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arrangement in series of small numbers up to, say, 100. It would be interesting to know whether a more rational arrangement would be gained by this process, but this is not important for teaching purposes.
Next would come the process of adding and subtracting grains of corn, or dots or little o's made on the slate. Methods of doing this are so familiar that I need not dwell upon them. The practice of multiplication and division in this way does not seem to need much exposition. We can repeat a row of any number of dots as often as we please, and count the product. We can divide any number into groups of any smaller number and find the quotient and remainder. All these exercises on the four rules of arithmetic need not take much time. My impression is that you will find, after a very little showing, that the child is able to perform the fundamental rules upon collections of grains of corn, or dots, without devoting much or long continued effort to the process.
The next step would be to extend the operations to continuous quantity as represented by lines and areas on paper or on the blackboard. The addition of lines consists in placing them, or lines equal to them, end to end, thus obtaining a line equal to their sum. Subtraction consists in cutting off from the longer line a length equal to the shorter one. Multiplication by a factor consists in adding together equal lines to a number represented by the factor. Division takes a twofold form. We may either divide a line into a given number of equal parts, thus obtaining a certain length as the quotient, or we may find how many times one line is contained in another, thus obtaining a pure number, or ratio, as the quotient.
Please understand that this system of visible arithmetic is not a substitute for ordinary arithmetic, but an auxiliary to it. Whether it is advisable to master it completely before beginning regular work with figures, or to carry on the two simultaneously, only experience can show.
However this may be, in teaching written arithmetic I would have the pupil make his own addition, subtraction, and multiplication tables, by the aid of countable things. Taking groups of six things—dots or grains of corn—the pupil finds the successive products of six by different factors and writes them down in order for himself. He thus knows exactly what the multiplication table means. About using it, I shall presently have more to say. The treatment of fractions in a visible way by dividing lines into parts is an extension of multiplication and division, and does not need development. I therefore pass on to the continuation of the method.
The next subject in order would be Ratio and Proportion. On my plan, the pupil reaches the first conceptions of this subject thru the eye by drawing a pair of lines of unequal length and then other pairs, shorter or longer, in the same ratio to each other. In this way he will see the equality of ratios to be independent of the special lengths of the lines. He can then be gradually exercised in forming for himself an idea of what a ratio means, or how equality of ratios is to be determined by multiplication or division. I would not have measurement with a rule applied, but only eye estimates. This, I may remark, is the general system by which I think we should begin in all cases. The reason for it is that in making eye estimates we depend more completely upon the eye conceptions than when we measure; but as soon as the conception is gained, we may proceed to measurement. Having got the idea of a proportion and the geometric mean, all this can be done without using figures or numbers. When the conception is well implanted, then proceed to numbers.
In connection with proportion would come graphical representation of all the quantities which enter into arithmetical problems. Take, as an example, questions of days' work in plowing a field. We draw a short vertical line to represent a man, or his power. From this line as a base we draw horizontal rectangle to represent the amount of land which the one man can plow in a day of ten hours. If we have several men, we add into one the lines representing them, and combine all the rectangles into one. Then we extend these rectangles to represent the days. To introduce the idea of compound proportion, we suppose a day of eight hours instead of ten hours making a rectangle shorter in proportion. I consider any problem in compound proportion solved when, and only when, the pupil is able to represent it graphically on this system. I am sure this process would be more interesting than that requiring the use of numbers.
The precise purpose of this course in visible arithmetic is so far from familiar that further enforcement of it may be necessary to its complete apprehension. It must be especially understood that exercises in formal reasoning do not enter into the plan. A power of visualization and giving a concrete embodiment to the abstract ideas are the fundamental points aimed at. If I should express the desire to have a pupil trained from the beginning in the mode of thought of the professed mathematician, I might meet the reply that this was expecting too much of the childish mind. Allow me, therefore, to put the requirement into a slightly different form. I wish the pupil trained from the beginning in the use of those helps to thought which the advanced mathematician finds necessary to his conceptions of the relations of quantities. If a mathematician has no clear conception of an abstract quantity how can we expect a child to have it? The mathematician conceives quantities by geometrical forms and the movements of imaginary visible points. Let us then train the child to represent the simple quantities he deals with by simple auxiliaries of the same kind, adapted to the state of his mind and to his special problems. What I wish him to use is not merely a tool, but a necessary help to thought. The visible arithmetic which I advocate bears the same relation to ordinary arithmetic that the geometric construction of complex variables does to the algebra of the mathematician.
Altho I have spoken of these graphic constructions as merely an auxiliary, I would, after denominate numbers are disposed of, be satisfied with the graphic representation of all solutions required. After this point, I would require very little numerical solution of problems, being satisfied when the pupil is able to construct a graphic representation of the solution. When he can draw proportional lines, explain discount by cutting off and adding fractions of a line to the line itself, and in general show that he can form a clear conception of the practical problems of arithmetic, I should consider that he
knew enough about it, so far as mere numbers are concerned. Everything beyond should be treated by algebraic methods.
Thus far, I have treated only of one main object of arithmetical teaching. But there is another purpose of a different kind, and that is facility in the use of numbers. The pupil must not only know the meaning of multiplication and division, and understand when each is required, but he must be able to cipher rapidly and correctly. My views of the best method of attaining this end are perhaps even more radical than those which I have already set forth. I think it can best be gained by short and frequent daily practice in the routine operations of the four fundamental rules, quite apart from the solution of problems. I would have something analogous to a daily five-minute run in the open air. The reiteration of simple problems after the pupil sees clearly how to conceive them is a waste of time. But this is not so with the exercises designed to secure facility. Leaving details to the teacher, I would outline some such plan as the following:
Let an entire class devote a few minutes every morning to reading or repeating aloud in chorus the addition, subtraction, or multiplication tables until it is ascertained that the large majority of the class has them well by heart. I should not make it a point to have them repeat the tables from memory alone, because I think the result is equally well attained by simply reading aloud. Another exercise would be that of adding columns of figures, following the method of the bank clerk, or of the astronomical computer. It would facilitate this to have the exercises printed on sheets beforehand. Twelve lines of figures would be a good number. The earlier exercises may begin with three in a line; when these are easily done, add columns of thousands, then tens of thousands, and so on. Do the same thing with exercises in multiplication and division. These may seem rather dull exercises, but we can easily add an element of interest by choosing some condiment of which a very little will suffice to flavor an otherwise long and tedious course. The mere act of repeating in chorus will give interest to the exercises. In addition, an element of interest will be given by noting from day to day the gradually diminish
ing time in which each pupil can complete his exercise and prove its correctness.
Thus far I have spoken only of methods of teaching. But I believe that, if the system which I advocate is intelligently pursued, it will be found practicable to greatly curtail the time spent in simple arithmetic, and thus rearrange the curriculum with the view of disposing of the subject of arithmetic and passing on to algebraic and geometric methods at a much earlier age than at present. In this connection attention may be invited to the report of the Committee of Ten, made in 1892, in which important changes in this direction were proposed. It must be admitted that in making such changes we shall be running counter to the ideas of the general public. When it is proposed to omit commercial and so-called advanced arithmetic from the school course, the reply will be that we are considering only the requirements of pupils preparing for a college course, and that business and commercial arithmetic is a prime necessity with the masses. There being in our country no body of men more influential than that here assembled in wisely directing public opinion on this subject, I beg leave to point out the fallacy in this plausible view. The experience of directors in our great enterprises shows that the best business mathematician is not the one who has taken a course in commercial arithmetic, but who has the best understanding of numbers and quantity in general, obtained by the more advanced courses of a mathematical character. A problein of practical business is best taken up by one who understands it. On the purely practical side that understanding can be better gained in one day by actual experience than by any amount of arithmetic in a course subject to all the drawbacks of being treated as an abstraction.
I once saw an interesting example of this. It was in connection with a building association on an old-fashioned plan, which, I fear, has gone out of vogue. It was a mutual benefit association in which accumulating results of monthly payments thru a term of years were to be equitably divided month by month among the members desiring advances. The mathematical principles involved, if investigated in detail, were so