Abstract
Multiplication and division of decimal fraction is one of the most difficult contents for 5th graders in the primary school. This study focuses on the division of decimal fraction, especially on the generalization from quotitive division to first application of ratio. The purpose of this study is to clarify the understanding process in generalization of quotitive division, especially the factors of difficulties in understanding the division of decimal fraction and the factors for overcoming the diffuculties. The main results are folloing; 1. The generalization of quotitive division is made by extending the idea of 'measurement'. 2. In the generalization, it is effective to use the operative material that can generalize the idea of measurement and that function as the semi-concrete material for number-line expression. 3. The primitive model on quotitive division include the properties of 'quotient is integer' and 'dividend is larger than divisor'. 4. Children feels difficulties in such situations that quotient is decimal fraction and that dividend is smaller than divisor. These situations are related with primitive model. 5. The stages that the idea 'how many times' is abstracted from how many pieces 'are folloing; (a) The idea of 'how many times' is primitively constructed in such a division that dividend is larger than divisor and quotient is decimal fraction. (b) The idea of 'how many times' is realized in such a division that dividend is smaller than divisor. (c) The merit of the idea of 'how many times' is realized when all division of decimal fraction is conceived by the idea.
