This paper aims to clarify the logic and the characteristics of the explanation of the meaning and rule of division of fractions in arithmetic textbooks published from 1893 to 1899. The most notable result of the analysis of the textbooks is as follows. Reciprocal numbers are an important concept of the division of fractions. In the above-mentioned textbooks, reciprocal numbers were accorded a position in the explanation of the rule of division. This fact indicated the possibility of a solution of the problem included in the explanation of the rule (a method of explanation based on the results of operation) in arithmetic textbooks published from 1886 to 1892. This fact shows historical progress, and gives us an important suggestion for the teaching of fractions today.

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A new movement of Arithmetical education occurred from Taisyo Era to early Syowa Era (1920s-1930s) in Japan. They say that this movement attempted to child-centered education in exchange for the teacher-centered education in previous Era. In this movement, the children’s life related Arithmetic was picked up and children learned Arithmetic by problem solving in the life. Such a teaching Arithmetic in elementary school has been called Living Arithmetic. In this article, to show the characteristics of Living Arithmetic in Yoshio Ikematsu’s study, Instruction of quantitative approach for life is focused. Ikematsu who was a teacher in the elementary school attached Tokyo Higher Normal School, assumed life to be extensions of time and space for children and a process of self-renewing for them. In this meaning, he placed Instruction of quantitative approach for life as the objective of Arithmetic instruction. He said that Instruction of quantitative approach for life included Understandings of quantity in life and Creations of quantity in life. In “Understandings of quantity in life”, children solve the problem on their life by cultivating a better understanding through considerations for life phenomena and quantity. In Creations of quantity in life, after solving the original problem, children consider the other problem on their life from the new view based on the methods of solving the original problem. It included extensions of time and space for children and this is a process of self-renewing for them. Those are clarified as two characteristics of Living Arithmetic in Ikematsu’s study.

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