was necessary than to compare one counter with another, all was well, because each unit entirely resembled every other unit; but if we were now to cut these ones into fifths, the fifths would not be of equal dimensions, nor could the child make a one out of any five fifths. Neither will it be sufficient to take any number of balls,