The article investigates the series of Das klein Zahlenbuch, published by Wittmann und Müller in a project for mathematics education, called ‘mathe 2000’. Focusing on teaching and learning materials which are various cards of insects, animals, pies, dices and so on, in the second part of the book, Zahlenbuch Teil 2 Schauen und Zählen (2004), the author discusses how the theoretical perspective is reflected to the practical aspect of those cards. As a result, in the playing book, children are expected to grasp numbers from 1 to 5 intuitively and to combine those numbers to understand bigger numbers such as 6 to 12. Through a variety of plays and games using the same cards, the multiple ways of expressing numbers are identified, which would foster children’s consciousness of numbers with the recognition of cardinal numbers.

　　The purpose of a series of our studies is to discuss effective methods of mathematical problem posing by using computer.　We have reported in the previous studies such mathematical activities by university students who are prospective teachers. In study (I), we studied the first practice of the problem posing by using computer after solving original problem, and through study (Ⅱ) to study (Ⅴ) we examined various ways of the effective problem posing.　

　　In study (Ⅵ), we pursued the study on the problem posing process by students, and requested students to make good-quality problems.　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 study the practice of problem posing by using computer after solving original problem which has some good characteristic to guide the developmental problem posing, and investigate the problem posing process in detail through interviews to students.　As in study (Ⅵ), 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 and interviews from the posed problems by them.　The results 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.

　　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”.

　　In this paper, for developing the Substantial Learning Environment with“Operative proof”that used counters and place value table, I looked back on past practices on operative proof, and found out these two issues.

　　One issue is that we need some devisal for students to plan operative proof by themselves. And other one is how to develop substantial learning environment to make the shift to the formal proof from operative proof.

　　I focused on the first issue, and to obtain a certain suggestion for this issue, I carried out a teaching experiment for the 7ht grade students on operative proof that used counters and place value table. As a result, I could obtain following suggestions.

　　For planning the operative proof, it is useful for students to have some successful experience with operative proof. This suggestion was obtained from the some devisal of order of the tasks in teaching experiment. And in the process of planning the operative proof, it is helpful for students to explain own operation orally and to compare own operation with others. This suggestion was obtained from the observation of student’s activity in the teaching experiment.

　　Future tasks are to consider other type operative proofs for ANNA numbers, and to design the concrete Substantial Learning Environments that used operative proof.

　　The purpose of this paper is to consider the role of questionnaires about school mathematics performance on the basis of the results of academic ability conducted by local government.　The authors have analyzed the relationship of the result of the problems and items on metacognitive knowledge in the questionnaire.

　　The following things become clear and are proposed from the result.

(1)　The mathematics performance is related to the strength of awareness about the metacognitive knowledge.

(2)　We propose an example of teaching and learning processes that introduced metacognitive support.

(3)　We propose the following items about metacognitive knowledge should be added to questionnaire.

　　●　You must reread a problem once again, when a problem cannot be solved.

　　●　You may solve by another method, when a problem cannot be solved.

　　●　It is better for you to think through drawing a figure, when solving a problem.

　　●　It is better for you to think through writing a formula, when solving a problem.

　　●　When solving a problem, it is better for you to think also by another method, if you can do it by one method.

　　The purpose of this paper is to attain elaboration of the Koyama’s (2010)“two-axis  process model” by designing mathematical activities that are understandable to students in lesson organization based on it, through an action research of“quadratic  function” in high school subject Mathematics Ⅰ.  

　　Entities of the mathematical thinking which are a vertical axis in the two-axis process model, were elaborated by quoting the“emergent  modelling” by Gravemeijer (2007).　In learning and teaching quadratic function, the author designed three learning stages based on the two-axis process model in each of the four activities by the emergent modelling and conducted lessons accordingly.

　　 As a result of qualitative and quantitative analyses of the lesson based on these frameworks, the following two points were found out:

　(1)　By setting up a task from a scene which is familiar to students, they show active attitudes towards problem solving.　They tend to try to understand the real phenomenon mathematically especially after  they have had experiences in observing changes and correlations in a table that represents two relevant numerical figures.

　(2)　In the process of generalizing ways of solving problems given by a teacher, students are more able to understand the general formula y=ax²+bx+c by exploring and projecting relations between y=ax² and y=ax²+bx+c, using function graphing software such as GRAPES. 

　　Kawasaki (2001) used the ‘Problem e’ below, to clarify the nature of geometrical sense.

　　　Problem e : Find α + β in the figure in which three congruent squares are placed side by side.

　　Independently, we found the formula Tan ⁻¹ 1/2+Tan ⁻¹ 1/3=π/4 (Euler’s formula) in Izumi (1961), and we recognized that the problem e is a geometrical version of Euler’s formula.　We felt that the problem e which is related to Euler’s formula has interesting geometrical aspects and the value in mathematics education.　In this article, we consider the nature of the problem e and its usefulness and possibilities as teaching material in junior and senior high school mathematics classes.

　　In Section 2, we considered a variety of solutions to the problem e and classified them by learning contents.　As a result, we came to the conclusion that the problem e can be used variously both in junior and senior high school mathematics classes and will be a valuable teaching material.

　　In solving the problem e geometrically,“the viewpoints of the sum of the measure of angles”are the key to the success. We considered “the sum of angles” from three viewpoints :

　　Viewpoint 1 : Adjoining two angles. This is the most natural way.　The examples of the solutions to the problem e from this viewpoint are shown in 2.1.1.

　　Viewpoint 2 : Reducing to the fact “the measure of an external angle is equal to the sum of the measures of two inner angles which aren’t adjacent the external angle”.　We recognized that this viewpoint is very useful. In fact, this viewpoint is a fundamental idea in this article.　The examples of the solutions to the problem e from this viewpoint are shown in 2.1.2 and 2.2.3.　We discussed in detail the geometrical proofs of Clausen’s formula which is more complicated than Euler’s formula in 3.2 of Section 3. Viewpoint 3 : Direct and effective way which doesn’t use the movement of angles to consider “the sum of the measure of angles”.　In viewpoint 1 and 2, we need to “move the angle adequately”.　We found the way from

　　viewpoint 3 which dosen’t need to move the angle by developing the way in 2.2.3.　We formulated this way as the lemma in 3.3 and we showed the proofs of the formulae which are more complicated than Euler’s formula.

　　In Section 4, we developed some problems which are related to the problem e from the viewpoints 2 and 3. As a result of the consideration in this article, the viewpoints and ways concerning the problem e will be a valuable teaching material that makes junior and senior high school students enhance their interest in geometry.

　日本を含む多くの国で近年実施された数学カリキュラムの改定，特に中等教育のそれにおいて， 統計教育の重要性が強調された。統計教育において，「ばらつき（variability）」は中心的役割を担うので， その目的を十分に実現するには，教員が有するばらつきに関連した「教えるための統計的知識（statistical knowledge for teaching，SKT）」， 考え（conceptions）とその指導や学習に関する信念が重要な役割を果たすことが予想される。本稿では，中等教員の有するSKT と考え，信念について調べるための概念的枠組み， そしてその枠組みに基づいた調査用具を提案する。それに加えて， 広島県の県立高等学校１校で数学科教員４名に対して行った調査の結果を報告し，そこから得られる興味深い傾向と示唆について議論する。

　　Process of generalization has epistemological direction from particular to general (thus, it should be separated from extension).　In this study, we consider about“how students know a generality in a process of generalization”, because if it appears to be generalized something by subject him/herself, sometimes s/he don’t know its generality (cf. Zazkis & Liljedahl, 2002, pp.393-394). Especially, we focus on the student’s intuitive knowing generality.　Because it is a very important basis of generalization.

　　We focus on Semadeni’s (2008) notion of stability of meaning by mental object.　His idea is, definition and especially mental object (it includes daily life’s experience, embodied objects, concept image, purpose of use a object, and so on) give stability of meaning for us.　For this reason, he concluded that one can reconcile the view that mathematical inference is a logical necessity, sure and certain and language, although crucial for concept formation and communication, is not a sufficient tool to deal with mathematical meaning.

　　However, his approach didn’t focus on learner’s conception.　So it’s not enough for our purpose.　Thus, we modified his notion; we focus on mental objet that arises from the short term (it means solve a problem) experiences.　These mental objects gives limited stability of meaning.　The stability promote and/or hinder one’s generalization.　We analyse a simple episode in Japanese junior high school by using this notion.　As a result, we concluded if one’s mental object reflects nature of a mathematical knowledge, his/her generalization is promoted and s/he knows its generality.

　　The purpose of this paper is to elaborate on social perspectives provided by recent studies in mathematics education, and to consider what makes the social construction of mathematics necessarily valid.

　　First, we review previous research on social construction in mathematics education and point out that they have not sufficiently dealt with the mathematical aspect of this social construction.　The application of Lakatos’s logic of mathematical discovery (LMD) to mathematics education might not adequately capture the inherent characteristics of mathematics, because an ideal LMD does not always occur in mathematics classrooms.

　　Second, we consider the terms“ necessary truth” and“ structure.”In philosophy, a necessary truth is distinguished from a contingent truth, in that the former is a truth that could not have been false.　As for structure, in mathematical logic, a structure is a collection of interpretations of mathematical statements.　For example, a positional notation provides a structure for natural numbers.　With special reference to this example, we consider that (i) if a universal set is finite, the truth of a mathematical statement may be established by checking all elements, but also that (ii) justification focusing on a mathematical structure make mathematical statements not only true, but necessarily true.

　　On the basis of the above points, we argue as follows: (i) justification focusing on mathematical structure makes a mathematical statement necessarily true; (ii) the valid social construction of mathematics emerges from such necessary truths; and (iii) in mathematics education, we must give students a chance to experience social construction paying attention to mathematical structures.

　　As for the study on power of expression, many studies has been made in mathematics education in former studies.　If these previous works are taken into consideration, the power of thinking, transmitting and reading a person’s thought should be realized to be“power of expression”in the future of the mathematics education. However, the framework which can grasp correctly about children’s“power of mathematical expression”has hardly been studied from rationale about the training method of“power of expression” .

　　Moreover, the way children’s“power of mathematical expression”is correctly grasped by researcher is one of the important issues.

　　Therefore, in this paper, the framework which can grasp correctly about children’s“power of mathematical expression”and can interpret a type of“power of mathematical expression”from children’s activity should be built from a theoretical aspect.　Moreover, I presume to explore about type of“mathematical power of expression”which children are holding by using the theoretical framework built in this paper.

　　In Germany, the national standard for elementary mathematics,“A Standard for mathematics education an elementary school”was released in 2004.　This is a symbol of the reform on elementary mathematics education in Germany.　The purpose of this study is to consider the aims of learning of“pattern and structure”which is the one of five contents domain of learning in the standard.　In this paper I considered the mathematics education meaning of learning of“beautiful package”in the innovative mathematics textbook“Das Zahlenbuch”(2012) of the elementary school in Germany that is embodiment of learning of“pattern and structure.”

　　Therefore, I analyzed the learning of“beautiful package”in the mathematics textbook“Das Zahlenbuch”(2012), and researched on the learning about“a beautiful package”for first grad in an elementary school.

　　Analysis of a textbook showed that the aims of learning of“beautiful package”is the same as the former, and the activity which carries out reason attachment of the pattern and structure is emphasized.　In the research, a useful suggestion was obtained from Japanese children’s reaction.

　　Recently, the importance of mathematical-activity has been especially increasing and, as its contents, the application–oriented mathematics, in which the functional value is emphasized, tends to be taken up.　However, it can be pointed out that the structure-oriented mathematics, in which the contents and the consideration of mathematics itself can attract learners, is also important for mathematical-activities. In our preceding study, we proposed the cyclic model of “structure-oriented mathematical-activity” which is designed to induce the learner’s proactive and dynamic considerations.　Especially, it is one of the cardinal significance of mathematical-activity to interest the leaners in mathematics itself, in other words, to induce the satisfactory attitude toward mathematics.

　　In this paper, we consider the processes and the obstacles to the achievement of the above significance using the cyclic model of structure-oriented mathematical-activity, from the viewpoint of the theory of didactical situations, and referring to the result of classroom practice of this cyclic model, develop some implications about materials for structure-oriented mathematical-activities.

　　The purpose of this paper is an examination about the issue of Mathematics　core Team in ZAMBIA to focus on teacher’s utterance in the lesson and teachers’description for the lesson.　The teachers, who belong to the core technical team for STEPS (Strengthening Teachers’Performance and Skills through School-based Continuing Professional Development Project) designed and one of the teachers did the examination lesson. Then science teachers, mathematics teachers, the board of education and managers in Copper belt observed the lesson.

　　The way of the study is as followed.　At first, I observed the lesson and the conference for the lesson, and I recorded with VTR.　Next, I made the protocol about this lesson, then I analyzed it.　Then I analyzed their papers made in the conference.

　　As a result, though math education in Zambia is under development, it has been making steady progress, thanks to Team KK, leaders of math teachers in their country.　There are JICA’s back up.　They open up the vista ‘real learners-centered approach’‘grow , up mathematics thinking’.　However, there is a gap between such ideal lessons and the daily lessons they have.　To fill the gap, they should know more about“contents”, “learners’characteristics”, and“PCK”.