The purpose of this study is to clarify the notion of "mathematizing" by Freudenthal. Freudenthal characterizes "mathematizing" by an activity as "organization". "Mathematizing" has not been enough interpreted from a viewpoint called as such an "organization" until now. Therefore, in this study, I define the notion of "mathematizing" based on the "re-invention" in the theory of mathematics education by Freudenthal. And I complement the notion of "mathematizing" by interpreting a meaning of "organization". In conclusion, "mathematizing" is an activity as act out a structure by working on reflective acting to organize experience.

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Numerous attempts have been made by researchers to answer "What is mathematical understanding?" and "How is it possible to capture individual understanding?" (e.g., Skemp, 1976; Beys & Herscovics, 1977; Herscovics & Bergeron, 1983; Pirie & Kieren, 1994). And a model of understanding in mathematics education has to consist with suitable components of the model and be used by researchers or mathematics teachers to analyze students' understanding. In this point, Koyama (1992) has studied mathematical understanding from a slightly different perspective and argues that we need to construct a process model of mathematical understanding. However, we need to pay attention to aspects of a cognitive function for promoting students deepen and expand their mathematical understanding. The aim of this paper is to define a cognitive function which could promote students deepen and expand their mathematical understanding. In this paper, the cognitive function is showed as an act which is observed clearly by researchers or mathematics teachers. As a result, I argue that if the act with conversations of students, "Why is that argument true?" or "Is that argument true or not?" are observed, students begin to deepen or expand their mathematical understanding. In this paper, with considerations of some previous researches, such act is explained in term of "proving act" by Harel & Sowder (1998).

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The purpose of this paper is to consider the negociation pattern of mathematics instruction by transformative listening of teacher. Transformative Listening : There is an attempt to interpret and make sense of what the speaker says, but always from the point of view of the listener. If a teacher is listening in a transformative manner they will characteristically have an open stance to the interrogation of assumptions they are making. As a consequence, I understood that there are six negociation patterns. A: (I)⊃(II)⊃(III) B: (I)=(II)⊃(III) C: (I)⊃(II)=(III) D: (I)=(II)=(III) E: (I)⊂(II)⊃(III) F: (I)⊂(II)=(III) (I) ideas in unaided solution (II) ideas in group discussion (III) ideas to put together thoughts ⊂ : Increase ideas ⊃ : Decrease ideas= : Equal ideas I thought that transformative listening of teacher would be four factors : multiplicity in children's ideas, fallibility in their ideas, prediction of their responses, teacher's metareasoning of their ideas. But most important factor is purpose of lesson. Eventually when the teacher is faced with the gap between purpose of lesson and practice of lesson, what ideas he make a choice among so many ideas decide on the negociation pattern.

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The purpose of this text applies the focus to "Replacement" that is a form of the sign model, considers semiotics, considers psychology, and designs a theoretical frame. And, the problem scene is analyzed by using the theoretical frame, and an educational role of a mathematical sign is clarified. First of all, it takes a general view from the early research of the relation between the sign and the concept for that. Next, the merit and the weak point in study are pointed out describing, and unifying a form model the replacement, and the mathematical guarantee of the expression transformation from a form of the sign viewpoint. And, it is described for the replacement to be able to have the metonymy from the research of rhetoric in the mathematics education, and to identify acknowledged structure to use the replacement with the reference-point structure in addition, and considers it from a theoretical frame of the reference-point structure. As a result, it has been understood to provide the function to make the signifier stand out acknowledging it in the replacement. And, there are two the educational role of the functions as the acknowledgment activity of the student and, the function for the professor to the student. The case not caught in the reference-point structure either existed, and in addition, the cause referred replaced being made an object, and pointed out the difficulty in study.

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The aim of this paper is to consider a constructive activity through "make up an example of" tasks. I named them "REIZUKURI", and presented a lesson plan to construct mathematical knowledge in the classroom (15-years-old students). According to previous research, asking students to provide examples stimulates class participation and produces an active atmosphere. Most of them were descriptions of developing and deepening mathematical knowledge in teaching. "REIZUKURI" addresses different mathematical objects and requires different mathematical knowledge, and it has three features: providing opportunities to deal with abstract mathematical notions, reversing the usual role of what is "given" and what is "to be found" and being open-ended. The lesson plan which I presented aims at constructing the concept of the factorization in prime numbers and the knowledge of how to examine the number of factors. This lesson starts by the task "make up an example of some integers that have only two factors". After students confirm the definition of the word "factor", they are told the definition of the word "prime number" by teacher through many examples concerning this task. And then, teacher asks students to make up an example of some integers that have only five factors, only six factors, only seven factors and eight factors. They will make up many examples by trial and error. I guess that presenting the task "make up an example of some integers that have only twelve factors" for the last time prompts their inductive reasoning and is achieved the aim of this lesson. In addition, the last task acts a important function as Being Mediative in the "Constructive Approach" (Nakahara, 1995).

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Fostering 'mathematical thinking' has been intended as a goal of mathematics education in Japan. Since mathematical concepts are all abstructive, 'mathematical thinking' is the only way to deal with them. Therefore, it's essential in mathematics. But it cannot be clear by its very nature. Rather, the auther intendes that too much clarification of it can lose the freedom of it. So, we must bring out 'mathematical thinking' from the children. This paper studies the potentiality of the sources of mathematical thinking in children. The Review of studies in past shows the esential part of 'mathematical thinking' and the part which has to be extended. Nakajima (1981) describes that the 'value' on which mathematics was created is the base of 'mathematical thinking'. But, Because the concept of mathematics effects on it, we must start with the concept of mathematics which is relevant. Wittmann (1995) suggests MATHEMATICS as the concept of mathematics which includes mathematical works in the broadest sense which covers all sciences and daily life. The auther think that if the 'value' which is the source of 'mathematical thinking' is derived from MATHEMATICS, the extension described above will be done. And the auther concludes that mathematical thinking can be fostered by the 'tearing unit', which is the essential part of Wittmann's mathematics education. As an example of mathematical thinking, 'Symmetric Thinking' is elaborated. The extension is illustrated. And a 'teaching unit' which fosters childrens' 'Symmetric Thinking' is designd.

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The purpose of this paper is to consider about the connection between mathematics and science as one of the way that develops science and mathematics. In this paper, the author views "inquiry" activities as a common content in mathematics and science, presents the learning cycle and the example to develop inquiry activities. The learning cycle consist of following four phases. ・Engage : This phase is that students understand the task presented by the teacher and have a question about the task. ・Explore : This phase is that students predict and pursue about their question . ・Explain : This phase is that students elaborate by explaining the results in Explore phase each other. ・Expand : This phase is that students apply in a new context on the basis of Explain phase. Using this cycle, students learn that there is a connection between mathematics and science, so it becomes smooth for them to use their knowledge from mathematics lessons and science lessons. Thus, the inquiry activity is a common learning process to use in mathematics and science.

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The purpose of this project is to research and develop the appropriate learning environments for children at the introductory level in elementary school. The aim of this paper is to clarify through the pilot lessons whether the numbers up to 20 can be introduced at the first term of the first grader in elementary school. The learning environments for the pilot lessons were designed based on the basic ideas of "Das Zahlenbuch" (Wittmann/Muller; 2004). The most of the children could see the number "16" as "10 and 6", "5, 5, 3 and 3", etc. at the end of the lessons. Such the facts from the pilot lessons tell us that introduction of the numbers up to 20 at the first term is possible.

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In this study, the significance for instructing the projection which was elicited in the current course of study and will be reappeared in the next course of study is reconsidered. In addition to this, the efficient of the instruction with using painted projection which was developed in the area of space geometry in the seventh grade. In the sketch and projection which was treated in Japan, the spatial elements, such as the shape, size and position are given to the students, but the color is not considered much. The materials which were developed have big characteristic. When the students have to express the solid which was painted by three colors (red, blue and green) through the sketch and projection superficially or guess the stereogram characteristics from the sketch and projection, they have to take the colors into consideration. Taking the colors into consideration make the students enhance and enrich the figure concepts. The teachers can instruct the students from inductive guessing to deductive guessing smoothly by making the students express the process and validity for guessing. Two results were got through three classes for seventh graders at junior high school attached a university. One is that the class in which this learning material was used makes the students enhance the change of viewpoints and the way of thinking for the unification. The other is that the class has the effect by activating the communication in which the reason was clear. Totally the class can cultivate the basic of the proof which the eighth graders will learn. Moreover, it is expected that treating the color which is not taken into much consideration in math makes the communication active and form the way of mathematical thinking as a result. This point is going to be researched.

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In order to carry out new discovery in any study, a new tool and technology are effective. The purpose of this paper is to propose the method of a new data analysis about achievement tests. If the correlation between the problems used for an investigation is metrically analyzed from various angles, a new understanding about the correlation of the investigation domain will be obtained. By conducting such analysis, the useful fundamental data for improving a curriculum and lessons based on results of the investigation can be obtained. By considering based on these data, it can reach to a concrete proposal. The OECD is performing international achievement tests (PISA) from 2000 every three years. The results of investigations of PISA raised various problems about school education of Japan, especially science and mathematics education. In order to profit this investigation in a curriculum or a lesson improvement, it is required to analyze the investigation data from a new viewpoint and to connect it to a concrete proposal. The method of the data analysis proposed in this paper is called "the network type data analysis" in order to put the foundation on the consideration about a network. The vertex set of this network is the whole investigation problems. The two arbitrary vertices are connected by an edge with the weight given by the mutual information between corresponding problems. The mutual information expresses the interdependence between two random variables metrically.

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From Meiji era, space geometry and the descriptive geometry had been taught separately in mathematics and drawing, respectively. Under the 1942/43 syllabi, descriptive geometry was widely adopted in mathematics course and used as a problem solving tool. But in the post-war secondary school, the projection skills have been treated just as one of the ways of expression of space figures. Here I considered an instructional plan of space figures utilizing skills of the descriptive geometry. It starts with a sketch of a cube, and finishes with a projection chart of two orthogonal cylinders. It also contains the developed drawing which is important to re-construct the original solid. I also showed a way of investigation on cross sections of a cube using projection, connected with its sketch. I would like to make this consideration more precise with other instructional materials including PC software and develop a curriculum on space geometry of secondary schools.

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Teachers' knowledge and classroom teaching are strongly related to each other. Every activity in the classroom teaching comes from teachers' knowledge of subject matters. Here, the transformation of teachers' knowledge of subject matters plays a significant role. Thus, this paper reports about teachers' subject matter knowledge that is categorized into content knowledge, pedagogical content knowledge and curriculum knowledge, and its transformation into classroom teaching by Mongolian primary school teachers, focusing on the concept of fractions. The study investigated 2 groups of teachers who have common professional degrees. The transformation was identified through a hermeneutical analysis based on the characteristics of teachers' subject matter knowledge and classroom teaching that are revealed through a questionnaire and classroom observation. It is concluded that when the teachers transform the subject matter knowledge into classroom teaching, they emphasize one of the three categories of knowledge; and in accordance to this emphasized knowledge, relation among the three categories of knowledge is modified in the process, and thus the modified subject matter knowledge is able to characterize his/her classroom teaching.

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The purpose of this study is to find out the significance of introducing the concept Learning-Teaching Trajectory, which was developed in the mathematics project in the Netherlands, namely TAL Project, into mathematics education of developing countries, and the focus of the study is on the case of South Africa in which the independent learning-teaching trajectory for the country, namely "Learning Pathway for Number (LPN)" is developed and applied for pre- and in-service training for primary school teachers in South Africa by the mathematics research project "Count One Count All (COCA)". There are two main significances of introducing the concept into South Africa. The first is to support South African teachers and improve their competencies in order to close the gap between the current competencies of the teachers and the ideal of new education policy, so-called Outcome-Based Education for the purpose of establishing democratic society denying the apartheid. The second is to develop LPN relevant to the context of South African children in consideration of what mathematics they should learn, what phenomena such mathematics occurs in, when, where and how such phenomena the children should meet and deal with them.

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The author analyses students' writing activities on number patterns in Zambia, applying SLE by Wittman into practice. In the previous research, deeper analysis should be conducted in order to grasp students' understanding of number patterns and other mathematical abilities including communication and discussion (Shibuya, 2008). Therefore, in this paper, the transcending recursive model (Pirie & Kieren, 1992) was modified and applied to describe the process of changing students' writing, which could offer some clues to approach their understanding and process of their writing deeply. According to three different students' categories expressed in the model, their characteristics were found out. Firstly, the author pointed out that the process of students' writing activities has slow improvement, return to the previous level of understanding, and instability of understanding. Secondly, the result of paper test does not always reflect on how well some students' activities were or how difficult others were. In other word, while some fast learners with high score on the test did not succeed in writing at all, other slow learners with low score successfully wrote their mathematical findings on their worksheet. The further research is needed to find out what sort of teaching support should be given when students can change their understanding and what should be the effective teaching and learning approaches toward improvement of students' learning in SLE.

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