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 test suitability of the levels of Subjectification by surveys of the reality of students. In the test, I check the following three points;

　(1) Whether at least one student corresponds to each level of Subjectification

　(2) Whether all students correspond to any of levels of Subjectification

　(3) Whether students who correspond to a certain level behave like students correspond to the level

　　 As a result, I argue that the validity of the levels of Subjectificaation is mostly certain.  And through surveys, I define differences in grasping knowledge between each level of Subjectification.

　Level1: Grasping knowledge as a formula

　Level2: Grasping knowledge as a formula, and Particular as a process

　Level3: Grasping knowledge as a process, and a formula as the process

　　 Finally in order to make up for these differences, I propose leading principles.

　　 The purpose of this research is to consider students’ mathematical ideas of how the mensuration formula is deduced and how these ideas are applied in the process of first revising how to deduce the area of a circle and then trying to deduce the volume of a sphere through the teaching unit of integral calculus.  In this unit students express the process of finding the volume of a sphere mathematically and interpret it by reviewing the process of obtaining the area of a circle.  Therefore we design and practice the unit of integral calculus in order to deduct mensuration formula for circle and sphere.  We consider typical student’s activities by the qualitative method in the teaching unit of integral calculus.  Then we clarify students’ mathematical ideas and how the students apply these ideas effectively.

　　 Students’ activities through classes can be summarized into four points.  First, they expressed the description of the arithmetic textbook mathematically, which helped them interpret the area formula of a circle more deeply and give new ideas for deducing the volume formula of a sphere.  Secondly, students interpreted the process of deducing the volume formula of a sphere by connecting mathematical ideas in reproducing the area formula of a circle with transition between two dimensions and three dimensions.  Third, an encounter with a circular argument made the students recognize that they should always bear precondition in mind.  At the same time, it gave them a good opportunity to explain about their ideas to others.  Finally, students attempted to apply the ideas of the area formula of a circle and the volume formula of a sphere which were produced in the previous stage by modifying and improving their ideas.

　　 The aim of our two consecutive papers is to advance understanding on the nature of proving activities required in the inquiry-based learning called Study and Research Paths (SRP), which is based on the paradigm of questioning the world proposed by Chevallard in the framework of Anthropological Theory of the Didactic  (ATD). In this paper, we first introduce the paradigm of questioning the world and the notion of SRP as well as the Herbartian formula, the media-milieu dialectic, the five phases of SRP, and the seven dialectics, and then explore the nature of proving activities in SRP through a theoretical consideration on the process of elaborating an answer to a given question. In the situations of SRP, the learners need to ask why-questions in order to understand the answer obtained from the media and use it to elaborate their own answer. We as a result reached at the claim that the situations of SRP trigger the proving activities to understand the mathematical structure which are difficult to be realized in the ordinary mathematics class based on the ‘old’ paradigm of monumentalism, due to the didactic contract spontaneously created in such situations. This claim is to be empirically verified in the second paper and in our future study. 

　　 The article discusses the challenges of a lesson study activity held in Serenje, Zambia, dealing with 1-digit multiplication, particularly focusing on teaching and learning including Kyozai Kenkyu and contents of discussion in lesson study.  Fourteen teachers for grade two and three gathered and planned two lessons.  Two trainers in the ministry were the facilitators of the lesson study.  The qualitative analysis was underpinned, utilising the transcription of lessons and lesson study, short interviews to children and the data of participatory observation in Grade 2 and 3 lessons.  First, the analysis showed that students did not understand how to count and recognise points shown in array diagrams in lesson.  Moreover, the array diagram was not effectively used for them to understand the concept of multiplication.Second, teachers did not succeed in guiding students’ better understanding in multiplication since they did not explain well when students did not understand in lesson.  Third, the analysis of the discussion revealed that some teachers did not fully understand the meaning of the order of two numbers in multiplication, discovered in the reflective discussion in lesson study. Conclusions for the improvement of lesson study are two points: Kyozai Kenkyu should be more focused in the whole group of participants by taking time; Discussion should hold the gap between the planning and the implementation in a concrete manner. 

　　 The purpose of this research is to propose the methodology to design and conduct a mathematics lesson through students’ activities with examples. In this paper, I made the lesson design model to enhance and use their activities with examples and then, conducted a series of experiment lessons based on it about a logarithm function in high school mathematics to analyze empirically.

　　 In principle, the model consists of three stages of students’ activities - operation, reflection, and application -  and a concrete teaching way to change them effectively.  Students form their own typical example at each stage  and change the stage by making use of it.  The teaching way with the model is necessary for a teacher to have the students’ activity stage increase.

　　 I designed a series of lessons based on the model and conducted it.  And then, I analyzed it qualitatively what typical examples students had and how students’ activities changed.  As a result, I argued that students had their own typical examples and changed their activities by using them.  Students had their own typical examples that the inequality is the example of inequality including exponent, because students calculated directly inequality 2^x > (6.37 × 10⁸)² in the class of first time. In the class of the eighth time, students solved the same problem using common logarithm and they had their own typical examples that the problem or inequality is the example that they solve the problem using common logarithm.  In the class of the ninth time, students changed the activities that they utilize a typical example of the eighth time as for it and consider the structure of other problems.

　　 I suggest that when students change the activities, they change the formation of their own typical examples, too.  Therefore, the result implies that students change the examples while being activities by learning of mathematics lessons and I could demonstrate the effectiveness of the model. 

　　 The aim of our two consecutive papers is to advance understanding on the nature of proving activities required in the inquiry-based learning called Study and Research Paths (SRP), which is based on the paradigm of questioning the world proposed by Chevallard in the framework of Anthropological Theory of the Didactic (ATD).  In the preceding paper, we explored the nature of proving activities in SRP through a theoretical consideration and pointed out that the situations of SRP would trigger authentic proving activities to understand the mathematical structure.  In this paper, we report the results of a teaching experiment conducted for undergraduate students of a teacher training university in Japan, and verify our claim.  A series of mathematics classes based on SRP was designed around the question of finding a method of calculating the cube root of a given number by using a simple pocket calculator.  We analyzed the data collected in this experiment from the following viewpoints: what kind of didactic contract is there in the realized situations, what kind of proving activities arise due to the didactic contract, and how the didactic contract promote or prevent the proving activities?  As a result, we found that the students were engaged in many proving activities by asking by themselves why-questions in the process of inquiry, due to the didactic contract entirely different from the one identified in the ordinary mathematics classroom.  We also identified that these proving activities play various functions of proof in the media-milieu dialectic.

　　 Inequality is one of the topics that high school students find difficulties to learn in Japan.  Some previous studies investigated the difficulties of learning inequality through identifying and considering, in terms of its teaching, different approaches to solve inequalities (Ito, 2002; Hattori, 2010, 2011).  Looking at inequalities in mathematics textbooks of Japanese secondary schools, one may find that the notion of inequality appears in different places and in different ways: as an object to be solved, as an object to be proved, as a method to show the magnitude relationship, etc.  We then consider that student’s difficulties result from its complicated nature that should be primarily uncovered.

　　 The aim of this paper is therefore to advance understanding on the nature of inequality with a view to clarify the origins of student’s difficulties in its learning.  We try to find an answer to the question “what is inequality?”  Relying on Anthropological theory of the didactic (ATD) developed by Chevallard (2006), our question in this paper is formulated as a task of elaborating a reference epistemological model of inequality by means of the notion of praxeology.  In order to accomplish this task, we first identify how the inequality is dealt with in different kinds of mathematics such as mathematics in the history and school mathematics in Japan, and then characterize its nature by the elements of praxeology (types of tasks, techniques, technologies, and theory).

　　 As a result, we first found a cultural factor that complicates the nature of inequality, that is, we use two Japanese terms, in a way very ambiguous, that correspond to the term inequality in English: futoushiki (literally translated to the expression of inequality) often used in secondary schools denotes the expression which represents a magnitude relationship (or inequality), while daishou-kankei (literally translated to the relationship of small and large) denotes the magnitude relationship.  While this distinction is not necessarily shared in the community of mathematics education in Japan, the former is often a representation of the latter.  This nature of inequality in Japan results in a twofold praxeology as a reference epistemological model: the one on the magnitude relationship (or inequality) including two types of tasks (solve an inequality and prove an inequality) and the other on the representation including three types of tasks (represents a magnitude, an interval or range, and a set).

　本研究は，児童の数学的概念の理解におけるオープンアプローチの影響を調査したものである。対象はジャマイカの地方にある２つの小学校の第４学年である。１つの学校は男女別のクラス編成である。授業はオープンアプローチな教授方法を用いて行われた。量的分析では，すべての男子が女子よりよい成績を示したが，性差は問題によって様々であった。

　　 The purpose of this paper is to describe what qualities and abilities that I have as an experienced mathematics teacher are displayed and how they function in the process of teaching unit development.

　　 I analyzed a journal in Research in Mathematics Education Vol. 14 published by Japan Society of Mathematical Education, of which I am the first author.  I used SCAT (Steps for Coding and Theorization) as this study method and analyzed the part of teaching unit development in this journal in the form of autoethnography.  As a result, I pointed out: (1) Although an experienced mathematics teacher pays serious attention to curriculum and textbooks in teaching unit development, he has a conviction that he should improve and develop the content of curriculum and textbooks in order to enhance mathematical views and approach suitable for the students he is teaching, and to make them realize the value of mathematical thinking; (2) An experienced mathematics teacher develops teaching unit where the students’ learning history (mathematical contents, knowledge and skill) and the value of mathematical thinking meet, and he analyze teaching materials from the viewpoint of feedforward; (3) An experienced mathematics teacher enhance the practicability of teaching materials in class by going back and forth between teaching materials and the image of class.

　　 Next, I created an analysis matrix for teaching unit development process, referring to the studies of PCK (Pedagogical Content Knowledge) by Shulman (1987), Grossman (1990) and Ball (2008).  Using this matrix, I analyzed how my PCK arises and functions in the storyline created in SCAT.  The analysis made it clear that knowledge of educational goal and curriculum content functions first and specific mathematical knowledge of each unit is added; furthermore, knowledge of mathematics and students begins to function followed by knowledge of mathematics and its teaching; and again knowledge of educational goal and curriculum content is employed.  Based on these results, I showed a structural model of my PCK in teaching unit development.

　　 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.  One famous practice in Japanese mathematics education is “Lesson Study”.  There are many teachers and researchers who try putting it into practice all over the world, but the strict definition of Lesson Study, the proper procedures of Lesson Study, how Lesson Study should be practiced, etc.  does not exist.  Nevertheless, Japanese mathematics teachers have been doing Lesson Study for many years without an explicit definition and characterization of the practice. This is one example of the culture of Japanese mathematics education which is based on Japanese philosophy.

　　 In this paper we examine three examples of fundamental aspects of Japanese mathematics teaching which are, at least partially, hyidden and implicit.  We examine these aspects by contrasting and comparing Japanese and US viewpoints in specific instances and support our claims with empirical evidence.  First, we begin with the ability to evaluate a high-quality lesson.  A US high school lesson was viewed quite positively by all Japanese researchers but US researchers had a neutral or negative evaluation of the lesson.  This difference may be caused by views that value different aspects of the lesson.  Many features of a lesson which Japanese and US researchers use to evaluate the quality of a lesson may be implicit and hidden.  Second, we examine the ability to craft a Lesson Plan.  Most Japanese teachers are able to craft a detailed Lesson Plan for Lesson Study, but many very capable US teachers struggle to craft a detailed Lesson Plan.  The ability to craft a good Lesson Plan is a key skill to be an excellent teacher in Japan, but it seems not to be in the US.  This difference may point to implicit, culturally driven, aspects of being an excellent teacher in the two countries.  Third, we examine the definition and characterization of Kyozaikenkyu.  There are several differing explanations about the concept of Kyozaikenkyu in the US.  It is unclear for US teachers, however, no Japanese teachers struggle doing Kyozaikenkyu, even though it lacks a strict definition.  It seems the concept of Kyozaikenkyu is unconsciously shared with Japanese teachers.

　　Finally, we use the model of Implicit Abilities of Teaching to discuss these three cultural characteristics. Thinking about implicit abilities, although still a hypothesis, may help to make sense of these fundamental cultural aspects of Japanese mathematics education.

　　 The purpose of the study is to consider a framework of proof reading comprehension specific to mathematical induction and to illustrate levels and difficulties of mathematical induction by means of the framework.  There are two main ideas used in this study: the idea of “Mathematical Theorem” posed by Mariotti et al. (1997) and the theoretical model of proof reading comprehension formulated by Yang & Lin (2008) and Mejia-Ramos et al. (2012).  In this study, the author will combine these two theoretical ideas in order to consider a framework of proof reading comprehension specific to mathematical induction.  Data are collected through the nineteen undergraduate students’ writing responses to a set of “scripted  statements and proofs” and “scripted  dialogue”.  As a result, different difficulties are characterized in terms of the three levels: meaning of terms and statements (first level), logical status of statements and proof framework (second level), and justification of claims (third level), in the following ways.  A difficulty at first level can be seen as a presupposition that a given statement is always true.  One of the difficulties found at second level is related to a weak understanding of connections between the statement and its proof.  Lastly, at third level, there is a misunderstanding of the proof of implication statement.  Some implications for further researches and teaching practices are also discussed.

　　 The purpose of this paper is to obtain implications to develop curriculum on probability from the point of view of formation of the student’s concept of probability.

　　 Firstly, I referred to the duality of classical probability and frequentistic probability from Otaki (2011) and the relation of the concept of probability and that of randomness from Kawasaki (1990).  Then, I pointed out that, to form the concept of probability, the followings are necessary: “understanding both classical probability and frequentistic probability, and mutual connection between them” and “understanding of the concept of randomness preceding that of probability, and connection between them”.

　　 Secondly, I considered the current curriculum critically from the point of view of formation of the concept of probability, and found that the current curriculum have some problems in this point of view.

　　 Finally, I suggest “spiral curriculum” from this point of view based on “Principles in Setting and Arranging Teaching Contents” in “A Study on Constituent Principles of Curriculum in Mathematics Education” (Nakahara, 2008).  This paper shows that spiral curriculum is effective to form the student’s concept of probability and is required in the current secondary education.

　　 A purpose of this paper is to elucidate the process for developing the regression line from the viewpoint of negation theory in order to clarify the proper learning and teaching of the regression line. For achieving this purpose, some base concepts related to the regression line is identified, and the relations between the regression line and them are characterized by negation theory.

　　 As a result, two kinds of negation are necessary as follows. First, through recognizing that a linear function cannot be applied to the real context that two variables do not change regularly, “the straight line not to ignore the variability of data” is considered as the regression line. Then, through recognizing that the regression model cannot be applied for problem situation to quantify the degree of the symmetric relations between two variables, the regression model is seen as “the model that the relations between two variables are not asymmetry”.

　　 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. The purpose of this paper is to provide suggestions for the future of statistics education in STEM using realizing modeling with Wittmann’s systemic-evolutional perspective.

　　 In order to achieve this purpose, I analyzed previous literature in terms of systemic-evolutional perspective. Two points emerged from previous mathematical modeling: (1) a model is refined by the repetition of modeling cycle, (2) all phases of modeling cycle are expressed in the form of “○○ in the problem solving process”. In this paper, I discussed the characteristic of modeling with systemic-evolutional perspective. The educational object of modeling is the problem posing process. In statistics education, this is constructing assumption of problem solving process aiming for common agreement with others.

　　 Based on the above-mentioned results, I suggested a backward emergent modeling from model-for to model-of. This is an effective way of considering the problem posing process at the unit level, and verifying assumption of problem solving process through the MEA’s problem solving process at the class level. Finally, I proposed a new paradigm of problem posing in which students inquire and refine the methodology to pose problems by themselves within phenomena involving the complex system.

　　 A diagram is a kind of an inscription.  In mathematics classroom, students think with different diagrams mathematically, so a diagram is considered a special type of an inscription, that is, it has an ability to enhance students’ mathematical thinking.  In this research, we focus on the meaning and function of a diagram, and the goal of this article is to analyze students’ mathematical thinking with a diagram from the several theoretical perspectives - semiotics and embodiment.  We assume that a diagram has a dual, complementary nature - cultural and cognitive - and it is the reason why we have chosen these theories.  By standing on these theoretical perspectives, we can really understand actual students’ mathematical thinking.

　　 The results obtained from discussions in this article demonstrate that a diagram has two important rules for construction and usage.  From the semiotic perspective, these are special characters to identify an inscription as a diagram; on the contrary, from the embodiment theory, these are based on or emerged from bodily actions as a certain typical pattern.

　　 Plus, we compared these perspectives by using the networking methodology.  Comparing with each other in the term of a basic principle, research question, methodology and result, we could describe some characters of mathematical thinking with a diagram such as:

● Mathematical thinking is a process to construct and use a diagram, while it is a process of diagramming.

● Mathematical thinking is directed by mathematical signs, while it is embodied.

　　 If we want to teach mathematics more effectively, we need more elaborated questioning with epistemological backgrounds for enhancing students’ mathematical reasoning.  The purpose of this paper is to develop a theoretical framework for designing teachers’ questioning in classrooms toward fostering mathematical know-hows.  The present paper extended a radical constructivist methodology: Conceptual analysis.  Through theoretical considerations, two important extended ideas were proposed for designing teachers’ questioning.  The first one is where it is difficult for learners to proceed on their learning trajectories, and the second one is what interpretation of social situations learners may make.  For these two viewpoints, the use of the two theoretical frameworks were proposed from previous theoretical studies: Instantiationdescription-chain model and radical constructivism with motivational assumptions.

　　  As a result, the paper developed and proposed a theoretical heuristics for designing teachers’ questioning, like Polya’s heuristics for problem solving.  In addition, as an example, the author tried to use the heuristics for designing teachers’ questioning, focusing on “an application task of quadratic functions” in Japanese high school mathematics.  Through the example-based consideration, the value and the limitation of the proposed theoretical heuristics is discussed.

　　 The purpose of this paper is to investigate senior high school student’s mathematical inquiry about the sequence of rational numbers coming nearer infinite √2 and the natural number and its limit value, which focus on discovering mathematical character thinked by the example of the sequence of rational numbers, such as thinked about the example, thinked through the example, thinked beyond the example. We investigated student’s mathematical inquiry by teaching units about the sequence of rational numbers by qualitative methods.

　　 As a result of our discussion, we obtained several insights:

　(1) By thinking back own problem solving process and various interpretations about the sequence of rational numbers coming nearer infinite √2 and the natural number and its limit value, a student’s mathematical inquiry is encouraged to consider several methods of discovering the mathematical character.

　(2) Substantial questions that students want to think deeply and prove are generated about convergence of the sequence of rational numbers and its limit value through comparing several sequences coming nearer to the same mumber and these recurrence formulas. Student’s questions accelarate to consider and discuss about the sequence of rational numbers and its fundamental truth.

　　 The aims of this study are to define curriculum knowledge by reviewing previous studies and to clarify the aspects of curriculum knowledge in a part of lesson plans written teacher’s idea for the lesson. The curriculum knowledge in this study consists of two parts, knowledge of intended curriculum and horizon contents knowledge. The knowledge of intended curriculum includes aims of education, subjects, topics and lessons, learning contents in the past and in the future, sequence of learning contents. Horizon content knowledge is mathematical horizon which consists of four elements, knowledge of advanced mathematical knowledge, nature of mathematics, mathematical activities and characteristics of mathematics. Particularly teacher’s horizon content knowledge impacts on lesson construction and methods in the case of actual lesson.

　　 Using this curriculum knowledge, a part of lesson plan as a product of transformation is analyzed. Three lesson plans are selected and author defined what kinds of curriculum knowledge there are. Firstly lesson plan are compared with teaching guide of course of study which is intended curriculum. After that the curriculum knowledges are chosen and defined. A part of lesson plan includes advanced mathematical knowledges, for example the idea of limits and analytic geometry and also such kinds of horizon content knowledge clearly impact on the aim of the lesson, the value of subject matter and way of teaching based on long span sequence.