This study clarifies the process that students comes to grasp knowledge as the thing which can be applied to various context. In this study, I name such a process“Subjectification”.  In this article, I define the process as  a social process as learning knowledge, students embody knowledge by noticing themselves seeing a  certain problem (figures, problem statements and numeric formulas) adequately”.

　　 The aim of this article is construction of the levels of Subjectification. By focusing on forms of students’ knowledge, I construct three levels of Subjectification.

Level 1: In case of instructing to use knowledge, students can use knowledge.

Level 2: In  case of a situation of learning knowledge or a context which students have already learned, students  can use knowledge.

Level 3: Even if students have never experienced the context, they can use knowledge.

　　 As the result of analysis of textbooks based on this levels, in order to reach higher levels, students need opportunities which they integrate different activities.

　　The purpose of this paper is to examine the effectiveness of lesson organization based on the Koyama’s (2010) “two-axis process model” of understanding mathematics for developing students’ understanding in high school mathematics classroom.　By using the principles and methods for lesson organization based on “two-axis process model”, the authors clarified levels and embodied learning stages for understanding the theorem of sines and cosines as a topic of “Figure and Measurement” in high school subject Mathematics I.　We did an action research with a pretest, mathematics lessons, and a posttest in order to examine the effectiveness of our lesson organization in terms of improving students’ mathematical understanding of this topic.

　　As a result of qualitative and quantitative analyses, the following two points were found out.

(1) About the understanding level of “general mathematical relations”, all students who understand how to prove the theorem can use it in solving mathematical problems, while the opposite is not the case.

(2) In order to shift their understanding up to the level of “general mathematical relations”, beforehand students need to understand the “relations of mathematical entities” at a certain level.

　　The first suggests that it is useful for teacher to make a distinction between understanding the theorem itself and using it in solving mathematical problems at the level of understanding “general mathematical relations” in teaching and learning of mathematical topic in “Figure and Measurement.” The second suggests that it is important for teacher to aim at students’ understanding of “relations of mathematical entities” in order to shift their understanding up to level of “general mathematical relations”.　These findings verify the effectiveness of our lesson organization based on the two-axis process model in high school mathematics classroom.

　　 This study clarifies the process that students comes to grasp knowledge as the thing which can be applied to various context.  In this study, I name such a process “Subjectification”.  In this article, I test suitability of the levels of Subjectification by surveys of the reality of students. In the test, I check the following three points;

　(1) Whether at least one student corresponds to each level of Subjectification

　(2) Whether all students correspond to any of levels of Subjectification

　(3) Whether students who correspond to a certain level behave like students correspond to the level

　　 As a result, I argue that the validity of the levels of Subjectificaation is mostly certain.  And through surveys, I define differences in grasping knowledge between each level of Subjectification.

　Level1: Grasping knowledge as a formula

　Level2: Grasping knowledge as a formula, and Particular as a process

　Level3: Grasping knowledge as a process, and a formula as the process

　　 Finally in order to make up for these differences, I propose leading principles.