In the current era, citizens are required to train risk literacy, which is the ability to grasp risk properly and make judgments appropriately. Since risk is represented as the product of the level of the damage and the probability, I believe it is the role of probability education that developing the ability to judge. Based on the above, the purpose of this paper is to identify the educational content of the probability towards development of risk literacy and decide its arrangement.

　　The results of this paper are the following three things:

・ From the definition of risk, expected value, expected utility and decision tree were identified as educational contents related to probability.

・There is statistical literacy as a subordinate concept of risk literacy, and it is learned in stages (Table. 1)

・ From the viewpoint of Construction of knowledge, educational contents were arranged according to the qualitative level of statistical literacy (Table. 2)

From the above, organizing such learning activities is expected to develop risk literacy, which is a decisionmaking ability in modern society.

　　According to the TIMSS, elementary school students in Japan are internationally highly ranked in arithmetic problem solving.  However, it is reported that the percentage of students who like learning mathematics and students who appreciate the value of learning mathematics are extremely low.  The major focus of this study is to develop classes to appreciate the beauty of arithmetic and to verify the effectiveness of these classes in an elementary school, on the basis of actual circumstances of Japanese children as mentioned above.  

　　The beauty of arithmetic has the following properties in the Course of Study in arithmetic: usefulness, conciseness, generality, accuracy, efficiency, development, and the beauty.  To help pupils appreciate them, the course study (part 1) suggested that “the self-assessment activities to transform mathematical ways of viewing and thinking and to value the beauty of arithmetic are important.”  So self-assessment activities using OPP was taken in as the way in detail.  Although a number of studies on the development of metacognitive ability using OPP have been made in actual teaching, the question here is the process of learning, which sets the selfassessment activities using OPP, which helps pupils appreciate the beauty of arithmetic.  

　　To begin with, on the basis of previous studies, the structural elements on the self-assessment activities using OPP to appreciate the beauty of arithmetic can be classified into four components: the setting of a situation to give the title, the setting of a situation to design an essential question, the setting of a situation to writing the learning records, and the self-assessment at the end of a class.  Then, a questionnaire was carried out it to clarify the actual situation of elementary school children regarding the appreciation of the beauty of arithmetic.  Next, a teaching material and a teaching process were developed which considered the process that children appreciate the beauty of the arithmetic.  As a result, it has been clarified that the setting of the selfassessment activities using OPP is valid in classes appreciating the beauty of arithmetic.

　　Currently in Japan all high school students learn indirect proof.  Indirect proof is expected to train the logical thinking ability of learners.  Previous studies about indirect proof have not been able to properly associate the structure of indirect proof with understanding, despite it’s importance.  Thus, the purpose of this paper is to construct a theoretical framework that understanding of the structure of indirect proof.  There are three main theoretical frameworks in this paper: a model of indirect proof (Antonini & Mariotti, 2008), deduction system of classical logic (NK), a framework of understanding of the structure of deductive proofs (Miyazaki et al., 2017).  As a result of textbook analysis using NK, we identified the following the structure of indirect proof (meta-theory specific to indirect proof): the low of contrapositive (proof by contraposition), introduction and cut of negation & cut of double negative OR introduction and cut of negation & disjunctive syllogism & low of the excluded middle (proof by contradiction).  In addition, we revealed that the learner needs to interpret the above inference rules using the set theory symbols and diagrams.  Based on this result, each level of a framework of understanding of the structure of deductive proofs are reconstructed in order to make it applicable to indirect proof.

　　The purpose of this paper is to embody the teaching and learning of mathematics for realizing students’ deep learning to foster their mathematical competencies and abilities in high school mathematics.  Mathematical competencies and abilities can’t be fully assessed by conventional paper tests alone.  Therefore, we focused on performance assessment as a method to evaluate students’ competencies and abilities nurtured in their authentic mathematics learning.  

　　First, we reviewed the concept and procedure of performance assessment.  Second, we practiced the performance assessment on two mathematics teaching units for 53 first grade students of the high school where the first author of this paper worked with two collaborative mathematics teachers.  Based on the evaluation of the students’ reports in the two performance assessments and the results of the questionnaire conducted after the each class, we examined how the incorporation of performance assessment in the end of teaching unit affected students’ mathematical competencies and abilities.  

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

(1)  Performance assessment in the end of teaching unit is effective for encouraging students’ deep learning of mathematics in a classroom.

(2)  Evaluating students’ competencies and abilities by using the shared rubrics on mathematical tasks in performance assessment will improve the level of mathematics teachers’ understanding of teaching units and  materials.

(3)  In order to do continuously performance assessment in a school, it is important for mathematics teachers to make carefully a year plan of incorporating performance assessment in the teaching and learning of mathematics  in advance.

　　The purpose of a series of our studies is to discuss effective methods of mathematical problem posing by using computer.  We have reported in the previous studies such mathematical activities by university students who are prospective teachers.  In study Ⅰ( ), we studied the first practice of the problem posing by using computer after solving original problem, and through study (Ⅱ) to study (Ⅴ) we examined various ways of the effective problem posing.  In study (Ⅵ), we pursued the study on the problem posing process by students, and requested students to make good-quality problems.  In study (Ⅶ), we investigated the problem posing process in detail through interviews to students.  The features of our methods are to give students enough time to create problems, and to provide situations in which students make conjectures on results and get the numerical calculation by using computer.  

　　In this paper, we study the practice of problem posing by using computer after solving original problem which has some good characteristic to guide the developmental problem posing.  In addition to the above features, we employ the method to divide the problem posing into two stages.  That is, we provided situations in which students made problems after solving original problem by using computers at the first stage and made a mathematical inquiry about their problems by group activities at the second stage.  From the activities, we investigate the problem posing process in detail through interviews to students.  As in study (Ⅶ), we require the students to describe how they contrived their own problems from the original problem or other sources and how they devised the problem with mathematical inquiry by group activities, and we analyze such procedures and interviews from the posed problems by them.  The results indicate that such activities enable students to extend mathematical inquiry.   

　　As in the previous studies, the practice shows some tendency that students who tackled making problem by using computer get some deep understanding for the mathematical properties related to the problems.  Students solved problems to each other, the solver and the student who posed a problem commented on problems each other, and they did the developmental problem posing by group activities. 

　　With regard to the improvement of the quality of education in SDGs, the necessity to review the curriculum from socio-cultural perspectives is set as one of the key issues.  Mathematics education has changed with times, along with that, the socio-cultural perspective in mathematics education also has kept in step with times.  Analysis of mathematics intended curriculum from various socio-cultural perspectives have been conducted so far, however, comprehensive analysis has not been done associating with those socio-cultural perspectives.  The purpose of this paper is to define the socio-cultural perspective of mathematics intended curriculum and construct a framework for the comprehensive analysis of mathematics intended curriculum with the defined perspectives.

　　 The author constructed the framework through the discussion mainly Bishop’s mathematical enculturation and key competencies by OECD.  In addition, the contents of basic structure of Mozambican mathematics intended curriculum was analyzed using the constructed framework for confirming validity of it.

　　 As a result of the analysis, the socio-cultural perspective of the mathematics intended curriculum of Mozambique was characterized to a certain extent based on the definition in this study.  Therefore, it can be said that the validity of the basic structure in the framework was proved.  On the other hand, it was also revealed that it cannot be characterized in detail by analyzing only basic structure of intended curriculum.  It would be possible to characterize it further in detail by analyzing the medium structure and the small structure.

　　What is needed in the future education is to learn mathematics in an interactive way, in communication with other students, in negotiating mathematical ideas and understandings rather than to get the best and correct idea of a mathematical problem. To improve the quality of mathematics class interaction, we planed and conducted a competition style group-based classes in the sixth grade mathematics lessons in a research project “School Support Project” in an elementary school for about four months from September, 2017 to December, based on the methodology of “Design Research”.

　　The purpose of this paper is to clarify the characteristics of negotiation and the mathematic learning observed in the competition style of group-based arithmetic classes. In order to achieve this goal, we planned and conducted a competition style of group-based lessons in a sixth grade class. The lessons were recorded with video cameras and the detailed transcripts were prepared for the analysis. One group, which was was selected in round 1 and lost in the battle, was chosen as the subject of analysis. Because activities of the Group were expected to have affected the system of the competition style of group-based lessons. That group’s activities were divided into four phases. Among them, finally identified three negotiation situations. The results of the analysis from the interactionist point of view, mainly based on Theory of Didactical Situation (Brousseau, 1997), were summarized as follows：

1. T he situation of competition style of group-based classes has the effect of strengthening the judgment by the students rather than relying on the teacher’s instruction, and this has the possibility of encouraging the negotiation in an independent and autonomous way.

2. In the competition style of group-based arithmetic class, it has the possibility to cause the adidactical negotiation, where the children would create their own ideas to win the match, rather than the ideas that the teachers expect, and improve their ideas while adapting to their own problems. The adidactical negotiation is characterized as an interaction with the purpose of trying to solve.

3. In the competition style of group-based arithmetic class, there is no fixed relationship between the student who explains and the student who explained as seen in the group activity in the Jigsaw method. Rather, the relationship between the two changes in the competition style of group-based arithmetic class. In this sense, it has the possibility of a symmetric negotiation.

　　And, the mathematics study here was an important part of the process of the mathematical processing. It is suggested that the situation of playing the game would encourage more independent, autonomous or interactive learning of the process of mathematics.

　　As the next Education Ministry curriculum guidelines have been announced, which is the largest revision ever made in the post war period, the time is coming to start the new curriculum based on “qualification and competence.”  It is emphasized in mathematics more than ever to cultivate the general-purpose competence by making the “mathematical perspectives and ways of thinking” work as it is expected to enhance the mathematical activities being aware of its involvement in the actual society throughout each school stage.  This research study aims at shedding light on the concrete features of the critical thinking the elementary school students in the upper grades demonstrate by paying attention to the critical thinking competence as the general-purpose competence and developing and practicing arithmetic classes to foster such competence.  To achieve those, this research, based on the perspective of critical mathematics education by Skovsmose (1994), focused on the socially openended problem (Baba, 2009) as a methodological aspect.  During the session “Purchasing of a car”, which we devised, students’ various values were expressed and we observed our students making full use of their diverse critical thinking consisting of mathematical judgments and social value judgments so as to get a solution.  In the problem given, mathematical tools such as tables and formulas were used by elementary school students.  This mathematical tool was used to integrate various social values and mathematical models based on the values. This suggests that the lesson practice featuring socially open-ended problem was successful in terms of fostering critical thinking skills in students.  While we acknowledge the existence of diverse values and the mathematics used with those values, fair critical thinking of students should be further cultivated.  In addition, in revealing the social values of children, it was suggested that, when socially open-ended problem is given to children, children are required to grasp the context of the problem as their own affairs.  For future work, the development of multifaceted methods to assess the critical thinking of students are required.

　　This research has two aims.  First, the author discusses how and why dialogue expands and deepens the learning of mathematics.  Second, the author discusses the necessity of a theoretical model of dialogue for learning, which consists of a learner, talking by the learners to someone as a dialogist and a learning group as a virtual learner, which is called “trialogue” in learning sciences.   

　　Dialogue, in this context is defined as a learning process that includes the following three characteristics: it is a one on one conversation, there is no winner in a dialogue;, the participants in a dialogue must come to mutual positive compromise; and dialogue is a process of participant presenting their own idea(s) and creating new ones.

　　The process of an individual learner contributing to a dialogue can be explained by a theoretical model of sign appropriation and use (Ernest, 2010), however, we will extend Ernest’s theoretical model to explain further the process of “trialogue”.  It is theoretically possible to analyse the change in an actual learner’s conception [of ideas], which is called the appropriation, by using this extended model.  This new extended model also enables a theoretical explanation of how the conceptual change of a learner occurs in someone who merely listens to the other participant’s dialogue.