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effect upon the instructor in such courses is to put him on his mettle and to induce him to give out his best, for he is conscious that his composite public is already possessed of a wider standard of judgment than is possible to the undergraduates of the home institution who have felt only home influence.
The presence of graduate work elevates the whole tone of instruction in the institution—in any institution—by making it imperatively necessary to have in the corps of instruction of the university, not merely the teacher who teaches, but much more largely than is either possible or useful in the college the teacher who also investigates. And in just this way it widens the opportunity of the undergraduate student by giving him, in addition to the instruction which under the conditions of the college curriculum is of necessity first of all an exposition of accumulated knowledge, more surely than before a premonition, at least, of the spirit and method of investigation which, thru its own discoveries, is adding to the ultimate total of knowledge. It has not only increased in quantity and intensified in quality the work performed by the undergraduate student, but by this demonstration of the meaning of receptive scholarship on the one hand, and of productive scholarship, on the other, it has immeasurably broadened his horizon and pointed out a way for future usefulness.
The students of the non-professional graduate faculties who are pursuing their chosen lines of advanced study and investigation and the teachers who, in directing them, are continually breaking new ground,“ represent," as President Butler writes in the annual report of 1903," the very heart of the University.” “In the Schools of Philosophy, Political Science, and Pure Science," he continues, " students and teachers are associated together in pushing forward the boundaries of human knowledge and in increasing the measure of human appreciation in some way, great or small. It is this spirit of investigation, of the scholarship which produces and not merely relates, that gives to these schools their tone, and to the university as a whole its best inspiration.”
WILLIAM H. CARPENTER COLUMBIA UNIVERSITY
METHODS OF TEACHING ARITHMETIC 1
As far back as October, 1892, I advocated in the EDUCATIONAL REVIEW (IV: 277-286), a system of implanting elementary mathematical ideas in the minds of pupils, with especial reference to the fundamental conceptions of number and quantity. While eminent masters in education had repeatedly pointed out the advantages to be gained by teaching arithmetic in a concrete form, their views had not up to that time taken effective practical form, at least in our own country. The purpose of my paper was to set forth a system of visible and graphical representation of arithmetical operations, illustrated by appropriate exercises, by which the advantage of apprehending mathematical ideas in a concrete form should be gained. I am not aware that this utterance excited any attention at the time, nor do I know whether it was a factor in the recent tendency of arithmetical teaching in the direction which it advocated. However this may be, the whole trend of recent experience and discussion among practical teachers is in the direction of the ideas advocated in the paper referred to; and a system practically identical with the one there found is now embodied in several recent arithmetics, even to the extent of proposing problems and exercises almost identical with those suggested in the paper. This fact encourages me to present a more complete development of the results of thought and observation during the thirteen years which have elapsed since my former utterance. To do this completely it is necessary to commence at the foundation and inquire into the main purposes which we should have in view in arithmetical teaching.
Without going into details, a very little thought will, I think, make it clear that the main end of mathematical teaching—we
Read before the Department of Superintendence of the National Educational Association at Louisville, Ky., February 28, 1906.
might say of teaching generally—is to store the mind with clear conceptions of things and their relations. In the case of elementary arithmetic, the things we first deal with are numbers. It follows that a clear conception of numbers and their relations is the end toward which our teaching should be directed. I think every reader who has carefully studied the mind of the apparently dull pupil will agree that the real difficulty is to give him an insight into the nature of the problem he is to solve. He may be able to repeat the words; but you find that they do not make a sufficiently definite impression on his mind. Clear and accurate conceptions of the relations of numbers are therefore to be generated.
To show what we mean by clear conceptions of number, we must stray into the field of psychology. We may conceive of the brain of a man as a microcosm containing within its narrow limits all that the individual knows of the structure, laws, and history of the entire universe. There are two universes, the microcosm within us and the macrocosm without us. The success of the individual, not only in all the applications of science, but in every branch of endeavour, depends on the accuracy and completeness with which the processes at play in the subject with which he is dealing are represented by corresponding processes in his own mind.
Admitting that everything known of external nature has its image in the mind of the man who knows it, I cannot but regard it as a defect in psychological nomenclature that there is no one general term used to express this mental image of an external object and nothing else. Passing over the question of nomenclature, let me make clear the thought. To take a familiar example, we all have an idea of the house in which we live. We can think of the building and the arrangement of its rooms, when it is out of sight, as if we had a picture of it in our mind's eye. This picture is not a flat plan, but rather a model embodying the arrangement of all the rooms in the house. What is true of the house is true of all human knowledge and of its applications. The engineer can in his mind erect bridges in which the actions of stress and strain shall correspond to those in the actual bridge; in the mind of the historian, events of human history and the motives which guided them may be repeated at any moment; in the mind of the chemist, compounds react as in the laboratory, and so on thru every branch of knowledge.
From this point of view my main contention is that the first and great object in training the growing child in arithmetic is to store his mind with clear and accurate conceptions of numbers, magnitudes, and their mutual relations, which he shall be able to apply with readiness in any actual case that may arise. That I have elaborated this point so fully is due to the fact that it should never be allowed to drop out of sight in our teaching. The latter must be arranged from the beginning with this great end in view.
Granting this, the next question in order is that of method. Here psychology can supply us with a guiding rule. However abstract may be the ideas which we wish to implant, they must originate in sensible objects. But they must not stop there, because generalization, conscious or unconscious, is to be aimed at from the beginning. Let me illustrate my meaning by taking the number 10 as an example. I think psychologists will agree that there is no such thing in the human mind as a conception of the number 10 otherwise than as a quality characterizing 10 distinct objects. A written or verbal symbol may be used for the number, but this is not a conception of it. The point is, that the word or symbol being pronounced or shown, the pupil should at once conceive of 10 objects as distinct from either 9 or 11, and should be able to handle that conception in all the ways that it can be handled.
Here, there is an obvious advantage in selecting such objects as have the least number of qualities to distract the attention from the fundamental idea of number. Hence, I prefer that the counting should be made upon small dots, circles, or other objects with few qualities than upon the more interesting objects which are met with in every-day life. In this suggestion I may seem to run counter to views which are entertained by very high authorities in education. There is, I admit, a very strong argument in favor of the view that the principles of arithmetic are best mastered when the child is taught to consider them as growing out of the problems that actually confront him in his daily walks. I fully agree that the practice thus suggested is one that should be carried out, but we must not depend wholly upon it. Perhaps I am a little old-fashioned, but I would not abandon the idea of applying the pupil's nose to the grindstone. I have no objection to the grindstone being interesting, and certainly do not wish to make it painful, but I want some drill in thinking of numbers and their relations as dissociated from the actual objects concerned. Just as rapidly as this power is attained in each and every branch, I am willing to see the interesting substituted for the instructive.
The idea of arranging subjects in order and completing one before passing to another is plausible. But experience shows that it has its limitations. The great principle which experience especially enforces is the educational value of frequent reiteration of very short and easy lessons. This is one of the main features of the system I am trying to develop.
Now, as my object is a purely practical one, it is necessary to have some idea, however brief, of the method by which the purpose in view can be most readily attained. The system I advocate may be called visible arithmetic. Taking up subjects in much the order of the traditional arithmetic, the first would be Numeration. Visible numeration consists in counting and arranging objects in tens and in powers of ten. At the earliest age where simple arithmetic can be commenced, whether in the kindergarten or the primary class in school, I should teach the child to count and arrange things in tens; then to arrange real or imaginary tens in hundreds, and so on. In accordance with the general principle which I have laid down, I would begin with rows, each comprising ten dots, or kernels of corn, and teach the counting thru ten such rows, making one hundred in all. We could then imagine the results of laying successive 100's in flat layers on top of each other, thus getting the idea of multiples of 100 up to 1000.
It would be psychologically interesting to see whether, in this way, we could plant in the mind what the psychologists call a "number-form” in a better shape than it commonly takes. I suppose we all have vaguely in mind from infancy a certain