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.  In this article, the analysis on the reflections of a teaching profession graduate student “C” in teaching practice was conducted as a case study in terms of “Time Aspect” for reflections on lesson practices.  Time aspect means the characteristics and property of reflection relating to time such as when reflections occur, the period of action (lesson practice) reflected by mathematics teachers and so on.

　　As the result, the characteristics and its change of C’s reflection on the learning activity of finding the area of a complex figure, pupils’ explanation and pupils’ whole discussion were found.  In addition, the factors of the change were recognised as the temporal and spatial interval from lesson practice, looking straight of objective facts in lesson practices, and understanding of intended curriculum based on experiences of lesson and reactions of pupils.

　　Previous Japanese studies on mathematics education have been focussed on primary education.  This is because learning has been prioritised over teaching.  Yet if we consider how contemporary society is lifelong learning, then it is upper secondary education that we ought to concern ourselves with and must uncover the prospects from the vantage point of didactics.  Thus, we aimed to establish a research method in the form of didactics of mathematics in terms of curriculum development in mathematical induction.  

　　As teaching is central to didactics of mathematics, the study of teaching materials becomes the most important core of this research method.  In order to bring together the various backgrounds relating to the study of teaching materials, we turned to Prof. Dr. Wittmann’s SLEs and Prof. Dr. Chevallard’s didactic paradigm of questioning the world.  While the former emphasises mathematical contents themselves as well as their systematisation, the latter takes an interest in methods of mathematical questioning.  As such, we developed an SLE and designed its implementation according to the didactic paradigm of questioning the world before conducting a teaching experiment.   

　　This paper concludes that such a study of teaching materials can itself be framed as a research method in the form of didactics of mathematics.  This is a case study on mathematical induction, where we developed ‘cut & paste problems’ by mixing isometric and isoperimetric problems and conducted a teaching experiment through questioning-the-world teaching.  The paper also reports on the results thereof. 

　　A purpose of this paper is to show the structure of the questioning in refine & elaborate. In this paper, we focused on argumentation as a viewpoint for conducting a theoretical consideration of the questioning in refine & elaborate. Argumentation is a technique or method of establishing a claim, which is a concept focusing on the process of collective discussion. First, we considered refine & elaborate as a phase aimed at overcoming epistemological obstacles in the generalization process, and identified the static and dynamic states of the conception. Second, we modeled the argumentation that leads to overcoming the epistemological obstacles, and showed the aspect using concrete examples. Then, to derive some questionings in refine & elaborate, we focused on the rebuttal and the qualifier showing the acceptability of argument. Because of the consideration, it was shown that the questioning in refine & elaborate is to actualize rebuttal and qualifier. And we revealed that questionings to actualize rebuttal R₁, rebuttal R₂, qualifier Q₁ and qualifier Q₂ are requested when organizing argumentation. Rebuttal provides “criteria” which are grounds for reference in determining appropriateness of the qualifier. Finally, we concluded that the questioning in refine & elaborate has a relationship of “rebuttal R₁ ⇀ qualifier Q₁”, “rebuttal R₂ ⇀ qualifier Q₂”.

　　At present, “matrix, linear transformation” is virtually not taught in Japanese upper secondary schools, and “complex plane” is the only unit where the concept of transformation can be involved, while it is pointed out by not a few mathematicians that the concept of transformation is important.  In fact, transformation (on plane) is also taught in junior secondary schools through the study of “motion of figures”, and is, we consider, a fundamental object in mathematics as well as function.  If so, is it possible to teach the concept of transformation sufficiently in study of “complex plane”? And what conception of transformation should be obtained in the study of “complex plane”?  

　　We considered the concept and the conception of transformation from the view point of APOS theory, and identified its conception which should be studied through the study of “complex plane” as a framework.  In APOS theory, mental structures, which are necessary to learn a specific mathematical concept, are studied. They consist of Action, Process, Object and Schema structures.  Referring to this theory, we identified in detail the proper conceptions of each stage of APOS in the study of transformation, and in particular we claimed that conceptions up to the Object conceptions of transformation should be acquired.  Also we surveyed the textbooks from the previous framework and pointed out that there exist some problems in constructing Object conceptions of transformation in the current teaching.  Then, we proposed a genetic decomposition, in which students could construct the required concept of transformation.  A genetic decomposition　is a hypothetical model in APOS theory that describes the mental structures and mechanisms that a student might need to construct in order to learn a specific mathematical concept.  Finally, we proposed teaching materials of mathematical activities which prompt students to encapsulate the Process conception of transformation into the Object conception of transformation.

　　The focus of curriculum in new times moves from content to skill and capability.  The two methodologies for developing the new curriculum, curriculum development and backward design theory, are paid attention to. How can the two methodologies function for developing skill-and-capability-based curriculum?  This paper attempts to design the statistics curriculum to develop statistical inference skill thorough considering the two methodologies critically.  

　　Skill-and-capability-based curriculum is developed in the following process: firstly, the overall staged development process of certain skill or capability is clarified; then the curriculum is designed based on the process; and the designed curriculum is implemented and improved by a teacher.  However, methodology of curriculum development can only function effectively for improving the teacher’s competency for implementing the designed curriculum.  Backward design theory can function effectively as guidelines for developing the curriculum based on the overall staged development process of certain skill or capability.  Thus, the addressed task to develop the skill-and-capability-based curriculum is to clarify the process and to design curriculum based on the process.  The overall staged development process of the statistical inference skill has been clarified in the previous study.  

　　The development process is regarded as a long-term curriculum in terms of backward design theory.  In addition, it is necessary to design a short-term curriculum to develop the skill gradually in the process.  In other words, the short-term curriculum is designed in terms of the three aspects, that is, learning goal, assessment, and learning experience and teaching, which are examined specifically.  Finally, the suggestion for developing skill-and-capability-based curriculum in school mathematics and the remaining addressed issues are summarized.

　　Seven companies publish mathematics textbooks for use in junior high schools in Japan. Introducing square roots in mathematics class, all of them discuss the relationship between the area and the side of a square. For example, there is a description of “If the area of a square is 2cm², how many centimeters is the length of one side of a square?” Hitotsumatsu (1990) casts doubt upon the description of “The solution of x² - 2 = 0 is + √2 and - √2 ” (p.7).

　　The aim of this paper is to develop a new approach to introduce square roots in teaching for readiness. That lies hidden in Square Dotty Grid and Isometric Grid. Our idea is based on the figures of “If a segment is multiplied by a and one more time running, then it is multiplied by 2 and 5 on Square Dotty Grid, 3 and 7 on Isometric Grid” We carried out classwork study and realized that the approach will succeed. 67 second-grade students in junior high school made three squares or three rhombuses using a diagonal’ length of a square or a rhombus, and responded to a question in worksheets during class. As a result of the survey, their notes gave us three suggestions. First, there is a possibility that students consider with the ratio in length between each one side of three squares or three rhombuses. Second, these teaching materials provide learning opportunities of inductive, analogical and deductive inference for students. Third, students may notice irrationalness themselves.

　　Many undergraduate students think that studying mathematics means memorizing mathematical formulas. As a result, they lack an understanding of the concepts of mathematics and parts of logic thinking.  Another common misconception is that mathematics is something that is not useful in everyday life.  

　　It is important to educate students so that they shift from memorizing mathematical formulas to using mathematics creatively.  For mathematics education in the undergraduate course, I did not pay attention to the raising of mathematical literacy.  The purpose of this research is to help students acquire mathematical literacy in undergraduate courses.  Mathematical literacy in this study refers to various cognitive abilities such as thinking, judgment, reasoning, logic, expression, etc., through the execution of mathematical modeling based on the definition of PISA by OECD.  In other words, mathematical literacy is the ability to formulate problems related to mathematics existing in the real world, solve the formulated problems using mathematical events, etc., and reinterpret them as real solutions to help in judgment and decision-making.  It is the ability to do.  It is necessary for students to realize that mathematics is deeply rooted in the real world and can help to solve problems.  Students have confirmed that mathematics is useful in society through mathematical considerations and practical classes incorporating examples that can be learned ambitiously into lessons.  I report a notable case.

　　New Course of Study guidelines have been made public, and a “talent and ability” based curriculum will finally be enforced.  The most pressing issues today are to identify the talents and abilities to be fostered through math education in the coming “talent and ability” based curriculum, to develop learning and teaching methods based on the curriculum, and to promote the improvement of lessons.  In this research, we focus, therefore, on critical thinking as a talent and ability to be nurtured through math education and one of the “21^st century abilities” (Katsuno et al., 2013).  The objectives of this research are the following: first, to position critical thinking as a thinking method for solving problems and clarifying the characteristics of math education; and second, to develop lessons using various problems from “combinatorics” as materials for fostering and strengthening the efficacy of critical thinking and the critical thinking abilities of children as a result of completing the lessons.   

　　The results of this research indicate the following findings.  For the first objective, critical thinking in a broad sense is shown to have an important effect on the process of solving problems in social contexts.  In terms of the characteristics of judgment standards, we can identify personal or social value judgments as well as mathematical judgments.  For the second objective, we focus on “socially open-ended problems”, as described by Baba (2009), aimed at developing experimental lessons for nurturing critical thinking in a broad sense.  In our prepared lesson, we find that students perform critical thinking and make alternative choices through compound-eye analysis based on their own social value judgments for developing “better” opponent matching in league competitions without accepting on faith the “ideal” opponent matching in league competitions completed by using mathematical induction.   

　　A future issue is to develop a method for evaluating critical thinking as described above.  Other issues include how to develop general-purpose critical thinking in the contexts of math education and investigating the characteristics of critical thinking as fostered in math education.  Further studies will be required to clarify these issues. 

　　The aim of this paper is to advance understanding on the characteristics of mathematical activities that take place in the multidisciplinary Study and Research Paths (SRP) based on the paradigm of questioning the world.  SRP formulated within the Anthropological Theory of the Didactic is the inquiry activity aiming at nurturing researcher’s attitude and allowing students to use any media (internet, book, etc.) and to learn mathematics according to its necessity during inquiry.  We carry out a teaching experiment for the students of grade 8 in Japanese junior high school, using the question related to the population explosion on earth.  A sequence of four lessons are designed and implemented in five classes.  The data collected in this teaching experiment are analyzed by means of different concepts related to SRP: the questions-answers dialectic is identified to clarify the multidisciplinarity of students’ inquiry, the media-milieu dialectic to see the dynamism of students’ autonomous activities, and mathematical praxeologies to characterize the mathematical activity in the multidisciplinary SRP.  The result of analysis implies first of all a great potential of implementing SRP in Japanese junior high school, in particular as a multidisciplinary activity which is strongly emphasized today by the Japanese Ministry of Education.  The analysis also shows that there are two different kinds of mathematical activities in the inquiry process.  On the one hand, it is the mathematical activity as a tool to construct an answer to the question of other discipline (in our case, social studies).  Mathematical praxeologies in such a case appear explicitly in distinction from the praxeologies of other discipline.  On the other hand, it is the mathematical activity which is in a mixture with the activity of social studies, wherein the type of tasks identified is not purely mathematical but interdisciplinary one including the elements of mathematics and social studies.

　　The aims of this paper are to propose strategies for supporting mathematical problem solving from a new perspective of academic abilities of mathematics and to build hypotheses regarding what opportunities prompt learners to apply mathematical methods in mathematical problem solving.  Since hypotheses should be falsifiable, it is important for our scientific endeavor to simultaneously achieve these two aims.  This paper will provide a starting point for the cycle of theory building and practical verification.  For this, we will analyze a conventional high school mathematical problem and conjecture what supports learners to shift their attention to a key aspect of the problem from a radical constructivist point of view.  

　　As a result, we will build the following two hypotheses about opportunities for applying mathematical methods along with two corresponding strategies for supporting learners to obtain such opportunities.  First, we will propose that an opportunity for applying mathematical methods dependent on particular mathematical content is to establish a cognitive status of noticing the effectiveness of the methods in the current context without proving it.  In order to provide such an opportunity, it can be effective to encourage learners to seek instances satisfying the given conditions.  Second, we will hypothesize that an opportunity for applying mathematical methods independent from particular mathematical content is to understand the cultural nature of mathematical activities.  In order to provide such an opportunity, it can be effective to familiarize students with customs in mathematics in an explicit manner.  It is concluded that dependence on mathematical content is a key property of designing an approach for teaching mathematical methods.  The distinction between different mathematical methods should be made in new discourses adopting a new perspective of academic abilities of mathematics.

　　The purpose of this research is to develop a method to evaluate ideas to solve problems at the time of mathematics classes at junior high school and to make a practical report. Therefore, the following method was developed. First of all, we identified the “mathematical way of thinking” that students would use when solving problems by learning of second-grade figures. And I created a unit guidance plan based on that “mathematical way of thinking” and worked on the evaluation from the perspective of “mathematical thinking”. Next, we decided on the criteria of evaluation more specifically when we conduct evaluation activities in each lesson. This criterion is also shown in the “personal evaluation card” used by students. I used this card to repeat the evaluation of “mathematical thinking” in class.

　　In doing so, students began to recognize the “mathematical way of thinking” that can be acquired in the process of solving problems. Regardless of the success or failure of the problem, the students were able to recognize the value of expressing their own ideas.

　　The purpose of this paper is to elaborate effects of definitions of logical implication on the unit structure of logic in mathematics textbooks.  Especially, this study focuses on the effects of their definitions on indirect proof. For this purpose, we carried out praxeological analysis of a Japanese high school mathematics textbook within Anthropological Theory of the Didactic.  The analysis consists of two parts.  First, we briefly review a definition of implication by inclusive relationships of sets.  Second, we identify what types of tasks appear how many tasks each type has respectively.  As a result, we found that the definition of logical implication justified solutions for almost all of tasks.  Especially, we can explain and justify the validity of proof by contrapositive by using the definition.  On the other hand, we showed that the definition did not validate of the method of proof by contradiction.  Furthermore, we suggested that above differences were caused by the following three reasons: 1) The current conception of implication based on the concept of set in Japanese school mathematics does not subsume the conception of non-implicational proposition, that is, singular proposition; 2) As long as following the description of the textbook, we cannot define negation for any propositions (we can define negation only for open sentences); nevertheless, 3) The textbook does not explicitly describe some other concepts, such as a logical consequence, required for explaining the validity of proof by contradiction.

　　The purpose of this study is to reveal teacher’s problems in the process of pedagogical reasoning based on Shulman’s model.  The participant in this survey was the grade three teacher in Metro Manila.  The questionnaires and interviews for the teacher are conducted before and after observing the lesson, comparison of dissimilar fractions.  Also the short test for the pupils was given by author at the end of the lesson.  We divided implemented curriculum into four parts which are teacher- intended curriculum, enacted curriculum, tested curriculum, and teacher recognized-attained curriculum.  The results compared with the objectives, learning contents, and way of teaching in each parts, it is showed that some differences existed between intended curriculum and teacher-intended curriculum, teacher expected curriculum and attained curriculum.  In the process of transformation it was found that the teacher followed teaching contents in teachers’ guide, however the way of teaching the teacher did was different from intended curriculum.  Particularly in the process of evaluation, it was described as misalignment with what the teacher intended to teach, what the teacher recognized about students’ understanding, and what the students have actually learned.  In conclusion, it is indicated the mechanism of causing the gaps among these curriculums.