Yutaka Ohara

The purpose of this study is to examine the effect of different instructional methods on the understanding of irrational number in senior high school students. The experiment consisted of three sessions : a pre-test, an experimental instruction session, and a post-test. In the experimental instruction session, two different groups were formed. In one group (SA method group), students rationalized the existence of irrational number and its calculation by rational number using successive approximation. In other group (GR method group), students rationalized the same with a geometric realization. By comparing the pre-test and post-test (questionnaire investigations), the following three points were obtained: 1) the SA method has the effect of promoting students' concern about the significance of root. 2) the GR method has the effect of encouraging students' recognition of the sum of square roots as a number, 3) both methods do not influence calculation skills, although they have an effect of encouraging students' recognition of the square root as a number.

The purpose of this study is to investigate how prospective teachers comprehend student's naive conception. For this purpose, firstly, we focus on theorem-in-action in conceptual fields theory (Vergnaud, 1990) and elaborate the notion in order to evaluate students' naive conception in detail. Secondly, we report the case study on liner function. Two types of data were collected: a questionnaire investigation for prospective teacher, and an interview on what they thought of students' naive conception. These results show that if prospective teachers identify both the factual and valuable aspects of students' naive conception, they might place them in teaching mathematics positively.