In this essay, I will explain my pioneering work for the international conferences between Japan and China, and those between Japan and Germany. These two pioneering efforts will have important meaning not only for my study group but also for Mathematics Education Society of Japan. At the same time, I will make a little comment on the mathematical cultures in China, and in some European countries.

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The purpose of this study is to solve the problem pointed out by many earlier studies and a lot of teachers that many children of primary schools are weak at spatial cognition. In particular, we focus our study on the stage of "Pre-Sketch Weltanschauung"(the stage where one can recognize for the first time in cognitive development that our living space is three-dimensional), because it is difficult for the students to acquire. In this paper, we made educational contents and educational plans, based on our hypothesis, for the students to acquire the Weltanschauung stage. And we verified them through real educational practices in primary schools. As a result of these practices, the following points became clear: Our plans and contents were useful for children to acquire the Weltanschauung stage. Finally, we made an educational plan for Geometry in elementary school based on the suggestions above.

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Mathematics can be viewed as a discipline that enjoys absolute "universality". However, whether mathematics possessed this same "universality" at its beginnings is doubtful. Abstraction and universality have gradually been incorporated at each step of mathematical development over time. The question thus becomes, "Why has universality been incorporated in a step by step manner throughout the history of mathematics?" Human activities, which have formed the basis of mathematics, must be examined to answer this question. One part of these human activities is mathematics itself, especially mathematical thought, from the standpoint of its theoretical development, as well as its philosophical and practical aspects. At the same time, mathematics and mathematical thought has played a vital role in other human activities, such as the arts precisely because of its "universality." The "Theory of Human Proportion" has a close relation to mathematics in the same manner as described above. This theory, which is a part of traditional European art theory, strives to establish the "canon" (rule) of the beauty of the ideal human body. The underlying principle is that "ideal beauty must be universal, " and many theoreticians of the arts have applied mathematical methods in order to crystallize ideal beauty. In this article, we shall trace the history of the "Theory of Human Proportion" from ancient Greece to the Renaissance from the viewpoint of the development of mathematical thought. Four mathematical methods shall be examined, specifically, the "fractional method" of ancient Greece, the "modulus method" in the Byzantine and Arabic style of the Middle Ages, the "geometrical construction method" in the European gothic style of the Middle Ages and the "quasi-decimal method" of the Renaissance. Mathematics shall be examined herein as an aspect of human activity with which relationships have been established with other aspects of human activities against the backdrop of the aforementioned four methods and their development. We endeavor to demonstrate that mathematics, through these relationships, is one of the most vibrant of areas of culture at large. Moreover, the illumination of the cultural and social aspects of mathematics should significantly contribute to our overall understanding of mathematics.

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We introduced blended learning, which is the mix of in-person and e-Learning system, into some lectures on mathematics in 2003. After some improvements, this blended learning has been introduced into a first course in linear algebra, which many new students have been taking since 2005. As a result, their scholastic ability in this course has been improved. This paper describes the nature of this learning and its effects. Moreover, we present the change of their scholastic ability and interesting data on this blended learning.

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The purpose of this study is to analyze children's brain activity during calculating. We examined concentration changes of hemoglobin in prefrontal cortex and performance time during performance of addition mushikuizan task-calculation of filling in the missing numbers-in fifth-grade children. It was found from the result that features of concentration changes of hemoglobin are different depending on performance time (difficulty) of the subject. Compared to case of university students, this result has a feature of great difference between individuals. Therefore it is important to examine result of children in considering education.

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The research activity which deepens the mathematical content of study itself, as well as the method of using the relation between mathematics and reality, is effective in activating learning with graphing calculators. Here, two examples are introduced:research of "movement on a plane" after studying high school "Math III" differentiation and research of "implicit functions" after studying "Math C" conic sections. I show through the student's reports that they deeply understand the contents of study, and try to apply earlier contents to solve a new problem.

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