This paper simply aims at development of teaching materials on mathematical proof based upon the theory of substantial learning environment proposed by Wittmann.  We think proof attitude and proving skills will become a core for new literacy in future days after modern times.  Mathematics education has to play a big role to realize it.

　　So far mathematics education research, however, paid much attention to the entrance of school mathematics, i.e. arithmetic teaching in Japan.  She missed the exit of school mathematics, especially high school mathematics.  In other words the issue has not been problematized academically on secondary mathematics throughout six years although almost all of students go up to senior high school after graduation of compulsory junior level.  Oncoming mathematics education research, therefore, should focus on the exit of school mathematics from the various angles as well as the entrance.  

　　On the other hand, senior high school mathematics still seems to be a cluster of teaching materials or collection of new instructional resources.  They are not always examined and surveyed carefully and academically.  They need to be scrutinized another angle besides mathematics because mathematics in senior high school stands at the exit of whole school mathematics.  That is to say institutional angle, cultural angle, and societal angle are inevitable.  Many students more than half launch out society with relevant citizenship after graduation of senior high school.  Moreover they must engage in lifelong learning and highly advanced information society whether they like it or not.  

　　In this paper, “Sylvester’s  partition theorem” is chosen as a throughout proof teaching material.  She will be prepared and considered in terms of substantial learning environments as an authentic teaching material first.  She will be analyzed by means of the core of mathematics education as a scientific discipline proposed by Wittmann second.  As consequences, some methodological techniques for developing instructional materials are identified.  The techniques imply not only developmental methods in mathematics education research, but also new direction of training-system for mathematics teachers in educational practice.

　　 The aim of this paper is to clarify what the experienced teachers in Japan have regarded important in constructing mathematics lessons.  Therefore, we interviewed two experienced mathematics teachers by using mathematics lesson documents from the historical resources, which may permit them to reveal the underlying principles of their teaching more easily.  At the result of these interviews, we found some principles that worked when the experienced teachers construct their classroom lessons.  Firstly, the lessons can be identified in terms of “the narrative coherence of a mathematics lesson”.  Secondly, the teachers shape their lessons by dividing them into two parts: before and after setting the problems.  Thirdly, there are four viewpoints in talking about the document; teacher’s own teaching style, children’s learning, contents of teaching and a general teaching style.  We found that experienced mathematics teachers have their own principles of the lesson construction and were able to talk about them through the interviews.

　　 The aim of this paper is to study the learning process of interpreting and utilizing Euclidean algorithm and its characteristics. For this purpose, students will review what they learned in the process of learning common divisor and the greatest common divisor, or GCD when they were elementary students at first. Then they are introduced to Euclidean algorithm and will be able to understand it. The authors designed classes to develop learning which makes use of interpretations of Euclidean algorithm, and analyze qualitatively students’ activities and their descriptions made in classrooms. The findings are as follows: First, activities to revise the learning of GCD with an elementary school textbook of arithmetic not only increase the probability of getting different ideas from learning to interpret Euclidean algorithm, but also attach new meanings to the learning and lead to finding new values by connecting and overlooking the learning of GCD. Secondly, activities to understand Euclidean algorithm proves they have various ideas and expression styles. Discussion among the students advances the mutual understanding of the ideas as well as further interpretations, which also promotes motivation to learn and help students understand more deeply in the end. Finally, experiencing activities to learn Euclidean algorithm is applied to the indeterminate equation, which helps students deeply understand the solution to the indeterminate equation.

　　 Secondary education under the old system consists of junior high school, girls’ high school, and vocational school.  In the field of history of mathematics education, although research on the preceding two kinds of schools has progressed, the last one has not been focused on yet.  Among the vocational schools, technical schools are considered to have some characteristics, such as the focus on applied aspect and the new contents of mathematics.  Technical schools were deeply affected by the development of local industry and the industrial policy of national government.  There were only about 20 techincal schools in the late Meiji era (1900’s).  The number increased to 480 at the end of World War II.  During this period, many techincal school mathematics teachers discussed about two issues, constructing a national curricurum and upgrading teaching contents.  In this paper, refering to about fifty mathematics programs actually used in the technical schools, I showed the following point.  First, mathematics programs of technical schools vary a lot according to courses, like mechanics, chemics, and so on, or school years.  Second, although there was no national curriculum for technical school under the old system, each school developed its own course of study and introduced “higher mathematics”, such as calculus and analytic geometry, in the late Taisho era (1920’s).  It was 1940’s at the mid World War II that junior high school, which is so called the main stream of secondary education, adopted the curriculum including calculus and analytic geometry.  

　　Textbooks actually used in the technical schools are not known well.  The author will continue to search these kind of materials to make clear the mathematics classes at that time.

　　 The article discussed how Japanese mathematics education for kindergarten should be shaped regarding a fine connection with primary mathematics education.  It learned from a German case in which Wittmann developed a project, accepting a holistic and consistent approach from kindergarten to university.  The paper examined the historical development of the early mathematics education, shedding light on the transitions from the oldest version of textbooks to the latest one, as well as the relationship between pre-primary and primary mathematics, focusing on the objective, contents and methodology shown in the textbooks.  The paper concluded that the future Japanese mathematics education for kindergarten could consider the inclusive education for faster and slower learners, preparing two different kinds of textbooks; create the concrete objects of what children should acquire in mathematics and contents linked with the objectives; and connect playing to mathematical activities from kindergarten to primary level.

　　 Hirabayashi(2006) discusses a traditional aspect of mathematics education in Japan from the concepts of GEI(art), JUTSU(technique), and DO(way).  These are traditional Japanese cultural concepts and they seem to be the fundamental philosophy of Japanese people; however, it is not easy to define these concepts with words. We can point out, at least, that Japanese mathematics education surely has its own cultural aspect, and some parts seem to not easily be understood by foreign researchers, partially because these cultural aspects are not easily described in words, and no explicit definition of these cultural ideas exists.  

　　Japanese mathematics teachers have accumulated and cultivated a lot of teaching abilities, which are surely existing in teaching practices but no theoretical framework to explain does not exist.  Ninomiya & Corey(2016) examined the framework of “Implicit Abilities of Teaching”.  In this paper, the framework of “Implicit Abilities of Teaching” and “Teaching Abilities Model”, in Ninomiya & Corey(2016), are used to examine Shido-an, or “Lesson Plans”, in mathematics education.  Reviewing the history of Shido-an in Japan, two major aspects about making Shido-an, which are “Formal-Explicit Aspect” and “Substantial-Implicit Aspect”, are found out.  “Formal-Explicit Aspect” of Shido-an making gives the information of what and how is the lesson, like Lesson Scenarios, whereas “Substantial-Implicit Aspect” of Shido-an making describes the results of the investigation for the lesson by teachers, and gives the focusing points for the discussion during Lesson Study. 

　　 Examining Japanese 9th grade mathematics lesson and its Shido-an about quadratic function, some results of discussion from “Substantial-Implicit Aspect” of Shido-an are found out.  Finally, it is found that Japanese teachers have cultivated not only “Explicit Abilities of Teaching” but also “Implicit Abilities of Teaching”, partially from “Substantial-Implicit Aspect” of Shido-an through the results of planning and investigating the lesson by teachers themselves.

　　In society that change suddenly, citizens need to acquire competency in making decision in everyday and professional situations.  On making decision, they need to rate choices and judgement using probability is needed at that time.  That is, competency of judgement using probability is important as competency of decision making.  Therefore, the purposes of this paper are to: ・ Propose the contents of learning that consider training of competency of judgement using probability in making decision. ・Derive the task based on the contents of learning for revision of teaching probability. The results of this paper are the following three things: ・ From the development of history of probability, there are suggested that a way of mutual understanding of the frequency idea and the subjective idea are important, and citizens need to do judgement using probability  that follow Bayes rule in everyday and professional situations. ・ It is effective that students learn Bayes theorem that is methodology of Bayes rule to train the competency given above. ・ From result of the research on actual conditions in this paper, because the treatment of Bayes theorem is inadequate in the teaching probability currently being used in view of training the competency given above, it is needed that the overall revision to the teaching probability from the Bayes theorem on down from the current.

　　As times and societies change, school education must also change. The focus of curriculum in new times moves from content to skill and capability. Curriculum development that focuses on developing skill and capability is required. In this paper, this issue is addressed in the case of statistics education. Specifically, this paper shows that statistical inference skill could not be gradually developed in the content-based current statistics curriculum in Japan. This paper claims the necessity for developing skill-and-capability-based statistics curriculum. To do so, first, five stages of development of statistical inference skill are identified and described based on findings obtained in previous research. The two big ideas of “sample and sampling distribution” and “variability,” which are essential ideas to the statistical inference, are particularly taken into account. Then, the goals and contents described in the current statistics curriculum in Japan are analyzed in terms of the five stages. As a result, it is revealed that the current statistical curriculum in Japan follows a continuous and systematic arrangement of statistical contents, but that it does not necessarily guarantee staged development of statistical inference skill.

　　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 like learning mathematics and students to 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 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 find them, the course study shows that “activities to mathematically understand and approach matters of everyday life are important.” Although a number of classes to solve a problem connected with matters of everyday life have been made in actual teaching, the question here is the process of learning, which returns to situations in everyday life at the end of a class, and helps pupils transform mathematical ways of viewing and thinking.  

　　To begin with, on the basis of previous studies, terms and conditions on classes to appreciate the beauty of arithmetic can be classified into four types: clarification of the beauty, the setting of an inevitable problem, the setting of a situation to examine and consider the beauty, and the presentation of the situations in everyday life at the end of a class.  Then, I designed a questionnaire and carried out it to clarify the actual situation of elementary school children regarding the appreciation of the beauty of arithmetic.  Next, I developed a teaching material and a teaching process with due regard to the process of appreciating the beauty of the arithmetic for children.  As a result, based on the above terms and conditions, it has been clarified that the setting of the situations connected to the matters of everyday life at the end of a class is valid in classes appreciating the beauty of arithmetic.

　　The purpose of this research is to clarify what humanity is formed through mathematics education practices.  To approach to this big purpose, it is necessary to focus on the particular students because they participate in the mathematics lessons in their own, different ways.  The author discusses their humanity theoretically and compare with each other Therefore, in this paper, the author aims to make the hypothesis about students’ humanity formed through secondary school mathematics education practices by focusing on student teachers.  

　　The author uses Paul Ricoeur’s identity theory which considers humanity in terms of habits to make the hypothesis.  Firstly, the author describes the habits formed in secondary school mathematics learning by referring to the Ricoeurs’ definition of habits.  Secondarily, the author makes the theoretical framework by adopting the hypothetical habits, and finally, interprets the students’ habits obtained from the worksheet descriptions and interview data by applying the framework.  

　　As the result of this research, the author made the hypothesis of the existence of humanity with a “confident identity”, which is acting as guaranteeing the correctness of logic.

　　In the current Course of Study for high school mathematics, upbringing of the mathematical power of expression is made much of. From national and international scholastic ability investigations, it is well known that there is a problem for Japanese children in speaking one’s thought with showing reasons definitely. Therefore, in this paper, the author focused on power to describe one’s thought with showing reasons definitely, namely power to express mathematically the process of thinking in order to suggest some means to overcome the identified problem. First, based on the review of previous studies, the power to express mathematically the process of thinking is classified into five elements; “Use of the mathematical expression”, “Being conscious of unconsciousness”, “Visualization of the consciousness”, “Clarification of reasons and the logic”, and “Sophistication of the expression”. And the author proposes the structure of these five elements. Second, based on the structure, six guiding principles are extracted for three stages of “Expression to promote thinking”, “Clarification of reasons and the logic”, and “Sophistication of the expression” in reference to previous studies on meta-recognition and explanation. Finally some teaching methods are proposed under each of six guiding principles for improving students’ power to express mathematically the process of thinking in high school mathematics lesson.

　　The purpose of this study is to reveal the roles of gestures that promote the interpretation of diagrams and language. We perceived of association between gesturing and proof construction by examining the process of constructing geometric proof from the semiotic viewpoint of gesturing. We analyzed the interviews of the four pairs of ninth graders who solved and discussed the four problems of proof including the understanding of generality of proof. We suggest that gestures have three roles.

　　First, gestures promote the interpretation of a diagram and language during the process of constructing a proof.

　　Second, gestures allow for the understanding of the generality of proving. By showing meaning of language as action, they can relate the different parts of each diagram together, and can interpret relationships between the diagram and language. Also, by showing one form of deduction in one diagram on another diagram, students can interpret correct deduction.

　　Last, there is a possibility to promote an inter-intra double semiotic process of interpretation through simultaneous use of gestures and language for diagrams. Gesture can be an important vehicle to convey one’s own massages to others.

　　STEM (Science, Technology, Engineering, and Mathematics) education has become famous all over the world, and statistics education also has to play its role in this education. That’s why the author focused on statistics education, and the purpose of this paper is to clarify the characteristic of statistics education in Japan.  

　　In order to achieve this purpose, the author compared statistics education in Japan with this education in New Zealand. The reason why the author paid attention to New Zealand is that Japan has to cope with globalization and internationalization from now such as New Zealand which has already had much experience on them. In the comparative analysis of textbooks, the author designed three lenses: the presence or absence of contexts within problems, the presence or absence of practicalities, and the presence or absence of contexts within expected solutions for problems.   

　　As a result of the comparative analysis, the following characteristic of statistics education in Japan was suggested: statistics education in Japan is endeavoring to add contexts to problems themselves, but it cannot be said that statistics education in Japan is endeavoring to add contexts to expected solutions for problems.

　　The purpose of this paper is to elaborate a prescriptive framework for designing mathematical activities necessary for shifting students’ indirect argumentations from naïve to proof-like.  The framework was derived from the following four theoretical resources: 1) Features and structures of indirect proofs; 2) Critiques of contents-general proof research; 3) Reflection on mathematical methods; 4) Mathematical literacy focusing on knowledge-how.  As a result, we proposed a general prescriptive framework for designing mathematical activities necessary not only for shifting students’ indirect argumentations from naïve to proof-like, but also for constructing mathematical knowledge-how.  More concretely, we made a conclusion that four questions should emerge from students’ mathematical activities in the following order: 1) How can we solve a particular problem? (The construction of an indirect argumentation); 2) Why can we solve in that way? (The construction of the method of indirect proof); 3) When can we solve in that way? (The construction of a situation where the method of indirect proof is applied); and 4) Why can we solve in that way at that time? (The construction of a list of heuristics to apply indirect proof methods).  Although our discussion starts from the particular topic of indirect proof, this paper succeeded in extending the general theory of mathematical literacy.

　　The purpose of this study is to describe how do teacher’s beliefs about curriculum knowledge work in the specific contexts, such as planning and practice through analyzing grade four mathematics class. From epistemological perspective, belief about curriculum knowledge is defined as value judgement based on curriculum knowledge, which appears as an action. In accordance with this definition, this study focuses on decision making situations and picks up three specific cases in the contexts with gaps between lesson plan and actual lesson; (1) students interests differs from the aim of today’s lesson, (2) students disagree to the answer teacher expected, (3) teacher picks up student’s idea which is not written on the lesson plan. The researcher used semi-constructed interview with video clip showing three specific scenes. The protocol was analyzed through identifying teacher’s interpretation, perception, decision making, and observable behavior.

　　The topic is the area of quadrilaterals. The aim is to let the students think how to find the area using the number of right triangles. We find three kinds of curriculum knowledge, which are curriculum knowledge about the sequence and relevance among grades and domains, knowledge about teaching materials (right triangles), and knowledge about conception of quadrilaterals and area. The results showed that teacher’s value judgement is strongly related to these knowledge. However, what curriculum knowledge influence on value judgement directly is knowledge about teaching materials, which are the function of a teaching material, values and goodness of a teaching material.