Mental arithmetic has its fascinations. It has, too, I am ready to admit, solid advantages. Its advantages, however, I apprehend are not precisely those which are sometimes attributed to it. There can be no doubt, I think, that it helps to cultivate the reflective powers; that it requires, and by requiring gives, the ability to confine the attention to continued mental processes. But for making expert practical accountants, which is generally quoted as its distinguishing benefit, I confess I am partial to the slate and pencil, and to that venerable parallelogram, the old-fashioned Multiplication Table, in the shape it came down to us from Pythagoras.
The reader will not, of course, understand me as wishing to discard Mental arithmetic. All that I mean to suggest is the inquiry, whether its advantages are not looked for in the wrong direction, whether they are not sometimes over-estimated, and whether this mode of teaching arithmetic, especially when pursued as a hobby, is not sometimes pushed too far, and made the means of curious display, rather than of solid and lasting benefit. In teaching mental arithmetic, too, for I would certainly teach it to some extent, I would suggest the expediency of teaching children, in performing these mental operations, to think in figures, in other words, to form conceptions of the arithmetical figures and signs, which are visible objects, rather than of quantities and relations, which are mere abstractions. Multiplication is a mere metaphysical entity. The sign of multiplication is a simple, visible symbol, addressed to the eye, and capable of being conceived by the mind with unmistakable clearness and precision. A child counting its fingers in the first steps of learning to add and to take away, is a pretty sight, doubtless. But it is painful to see a person grown to man's estate, and in other respects well educated, as I have very often seen, still dependent upon the same infantile contrivance,—still counting fingers when required to add long columns of figures. Count the fingers, if necessary, in order to get the child under way. But the sooner the leading-string can be dropped, and the child can be made to picture in his mind the pure figures and signs, their combinations and results, without reference to fingers, or apples, or cakes, or tops, the better for his arithmetic, and the better for his mental cultivation.
The subject has a painful interest for the Sabbath-School Teacher. The teacher of the infant school, indeed, has some opportunity for employing this principle of pictorial representation, in teaching the little ones of his charge. The infant school-room usually has conveniences for maps and picture cards and diagrams, and even blackboards; and most infant school teachers wisely avail themselves of the opportunity afforded. But go into the main school-room—what can the teacher do? Twenty, thirty, forty classes huddled together into one room, compact as sheep in a pen, how can the individual teacher, if disposed, use adequate visible illustrations for the instruction of his class? Where shall he place his blackboard? where shall he hang up his maps? where shall he suspend his models? where shall he exhibit his specimens? The utmost that can be done in most of our schools, as at present provided for, is to have a few maps on the distant walls of the room which the superintendent may refer to, whenever he chooses, and which all the children may see who can! The time must come, however, when the teaching of religious truth will be considered of as much importance as the teaching of arithmetic or of chemistry, and the Sabbath-School will have the same facilities for imparting instruction as the week-day school. But that time has not yet come. In the meanwhile, let the teacher carefully avail himself of whatever subsidiary aids are within his reach. No teacher should ever present himself before his class without a Bible Atlas and a Bible Dictionary in his hand. Many of those things with which his class ought to be made acquainted, are here not only described, but delineated, with equal accuracy and beauty. Thanks to the booksellers and the religious publication societies, the scenes of sacred history, and indeed religious topics generally, have been illustrated in cheap pictorial cards, both large and small, and with admirable fidelity and skill. These form a part of the indispensable furniture of the Sunday-School teacher. They are to him as necessary as are experiments, or a cabinet of specimens, to the lecturer on the physical sciences. The Sabbath-School teacher should be continually on the look-out for publications of this kind, not only for instructing and furnishing his own mind with definite ideas, but for exhibition to his class. A wise teacher will not only have something to say to his class, but also something to show. The ideas which the child gets from looking at really instructive pictures and maps, never leave him. How much also our intelligent apprehension of the scriptures is increased, by a knowledge of topography, and by associating each event in the sacred narration with the place in which it occurred?
It may be proper to say, too, in this connection, that it is with a view to the principle now under consideration, that in preparing books and papers for the young, authors and publishers feel justified in giving so much labor and space to pictorial illustration. When, indeed, such illustrations are merely for display, they deserve the contempt which they often receive. But when these pictorial illustrations have a definite meaning and design, when they teach something, when they connect in the child's mind sound religious truth with distinct and easily remembered visible forms, they are a really valuable aid in the inculcation of doctrine.
The power of attention, like all the mental powers, is by nature greater in some than in others. Still, there is no power more susceptible of improvement. The importance of its cultivation cannot well be over-stated. It affects not one study only, but all studies; not one mode of study only, but every mode of study, by text-book or by lecture; lessons to be recited by memory, or those by question and answer; not even study only, but conduct and manners, the regulation of the heart and the formation of the character. The precise measure of a child's success, in every thing that pertains to his character and standing as a scholar, will in nine cases out of ten be his power and habit of attention. There are indeed lamentable cases of wilful and intentional disorder. Yet every teacher knows that by far the greater portion of the things which interrupt and disturb a school arise from thoughtlessness and inattention. There are also equally undoubted cases of ignorance that is no crime. Yet the great majority of those who fail in their studies, fail simply because they do not attend. To attend, however, means something more than merely to be bodily present, more even than to have the ears open and the eyes fixed in the direction of the speaker, when a thing is said, or done. An old lady used to sit in the same aisle with me in church, and unfortunately lived opposite me in the street, who was neither deaf nor blind, and who was never absent from church, and yet she sent over invariably on Sunday evenings to know what it was the minister said about that meeting on Wednesday night, or that meeting on Friday night,—she did not rightly understand!
But it is not necessary to go to church, to find those who "having eyes see not, and having ears hear not, neither do they understand," who look without seeing, and hear without comprehending. Publish a notice in your school, making some change of hours or lessons, or giving any specific direction. No matter how simple, or how plainly expressed, the notice may be, or how particularly attention may be called beforehand to the announcement about to be made, where is the happy teacher who has been able on such an occasion to make himself understood by all? Teachers and preachers and speakers of every name have generally very little idea how much they are misunderstood. Let me give some instances.
In my own Sunday-School, I had neglected one morning to bring with me the teacher's class-books. After opening the school, I rang the bell as a signal for attention. There was a general hush throughout the room. All eyes were turned to the desk. I said: "Your class-books unfortunately have been left behind this morning. They have been sent for, however, and they will soon be here. As soon as they come, I will bring them round to the several classes. In the meantime, you may go on with your regular lessons." The bell was then tapped again, and the routine of the school resumed. In about a minute, a girl came up to the desk, with, "Sir, teacher says, will you please to send her class-book; it was not brought round, as usual, this morning, before school opened!" Here was a class of ten girls, averaging twelve years of age, and not one of them, nor their teacher, had heard or understood the notice which I thought I had made so plain!
Here is another instance. At the examination for admission to the Philadelphia High School, as a means of testing among other things how far this very faculty of hearing and of attention has been cultivated, the candidates are required to copy a passage from dictation. These exercises are always preserved for reference, and in order to show the fairness of the examination. On one occasion, when I was Principal of the School, I took the pains to copy out a few of the exercises, in order to show the singular freaks into which an uncultivated ear may be led. One or two specimens will serve to illustrate the point. The first clause with its variations, was as follows:—
In another part of the passage occurred the following: