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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 a3, 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

ab

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.)

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. We 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,

say six, and calling the whole six one, to call each ball one-sixth, since this method, though advisable at a later stage, would only introduce confusion at present. It is, therefore, desirable that the unit should now be perfectly simple, and capable of division into parts exactly like itself; for which purpose we shall adopt length as the object of measurement. We must also observe that the preliminary notions are now to be almost entirely created, while they did exist, though in a rude form, when the pupil first began the study of whole numbers. He has been accustomed to the consideration of several things of the same kind, but rarely to that of the division of one of these objects into equal parts. His half has, most probably, been merely a division into any two parts whatsoever, and he can accordingly, with perfect consistency, talk of the larger and the smaller half. There is, therefore, more preliminary work, and, as in the case of whole numbers, palpable means of instruction should be adopted. We should recommend the following simple apparatus, making no apology for putting the parent to further expense and trouble: if there be any one of this class who does not think the education of his children the very first object of his life, next to procuring subsistence for them, and arithmetic a most im portant part of that education, we advise him to laugh at us, and put this paper aside,—we are not writing for him. Let from twelve to twenty slips of cheap wood be procured, each exactly one foot in length, and about. two-tenths of an inch in breadth and thickness. Let the first of these be divided by a line, or a scratch extending all around it, into two equal parts; the second into three equal parts; the third into four equal parts; and on up to the eleventh, which will be divided into

SO

twelve equal parts. These will be enough at first, but we should further recommend several other slips, divided respectively into 36, 60, 84, 90, 100, 120, and 180 equal parts; which numbers are chosen on account of their having a great number of divisors, considering their magnitude. On the small ends of each rod let the number of parts be marked into which the foot is divided. The following is a representation of the rod in which the foot is divided into quarters :

A common pair of compasses will also be necessary. It might be thought that one rod would serve the purpose of four, if the sides were differently divided, but this is not the case, as will be seen hereafter; and also that the halves and quarters might be placed on the same rod, and so on; which, however, would be directly against our object, as tending to something like taking for granted one of the simple propositions, of which we wish to furnish a palpable proof.

These rods having been laid side by side, the pupil is called upon to observe that they are all of the same length, being that which he has used and called a foot, if the suggestion of our last article (page 55) has been adopted; if not, that method should be now put in practice previously to going any farther. He should then have his attention drawn to the fact that each foot rod is divided into equal parts, that is, that the subdivisions of each rod are equal, but that those on different rods are unequal. This he should ascertain by measurement with the compasses. Again, he is to

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