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ellectual interests and standards of men and of is women are by nature more intuitive and ir school training should supply the harder, more ement; should give balance, and steadiness; should

k, if not as men do, at any rate so that they men's way of thinking. This is of course a sound only question is, how much mathematical pred for this end. It is a matter of syllabus,

would suggest, as a solution of the problem, that on be compulsory in girls' schools for say a euse, from twelve to fifteen, but that it should e taken up to Matriculation standard. Anyone Mahematics knows that the subject does not at ses regularly in difficulty, like the inclined cv books. It is rather like a great mountain, with very here and there, and with long slopes Most of us find that og sy going in between. coco ore of these precipices somewhere or ve never get up it. The average school e meervice that she cannot climb (without

assistance from her teacher), generally per Fourth or Lower Fifth Form. It edergy and time to let her go no further. ere recommended would include good netry, and simple algebra of the kind Royal Naval College at Osborne, are in this series-for the ordinary ach of the elaborate manipulation ang in school is quite useless to the

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is a real part of education. It is not an easy subject, and, taught with constant reference to the actual sounds on the voice or an instrument, is found to be an admirable intellectual training. Most girls have to learn the piano: many have a real taste for music, and if they took harmony in school instead of algebra, it would be much more real to them, since it would be related to something in their daily life, and in their thought and interest. The study of harmony, in a serious and definite fashion up to a good Matriculation standard, would not only train a girl's intelligence and reasoning power, thus doing for her what Mathematics does: it would also have a valuable effect on her musical taste, and would enable her to appreciate better music, and to enjoy music in a higher way.

For the exceptional girls, as we have said, more advanced mathematical study would be retained in the school curriculum and in the Universities, just as it is now, though in College Scholarship competitions more use might be made of the school record. But in the case of these exceptional girls, who are making Mathematics their main study, there is one serious danger to be guarded against,-narrowness, hardness, ossification. Women lead narrower lives than men, and follow with greater intensity and rigidity any purpose they have set before themselves. Thus a young woman at College, who is following a mathematical honours course, is much more likely to be hardened and narrowed by it than is her brother. In her leisure she will play games, but will hardly have enough surplus intellectual vigour spontaneously to read literature, or to talk politics and philosophy. If these women are to do all they should later on as teachers, administrators, organisers, they must have in their College life humanising influences. To take another tripos, or an Arts course at another University; to make oneself read poetry and history: to do social work; to develop the whole nature of the woman through relationships with friends and family: all these are correctives of that petrifaction, that stoniness of heart, which higher mathematical study may induce. The schools would be encouraged to do their part, if a high standard of requirement in a literary subject were compulsory for all mathematical scholarships, just as good Latin is required for historical scholarships.

For the ordinary girl, a moderate amount of mathematical study; Latin or Harmony as an alternative to Mathematics in Matriculation examinations: for those who will never go to College, and whose gifts are practical rather than academic, a very limited course of generalised arithmetic and elementary geometry for the exceptional girl Mathematics as it is now, but with literary study compulsory to the end: such, briefly stated, are the lines on which the present writer would solve the problem of the place of Mathematics in the education of girls.

SARA A. BURSTALL.

HIGHER MATHEMATICS FOR WOMEN.

I have been asked, in connection with the work of the International Commission on the Teaching of Mathematics, to write a paper on the above subject, and my attention has been specially invited to the question whether, for instance, the study of Mathematics by women should have a special bearing on the subsequent study of Economics, or of statistical inquiries into sociological questions, and therefore perhaps differ in kind from the Mathematics required by, e.g., a future engineer. I understand the expression Higher Mathematics to be used, not in the sense that mathematicians might give it, but merely to mean Mathematics as studied at the Universities by those who are at least to some extent specialising in the subject as distinct from the Mathematics which form a part of general education at secondary schools.

It will perhaps conduce to clearness if I state at once that, for reasons set forth below, it does not appear to me necessary or advisable to have separate programmes of mathematical study for men and for women at the Universities. If an inherent difference in mathematical ability between the two sexes were established it might seem at first sight to furnish a reason for a difference of programme. But experience has shown that some women have sufficient ability and sufficient liking for Mathematics to justify their making it their principal subject of study at the University, and this being so, the inquiry how their powers compare with those of men, either on the average, or as regards the highest degrees of mathematical ability becomes irrelevant to the question before us. For the great differences in ability among the men who specialise in Mathematics afford a range amply wide enough to include all the women who could on any hypothesis be reasonably advised to do so.

In considering the nature of the demand for mathematical education on the part of women, I may take as fairly typical the present (1911-12) students of Newnham College at Cambridge. This is convenient, both because information about them is easily accessible to me, and because the extent to which specialisation is carried at Cambridge, and the consequent sharp differentiation between the different Triposes, makes classification comparatively easy. I should explain that, with the exception of a few graduates from other Universities, and a few other students taking special courses, the women students

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at Cambridge, whether at Girton or Newnham, are reading for one or other of the Tripos examinations, i.e., examinations which, for men, lead to degrees in Honours. Out of 213 students at Newnham in November 1911, 30 were reading Mathematics; 2 Engineering; 35 Classics; 3 Moral Sciences; 1 Law; 33 Natural Sciences (including Geography); 39 History; 61 Medieval and Modern Languages; 9 Economics. Over 14 per cent. of the students therefore were working at Mathematics, and this proportion is somewhat less than has been usual in other years. Of the 30 reading Mathematics, 9 were in their first year, 13 in their second year, and 8. in their third year, at the University. A certain number will not continue to work at Mathematics through their whole course, but after obtaining Honours in Part I.* of the Mathematical Tripos at the end of their first or second year, will turn their attention to some other subject, e.g., Science, especially Practical Physics for the first or second part of the. Natural Science Tripos; or Geography for the Diploma in Geography; or possibly Economics for the Economics Tripos. They might even take some subject unconnected with Mathematics, such as History or Languages. The proportion of students of Mathematics is usually much the same at Girton as at Newnham; but probably the proportion at Cambridge is in excess of that among women students at the Universities generally, as the prestige of Cambridge in Mathematics no doubt tends to attract mathematical students there.

The next question to be considered is what leads these women. to make a special study of Mathematics. It is almost always, I think, that they have liked the subject and succeeded fairly well in it during their school education, and have consequently been advised by their school teachers to go on with it at the University as a subject they are likely to do well in. If, as is frequently the case, they intend to become teachers, they think it is the subject they will like best to teach, and which they are most likely to teach well and to find profitable employment in. The estimate thus formed of their powers and tastes, either by the students themselves or their teachers, does not always prove correct. It happens occasionally that facility in manipulating algebraical formulae has been mistaken for mathematical grasp. It happens, occasionally, that taste for the subject diminishes as the difficulty increases. It happens also, sometimes, that the amount of previous preparation required for success in the Cambridge courset by any but persons of unusual ability is under-estimated, and that a student who has come to the University with the intention of studying Mathematics, but insufficiently prepared, has to be advised to change her plans. It does not often happen that, on the other hand,

* The schedule of subjects for this examination is given as an Appendix to this paper.

.

It must be remembered that in the Cambridge Honours course there is a strict time limit.

a student who comes up intending to devote herself to some other subject, changes to Mathematics.

If I am right in the above diagnosis of the motives that lead women to the study of Mathematics at Cambridge, the subject is in almost all cases studied mainly for its own sake— not because it is useful as an adjunct or stepping-stone to something else. And educationally it is very important that this should be so. A subject which is studied, not for its own sake, but because it is useful for something else, is almost always degraded in the process, and loses much of its educational value, whether the ultimate object be merely to pass an examination or to acquire the minimum knowledge necessary for dealing with some other and different subject of study. It is, no doubt, quite possible to learn Mathematics after a fashion, and up to a certain point, purely as an instrument for some ulterior purpose. But probably for everyone, and certainly for anyone with any mathematical ability, the loss from such a method of study would be great. There would be loss in mental training, loss in knowledge, and loss in interest. A recent writer in the Engineering Supplement of "The Times" advocating a continued attention to Mathematics for its own sake on the part of engineers, said :-"Its importance in education is to form deliberate judgment, to assign a measure to results, to "disclose fallacies, to discourage narrowness, to reveal the unity of whatsoever things are true, and generally to exalt the mind." I should be inclined to add, to stimulate the imagination. The study of Mathematics reveals to the eyes that can see whole vistas of knowledge, whole aspects of the universe, unsuspected by those without mathematical perception.

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I would not, therefore, limit the opportunities of pursuing the study of Mathematics for its own sake in any way, either for women or for men. It must not, of course, be forgotten that among those who study Mathematics at the Universities a fewone here and there may have sufficient power and originality to carry the subject in some direction or other beyond the present limits of our knowledge. But quite apart from this, it is important to keep before everyone, and before women especially, the value of knowledge for its own sake. I say before women especially, because I think the education of women has suffered more than that of men from what I may call, for shortness, the commercial or utilitarian point of view. The old bad education so prevalent in the girls' schools and the private schoolrooms of 50 or 60 years ago--drawing-room accomplishments, needlework, a little arithmetic and something of general information-was a kind of degenerate technical education for domestic life. The standard was low because neither thoroughness nor a high standard were needed for the "The Times," Engineering

The Teaching of Mathematics, Supplement, October 5th, 1910.

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