It is still very difficult for children to understand the meaning of division although various efforts for the improvement of teaching division have been implemented. Therefore overall researches which both capture difficulties and misconceptions of various kinds of division systematically and propose better way of teaching and learning of it to overcome them are required.

　　 From this perspective, as the first step of our research, we developed six sets of test of division, for fifth and sixth graders in the elementary school and first graders in the junior high school, which could reveal difficulties of understanding the meaning of division systematically. In six sets of test, various factors of story problems of division which will be expected to affect the percentage of correct answers, such as effect of the order or value of the dividend and the divisor, effect of figures, number lines, word expressions, key words and so on, were taken into account. It is the main reason for us to develop new tests why most of previous test or studies of division only focused on specific grade or specific kind of division such as division with fractions which children have difficulties.

　　 We conducted the longitudinal and cross-sectional survey with six tests for children at both elementary schools and junior high schools located in three prefectures: Okayama, Hiroshima and Kagoshima. This article reported results of children’s performance of solving problems of “partitive division and the extension of its meaning”, and analyzed children’s difficulties and misconceptions of them in terms of various factors of story problems of division.

　　 Conventionally, “assessment” of education was considered as summative assessment like test results and performance etc. But now, it has been strongly desired to improve the teaching on grasping the actual situation of students. Today, while there are many discussions about the teachers’ teaching, on the other hand there is no teachers’ assessment. Even if many researches such as wrong answer analysis were conducted in mathematics education research, it does not have a position as the teacher’s competency. Therefore, there is need to clarify the assessment literacy of teachers as teacher education research.

　　 In this research, it was firstly considered the background that has not been aware of the assessment literacy on teachers so far. It was clarified that the necessity for conversion to the continuity of the teaching and assessment through focusing on value judgement and problem-solving of educational assessment.

　　 Assessment literacy is not simply points to only the “knowledge  of student”, it was revealed to regard as a composite knowledge associated with subject matter contents and teaching methods. Then, in the conventional teacher education research it was pointed out that the assessment literacy has not been sufficiently discussed. Finally, it was revised the theoretical framework of assessment literacy based on the model of Abell & Siegel (2011).

　　 Contemporary society is filled with statistical data and information, and requires all citizens to become fully statistically literate. School education, especially statistics education, must respond to the request from the society. However, statistics education in Japan cannot yet handle it adequately. This research attempts to reflect on statistics education research in Japan and then present its tendency to provide a sound basis for meeting the request. To achieve this purpose, the titles of every statistics education research paper appearing in five kinds of publications are analyzed and categorized. The results of the analysis present that statistics education research in Japan is interested considerably in the pedagogical facet of didactic phenomena. This tendency turns out to be distinctly different than both that of mathematics education research in Japan and that of international statistics education research.

　　 The aim of this paper is to suggest a new aspect in REIZUKURI strategy.  The strategy with “Make up an example of” tasks induces students to do mathematics.  Their examples are about mathematical concepts and principles.  A teacher asks multiple examples to them repeating the term of “Another?”.  In this paper, observing examples and processes to make up an example, it is suggested that the strategy has a potential variety of integration.  There are three types of integration of a high order, intensive integration and extensive integration (Katagiri, 2004).  Each type can be assumed at the task of “Make up an example of two real numbers whose sum and product are equal”.  Processes to make up an example are adopted a conjecture with trial and error, equations, the quadratic formula, a fractional function, and so on.  If these form a group, it conforms to integration of a high order.  The relationship between simultaneous equations in two unknowns and a quadratic equation conforms to intensive integration in particular.  As year follows year, examples of two real numbers whose sum and product change from whole numbers to fractional numbers and irrational numbers.  Therefore, it conforms to extensive integration.  In addition, it is expected to facilitate the connection about contents of learning among schools.

　　 With regard to the next revision of the Guidelines for the Course of Study, various arguments are being made about the need for a talent and ability-based curriculum reform. Among them, one of the urgent issues is how to foster children’s critical thinking skills, which are regarded as generic skills, in the context of subject education. This study explores the abduction performed by students in math classes that fosters critical thinking and elucidates the position of such abduction in the process of critical thinking, as well as the types of roles it can assume. The research results elucidated that abduction could induce the appropriate inference at an early stage of the critical thinking process. Defined as “reasoning through conscious deep thinking” (Yonemori, 2007, p.61), abduction has an innate affinity with critical thinking and plays an important role in the process of critical thinking. Based on suggestions in preceding studies on abduction, future research is expected to elucidate the characteristics, logical position, and components of critical thinking that can be fostered through math teaching in order to develop and assess teaching methods that foster such critical thinking and to suggest practical teaching models.

　　 The purpose of this paper is to investigate senior high school student’s mathematical inquiry and its phase about the equation of the polar line of ellipse, which focus on comparing mathematical characters between circle and ellipse, reading the form of symbolic expression and reviewing the method of producing the equation of the polar line of circle, connecting with the method of producing the equation of the polar line of ellipse to the polar line of circle, considering mathematical characters of the polar line of ellipse repeatedly.  We investigated student’s mathematical inquiry by teaching unit about the equation of the polar line of ellipse by qualitative methods.  As a result of our discussions, we obtained several insights:

(1)  Reading the equation of the polar line px/a² + qy/b² = 1 by paying attention to the pole (p, q), points of contact   between ellipse and its tangent, the points on the polar line is no easy matter for many students.

(2)  Considering the equation and symbolic expression from many angles and reading the figure about the polar line of ellipse from different angles become a foundation of producing the equation of the polar line.

　 Reading symbolic expression can accelerate to understand the way of dicovering the equation of the polar line by connecting with reading the figure, especially the pole and the polar line in seeing continuous move by GeoGebra.

(3)  Discussion with others and reflection on own writing urge to explore about relation between the circle and the ellipse, the equation of the polar line of ellipse.  Reflecting on own writing and thinking process about the  polar line of ellipse accelerate student’s further mathematical inquiry.