The purpose of this theoretical discussion paper is to elaborate the concept of “teacher” in the context of lifelong learning of mathematics. In the existing literature on lifelong learning of mathematics, the absence of a teacher was seen as characteristic of good lifelong learning of mathematics. However, empirical research on voluntarily learning of mathematics was often conducted in the environments where mathematics teachers exist. An issue on the naïve use of the term “teacher” is raised.

　　This paper consists of the following four main sections. First, we introduce the concept of “mode” for capturing the fact that the same learner takes different learning styles. Second, we introduce the first philosophy of mathematics: Ethics. It is a key to reflection on the fundamental concept of “teacher.” Third, from this ethical perspective, we introduce the concept of “respectful learning” and argue that it provides the next theoretical orientation in research on lifelong learning of mathematics. Finally, we introduce the concept of “boundary object” and propose the following promising research questions in this field: For the purpose of realizing respectful learning, how and what can function as a boundary object? What conditions are necessary in learning environments for making something a boundary object?

　　The purpose of this paper is to identify the possibility that the norm that it is good to talk evenly in group learning creates a new difficulty in learning mathematics.

　　The research method is to collect data by first designing and implementing group learning based on the norm of talking evenly using Wiggins’ (2017) spider discussion.  Second, we defined the ideal group learning as a group in which ideas were emergent, and then collected examples of group learning in which ideas were emergent.  Then, we examine the bias in the number of conversations between students in these two group studies.

　　As a result of collecting the data from the classes, we found that the number of conversations among the students was uneven, although they were able to engage in activities with an awareness of the need to discuss equally.  In addition, in the case of the group that generated ideas, the conversations between the two students were biased, and they were not able to discuss equally.

　　These results indicate that there is not necessarily a relationship between the number of times students talk to each other and the depth of learning.  In addition, by examining the causes of the unevenness in the number of conversations among the students, we found that the norm that it is good for students to discuss evenly may create a new difficulty in learning by hindering the fluidly changing roles of students in group learning, making it difficult for the participants to have opportunities to generate their own ideas.  We found that this may create new learning difficulties that prevent participants from having opportunities to develop their own ideas.  This suggests that human relationships are involved in the understanding of mathematics.

　　In the future, we will examine the validity of this new difficulty in learning based on the suggestion that human relationships are involved in the understanding of mathematics.

　　The purpose of this paper is to clarify how in-service teachers in elementary schools are improving their math lessons while being influenced by a wider range of social factors.

　　To that end, while we conducted School Support Project at a public elementary school for about four months from September to December 2020, we did reflections on daily class with the two teachers (Teacher G and Teacher H) of the elementary school.  We also participated in the in-school training where the teachers had sessions to reflect on their teachers’ daily practices, including research lessons.  These lessons and reflective sessions in-school training were recorded with video cameras and IC recorders.  Detailed transcripts were prepared based on the collected data.

　　We analyzed the math lessons conducted by each teacher using the patterns of “MATOME” of Iwasaki and Steinbring (2009) as a theoretical framework.  As a result, it was revealed that the two teachers had transformed in different (opposite) directions, despite participating in the in-school training in the same way. Therefore, we attempted to understand why the two teachers underwent such different transformations. Specifically, we analyzed and considered the factors that changed the teachers’ lessons from the viewpoint of Engeström’s “activity system” based on the transcripts of dialogue records of reflections with the two teachers.

　　As a result of the analysis and consideration, it became clear that the transformation of the two teachers was strongly influenced by the components located outside, such as the COMMUNITY, RULES, and DIVISION OF LABOR.  In particular, the mechanism of the transformation of two teachers could be described as follows: That is, these components located outside the activity system are contradictions within the components or between them of the activity system of the two teachers.  Then, resolving these contradictions, the teachers try to improve their educational methods as INSTRUMENTS.  As the result, the teaching method is changing.  It should be noted here that although the transformation of the two teachers looks like strongly influenced by social factors such as the COMMUNITY, RULES, and DIVISION OF LABOR, but each teacher is subjectively improving his/her lessons.  When considering the transformation of math lessons by teachers, the results of this study strongly suggest that it is important to see each lesson not as an isolated lesson, but as an activity performed within the school’s entire social system.

　　In Anthropological Theory of Didactic, it is often questioned what is the condition under which specific knowledge can “live” in a given institution, and also what will happen if the condition varies.  In this paper, we focus on the logical concepts such as “necessary condition” in secondary mathematics.  It has been reported that notions of “necessary condition” and “sufficient condition” are often just memorized in a contextless way by students and lose their raison d’être in studying.  We point out that this loss of their raison d’être comes from the missing of the niche of logical notions.  Therefore, in this paper, we propose a practical course of lessons in which the raison d’être of the logical notion “necessary condition” can be actualized.  The theme of the class is to search for similarly-dissectable triangles, that is, triangles which can be dissected into two figures which are similar each other.  Then we show the results of this practice and discuss the possibility that logical notions can “live” in the institution of secondary mathematics.

　　It is often attractive in mathematics to discover a regular pattern or a rule in mathematical objects.  And, especially in senior secondary mathematics, a discovery of a pattern should be followed by its proof.  However, it seems that most of students do not find it interesting to consider why the discovered rule holds, that is, how to prove it.  In light of Komatsu’s argument about “explanatory proof”, we propose the notion of “explanatory understanding of proof”, which can, we believe, induce students’ mathematical interest. “Explanatory understanding of proof” means the situation where students realize the discovered pattern depends on the specific properties of the objects involved in the statement and moreover students can have a perspective which can tell whether the pattern remains true or not in the similar relating statements.  Then, we propose a mathematics class design, in which students are expected to reach “explanatory understanding of proof”, so that students can find the lessons mathematically interesting.  The keys to this design are to include imitating activities in classes, and to prompt students to apply the understood principle to other situations.  We also practiced the designed class and analyzed the results.  These results indicate the partial effectiveness of our design and simultaneously imply suggestions for the future development of mathematical classes in the same line.

　　The spread of COVID-19 has caused changes in children’s everyday life and school life, including temporary closure of elementary schools, junior high schools, high schools, and special needs schools, and staggered school attendance.  This may result in differences in children’s achievements and affective aspects compared to before and after the spread of COVID-19 in 2020.

　　In mathematics education research, the relationship between students’ achievement and affective aspects has been examined.  On the other hand, it is known that students’ socioeconomic status has some influences on their achievement in school education.  In other words, socioeconomic status may affect the relationship between students’ achievement and affective aspects.  This study examines the relationship between the achievement and affective aspects in mathematics from the perspective of stratification of socioeconomic status, analyzing fourth-grade elementary school students in TIMSS 2019.

　　As a result, (1) the affective aspects of the interest and confidence in mathematics are affected by stratification of socioeconomic status, and the aspects of the teacher support and anxiety in mathematics are hardly affected by stratification of socioeconomic status.  (2) there were no significant differences in the relationship between mathematics achievement and affective aspects: interest, confidence, teacher support and anxiety in mathematics from the perspective of stratification of socioeconomic status.  (3) the affective aspect of teacher support was found to have a small positive effect to the students having a lower sense of working on problems in mathematics class.

　　Thus, since there is not significant effect from stratification of socioeconomic status in the relationship between mathematics achievement and affective aspects, the efforts made in school education may become a major factor in the relationship between them.  Based on the results (3) and the fact that the mainstream of mathematics education in Japan is problem-solving learning, we would like to make efforts to clarify the characteristics of students who have lower sense of working on problems in mathematics class and to explore the way of supporting for such students as the future tasks.

　　The purpose of this paper is to develop a socially open-ended problem for high-school mathematics lessons with a focus on the nurturing and evaluation of the students’ critical thinking and to verify its effectiveness through practical lessons.  This study aims to nurture critical thinking, defined as a broad sense to include mathematical problem solving in the real world.  Here, the definition of “critical thinking” is “to propose solutions based on a mathematical model built on the students’ own values by identifying the problem beneath the superficial differences” and the “effort to improve the students’ own solutions by determining appropriateness and reliability of other people’s solutions” (Hattori, 2020, p.162).  The study developed a lesson for the secondyear students in high school, with “propose a mayonnaise nozzle” as the theme.  As a framework to evaluate students’ critical thinking, the study also developed rubrics.  Through the practical lesson, students made full use of their mathematical knowledge they have acquired so far and solved the problem based on their individual values.  In particular, the students were identified with an attitude to improve their solutions by observing and determining appropriateness and reliability of other people’s solutions.  The study also suggested the possibility of evaluating critical thinking demonstrated by students through rubrics to a certain extent.  On the other hand, “reviewing the definition of critical thinking in a broad sense in the context of mathematics education”, “necessity to analyze not only the worksheets but also multiple aspects, such as presentations, etc., in the rubric-based evaluation”, and “methods employed by the teacher in lessons where social values are shared” were identified as the issues to be addressed in subsequent studies.