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The best method of briefly treating this important subject is to approach it historically and see how Mathematics came to be a part of girls' education. The facts are by no means obscure, since they have occurred within the lifetime of many who are now teaching in girls' schools.

Fifty or sixty years ago, as we all know, girls did not learn. Mathematics, and if we go back to the days of Queen Elizabeth and Lady Jane Grey, when young women of the upper classes did receive a good education, we find classics, not Mathematics, as the staple. Indeed the introduction of Mathematics as an ordinary and compulsory subject (not an extra) in all boys' schools is comparatively recent. The importance of the subject has grown with the developments of science, and the value in active life of Applied Mathematics.

In the eighteenth century and earlier, girls were of course taught very elementary arithmetic for the practical purposes of housekeeping, just as the boy who was going into business was taught in a Writing or Commercial School separate from the old Grammar School. That this arithmetic teaching was purely for practical purposes is clear from the absence of it in the curriculum of those young gentlewomen who could rely, in their future life, on having a housekeeper or land steward to manage their affairs.*

Emphasis on Mathematics as an essential part of a girl's education is one of the most characteristic features of the reform movement that followed, in the "seventies," on the publication of the Taunton Commission Report. There, it will be remembered, were exposed the inaccuracy and superficiality of the current methods, and it was thought that the best way to reform these evils was to improve and develop the teaching of arithmetic in the girls' schools, and to introduce geometry and algebra. These subjects, it was thought, would make girls accurate and thorough; would teach them to reason; would develop their intelligence; and would be the cure for the evils which the Taunton Commission revealed. Some of the best schools, which were at that time feeling their way towards an improved condition

*In the report of the Schools' Inquiry Commission (Lord Taunton), Vol. IX., p. 793, we find quoted the statement of a Schoolmistress as to the indifference of parents even as late as 1865. "If I report that another (girl) does not try to improve herself in Arithmetic, her mother says 'It really doesn't matter; her husband will be able to do all her accounts for her, you know.'"

of things, eagerly seized every opportunity of obtaining good teachers of Mathematics-men or exceptional women. Qualified women teachers of Mathematics were available at a very early stage, since it was at the University of Cambridge that the pioneers established important women's Colleges as early as 1871.

The example of Mrs. Somerville undoubtedly influenced public opinion, and those who were in the movement themselves will remember how Mathematics for girls became a kind of test subject, which could be used to prove that women possessed intellectual powers not inferior to those of men and that, therefore, if girls could take good places in mathematical examinations, opportunities of higher study at the Universities and elsewhere should be afforded to their sex. Such opportunities were given. At the same time, in the examinations which were compulsory for admittance to University study, Mathematics had taken a place side by side with classics, and so all the girls who wished to go to College were obliged to study geometry and algebra. There was and is a movement against compulsory Latin and Greek in University Entrance Examinations for boys; this movement has prevented classics being made as much a matter of course in the modern curriculum for girls as Mathematics is. Latin is studied by girls with success; but not by all girls. Mathematics has become inevitable.

Of late years a reaction has shown itself. The schools have now some practical experience of what teaching Mathematics to Matriculation standard involves for the average girl. The old fashioned faculty psychology can no longer be unquestionably quoted in an argument to the effect that "Geometry trains the logical faculties"; which are, it is thought, deficient in women. At the same time there has been a revolt, led by the engineers, against the actual methods and material of mathematical teaching in the schools which have been found too academic and unpractical for the needs of real life, that is, the need of boys. Euclid is dethroned. Practical Mathematics of various kinds is introduced, and school Mathematics is more definitely directed towards the demands of careers which do not concern girls. At the same time we are beginning to believe that the curriculum should be differentiated for different types of pupils: vocational education is urged, and even where the truth of it is not accepted, academic studies are so followed in school as to have a direct bearing on future occupations.

These things directly affect the place of Mathematics in the education of girls. It is closely related to the characteristic industries and activities of men, engineering in all its forms, war, navigation, building, land surveying, finance, and some of the higher forms of commerce. It has no such relation to the characteristic activities of women: it is not required for any vocation which the ordinary woman will follow. Its only value is as a mental discipline, a point well brought out in the

article on Mathematics by Miss Sheldon in the recently published book of the Head Mistresses' Association, "Public Schools for Girls." The difficulty and dislike felt by many girls for the subject are probably due to this fact. "What is the use of it"? they say. It is divorced from reality, from experience: it makes no appeal to their special instincts as women. Biology, on the other hand, does make such an appeal, so that botany and (wherever it is taught) zoology are very successful subjects in girls' education. The same is true of history and literature, particularly English literature: subjects traditional in girls' education, and directly related to the emotional and altruistic interests of women.

Another matter must also be considered,--the crowding of the curriculum which has become so serious in boys' schools, and which is far more serious with girls. They are physically not so strong at the secondary school age; many of them have domestic duties which do not fall on the boys; many are expected to study music, and to give time to social duties in a fashion that is not expected of their brothers. Thus the girl with less physical energy, is required under existing systems to do more, not less, than the boy. This cannot go on something must be given up. Since Mathematics was originally arbitrarily introduced into the girls' schools, and has no direct relation to the future activities of girls' lives, it is the opinion of many, including the present writer, that the amount of Mathematics demanded of girls must be materially reduced, and that it should be no longer compulsory, as it is at present, in all examinations of matriculation standard.

At all events, whether we take this view or not, it is clear that the problem must be reconsidered from the beginning. The present writer would venture to put forward her contribution. to the discussion as the result of more than thirty-five years' experience as a student and teacher of Mathematics, and as one who is responsible for the curriculum of a large girls' school. It appears to her that in this matter girls fall into three groups, and that to state this briefly in detail is the best way of formulating what her experience leads her to think.

(1) There is a small but very important group of girls and women who have a real taste for Mathematics; who enjoy this study; who find it comparatively easy, and who, it is not too much to say, revel in it, as a delight, as a new world of thought and power. Its highly abstract character appeals to those faculties which make women delight in the study of philosophy, and also though this must be said with diffidence-to those more spiritual qualities that make religion so much to women. For girls of this type Mathematics is an admirable training. Some of them possess powers which may enable them to do research. Mrs. Somerville presumably was one of these.

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Others have gifts for administration and organisation, and the sort of training that they obtain by studying for such examinations as the Mathematical Tripos at Cambridge makes everything of the kind that they may have to do in after life easier. Mathematics, in such cases, has a distinct relation to the occupations of later life.

Now the opening of opportunities of mathematical study at school and college to such women as these has been of the highest importance. It has made them happier as human beings, and more useful as citizens, and it would be a loss and an injury if anything were done to prevent such women from enjoying in the future the opportunities they have had during the last forty years. What was won for us by the pioneers we

can never consent to surrender. Those mathematical studies must be preserved in the schools and, a fortiori, in the colleges for the few, who will return to the community in adult life more valuable service of one kind or another, because of their mathematical education.

(2) Secondly, there is another small group, rather larger than the first, but at the other end of the scale, of girls and women who cannot do Mathematics at all, or, if they are able intellectually, do it with a large and wasteful expenditure of power. To get such girls through compulsory mathematical examinations on their way to College is like making diamonds in a laboratory. You secure the results, but they are wretchedly small, and cost much more than they are worth. There are able boys who find Mathematics equally difficult; certainly Lord Macaulay found the subject repulsive. But this paper is not concerned with the needs and difficulties of boys.

It may be thought that the numbers in this group are so small that it is not worth while, for their sakes, to limit the compulsory Mathematics for examinations. Everyone must form his own opinion in this matter, but the writer has seen enough of hard cases coming under this rule, to make her an ardent advocate for the removal of the mathematical barrier in College Admission Examinations.

(3) In the third group comes the great bulk of girls who can and do study Mathematics with some satisfaction to their teachers and examiners, if not to themselves, and who pass their examinations more or less successfully. About this there is room for considerable variety of opinion. Undoubtedly mathematical study is of value for the average girl; it does train accuracy and thought, power of grasp, and a certain sense of form. The value of the logical training is more doubtful. What is really wanted for girls is the training in logic as applied to concrete things, and this is found perhaps more usefully in the study of history or of the new geography, as well as in science.

Undoubtedly the average girl finds Mathematics harder than the average boy, and is less successful in it, in spite of her greater industry and earnestness about her work. The testimony

of teachers of mixed classes, and of the examination lists, is quite clear on this point; it is really a commonplace. When an equal degree of success is secured by the average girl it has~ cost a great deal more. This point should be emphasised.There is only a certain amount of power available in school, just as an engine will only develop a certain number of units of work, since a pound of coal can only give out a certain number of units of heat. It is found in practice by those who teach girls that the mathematical work takes up a greater number of units of energy than anything else. Girls with less physicalenergy, or vigorous girls at times when their energy stock is low, find mathematical work suffers first; conversely, to take the subject out of a girl's course of study, when she is over pressed, gives relief at once. Again, any serious or even trivial occurrences which may tell against effective school work, such as, heat, fog, bad ventilation, slackness of discipline, emotional stress, all tell on the Mathematics first; and, since girls are on the whole naturally more susceptible to physical and emotional disturbance, they are much more likely not to do themselves justice in mathematical examinations, and to fail even when they deserve to pass.

It may be said of course that a subject is valuable in proportion as it is difficult, and that the training in courage and perseverance gained by going through with a difficult matter against obstacles, is of the highest value. So it is, but even that may be bought too dearly, and it is bought too dearly by many a conscientious girl. One cannot but feel that the time and energy given to mathematical study, by the average girl who matriculates, could be spent to better purpose in other ways. Has she, for her four or five years' study with a lesson every day, and homework every day, gained anything that is adequate to the expenditure that has been made?

To this question some authorities will answer "Yes," though they admit that the progress made in actual knowledge and skill is not at first sight adequate. They hold that the girl has won something by her striving; has gained in reasoning power and intelligence by the hours she has spent in grappling with problems she hardly could solve. She will be a more useful woman; one less likely to make mistakes; more amenable to right reason, because she has persevered in her mathematical studies up to Matriculation standard, poor as her result seems to the mathematician. Such a belief is of faith, rather than susceptible of definite demonstration, but it is held strongly by experienced teachers, whose opinion is entitled to respect. Miss Sheldon, a mathematical specialist, takes this view, in the article already referred to, recommending that the average girl should take Mathematics to the Matriculation standard (approximately) but not beyond; she should not, e.g., take trigonometry.

It is also the view of certain educated men, not teachers or mathematicians, that we must avoid any serious divergence

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