The purpose of a series of our studies is to discuss more effective method of mathematical problem posing by using computer. We have reported in the previous studies such mathematical activities by university students who were the prospective teachers. In study (I), we practiced the problem posing by using computer after solving original problem, and through study (II) to study (V) we examined various ways for the effective problem posing. The features of our methods are to give students enough time to create problems, and to provide situations in which students make conjectures on results and get the numerical calculation by using computer. In this paper, we report the practice of problem posing by using computer after solving original problem which has some good characteristic to guide the developmental problem posing. In addition to the above features, we requested students to make the good-quality problem. We require the students to describe how they contrived their own problems from the original problem or other sources and how they devised the problem with mathematical inquiry by using computer, and we analyze such procedures using the posed problems by them. Results of analysis indicate that such activities enable students to extend mathematical inquiry. As in the previous studies, the practice shows some tendency that students who tackled making problem by using computer get some deep understanding for the mathematical properties related to the problems. Students solved problems to each other, and the solver and the student who posed a problem commented on problems each other. As a result, there was communication between students. We observed more positive learning attitude of students than the usual classes. It is asserted that the opportunity in a class to discuss problems posed by using computer is very important in such activity, in particular for the prospective teachers.

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　　There are crucial issues in the transition from the primary school to the secondary school mathematics curriculum. One of the main issue of the transition is concerning with the notion of variable. The purpose of this paper is to address the following question: how we should interpret the evolution of students' conceptions of mathematical expression with variable as a whole?
　　For attaining this purpose, firstly, the conceptual change approach and the theory of reification are discussed. Secondly, as a preliminary analysis, the notion of variable can be considered in terms of the theory of mathematical symbolism; for example, some semiotic viewpoints for mathematical expression, the syntactic definition of mathematical expression, and the difference between constant and variable and quasi-variable.
　　As a result of such preliminary considerations, the evolution of conception of mathematical expression can be characterized as both "horizontal development" and "vertical change". The "horizontal development" can be explained as transition from the interiorization to the condensation in the same semiotic level. The "vertical change" can be explained as transition from the condensation to the reification among the different semiotic level. In the final place, the horizontal and vertical evolution of conceptions can be identified in two different didactic situations with the help of the following scheme.

Fig. 1: An interpretative framework for conceptual change on the notion of variable

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　　The aim of this paper is to characterize a construction of children's conception on the elementary concept of probability. Firstly, three dualities on classical probability and frequentistic probability are identified. A construction of children's conception on the elementary concept of probability is accomplished as a cooperation of these dualities.

table 1: Dualities on classical probability and frequentistic probability

Secondly, a model of a cooperation of dualities on two concepts of probabilities is elaborated, using analogy with "symbiogenesis" which is a concept for interpreting cooperation on evolution of living things in the theory of evolution. This model is laden with three points of Table 1. Phase 1 indicates the condition of learning classical probability or frequentistic probability. Phase 2 indicates the condition of learning two probabilities and of occurring some misconceptions. Phase 3 indicates the condition of understanding two probabilities.

Figure 1: A model of symbiogenesis of probabilistic conception

Finally, it is suggested that a conflict between dualities in phase 2 of Figure 1 is implicit in the present curriculum on probability in Japan, and then that a means for leading phase 3 is insufficient.

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The purpose of these studies is to clarify functions of mathematical proofs in school mathematics, especially constructing processes of mathematical conception. We can find a theorem which describes a relation between two sets. In this paper, we consider such kind of theorem, which is limited to the case where the relation is a function between two sets. In this case, we can prove the theorem by constructing a function between two sets. In this constructing process, we make a family of sets and functions with an indexing set - a sequence of sets and functions. This indexing set is usually an ordered set, natural numbers, real numbers, for example. From this setting, we can construct two kinds of limits. One is a projective limit; the other is an inductive limit. We can construct a function from these structures as a function from the inductive limit or a set of a proper restriction of this, to the projective limit. Form these constructions, we can reform other constructions by embedding the indexing set such as natural numbers into other ordered set such as a positive real numbers. As a result of this reform, we can get a more suitable process of proving the theorem, a more useful method for constructing the function or a more important function between the two sets. This means that ordered sets has important role in constructing process of functions that is proof making process.

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In Ecuador, the student academic achievement is not high. It was revealed by international survey on academic performance for Latin and Caribbean counties called SERCE (the Second Regional Comparative and Explanatory Study), as well as the PISA test which the Municipality of Quito conducted on its own to 282 students of the third year of junior high school in September 2006. As is well known, PISA is international surveys to measure student application ability, and it's constructed by OECD. The result of the PISA test which the Municipality of Quito conducted revealed that the one of the PISA mathematical contents which is "Uncertainty" was the most difficult for the third year students of the junior high school. This fact is usually interpreted as a sign that the "Uncertainty" is the most difficult mathematical content for them to acquire and exert their knowledge and skills. This, however, does not necessarily mean that in the content "Uncertainty" the students will not be able to improve their performance in the near future. In other words, finding out the mathematical content which the students cannot perform highly at this moment is totally different thing from a possibility of improvement of their performance in their future. Then, how can we recognize the mathematical aspect which will be related with the improvement in their performance? To resolve the research question mentioned above, this present study examines differences of the test item difficulty and discrimination between the third year students of the junior high school and the first year students of the senior high school, by making use of the test theory called Item Response Theory (IRT), using the data collected by reconfiguration PISA test of Quito Municipality in February 2010. Sample students were selected from 6 Quito municipal schools who took the PISA test in September 2006, and its number was 514. The study revealed that the mathematical aspects in which the students have possibility to improve their performance are correlated with the degree of the use of mathematical calculation and algorithm. In addition, we got implication that reading ability influences the student mathematical performance improvement.

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In this paper, we clarify the inhibitory factor of flexible idea in present mathematics instruction. And we propose the problems of making a student's flexible idea. We analyzed the inhibitory factor using an achievement test and a creativity test about the area of the figure. As a result, we found that the learners have functional fixedness in the mathematics problem solving. We call it "Temporary plateau of thinking in problem solving". We made two kinds of problems of breaking out of the "Temporary plateau of thinking in problem solving", problem which put restrictions on the solution and problem which required that many methods should be considered. We compared the usual problem with the problems of breaking out of the "Temporary plateau of thinking in problem solving". The results are as follows. - When compared with the usual problem, the problems of breaking out of the "Temporary plateau of thinking in problem solving" made many flexible ideas. - When compared with the usual problem, the problem which put restrictions on the solution was still difficult for students. - If the teacher gives appropriate teaching, the students can break out the "Temporary plateau of thinking in problem solving".

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The purpose of this research is to develop instruments to measure pupils' potential ability for learning mathematics and to study theoretically and empirically the ways to utilize the instruments for improving mathematics education. In the previous study (Nakahara et al., 2010), we found that teaching of potential ability to 4th graders had a positive effect on improvement of potential ability. Following the study, the purpose of this study is to investigate the effect of the teaching of potential ability on a certain kind of mathematical thinking ability that is roughly defined as problem solving ability for textbook-level "process problems" (Charles & Lester, 1982). In this study, firstly, 4 sets of three worksheets for teaching of potential ability were constructed according to the theoretical framework of potential ability. Each set of three worksheets was related to one of the four factors of the "mathematical thinking" of potential ability (i.e., pattern, logic, operation, flexibility of thinking), and each worksheet had one or two mathematical problem(s). For experimental group, the classroom teacher conducted 15 minutes teaching in the morning per a week using the worksheet. And, before and after the 12 week teaching period, we conducted mathematical thinking ability test for both experimental (N=121) and control (N=121) groups. The two-way ANOVA showed significant interaction between the teaching of potential ability and the improvement of mathematical thinking ability. Including the result of our previous study, the positive effect of teaching potential ability on both the potential ability itself and mathematical thinking ability was found. Finally, we discussed how the teaching of potential ability could influence the improvement of mathematical thinking/problem-solving ability and how to consider alternative ways of teaching to improve it reflecting results of our studies.

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The initiative of teachers in developing countries is essential for the development of their own education, and should be considered in international cooperation in the area of education. Therefore, the focus of this research is on teachers' reflection on their mathematics lessons. This paper reports the result of analysis on the characteristics of two (young and experienced) primary mathematics teachers' reflection in a rural area of Zambia. Qualitative data analysis on their writing regarding their own mathematics teaching was utilized, and the following characteristics were identified: The young teacher tends to regard her pupils as a whole, and her evaluation focuses more on their performance of exercises. Based on this evaluation, she tries to improve her lessons in terms of simplicity and concreteness in order to give her pupils mathematical knowledge efficiently. While, the experienced teacher tends to regard her pupils as individuals, and to evaluate and even analyse their misunderstanding considering their previous experience, home background, mental state and so on. Then, she tries to support pupils who don't understand learning content by introducing small group activities and peer teaching so that those pupils could be free to ask for help or to question to friends. However, both teachers had a limitation of reflecting toward realizing mathematics teaching which is intended in Zambian syllabus and textbooks, due to the view of traditional mathematics education. The syllabus and textbooks put emphasis on acquisition of mathematical knowledge, interest, skill, communication, problem solving, application and so on through learner-centred, activity based, participatory and context based approaches.

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In this research, I focus on the organizing activity by Freudenthal as mathematical activity on the secondary education for making improvement. Freudenthal mentioned that local organization is necessary logically. In mathematical education in our country, local organization is taking place in geometrical region and it is connected with B closely. However, Compared with geometrical region, organization in algebraic region was not elaborated. I consider local organization in algebraic region while turn up it in geometric region, identify of assignment is this paper's aim. In conclusion, I suggested that there is an assignment that should establish level between algebraic operation and local organization that Freudenthal and Okada (1977) mentioned. It is reason that the fundamentals for organization as weighting to master way of calculation in algebraic region are poor unlike geometric region, in algebraic region.

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Linguistic aspect of mathematics education is one of crucial points in multi-cultural countries. However, there are few researches focus on this point and no research has been done as diagnosis so far in developing countries. Thus, the purpose of this article is to develop the diagnostic evaluation method and evaluate it in Zambia which is one of such countries with reviewing the Newman's Error Analysis and the research which applied it in Zambia. I conducted the interview survey to grade 3 to 7 pupils at 2 primary schools in Zambia with equipping the research tool which was developed for this research which was developed considering on the context of Zambia. As the result, several difficulties pupils face were observed. There were pupils who read problems smoothly, but could not solve it or explained the situation of problem properly but still could not solve it and so on. However all pupils including those who could not even read a word problem did solve it at certain stage which is "listening", "taking explanation" or "looking at real objects". Finding such discriminated stages might imply the sensitive linguistic situation in Zambia and connect to educational suggestions.

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In our country, it has been an important and difficult task for many of the teachers to achieve 'the class where the students thinking is dominant' or 'the class where the students work actively or independently'. One of the causes seems to be that we understand the class intuitively and the characteristic is not clear. We need the view point or framework which enables us to see it in the mathematics class interaction. The research focuses of this paper are following two points: (1) To construct the framework to see the feature of the class described above. (2) To examine the effectiveness of the framework and to identify some important elements to realize the class. To be more specific we regard 'actively or independently' described above as "Devolution of the Intellectual Responsibility" and further "adidactical situation" which are the central concepts of French mathematics didactics. We assume that there are some steps or levels from the didactical situation to adidactical situation, we developed a framework 'Didactical Situation Model (DSM)' based on mainly two ideas. One is a model developed by Mellin-Olsen (1991) identifying the level of knowledge control in a didactical situation. The other is the types of "MATOME" identified by Iwasaki and Steinbring (2009). To examine the framework DSM, 5 mathematics lessons were taken from a sixth grade mathematics classroom. Analysis of the third lesson revealed that: (1) The DSM enables us to catch "the Devolution of the Intellectual Responsibility" in the each of the didactical situation in the lesson more concretely. (2) For realizing "the Devolution of the Intellectual Responsibility" in the interaction, it is important for teachers to evaluate the student's idea from the view point of the problem concerned instead of evaluating student's idea from the view point of teacher's expectation. (3) To achieve 'the class where the students work actively or independently', it is extremely important that students can have the goal level of control of the DSM.

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We have conducted a research project "School Support Project" in an elementary school for about four months from September, 2010 to December, based on the methodology of "Design Research". One of the purposes of the project was to reform the sixth graders' mathematics lessons on the topic of division of fractions. A main concern of the lessons was to discuss with the students the calculation methods. We tried to reform the mathematics lessons based on the viewpoint "Mathematics as the science of patterns" known for the name of Erich Ch. Wittmann in mathematics education. Based on the view point, the word problems for introducing the division of fraction were selected from some practical research reports carefully. The learning trajectory was outlined based on the "Epistemological Triangle" devised by Heinz Steinbring. The research focus of this paper is to examine the effectiveness of the view point "Mathematics as the science of patterns" with reporting what happened in the classroom. The lessons were recorded with video cameras and the detailed transcripts were prepared for the analysis. The results of the analysis were summarized as follows: ・We were able to find many students who found patterns, checked them and formulated as calculation methods of division of fractions in the class. ・We could infer that the viewpoint of "Mathematics as the science of patterns" played an important role to reform the sixth graders' mathematics lessons on the topic of division of fractions. ・Therefore, the viewpoint of "Mathematics as the science of patterns" could work as a viewpoint by which primary teachers reform their everyday mathematics teaching. ・The word problems for introducing the division of fraction and series of equations generated from them were important to conduct the lessons compatible with the view of "Mathematics as the science of patterns". Because they played an important role as a Reference Context for the students. ・Moreover, it was important for teachers to organize the classroom interaction carefully from the view of "Mathematics as the science of patterns", especially the five realized viewpoints which are compatible with the view of "Mathematics as the science of patterns". ・The most important role of the viewpoints of "Mathematics as the science of patterns" is that it gives an elementary teacher who does not specialize in mathematics a basis to evaluate what is the essential element of the teaching of mathematics, and it enable him/her to reform his/her everyday teaching by oneself.

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The purpose of this paper is to suggest that the teaching focused on the extention of multiplication of fractions on the sixth grade is effective to promote learner's understanding of its meaning. In the teaching of multiplication of fractions, we adopted the following four activities in the lesson. (1) The first lesson was designed for the extention of the meaning of multiplication of fractions in comparison with multiplication of integers. (2) The classification activity of mathematical word problems was introduced to promote learner's awareness of the mathematical structure on multiplication of fractions. (3) The number line was introduced to support leaner's understanding when the ratio concept became the theme of the lessons. (4) We introduced the activity to organize the relationship among multiplication of integers, decimal fractions, and fractions. Through the implementation of these experimental designs into the lessons of the multiplication of fractions on the sixth grade, the performance of many children was improved. The result of the post evaluation showed that they were able to understand the meaning of multiplication of fractions. But, there were some difficulties for each learner to realize the extention of its meaning. We need to improve the lesson designs according to learner's situation of understanding. Especially, the standardization of the lesson of multiplication of fractions and the way of using the number line are two major tasks for now.

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The conception of mathematics held by the teacher may have a great deal to do with the way in which mathematics is characterized in classroom teaching (Dossey, 1992). View of mathematics is distinguished precisely into epistemologies of mathematics and of mathematics education (Sierpinska & Lerman, 1996), but here I limit the discussion to epistemology of mathematics. The purpose of this study is to explore the educational effect of teachers' views of the relationship between the competencies and the nature of mathematics. The DeSeCo Project lays out conceptual framework for key competencies in an international context (Rychen & Salganik, 2003). The point I would like to make is that teachers' views of mathematics influence on teachers' views of key competencies. Through discussing, although the conceptual framework of the DeSeCo Project classifies competencies into three broad categories, there is only one category which called "use tools interactively" competencies would be linked to mathematics education. This "use tools interactively" competencies is consistent with external conceptions of mathematics rather than internal conceptions. But if we try to take key competencies as a whole, and think of as a goal in mathematics education, then teachers need for a having internal conception.

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Many studies show that analogy plays an important role in mathematics education and mathematical problem solving. The purpose of this study is to investigate how to use analogy in junior high school mathematics instructions. In this paper, I classify the type of studies which is related to analogy in mathematics education. Among them, the result of an experimental lesson study (conducted Richard, L. & McDonough, I. 2009) suggests that supported analogies can improve students learning and problem solving abilities. The supported analogies are consisted of four strategies, that is source presented visually, source remains available during target analog instruction, visual alignment between the two analogs, and gestures between the two analogs. I discuss about some roles of this supported analogies playing in the junior high school mathematics instruction. Then I develop some teaching materials in which the above supported analogies can help junior high school students learning of some topics that are considered for them to understand.

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