Hiraku Tohyama (1909–1979) was a mathematician who developed two theories on arithmetic education. One is “suido-houshiki,” the systematic method of instruction for performing calculations on paper, and the other is “ryo-no-taikei,” the method of curriculum design wherein several types of quantities are arranged in the right order. This paper challenges the widespread understanding that Tohyama believed that arithmetic education should be provided in accordance with these theories. Around the 1950s, Tohyama clearly distinguished between mathematics and art on the ground that mathematics is logical and systematic. He intended to criticize those who advocate mathematics education that is suited to children’s everyday life . Tohyama developed the two abovementioned theories while editing arithmetic textbooks in the late 1950s. Because he believed that textbooks were indispensable to teachers, he wished to produce textbooks that were systematic. However , in 1963, he started to insist on the close relation between science and art and began to raise doubts about the effectiveness of lessons that were based on plans and textbooks. This radical shift in his educational thought occurred after he witnessed unexpected active participation by children in learning in the classroom. In conclusion, that teachers should use systematically organized textbooks was not a strong belief that Tohyama held throughout his life; rather, it was a temporary opinion. He finally concluded that instructional theories such as “suido-houshiki” and “ryo-no-taikei” were fallible.

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The real aspect concerning the past educational efforts in various regions in Japan is not clarified enough up to now. Last year a variety of materials of Wakayama Normal School in the Meiji era were found, organized and opened. These mateirals consist of three kinds of documents, which find out the actual contents of mathematics instructions, especially geometry. They are geometry test problems, the school syllabi, and the weekly record of lessons, which were made around the time of the first national syllabus in 1910, the late Meiji era. Three aspects of geometry education which contains the curriculum, textbooks and test problems at that time are examined and several interesting facts were found as follows. First, geometry was taught at the normal school as strictly as secondary schools in the Meiji era. It means that normal schools had a part of the role of the secondary education. Second, before the first national syllabus was established, normal schools flexibly made changes in the curriculum according to the actual conditions of their students. After the establishment of national syllabus, the teachers swiftly prepared teaching plans under the syllabus, and continued to reform them.

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For the logic of explanations of the meanings of four arithmetic operations, there is a need for explanations with continuity, that is, which can be extended to operations of rational numbers (fractions, decimals), and do not have individual and limited characteristics only applicable to integer operations. The process of trial and error to develop these meanings was tangibly expressed in textbooks during the period of textbook authorization system in the Meiji Era (from 1886 to 1903). This paper mainly targets arithmetic textbooks during the second period of textbook authorization system in the Meiji Era (from 1900 to 1903), analyzing the logic and characteristics of explanations of the meaning of division of fractions. In doing so, the main focus is set on whether there is formation of continuity with the logic of explanation of the meaning of integer division. As a result of this analysis, directionality is seen in formation of continuity. At the same time, unique tangible examples of this directionality were found.

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