The paper focuses on the impact of human relationships in small-group mathematical activities for more effective mathematics learning. The author believes that learning mathematics in small-groups and improving caring competencies, which involves building relationships to care for others and to be cared for by others, are mutual in terms of beneficence. Therefore, the purpose of this paper is to make clear how the improvement of caring competencies contributes to the development of knowledge in small-group mathematics learning in order to comprehend the relationship between mathematics learning and caring competencies.

　　First, the author defined the caring competencies emerged in the mathematics context by reviewing previous studies. Then, by focusing on the small-group activities in which the emergence of 8 graders’ ideas was identified to analyze the situations in which the caring competencies were demonstrated, the author examined how the caring competencies contributed to the development of mathematics knowledge. The results implied that it was difficult for the students to put their own mathematical ideas aside and stand on different viewpoints of other in group members. On the other hand, the interaction with others’ideas while using the caring competencies provided an opportunity for the emergence of mathematical ideas, which updated students’mathematical knowledge.

　　In addition, the author applied the framework developed by Krummheuer (2011) and categorized the students’ statements in small group activities in order to analyze their roles in the group. The results show that it is difficult for the students to state their new ideas in small groups. In order to solve this problem, the research findings suggested that the students in small groups need to switch their roles spontaneously and rebuild new human relationships that enable themselves to care for each other’s statement.

　　The conclusions of this research proved that developing caring competencies can contribute to the following aspects:

● Solving the difficulties to commit others’ideas, which contributes to emerging new ideas.

● Realizing ideal small-group activities where roles are not fixed and students switch roles voluntarily to state.

　　The purpose of this study is to clarify how metarules for the endorsement of numbers change in development of numerical discourse from real numbers to complex numbers. Commognitive theory is chosen as a theoretical framework in this study. To achieve this purpose, the data was collected in two lessons in which two second grade high school students participated. It was analyzed through four interrelated characteristics (keywords, visual mediators, endorsed narratives, and routines). As a result, three metarules and commognitive conflicts were identified: MR1; Endorsing objects, which can be realized in real models, as numbers, MR2; Endorsing objects, which can be expressed on a number line, as numbers, MR3; Endorsing objects, which can be expressed on a complex plane, as numbers, CC1; A conflict between real numbers and complex numbers, CC2; A conflict relating metadiscourse for discourse for defining numbers, CC3; A conflict between real numbers and ordered pairs.

　　The contributions and an implication of this study are as follows: (i) identified three metarules for endorsement of numbers and their changing processes, (ii) indicated that one of the opportunities for the change of metarules is to introduce new visual mediators, (iii) indicated the importance of interactive consideration of the duality of complex numbers (algebraic-geometric).

　　The aim of this paper is to understand the place of programming in mathematics learning.  In this study, we hypothesized that one of such places is the inquiry activity, and we designed lessons with the Collatz problem, an unsolved problem in mathematics.  The lesson design was based on the ideas of the paradigm of questioning the world, specifically Study and Research Paths (SRP).  SRP formulated within the Anthropological Theory of the Didactic (ATD) is an 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 2 in Japanese high school, using the question related to the Collatz problem.  The data collected in this teaching experiment are analyzed by means of different concepts related to SRP.  We used a QA map that can describe the questions-answers dialectic in the inquiry activity, and we made several characterizations of the QA map to describe the dialectic between mathematics and programming.  The result of analysis showed that students engaged in inquiry activities similar to those of researchers, and they were actively using the program in their activities.  The analysis also shows that there are two different characteristics of the program developed in the inquiry process.  On the one hand, it is to ignore the economy of the program when writing it.  On the other hand, it is the use of media references to codes that have not yet been learned.  These characteristics reveal that students use programming as a tool in their mathematical inquiry.  We conclude from these results that one place for programming in mathematics learning is the mathematical inquiry.

　　The aim of this paper is to investigate the potential of the mathematical question we call “the problem of grid-point on a circle” (specifically “How many integer solutions does the equation x² + y² = r² has?) as a mathematical inquiry theme. We investigated in detail the mathematical background of this question with an analytical tool, called Question and Answer map, which has been developed within the research community of the Anthropological Theory of the Didactic (ATD). The results of analysis show that inquiring this question leads to different mathematical ideas, such as Pythagorean numbers, Fermat’s theorem on sums of two squares, Jacobi’s two square theorem, and so on. In addition, through this analysis, we confirmed the power of Question and Answer map in the epistemological analysis of a given mathematical theme as well as the didactic analysis that implies the clues for teachers to carry our inquiry-based lessons.

　　The probability we consider in our daily lives is conditional probability. For this reason, the teaching of conditional probability is important in school mathematics. However, some problems have been pointed out in teaching conditional probability. For example, although previous researches have pointed out the problems of tasks and lessons for the introduction of conditional probability, they have not led to the design and verification of specific tasks and lessons. Therefore, the purpose of this paper is to design and verify tasks and lessons for the introduction of conditional probability in school mathematics. The theoretical framework is the process of formulating the concept of conditional probability from the perspective of negation.

　　As a result, we first identified the hypotheses about tasks and classes. The hypotheses about tasks are: “If we use a question that asks for the probability of cause, which is the opposite of the probability of outcome, as an introductory question to conditional probability, students may realize that they cannot derive an appropriate model given the information they have”, and “If the question involves several people with different information about the same event, students may realize that probabilities apply to our information about the event”. The hypothesis about the lessons is that ［1］ students recognize that using unconditional probability, students recognize that they cannot derive an appropriate model in consideration of the information obtained, ［2］ they reinterpret the probability that does not ignore the condition as conditional probability, and ［3］ they avoid confusing unconditional and conditional probability by characterizing the former as a model that has no conditions related to the event. This course of instruction is effective as an introduction for students to formulate the concept of conditional probability. We then designed and implemented a task and a lesson to illustrate these hypotheses, targeting 10th grade students who had not studied conditional probability. As a result, we were able to illustrate three hypotheses.

　　This study is conducted to identify the characteristics of the processes of student valuing development and the influences of teacher valuing in mathematics education in Japan as a part of International Comparative Study “The Third Wave”.

　　Thus, the purpose of this paper is to clarify the characteristics on the the development of students’ valuing by analyzing the descriptions on the reflection sheets and the questionnaire “What I Find Important (in mathematics learning) too (WIFItoo)” in the lesson practice of the unit “Function y=ax²” implemented by mathematics teacher “A” for her students in Miyazaki Prefecture.

　　As a result, we were able to derive the following “Instantial Valuing” that are formed instantaneously by learning in each lesson: the sub-dimension Process concerning the mathematical learning process, the subdimension Application in problem-solving situations, and the sub-dimension Facts/Truths concerning the relationship between the mathematical world and the real world.  In addition, we were able to derive the subdimension Comfort related to environmental factors in learning mathematics and the sub-dimension Exposition related to independence in learning mathematics as “Relatively Stable Valuing” that are formed through mathematics learning experienced over a long period of time.  It was also confirmed that the former tended to be expressed in the descriptions on the reflection sheets, and the latter in the descriptions in the questionnaire.

　　As for the influence of the teacher on the formation of students’ valuing, we found that the teacher’s comments in mathematics lesson had a strong influence on the formation of “Instantial Valuing” such as Process, Application, and Facts/Truths, while the teacher’s comments had little influence on the formation of “Relatively Stable Valuing” such as Comfort and Exposition.  On the other hand, the possibility that the teacher’s comments have little influence on the formation of “Relatively Stable Valuing” such as Comfort and Exposition was derived through comparison between the descriptions on the reflection sheets and the lesson records.

　　In order to realize the lessons where students think autonomously and developmentally, the teacher’s consciousness about how to grasp the problem solving and thinking that the students do will influence.  With reference to Ernest’s（2015）  Fallibilism and Absolutism, we developed a questionnaire on teaching students to think autonomously and developmentally, and from the case studies of two teachers, we caught the trend of “classroom practice emphasizes teachers’ planned developmental guidance and short-term academic improvement, fixing instruction and support to control learners’ thinking and attitudes”.

　　These findings are findings within the scope of case studies, and are not findings that sufficiently demonstrate versatility.  In this study, we conducted a questionnaire survey on lessons where students think autonomously and developmentally for elementary, middle school and high school teachers nationwide.  The purpose of this paper is to analyze Akita prefecture data and obtain suggestions for guidance.

　　As a result of analyzing the data of 124 elementary school teachers, 71 middle school teachers, and 35 high school mathematics teachers in Akita prefecture, the following 2 points were clarified.

　　1. As  for teachers’ consciousness, there is a tendency for Awareness of Developmental Mathematics to be seen when the lesson is planned, while to increase absolutely Awareness of Mastery Mathematics  when the lesson is practiced.

　　2. In  the composition of each factor, “G.  Numerical values, conditions, content, and sequence of problems” was not included, indicating that teachers have an absolute, fixed view of mathematics and its teaching  as the subject of instruction, which may be considered a manifestation of their Awareness of Mastery Mathematics.

　　In the future, following the Akita prefecture data, we will proceed with the analysis of national wide data and clarify the trends of students and university faculty members.  In addition, we will proceed with the analysis of whether there is a difference depending on the number of years of teaching profession, or whether there is a relationship with the good results of the national academic ability / learning situation survey.  Furthermore, we will consider the training method using this survey.

　　The purpose of this study is to clarify how the five international students viewed Japanese mathematics education through a year of lesson observation at attached elementary school by analyzing the results of their papers submitted to the university’s research bulletin.  The following three points can be cited as the main results of the study.

(1)  The characteristics of Japanese mathematics education have been considered from the viewpoint of a highly heterogeneous “stranger” by practitioners in developing countries.

(2) T he common understanding of what has been mentioned in previous studies, such as the problem-solving lesson process and the quality of Lesson Study, as contents of interest in Japanese mathematics education by  practitioners in developing countries, was confirmed.

(3)  The characteristics of Japanese mathematics education, such as teaching in the mother tongue and mathematics education away from examinations, were clarified as advantages that had not been recognized  before.

　　Thus, using the stranger theory as a background, the characteristics of Japanese mathematics education were identified in terms of points that have been pointed out in previous international comparative studies of mathematics education, as well as characteristics not seen in previous studies.

　　The purpose of this review paper is to review an international trend on how international journal articles in mathematics education should be written, and to draw implications for conducting mathematics education research in Japan.  We tackle with the research paper format problem raised in M. Niss’s paper “The very multi-faceted nature of mathematics education research” (in For the Learning of Mathematics (FLM), vol. 39, no. 2, pp. 2-7.) and investigate how a paper format is understood in Educational Studies in Mathematics (ESM), in Journal for Research in Mathematics Education (JRME), and in FLM.  The result of this investigation is reported through a narrative review methodology: We reviewed reaction papers to Niss’s paper in FLM, editorials in JRME and in FLM, and chapters for ESM, JRME, and FLM in Compendium for Early Career Researchers in Mathematics Education.  As a result, we show as follows: (1) ESM can accept any type of papers, but explicitly require the use of appropriate theoretical frameworks to empirical research paper; (2) JRME takes a stronger position on the use of appropriate theoretical frameworks than ESM; and (3) FLM emphasizes conversation between researchers and primarily publishes essays on mathematics education.  The following two implications from the review survey is discussed: (1) A theoretical framework is a construct, which allows researchers to make any decision in all the research processes, and thus, an appropriate use of theoretical frameworks helps us to improve the qualities of our manuscripts; and (2) It is also important to write a meta-research paper without any theoretical framework, which should be published in a timely manner.