This research aims at clarifying the transition process from elementary mathematics to secondary one through the design experiment methodology. In this paper, we sorted out students' dynamic views which enabled them to enhance their understanding of inclusion relations between figures through analyzing our data of classroom lessons "constructing the figures using the operative sheets" based on the grounded theory approach. We could find the follwoing seven dynamic views as a result of our analysis. (A) Concrete manipulation, (B) Idealization and mental operation, (C) Gesture, (D) Grasping the movement of some point during the whole figure change, (E) Recognition of invariants, (F) Reverse operation, and (G) Simultaneous identification of invariants and variables. Then, we characterized the dynamic views of (A) concrete manipulation and (B) idealization and mental operation as the figurative image schema, the dynamic views of (C) gesture, (D) grasping the movement of some point during the whole figure change and (E) recognition of invariants as the opeartive image schema, and (E) reverse operation and (G) simultaneous identification of invariants and variables as the relational image schema. We consider that these characterizations may suggest the possible categories of the image schemata for the recognition of geometry, although we don't claim that these are all cases. Also, we may know why some students could learn the inclusion relations between figures and the others not, if we observe children's thinking in geometry in terms of the framework.

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The purpose of this paper 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". Firstly, we surveyed Kunimoto's researches concerning "Pre-formal proofs" and Wittmann's researches concerning "Operative proof" in mathematics teaching. And we pointed out that "Operative proof" is included in "Pre-formal proof". Moreover, through considering Wittmann's papers concerning "Operative proof", we pointed out the following essential points of the concept. ・"Operative proof" is the proof integrated into one of the mathematical activities. ・"Operative proof" is the proof based on the results of the operations given to the mathematical objects expressed appropriately. Secondary, we considered "Das Zahlenbuch" from the point of view of how operative proofs are embodied in the classroom situation and showed that counters and the place value table are nicely used in "Operative proof". Finally, we considered the possibilities of "Operative proof" with counters and the place value table referring to "Das Zahlenbuch" and pointed out the following two roles of "Operative proof". ・"Operative proof" has the possibilities of supporting the formal proofs of advance levels. ・"Operative proof" has the possibilities of promoting to discover another mathematical patterns. Our further tasks are to research experimentally how children work with the "Operative proof" and to develop investigate the Substantial Learning Environments with "Operative proof" for children.

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In this research, I set "The thing is left after all you've learned has been forgotten" as "Ability to consider and to express mathematically" in mathematics education, that is "mathematical idea", purpose in this research is improving lesson for learners can acquire it. For such occasions, I checked theory of substantial learning environment by Wittmann and organization by Freudenthal, and show points to remember for designing substantial learning environment in algebra area, then I set out and put in practice. Purpose of this paper is show suggestion that get through analysis and prospect of lesson, and assignment for refining theory. As the result, show that mathematical thinking under thinking gives difficulty for learners, when they experience thinking of local organization.

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The aim of this research is to form of the model on process of creative thinking. A viewpoint of creative thinking is risen quality of mathematical thinking by all member of the class. The model on process of creative thinking is formed by these precede studies: The process model of understanding mathematics for building a two axes process model by KOYAMA, Episode analysis by Schoenfeld, Through the recursive eye mathematical understanding as a dynamic phenomenon by Pirie & Kieren. This model has horizontal axe that means progress of time. And vertical axe means eight level of subject of mathematical thinking's quality. I insist that when The present episode changes next episode, discussion by all member of the class has some children's question. And therefore I built the model that we can write situation of episode movement by children's question. The main results on this study are followings; 1) The quality of creative thinking and the subject of creative thinking were become higher than before discussion by all member of the class. 2) From this case study, children's question is driving force that creative thinking of children is more high quality.

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The purpose of this paper is to set the framework of learning activity's theory which students exploit mathematical knowledge into their own daily life. For this purpose, the author considered that how people exploit mathematical knowledge into their own daily life, and defined attitude exploiting mathematical knowledge into their own daily life as "to be changed their own daily life activity by mathematical knowledge". In addition, the author proposed mathematical learning's hierarchy based on Bateson's theory (Bateson, 2000) and Engestrom's activity theory (Engestrom,1999). From these notion and analysis of "national scholarship and learning situation research", the author thought that domestic mathematical education is not "exploiting mathematical knowledge into their own daily life" but "exploiting living scene into mathematical field". In order to develop students' ability which they can exploit mathematical knowledge into their own daily life, it is necessary to develop learning which changes students' living activity system.

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In this paper, I clarified the relationship among the power to analyze the teaching contents, the power to construct the teaching plan and the power to practice the trial lesson. I took the students of mathematics department in university of education as the object of study. I measured the power to analyze the teaching contents using the test. I measured the power to construct the teaching plan using the teaching plan made by student. I measured power to practice the trial lesson observing a trial lesson practiced by student. The results of analysis were as follows: - All of the students' the power to analyze the teaching contents, the power to construct the teaching plan, and the power to practice the trial lesson are low. Those were about 30 to 40%, when it converted by percentage. - The students' power to practice the trial lesson has strong relationship with the power to analyze the teaching contents. But the students' power to practice the trial lesson has no relationship with the power to construct the teaching plan. - The student with high power to analyze the teaching contents has the tendency for the power to practice the trial lesson to be also high.

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Suppose that the two right triangles ABC and DEF satisfy the following: ∠A= 90°, AB=AC ; ∠D=90°, ∠F=60° and BC=DE. We consider geometric figures in the situation such as a vertex of one triangle is lapped over some vertex of the other, and one triangle is fixed and the other turns around the lapped vertex. Except for congruence, we pay attention to such geometric figures that satisfy the conditions (1) and (2): (1) The sides of those triangles are not overlapped. (2) A vertex of the fixed triangle lies on a side of the moving triangle, or a vertex of the moving triangle lies on a side of the fixed triangle. In Section 1, we consider one of such geometric figures. In Section 2, we consider the case of A=F and give the explicit expressions of the areas of the overlapped geometric figures. By examining all the cases in Section 3, we can find the twelve geometric figures and we list those figures in Subsection 3.7. This article becomes useful for a motivation expecting and prompting student's mathematical activities, for an example in teaching of ways of seeing and thinking by using mathematics, and for a reference in designing topic-based learnings.

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This paper is part of doctoral study which aims to improve mathematics learning in Zambia. In this research, the author adjusted Substantial Learning Environment (SLE) that can foster two different abilities of mathematics: basic calculation skills and other mathematical abilities, into Zambian mathematics lessons. This paper qualitatively described Grade 5 students' learning process in number brick, one of SLEs. The author analyzed the protocols of interviews in which three students solved questions on number brick for seven times after each lesson. After explaining about each student's learning, the author identified four learning characteristics which were observed in these three students: eclectic style of calculations; discovery of number patterns and rules; instability of learning; and language difficulty. Eclectic style of calculation was influenced by the previous teaching and learning, but parts of instability of learning and discovery of number patterns and rules were influenced by the teaching in our lessons. The analysis indicated that their learning process was related to the teaching in our lessons; therefore, the author will continue to analyze the other part of lessons and students' learning process, and in the end examine the quality of the teaching and learning situations in depth.

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