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complex that only a professed mathematician would be able to construct or apprehend their theory. Yet, when the problem was faced as an actual one, the whole process was gone thru with by every-day business men and laborers without the slightest difficulty. Not one of these could have explained the process to a mathematician, but he did it correctly when the concrete case was before him.
We should also try to dispel the current notion that the use of algebraic symbols belongs to a more advanced stage of study than arithmetic. We have advanced a little in the right direction since the time when the signs + and — were considered as belonging only to algebra, and therefore not used in arithmetic. I trust that we shall go further and not only introduce symbolic operations, but the simpler literal symbols at a very elementary stage in the course. Careful examination will show that in its simplest form, algebraic operations are even simpler than arithmetical ones.
Having suggested all these innovations, allow me to sum up in briefest compass the practical conclusions which I draw froin a survey of the field :
I. I do not propose that we shall try to train a pupil in abstract mathematical reasoning until he reaches the stage where pure geometry can be advantageously taken up. But, from the very beginning, he should be trained in the faculty of mental insight. This can be done by problems like this, to be answered by thought without making a drawing. Of three houses, A, B, and C, B is 100 meters north of A and C is 100 meters west of B. What is the direction of C from A, and about what would you suppose its distance to be?
II. I regard time spent in the schoolroom poring over problems and trying, perhaps vainly, to see how they are to be solved, as time wasted. Much waste in this way is indeed unavoidable, but our policy should be to reduce it to a minimum by explaining the problem whenever the pupil does not readily see into it for himself.
III. Of course we should train the mind in seeing how to attack a problem. The objection may be made that whenever we help the pupil in this respect, we diminish his power of helping himself. I admit this to a certain extent, but my solution is that we should devise such problems that the course of thought they require can be seen without spending time in vain efforts. Please let me cite once more the analogy to outdoor exercise. We should all agree that if we coupled the exercise of taking an outdoor run with the requirement of finding out at every few steps what path was to be followed, and put an end to the exercise if this right path could not be found, it would materially detract from the good of the exercise. Let us, then, promote facility in calculation by exercising the pupil in purely straight-ahead work, without requiring him to stop and think what is to be done next.
IV. I have found in my own experience that words are as well and more easily memorized by repeated readings than by the same amount of repetition from memory. If this principle is correct, then we lose nothing by having a multiplication table before the pupil every time he repeats it, so that he shall read instead of memorize it. I do not present this view as a demonstrated fact, but as one worthy of being tested.
V. The plausible system of learning one thing thoroly before proceeding to another, and taking things up in their logical order only, should be abandoned. Let us train the pupil as rapidly as possible in the higher forms of thought and not be afraid of his having a little smattering of advanced subjects before they are reached in regular course. Let us remember that thoroness of understanding.is a slow growth, in which unconscious cerebration plays an important part, and leave it to be slowly acquired. A teacher aiming at thoroness might have kept Cayley or Sylvester working half his life on problems of advanced arithmetic without reaching his standard of thoroness. Let us promote the development of higher methods in the earlier stages by introducing algebraic operations immediately after the four fundamental rules.
VI. Separate the actual exercises for acquiring facility in arithmetical operations from the solving of arithmetical problems. If I am right, it will be more conducive to progress to be satisfied with the graphic representations of problems, without the arithmetical operations of solution, than by actually going over the solution itself.
VII. If I am not straying too far from my theme, I may devote one moment to the extension of the ideas I have advocated to the mensurational side of geometry and physics. As a part of the arithmetical course, let us teach geometrical conceptions, the aim being a correct apprehension of lines, lengths, angles, areas, and volumes as they actually exist in the objects around us, and are to be conceived in thought when these objects are out of sight. Valuable exercises in this respect will be endeavors to estimate a result in advance of calculating it. If a freight car is the subject of measurement, either in thought, or by a picture, let the pupils form the best judgment they can as to the number of cubic meters or of tons of water the car will hold before making the computation. Practice in estimating lengths, angles and magnitudes generally, by the eye, should be part of the elementary course.
I may conclude with a brief statement of what we may hope for by the system, of which the general principles have been outlined. It is, in the first place, the more rapid command of the nature of all those problems which are to be solved by numbers and in the second place facility in performing the necessary operations. To put the system into successful operation requires a well-chosen collection of problems, explanations, and exercises quite different from those with which we are familiar. It also requires, both in the designer of the exercises and in the teacher, a clear idea of the end to be kept in view, and of the best means of attaining it in each special case. Here, my own part must end with the submission of my ideas to the most conipetent authorities represented in the present assemblage, hoping that the plan will be deemed worthy, at least, of the test of a fair trial.
SIMON NEWCOMB WASHINGTON, D. C.
THE EDUCATION OF WOMEN 1
Woman's claims for higher education have been heard, and she is not only availing herself of present opportunities, but is seeking fresh ones. Oxford and Cambridge date back to the twelfth century, but the founding of Newnham was in 1875. Harvard was founded in 1636, but it was not until 1894 that Radcliffe granted degrees to women. Yale began its beneficent career in 1701 and Princeton in 1746, but Mt. Holyoke Seminary did not open its doors until 1837, and college work was not established there until 1888. Vassar's noble work for women dates from 1865, while Barnard is comparatively a child among the women's colleges, since it is only fifteen years old. Wellesley and Smith began college work in the same year (1875), while Bryn Mawr began a decade later. Whatever advantages women have enjoyed from these and other institutions of learning, have been won in the face of prejudice and deep-rooted and time-stained opposition.
Now, it would seem to appear, after an examination of the curriculums of men's colleges and women's colleges, that the latter have been patterned too closely after the model established for the education of men. Students, faculties, and trustees in institutions for
for the training of young women have been too prone to copy the newest departures in one or more institutions for males. Rarely has there been any thought of adjustment to sex. The idea of the advisability and desirability of a distinct line of differentiation in the making of the courses of study does not seem to have obtained, or if it has obtained, the necessary means and opportunities were not at hand for carrying out varying courses of study. The notion has seemed to be too generally prevalent that whatever was good for a young man
1 From an Address delivered at the Convocation of Adelphi College, Brooklyn, N. Y.
to learn was equally good for a young woman; that whatever studies the young man might successfully pursue, a young woman might pursue with equal success. Mathematics and the · sciences based upon mathematical foundations are to be found in women's colleges. Is there no indication that a distinct line of differentiation should run between the two courses of study for each? Has nature endowed both man and woman with equal gifts of heart and mind and with equal richness of gifts? Has nature blazed a path which each should follow in coming into the possession of his or her powers? If there be a natural line of division, then we fly in the face of nature if we give the same course of study to both. If nature has given us a clew and has indicated the desirability of a different kind of training for woman, then do we not unsex her when we attempt to force upon her the same education which fits a man for the struggle of life? Shall we attempt to make her a mannish woman? The best training for a young woman, it has been found, is not the old rigid college course which has been found unfit for young men and which has been discarded for the elective system. Most college men look forward to getting professional training as few women do. Women take up higher education because they like it; men, because their careers depend upon it. To the man the studies are more objective and prepare him directly to face the world as it is.
Let us keep clearly in mind in examining this question that there is a womanly type of woman. There are ladylike men; there are masculine women. Each variety is a differentiation from type. We must hold clearly in mind the type of woman we are to educate. We must establish an ideal which we are to attempt to realize. This ideal must be established in the light of future usefulness. It must be made with an entire life's work in view. It must be made for a sex, or at least for that part of the sex mentally equipped for higher education, and not for the choice few who are unusually gifted. It must be made, not for the future authoress, nor musician, nor painter, nor teacher, nor manager of a business concern. Higher education will enable all these classes to realize them