The professional training of the Mathematical Teacher after his degree course is finished is a principle which has come to stay, and it therefore behoves all who are concerned in the practical administration of this principle to do their utmost to make the course of professional training as valuable as possible for the future teacher. So far as I know nothing has as yet been done by the Mathematical Association as a body in helping forward this movement, but surely the Mathematical Association will fail in its fundamental aims if it does not take a very active part in helping to decide what are the best courses of instruction to give to the student-teachers while in training. The object of the present paper is not to lay down dogmatic assertions as to what ought to be done, but rather to put forward suggestions, and thus to act as a nucleus round which the discussion can revolve. No attempt is made to deal with the training of anyone except the specialist type of mathematical teacher,—the person who in time to come is destined to teach in the Advanced Departments of the Secondary Schools in addition to taking his share in the general mather matical work of the school, and who, if weighed in the balance and not found wanting, will be called upon to act as Head of a Mathematical Department, and thus to guide and mould the mathematical destinies of the institution to which he is attached. In almost every University nowadays there exists a Department of Education, and within that Department of Education a School of Mathematical Pedagogy. It is for us to discuss now what ought to be the type and scope of instruction given in this School of Mathematical Pedagogy.

There remains the further extension to what may be called “multiplicate proportion” (duplicate, triplicate, … n · plicate). This may be regarded as a limiting case of compound proportion, being a relation between variable quantities P, X of two different kinds such that measures p, x of these variables are in the proportional relation p α xn. (The definition can, of course, be expressed—like the definitions of direct and inverse proportion—in terms of ratios of any two quantities of the one kind and the two corresponding quantities of the other kind.)

Since due recognition is now being more and more widely given to the importance of Vector Analysis for three-dimensional work in mechanics, geometry and mathematical physics, the time is opportune to consider the place which that subject merits in a University degree course. A student who undertakes anything like research work in applied subjects is considerably handicapped if he has had no training in Vector Analysis; but the need for it is felt much earlier than the stage at which research is generally begun. Every University course contains subjects which are less useful than Vector Analysis; and on the ground of utility alone there is no question that the latter deserves a place in the curriculum. It will, however, be found quite unnecessary to displace anything else, because the time occupied in teaching the essential parts of vector algebra and calculus will be saved, even during a three years’ course, by the application of this method to the three-dimensional parts of mechanics and mathematical physics.

In our profession we often come into contact with those who do not understand Pure Mathematics. Some of these are respectful as if entering a shrine; but others, of the baser sort, are contemptuous. These have found the way of progress dark and difficult. Their guides, perhaps, have not inspired them, and symbolism seems far removed from the needs of ordinary human life. They deny that it can have an ideal, that it can have any claim to beauty. They admit that it has some purpose in its application to the practical affairs of life, but that it can have an end in itself is incomprehensible. It is associated in the minds of some with the removing of interminable brackets, and with the chasing of elusive unknowns. Since it deals usually with the variable and not with the particular, and since it is concerned more with deductions from data than with the truth of these data, it has been described as the subject “in which we never know what we are talking about, nor whether what we are saying is true.” It is usually considered unwise to express lack of appreciation of the works of a great painter or sculptor or poet, but there have been men of intellect who not only have expressed ignorance of scientific method, but have also been inclined to exaggerate that ignorance. The questions that such a criticism of the subject naturally raises are: Is this hostility justified? Can it be wholly attributed to feebleness of insight? For it is possible that although Mathematics need not fail to supply the necessary stimulus to intellectual thoughts and aspirations, yet its exponents may fail in the interpretation and expression of its ideals; and it must be admitted that the way of mathematical learning can be, and often is, a dull and cheerless one, especially for him whose aptitude is weak. The inherent beauty of mathematical reasoning may be marred by methods that are brutal, and its outlook warped by unduly insisting on the necessity of the moment.

In a recent number of the Mathematical Gazette (March 1920) I proposed the following problem for solution:

Some of the figures in the long division sum have been replaced by crosses, and it is required to restore them. Four solutions are possible—of course in scale ten. It is understood that the sum is to be worked as in elementary arithmetic and that there is nothing fictitious. There must be no line of zeros, neither must the left-hand figure of any line be zero. In the divisor the digits figure ought not to be zero (for such a zero would ordinarily be cancelled before working the sum), but it is not enough to assume this: proof must be possible from the data. In the Four Fours the proof is instantaneous, because there is no remainder and the right-hand figure of the dividend is 4.

As a student in the Department of Textile Industries of the University of Leeds—then the Yorkshire College—I was taught to emulate the successes of my forefathers in the spinning of yarns and in the weaving of fabrics,—and also to perpetuate some of their errors. Empirical work, without any discrimination or “imaginative insight,” was the fundamental basis of practically all instruction.

I draw your attention to two dates of some importance in the teaching of Geometry. In 1903 Cambridge ruled that “Any proof of a proposition will be accepted that appears to the examiners to form part of a systematic treatment of the subject”; in 1909 the Board of Education issued Circular 711. These two dates determine three periods of time.