Japan’s curriculum of secondary school mathematics was revised in 1942, in which descriptive geometry was included in solid geometry for the first time.  There are two streams of background of this reform.  One is, of course, educators’ efforts to reform geometry teaching including international movement.  Another is Japan’s long tradition of teaching mechanical drawing (called “Yokiga”, descriptive geometry in mathematics) in the school subject “arts” in secondary general education during late 19^th to mid-20^th Century.  Here the latter is examined, to show how the subject had been taught and learned.  Curricula, textbooks, and actual notebooks of students are referred to describe the transition and characteristics of teaching Yokiga.  

　　The reason why Yokiga was incorporated into general education in the early Meiji era as part of Western painting education was that they had the practical purpose of drafting and cartography and were positioned as the basis of science and technology.  Yokiga had been strongly influenced by trends in arts education, such as so-called “national preservation campaign” in 1880s and “free drawing movement” in 1920s.  Actual school reports and national survey also showed the fluctuation of the number of textbooks adopted by secondary schools.  Student notebooks on 1889 and 1940 are examined to show Yokiga was carefully taught and learned in the real classes.   

　　Yokiga that had been taught from early Meiji era played a significant role in the reform of solid geometry within the curriculum of 1942.  Although these contents are not currently treated in secondary schools, but we would learn many suggestions from this cross-curriculum study even today.

　　The purpose of this research is to clarify the viewpoint that individual view of mathematics gives to the research area of lifelong mathematics learning.  For this purpose, after an overview of the research area of lifelong mathematics learning and the arguments related to view of mathematics on research in mathematics education, I construct a theoretical framework for individual views of mathematics in lifelong mathematics learning.  After examining the actual situation and interpreting the results using the theoretical framework, I consider the position of an individual’s view of mathematics in the theory of lifelong mathematics learning in mathematics.  

　　In conclusion, I obtained the following.  Since the view of mathematics differs at the individual level, it is an index that characterizes that individual.  It is natural to regard it as a complex view of mathematics, and a complex view of mathematics is formed during the course of lifelong mathematics learning in mathematics. School education plays a role in fostering a mathematical view that does not hinder subsequent mathematics learning.  

　　When studying “mathematical” lifelong learning rather than general lifelong mathematics learning, it is inevitable that subjectively conscious discussion of mathematics learning is inevitable.  It depends on in other words, when studying the practice of lifelong mathematics learning in mathematics, it is essential to interpret through the subject’s view of mathematics.  In the theory of lifelong mathematics learning, focusing on an individual view of mathematics guarantees that the research target is lifelong “mathematics” learning.

　　Abstract  In this research, we consider the “subject” in mathematics as a “sign”―in Peirce’s semiotics―that directs our attention to different things, and we expose its evolutional development in a mathematics class from the perspective of its ontological status.  We analyzed subjects through a lesson using the square root in the 9th grade while comparing them with Euler’s mathematical activity in “Analysin Infinitorum” from the viewpoint of Peirce’s “ten classes of signs.”  The results highlighted the following points.  

　　First, we described the evolutional development of subjects in a mathematics class and Euler’s mathematical activity using the ten classes of signs.  Because both cases are similar in that the subjects develop almost evolutionarily in the order of the classes, it may be inferred that the mathematics class has a structure similar to Euler’s mathematical activity.  Second, both cases, however, have a different point.  In the mathematics class, there is occasion for the class to change from “legisign (sign as law)” to “sinsign (individual sign).”  In other words, the subject in mathematics exists within the classes, including the sinsign, without existing for many members when the sign is appeared under the law or concept which exists for only some members in the mathematics class.

　　The purpose of this research is to clarify the actual conditions of learning by conducting a case study dealing with the condition for the formation of triangles in the introduction of the unit “triangle” in the third grade of elementary school. This case study focuses on one extracted child and qualitatively analyzes the learning aspects. The qualitative analysis uses the theory of geometric discourse development (Wang, 2016). The basis of this theory is van Hiele’s theory of learning levels of geometry (van Hiele, 1986) and the theory of commognition (Sfard, 2008). The development of geometric discourse is divided into four levels. The emphasis is placed on geometric objects and its routines that support the transformation of discourse.

　　In lesson study, there were three group interactions and three whole-class interactions. Then from the utterance record and video record of the extracted child in lesson study, the mathematical use of the extracted child and its routine were interpreted. As a result, the following three points were clarified, and it was demonstrated that it is possible to learn the condition for forming triangles in the third grade of elementary school.

1: The geometric discourse of the extracted child changed from level 1 to level 2.

2: The possibility of realizing discourse using deductive reasoning was suggested.

3: The level 3 discourse was partially realized by using a slightly abstract object as the object of inference and a common narrative.

　　The purpose of this research is to catch a successive change in a figure consciously in the unit of the junior high school for 2 years “combination of a figure”, relate to read the system, develop an investigated learning activity and consider its learning effect qualitatively through class practice. So the professor experiment which made the equality of which I consist between 3 sides of length with changing a figure successively and the mathematical activity that the change is interpreted an axis was made. Student’s activity and description in its class were analyzed qualitatively.

　　As a result, the following 3 points became clear.

(1) Understanding to the manner of the change in a relational expression of which I consist between the length of BD and CE,DE deepens through the activity that I try to be conscious of a change in the length of the vicinity and read the system though it makes them contrast with a successive change in a figure many times.

(2) They become able to make a change in the length of each vicinity in one equality more conscious in the process which tries to check a successive change in a figure to check a change in equality.

(3) The activity that I try to check a change in the length by a ch art has a possibility it gains by the process going to be conscious of a change in the length of the vicinity and check a successive change in a figure. A session of this professor experiment, there are a theorem if disobeying, by which a theorem of the angle at the circumference is Thales and a possibility that trigonometrical function leads to an insight of future learning.

　　The purpose of this study is to clarify a difficulty of junior high school students’ understanding and a possible form for teaching about meaning of probability from more various viewpoints based on modelling than the relation between statistical probability and mathematical probability.

　　As a result, first, we constructed a new framework by putting Pfannkuch & Ziedins’s (2014) framework and Ikarashi & Miyakawa’s (2013) it. Second, we pointed out that a factor of the difficulty was that many students cannot relate not randomizer in physical world to model probability in mathematical world well which previous studies have pointed out, but randomizer in hypothetical world to model probability in mathematical world well. Third, we pointed out that probability teaching which National Institute for Educational Policy Research (2015) proposed has not really contributed to students’ understanding about meaning of probability. Fourth, we identified two activities that should be set in teaching about meaning of probability. One is to make students aware of hypothetical world, and the other is to make students’ perception about probability coming and going among three worlds (physical world, hypothetical world, and mathematical world).

　　The purpose of this research is to clarify the process of mutually constructing operational and formal proofs in the proof learning of literal expressions involving negative numbers and subtraction. Operational proofs in the previous studies have been limited to the mathematical phenomena involving additions and natural numbers that can be expressed as quantities. Therefore, we examined the proof learning of expressions involving the concept of subtraction and negative numbers cannot be explored. However, considering the actual situation of the students, it is not a little difficult to proceed with the proof in an abstract manner immediately away from the concrete, and it is difficult to accompany the concrete image and operation with the understanding of the proof of the character expression. It is thought to contribute. We propose two things to make this clear.

　　First, as the meaning of operational proofs, “the process of deriving formal proofs based on operational proofs” and “the process of exploring operational proofs based on formal proofs” have profound implications for formal proofs. Recall a learning process that encourages understanding.

　　Second, through the process of constructing operational proofs and formal proofs, it is possible to perform operative proofs on proofs of continuous character expressions involving the concept of negative numbers. It turned out to be promoted. The problem of the sum of two numbers is pushed up to concreteness by analogy of the operation, and the problem of the difference of two numbers can be regarded as a higher-order concrete example of formal proof. It has been clarified that the mutual construction of operational and formal proofs in these two problems can lead to the search for proofs of character expressions involving the concept of negative numbers, such as the problem of the difference between two numbers.

　　 The purpose of this paper is to clarify the conditions and constraints on cross-border lessons (CBLs).  In related international classroom research thus far, the focus has mainly been on exploring the possibility of cooperative distance learning or fostering knowledge and attitudes via cross-cultural communications. Additionally, in the international comparative study of lessons, the cultural characteristics of the specific teaching and learning activities of each country have been considered; hence, the focus has been on the implemented lessons.  In contrast, this paper characterizes topics discussed and agreements reached through collaborative task design for CBLs between Japan and Russia.  

　　First, we describe the proposals being discussed between the two countries with the developed coding system.  Then, we analyze the task design process as networking these codes.  Through this analysis, we identify the conflicts manifested in the discussions between the two countries as follows: how to relate the correlation of data (graph), how the trend in data (graph) can be interpreted in terms of energy and environmental issues, what is necessary for the classroom setting(s), and how to distribute lesson materials to students.  

　　By considering the causes of these conflicts and strategies for solving them, the following conclusions are presented.  

　　Regarding the constraints on lesson design for CBLs, since discussions between the two countries usually involve time differences, it is inevitable that they occur via e-mail.  As a result, discussions about lesson design have been intermittent rather than instantaneous.  Further, in designing tasks for CBLs, cultural differences inevitably arise with respect to the teaching objective.  Especially in the case of Japan and Russia, the following cultural differences are pointed out: Japanese teachers emphasize students’ activities for obtaining mathematical conclusions using mathematically modeled graphs, whereas Russian teachers expect to have students develop interpretations and evaluations on mathematical conclusions according to social contexts.  Constraints in the classroom setting and customary teaching strategies are also pointed out as differences; in Japan, the learning process is recorded on the blackboard, and in Russia, the focus is on accessing learning resources with a single monitor.

　　When considering these constraints, it is necessary to state clearly the following in the lesson plan to resolve these conflicts as conditions of CBLs: subordinate items to make the teaching objective(s) more specific and ensure that classroom settings and classroom culture (e.g., teaching strategies) can be shared between the two countries.