The purpose of this study is the realization of learning/teaching that links pre-existing knowledge to new knowledge．For this purpose, to exploring the derivation of a theorem is valid. And this is appropriate to the thinking of emphasizing a learning process in recent years. Therefore, this study has done some teaching materials research that focus on the derivation of a theorem. In this paper, some of the results of the research are arranged from the viewpoint of "seeing the determining condition of triangles as a function." And, it’s proposed as a new teaching material. In addition, this shows "usefulness of the derivation research." On the other hand, the aim of this paper is also to show "an example of a spiral-learning which progresses by considering one viewpoint as an axis."

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The programs dealing with the application of mathematical modelling to the solving activities in real-life problems have been conducted mainly in the United States, Australia, and several European countries such as Holland, Germany and so on. The students who have participated in these programs vary in age, academic ability, and other things. This present program deals with a solving activity in scientific problems for junior high school students. It was carried out as the trial step of the mathematical modelling challenge program we have been carrying forward in Japan. In this thesis, we report the content of this program, the feedback from the students and what we gathered from it. “The movement of a cart rolling along the slope” is taken up as a teaching material and some third-year students at junior high school (ninth graders) tackled it. As a result, it turned out that these scientific problems were effective as the mathematical modelling challenge. But, on the other hand, it was confirmed that Japanese students lack modelling skills in thinking flexibly considering the various conditions and uncertain numerical values arising from complicated real-life problems. At the same time, it is suggested that the activity like this can foster the students’ attitude to try to find answers from a broader point of view instead of just staying within the area of functional formulas which they have already learned.

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This paper focuses on the effective learning to understand the transformation of equation about speed utilizing the RTMaC lesson study in the Japanese elementary school. In Japan， children activities in the school mathematics tend to be inclined to numerical calculation. Because of that， it is generally said that the children have less opportunities to understand various concepts of mathematics. This would be considerable issues of the children developing further mathematical thinking. One such example is the transformation of equation about speed. In response to this， we developed teaching materials and methods， and had the classroom practices. As a result， we confirmed that the teaching materials and methods elevate the students to understand the concept ， and it leads to the understanding of the transformation of equation about speed.

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Suppose that we drop a needle of length l at random onto the plane with a set of parallel lines a distance d apart. Buffon's needle problem is to calculate the probability that the needle lands crossing one of the lines. We generalize the problem into a higher dimension as follows. Suppose that we put a needle of length l at random in the space with a set of parallel planes a distance d apart. Then we calculate the probability that the needle is crossing one of the planes.

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In recent years, the number of students from Vietnam is increasing sharply, and many excellent Vietnamese students have entered Japanese universities. In addition, the results of the International Mathematical Olympiad are better than those of Japan. Therefore, in order to research the status of mathematics education in Vietnam, we inspected mathematics classes at high schools in Vietnam and we compared contents of textbooks in Vietnam with those in Japan. By these results, it turned out that mathematics education at high schools in Vietnam, in comparison with those in Japan, had the following features:
(1) Learning terms of some units in Vietnam were different from those of Japan.
(2) Exercises in textbooks were more difficult.
(3) Some mathematical symbols were introduced earlier.
(4) Students always used a scientific calculator.

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Currently, the Education of probability has not been implemented in the curriculum of elementary schools in Japan. However, it is common to have implemented the probability education from the stage of primary school abroad. When considering the development of the probability concept in children, it is considered that it needs to make the appropriate probability education in a kindergarten, an elementary school. This paper focuses on a probability teaching materials that target elementary school that has been developed abroad were examined from the point of view of the "level in the recognition of the probability concept" about the possibility and its significance.

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The authors have been teaching not only basic subjects but also specialized subjects at nurse and medical technologists training institutions. Results of final exam of the subject and a short quiz of each unit can also be enjoyed acceptable level. But over time, learning content as the coming school year increased to proceed, students forget the contents of the past. Accordingly, the authors presented some national examination self-learning softwares for nurse and medical technologists for the first time in Japan. This paper introduces development of the improved software, including the perspective of engineering.

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