This paper focuses on a design experiment methodology in mathematics education which has been developed as a methodology for establishing a close and dynamic relationship between theory and practice, and discusses the comprehensive characteristics of the methodology. The design experiment methodology intends to develop (local) theories in mathematics education through designing, practicing and systematically analyzing daily classroom lessons over a relatively long period, where a researcher is responsible for students' mathematical learning in collaboration with a teacher. However, the methodology has also been questioned as to its scientific quality by the positivist scholars, since it explicitly deals with classroom practices that can be characterized as complex phenomena. Thus, this paper tries to place the design experiment methodology especially from a scientific point of view. The points discussed in this paper are the following.

　1. The design experiment is an effective methodology for realizing mathematics education as a design science, and it intends to create a fruitful relationship between theory and practice.

　2. The design experiment aims to construct a class of theories about the process of learning and the means that are designed to support that learning through (a) designing and planning the learning environments, (b) experimenting the design and the ongoing analysis, and (c) the retrospective analysis.

　3. The design experiment is an interventionist methodology that goes through an iterative design process featuring cycles of invention and revision.

　4. The design experiment has its intention of producing a theory which premises a social and cultural nature of the classroom, active roles of teacher and students, and learning ecologies and complexities of the community. Thus, it is opposed to an orientation of theory-testing that the positivist studies adopt.

　5. The design experiment has been critically discussed in terms of the traditional scientific criteria like generalizability, reliability, replicability and others.

　6. We can indicate four points as our tasks for enhancing the scientific qualities of the design experiment;

　　・Implementing consciously both processes from scholarly knowledge to teaching, and conversely from craft knowledge to researching and scholarly knowledge,

　　・Analyzing practical data in a systematic way and unfolding a logic of the analysis,

　　・Assessing and evaluating the design experiment using the revised scientific criteria, and

　　・Placing some philosophy which the design experiment is based on.

　　 The purpose of this paper is a proposal of a new viewpoint of mathematics in Mathematics Education and an improvement of Japanese teaching situation.

　　 We overcome an old image of mathematics and I advocate a new image of it.

　　 The proposals are followings.

(1) Seeing mathematics as "science of pattern"

(2) Teaching children to do mathematics (mathematics as process) instead of absorbing a ready-made mathematical knowledge (mathematics as product)

　　 We should change a paradigm of Mathematics Education and do an authentic mathematics teaching.

　　 In classroom teacher and students construct a culture, which simultaneously influences students' thinking and learning. This culture is said classroom culture. Thus the essential feature is the 'reflectivity'. And classroom culture has norms of teaching and learning. This is the 'nomativeness' of classroom culture.

　　 Social construction of classroom cultures is the important problem in actual classroom situation. Thus I discuss the features of sound and unsound classroom cultures according to these perspectives. It becomes clear that the 'mechanistic-technomorph' culture corresponds to unsound one and the 'systemic-evolutionry' culture corresponds to sound one (Wittmann, 2001). They contain the aspects for mathematics and how to use a textbook in teaching mathematics.

　　 But we should grasp that systemic cultures contain mechanistic ones. In other words, it is important to transform mechanistic classroom cultures to systemic ones gradually. It is said 'enculturation' in mathematics education (Bishop, 1988).

　　 This paper is a part of "Research on Cultural Value in Mathematics Education". I grasped and surveyed "mathematics", particularly "the development of mathematics" as cultural elements, and brought their cultural nature into focus.

　　 What's more, I went on to examine the present status of Japanese mathematics education from the viewpoint of the cultural nature of development of mathematics in the world.

　　 The following issues emerged;

　(1) Present Japanese mathematics education doesn't grasp the development of mathematics as an organic whole paying attention to its cultural nature.

　(2) It isn't enough to discuss on the grounds that we did away with traditional Japanese mathematics "Wasan" and adopted only European mathematics from the various mathematics of the world.

　　 We should also reflect upon the extent that the learning material is explained haphazardly in the texts etc. without solving those issues.

　　 Concerning issue 1, it goes without saying that we must get down to reflecting concretely the development of mathematics into the development of mathematics education.

　　 In this paper, I considered issue 2 particularly by clarifying the property of "Wasan" and the decision to accept European mathematics in the Meiji Period and I explored a course for the future Japanese mathematics education. Consequently, I insisted that we need to consider "Wasan" under the spotlight of cultural sociology and that we must proceed to study what culture we appoint to the spirit base of future Japanese mathematics education.

　　 The purpose of this paper is to identify two means of objectification -semiotics and semantic- in constructing the knowledge system, to describe a process of generalization with three steps -realistic, contextual, semiotic-, and to represent the constructed knowledge system as a semantic network. From a cognitive linguistics perspective, it is considered that mathematics is regarded as an opened system and natural language and gesture have an important role in learning.

　　 The roles of natural language identified through a series of lesson observation are following:

　(a) Linguistics Template

　(b) Label and Activator

Specially in the point (b), although natural language has ambiguity (polysemy, diversity, …), it as a label makes learners activate a naive idea. And, considering the ambiguity in communicating with others, learning would progress. Not only the semiotic means of objectification but the semantic means is important for progress of generalization and it is realized by asking, for example "What is the center?", that the semantic content of natural language with two roles is clarified. In this paper, 7^th students had understood a circle as a symmetrical figure and a plane figure separately before it is realized that the axis of symmetry is a diameter of a circle too.

　　 In this study, we construct the problem classification model of the cognitive domain for formative evaluation in school mathematics. As a result, we constructed the problem classification model which has the hierarchical structure that is composed of 8 factors and 36 items. The feature of this model is shown below.

　(1) It is easy to set the difficult-degree in the problems.

　(2) It is suitable to grasp the achievement level of an educational objectives and it is convenient to guide students after the achievement test.

　　　This is useful for teachers to know what problems they imposed on their students, in order to measure the achievement of educational objects. Here is focus of our paper.

　　 We can see analogy as a cognitive process in which one finds some similarity among two different objects and transfers some information of one side to another. Although mathematics is presented in certain reasoning which is made from purely logical formations, we must rely on the uncertain reasoning of analogy during we learn mathematics.

　　 Analogy consists of two basic elements: base domain and target domain. Our knowledge about base domain is applied into target domain. In this framework, analogy can be seen as a mapping of information from base domain to target domain.

　　 In this paper, I make the following remarks to analyze analogy in mathematics education.

　a) There are two kinds of base domain: model and analog.

　b) We must consider user and producer of analogy. A base domain of producer is a target domain of user, and vice versa.

　C) Associations of base domain are mapped into target domain, then new concepts might occur in the target domain.

　　 Based on the above remarks, I illustrate some roles of analogy in understanding mathematics.

　i) Analogy develops new mathematical concepts.

　ii) Analogy makes misunderstanding of mathematical ideas.

　iii) Analogy creates some mathematical words.

　　 The purpose of this paper is to investigate the actual conditions of teacher's listening on introduction of lesson. There are three forms of listening of teacher; the evaluative, interpretive and transformative listening of teachers, which I adapt for analysing the listening of teachers.

　(1) Evaluative Listening: If a teacher is listening in an evaluative manner they will characteristically have an evaluative stance. For a teacher, student's contributions are judged as either right or wrong.

　(2) Interpretive Listening: Interpretive listening is characterised by an awareness of the fallibility of the sense being made. If a teacher is listening in an interpretive manner they will characteristically have an active interpretive stance.

　(3) 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.

　　 Problem posing on introduction of lesson are made by teachers instruction (questioning, listening) and childrens responding. This series of exchange call QRL-System in this paper.

　　 There are two types of problem posing on introduction of lesson; immediate posing, passive posing. There are five styles of posing in some types; (1) immediate posing of problem (2) guidance to subject matter (3) explicitness of subject matter (4) to be overt of unknown things (5) subject matter from vague things. In this paper, these styles are told to be built on seven lessons.

　　 Teacher's listening on introduction of lesson makes a chance of problem posing. Teacher's listening on introduction of lesson play an important roles on mathematics instruction.

　　 In this research, I have two aims:

(1) Clarifying the effectivity of “Double and Half” and the way to make use of it in lessons for “Ratio”.

(2) Designing lessons to promote comprehension for “Ratio”, which make use of “Double and Half”.

　　 In view of two aims, I have carried out the experimental lessons, which intuitive recognition of “Double and Half” and “similarity” were considered. The subjects were the class A (35pupils) and the class B (34pupils), and were 5th grade of the elementary school in Kobe. In the lottery, the class A compared “lots : blanks = 3 : 2” with “lots : blanks = 2 : 1”. And the class B compared “lots : blanks = 3 : 2” with “lots − blanks = 2”, also did “lots : blanks = 3 : 2” with “lots : blanks = 2 : 1”.

　　 As results of analyses, I have clarified the following contents.

(1) Presentation of “lots : blanks = 2 : 1” promoted comprehension for “Ratio”.

(2) Struggle for comparison of subtraction was effective in presenting “lots : blanks = 2 : 1”.

(3) The Inquiring Similarity Lesson was effective in promoting comprehension for “Ratio”.

　　 When information X_K is presented pupils, there are two cases where they interpret it as information X_K or as information U_K(=aX_K + e: aX_K is information similar to information X_K and e is errors. ).

　　 In this paper, we we have assumed the anticipatory schema F(X_K)(or F(U_K)), the automatic procedural schema A(X_K)(or A(U_K)), the promotional conscious schema C(X_K)(or C(U_K)), and the restrained conscious schema C'(X_K)(or C'(U_K)) on the grounds of the schema theory. Degree of influence of respective schemata is represented by P(F(X_K))(or P(F(U_K))), P(A(X_K))(or P(A(U_K))), P(C(X_K))(or P(C(U_K))), and P(C'(X_K))(P(C'(U_K))). Degree of behavior and judgment is decided by how to combine 3 schemata, and it is represented by P(E(X_K))(or P(E(U_K))). There are 15 types of P(E(X_K))(or P(E(U_K))). These types are showed in the following Table 1.

　　 We made an investigation to see if there were 15 types. The subjects are 53 pupils of Kobe municipal Kielementary school and they are 5th grade.

　　 The problem of the investigation is the following.

　　 “The weight of 0.96 liter of corn is 0.48 kg. Question 1. What liter is 1 kg of corn? Question 2. What kg is 1 liter of corn?” There are 5 subquestions and 5 reflectional columns on Question 1 and Question 2, resppectively.

　　 As results of the investigation, we could specifypupils who fulfiled 8 types of 15 types.

Table1. 15 Types of P(E(X_K))(or P(E(U_K)))

　　 The study of solids is a natural science of spatial figures, and solids are the most suitable objects to apply scientific and analytic methods in school mathematics. On the other hand, the study of solids has several difficulties. In this paper we introduce the concept of acuteness of a solid, and a method measuring acuteness of solids. We expect that they bring us lessons of abundance in school mathematics.

　　 Let X be a convex polyhedron, and A a vertex of X. Suppose that k faces meet at the vertex A and whose interior angles at A are θ₁, θ₂, …, θ_k. Then we define the the deficit of the angle c(A) at the vertex A by 360° − (θ₁ + θ₂ + … + θ_k). By the hypothesis of convexity, we have c(A) ≧ 0. This c(A) measures the acuteness of the polygon X at the vertex A, and has several nice properties. For example, by using the Euler formula for convex polyhedra, we see that the sum of the deficits of a convex polyhedron is equal to 720°.

　　 The purpose of this research is to consider the lesson design of the arithmetic in a combined class. In this paper, as the preparation for the lesson design, we performed the experiment lesson in the combined class of the Kagoshima University department-of-education attached elementary school and analyzed the aspect of the interaction between the children of different grades.

　　 The experiment lesson was carried out as group study by the group which consists of four children including the child of different grades. The subject matter of the lesson was an Arithmogons (calculation triangle). The situation of the experiment lesson was recorded on video, and we analyzed the content of an utterance.

　　 The following became clear from the analysis result.

　　 First, it was shown by this experiment lesson that study of the child of the different grade in a combined class is possible enough. With the first half of an experiment lesson, it was often seen that a child of one grade spoke and the other grade child was only hearing it. However, in the second half of a lesson, it was often seen that the children of different grades spoke similarly about the same theme. This shows that the joint study by the children of different grade is possible enough.

　　 Secondly, as an aspect of an interaction in group study, the following three types were able to be observed.

　　 Type1; The argument in a group is distributing and the contents are not accumulated.

　　 Type2; The assignment of the role of trial and monitoring is performed within the group.

　　 Type3; An argument within a group is advanced together and the contents of the argument are accumulated.

　　 From this result, it became clear what kind of help a teacher should do with these types of the interaction. And this has given an important suggestion to the lesson design of the arithmetic in a combined class.

　　 E. Ch. Wittmann advocated the concept of “Substantial Learning Environment”. In order to crystallize this concept, “Das Zahlenbuch” was edited by Wittmann and others. There are seven ground conceptions as editing principle in “Das Zahlenbuch”. “Aktive-entdeckendes und soziales lernen” (Active-finding and social learning) is one of them and is indispensable to compose the mathematical lesson.

　　 The purpose of this paper is to reveal the concept of “Aktive-entdeckendes und soziales lernen”. Then I developed the mathematical material of negative numbers for 7th grade students which is based on “Aktive-entdeckendes und soziales lernen”. The mathematical material is “21 Number Cards” which are whole number from -10 to 10.

　　 Finally I analyzed the students activities of the lesson and reported the results and identified meaningful of the lesson.

　　 Conceptual change theory has been widely used to explain students’; understanding in a series of developmental studies referring to science education (West & Pines, 1985). This theory was developed by drawing on the philosophy and history of science, in particular Thomas Kuhn’s account of theory change and Imre Lakatos’s work of the scientific research programme. And it mainly used to explain knowledge acquisition in specific domain, with characterizing role of reorganization of existing knowledge in processes of learning. Conceptual change researches in mathematics education have been addressed by drawing on the history and philosophy of science with similar to science education (Vosniadou & Verschaffel, 2004). As has been discussed in mathematics education domain, we need to take the specificity of mathematical knowledge into account with a deep epistemological analysis of what the concepts considered consist of as mathematical concepts.

　　 The purpose of this paper is to consider a fundamental condition in conceptual change that implies an essential activity in the teaching situation of irrational numbers from the epistemological points of view. For attaining this purpose, firstly, the problematiques of teaching situation of irrational numbers can be summarized. Secondly it can be considered about relationship between conceptual change and history of mathematics, focusing on interrelationship between “content” and “form”. Thirdly it may be pointed out that the discovery of incommensurability and the relativity of standards of mathematical rigor lead to the shift on human attitudes toward mathematical objects. In the final place, since “the construction of meaning under a new mathematical method” and “the shift on attitudes toward mathematical objects”; are the fundamental condition underlying students’; activity that can be necessary for the teaching situation of irrational numbers, the teaching situation of using Euclid algorithms can be described as an approach to such kind of activity.

　　 In this paper, I consider about the relation between the understanding in learning mathematics and mathematical problem posing. Students often pose problems in problem solving, in order to get a better understanding as acts to reconstruct problems. Problem posing comes out when students shift levels of understanding in learning mathematics. The significance of problem posing is that it promotes reconstruction of the problem and generate “folding back” which is claimed in the theory of “transcending recursive model” (Pirie, S. & Kieren, T., 1994). Therefore, as the progress of problem posing encourages effective thinking of the students in learning mathematics, and also as the growth of understanding enhances the way of problem posing, both sides grow complementarily.

　　 Moreover, I eraborated the “transcending recursive model” as a prescriptive model in teaching, by embedding a problem posing in it and by thinking of the modality of teacher inventions. This “extended transcending recursive model” could be a base of making a teaching plan.

　　 The purpose of this study is to acquire the fundamental data for the curriculum development based on the realities of children's probability concept.

　　 In this paper, the author investigated the Common Cognitive Path (CCP) in the children's probability judgments. As a result, the following CCP were found in the sixth grade children in the problem to compare likelihood of drawing out a red ball from the box contains red and white balls.

　　 (1) The total number of balls in each boxes is the same, but the number of red balls is different → (2) The total number of balls in each boxes is different, but the number of red balls is the same → (3) The total number of balls in each boxes is different, but the ratio of the number of red balls to white balls in each boxes is 1:1 → (4) The total number of balls in each boxes is different, but the ratio of the number of red balls to white balls in each boxes is the same, but the ratio is not 1:1

　　 Then the author proposed the methods and techniques of instruction based on the CCP, and also constructed the rubric to evaluate grounds for children's probability judgments.

　　 As a result of instruction based on the CCP, the ratio of children who judge likelihood objectively based on the idea of the ratio of the number of red balls to white balls increased.

　　 This shows the achievement of criteria on the rubric assessment. This study shows the necessity and the possibility of the curriculum development based on the realities of the probability concept of children in the elementary education stage, and it also gives the suggestions for the curriculum development that connects with the curriculum of probability in secondary education in Japan.

　　 I conducted a practice of “periods of integrated learning” in upper secondary school students, while taking advantage of the educational effects and positive features of debate-based education. The aim of this practice was to enable the students to gain statistical knowledge. Subsequently I examined the effectiveness of the method. At the same time, I investigated an ideal method by which debate-based statistical education can be made to effectively cooperate with other subjects, such as “information.” The following results can be pointed out. First, debate-based statistical education in upper secondary school students is effective as a method for voluntarily analyzing data from various perspectives. This is based on the game-like feature of debates. In addition, the form that promotes the study involving the points of view and interpretation of statistical data works effectively; it is also closely associated with the examination of different points of view, which is brought about through debates. Second, statistical education through the cooperation between “periods of integrated learning” and other subjects — for example, “information” — can be expected to have the effect of resolving the problems that due to the shortage of class hours as well as the effect of making use of a particular characteristic of each subject. For example, the methods of computer operation fall under the domain of the information course, while concepts such as statistical ideas and representative values fall under the domain of the mathematics course. However, it is important for the two to complement each other in both classes.

　　 The continuous research on mathematical attainment is a part of the International Project on Mathematical Attainment (IPMA) in which such countries as Brazil, Czech Republic, England, Hungary, The Netherlands, Ireland, Japan, Poland, Russia, Singapore and USA are participated. The aim of this project is to monitor the mathematical progress of children from the first year of compulsory schooling throughout primary school and to study the various factors which affect that progress, with the ultimate aim of making recommendations at an international level for good practice in the teaching and learning of mathematics.

　　 In Japan, the total of eight different public primary schools have agreed to participate in the project. We asked all two-cohort children and their classroom teachers from these schools to be involved and to take mathematical attainment tests for six years. We already carried out six tests, i.e. Test1, Test2, Test3, Test4, Test5 and Test6 to about 300 children of first cohort for 6 years.

　　 In this paper, we considered upon the evaluation about the percentage on Test6 by means of comparing the performance of them with that of pupils in Singapore. As a result of consideration, we founded out the following:

　・More efforts should be made in the teaching and learning of advanced contents such as finding out rules in sequence on numbers, percentage and computation of decimal numbers.

　・We should reflect on our curriculum from the point of the progress on pupils' learning ability on some contents such as “even and odd numbers” and “multiplication of decimals”.

　　 Once I saw a handout that had been used in 1945 where difficult “Construction problem” queued up. This is fairly different from the syllabus at that time being tried a bold reform to learn the connection of mathematics and the real world. After the war, such a content rapidly does “Revival” oppositely in the new system high school. In this paper I considered about such an imbalance with innovation of a curriculum and an actual lesson.

　　 There are some aspects to see how educational activity was done in the old system secondary schools. There exist researches on curriculum reform with its textbooks during World War II and on the transition of entrance examination, but I found few papers on the materials used in the real classroom, or on the magazines for preparing entrance examination. I mainly examined actual notebooks and handouts of students of those days, the upper school entrance examination, and the motion of the taking-an-examination community.

　　 Although the consideration is the limited range, it became clear that the influence which the 1931 syllabus gave teaching construction of a real classroom scene was small, and that a dramatically big change was produced in the 1942 syllabus. The contemporary succession which raises children’s spirit of inquiry is asked about the trial in the 1942 syllabus, as well as “traditional” constructing instruction as a new way of “integrated study” of geometry.

　　 “Hatsu Gaku #156” was an official document presented in April 1947 by the Ministry of Education of Japan (Monbusho). This thesis determined the curriculum for the Japanese new upper secondary schools (Kotogakko). Until recently, it was not evident how “Hastu Gaku #156” was created by the Japanese Curriculum Revision Committee because historical materials were lost. Recently, former restricted documents of the GHQ/SCAP were unclassified and opened to public viewing at the Japanese National Diet Library. The purpose of this article is to consider the process of the creation of “Hatsu Gaku #156” by using these documents.

　　 In December 1946, discussion of the upper secondary school curriculum was started between CI&E (the Civil Information and Education section of GHQ/SCAP) and Monbusho. Monbusho decided that “Kotogakko” should consist of college preparatory courses. CI&E criticized Monbusho. CI&E thought that new upper secondary schools should have a broader focus. Through a serious of conferences in December of 1946, CI&E forced their ideas and opinions on the Monbusho.

　　 In February 1947, Monbusho and CI&E came to an agreement with regards to upper secondary school curriculum. “Hatsu Gaku #156” was completed in March 1947. “Hatsu Gaku #156” integrated comprehensive courses and a unit credit system, among other things. In April 1947, Monbusho officially declared “Hatsu Gaku #156” to determine the curriculum of Kotogakko.

　　 In this article, some combinatorial properties in figures are discussed. Here, a combinatorial property means an invariant property which has one of the following features: An invariant property obtained by adding, subtracting, multiplying, dividing or composing some of them for some plural values related to figures; an invariant property obtained by decomposing or composing figures.

　　 Practically, we consider the invariant property in the sum of interior angles or exterior angles of polygon as a typical example of combinatorial properties. By attaching importance on the invariance, we can develop combinatorial properties various way in figures, like the sum of the deficits of solid angles in a polyhedron. We show how such development proceeds, taking the case of closed polygonal lines. Then, it involves the consideration of directions and extends to the consideration of a combinatorial properties including the winding numbers of polygonal lines or closed curves.

　　 Through those cases, we extract important elements to develop such materials: To apply some kind of sums or subtractions which are significant to figures; to utilize various decompositions or compositions of figures; to consider the rotations or directions of plane figures.

　　 The purpose of this paper is to discuss effective methods of mathematical problem posing by using computer. We practiced such mathematical activities by university students who were the prospective teachers. In study (I), we reported the practice of problem posing by using computer after solving original problem. In study (II), we reported the practice of problem posing by using computer which assign students planning problems freely from the first. In study (III), we reported the practice of problem posing by using computer after solving original problem which is different from the problem in study (I), and examined the effective way of making use of problems posed by students. A feature of the method is to give students enough time to create problems. And another feature is to provide situations in which students make conjectures on results and access the numerical calculation by using computer.

　　 In this paper, we report the practice of problem posing by using computer after solving original problem which is different from the problem in studies (I) and (III). Actually we examine how to pose original problem in the problem posing by using computer and what the relation exists between the original problem and the problems posed by students. 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. When students solved problems to each other, the students who are the solvers and the makers commented on problems each other, which showed communication between them. We observed more positive learning activities than the usual classes.

　　 It is emphasized that the opportunity to discuss problems posed by using computer in a class is very important in such activity, in particular for the prospective teachers.

　　 In Colleges of Technology, which conducts engineering education, mathematics is necessary for the students to learn engineering major subjects. It has been claimed that the students can properly use mathematics in theoretical part of the major subjects. They are required not only for calculation abilities but also for understanding mathematical concepts in engineering. I would like to point out two requirements for the students properly to use mathematics in the major subjects as follows;

　(i) Students need to consider mathematics in the major subjects shares the same content with that in regular mathematics class.

　(ii) Students need to image the mathematical concepts.

　　 Based on theses requirements for the students, I would like to propose what we can actually do in regular mathematics class for the first and second year students from the perspective of initial engineering education at National Colleges of Technology as follows.

　(i) I would like to propose teaching materials that students can readily conceptualize what integral is in two credit hours before studying integral.

　(ii) I would like to propose visual teaching materials that students can readily verify results of indefinite integral from graphs and figures.