This study tries to clarify what the epistemologies in mathematics education may influence on the conceptions of teaching and learning and research through examining the historical development of researches of epistemology and drawing the further research tasks. In this paper, we discussed and clarified the followings. First, on the relations between the epistemologies and the teaching practices in the classroom, we showed various paradoxical situations caused by the epistemologies of objectivism, Cartesian dualism, and representational views of mind. Second, on the relations between the epistemologies and the conceptions of the teaching and learning of mathematics, we examined the debates between radical constructivist and socio-cultural researchers. Based on them we clarified that the radical constructivism and the socio-cultural views were approaching with each other, in particular in their conception of "social" and "intersubjectivity", with drawing the view of interactionism with each perspective, and that there was an insurmountable barrier between the mechanism of reflective abstarction and that of internalization in which the social practice in the sense of "thick" may influence on the recognition of the individual. Third, on the relations between the epistemologies and the research methodologies, since they are directly related with each other, it may be necessary to specify the relations. Moreover, we as researchers are requested to cope with the different worlds of reality in the sense of small letter, which is suggested by the metaphor of Bricolage. Lastly, as the future tasks of the epistemology studies, we indicated the problem of coordination or integration between constructivism and socio-cultural views, the problem of characterizing social interactions and cultures in the mathematics classrooms based on the epistemologies, the problem of situating the view of mathematics in each epistemology, the problem of adopting and organizing various perspectives, and clarifying the epistemological bases of Japanese mathematics education. We consider these tasks essential to an establishment of the identity of mathematics education.

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The paper discusses how two Zambian teachers at the basic level successfully changed their teaching and their views on students, teaching materials and their own teaching during lesson development in collaboration with the author. As a result of examining the continuous cycles of planning, implementing, and reflecting lessons based on SLE, the author concludes that both of the teachers drastically improved their teaching techniques that met students' learning and developed their reflective observations on students' learning. On the other hand, there were insufficient mathematical content-based discussions. That was not only because of teachers' faults, but also because of other related issues such as inflexible curriculum, students' abilities, and the ways of collaborations by the author.

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This paper reports part of an ongoing developmental study into conceptual change in the teaching and learning of mathematical expressions. Especially, in the transition from elementary to secondary school mathematics, the teaching and learning of variables can be a crucial didactic issue. The purpose of this paper is to design a teaching experiment in order for sixth grade students to develop their conceptions from the operational to the structural through the teaching and learning of variables in functional relation. To attain this objective, the theory of reification invented by Anna Sfard is accepted as the theoretical framework in this study. Also, we attempt to design a particular conceptual change situation with the help of the teaching experiment methodology. Although the teaching experiment methodology has three main phases (instructional design and planning; ongoing analysis of classroom events; retrospective analysis), we focus on the first phase in this paper. By way of conclusion, the following points are discussed as theoretical orientations for designing conceptual change situation on the notion of variable. ・In order to characterize conception of variable in the experimental lesson, different mathematical representations and their functions can be positioned in three hierarchal phases: interiorization, condensation, and reification. ・The three hierarchal phases is schematized in terms of the structuralizing the operational representations through the interrelations between numerical table and mathematical expression. ・The teaching experiment aimed at the reification process of variable in functional relation is designed in terms of the structuralizing the operational representations. These three points can be very crucial perspectives, when we will analyze and illustrate the teaching experiment. As a future task, we would like to report the actual teaching and learning process in the classroom events.

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Nowadays, the improvement of teaching toward students oriented activity in the high school mathematics class is strongly required to enhance the mathematical thinking of students. The author has a belief that the essences of teaching matters should be found out by students themselves in the class, and the teacher should prepare appropriate situations to realize such findings. In particular, it appears often at the introduction of teaching contents. In this paper, we study a practical plan to realize the aspect stated above in high school mathematics classes, through teaching materials and actual practices. We put the theoretical base of our assertion on the constructive approach proposed by Ito (1993) and other authors. The main findings are as follows: (1) We show that it is effective to provide opportunities that students consult or discuss with neighboring students each other about the focused problem they are facing in the class, and that such effective communication by students is possible if the teacher prepares proper situation and set appropriate time allocation. (2) We analyze what are the important factors on teaching materials to be used at the introduction, and we show several concrete samples of teaching materials that satisfy such factors. The samples have been examined in actual classes and proved to inspire students' mathematics thinking at introduction. (3) We report one practice of a class by the author, which incorporates the factors in (1) and (2), and analyze the assessment of it through the questionnaire to the students. As a result, high percentage of students give positive answers to the planned class activities by expressing their satisfaction for good understanding.

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The purpose of this paper is to reveal the tasks to realize teaching and learning aiming to foster mathematical literacy and approaches to solve those. To achieve, firstly it was described that today' mathematical literacy embraces both of application-oriented and structure-oriented, and it is emphasized application-oriented mathematical methods. Then teaching and learning incorporating structure-oriented into application-oriented mathematical methods was thought as teaching and learning aiming to foster today's mathematical literacy. But, the connection of application-oriented and structure-oriented became the task to realize this teaching and learning because a purpose of both of them is different. To solve the task, the author referred to Lesh' researches which focus on teaching and learning (what is called Model-Eliciting activity) that creates mathematical structures during iterative cycles of mathematical modeling. According to this activity, it was thought that the above-mentioned task on the connection will be able to be solved by "developing models". However it was also thought that it is difficult to realize teaching and learning by only this view. The reason is that teaching and learning in mathematics education should be thought from view to both of mathematics and students but this is only view to mathematics. And it is not necessarily said that Lesh' researches enough clearly describe this activity from view to students. Thus, it was pointed as teaching and learning aiming to foster today's mathematical literacy that the argument about students in developing models is not nearly enough. And an approach to solve this is to relate research areas with each other from view of developing models, which will be the foundation of arguments about students in developing models.

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The purpose of this paper is to clarify the tasks of the learning and teaching to arithmetical activities. For attaining this purpose, firstly it is described the philosophy and concrete example of arithmetical activities and the problem for them. In addition, it is described the critical study of problem solving as the method of learning and teaching before. Based on them, it is described the problem to find out the new method of the learning and teaching and the view of the solution to the problem. Focused to learning, arithmetical activities are divided into two parts on the point of purpose and method: "purpose model" and "method model". For the nature of arithmetical activities and in today's regard, it is necessary to learn arithmetical method. However, the status of learning and teaching is no more than about arithmetical contents, so not method so much. Consequently the problem is that find out the method of learning and teaching for "purpose model" arithmetical activities. For the approach to this problem, it is necessary to clarify the process of children' thought on arithmetical method and the relation between arithmetical contents and method, therefore study with vision of "method model" and mathematical activities. The author will work on the task with these points.

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A claim of this research is to place extension to arithmetic with algebraic characteristic in the aim of arithmetic education. As for that approach, the author focused on algebraic reasoning. In such a point of view, it becomes the problem in arithmetic how algebraic reasoning is promoted. The nature of algebraic reasoning is purposeful generalization and "symbols as objects" is located as an important stage in the process of generalization. By consideration of case study on division with fraction, even if "symbols as objects" was not accompanied, algebraic reasoning worked, but it turned out that a student shows difficulty to an under-standing of generalization. Algebraic reasoning is accompanied by "symbols as objects", and going to successful generalization became clear from this. At the same time, it is clarified that successful "symbols as objects" in arithmetic is so difficult. Therefore, "a setup of learning activities suitable for a grade stage" and "recognition of quasi-variables" were key ideas that will solve the tasks of "symbols as objects".

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The purpose of this research is to consider the concept of "Operative proof" that E. Ch. Wittmann proposed as a mathematical learning activity in 1996 and to construct foundations for developing the Substantial Learning Environments with "Operative proof". Especially, for developing the Substantial Learning Environment with "Operative proof" that used counters and place value table, in this paper, I investigated how children operate counters and place value table. I interviewed children to examine their reaction for counters and place value table. This interview is composed of the following questions. 1. Reading the representations of numbers with counters and place value table, and expressing the number on the place value table with counters. 2. Understanding the meaning of the operation of counters, and explaining it. 3. Using the operation of counters to solve arithmetic problems. And this was executed for 45 children of the fifth grader at an elementary school. The following suggestions were obtained from results of the interview for the design of the Substantial Learning Environment. ・Children don't need special guidance to understand the representations of numbers with counters on place value table, but they need some guidance to understand the meaning of the operation of counters. ・At the first step of the learning activity, teacher should guide children how to use counters and place value table. ・For the learning activity with operative proof, children need not only formal computation practice like calculation by writing but also basic learning activity to see number flexibly. Future tasks are to design the concrete Substantial Learning Environments that used operative proof by making the best use of these suggestions.

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The purpose of this study is to reveal the factors which help to establish the performance assessment and the viewpoint of the instruction for cultivating the ability to utilize the knowledge contained within a teaching material. First, this study describes the performance assessment, the rubric and the outline of the unit. Next, the relationship between the result of the performance assessment and that of the unit test is investigated. As a result of the investigation, it is not shown that significant correlations between the way of thinking in mathematics and the results of the performance assessment were found. After that, however, the relationship between the result of the performance assessment and that of the questionnaires for learning is explored. The results show that it is important to enhance the convictive attitude (students do not ask some questions to others easily, but they get some advice from others after they think and pursue their own ideas or solutions) in order to cultivate the ability to utilize the knowledge.

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　　Iwasaki & Steinbring (2009) identified the different types of "MATOME", which is one of the most important Japanese teaching concepts, in a mathematics class. They indicate that MATOME, especially in its traditional and classical form, which is a widely shared cultural expectation in Japan, would be a blind spot of Japanese mathematics teaching and suggest a possibility of the alternative version of MATOME: Student's MATOME. Confronting these difficulties as a reflective practitioner, it could be the starting point for Japanese teachers to have a better understanding about the mathematics class interaction where MATOME occurs. The research focus of this paper is to examine how students' MATOME / teacher's MATOME occur in mathematics class interaction, and to find out the causes behind the occurrences.
　　We conducted the school support project with some lesson studies in a third grade classroom with the class teacher Ms. M for improving mathematics classes over a four-month period from September to December, 2011. For the purpose mentioned above, two classes are taken from the third grade mathematics class in October and December, 2011.
　　We identified the situations where students' MATOME / teacher's MATOME occured in the two classes by analyzing the interaction from the view point of the different types of "MATOME". And then we examined features of the interaction where students' MATOME / teacher's MATOME occurred by comparing them, and further considered the causes. The results are summarized as follows.

[table]

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In previous studies, it has been reported that misconceptions have some resilience. The resilience is recursivity with misconceptions or fragility with conception, and one of the important natures in mathematical recognition process. Many studies in mathematics education, however, have not regarded this nature as the subject matter of researches. The purpose in this paper is to address the resilience of misconception. Particularly, this paper focuses on misconception of the law of small numbers (MLSN), which makes learners believe that a small sample will be representative a population.
　　For this purpose, a tetrahedral model of static aspect of conception (TMSC), which is based on Steinbring's framework (i.e., the epistemological triangle) and on Mizoguchi's model (i.e., the C(C,N,E) model), is used as interpretative framework. TMSC consists of four components which are 'object/reference' (O), 'sign/symbol' (S), 'notion' (N) and 'conviction' (C) (the C and N in these four letters have different meaning from those in the C(C,N,E) model). N means learner's concept, ambiguous idea, knowledge, or mental model. C means learner's attitude towards mathematics or mathematical knowledge. O means learner's practical experience on object or reference context, which are laden with N and C. S then means learner's practical experience on sign or symbol, which are laden with N and C. One solid line and three broken lines show whether four components connect directly or indirectly with each other. TMSC as a whole means that O, S and N are nothing but connected indirectly by C. With this model, misconceptions are characterized as "non-shift of only C." MLSN is then identified by TMSC: [O] small sample; [S] population; [N] the law of large numbers; [C] determinism.

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

　　The resilience of MLSN results from a kind of firmness. One of sources bringing firmness to misconceptions is a traility of C. In case of 'MLSN', which has probability as O, the C has three aspects as the traility: 1) nature of knowledge of probability; 2) attitude towards phenomena; 3) symbolic notation of number. The shift of C in 'MLSN' can be characterized as Fig. 3. That is, transitions of three aspects of C need shifting C itself, so it is difficult. The determinism as C in 'MLSN' means that a transition of one aspect of 1)〜3) does not occur at least.

Fig. 2.　A traility of ‘conviction’: ‘MLSN’

Fig. 3.　An interpretative framework for shift of ‘conviction’: ‘MLSN’

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The purpose of this study is to clarify the elements which enable students to connect between inductive activity and deductive one in the process of teaching Pythagorean Theorem. For the purpose, we designed the learning trajectory of Pythagorean Theorem based on mainly two points of view: H. Steinbring's idea of "Epistemological Triangle" and the viewpoint of "Mathematics as the science of patterns" known for the name of Eric Ch. Wittmann. Three lessons were done according to the learning trajectory in our research project "School Support Project", which was conducted in a junior high school for about four months from August, 2011 to December. The lessons were recorded with video cameras and the detailed transcripts were prepared for the analysis. We analyzed the process of the lessons and tried to find the important elements which supported the students mathematical activity, especially ones which made connection between discovery and demonstration of Pythagorean Theorem. The important elements clarified from the analysis were summarized as follows: (1) It is important to develop the students own construction, which are various and individual one, in the stage of inductive activity. The students could use the constructions as referent context in the stage of deductive activity spontaneously. (2) It is also important to start the lessons from the specialization of Pythagorean Theorem, especially the case of the base length is 1, and to develop based on the viewpoint of "Mathematics as the science of patterns". The students could develop the specialized idea more generalized one by using "What if not?" strategy automatically. The typical patterns of inductive activity which had been done in the previous stage would be repeated in the next stage independently.

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The purpose of this paper is to suggest that the teaching focused on the extension of multiplication of decimal fractions at the fifth grade is effective to promote learner's understanding of its meaning. In the teaching of multiplication of decimals, we adopted the following three activities in the lesson. (1) Teaching of a number line before learning about the multiplication of decimals. (2) promote the expansion of teaching the meaning of multiplication of decimal. (3) Using a combination of number lines the multiplication of decimals and the study of percentages to promote learner's awareness of the mathematical structure on multiplication of decimals. Through the implementation of these experimental designs into the lessons of the multiplication of decimals at the fifth grade, the performance of many children was improved. The result of the post-test showed that the students were able to understand the meaning of multiplication of decimals. But, there were some difficulties for each learner to realize the extension of its meaning. We need to improve the lesson designs according to a learner's situation of understanding. Especially, the standardization of the lesson of multiplication of decimals and the way of using the number line are two major tasks for now.

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Since the Meiji era, a lesson study has worked in the elementary school teacher communities in Japan. Japanese lesson study, called jugyokenkyuu is a professional development process that Japanese teachers engage in to systematically examine their practice. However, because there is a national curriculum and this curriculum has the force of law, Japanese lesson study tends to focus on only teaching method. Even so, lesson study is accepted around the world as a good model for mathematics teacher education. In recent years, qualities that are required of teachers are distributed in two major compartments. One is personal abilities for living through that period of time. The other is abilities for getting along with people. Jugyokenkyuu will contribute to develop abilities for getting along with people, but it seems that jugyokenkyu will not contribute personal abilities for living through that period of time. For this reason, we should construct new lesson study in japan. For putting this into practice, we need to motivate teachers. In the current national curriculum in japan, there are content-free subjects of mathematics. Mathematics teacher should make a curriculum for a lesson. In conclusion, I suggest that new lesson study needs to aspire to becoming a research of curriculum development. And to do that, teachers had better utilize frameworks of Substantial Learning Environments by Wittmann (2001). In addition, for ensuring validity of research, it is useful for using the teaching experiment methodology.

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Generally in Japan, teaching of argumentation in geometry region purpose to acquire matter of argumentation. While on the other hand, aim of this research is to find purpose and assignment of teaching of argumentation in algebraic region in late-stage secondary education. In this paper, I focus on the universal and constitute the lesson. Actually, I practice and analysis it. In make up the lesson, I prevent to jump from singular proposition and particular proposition to universal proposition, because force students to check the universal. The effect of analysis of the lesson and reports, in argumentation that conscious, students jump from singular proposition and particular proposition to universal proposition as jump in argumentation of geometry in prior secondary school and do like same it. So this paper concludes that we need to resolve it and the universal is purpose in teaching of argumentation in late-stage secondary school.

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