The possibility of a new and stimulating course in mathematics in the Junior High School is the most encouraging feature of our recent advance in the teaching of the subject. Such a course enables the schools to break away from the present initial course and prepare the pupil for the best that mathematics has to offer in the high-school period. It enables them to take the best that other countries have to offer and to introduce it by the natural steps that psychology shows that the youth is ready to take. It allows mathematics to relate itself to the interests and apparent needs of young people instead of being presented as a purely abstract science, since the early stages of a rational course can easily be made much more apparently practical than is the case with the present course. The two standard courses in algebra and geometry, as they have come down to us, are satisfactory so far as technique and logic are concerned, and we shall make a great mistake if we lose the best features which they possess. They are not, however, good introductory courses, and we shall do very much better if we can first establish a taste for mathematics, showing numerous applications of the science and various recreations that come along with them, and then offer a set of strong electives in the Senior High School for those who show an aptitude for the subject.