Abstract
The purpose of this study is to propose a way for changing the current state of teaching and learning mathematics from memorizing knowledge and skills to cultivating students' mind, for example their interest or attitude toward mathematics in high school. In this paper, I suggest the so-called "open-activity" and develop cultivating concrete instances of it in high school mathematics. "Open-activity" is to work on an open problem that Pehkonen defines. Strictly speaking, it is a set of students' divergent and convergent acts. Students chose some documents, pose conditions for their starting situation and goal situation of an open problem, compose and solve their questions. I set three frames of "open-activity" based on this conception and developed three concrete instances; "filling circles problem", "the cross section problem of the tinplate" and "triangular five centers problem".
