Since PISA2003 many discussions about mathematics education take place in Germany. Soon, KMK (Kurturministerkonferenz) revise curriculum at Elementary(till 4^<th> grade), middle Stage (10^<th> grade), secondary School (Hauptschule: 9^<th> grade) and announce standards.
　　The characteristic is "from material building to mathematical formal building". i.e.
(1) Attach more importance to mathematical general competence than material competence. Ex. "to solve problem mathematically", "mathematically modeling", "to make use of mathematical expression", "to argument mathematically", "to communicate mathematically", "to treat symbolic, formal and technical elements of mathematics"
(2) Attach mathematical ideas

[table]

(3) Present three levels of problems first level: reproduction second level: making a relationship third level: generalization and reflection Implication to Japanese Mathematics Education
(1) Attach important to learning environments
(2) Developing learning environments
(3) Bring up many competences in one learning environment
(4) Practice intelligently or productively
(5) Attach importance to longer performance
(6) Improve a quality of lessons
　Finally I hope Japanese mathematics teaching is more open and long-term than earlier.

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This paper aims at improving mathematics teaching and learning in Zambia. Number brick, which is one of Substantial Learning Environments (SLEs), was applied into mathematics lessons. Number brick gives students a chance to enhance two different abilities of mathematics: basic calculation skills and other higher-order thinking skills. This paper qualitatively discusses the whole-class learning situations in Grade 5 lessons. The author mainly described two learning situations in a series of lessons. The teacher helped students' incomplete answers by encouraging them to say some more mathematical expressions. On the other hand, students showed some difficulties and instability in simple calculations which they had already learned. Furthermore, in the analysis of students' verbal expressions, the verbal responses were limited to a few words. The students successively used a verbal format which the teacher introduced in order to help their language difficulty. In other cases, on the other hand, they were not able to reason or explain their mathematical ideas and the teacher failed to help them. In conclusion, students were given an opportunity to foster their higher-order thinking skills in lessons; however, we found some challenges in teaching such as: responding students' incomplete or wrong answers, and deliberately creating learning situations where students can be faced to their mistakes in their learning.

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This paper is a part of "Research on Cultural Value in Mathematics Education". The author has aimed to clarify the problem that the present school mathematics education has by catching the mathematics development as the organic whole, clarifying the culture seen there, and applying this aspect to historical development of the mathematics education of our country. This paper considered what the concept of quantity on "Decimal" cultivated by Japanese mathematics "Wasan" culture was. And, this paper considered how to have transformed the concept of quantity in Japan by receiving European mathematics based on "Fraction" at how to catch the quantity. The transition of the handling of "Decimal" and "Fraction" in the arithmetic textbooks at the last years of the Tokugawa shogunate and the Meiji era period was examined. First of all, the receipt of "Fraction" of European mathematics in study was examined. The modality of two receipts of "Fraction" of European mathematics was confirmed. One is the receipt of "Fraction" being based on the idea of the fraction caused from the result of the multiplication and division calculation, and with "Concrete number" including "Decimal". Another is the receipt of "Fraction" being based on the idea of the fraction caused by the equal dividing operation, and without putting "Decimal". Next, the modality of the receipt of "Fraction" of European mathematics in educational was examined. When the education of the arithmetic at the Meiji era period was developed, development being based on the receipt that was based on "Decimal" deeply related to Japanese mathematics "Wasan" culture and "Concrete number" did the main current such as becoming the model of the authorized textbook. On the other hand, development based on the receipt being based on the idea of the fraction caused by the equal dividing operation saw faddish. However, it can be thought that it was excluded from the authorized textbook, and removed from the main current of the development of the arithmetic education gradually. As a result, it is possible to conclude it as follows. By receiving European mathematics, the concept of quantity of taking root in Japanese mathematics "Wasan" culture managed by "Concrete number" including "Decimal" was transformed to the concept of quantity of developing in the direction where it comprehends past "Decimal" and "Fraction" connected with algebra, through to place based on the treatment by "Concrete number" including "Decimal", to catch "Fraction" of European mathematics based on the fraction caused by the division operation, and to catch "Algebra fraction" caused as a result of the multiplication and division calculation.

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The purpose of this paper is to consider toward creating the lesson of mathematical problem solving. In this paper, we first considered metacognitive knowledge that one of the metacognition's side. As a result, metacognitive knowledge is belief, and we made certain of importance of metacognitive skill for constitution of metacognitive knowledge by advanced researches. Secondly, traditional way of developing metacognitive knowledge is supporting children. In other words, this way is not teaching. And the question is how metacognitive knowledge is constructed in classroom. So, we considered the model by Cobb & Yackel (1996), and norms by using in this model. As a result, we cannot see from the model what reflexivity between individual perspective and social perspective. There fore, we focus on the reflexivity, and considered model of lesson by using norms and metacognition. Finally, we considered relationship the share of metacognitive knowledge and the constitution of norms. And we suggested new section that the discussion of metacognitive knowledge, in mathematical problem solving lesson.

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In this research, I focus on organization by Freudenthal as essential mathematics activity for ideal situation of secondary mathematics education. However Freudenthal didn't make organization clear. So in this paper, at first, I define organization provisionally. It is that "Associating experiences of thinking for mathematical objects by high order thinking, and making up mathematical essence of the thinking." Next I characterize it by insistence that operational conception goes ahead structural conception by Sfard (1991). In this paper, focusing on coming up irrational number, considering channel of thinking. In conclusion, I conducted that organization is a thinking activity in changing from condensation to reification is organizational.

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Recently, the lesson plan from a constructivist perspective is central problem of mathematics education. Therefore teachers must have belief from constructivist perspective in mathematical teaching. In this paper I consider the impact of mathematics teacher's view of mathematics on practice. View of mathematics as conception of mathematics has been brought to light by Dossey (1992). Teachers need to take Aristotle's, and Fallibilist's views of mathematics. We can assume the hypothetical teaching trajectory for the way to consider the impact of mathematics teacher's view of mathematics on practice. It is formed the basis for Simon's hypothetical learning trajectory, and thought of the hypotheses about the teacher's prediction as to the path by which learning might proceed. The hypothetical teaching trajectory is also made up of three components: teacher's supposition about the learning goal, the learning activities, and the learning process. In this paper, while considering the factor of mathematics teacher's view of mathematics on practice, I constructed a framework of teacher's belief System based on the View of Mathematics for the hypotheses of these components.

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The final purpose of this research is to develop the school mathematics curriculum of geometric figure in compulsory level based on the theoretical framework in order to mediate between elementary and secondary school mathematics, according to Okazaki & Iwasaki (2003). For attaining this purpose, this paper describes a experimental study on the fourth grade's lesson on geometric figure, whose focus is to promote the awareness of the properties of geometric figure. The characteristics of children who are promoted the awareness of the properties of geometric figure are as follows in concrete terms: (1) to be able to give the properties of geometric figure which is presented, (2) to be able to remind the geometric figure and to give other properties of it when some properties are presented. We developed the teaching-learning materials to promote the awareness of the properties of geometric figure and implemented those materials in the fourth grade's lesson. The effectiveness of the developed materials and the lessons were evaluated by the use of the performance task right after lessons and six months later. The result of the evaluation shows us that the developed materials and the lesson design are effective to promote the awareness of the properties of geometric figure and have a lasting effect for a half year.

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