The textbook “secondary mathematics, category I” and the textbook “mathematics, category I” were made in japan during World War II. The purpose of this paper is to make the succession and the new introduction to principles for structures of teaching materials in the “secondary mathematics, category I” clear through the comparison of the “mathematics, category I”. For this purpose, three principles for structures of teaching materials were adopted on the basis of editors’testimonies, and structures of teaching materials with regard to the unit “quadratic equation” in the “mathematics, category I” and the unit “quadratic function” in the “secondary mathematics, category I” were analyzed. Through the comparison, the following points are suggested: The “secondary mathematics, category I” took over two principles for structures of teaching materials from the “mathematics,category I”. On the other hand, the “secondary mathematics, category I” accepted a new principle for structures of teaching materials. It was appeared in the “secondary mathematics, category I” that editors intended to emphasize functional thinking and to create “an educational curriculum designed newly” by a chain of questions.

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This paper examines the development of Perry movement in England, focusing on Cricular711 (1909). This circular published by Board of Education was the first circular on mathematics teaching. I examine this circular from three aspects, first the position in the history of mathematics education, second the contents of this circular, and third the effect of this circular to mathematics education. Firstly this known as a quite important document around twenty century and also today’s research. Secondly, this circular discussed Geometry and graphic Algebra, which were the attract issues in school mathematics. For example, in this circular three stages in teaching Geometry was discussed. Thirdly Charles Godfrey(1874-1924), who was a mathematics teacher and textbook writer, lectured this to mathematics teachers in Mathematics Association. In this lecture, he criticized the circular form some points. For example, this circular did not consider the purpose of teaching mathematics so much. However, he agreed with the ideas shown in this circular and explained to the teachers. As a result of the examination from three aspects, I traced the development of teaching mathematics in England around twenty century. Indeed, Circular711 encouraged some ideas shown in Perry Movement and the change of mathematics education.

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This paper focuses on a reform conception of elementary school mathematics education proposed by Walter Warwick Sawyer in the era of the new math movement in the U.S.A. Sawyer was a mathematician famous as both an author of many popular introductory mathematics books and an advocator of the introduction of pattern discovering activities to elementary school mathematics education. This paper also examines the “Math Workshop for Children”, the instructional material for elementary school mathematics education developed under Sawyer’s supervision. Three conclusions can be made. Firstly, Sawyer’s reform conception was to relate concrete manipulation of numbers to abstract algebraic rules through the experience of pattern discovery. Secondly, “Math Workshop for Children” aimed to help children grasp abstract algebraic rules such as the commutative or distributive rule through the experience of pattern discovery. Thirdly, the “Cross Number Puzzle”, one of the materials for pattern discovering activities included in the “Math Workshop for Children”, was introduced in the textbook written by the University of Illinois Committee on School Mathematics, the first mathematics curriculum development project which advocated introducting discovery learning into mathematics education in the new math movement.

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