LETTER XVI

MY DEAR M

to discipline mind.

ARITHMETIC.

Some studies are mainly intended The knowledge they impart may not be of immediate practical utility; whilst the exercise of mind employed in its acquisition may be invaluable. We want it not so much for its own sake as to increase our power to acquire other things-the command of attention, the quick perception of relations, the prompt recognition of conclusive reasoning, the habit of following a process patiently, and conducting it with precision. The medium through which we acquire these powers answers its purpose perfectly in proportion as it is entirely divested of any attractive interest in itself; that is, as it is purely abstract. This is just the case with the science of mathematics, from the lowest to the highest of its branches. One question in a hundred of those solved by the student will never be presented to her again; but the processes of reasoning through which she passes in the course of their solution, give her power and acumen, which will be available thenceforth at all times and on every subject. "We learn mathematics," says Locke, "not to make ourselves mathematicians, but reasonable men."

Now, these powers and habits of mind are exceedingly valuable; they constitute in fact the elements of

mental greatness; and as arithmetic is the best exercise for creating such powers and habits, it becomes us to bestow all necessary pains that we may fully realize its advantages.

First, I would say, do this by securing the use of the reasoning powers in the several operations, rather than the exercise of the memory. The methods of performing arithmetical operations now in use, being adapted to a most extensive and complicated state of mercantile transactions, has become so completely technical, that it may be readily seized by the memory, and the several processes performed almost without concurrence of any other faculty of the mind. The greater number of operations are performed on the result of previous calculations, formed into tables and committed to memory; and they may be applied to the work in hand without any very laborious exercise of the judgment. The rules, too, laid down in common arithmetic books for the performance of ordinary operations, are generally clear and decided enough to conduct the pupil through the solution of the proposed questions, and to a superficial mind may appear to supersede the necessity for any further instruction. But the value of a thorough acquaintance with these rules, and readiness in their application, is small when compared with a knowledge of the foundations of the art, the universal principles on which such rules are constructed. How far you may give your pupil the benefit of such knowledge, is a question to be determined by capacity and circumstance; but if you would teach arithmetic efficiently, you must yourself be thoroughly acquainted with them. I may say, however, with some confidence (experience is my authority), that the rationale of arithmetical pro

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cesses may be made intelligible to learners; and that arithmetic is not only a more beneficial, but a more agreeable study when this method is pursued. I think that the age at which such a plan is impracticable, is an age at which arithmetic should not be taught at all; it certainly is a study not adapted to a very early age. But if a very young child must needs be introduced to the science of numbers, I would have the exercises of calculation conducted entirely by the use of tangible objects, until the period when the mind shall have acquired the power (not one of the first to be developed), of regulating its operations by signs entirely arbitrary and conventional. Much may be done in this way. You will scarcely imagine till you have made the experiment variously, how far the working even of complicated questions may be conducted by diversity in the arrangement of a number of cubes, counters, &c. The great advantages of this plan are that the reasons of the several conclusions are self-evident; the ideas of numbers, their relations and proportions, are presented for the first time in a decided form cognizable by the senses, and tested beyond doubt at each successive step. If these were not secured, there is still a negative advantage resulting, which it is of immense importance to realize—the learner is thus saved from acquiring a ready use and application of terms which convey to her mind at best very indistinct ideas, and of associating them in processes which she does not understand. There is nothing so difficult to learn as the use of a term with which the mind has acquired a useless familiarity.

In the second place, I would recommend that your pupil's certainty of the correctness of her solution should

never depend solely on your word, or on its conformity with an answer given in her book. Let her carry the question through two or three different modes of operation; and let her fully understand wherein they differ, and whereupon an identical result must ensue. If you are mistress of the theory on which such operations are founded, an endless diversity of methods in carrying it out will occur to you; or if your ingenuity fail, open any good treatise on the scientific principles of arithmetic, and you will be furnished at once with what you want.

There is much in the practice of calculation, as it obtains among half-civilized nations, which is suggestive of mode for the initiation of the young into the mysteries of science; and there is a great deal in the history of its development which may be made interesting to children, and lead them unconsciously into some of the essential principles of computation. For instance—the fact that every nation of the earth, almost without exception, has adopted the decimal radix, and that this evidently arises from the natural circumstance of the fingers being the first objects presenting themselves for numeration—hence the term "digit." This fact receives interesting illustration from the practice of the Chinese, who adopt a method of reckoning on the fingers, by which the hand is made capable of expressing all numbers under 100,000. By placing the thumb nail on each joint of the little finger, either at the back, or the front, or the inner side, they express the nine units; this method is repeated on the next finger for the nine tens, on the next for the nine hundreds, and so on. There are many other interesting and in structive varieties in the practice of indigitation among

the ancient nations of the East; and some of the devices might, with very great advantage, be introduced into modern instruction. Then comes the fact, that in all of the few exceptions to the use of the number ten, as the point at which the numeration of units shall stop, and counting proceed by the numeration of groups of ten, till ten groups are numbered, and then by groups of ten tens, &c.; the numbers five or twenty have been chosen, the one evidently suggested by the number of fingers on one hand, and the other by the number of fingers and toes collectively. This the learner will readily comprehend, and she may be led to perceive the superior excellence of the decimal over the quinary and vigesimal radices, as a medium between the frequent change of terms incurred by the use of the former, and the too great comprehensiveness of the latter.

A history of the practices of the various nations will furnish you with many plans for the illustration of these primary principles. Take, for instance, that of the Malagasi. When they wish to count a great number of objects, such as the number of men in a large army, they cause the men to pass in succession through a narrow passage, before the individuals employed to count them. For each man that passes they deposit a stone in a certain place. When all have passed they proceed to arrange the stones in groups of ten; then in heaps, each containing ten groups or a hundred stones; and in the same way dispose the groups of hundreds so as to form thousands, until the number of stones is exhausted.

Now, what would you think of giving your pupil an exercise of this bulky and cumbrous order? Suppose you set a large number of counters before her,