Conclusion and DiscussionsThe resea

Conclusion and Discussions
The research findings found that there were 7 kinds of stu-dents’ Mathematical Communication by students’ Gestures including 1) rigorousness by students’ beat gesture-Mathematical Communication in which the students wanted to emphasize the statements or pictures as their own ideas, or communicate with their friends or teacher; 2) rigorousness by students’ metaphoric gestures—gesture referring to content or concepts of the lesson; 3) economy by students’ deictic gesture—this kind of gesture could be easily performed economically with common under-standing; 4) economy by students’ iconic gesture—this kind of gesture was to draw a picture to communicate economically with each other for common understanding; 5) freedom by students’ deictic gesture—the students used various kinds of point-ing, and had freedom to express their mathematical ideas; 6) freedom by students’ iconic gesture—the students could use many kinds in drawing picture as well as have freedom to ex-press their mathematical ideas; and 7) freedom by students’ deictic and iconic gestures—the students used both of pointing, and drawing picture to communicate their ideas as well as had freedom in express their mathematical ideas, and the most commonly used the characteristic of economically mathematical communication by deictic gestures, and Students’ self learning through problem solving while the teacher take notes students’ idea for later discussion in Open Approach. The teachers used gestures for observing students’ learning in the classroom and students had happiness in the classroom. Furthermore, the schools in Lesson Study and Open Approach context, the stu-dents had opportunity in learning based on their potentiality, being able to think, perform, and express. They preferred to express divergent think.