In results of Germany students (11-15years) of PISA2003 is not good, under France, Finland and so. Educational discussions come out in mass-communication and the public. KMK makes publish "educational standard" and the Ministries of Education, Science and Culture of each State make a new publish their Curriculum. Essential reform is from "substantial training" to "mathematical formal building of character". In Germany Mathematics Education emphasis mathematical competencies rather than contents. The Characters of this reform in Secondary School I are follows. (1) Stimulating problem posing (2) Practice intelligent training (3) Solving Fermi inference problem (4) Harmonic achievement of application and structure-oriented teaching

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From the five representation styles, representation has not only its function but also unique features, and there are different degrees of abstraction. On the other hand, it should be said that writing contributes to the learning of mathematics. So, focusing on "writing", I found out, from the two difficulties such as optionality and motivation (significance), the situation of not being able to write as well as not to write. And I propose some view for guidance to promote students to write positively. (1) Students should be encouraged to write their own ideas and allowed the use of various kinds of expression freely, (2) Teachers should prepare contents which are "funny" and "understandable", and promote autonomous motivation, and make them write actively gradually, (3) In mathematical activity, having "may cause sensory Bilingual", both teachers and students should be engaged with objects which their inquiring mind go to, and through which they use some expressions and write positively. (4) In mathematical activity, by enjoying a higher order and seeking some kind of value, students should use a new expression on a higher degree. Finally, I insist it is important, by looking back what they wrote, students should get a viewpoint where they can find a value about writing.

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This paper discussed the possibilities and limitations of applying theory into practice in case of SLE which was applied and reformed in Zambian mathematics Grade 5 and 6 classrooms. The author took out the four major points for its reconstructions in the process of reconstructions of SLEs in lessons. The four points included support for understanding of the starting point; acceptance of small step learning; limitations of flexibility of the material; and language support. They were not necessarily harmonized solutions with the theory of SLE; however, the teacher and the author who implemented 23-24 time lessons had applied SLEs into classroom in order for pupils to understand 'better' than the previous lessons. The author concluded a series of actions would be relevant in the context despite challenges identified throughout the practice.

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The purpose of this paper is to clarify the tasks about mathematical contents in the current curriculum. From the viewpoint of mathematical literacy, it focused specifically on the function. For attaining this purpose, firstly it is described that today's mathematical literacy include both of mathematics of application-oriented (functional) and structure-oriented (theoretical) in which it is emphasized application-oriented (functional) mathematical methods. In this regard, the function is very important to today's mathematical literacy. The function in the school mathematics is defined by a point to see a change dynamically. Therefore, the definition is application-oriented. However, the process is structure-oriented. For example, students are learning with an emphasis on the nature of primary and exponential functions such as a typical function. In addition, the end of the curricula is calculations of the differential calculus and integral calculus, and this is inconsistent with an aim of the function learning. The conclusion of this paper is that the function domain must change the curriculum from structure-oriented to application-oriented as the nature of function. For that purpose, it is necessary to emphasize the process in which it results to a function and returns a function to "real world" in unit composition and a class. The author will work on the tasks that clarified in this paper in future.

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This research aims at clarifying the transitional process from elementary mathematics to secondary one, especially arithmetic to algebra, and from the viewpoint of Early Algebra, we focus on algebraic reasoning at elementary grades. However, it does not yet become clear enough. The purposes of this paper, therefore, are to clarify algebraic reasoning through the semiotic analysis of classes of the multiplication and division with fractions, obtain the implications to promote algebraic reasoning, and obtain the implications for the classes. The results are followings. (1) Algebraic reasoning was identified by the semiotic analysis of the classes from the viewpoint of generalization and justification. The characteristic of the reasoning depends on deductive reasoning grounded on a property of numbers and operations or a pictorial expression. (2) From the viewpoint of justification, we regard deduction using a model and a specific case as deductive reasoning, in the case of deduction using a model, it is important that the model is made a tool, in the case of deduction using a specific case, it is important to examine whether become the generic example or the representative special case, and to draw various deductive explanations. (3) About the implications for the classes, at the time of classes of the multiplication at the second grade and the unifying partitive division with quotative division at the third grade, it is important to make students understand that the relation of multiplication to division is reverse operations from concrete operations. At the time of classes of the division with fractions, it is natural to understand the method of calculation from a property of operations.

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　　In prior studies, it has been pointed out that misconceptions play important roles in processes of mathematical teaching and learning. The research objective of this article is the development of the theoretical framework in order to inquire misconceptions from the viewpoint of conception model.
　　Regarding to this objective, the model of condition of conception is constructed. This model is "tetrahedral model of static aspect of conception", which consists of "object / reference context as 'event'" (O), "sign / symbol as 'event'" (S), 'notion' (N) and 'conviction' (C). This is based on Steinbring's framework "the epistemological triangle" and on Mizoguchi's model "the C(C,N,E) model" (these letters C and N have different meaning from those in the tetrahedral model), and shows that O, S and N are nothing but connected indirectly by C. N means learner's concept, ambiguous idea, knowledge, or mental model. C means learner's attitude towards mathematics or mathematical knowledge. 'Event' means learner's practical experience which is laden with N and C. Then, O is object / reference context which is laden with N and C, and S is sign / symbol which is laden with N and C. The tetrahedral model can identify relatively misconception, because this model can describe context of mathematics teaching and learning through N.

Fig. 1. The tetrahedral model of static aspect of conception

　　There is probability misconception of "the law of small numbers". This misconception makes learners believe that a small sample will be representative of a large sample. The misconception of the law of small numbers is characterized as Fig. 2. "Small sample" and "fraction numerals as probability based on equiprobability" cannot be linked by "classical probability and frequentistic probability". They are connected by determinism. This argument is justified by interpreting history of probability and resilience of misconception.

Fig. 2. The tetrahedral model of static aspect of conception: the law of small numbers

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This study aims to develop teaching materials and instruction methods which help pupils acquiring the spatial thinking ability. Through experiment lessons, I found the followings as pupils' problems concerning the spatial thinking. 1) Pupils understand the projective view of solid. 2) Pupils have the difficulty to keep the front face of solid in one direction when they draw their figures. 3) Pupils often ignore the relative size of solids when they draw several solids in one figure. 4) Pupils often reverse the figure of solid. 5) Pupils sometimes draw the figure seen from diagonally above of solid. 6) Some pupils cannot consider two figures simultaneously in the process of making solid. 7) Pupils do not check the possibility of an alternative solid after they make one solid.

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The Programme for International Student Assessment (PISA), which is coordinated by the Organization for Economic Cooperation and Development (OECD), is a worldwide evaluation of 15-year-old school pupils' scholastic performance. Japan attained the highest level on PISA Mathematical Literacy Test in 2003. However, Japan could not reach the highest level in the area of the Probability and Statistics, called 'Uncertainty', on this test. At that time, the Japanese course of study did not cover the contents of probability and statistics totally. That is why we should accept the fail in that area. Then, what kind of answer patterns did the Japanese students have on PISA2003 Mathematical Literacy Test, especially in the area of "Uncertainty"? If we get the picture of their answer patterns, it would be possible to make a deep analysis of secular change between the results in PISA2003 and the future results in PISA2012 in the future. The aim of this study is to reveal the answer patterns of Japanese students on PISA2003 Mathematical Literacy Test. Especially by focusing on the "Uncertainty", domain compare 13 countries and areas (namely, Australia. Canada, Finland, France, Germany, Hong Kong, Ireland, Italy, Japan, Korea, Netherlands, New Zealand and the United States). Analysis revealed that Japan has an unusual answer pattern in terms of items functioning. Particularly, "Uncertainty" shows a marked tendency to possess unusual answer pattern.

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In the Republic of Zambia, Lesson Study Support Project was started by JICA technical cooperation in 2005. The Lesson Study contributed to disseminate methodology of Lesson Study to stakeholders. However, teachers feel compulsion toward Lesson Study because of governmental directive. Therefore, quality of Lesson Study has been still low in Zambia. Baba & Nakai (2009) have pointed out necessity of accumulation for lesson practice through Lesson Study. Therefore, it is time for examining the quality of the Lesson Study report for five years. In this study, author focused on the Lesson study report called research bulletin. Research bulletin is one of the deliverables on Lesson Study. Through analyzing its contents, I clarified current problems of Zambian Lesson Study. Through the result of analysis, it was clarified that it is important for research bulletin to writing process than deliverable. Zambian research bulletin on Lesson Study was only information of Lesson Study implementation. Especially it is serious that conscious mind of teachers is intractable through Lesson Study. Moreover, submitting research bulletin is purpose of Lesson Study in itself. Therefore, it is necessity to develop the system that the education board evaluates the research bulletin of each school.

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So far the study of metacognition in mathematics education has focused on metacognitive skill. From definition of metacognition, metacognitive knowledge is incestuous relationship cognitive activities, though metacognitive skills are not held context of cognitive activities. Therefore, study of metacognition in mathematics education should focus on not only metacognitive skill but also metacognitive knowledge. And purpose of the study should be to development usable metacognitive knowledge in mathematical problem solving. This paper has considered the state of the metacognition in mathematics education. Therefore, the metacognition research in mathematics education was surveyed, and it checked mainly being focused on metacognitive skill these days. On the other hand, about metacognitive knowledge, it is a target to classify mathematics education like cognitive psychology and to elaborate. Research in consideration of norm nature called the metacognitive knowledge which should be formed in mathematics education is seldom done. And although we surveyed the research on training of metacognition, the focusing of almost all the researches was carried out to how meta-cognitive skill is worked during problem solving. Therefore, the danger that subjective metacognitive knowledge would be formed was pointed out. As a conclusion, we show two points as extension of metacognition. ・About value decision of metacognitive knowledge, pupils do not subjective but do intersubjective. ・About monitoring in metacognitive skill, pupils monitor not only self but also others and classroom. Consequently, we suggest that teaching and learning of metacognition should work on not only individual solving but also collective solving (Neriage). And teachers do the control discussion and the metacognitive support to classroom in Neriage.

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The final purpose of this study is to develop the curriculum of geometric figure in the upper grades which promotes the mediating children's empirical recognition of the geometric figure to logical one. For attaining this purpose, this paper describes the usefulness and the theoretical framework of curriculum development to promote the awareness of the set of the relations among properties of geometric figure by applying Lampert's theory. Lampert insisted that the mathematical knowledge was made up of three mental components related to the learning content, and illustrated the interactive relationships among those components by using the idea of awareness and unawareness. We applied this theory to the lessons of geometric figure. The framework that we constructed is composed three components; concrete and intuitive understanding, (among the geometric figure) relational understanding, and logical understanding. The main purpose of teaching and learning of the geometric figure in elementary school is the understanding of properties. So we described the relationships and the mechanism among the three components for understanding the properties by the awareness and the unawareness of properties. Furthermore, for evaluating the framework, we designed the fifth grade's lessons "geometric congruence" as an application example and evaluated it. As a result, we could confirm that the three components and the interactive relationship we insisted in this paper became the theoretical framework to design the lesson which promoted to change children's understanding about geometrical figures from empirical one to logical one.

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The objective of this study is to identify current issues concerning undergraduate and postgraduate teacher-training curriculums and to provide guidelines for solutions and improvements in order to make sure that the contents and methods of Senior Seminar for Prospective Teachers, which will commence in 2013, are appropriate and of high quality. This is an empirical study for issue resolution with the purpose of developing educationally-effective curriculums for teacher training which involve both undergraduate Teacher-Training Programs and Postgraduate Advanced Teacher Training Courses. For the goal of Advanced Teacher Training Courses established in 2009, we are intended to develop effective programs relevant to today's task of training prospective teachers to acquire high degree of professionalism and practical teaching skills.

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