In this paper I consider an arithmetic in "Das Zahlanbuch 1" and bring out the following points. (1) In "Das Zahlenbuch 1" the authors are conscious of a dynamic structure of operation. And it has a substructure. Ex. from simple formulas to complex formulas. (2) They make much of "power of five". Since objects under five are optically grasped simultaneously, children must not count objects. (3) Children understand knowledges under rich relationships. (4) We have to understand mathematics as a "science of patterns". (5) Authentic situations are more important than earlier. (6) From a point of methodology, openness of teaching is guaranteed both the property of subject and the development of children. (7) Structure-oriented and application-oriented teachings are achieved harmonically.

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Expressing the learners' ideas, or cultivating students' ability of representation, have been focused on as the important mathematical activities in these days. Some of such activities are Learning Playback (Nakamura, 1989), Balloon Method (Kameoka, 1990), Reflexive Writing Activity (Ninomiya, 2005(a)), and so on. These "explaining one's understanding" activities tend to be more focused on. In this paper, inevitability of such tendency is examined. From the basic idea of Hirabayashi (2001(a)), ecological viewpoint is adopted for both methodological and contextual examination. Through the outcomes of Uchida (1986) and Ninomiya (2005(a)), reflexive nature of thinking and representing is pointed out, and it is found that expressing the idea makes the learner think deeper. From such reflexive nature, the nature of "Product" is shown, as "knowing the achievement of what they have learnt". Within such vision, the theory of Meta-learning, by Bateson and Mellin-Olsen, and the theory of Implicit and Explicit Mathematics, by Chevallard are examined. From the viewpoint of learning activity, Meta-learning, or learning 2 in Bateson's term, may result the outcome of learning as "knowing the achievement of what they have learnt". On the other hand, From the viewpoint of mathematical knowledge, both implicit and explicit products should be aware of, for the ideal learning outcome, as "knowing the achievement of what they have learnt".

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The purpose of this paper is addition of significance to mathematical activities as the teaching and learning to foster mathematical literacy. For attaining this purpose, firstly it is described that today's mathematical literacy include both of mathematics of application-oriented (functional) and the structure-oriented (theoretical) in which it is emphasized application-oriented (functional) mathematical methods. Purpose, contents, methodology, and evaluation in mathematics education are inseparably and interdependence related. Therefore, change of the purpose requires reconsideration of contents, methodology, and evaluation. Current "problem solving" is considered from the perspective of mathematical literacy, and following two research tasks will be emerged: "abstraction" and "relating with generalization and reduction". In this paper, for solving them, the solutions are explored from sides of the global viewpoint (curriculum) and the local viewpoint (teaching and learning), based on Shimada's mathematical activities. The following is suggested as a result. In the former, the principle of curriculum construction was suggested. In the latter, the necessity as follows for two points is suggested: introduction of real world problem with necessities to solve, and relating structure-oriented (theoretical) mathematics as a means for solving.

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The purpose of this paper is to consider how metacognitive activity in process of self-solution have an impact on metacognitive activity in process of polish up. First, I checked on that essences of problem solving have to be engage for children's liberty and diversity. Next, I categorized children by difference of strategy or case in process self-solution, and divided process of polish up into three steps from Koto (1992)'s theory. Then, I considered metacognitive activities of classified children in process of polish up. Therefore, a metacognitive activity in process of polish up is so unrelated to the difference of a metacognitive activity by two classifications in process of self-solution. However, the result of the one and the activity referred to for the "self-evaluation" is chiefly thought as a difference.

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The aim of our continuous studies is to investigate the function of "self-referential activity" in mathematical problem solving. The term "self-referential activity" means solver's activities that she/he refers to her/his own solving processes or products during or after problem solving. Since study (VII), we turned our focus to the self-referential activity after temporary termination of initial problem solving, i.e., "looking-back" activity in problem solving, and theoretically identified six functions of looking-back activity in problem solving. And, in study (VIII) and (IX), we examined some kinds of looking-back activity and the effectiveness of looking-back treatments for developing solver's solution in a paper-and-pencil environment. The aim of this article is to investigate problem-solving and looking-back activities for solving "Paper-Rock-Scissors Problem" in interview environment. In order to investigate them, two types of paper-rock-scissors problems were offered to subjects, which had same problem structure but the second problem was more complicated and relatively difficult. And, the interview was conducted with the method of suggesting a specific type of looking-back activity corresponding to the subject's solution. From the videotaped interview data, the situations of emergence of looking-back activity and the aspects of contribution to the development of solution were intensively analyzed. The main results are the followings; 1) the cases were found that the development of solution was provoked by looking-back activity after problem solving as well as during problem solving. 2) the case was found that looking back previous (easier) problem and/or their own first solution during second (more difficult) problem solving activity contributed to the understanding of the second problem situation and the development of solution. 3) "direct suggestion from others", "making the problem more difficult/advanced", "discrepancy between estimation/expectation and result", "verification of consistency among multiple answers or solutions" and were supposed to be the trigger of emergence of looking-back activity.

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The purpose of this study is to clarify that how image functions in student's problem solving. In Chapter 2, the term <image> and several related notions are defined. In Chapter 3, interview research to university students are described and analyzed. The process of identification of <image> is also developed. In conclusion, four functions of <image> are proposed in Chapter 4. These are as follows: problem comprehension, promoting thinking, communication, appreciation. This study suggests that a teacher must interpret the learner's activity from the viewpoint of the functions of <image> submitted in this study.

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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 schools. The aim of this paper is to clarify through pilot lessons whether the "halbschriftlichen Rechenstrategien" is adequate to the second grader in elementary schools. 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 choose the good ways of calculation and could calculate the tasks of addicion by the end of the lessons. Such the facts from the pilot lessons tell us that the halbschriftlichen Rechenstrategien is usefully the second grader children in elementary schools.

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The purpose of this paper is to suggest the educational functions of geometric construction as an educational material on elementary school level mediating between elementary and secondary school mathematics to promote children's empirical recognition of the geometric figure to logical one. It is a general knowledge that geometric construction in elementary school mathematics has the educational functions to make children understand and apply the properties of geometric figures. Moreover, in this paper we stress that it has the educational function to change their recognition about the geometric figure. While they learn a geometric figure as the set of properties in elementary school mathematics, they need to grasp a geometric figure as the set of the relations among properties in secondary school mathematics. This conceptual gap between elementary and secondary school mathematics cases serious difficulties for the learning of proof in secondary mathematics. We focus on the educational functions of geometric construction to mediate these two sides, according to Okazaki & Iwasaki (2003). We designed the experimental lessons to confirm the educational functions of geometric construction for the fifth grade's teaching whose topics was the geometric construction of a rectangle by using the ruler and compass. When designing the teaching plan, we adopted the idea of Backward Design by Wiggins & McTighe (2005). The performance task was developed to evaluate the learner's cognitive changes about the geometrical figure. Thorough the quantitative and qualitative evaluations of the experimental lessons, it was suggested that the geometric construction had the educational function to promote learner's logical cognition about geometrical figure. The educational functions of geometric construction on elementary school mathematics that we confirmed through this study are as follows. (1) to promote the use and understanding of the properties of geometrical figures in the problem solving. (2) to give learners the chance to recognize geometric relations itself. (3) to change the recognition of geometrical figure from empirical to logical.

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In this paper, we propose method which will improve the analysis power of teaching contents for students of mathematics department in university of education. We used this improved method in order to raise the students' analysis power of teaching contents of the lesson in the university. The results were as follows: - This method had the effect on increasing the students' knowledge about the objects, the content, and the point of concern in teaching. It also had the effect in strengthening the relationship among the knowledge about each of the three aspects. - This method did not have a clear effect on increasing the students' knowledge about the structure of teaching contents. Neither did it have a clear effect on strengthening the relationship among the knowledge about the structure of teaching contents and other knowledge. - Students had a tendency of making the teaching plan for one (1) lesson without considering the relationship of teaching plan of whole unit. - In the class observation, students could assess the analysis power of teaching contents directly related to the class. However, students could not assess the analysis power of teaching contents which teacher has as background knowledge of teaching.

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