The accompanying report deals with the mathematical training of prospective teachers in elementary schools as described in the reports submitted by the International Commission of the Teaching of Mathematics to the Fifth International Congress of Mathematicians, held at Cambridge, England, in August, 1912. A comparative study of the facts presented in these reports is of interest for those engaged in the training of teachers in this country, if only because they indicate that the standards elsewhere are as chaotic as they are here. The requirements of the normal schools, or corresponding institutions vary from a review of the elementary-school arithmetic to the mathematics required for entrance to colleges and universities; or, from another point of view, from an emphasis on the purely professional needs limited to the immediate requirements of the elementary schools, to academic and cultural aims founded on the principle that the more a teacher knows about the subject, in addition to the purely professional training, the more successful will be his teaching. But, great as is the variety of standards and aims in Europe, almost every standard finds a parallel in this country, owing to the absence of uniformity--a condition almost paralleled in England and Switzerland. But, if a generalization may be permitted, it would be true to say that the academic standards in the best systems are higher in the more advanced countries of Europe than they are in the United States. Countries included in this report are: (1) Belgium; (2) Denmark; (3) England; (4) France; (5) Germany; (6) Hungary; (7) Italy; (8) Russia; (9) Sweden; (10) Switzerland; and (11) the United States. A bibliography is included. (Contains 2 footnotes.) [This bulletin was written with the editorial cooperation of David Eugene Smith, William F. Osgood, and J.W.A. Young. Best copy available has been provided.]