Five years ago alert teachers of mathematics were thinking about the role of their field in the preparation for a war which for us in the United States was then two years in In the early stages of the "national defense55 movement there was widespread recognition of the low level of achievement which seemed to characterize the majority of our students. There was also a tendency to say that boys in various branches of the military services needed "a course in algebra/5 or "a course in trigonometry,55 or some similar sort of "blanket55 prescription. Then, as the situation became urgent, the "refresher55 and other pre-induction courses took the spotlight. Later, as more of the time and the services of capable consultants on secondary mathematics were made available, the character of the authoritative recommendations changed. Reports and articles became increasingly definite concerning the kind of mathematics actually needed for specific purposes, so that today the minimum content is defined much more precisely than before.As planning for the post-war period comes to the focus of attention there is substantial agreement among the writers on several points. Especially significant is the current emphasis on the importance of having the student acquire a genuine understanding of the material. In the literature, at least, it seems to be recognized that "drill on the fundamentals55 is not the sole remedy for low achievement. The ends or goals to be sought, expressed in terms such as "mathematical literacy for all who

Learning mathematics, like learning language and the aesthetic subjects, is an active process and should call into play our imaginative and creative powers so that the subject is a delight to pupils of all ages. Investigations of every aspect of mathematics—number knowledge itself and written calculations, as well as measures, shapes, and representations of all kinds—are included. In the title of this article, the word experience could very well be changed to investigations. Investigations, to me, cover a wider range of activities than experiences, which arc often thought of as practical experiences.

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Much of the adverse criticism of the teaching of arithmetic in the past can be summed up in one sentence. The complaint has been that too often the learning has been “mere formalism.” In this eontext the term “formalism” means that mathematics becomes a game played with symbols according to fixed rules. Little or no attention is paid to the “meaning” of the symbols; that is, the nature of the objects, operations, and relations that are symbolized. Lack of understanding by students, lack of motivation, lack of satisfactory achievement, lack of ability to apply mathematics in problem solving, and other more specific complaints are often attributable, in the last analysis, to excessive formalism.