This paper deals with teaching and learning of practical mathematics in the Meiji era. Until early 1870s, the Japanese people learned traditional Japanese mathematics, Wasan, and mathematics was regarded as a practical science. After the Meiji Restoration, since the introduction of modern educational system in 1872, mathematics has been taught in Japan mainly in Western style in all levels. At technical institutions, applied (or practical) mathematics was taught after a standard course of elementary mathematics, not intended merely for applications. At the Imperial College of Engineering in Tokyo (Kobu Daigakko), a predecessor of the School of Engineering of the University of Tokyo, applications of mathematics to engineering problems as well as (applicable) higher mathematics through engineering problems were taught. Professors Ayrton and Perry taught such topics by using squared papers extensively. After returning to England, Perry wrote several books based on his idea of practical mathematics which was generated by his teaching experiences in England and Japan. Perry's idea had influence in Japan since the second half of the Meiji era, though very small at first. The first Japanese advocate of Perry's idea on the teaching of mathematics was Ariya Inokuty, professor of mechanical engineering at the School of Engineering, the University of Tokyo. Among the books on practical mathematics in the Meiji era, “Practical Higher Mathematics” by K. Aichi and T. Tsunoo, published in 1909, has several features. It begins with an introduction of graphs. However, this is not a book based on the idea of Perry.

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This study aims to clarify the actual situation of the training to become a secondary school mathematics teacher for a soldier and a sailor before World WarII. In Japan, two kinds of secondary school teachers training for a soldier and a sailor were done before World War II. One of the above teachers training was the measure to give an officer, who retired according to the disarmament since 1922, to work as a mathematics teacher. A short course, which was a half year course or 18 months course, was held for the retirement officer between 1924 and 1935 by the Higher Normal Schools, and 149 retirement officers obtained the mathematics teacher license. And the other was the measure to the disabled soldiers and sailors who entered the Japan-China War and so on. The secondary school teachers training institutes, which were two or three year course, for the above soldiers and sailors were founded at Tokyo and Hiroshima between 1939 and 1948, 67 disabled men obtained the mathematics teacher license. They contributed to the insufficiency of the secondary school teachers which occurred after the end of World War II, however it was little.

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The old curriculum in secondary schools was widely revised in wartime and descriptive geometry was adopted into the teaching of solid. Originally, school solid geometry was founded in 1880’s by famous mathematician Dairoku Kikuchi, and descriptive geometry called “Yokiga”, also in 1880’s by a school teacher Sakugoro Hirase. Since then, they had been taught separately in different school subjects of “mathehmatics” and “drawing and arts.” Many mathematics educators had made efforts to take descriptive geometry into the teaching of solid. Their goals vary a little, such as “to make practical geometry,” “to connect with other subjects,” or “to help students work,” but the discussion was not active mainly because the entrance examination of upper schools did not contain solid geometry. The new textbook “Mathematics 4 and 5, Category II” published in wartime, not only took descriptive geometry into the teaching of solid, but also utilized the skills of projection chart. A proof of the conic section was shown by the cross-section of a cone and plane using those skills of projection chart. This was really epoch-making, but our current junior/senior high school curriculum doesn’t have sufficient connection between solid and descriptive geometry.

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The purpose of this paper is to evaluate the mathematics education in right after the end of World War II as a phase of continuous process from the pre war period to post war period. In the prior literatures of history of mathematics education, the period was placed as an age of discontinuity between pre and post war. However focusing on the instruction of multiplication and division by fraction, there were aspects of continuation for instructional improvement. From such a point, we review the improvement in pre war period and identify the continuation of the instructional improvement in the period. As a result, the changing from teaching the method of calculation to helping students to understand the meaning of calculation were pointed out as an instructional improvement in pre war. The point can be seen in right after war period, then it continued to post war period. And in post war period, the meaning of multiplication and division by fraction were explained with the proportional relationship of numbers. Then multiplication is the calculation to find the quantity expressed by proportion, and division is the calculation to find the proportion and base quantity. Such an explanation was organized in right after war period and became the foundation of instructional explaining in post war. From the point, the instructional improvement in pre war continued in right after war period and post war period.

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