In Teacher Education we need to carry out the vicious circle of next each aspect that recognizes and overcome it. (1) mathematical aspect (2) psychological aspect (3) didactical aspect (4) practical aspect In each aspect we should change our conventional thuoght fundamentally. (1) Students should learn mathematizing instead of consuming the finished product mathematics. (2) Students learn about children's ways of thinking and learning. (3) Students know Not guidance and receptivity, but children's activities and thier organization in future teaching'. (4) Students are aware of the different conception of a traditional approach and a modern approach. I and my students do mathematics lessons in elementary schools. When we do the lessons, we always considered the following points to plan the lessons. (1) Openness of the lesson. (2) Autonomy of children is guaranted. (3) The lesson is a rich learning enviroment of content. (4) The lesson is always developing. (5) Children often explain the reason (operative proof). (6) Improve a mathematical ability of children comprehensively. Finally I propose following points. (1) We establish relementary mathematics as process] based on quasi-empiricism. (2) Development of curriculum and teaching from a point of systemic view. (3) Accumulate many experimental lessons and study an appropriate time and an adequacy of substantial learning enviroments. (4) Teacher inspects the curiculum critically. The teacher becomes thoughtful and think critically.

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Teaching and learning of long division in Japanese elementary school more or less are based on the mapping theory. An analysis of the ten minutes episode on a typical introduction to the algorithm of division (3-digit number÷1-digit number) is presented to explore problems of that theory. The result is that behaviorism dominates the teaching and learning long division in Japan. Therefore the contextualist theories of truth on constructivism should receive more attention in our country. Even in our lessons, contexts in problems are also signified. But the problem is whether students make activity in those contexts. As in fact expressions are shown at first in a process of solving problems, students have not enough time to think the problem in the contexts. On these points of view, Lampert's lessons suggest our reconsideration to change learning theory to compensate our lessons for them, because they have a large gap. Gravemeijer (2002) points out faults of mapping instruction, which are also our problems to be conquered in Japanese lesson. What to be reformed is clear. If external factors are remedy, it is possible to improve teaching and learning of long division from Constructivist viewpoint in Japan.

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According to Third International Mathematics and Science Study (TIMSS) by The International Association for the Evaluation of Educational Achievement (IEA), Japanese children's mathematics record is ranked among the tops. However, the mean value in the ratio of the children - who think 'studying mathematics is interesting', 'in the future, I want to work with mathematics' and 'mathematics is important in daily life', is each less than the international average. Therefore, the ratio of the children who think well of mathematics is very small. The Course of Study revised in 1998, 1999, which is based on the above situation, aims at mathematics education which regard children's subjective learning to be important, is related to real life, and develops children's awareness of values of mathematics and the significance of learning it. However, according to recent international studies-TIMSS-2003 and PISA-2003, the situation has not changed. To improve such situation is considered difficult in usual mathematics lessons based on the view of learning which focus on mathematical knowledge. In response to the limitations of the current mathematics lessons, the aim of my research is to design mathematical learning environments based on the theory of situated learning (Lave & Wenger, 1991) - which focuses on mathematical activities - and propose it as a new model of learning mathematics. The purpose of this paper is to examine the practice in the mathematical learning environment - 'Let's share confectioneries'. As a result, the environment points to the possibility for children's subjective mathematical activities relevant to real life, and the awareness of values of mathematics as a product of the activities.

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　　In this paper, I have analyzed six examinations of conceptual and procedural knowledge on "Velocity". 3 examinations on conceptual knowledge as well as 3 examinations on procedural knowledge consist of "Velocity Context", "Duration Context", and "Distance Context", respectively. "Velocity Context" is judged by duration and distance, "Duration Context" is judged by distance and velocity, and "Distance Context" is judged by velocity and duration. The subjects are 211 pupils of 5 elementary schools in Kobe and are from fourth to sixth grade.
　　I have conducted Guttman's scalogram analysis to six examinations, and validity of them is statistically confirmed by CR, MMR, and PPR. Also reliability of them is evidenced in terms of coefficient of reliability by KR-20. And then results of 6 exam- inations are shown in Table 1.

Table 1: An Outline of the Subjects' Scores
Note: For Table 1, Xi, Yi, and Zi are respectively “Velocity Context”, “Duration Context”, and “Distance Context” in conceptual knowledge,. And xi, yi, and zi are respectively “Velocity Context”, “Duration Context”, and “Distance Context” in procedural knowledge.

　　As results of analyses on the basis of Table 1, I have clarified the following content in correlations.
(1) For the subjects' Scores, there is a weak correlation between Xi and xi, or Yi and yi, but not much between Zi and zi. As results of 2 factors' analysis of variance on the basis of Table 2 and Table 3, which was made out of Table 1, I have cleared the following. And asterisk in Table 3 shows that it is significant at 5% level.
(2) There are significant in the main effect of Factor B and the tow-way interaction of Factor A×Factor B, but not in the main effect of Factor A.
(3) As the result of multiple comparison tests about Factor B, subjects are easiest to understand "Distance Context" among 3 contexts.
(4) As results of multiple comparison tests about the simple main effect on Factor A and Factor B, subjects are easy to comprehend in order of "Yi→Zi→Xi" and "zi→xi→yi". And also xi, Yi, and zi are more predominant than Xi, yi, and Zi, respectively.

Table 2: AB add-up about means

Table 3: Results of analysis of variance about Factor A and Factor B

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　　In this research, I have two aims:
(1) Which of two relationships among comparative objects do subjects pay attention to?
(2) What kinds of stages of development are there in comparative strategies?
　　In view of two aims, I have carried out the investigation for 5 Lottery Problems. The investigation was carried out in the second term. The subjects are 158 pupils of six elementary schools in Kobe and they are 5th and 6th grades. Also 87 subjects of 5th grade aren't taught ratio in this term and 71 subjects of 6th grade have learned it. The following ploblems were carried out:
　Question.
　There are the A box and the B box with lots and blanks, respectively.
　The A box has P lotteries total with Q lots and R blanks.
　The B box has S lotteries total with T lots and U blanks.
　Which box will have a greater chance of drawing a lot?
　(1) Please mark the one among 3 choices.
　　A box will have a greater chance. Both boxes will have the same chance. B box will have a greater chance.

[figure]


　(2)Why?
　Note. Question1: (P,Q,R,S,T,U)=(3,5,8,4,4,8), Question2: (P,Q,R,S,T,U)=(5,10,15,3,6,9),
Question3: (P,Q,R,S,T,U)=(4,8,12,8,4,12), Question4: (P,Q,R,S,T,U)=(6,7,13,4,3,7),
Questions5: (P,Q,R,S,T,U)=(5,8,13,4,7,11)
　　As results of analyses, I have clarified the following contents.
(1) Most of 5th and 6th grade pupils pay more attention to "Part-Part" representation than "Part-Whole" representation.
(2) 5th grade pupils pay more attention to "double or half" than 6th grade.
(3) There is the period of transition from "Comparison by double or half" to "Comparison by Ratio".
(4) There are 4 stages of development on "Comparison by Ratio".
The first is the incomprehensible stage on "Comparison by Ratio".
The second is the stage to compare by "Specific Ratio" of "double or half".
The third is the stage of the period of transition to use "Ratio" or unuse it with dependence on "Numbers".
The fourth is the quite comprehensible stage on "Comparison by Ratio".

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"Ratiber und Goldschatz" (Thief and Treasure) is a learning environment in an innovative german mathematics textbook, Das Zahlenbuch, which was edited by E. Ch. Wittmann and others. I developed the mathematical lesson of negative numbers for 7th grade students which is based on "Railber und Goldschatz". In this paper, I analyzed the students activities of the lesson and reported the results. In the lesson, the most of the students learned activity and can answer the questions like "-2+3, 4-6 etc".

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This is a part of the series of research on the process of understanding mathematics based on the "two-axes process model" that consists of two axes, i.e. the vertical axis implying levels of understanding such as mathematical entities, relations of them, and general relations, and the horizontal axis implying three learning stages of intuitive, reflective, and analytic at each level (Koyama, 1993, 1997). The purpose of this paper is to analyse the process of understanding in the lesson on "stellar polygon" at 2nd Grade in a lower secondary school attached to Hiroshima University in order to see the development of students' understanding of the properties of "stellar polygon" and to get some implications for the teaching and learning of mathematics in lower secondary school. The objectives of the lesson on "stellar polygon" were to improve students' understanding of how to find out the measure of an angle in a stellar polygon and to promote their mathematical thinking and attitude toward mathematics through generalising the mathematical tasks on regular polygons to regular stellar polygons. As a result of qualitative analysis of the data collected in the observation and videotape-record during the lesson, we found out the four suggestions for improving the students' understanding of mathematics in lower secondary school. First, there were three different approaches to the same task 1 in a regular nine-pointed star polygon. It means that mathematics teacher should pay attention not only to the correctness of the answer to a task but also to the process of reasoning to the answer in mathematics lesson. Second, when the students worked on the complicated task 2 in a regular nine-pointed star polygon they folded back to the image-making level. This fact suggests us the importance of teacher's decision making on what kinds of learning situation should be set up for helping students improve their mathematics understanding. Third, a questing about the reason by the teacher and the response by a student changed the development of the lesson to a higher level of understanding. It suggests the importance of social interaction among a teacher and students in mathematics classroom. Finally, the students had difficulties when they worked on the more generalised task 3 in a regular n-pointed star to formulate the measure of an angle. The fact suggests that students should reflect what they have done in the task 2 (reflective stage) and do more activities for integrating them (analytic stage) in the "two-axes process model" of understanding mathematics. Being based on these suggestions, when a teacher intends to help students develop their understanding, according to the levels of understanding and three learning stages involved in the "two-axes process model", it is suggested to set additionally the alternative tasks such as task 1' of geometrical figures in a regular nine-pointed star polygon, task 2' of the measure of an angle in a regular nine-pointed star polygon, and task 3' in a regular 12-pointed star polygon before posing the generalised task 4' in a regular n-pointed star for asking students to formulate the measure of an angle in general.

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My purpose in this paper is to obtain didactical implications for effective communicative activities in mathematics learning, mainly from the analysis of "Converse of Pythagorean Theorem" lessons in a public junior high school, Hiroshima city. To develop the rationale for this analysis, this paper uses the epistemological frameworks of Interactionisim and design research. According to Cobb (2000), "research of this type involves both instructional design and classroom-based research (p.57)", and "it is this theory that makes the results of a series of design experiments potentially generalizable even though they are empirically grounded in analyses of only a small number of classrooms (p.59)". This research finally shows us three points for improvement of teaching and learning mathematics concerning mathematics teaching and curriculum development as follows; (i) Mathematical ideas emerged from classroom, when classroom discussion and group discussion became reflexive. Moreover, signs used in discussion helped emergence of mathematical idea. (ii) The role of teacher is to make student speak of reasons, and compare their mathematical ideas, in order to develop effective communicative activities (iii) This paper illustrates student's understanding of converse of proposition and indirect proof with the practical cycles of planning, instruction, and analysis.

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Tanaka, (2005) pointed out two roles of Dynamic Geometry Environment (DGE); one makes students to be able to conjecture with gathering data inductively, and another gives a perspective for deductive inferring to make the conjecture universal. In this research, we consider that the teaching and learning processes are based on the above roles. Hence, our target is to consider students' expected activities and teacher's supports. The purpose of this paper makes clear them. A didactical experiment is analyzed, which focuses on the first role of DGE. We analyze both student's activities and teacher's support through this experiment. In conclusion, we propose three cases for student's activities and teacher's support corresponding to above analyses. ・Case1: When students conjecture inductively, but it is not to coincide with teacher's intention, it is not a path to solve the problem directly; a teacher suggests modifying students' inductive activities. ・Case2: Students make a conjecture without conviction by using functions of Dynamic Geometry software; a teacher suggests for students to make sense of their conjecture. ・Case3: Students are not specifying the method of inductively gathering facts; a teacher suggests for them to describe the procedure how to gather data.

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In this paper, I describe the relation between the creativity and the understanding of the learning contents in "The Space Figure". The experiment involved 145 participants from junior high school students grade 1. For the measurement of the creativity, I made a creativity test of "The Space Figure" and used the test score. For the measurement of the understanding of the learning contents, I made a scholastic achievement test of "The Space Figure" which has three categories. Three categories of the scholastic achievement test are composed by the problems about the understanding of a basic computation skill, the understanding of the procedure when the students solve problems and the understanding of the structural relation of the learning contents. I examined the correlation coefficient and the causal relationship between the creativity test score and the scholastic achievement test score. The results are as follows: - The students who get higher scholastic achievement test score get higher creativity test score. The contrary of this proposition is not realized. - The students who get higher score of the basic computation skill get higher scores of the divergence, the fluency and the flexibility of the creativity. - The students who get higher score of the understanding of the procedure when the students solve problems get higher scores of the divergence, the fluency and the flexibility of the creativity. - The students who get higher the score of the understanding of the structural relation of the learning contents get higher scores of the divergence, the fluency and the flexibility of the creativity. - The understanding of the structural relation of the learning contents in three categories has the biggest influence to the fluency and the flexibility of the creativity.

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We have developed the system that classifies the problems in school mathematics for criterion-referenced evaluation into each viewpoint of learning situation by using neural network. We classified the school mathematics problems in four viewpoints for the criterion-referenced evaluation by using this system. The following matters were clarified; (1) As for the specialists of mathematics education, the average of the identification rate in this system is 84.6% and the system is highly accurate. (2) As for the novice of mathematics education, the average of the identification rate when using this system is higher 11.4% than not using this system. (3) The average of the identification rate using this system is 36.4% higher than using a multiple discriminant analysis. As a result, by using this system, the teacher was able to classify the problems of school mathematics into four viewpoints of the evaluation more accurately.

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A purpose of this research is to show current direction of literacy. In addition, as concrete context, an environmental problem is introduced as a focus for one of the literacy perspectives. From historical viewpoint of literacy, literacy theories have had two aspects; one is literacy as "culture", and the other is literacy as "reading and writing". OECD/PISA literacy and AAAS/SFAA literacy which are well-known literacy theories are considered in these two viewpoints, literacy as "culture" and literacy as "reading and writing", as a frame. As a result, present direction of literacy can not only be narrowed down to "reading and writing" but also extending to "culture". In this research, environmental problem is set for one focus for literacy that includes culture. And, approach for an environmental problem in mathematics education is considered based on D'Ambrosio (1994)'s work. That is, it suggests that curriculum constitution based on progressivism is important and mathematical modeling activity is effective approach.

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The purpose of this study is to acquire the fundamental date in the curriculum development based on the realities of children's probability concept. In the investigations by Fischbein & Schnarch, and the author, it is considered that children in the elementary education stage judge the probability subjectively, and present the aspect of complex misconception. Especially, the incidence of the heuristics of Compound and Simple Events increases with the development of the age in the longitudinal and cross-sectional study. Then, the teaching and learning materials and the evaluation problem to support conceptualization of the sample space are proposed. 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. It also gives the suggestions in the curriculum development that connects with the curriculum of probability in secondary education in Japan.

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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. The purpose of this paper is to analyze the data of three tests, i.e. Test 5, Test 6(1) and Test 6(2) to investigate children's progress of mathematical attainment in order to find out some suggestions for improving the teaching and learning of mathematics at these primary schools. In our previous paper (Koyama et.al., 2002), according to the percentage of correct answer to each test item, we made such categories as high [H], medium [M] and low [L] attained items. We defined the fixity of mathematical attainment such that for three tests if a child's changing pattern of correct (1) or incorrect (0) on an item is [1→1→1] or [0→1→1] then the child's mathematical attainment on the item is fixed. As a result of analysis in terms of these categories and the fixity of mathematical attainment, we found out the following: ・There were three different types of [L→H], [L→M] and [L→L] from Test 5 to Test 6(1) based on the progress of each some test item in Test 5 which is learned at the fifth grade. ・There were five different types of [H→H], [M→H], [L→H], [L→M] and [L→L] from Test 6(1) to Test 6(2) based on the progress of each some test item in Test 6 which is learned at the sixth grade. ・We found that the three basic items in Test 6 which are learned at the fifth grade was fixed among children. These results suggest that the teaching and learning of mathematics at the fifth grade was partly effective and that more efforts should be made in the teaching and learning of contents such as area and volume, proportion and even and odd number.

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USA-Japan Comparative Research on Science and Mathematics Education aimed at obtaining implications to improve quality of mathematics and science education through comparative study method. One of three groups, which were formed under this research, focused on relationship between school and society, and conducted the third-year field survey from Nov. 29^<th>, 2004 to Dec. 6^<th>, 2004. It visited National Council of Teachers of Mathematics and National Science Foundation in Virginia, Association of Children's Museums and Association of Science-Technology Centers in Washington DC, American Museum of Natural History in New York, and Family Math Program in San Francisco, in order to survey on the linkage programs between school and society. From this year survey, there are two major findings. The first one is that there exist two types of education whose major concerns are equity or excellence. The equity corresponds with children museums and the excellence with science museums. The second one is difference in the societal background between USA and Japan in term of advocacy and high attrition rate of teachers. As a result of the three years survey, we realized that it is necessary to set a long-term direction of education like Science for All Americans (AAAS, 1989), and it is the most important point that we can learn from USA despite of the different social backgrounds. This fundamental direction should serve as a starting point of considering a model of new education in 21^<st> century. It should liaise with social education institutions such as museum beyond the boundary of formal education system, and include promotion of scientific literacy in the wake of life long learning society. Finally as a way of conclusion for three years survey, there are 7 more suggestions as follows: (1) Appeal to the society from the academic association (3^<rd>) (2) Establishment of Research Institute of Science and Mathematics Education (2^<nd>) (3) Establishment of synergy relation between school and museum (2^<nd> and 3^<rd>) (4) New role of museum (3^<rd>) (5) Establishment of system of out-of-school science and mathematics education except museum (3^<rd>) (6) Action toward parents (3^<rd>) (7) Budget allocation (3^<rd>)

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This study focused on Chinese students in Japan to consider how Japan had had influences on the process of changes in western mathematics in China. In order to clarify the actual situation on mathematics education which Chinese students took in those days, this study considered the feature of curricula and the actual situation on mathematics education in Tokyo Higher Normal School (THNS) and conducted a tracing survey to graduates from THNS on their professional career after returning to China. The products of this study are below. (i) After Gokotokuyaku in 1907, most of the Chinese students moved to Higher Normal School and THNS accepted many of them. (ii) The increase in the number of mathematics lectures in Faculty of Mathematics, Physics and Chemistry (FMPC) in THNS shows how professional mathematics education was strengthened. (iii) Lecturers in Mathematics and Education Departments consisted of graduates from Tokyo Imperial University and THNS. They had studied abroad and written many books on mathematics education. (iv) As the result of tracing survey for the Chinese graduates from FMPC, mainly from mathematics department, they become not only mathematics teacher in Higher Normal School but also translator for mathematics literature and civil servant at education authorities.

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　　Within a study that seeks to understand the process of learning mathematics in secondary schools, this paper describes 'participants', 'mathematics' and 'activities' as core elements of regular classrooms. The interdependent relationship among these elements forms the basis of a proposed interactive model. The potential linkages in this model are illustrated using 'divisibility tests' from the Kenyan secondary school curriculum. This hypothetical illustration illuminates a missing phase in our study, that is, extended lesson-observation in ordinary classrooms. However, the interactive model through its six interactions-three identities and interaction between each pair of the elements, provides a basic reference for the future theoretical and empirical needs of the study on regular mathematics classrooms.

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　本研究の目的は,バングラディシュにおける初等教員研修校での数学教育の現職教員研修の効果を把握することである。本目的を達成するため,質的,量的両側面より調査を実施した。具体的には,代表性を考慮して選出した2校の研修校における2004-2005年度の全研修生(267名)を対象に,1年間に渡る研修の前後で質問紙調査を行い,そのデータをSPSSによって統計的に分析した。その結果,研修の前後での変容について,次の諸点が明らかになった。現職教員研修の効果として,全般的に教科の知識や教授技術の向上が認められた。しかしながら,数学教授への態度という点では,統計的に有為な効果は見られなかった。また教師の個人的属性(学歴,年齢,教員経験年数)と彼らの教科知識,教授技能,態度に相関があるかを調べた。学歴はそれら教師の能力に大きな影響を持つものの,年齢,教員経験年数はそうでないことが分かった。その他にも,教授技能は教師の態度と強く相関があること,教科知識は教師の態度との関係が最も弱いということが分かった。

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The purpose of this paper is to study effective method of mathematical problem posing. We practiced such mathematical activities by students in high school (Grade 3) who were not necessarily good at mathematics. In the previous study, we reported two practices of problem posing on the fields of analysis and geometry. A feature of the method is to provide situations in which students plan problems freely from the first using enough time to create problems. In this paper, we report the practice of problem posing on sequences and examine the effective use of problems posed by students. The practice shows that students who tackled making problems learn many mathematical contents related to the problems. We extended some problems posed by students in a class to show their relations with the golden number, Fibonatcci number and Pytagoras number. Each student solved one problem posed by another student and commented on problems each other, which established an effective use of problem posing and communications between students. We observed more positive learning activities than the usual classes.

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It has been an important subject how we plan to establish the relationship between mathematics and other fields in the mathematics education. In such practice, we assert that it is important to have some practical viewpoint which is significant mathematically and also integrates some common aspects often appeared in other fields. In this paper, we call the realizations of graphs in space the frames, and study the movement of frames as a teaching material of figures. The movement of frames is considered as a mathematical model of various mechanical devices, and thus it is used as such viewpoint to understand the utilities of mathematics realistically. We show rich contents of the frame movement as teaching material by two concrete models. One of them is the movement of linkwork, and to clarify the principle of the movement of linkwork of four links needs the practical use of integrated mathematics. Also, to clarify and apply the principle of the line movement of a linkwork becomes a good teaching material to learn how mathematics is important in such process. Another of concrete materials is the movement of the frame composed of squares arranged rectangularly, which is a mathematical model of the stability in the framework of building. The material has a new aspect of mathematics problem mathematized from the realistic situation, and it is an appropriate one to show a harmonic relationship between the theory of vector spaces and that of graphs and also to learn the dimension of the movement of figures.

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I investigated the contents of the mathematics which is used about the electrical engineering. In the electrical engineering, mathematics is used in a lot of scenes. It appears important to introduce the mathematical formulae and computation in basic electronic experiments into my teaching method. I think that there are two problems when the student learns engineering. (i) The teacher of the mathematics doesn't tell the scene how mathematics is specifically used for about the engineering too much in their class. (ii) The teacher of engineering doesn't tell the explanation of the mathematical theory too much in their class. Therefore, I propose to implement the following "cross-curricular class". (i) The teacher of the engineering describes the contents of the experiment on the engineering and how to make a mathematical formula. (ii) The teacher of the mathematics describes the mathematical formula and the computation which was used there. In this way, both cooperates and makes a class. By the questionnaire, it found that the student expected such a "cross-curricular class".

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