Reprinted with permission of the Mathematical Association of America
In recent months I have been present at many informal discussions in regard to the required work of a college course. The background of such discussions has been the woman's college, and the aim of all the participants has been to find the best method of training the woman student for her part in the work of the world.
Some argue that with the large amount of required work in college a student has too little time in which to sample new courses or to devote herself to a subject for which she shows an aptitude. Their demand is: "Lessen the required work." Others argue that the required work of a college course is seldom more than sufficient to give the solid foundation of general culture upon which the structure of specialized subjects may be safely and permanently reared.
Some of the arguments of the opponents of required mathematics have a curiously familiar sound. To be sure they are not put as crudely as in the world outside the college. There no one hesitates to ask: "What is the use of mathematics for a girl? She will never need more than the elements of arithmetic." The opponents of the subject in a woman's college, especially if they are themselves women, are somewhat wary of asking that time-honored question. They know that they are supposed to claim the same stiff intellectual training for women as for men, so they avoid stating the question in bald terms of sex. Instead they say: "Every student who enters our college has already had several years of mathematics. She has enough for all practical purposes. [This is nothing but the old query revised.] She has already found out whether or not she has an aptitude for the subject. If she cares for it, by all means let her elect it, but if she has no aptitude for the subject, let her take something that is really worth her while instead of wasting her time on a subject she will never use."
My belief is that in spite of her school training, however thorough, the student needs some college mathematics if she is to have an education that will send her out into life with the best general equipment. Mathematics as taught in college is viewed from an angle different from that used in the schoolroom. This statement does not apply to solid geometry which is only an extension of the plane geometry of the school, and which we hope some day to see put back in its proper place in connection with plane geometry. Take, however, college algebra and trigonometry. Both of these subjects make use of material in the way of ideas and methods that the student has already worked with in school, but this material is handled in a very different way. In her algebra the schoolgirl is concerned almost entirely with processes. She needs but little theory. This little is sometimes explained to her, sometimes she has to remember and reproduce it; but even in the latter case the average student seems to have acquired little grasp of the underlying principles. The consequence is that almost every college freshman has a definite idea of logical reasoning as connected with the subject of geometry, but has little idea of it in connection with any other part of mathematics.
In college the freshman, while once more dealing with symbols and processes that were familiar in school, is now concerned with them from the standpoint of logical combinations. She is applying to them the methods that she had thought confined to geometry, deducing the laws that rule in the subject of her study, and expressing these laws in exact mathematical language. Unless much emphasis is laid upon this side of mathematics in the freshman classes, the claims of its enemies would seem to me to have some foundation. The application of the theory to special problems is necessary as making the subject more vivid and as preparing the student for the application of abstract reasoning to all manner of problems stated in advance; but the true value of the course lies in the training it gives in applying logical processes to the mathematical concepts already familiar from school days.
The average student may not like to find that the mechanical work of her school algebra is replaced in college by this demand upon her reasoning powers, yet she is ordinarily able to master the required amount of freshman mathematics in spite of the extra difficulty. On the other hand there are always a few students to whom such a treatment of mathematics offers an insuperable difficulty,—students who seem to have no idea of logical sequence. In their statement of a geometrical construction no attention is paid to the logical order in which the lines must be drawn, in their statement of a proof the effect precedes the cause, and their minds are so constructed that it seems impossible to convince them of error—one order seems to them as good as another.
This latter class of students is always cited by our opponents as the strongest argument for dispensing with a general requirement of mathematics. "Why torture such a student," they say, "with a subject for which she has no fitness? Why should she not take the subjects for which she shows some aptitude, and let mathematics severely alone?" They usually omit to name the college subjects that do not require at least some modicum of reasoning ability. My feeling, on the contrary, is that no matter what other subjects such a student has to omit, mathematics is one subject absolutely essential to her training. Such a student needs to develop her reasoning powers, and freshman mathematics gives her the best field for practice. Other subjects also require logical ability, but often the logical framework is so obscured by the newness of the material, the unfamiliarity of the nomenclature, and the large number of strange concepts, that the value of the subject from the standpoint of logical training is quite lost.
As for the examples that the opposers of required mathematics quote from time to time of women brilliant in other lines of college work who found themselves totally unable to do even enough work in freshman mathematics to gain a passing grade, I cannot deny that such women may exist, but I wish to record my deep conviction that the majority of these cases have been diagnosed incorrectly. I have seen so many students of mediocre abilities fight their way through the difficulties of required mathematics by sheer common sense and will power, that I am sure that most of the cases cited so solemnly are cases of "I will not" and not of "I cannot." Because they did not want to master a difficult subject, they were willing to profess incompetence in order to get their own way.
Here the question of sex enters in. Parents and guardians who would suffer keen mortification if the boy for whose education they are responsible were in danger of being rejected by his college because of his failure in required mathematics will condone any shortcomings of the girl in that line with a deprecating, "You know that one does not think so much of a failure in mathematics for a girl." But if sex must be considered in this matter, why not consider it from this other standpoint, namely, that the woman is prone to look at everything from the personal side? Her own feelings and her background, or lack of background, color the medium through which she views such subjects as history and literature, and affect her judgments of the facts. The personality of her instructor is a factor in inclining her either to believe or disbelieve his interpretations of the theories of economics or philosophy. Of course there are fundamental laws in all these subjects so well established that the personal element cannot enter into their consideration, but there is also a large body of conclusions from these laws, and it is these conclusions from laws that are sometimes only partially understood that are now in question. Herein lies a great advantage of mathematics; it furnishes the woman student with a subject in which the validity of the conclusions drawn from its laws can easily be tested, and in which the personality of the instructor and the bias of the student can play no part.
The foes of the mathematics requirement then say: "Suppose we grant that our students need training in reasoning, and that as women they need especially training in reasoning upon an impersonal subject. Why not then require a course in a science that shall be the equivalent of the mathematics course in these two respects? Any science gives a student the opportunity to acquire the habit of assembling data, rejecting the extraneous elements, and forming the fitting conclusion. No one can bring any personal prejudice into the interpretation of the phenomena that would be discussed in a required course in any established science." Such a suggestion seems quite to overlook the fact that the only sciences that furnish a training at all equivalent to that of mathematics are those that have mathematics as their foundation. Without a preliminary training in mathematics that almost necessitates the inclusion of trigonometry the most rigorous of sciences can only be treated from the more or less popular point of view. The training so acquired is no doubt valuable, but it cannot satisfactorily replace the required work in mathematics.
Another argument that is often brought against this college requirement is that it gives a subject in which the students have already had several years' experience an unfair advantage over others that are not begun until after college is entered, for each year of required work in a familiar subject postpones the opportunity for them to become acquainted with new subjects in which they may later desire to specialize. This argument does not take into account the fact of which I have already spoken, that even in freshman mathematics the methods are used that must be used in any further study of the subject, so making a break between the character of the work in school and that of the work in college. There are sometimes freshmen whose work in all school studies has been so good that no one subject seems to stand out as more particularly suited to them than another, who first realize that mathematics is the subject that they want for special work when they become acquainted with it afresh in their required work. Without this requirement these students will probably be lost to the subject. They may of course elect mathematics, but, under the false impression that the rather mechanical methods of mathematical work already familiar to them are to be continued in college, they usually prefer entering upon some new subject. Given the recognized attractions of the new and untried, mathematics, if not required, will be at a disadvantage as compared with other subjects.
In addition to this unfortunate effect upon the department of mathematics there are other deleterious effects upon both school and college that seem to me bound to follow the dropping of mathematics from our required work. With the best of intentions on the part of the high school teachers the grade of work is lower in a subject that is necessary for entrance to college but that is not to be tested in the college class room than it is in a subject that will be so tested. In this I speak from experience. I once saw the effect upon the preparation in algebra, when the college with which I was connected eliminated college algebra from the required work, retaining solid geometry and trigonometry. The school work in geometry was kept up to the standard, for that work was expected to have its classroom test. The college work in trigonometry, however, revealed very clearly the fact that, while the required subjects of algebra had been nominally studied in school, the work had been done very superficially. Furthermore there will be a disastrous change in the quality of the teaching, in so far as it is done by women, if more and more women go out to teach in the schools who have had no mathematics beyond that of their high school course. The lack of background is at least as serious a fault in the teaching of mathematics as in the teaching of any other subject.
In the college itself there will probably be a marked effect upon the science departments. If the student is permitted an unfettered choice of her required science—and no one seems to hesitate a moment as to the necessity of such a requirement—the non-mathematical sciences (if such sciences truly exist) will be overrun with more students than can be handled easily, while the sciences known as mathematical will have only the few students who have brought from their school days a love of mathematics and no dread of its symbols. If the curriculum committee meets this situation by requiring the choice of one of the mathematical sciences, even then there will be a difficulty because of the poor equipment in mathematics possessed by the students. Under such circumstances the science requiring the least amount of mathematical knowledge will naturally be given the preference, and the instructors in even the most mathematical of the sciences will have to confine themselves to a rather popular treatment, if they wish to have a fair proportion of the students elect their subject. In any case much of the value of these sciences as aids to exact thinking will be lost. Of course, for any further work in the selected science some mathematics is necessary, and under the supposed arrangement the student must take in her maturer years in college the fundamental work that she now acquires in her freshman year.
With the two apparently contradictory tendencies at present noticeable—one, to minimize for women even in the science courses in college the necessity of any mathematical training beyond that of the high school course; the other, to encourage these same students to place more and more emphasis upon their work in science, especially in the line of laboratory research,—it is evident that the majority of women workers in science will soon be forced to limit themselves to those fields in science that can be cultivated by means of the very simplest mathematical tools. These fields may be wide and they may be fertile, but by permitting this limitation women are denying to themselves the equality of opportunity with men that has been won for them at such a cost by the pioneers in the struggle for the right of women to share in the higher education.