Modern complicated society asks school education including mathematics education not only to make students understand contents of subject, but also to form humanity. Considering the change of future society, it is necessary to identify immutableness and fashion of “usefulness in the study of mathematics” which Perry. J stated in 1901. Moreover, suggestion of teaching should be given from the view point of mathematical activity. In this paper, the author examined today’s significance of Perry’s aspect of mathematics education perspective by reflecting history of mathematics education in Japan and mathematical literacy, and gave suggestion of teaching by focusing on RME.

　　 In our country, Perry’s aspect of mathematics education was accepted by leader such as Ogura Kinnosuke. Perry’s aspect of mathematics education, however, was near to neglected by worldwide movement such as modernization. Today’s society in which flexible ability such as mathematical literacy are required, Perry’s statement which aims to balance understanding of mathematical contents and forming humanity is worth to be reflected in today’s society. And theoretical coherence of Perry’s statement and RME was shown by compareing Perry’s statement of mathematics education with that of Freudenthal.

　　 The purpose of this paper is to clarify a process of teaching and learning about data analysis and probability from the perspective of mathematical literacy.  For this purpose, firstly it was pointed out that it was necessary to learn descriptive statistics and probability comprehensively while utilizing those as inferential statistics in order to learn statistics and probability as mathematical literacy.  That was why to connect between statistics and probability positively.  And I considered teaching and learning incorporating structureoriented mathematical processes into application-oriented processes as a form of teaching and learning to foster today’s mathematical literacy.  Such process of a teaching and learning mathematics while utilizing mathematics for solving a problem of the real world matched Lesh’s “model  elicit activity” and “model  exploration activity”. In considering lesson based on those activities, it was clarified that a probability concept was developed as “validity” to evaluate and deny the statistics concept that was “means” of the data analysis.  Furthermore, it  was thought that such a process of the learning and teaching to connect statistics and probability could be realize by “refining  models” to evaluate and deny models from a viewpoint of sharability with others and reusability in other situation.

　　 The purpose of this study is to identify the characteristics of Japanese prospective mathematics teachers’ reflection on their own lesson practices by qualitative data analysis based on their comments in post lesson discussions.  After studying previous studies on pre-service training for prospective mathematics teachers in japan, especially on teaching practice, and on mathematics teachers’ reflection on lesson practices, the analysis on the reflections of a teaching profession graduate student “A”  in teaching practice was conducted as a case study.  As the result, it was found that the student’s reflection was gradually focused through the teaching practice on the notice to the gap between his intended and implemented mathematics lessons, the difficulties to organize students’ learning activities and facilitate their mathematical thinking and expressions and the selfawareness on the lack of depth of study on teaching content as a mathematics teacher.

　　 The purpose of this paper is to clarify conditions to realize the lesson for students’ active learning in the high school mathematics subject of ‘Properties of Integers’. From the viewpoint of the theory of didactical situations  (TDS), this study regards the lesson for students’ active learning as the lesson in which adidactical situations realize. 

　　 In this paper, the purpose was achieved through the following 3 steps.  First, some milieu was designed from the viewpoint of adidactical situation and fundamental situation in TDS, and the lesson was planned.  Then, a priori analysis on the planned lesson was performed based on TDS.  Secondly, the author myself practiced the lesson and performed a posteriori analysis of the lesson.  Thirdly, comparing the result of the a posteriori analysis with the one of the a priori analysis, it is considered what kind of differences were in the result of both and what made the differences.

　　 The result of the consideration showed that the following points can contribute to realize adidactical situation for the lesson on ‘Properties of Integers’:

・ Setting the situations where students cannot examine all the cases by division in order to reflect the  result and process by division;

・The new additional milieu promotes the emerge of mathematical knowledge.

　　 The purpose of this paper is to propose what a mathematics teacher education for Upper secondary level should be.  In our country, research on teacher education has not been mainly focused on in the mathematics education research.  Therefore only small numbers of research papers on this topic have been reported.  On the other hand, mathematics teacher education has been taken up from the international point of view.  However, the main target for mathematics teacher education is not upper secondary level, but lower secondary level and primary level.

　　 In the first parts of the paper, the direction and approach of mathematics teacher education for new age was considered.  In the middle parts of paper, we illustrate the difference in quality between Academic-oriented lesson study and Meritocratic-oriented lesson study.  The quality is characterized by three parts: the viewpoint regarding subjects, the relation to researcher, and the position in the school system.  In the last parts of paper, we consider to propose the method of curriculum development through Lesson Study.  Mathematics teachers should be interested in three points for curriculum development: in the subject matter, in the learner, and in the learning object.

　　 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.

　　 This paper is intended to examine harmonious treatment of “mathematical culture” and “culture of mathematics”.  I involve the formulating “personal learning” (i.e. “humanistic learning”) that is seen from the Societal component perspective by Bishop at high school education.  

　　 First, I identified “mathematical culture” and “culture of mathematics”, after that, I defined “mathematics” as used respectively.  There after, I state that “impersonal learning” (Bishop, 1988) is caused by the uneven distribution of the “culture of mathematics”-centeredness.  Furthermore, from Bishop’s view, I assume that at high school level, negative effects on mathematics learning are caused by uneven distribution of “culture of mathematics” -centeredness.  I confirmed the following “principle for formulating the humanistic learning”;

　　 By relating to the fact that how human has been made mathematics, 

　　 (1) students constitute a personal meaning through activity.

　　 (2) human activities from the past to the current had been leaded to mathematics.  Lessons can be learned with more understanding of the non-standard mathematics using a contextual “teaching material”.

　　 The Societal component is one of the three components (i.e. Symbolic, Societal, Cultual) of mathematical enculturation curriculum developed by Bishop; it is based on the paradigmatic- typical inspired effort through project teaching.  It consisits of complementary values called “control” and “progress”.  In this paper, in order to deal with harmony of culture of mathematics and mathematical culture, I decided to ask the social components by Bishop.  Because there is a need to promote the development of progress which lacks in high school mathematics education.

　　 Finally, I discuss a case of “The application report of probability and statistics”.  This project teaching is a piece of personal research undertaken by the learner, using reference material, and written up in the form of report.  The title of student report is the “cutting corners degree” of each television station as seen from the rebroadcasting number?”.  This case study can be summarized as follows:

・ Not only in the past of society, the project’s theme also needs to be assumed related to the current and future of society.

・Personal meanings become enriched by being publicly shared by all the presence.

・Non-Mathematical criteria should enter into the project environment.

・ Not only authentic culture of mathematics, there is a need to look also to mathematical culture as familiar subculture.

　国内外の研究では，モンゴルの中学校における数学教育の課題の一つとして図形領域の到達度が低いことが挙げられている。その原因として，Javzmaa（2009）は，教師の図形領域に関する教えるための数学的知識（MKT）の欠如を指摘している。本稿では中学校教師のMKT について概念イメージと概念定義の観点から明らかにすることを目的とした。そこで質問紙調査を実施し，結果として，モンゴルの教師のMKT の特徴としては，内容と教授法に関する知識（KCT）と内容とカリキュラムの知識（KCC）に依存していることが明らかとなった。そしてまた，モンゴルの教師の多くが数学に対してPlatonist belief を有していることが示唆された。

　　 In the current national course of study for Mathematics 1 and Mathematics A, task-based learning was newly added.  Since many students in Japan are not interested in mathematics, it is important that we develop teaching materials for task-based learning that interest them.

　　 In this paper, we describe a series of studies on task-based learning as follows:

　　 First, why we have selected Euler’s polyhedron theorem and Euler’s number as the theme of teaching materials for task-based learning.  It is important to develop them with the goal of giving each student the best curriculum possible.  We are conscious of the meaning of task-based learning, namely, relationship of real lives, development, interest and concern of students, and mathematical activities.  To ensure this outcome, we feel that Euler’s polyhedron theorem and Euler’s number are sufficient for the purpose of task-based learning.

　　 Second, how we have developed teaching materials for task-based learning since 2012.  We have developed them by assuming the students’ questions about this theorem, for example, what is the figure two at the righthand side of the formula in this theorem, and have practiced them for college students.  As the result, it turned out that some teaching materials were useful in terms of cost-effectiveness, the degree of interest, and the level of difficulty.

　　 Third, we describe the results of practice for high school students.  We had an opportunity to practice a teaching material for high school students, and selected Euler’s number of figures made with lines and triangles by students as the teaching material, but it was not always equal to two.  Interpreting the results of questionnaires, this teaching material for task-based learning is useful for high school students.

　　 This research aims to develop the theoretical framework for designing teaching and learning process of mathematics which focuses on significant roles of social interactions: negotiation of meaning and negotiation of representation, and to propose the improvement of mathematics lessons of concrete teaching materials in terms of the theoretical framework.  Main findings of this paper as the first step to these two purposes of this research are summarized as the following four points. 

　　 Firstly ‘model  of mutual development of mathematical meaning and mathematical representation’ was developed for both the analysis of social interactions among children and designing mathematics lessons emphasized on social interactions.  This model had three characteristics as follows.  The first is that this model has four theoretical backgrounds: 1) basic unit of mathematical meaning and mathematical representation based on symbolizing, 2)‘signifier  - signified’ relation of mathematical meaning and mathematical representation among basic units, 3) Three levels of mathematical meaning and mathematical representation, and 4) appropriation and use of mathematical representation symbolized mathematical meaning.

　　 Secondly three principles for designing mathematics lessons based on ‘model of mutual development of mathematical meaning and mathematical representation’ were also proposed.  These principles had the epistemological background of the coordination of constructivism, interactionism and socioculturism.

　　 Thirdly important aspects of the improvement of the unit ‘addition’ in second grade were proposed based on both ‘model  of mutual development of mathematical meaning and mathematical representation’ and three points for designing mathematics lessons by the analysis of elementary school textbooks.  An alternative new plan of ‘addition’ was also developed from points of the improvement. 

　　 Finally the teaching experiment of ‘addition’  based on the new plan was conducted at the second grade classroom in the elementary school.  As the result of both qualitative analysis of seven lessons and the analysis of the post-test, the effectiveness of ‘model  of mutual development of mathematical meaning and mathematical representation’ and the new plan of ‘addition’  were verified respectively.

　　 The purpose of this paper is to examine the students’ use of examples and a changing process of their activities with examples.  The author planned and carried out lessons about the equation of straight line in “Mathematics Ⅱ” in high school.

　　 The analysis identified a changing process with three stages by analysing a series of students’ activities qualitatively: operation, reflection, and application.  The result implies that a way for students to interpret their activities affects the subsequent reflection in changing from the operation to reflection, and that it is necessary for a teacher to have students reflect their interpretation of activities.

　　 In order to commemorate its 20^th anniversary, the Japan Academic Society of Mathematics Education (JASME) compiled in 2014 a special issue, in which all the articles published between 2004 and 2013 were collected. The issue is supposed to represent the sense of themes dealt with by the academic society as a whole. Categorization within the issue represents the characteristics of both the compiled articles and the academic society from inside, while comparison with collection of similar articles represents the characteristics from outside. The purpose of this research is to conduct a meta-analysis on this 20^th anniversary special issue from the perspective of international handbooks, whose characteristics are time-spanning, comprehensiveness, theoretical diversity and criticalness. Within the articles compiled in the special issue, the following five themes were identified and chosen as focus, based on the criterion of having relatively less attention: Globalization and internationalization; Technology; Social, cultural and political aspects; Assessment; and Policy decision makers. As a result of the analysis, the following points were found: (1) the special issue has no category related to “society” and“culture”, and very few articles with those or derived keywords; (2) a similar trend was found in the case of “technology” and “assessment” ; (3) the focus of discussion of the Journal of JASME is on activity in the classroom; (4) the keywords “global” and “international” are very unique; and (5) some articles are written in the context of international cooperation. On the other hand, the handbooks analyzed in this study have taken up and critically reflected themes such as social, cultural and political aspects of mathematics education, including globalization and internationalization, technology, and assessment.

　　 The author has been engaged in the study of structure-oriented mathematical activities.  In structure-oriented mathematics, there is emphasis on constructing mathematical concepts and developing mathematics.  Therefore, in the secondary education, proving should be the main method in those activities.  On the other hand, as De Villiers (1990, 1999) pointed out, the function of proof has been seen almost exclusively in terms of verification, although verification is only one aspect of proof, and to consider proving as mathematical-activities we should take the various functions of proof into account.  

　　 In this paper, we consider the mathematical activities involving proving in high school from the viewpoint of functions of proof argued by De Villiers (1990, 1999), and obtain some implications about developing such activities as structure-oriented mathematics.   　 

　　 Two previous studies, the mathematical activity model by Hamanaka & Kato (2012, 2013) and the mathematical activities involving “proofs and refutations” by Hayashi (2013) are considered and it is shown that the former focuses on verification and explanation, the latter on communication.  

　　 Then, as the mathematical activity focusing on proof as a means of discovery, we propose the activity taking up the mathematical classification problem.  Also its sample material is shown and the characteristics and the meanings of such an activity are considered.  

　　 Generalization is very important knowing process in mathematics learning.  However, its purpose, meaning and usefulness (we call, the function of generalization) are not so clear for not only students but also mathematics educators.  In fact, even very young children has ability to generalize something (Vinner, 2011), nevertheless generalization does not appear just by performing in mathematics class (cf. Tatsis & Tatsis, 2012).  This problem should not simply be attributed to students’ cognitive ability, but also that they may not realize the function of generalization.

　　 In our previous study, we identified six functions of generalization: purification, valiablization, unification, discovery, association, and socialization.  In this study, we focus and analize their epistemological ordered structure between two functions.  Of course, their actual relationship in mathematics learning is more complex and tangled.  However, relationship between two functions is the most simple structure.  Thus, our analysis provide the basis for understanding the function of generalization in learning mathematics.

　　 Purpose of this paper is to identify the viewpoints for developing a model which represents aspects of a lot of statistical knowledge of children and to develop the model.   

　　 For this purpose, the author focused on Polanyi’s classification of knowledge in terms on symbols and words (Tacit Knowing・Explicit Knowing) and Hirabayashi’s classification of knowledge in terms on a complex organic system (Method Knowledge・Content Knowledge). 

　　 As a result, the author identified two viewpoints for developing a model: from Tacit Knowing to Explicit Knowing (Problem Solving), from Method Knowledge to Content Knowledge (Problem Posing).  Then, the author developed a matrix model representing aspects of statistical knowledge of children (Fig. 1) through integrating following two classification: Polanyi’s classification of knowing in terms on symbols and words (Tacit Knowing・Explicit Knowing), Hirabayashi’s classification of knowledge in terms on a complex organic system (Method Knowledge・Content Knowledge). At the same time of developing this model, I explained this generality for reasons of Hirabayashi’s mathematical analysis activity: <Vague Unity> → <Analysis> → <Defined Unity>.