The aim of this article is to examine two perspectives of the recent research in mathematics education; they would respectively be expressed in two key-words: "culture" and "ecology". The former perspective may be referred to the recognition that mathematics and its education, as a domain of human cultures, have charactors good or bad which are common to all other cultures. The latter ecological perspective would be extended to the argument that even mathematics education should be treated as a complex organic system in its research. These perspectives come from the reflection that in hitherto ways of mathematics teaching, mathematics is apt to be regarded as the absolute entities and its education often falls into the mechanical training. However, in the recent mathematics eduction, especially in elementary schools, the classroom of mathematics comes to proceed in the process of negotiation among pupils being guided by teachers. The auther would like to confirm the legitimity of this way of mathematics education. It would be also suggested why it is desirable to make pupils acquire the knowledge as a whole without making it in pieces and craming.

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Ethnomathematics reveals that mathematical activity exists everywhere in the society. Children practice ethnomathematics even before they learn mathematics in school. Thus they feel implicitly strain between two types of mathematics. Besides this conflict in mathematics, there is another type of conflict, which is related to the educational objective. Kenyan society has been tackling this problem of cultural conflicts since independence, and development of verb-based curriculum is in line with this effort. The essence of verb-based curriculum is a structured list of children's mathematical activities. Since this curriculum has focus on activity itself, the context in which the activity develops, is important and ethnomathematics, therefore, is required for the curriculum. And ethnomathematics necessitates a structure, which ensure the mathematical development in the classroom. Both aspects of context and structure are important factors in this curriculum. To apply this principle of verb-based curriculum to the Kenyan education, the author, in this paper, listed up and analyzed all verbs in the present Kenyan syllabus for primary education. 53 verbs have been identified and analyzed for children's learning activities in mathematics education. Comparing this result with the analysis of Japanese syllabus in the previous study (Baba, 1999), their universality and specificity in mathematical activities have been found out. While the universality is that more than 30 verbs of Japanese and Kenyan syllabuses correspond with each other, the specificity is that the former has more inclination towards children's internal activities and the latter has more inclination towards economic and procedural activities.

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In this paper, we describe the degree of creativity development of elementary school students by using "the quadrilateral" that is a basic figure as the teaching materials in arithmetic education. The experiment involved 428 participants from students grade 3, 4, 5 and 6. The degree of creativity development is evaluated from both side: the creativity ability and creativity attitude. For the creativity ability measurement, a new creativity ability test problem was created. Furthermore, we examined the degrees of creativity ability development and the relationships among the scores of each evaluation viewpoint by using the test score. For the creativity attitude measurement, we used a Creativity Attitude Scale (CAS) that was developed by Saito, N. We examined the degrees of creativity attitude development and the relationships among the scores of the creativity attitude factors by using the scale score. Moreover, we examined the correlation coefficient and the causality relation between the creativity ability test score and the creativity attitude scale score. The results are as follows: - The higher grade students (grade 5 and 6) achieve higher score compare with students grade 3 and 4 in the creativity ability test. However, the scores of the flexibility and the originality are very low in any grade. - The difference of the creativity ability development is seen between students of grade 4 and grade 5. The creativity attitude is the same characteristic, too. - The lower grade students (grade 3 and 4) achieve higher score in the creativity attitude scale compared with the higher grade students (grade 5 and 6). - The students that get higher creativity attitude scale score achieve higher creativity ability test score. However it doesn't occur conversely.

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In this paper, we describe the relationship between the creativity attitude and the scholastic achievement of mathematics learning. The experiment involved 103 participants from the first grade junior high school students. For the creativity attitude measurement, we used a Creativity Attitude Scale (CAS) that was developed by Saito, N. For the scholastic achievement, we used the 5th test score with "the plane figure" as the test content, and average mark from five tests that are performed by the students within a year. We examined the correlation coefficient, the causal relationship and the regression line between the creativity attitude scale score and the scholastic achievement test score. The results are as follows: - The correlation coefficient between the creativity attitude scale score and the scholastic achievement test score is very low. - There is a casual relationship between the creativity attitude scale score and scholastic achievement test score i. e. the students that get higher creativity attitude scale score also get higher scholastic achievement test score. However it does not occur conversely. - As for the students that get higher creativity attitude scale score, the scores of all factors that composed the creativity attitude get higher. - The level of the logic, the identity and the precision within seven factors of the creativity attitude have great influence on the scholastic achievement.

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In the atomistic perspective which teaching materials are divided in detail teaching takes many small steps. Students may acquire a certain amount of knowledge and skills, however they may be in danger of learning without knowing what they are doing as a whole and why. So this study aims to rethink mathematical learning from the holistic perspective, especially the 7th grade's algebraic contents. In order to design classroom lessons from the holistic perspective, we need to know beforehand what sort of problems encourage students' mathematical ideas to emerge and how they develop their ideas through solving the problems. The main purpose of this article is to make it clear how 7th grade students who have not learned the contents of letters and algebraic expressions can emerge and develop their algebraic ideas through solving problems in the context of the linear equation. We conducted several case studies and analyzed students' problem solving activities. The results are as follows; 1. Students could create naive algebraic ideas through solving problems presented by Bednarz and Janvier (1997). 2. The interplay between using trial-and-error method and setting up the expression could help them to understand local relationships and to create algebraic ideas. 3. The idea of uniting several terms emerged in two situations. One was the situation of simplifying the numerical calculation and the other was that of understanding the structure of the parts in the expression that they set up. 4. It was effective for the emergence of the idea of distributive law that they conceived the problem situation and the numerical computations structurally. The idea became explicit when the explanation was sifted from numerical calculation to algebraic expression using letters and when they reinterpreted the expression as the number of pieces of one letter. 5. Students could create a feeling of necessity and appreciation of using letters through the problem solving in the context of linear equation before they learn algebraic expressions.

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Recent research concerning interactions in mathematics classroom suggests that a starting point for improving the everyday mathematics classroom is to have a better understanding of its unique cultural practice. The focus of this research is to gain a better understanding of mathematics classroom from students' perspectives. In particular, it aims to analysis the relationships between the nature of teacher-student or sutudent-student interactions and the students' mathematics learning. The research methodologies used in this qualitative study were hermeneutic and micro-ethnographic case studies. Four mathematics lessons were taken from an 8th grade mathematics classroom. The context was the teaching of basic triangle congruence theorems. Five students were interviewed after the lessons. This paper reports the class interactions from the perspective of one student, Yama, who was one of the more active students in the mathematics class. Yama developed his mathematical understanding in the class interactions by taking advice from the teacher and the other students, especially one of the active participants in the class, Yoshi. Yama developed his mathematical understanding and ways of participating in class through the interaction processes. An analysis of the class interactions from Yama's perspective revealed that: (1) It was very important for him to interact with another active student, Yoshi, who played the role of his adviser. They were learning from each other. Their fruitful relationship developed since they were able to complete the same task interactively in the process of the class interactions. Yama developed his mathematical understanding by reacting to Yoshi's suggestions. In this sense, the mathematical understanding itself that he developed was the result of the interactions. The fact that their fruitful human relationship influenced the mathematically rich interactions is worthy of attention. (2) Yama's "thinking with announcing" acts contributed to developing his mathematical understanding through his participation in the class interactions. The interview with the students revealed that the "thinking with announcing" activity was shared with some students in the classroom and was considered by them to be a natural process. The "thinking with announcing" represented the emergence of a classroom microculture. The classroom microculture and his mathematical knowing seemed to be developed interactively.

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The purpose of a series of our studies is to examine the role of "self-referential-activity" on mathematical problem solving. The term "self-referential-activity" means solver's activities that he/she refers to his/her own solving processes or products during or after problem solving. In study (I), we proposed the theoretical framework for analyzing self-referential activity. And, in study (II) and (III), we elaborated the variable "OG/NOG" and "M-SE/SE-C" respectively. In this article, we examined a student's problem solving process observed in a class which purpose was to construct a square from two given smaller squares (e.g. 7cm and 3cm). The student's solving activity was analyzed from our theoretical framework. As a result, the solving process was characterized as "Fail to pursue an approach but continue to pursue it" activity. We had theoretically assumed the existence of this kind of Local-Bad-Judge in study (IV), so this finding supports our theoretical assumption and its powerfulness of our framework. In addition, we discussed the possibility of translation from unsuccessful situation to successful one, and from these perspective we proposed teacher's suggestions or comments for the student to improve these kinds of unsuccessful situation to successful ones. The variables in our theoretical framework help us to consider such suggestions or comments. And, it is suggested that such suggestions or comments do not always need to be general one but be oriented to current situation.

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The purpose of this paper is to investigate the developmental aspects of analogical reasoning from 3rd to 5th graders, which is focused on the function of cognition/understanding. In order to analyze their analogical reasoning, five viewpoints are established as follows: (1) Representations of the targets; (2) Choices of the bases; (3) Mappings between the targets and bases; (4) Adaptation processes; and (5) Knowledge related to both the targets and bases. As a result, the following implications are found out. Firstly, when children fail to represent the targets, they fail to reason by analogy successfully. Secondly, children's knowledge related to both the targets and bases has an influence on successful analogical reasoning. Thirdly, the major point of successful analogical reasoning in this investigation is the adaptation processes. It is classified as follows: (a) Applications of relational features of the bases to the targets without any transformations; (b) Transformations of the targets for applicable; and (c) Transformations of the bases for applicable. This classification becomes difficult in turn. Finally, while 3rd graders tend to pay attention to similarities between the targets and bases, 4th and 5th graders pay attention to not only the similarities but also differences between the targets and bases. 4th graders, however, tend to give rise to cognitive conflict between the targets and bases because of the differences.

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There are many students who have suitable knowledges about geometry but can't solve geometrical problems due to lack of the sensorial aspect of geometry. Generally speaking geometrical sense has a crucial feature for the teaching of geometry in the following days. But the meanings of geometrical sense is very vague for us. The main purpose of this study is to clarify the meanings of geometrical sense in the teaching of geometry. To clarify the nature of geometrical sense I classified it from phenomenological viewpoint as below. <Geometrical Sense> ◎External Sense: geometrical perception about external object or figure ◇Perceptual Intuition: perception of concrete entity (visual perception about the spatial aspects of geometrical figure: shape, size, position) ◇Essential Intuition: perception about ideal meaning of geometrical figure (linguistic perception about the essential meanings of geometrical figure or geometrical relation) ◎Internal Sense: perception about individual situation to the external object ◇Value Judgement: perception about "goodness" (directional and intellectual perception for judgement and selection of geometrical information) ◇Emotional Sensibility: perception about "beauty" (individual perception about sensibility for geometrical figure: beauty, stability) The characteristics of geometrical sense are explained by next five points of view: individuality, unconsciousness, interest, simplification and perception. Especially "perception" is an essential feature which characterizes geometrical sense. So I tried to enumerate the perceptual functions of geometrical sense and analyzed each of them by case-study. <Perceptual Functions> ・Plane Perception Part and Whole: "Gestalt" and "Figure and Ground" Synthesis and Analysis: "Perceptual Construction" and "Perceptual Transformation" ・Spatial Perception "Perceptual Constancy" and "Back and Forward Perception" To promote the growth of geometrical sense is an essential aim of the teaching of geometry both in elementary school and in junior high school. Especially in elementary school, the sensibility for "beauty" of geometrical figure and the "richness" of geometrical sense are taken notice of. And in juniore high school, the "sharpness" of geometrical sense is particularly considered.

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Treutlein P. (1845-1912), who was an active mathematics teacher at secondary schools in Germany and developed a new geometry curriculum for secondary schools, gave a great deal of influence on geometry teaching in Japan. His book, Die geometrische Anschauungsunterricht als Unterstufe eines zweistufi gen goemetrischen Unterrichts an unseren hoheren Schulen (1911) was translated into Japanese in 1920 and accepted as a new model of geometry teaching curriculum in Japan. Treutlein called his own geometry curriculum 'der geometische Anschauungsunterricht' (the geometrical intuitive instruction), which was the geometry teaching for lower levels of secondary schools at that time. The main goal of his 'der geometische Anschauungsunterricht' was to develop 'das raumliche Anschauungsvermogen (spatial imagination)' of pupils through verious geometrical tasks: cutting solids in mind, making their models, making new figures with some pieces of another figures, estimations and measuring of lengths and eares, construction and so on. The purpose of this paper is to consider the question what the meaning of `das raumliche Anschauungsvermogen' is. In order to interpretate his verious tasks in his geometry teaching and to make clear the histrical process of the development of the geometry teaching in Japan, it is very important to consider the qustion. The analysis of his own statements about geometry teaching and some tasks in 'der geometische Anschauungsunterricht' in this paper made it clear that 'das raumliche Anschauungsvermogen' was the ability to imagine and manipulate figures in mind, and that the figures meant both plane and solid figures. Although it seems that spatial imagination is usually only about solids in these days, spatial imagination for his geometry teaching means the ability to imagine and mamipulate figures both in plane and in space.

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The purpose of this paper is to study how Minoru Kuroda (1871-1922) understood the thought of function and the education of function. Minoru Kuroda was worried that students lost interests in Mathematics and that they reduced the will to learn. So, he emphasized theories and applications in Mathematics. He considered that theories in Mathematics should be learned in applications. As a result, he accepted the thought of function considered by Felix Klein. And Minoru Kuroda considered that the thought of function must be trained in both algebra and geometry. After due consideration of the present condition of Japan in those days, particularly he accepted Klein's geometrical representation. And the concept of coordinates was accepted in algebra, and so analytical geometrical treatment and a point view of function were accepted in geometry. But he could not put the culture of mathematics, calculus and astronomy into a field of the vision about the education of function.

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A method of mathematical problem posing by students in high school or university is discussed, which emphasizes students' experience to synthesize mathematical ideas through it. A feature of the method is to give students enough time to create problems, and another feature is to assign students planning problems from the first. Actually, I report practice of problem posing that way. One practice of problem posing is for some university students, and it shows some tendency that students who buckle to their problem posing get some deep understanding for the mathematical theories related to the problems. An analysis also shows that the problem posing seems to have some special characteristic which might not be learned through the usual classes. Another practice of problem posing that way is for some high school students, who were observed to learn many mathematical ideas through the activity. Some students say that they felt difficulty in making problem for the first but gradually felt interested in considering around various problems. The practice shows that opportunity in a class to discuss problems made by students is important in such activity.

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An extension of real numbers to complex numbers has a certain gap, since the latter is the 2-dimensional vector space consists of real part and imaginary part. In this paper, we compare the geometric meanning of two notations a+bi and (a, b) for a complex number. We may consider the former as a geometric vector of the 2-dimensional vector space spanned by orthonormal vectors 1 and i, where 1 is the unit element and ii=-1, the latter as a number vector of the 2-dimensional vector space spanned by (1, 0) and (0, 1). Therefore, we can easily see that two definitions are the same essentially. The multiplication by i for number w in the complex number plane is a rotation by π/2 around the origin, however this multiplication is not a homomorphism with respect to the multiplication of complex numbers. The multiplication by i is the homomorphism for a ternary compositions u-v+w and uv^<-l>w (v&bne;0), which due to E. Cartan (1927). The complex number relates a matrix of left multiplication and also cosine, sine function. Therefore, for understanding of complex number, it is useful to consider it together with matrices and trigonometric functions.

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