The purpose of this paper is to show the results of my previous research on humanizing mathematics education and to construct the principle for humanizing mathematics class. In my previous research based on the "theory of situated learning", I established the process of mathematics learning. But in the "theory of situated learning", there isn't a perspective to analyze how students participate in a cultural practice of mathematics and develop themselves. To have the perspective, I use a concept of "awareness". I think that to educate students' awareness is to develop themselves. Therefore, it is necessary for a teacher to place the way for students to educate their awareness in the previous established process of mathematics learning. Also, I think the former is humanizing mathematics and the latter is understanding the mathematical natures children have. In these circumstances, I think that to combine the former and the latter could be the classroom principle for humanizing mathematics education. Based on this classroom principle, I analysed an arithmetic classroom. As a result, I made the next point clear. It is the humanizing mathematics education that individual student makes one's awareness clear and educates it through the interaction with others in mathematics learning, in other words, through the collaborative mathematics learning, individual student becomes aware of mathematical relations and values, and develops oneself.

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Modern Mathematics has its origin of development in modern Western Europe, and has made a dramatic progress together with industrialization and development of modern education there. On the other hand, the developing countries, which are now experiencing drastic process of modernization and Westernization, propose for indigenous and natural development of society, but at the same time are obliged to import models from the developed countries. The same model-borrowing applies to mathematics education. Considering such a situation, the author proposes for utilization of ethnomathematics, which was initiated in 1984 by D' Ambrosio, a Brazilian mathematics educator, for the sake of endogeneous development of mathematics curriculum in the developing countries. More concretely, while the traditional curriculum is called noun-based in such a sense that it focuses on knowledge as a product of activity, which is expressed in noun form, the proposed curriculum is called verb-based, aiming at development of ethnomathematical activity itself, which is expressed in verb form. It means that this curriculum expresses development of mathematical activities, which are found in children's environment, in terms of verbs. Furthermore in this research, the semiotic chaining in Presemeg (1997) is applied to analyze the development of mathematical activity.

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If we regard students as decision-makers who decide to participate in teaching situations and they not decide to, then it is a very important task for mathemtics education how to help the students relate themselves to any new mathematical ideas or ways of thinking. From this perspective, clarifing the meanings of the new mathematical ideas we intend to teach or to make clear what are these for us becomes more important than simplifing them. In this paper, the concept of "Grenze" will be considered. It have been introduced by German mathematics educators in IDM. They regard it as the best characterizing of metaknowledge. It could be regarded as the kernel of what are any new mathematical ideas or ways of thinking. The goals of this paper are as follows: (1) To make clear the meanings of the concept of Grenze and it's epistemological roles. (2) To show that a Grenze was realized in one of the mathematics classrooms which engendered eager participations and interests. (3) To identify the facts that need for realizing the Grenze by analysing the teacher-students interactions in the classroom. For the goal (1), the theory of proportion, especially the definition 5 in THE ELEMENTS BOOK V was considered. The main results of the consideration are summerized as follows: The meanings of Grenze can be characterized as a developmental relationships between unknown new ideas and known old or more familiar ideas. We need a context for constructing the developmental relationships. If we can construct the context and recognize the Grenze of our familiar ideas, then we can have a good perspective about the new ideas' novelty without pre-understanding of the new ideas detailed. So Grenze could play an important role in an introductry phase of didactical situations. Especially, if we regard the students as decision-makers, the role must become crucial. For goals (2) and (3), microethnographical case studies were utilized. About 10 minits introduction episodes including teacher-students interactions were taken from an 8th grade mathematics class, where the students' eager participations and interests were observed. The content which the teacher tried to introduce was basic triangle congruence theorems. Analysis of the episodes reveals that the Grenze was realized through teacher-students interactions in the classroom. The main facts that need for realizing the Grenze were as follows: (i) In the mathematics classroom, we observed a classroom culture which gave students a standard of valuable argumentations. The students seemed to find that it is reasonable to explain why it (geometric relations) is true in terms of other well-known geometric relations or results, that is to say, they were in the classroom micro-culture of mathematical argumentations. (ii) The students were permited to make free use of their familiar ideas of triangle congruence and encouraged to argument based on these ideas.

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I consider effective negotiation in mathematics education to be negotiation through which mathematical meaning emerge. I name such a negotiation Emergent Negotiation. The purpose of this paper is to clear the process that mathematical meaning emerge through Emergent Negotiation. In this Paper, I analyze four episodes that I regard as Emergent Negotiation. And based on this analysis, I try to clear the process that mathematical meaning emerge. The results are as follow; 1. The processes of transition from based idea to emergent idea are classified into two categories, "introduction of the new reference context", "modification of the old reference context". 2. There are two factors in this transition, factor in related to cognition and to society. On the factor in related to cognition, I give four mathematical thinking: deductive thinking, orderly thinking, thinking about unit, and thinking about function.

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It is considered that fraction concepts are difficult to teach/learn in Japan's elementary schools. There are a lot of factors including the following: ・A variety of the meanings, the complexity of the meanings, and the complicated notations for fractions (Ishida, 1985) ・The continual argument about introductory lessons for fractions ・The appearance of new meanings for fractions in the third grader's mathematics textbooks and some theories In addition, children's everyday concepts of fractions have been taken into little consideration in teaching/learning fraction concepts. In other words, the following have not yet been explained. 1) What everyday concepts of fractions do children have, before they begin learning fractions at school? 2) How are children's fraction concepts formed through fraction lessons, based on their everyday concepts of fractions? 3) What teaching/learning are suitable for fraction lessons when we consider children's everyday concepts of fractions? This paper answers the first problem. That is, the purpose of this study was to clarify children's everyday concepts of fractions. Here, everyday concepts, and mathematics concepts corresponding to the scientific concepts are defined by Vygotsky's theory. Three investigations were conducted to clarify children's everyday concepts. In the first investigation, six third grade children were interviewed to collect information for a questionnaire (Yoshida, 2000). In the second one, twenty-one second grade children were interviewed using not only a questionnaire but also some concrete objects such as ribbons, Origami (folding papers), kumquats (small oranges), and sliced cakes and cookies made of cardboards etc. Finally, thirty-nine third grade children were given questionnaires as a test at the same time without any concrete objects. All children have not learned fractions at school at all. The first and second investigations were recorded on videos and tape recorders. The results from the second study, particularly, reflected the children's everyday concepts of fractions of the three investigations, because it was conducted in the most realistic everyday context using concrete objects. Moreover, the children were younger. The questionnaires for the second and third investigations were the same but different from the first one. They were based on the result from the first study, and were classified into six categories: (I) equipartition, (II) a view of unit '1', (III) quantity sense, (IV) equivalence, (V) order, and (VI) comparison between two quantities. Finally, children's everyday concepts of fractions were clarified for each category. And furthermore, the children's everyday concepts of fractions were integrated concisely into the following: EF1) Equipartition concepts that are used when children partition the whole equally by regarding the whole as a unit '1' EF2) Construction concepts that are used when children construct a unit '1' subjectively corresponding to an integer EF3) Identification concepts that are used when children identify a unit '1' subjectively corresponding to an integer according to the existing scale EF4) Identification concepts that are used when children identify the objects which has no scale, severing the connection with the whole EF5) Quantity concepts that are used when children consider the quantities in a real context EF6) Fundamental concepts for equivalent fractions that are used depending on the contexts EF7) Basic concepts for ordering fractions that are used in everyday concepts

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　　In this paper, I have described lessons on velocity, which stand on the ground of a graphical schema (Dorfler, W., 1991, p.74) of the process of generalization. This model proposed by Dorfler, W. is as follows.

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　　 I have subdivided child's knowledge on velocity into conceptual knowledge and procedural knowledge. Simultaneous activation of these knowledge has been my task since I have grappled with a study of child's knowledge on velocity. Hence I considered to try to make advantage of this model.
　　To connect one item with another item on this model, I have make use of views of Toda, K. and Mach, E..
　　Toda, K. suggested the following differentiation of stages of problem solving in 1954.: When one is confronted with a personal task, which is appropriate for the mathematical thinking and the mathematical process, one (1) perceives it as one's task, (2) makes a problem out of it, (3) gives the mathematical presentation to this, (4) recons out it, (5) interprets the result as the answer of the problem, and (6) considers it as the answer of the task. When I consider these six stages, Stage (1)and Stage (6) are concerned with competence analyzing tasks, Stage (2) is related to problem making ability, and Stage (3)〜(5) are related to word problem solving ability, I think. Also I considered the process of the thinking experiment of Mach, E. and thought the following process of the thinking experiment about the elementary school.: (1) Establishment of ideal/abstract conditions (Establishment of the precondition), (2) Establishment of factors (constant and variable) (Establishment of observational conditions), and (3) Establishment of the changeable method of factors. And after (1)〜(3), the following process follows.: (4) Establishment of the hypothesis and (5) real experiments. Moreover lessons were going well under these two views.

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The concept of geometry is being investigated very actively on the basis of cognitive theory. But there are very few reports about the methods to develop the curriculum of the teaching of geometry as well as to describe the process through which the concept of geometry is recognized. So to reflect upon the teaching of geometry, I intended to investigate the recognition of the concept of geometry by use of phenomenology. The main purpose of this study is to clarify the meaning of ideality and objectivity on the concept of geometry. First I reviewed the results of a series of previously reported studies on the concept of geometry. Then I listed up the problems worthwhile to investigate. What is ideality and objectivity on the concept of geometry? What characteristics dose ideality and objectivity has? How is ideality and objectivity recognized? How is ideality and objectivity taught? The characteristics of ideality on the concept of geometry are as follows: nonentity, identity, repeatability, meaningfulness, consideration, recognition along stages and dependence upon perception. Ideality on the concept of geometry is recognized along the following three stages: sensitivity, imagination and ideality. There are two types of recognition from sensitivity to ideality, that is to say, from sensitive entity to idealistic entity. One type of recognition is so-called "abstraction" from object to shape, and the other is so-called "generalization" from diagram to figure. In this study I used the term "objectivity" on the concept of geometry from the idealistic point of view. The characteristics of objectivity on the concept of geometry can be expressed as follows: intersubjectivity, identity, evidence, dependence upon anyone, anytime and anywhere, repeatability and linguistics. Objectivity on the concept of geometry is recognized along the following three stages: dialogue, linguistics and objectivity. Special attention should be paid to the recognition from "ideality" of individual concept of geometry to "objectivity" in the teaching of geometry. Then "imagination" of treating sensitive entity, "dialogue" among classroom members and "linguistics" which is objective in mathematical world should be emphasized.

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The major aim of this article is to examine the validity of "The components and its levels of spatial thinking" by author through a written questionnaire paper research. The result from research gave the validity to "The components and its levels of spatial thinking" in terms of existence and hierarchy of levels. Additionally the following implications are found out: -Regarding with the tendency of attainment of levels, the term of development falls into following three divisions: 3^<rd> to 5^<th> grade; 6^<th> to 7^<th> grade and over 8^<th> grade. -In progress between lower levels, representation and spatial reasoning is easier and imaging and visualization is most difficult for students. Especially, students do not necessarily develop the ability of imaging and visualization in this progress. -Mental operation with transformation of object is the feature in upper levels. -Students are able to sketch the given object in an early stage. Interpreting of the projection and net is in same level of thinking, and is more difficult than doing of the sketch. Regarding with any representation, the construct is more difficult than the interpretation. And, the projection is little more difficult than the net for students, and they are able to deal with these representations at 8^<th> grade.

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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 study (V), we discussed some possibilities of transition from unsuccessful situation to successful one or of keeping successful situataion in problem solving process. And, from this perspective we proposed some teacher's suggestions or comments that enable solver to move from unsuccessful situations to successful one or to keep successful situation. Furthermore, we assumed that such suggestions or comments do not always need to be general one and be oriented to current situation. In this article, we examined the assumption by exploring college students' problem solving processes. The task that subjects solved was called "Making-Square-Problem" which purpose was to construct a square from two given smaller squares (e.g.7cm and 3cm). The subjects were asked to solve the task and an interviewer gave some suggestions or comments that we proposed in previous study, and the problem-solving activities were analyzed from our theoretical framework. As a result, the suggestions or comments proposed in previous study were effective on improving solvers' problem solving activities or on keeping successful situation. And it is suggested that such suggestions or comments are effective not only at the phase until the end of problem solving but also at the stage after the end of problem solving.

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The purposes of this study are to explore the meaning and the method of the reflective activities and to propose "reflective learning" as the practice example of reflective activity. Although many papers argue the importance of "looking back" and "reflection" in arithmetic and mathematics learning, they have not made reference to the concrete classroom activity of "looking back" and "reflection". Therefore, the autors defined "reflective activity" as the activity which looks back upon contents of learning after the end of certein learning, and we summarized the meaning of it to the following five: (1) Contents of learning are realized again, and deep understanding is brought. (2) Contents of learning are integrated, and re-composition is brought. (3) Self-evaluation about contents of learning is brought. (4) Lesson induction about contents of learning is brought. (5) The connection of learning is brought. The method and the feature of reflective activity were considered from three viewpoints (the individual reflective activity, the tutoring reflective activity, the collective reflective activity) supposing the actual teaching and the educational guidance where reflective activity is performed. Then based on the results, we proposed "reflective learning" as the practice example of reflective activity after the end of small unit.

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The purpose of a series of my studies is to construct the process and principles of teaching mathematics using "errors" in order to change mathematical belief of students. In this paper I practically examine the validity of the process and principles of teaching mathematics through an experimental class on "the barycenter of quadrangle" for eleventh grade students. It is based on Lakatosian theory (Quasi-empirical view of Mathematics (1978), The logic of mathematical discovery (1980), The methodology of scientific research programmes (1986)). As a result of this practice, the validity of teaching process was suggested, because many students realized the significance of using an "error". And the validity of teaching principles for posing problems (Ia, Ib, Va) was suggested, but the validity of principles for the role of teacher in class was not necessary suggested. Moreover I suggested three implications so as to more validly use the process and principles of teaching mathematics using "errors".

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The purpose of a series of my studies is to gain a suggestion for the processes of Teaching and Learning Mathematical Concepts from the knowledge of the Theory of Embodied cognition (Lakoff (1993), (2000), Johnson (1987)). In this paper I investigate the actual situation of 7-8th, 10-11th grade student's embodied explanation of the concept of function. This investigation is based on the hypothesis of Arima(2001c) and the investigation of Sekiguchi (2001). As a result of this investigation, it was found out that students use "directional-schema" and "Pathsschema" for explanation of the concept of function. Because some students use metaphorical expressions and meta-representations.

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In this article, a type of writing activity, which has a Reflexive Nature, and enable students to be aware of "others" or the social aspects under the learning environment, is discussed. First of all, the relationship between cognition and writing is examined through the results from psychology, and it is suggested that writing activities effect on the learner's cognition or understanding. Then, considering the concept of "Reflexivity", the terms of "Mathematical Writing" and "Reflexive Writing" are defined. Reflexive writing is one type of mathematical writing, and it is formed with (1) student's own answer or solution, (2) ideas or comments from Second Personal Others, and (3) comments from First Personal Other. Because the reflexive writing activity and student's learning may develop their mutual interaction, the nature of their relation is reflexive. Also, reflexive writing is a reproduction of the class activity, which is produced from each student's viewpoint, it is the reflection of each class activity, and the nature of their relation is also reflexive. Furthermore, the social aspect of mathematics learning, especially the concept of "Others" is examined. The three types of "Others" in the learning environment are identified, and a framework of learning, "Reflexive Writing Activity", which consciously emphasizes the existence of "others", is presented. Moreover, one type of learning method in this framework, "Characters Method" has put into practice for some implications. In conclusion, Reflexive Writing Activity is not only the effective learning method which is helpful for both knowledge-understanding and the rich-fruitful learning environment, but also the aims of mathematics learning which should be regarded as one of the fundamental abilities for students.

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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, Holland, 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. At the present we have carried out three tests, i.e. Test 1, Test 2 and Test 3 to the about 500 children of first cohort for two years. The purpose of this paper is to analyze the data of these tests, to investigate children's progress of mathematical attainment and to present a way of seeing the fixity of mathematical attainment in order to find out some suggestions for improving the teaching and learning of mathematics at these primary schools. First, 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 found out there were five different types of [H→H ], [M→H], [L→H], [L→M] and [L→L] based on the progress of each test item from Test 1 to Test 2 or from Test 2 to Test 3. For example, the type of [H→H] means that for those test items in this type children had done well at the first test and did so at the second test a year later. The type of [L→H] means that for those test items in this type children had not done well at the first test and became to be well at the second test a year later. It reflects a positive effectiveness of the teaching and learning of mathematics for one year. Using these types, we found out that the teaching and learning of mathematics at the first grade was more effective than that one at the second grade in these schools. Second, 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. We found out that four items in Test 1 were insufficiently fixed among children and suggested that more efforts should be made in the teaching and learning of mathematics related these items.

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In this paper we describe the relationship among the creativity power, the creativity attitude and the scholastic achievement of mathematics learning. The experiment involved 80 participants from the first grade junior high school students. For the creativity power, we made a new creativity test of "the plane figure" and used the test score. For the scholastic achievement, we used the test score with ""the plane figure". For the creativity attitude measurement, we used a Creativity Attitude Scale (CAS) that was developed by Saito, N. We examined the correlation coefficient and the causal relationship among the creativity test score, the creativity attitude scale score and the scholastic achievement test score. The results are as follows: - The correlation coefficient between the creativity test score and the scholastic achievement test score is middle. - The correlation coefficient between the creativity test score and the creativity attitude scale 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 the diffusion, the fluency and flexibility of factors that composed the creativity attitude get higher.

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In this paper we describe the relationship among the creativity power, the creativity attitude and the scholastic achievement of mathematics learning. The experiment involved 130 participants from the second grade junior high school students. For the creativity power, we made a new creativity test of "the linear function" and used the test score. For the scholastic achievement, we used the test score with "the linear function". For the creativity attitude measurement, we used a Creativity Attitude Scale (CAS) that was developed by Saito, N. We examined the correlation coefficient and the causal relationship among the creativity test score, the creativity attitude scale score and the scholastic achievement test score. The results are as follows: - The correlation coefficient among the creativity test score, the creativity attitude scale score and the scholastic achievement test score is very low. - There is a casual relationship among the creativity test score, the creativity attitude scale score and the scholastic achievement test score i.e. the students that get higher creativity attitude scale score also get higher the creativity test score and the scholastic achievement test score. However it does not occur conversely. - As for the students that get higher creativity attitude scale score, the scores of the diffusion, the fluency, the flexibility and the originality of all factors that composed the creativity attitude get higher. - As for the students that get higher score of the flexibility, the scores of the diffusion, the fluency and the originality of factors that composed the creativity attitude get higher. - As for the students that get higher score of the originality, the score of the diffusion, the creativity attitude scale score and the scholastic achievement test score get higher.

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The purpose of this study is to develop the instruments of measuring pupils' potential ability for mathematics learning. Through the pilot tests and discussion, we elaborated an instrument composed of 20 problems which can be categorized into five components: logical reasoning, pattern recognition, manipulation, flexible thinking and application. By using this instrument, we investigated the potential ability and achievement of lower secondary school students: 73 1^<st> graders (A school), 62 1^<st> graders (B school), 73 2^<nd> graders (A school) and 59 2^<nd> graders (B school). As a result, we could find the following points. 1) The average of potential ability of different school students were almost same. This point suggests the validity of this instrument. 2) The 2^<nd> graders' potential ability was remarkably higher than that of 1^<st> graders of the same schools. This point suggests that potential ability can be advanced by their learning. The advance about the components of logical reasoning and application was more remarkable than that of flexible thinking. 3) The average of achievement of different school students were very different. Because the difference of potential ability was not so remarkable, we could grasp the difference of the effect of mathematics teaching. 4) The difference (achievement-otential ability) and the proportion (achievement÷potential ability) can be introduced as an index of the effect of mathematics reaching.

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One of the perspectives to analysis mathematics classroom is to investigate teacher's metacognitive activities. The purpose of this study is to construct the framework of teacher's metacognitive activities underlying mathematics teaching and to analyze the characteristics of teacher's metacognitive activities. Then it is able to create the program of teacher education on the basis of teacher's metacognitive activities. For the purpose of this study, this article proposed the framework to investigate teacher's metacognitive activities, on the basis of Artzt & Armour-Thomas (2001). To put it concretely, firstly teacher's metacognitive activities were compared with metacognitive activities on mathematical problem solving. So teacher's metacognition has two aspects, metacognitive knowledge and metacognitive skill. Especially this article focused on teacher's metacognitive skill that is consisted of monitoring, self-evaluation and regulation. Secondly, it makes clearer understanding about teacher decision making to investigate teacher's metacognitive activities underlying mathematics teaching. Using the framework to investigate teacher's metacognitive activities, two mathematics instructional practices were analyzed. As the results of these practices, the following metacognitive activities were founded. The teacher monitored students' comments and activities, reflected some contents that students' had already learned.

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As L. S. Shulman (1986) have noted, there are content knowledge, pedagogical content knowledge, and curricular knowledge in teacher's knowledges. For Shulman, content knowledge is knowledge about the subject, for example mathematics, and its structure. Pedagogical content knowledge includes' the ways of representing and formulating the subject that make it comprehensible to others' and' an understanding of what makes the learning of topics easy or difficult; the concepts and preconceptions that students of different ages and backgrounds bring with them…' Finally curricular knowledge encompasses what might be called the 'scope and sequence' of a subject and material used in teaching. In this paper, I describe the knowledges in mathematics teacher, in paticular pedagogical content knowledge and a way to overt them. Pedagogical content knowledge is made up of several components; knowledge about the subject-matter, knowledge about students and knowledge about ways of presenting the subject-matter. Because they value quality above quantity, I concentrate on the relationship that knowledge about ways of presenting the subject-matter is due to knowledge about the subject-matter and knowledge about students. Shulman (1986) distinguishes between two kinds of understanding of subject-matter that teachers need to have - knowing "that" and knowing "why". I think that all three pedagogical content knowledges have two sides, knowing that and knowing why. To ask mathematics teacher a question on children's responses and subject-matter overt his or her knowledge about ways of presenting the subject-matter. After all knowledge about ways of presenting the subject-matter is due to 'knowing why' side of two other knowledge. The ways to overt pedagogical content knowledge in mathematics teacher connote the following three necessary condition; (1) to create conflict condition (2) to encounter children's errors and misconceptions (3) to be haven a keen awareness of a context in knowledges.

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The purpose of this paper is to study how Tsuruichi Hayashi (1873-1935) understood the thought of function and the education of function. Hayashi considered that, for changing the exsisting mathematics education, we had to depend on the modern principle and accommpish a new mathematics education. He thought that culutivating the concept of function was a driving force of the revolution, and that it had to be generally done in all places through all of the mathematics education. He insisted that the use of graph was very important as a usufull way for culutivating the concept of function. He regarded elementary caluculas as the development of the concept of funntion, and he thought that the speed was represented as a differential coefficient. We can classify the teaching contents in his texts into five periods, in the followings. In the first period, from 1907 to 1912, the education of function has been hardly given to students. In the second period, from 1913 to 1921, the education of function began to be given to students. But, the teaching contents were not yet enough. In the third period, from 1922 to 1924, the teaching contents have been enriched. Especially in geometory, the figure elements were regared as movable. In the forth period, from 1925 to 1930, the form of the teaching contents was arranged a little. In the fifth period, from 1931 to 1935, the order of the teaching contents has been standardized. But, elementary caluculas was not taught in any period.

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The purpose of this paper is to discuss more effective method of mathematical problem posing by using computer. The practice of problem posing is for some university students who want to become a teacher in the future. A feature of the method is to provide situations in which students make conjectures on results and get the numerical calculation by using computer. And another feature is to give students enough time to create problems. Actually, in the practice, some students had difficulty in making problems by using computer, but many students showed much originality in their problems, which might not be made without using computer. We observed the active learning activities which might not be observed in the usual classes. It is asserted that the opportunity of problem posing by using computer is very important.

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The purpose of this research is to define the framework of systematic understanding of numbers, and to investigate systematic understanding of numbers under the framework. I looked back upon old government guidelines for teaching, and defined following framework of the systematic understanding of numbers. 1. Understanding of the structure of a set of numbers ・Understanding about a calculation rule ・Understanding of the operation in a set of numbers ・Understanding of the linear order relation in a set of numbers, and understanding of density of a set of numbers 2. Understanding of a relation between sets of numbers ・Understanding of a inclusion-relation of sets of numbers 1 and 2 are complementary rather than independent. Under this framework, I investigated the degree of achievement of systematic understanding of numbers and obtained the following results. ・The gap on a concept exists between real numbers and complex numbers. ・The image to numbers and the definition of numbers are not in agreement. ・Since expression of numbers and the concept of numbers are not distinguished, understanding is ambiguous. A future subject is to consider systematic instruction of number concepts in school mathematics on the basis of these results of an investigation.

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We have argued about the formalization of the locus of Apollonius using analytic geometry in the articles). It can be expressed in two ways as follows: one is done by means of finding out "a center and a radius" of the circle determined by the locus of Apollonius; the other by finding out "both ends of the diameter" of its circle. We presented three typical methods on determining "both ends of the diameter" of the circle in the elementary geometrical style. Besides, we thought over the significance of teaching the formalization of the locus of Apollonius in senior high school. It is usual, when dealing with this locus in senior high school, to decide "the center and the radius" of the circle using analytic geometry. But it is always the case to find out "both ends of the diameter" of the circle using elementary geometry. We scarcely find how to decide "the center and the radius" of the circle using elementary geometry. Can it be possible to treat the method in senior high school teaching? This question motivated us to write this article. This article aims at making clear the followings: (1) to construct any point satisfying the condition of the locus of Apollonius in the elementary geometrical style, (2) to construct "the center and the radius" of the circle of Apollonius in the same way, (3) to present some properties of the circle of Apollonius. We described five individual methods on the construction of "the center and the radius" of the circle using elementary geometry, in the second part of this article. This is an important part of this article. Especially, the fourth method of this construction is equal to a means of finding out "both ends of the diameter" of the circle in the elementary geometrical style. The circle of Apollonius is one of the most excellent materials of teaching geometry in senior high school.

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In this paper, we show an aspect of the winding numbers of closed plane curves as a developmental instructional material. Our assertion is that to examine the winding numbers of closed curves gives us a good chance to experience mathematical activities including mathematizing intuitive rotations of closed curves, observing various properties of winding numbers, expecting some laws on them from the observation, verifying such expectations and generalizing results. As a model of practical treatment on the material, we describe a process to induce a relation between the winding numbers, a relation that the winding number of a closed curve is equal to the sum of winding numbers of complementary domains of the closed curve minus the appropriately defined weights of the intersection points. The treatment of the winding number as an instructional material will reinforce a recognition of the importance of orientations on geometrical figures.

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