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.