The purpose of this paper is to summarize the findings of previous studies on mathematics teachers’ values alignment strategies and to develop a framework for future analysis of the strategies.  As a result, I proposed a 3×3 matrix for the classification of mathematics teachers’ values alignment strategies, in which the treatment of teachers’ and students’ valuing is positioned as two dimensions.  In addition, the position of the strategies proposed so far in the matrix and the possibility of deriving new strategies were also discussed.  

　　As for the future tasks, first of all, the values alignment strategies of Japanese mathematics teachers should be clarified.  Next, the relationship between teachers’ values alignment strategies and students’ valuing formation is one of the issues to be addressed.  The content of the valuing in the values alignment should also be examined.  In the future, we would like to examine whether it is possible to acquire new valuing by leaving it only to teachers and students concerned, and what is the necessity of external intervention.

　　In mathematics education, metacognition is considered to promote problem solving, and Schoenfeld (1987) pointed out that teaching metacognition in the classroom improves the ability to solve basic and complex problems.

　　For this reason, there have been many studies on metacognition in mathematics classes.  These studies can be divided into two main categories: those that train through descriptive representations, such as Shigematsu et al.’s (1998) math writing and Kameoka’s (2017) Fuki-Dashi method, and through learners’ interactive activities, such as Kato’s (1998).   

　　However, in many previous studies, teachers didn’t explain to children how to use metacognition, and tell to solve problems with metacognition in mind.   

　　In this case, it has been pointed out that indirectly teaching metacognition has only a small effect on students’ behavior (Charlotte Dignath, Gerhard Buttner, 2008).  In addition, children’s discussions are not about metacognition but only about what specific activities they have done (John Mason and Mary Spence, 2000).  So, teaching metacognition in the classroom requires that the teacher explains metacognition and that children are aware of metacognition.   

　　In this study, I will clarify the relationship between problem solving and metacognition.  Then, we aim to clarify how to explain metacognition to children and make them aware of it, and how to develop metacognition.

　　The purpose of this treatise was to develop elementary school mathematics lessons that foster statistical problem-solving skills.  In Japan, statistical problem solving is performed using the PPDAC cycle, which is carried out from the 5th grade, but overseas, statistical education linked to probability is provided from the lower grades of elementary school.  In Japan as well, in order to develop the ability to solve statistical problems from the lower grades of elementary school, two units were developed using rock-paper-scissors play, which is a familiar phenomenon for children, in order to secure the number of lesson hours, which is an issue.  I practiced at.  As a result of protocol analysis, descriptive analysis of children, and comparison of groups in the two units to obtain confidence intervals, it was possible to analyze that there are superiority and non-inferiority between the units of both groups.  The results showed that both unit configurations were generally effective.  This suggests the possibility of developing statistical problem-solving skills from the lower grades.

　　In this paper, we discussed the implicit assumptions about elementary event of probability in Japanese high school, because the treatment of elementary event of probability in Japanese high school is different from it in probability theory. First, from the descriptions of elementary event and probability in probability theory, classical probability, and high school mathematics A textbooks, we identified the implicit assumption about elementary event of probability in Japanese high school as“each elementary event is equally likely.” Next, we posed the research question:“When asked about elementary event without the goal of defining probability and without knowing that they will later consider equally likely elementary event to define probability, will students be convinced that the answer is only if each root event is equally certain?” As a result of analyzing a class for high school students, it was confirmed that students were not convinced of this. Then, we pointed out that this result could happen even if we enhanced the explanation with examples or followed the development of other textbooks, and discussed the teaching of elementary event and probability to overcome this challenge.

　　The theory of complex functions is equipped with many beautiful aspects, which cannot be seen in the theory of real functions, and we have been interested in introducing such aspects of complex functions. However, such an approach has many difficulties.  In fact, though the current senior secondary mathematics includes study of complex number and complex plane, the focus of their study tends to be in algebraic treatment, rather than dynamic aspects of complex number and their operations, which deeply relates with complex functions.  There exist previous studies about this issue, which focuses on dynamic aspects by recognizing complex number operations as transformations in complex plane, which is virtually complex functions.  First, in this paper, we identify and clarify the view of complex functions that is possible and realistic to treat in senior secondary mathematics.  Complex functions involve transformations of 2-dimensional regions, which is difficult to seize for students, while it is not so difficult to consider images of finite 1-dimensional figures by a complex function.  Then we focus on considering how the image of 1-dimensional figure changes, when the original 1-dimensional figure changes from one considered figure to another: we think these considerations are essential in considering complex functions in secondary mathematics.  Secondary, it is important to show the rationale of this new view.  For this purpose, we developed the teaching material and teaching practices relating the proving principle of the fundamental algebraic theorem, which cannot be achieved by usual view but can be achieved by the new view of complex functions.  As the result, we could see the possibility of this practice to a certain degree and obtained some suggestions for further improvement.

　　This study aims to identify the difficulty in understanding the process to solve a locus problem which could be solved by using equations. For this purpose, we use APOS theory (Arnon et al., 2014) as a theoretical flamework. APOS theory is a constructivist theory developed by Dubinsky. Using this theory, we can analyze the quality of an equation conception which a subject applies to a locus problem.

　　Our work is divided into three steps. At first, we describe a Genetic Decomposition of an equation, that helps us to identify students’ differences in terms of their conceptions and to investigate how to use their Schema in dealing with mathematical problem situations. Secondly, we analyze problems which are in the Japanese textbook. By analyzing those problems, we point out that it is difficult for students to think an equation as Process, or to focus on any order pair that satisfies an equation while solving a locus problem. Lastly, we propose the instruction model to overcome the difficulty, that is, to help the students to think an equation as Process, when they solve a locus problem. In this proposal, the locus problem solution, by using equations, is contrasted with the one based on elementary geometry.