In this paper I argued a neccesary of a paradigm shift from the mechanistic and atomistic (MAME) to the active and creative Mathematics Education (ACME). In MAME teaching materials are divided in detail and teaching is done to make stimulus-response patterns. Then children have to learn mathematics passively. On the contrary in ACME children learn actively, spontaneously and cooperately. They are doing mathematics. They are involved in relation-fulled situation, find or pose several problems from it and solve them by thierselves and cooperately. Then they develop and expand them. According to management theory, ACME is primary and MAME is secondary. When children are forced in MAME situation over a long time, they form unuseful knowlegdes and skills and arm aptitudes. In fact we have seen many children who dislike mathematics and can't find a value of mathematics. I think the following things are indispensable. (1) Unifying teachnig materials for 2 years (2) Constructing teaching materials around basic mathematical ideas (3) Reorganizing textbook suitable to ACME (4) Improving a teacher training system (5) Improving an entrance examination system

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Although radical constructivism has its proper intention and effectiveness in mathematics education, it has the crucial problem of solipsism, which could not develop leaving the principles original. The researchers taking sociocultural perspectives in constructivism to solve the problem are called complementarists. But Lerman criticizes them, arguing about intersubjectivity in mathematics learning from sociocultural perspective with Vygotskian theory. No doubt his discussion is logical especially about intersubjective problem. But constructivism as the educational view point can't be excluded because it inherits the naturalism philosophy of education. For example the autonomy or creativity and enculturation or sociocultural practice should be balanced in mathematical education, even if they coincide or confront. Of course it is better that theoretical program in research would have no contradiction. But we can't avoid the inconsistent principles between constructivism and sociocultural approach and simultaneously the approach with them for the sake of their usefulness in mathematical education. I support complementarity from the view point of practice and theory in mathematics education, which Cobb suggested as a theorical pragmatism.

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Wittgenstein pointed out that the idea "the rule is understood by the interpretation" is misunderstanding. And, he insisted that we should admit the existence of the grasp of the rule which is not interpretation. So we can do some acts by following the rule not because of consisting of specified rule of the game, but just because of doing the practice customary based on no ground, that is without hitting other possibilities. It is not the agreement of the interpretation, but the agreement of the judgement that is essential to play the language game and to talk with each other. From this point of view, the customary practices is essential to discussion and it is necessary to form the mathematical customs before the mathematical discussion. So we should reconsider the discussion in mathematical class from this respect, and ask "Is the mathematical discussion; the interpretation game that should be based on the agreement of the judgement, actually made in the mathematical class ?". It is most important educational task for mathematics teachers of the lower classes in primary school to form some customary practices or techniques that are more useful for children to learn further mathematical activity. Therefore activities of concrete operation of matters or drawing pictures or figures are given priority to busy discussion or calculation drill or detailed explanation, especially in the primary school lower classes.

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Both critical mathematics education and ethnomathematics can be interpreted as the reactions to the modernization theory. The former protests within the developed countries against the exogeneousness of development supported by the theory. And the latter within the developing countries questions the applicability of monistic view of development which the theory implies. However, both criticisms beyond their peculiarity may be deeply rooted in the problems which each society suffers from, that is to say, the critical view of the controlled majority toward the controlling minority and of the prejudiced minority toward the innocent majority. These terms, majority and minority, here are relative concepts and they can contain majority-minority relation at the other level within themselves. Ethnomathematics so far has gained a certain level of momentum in the developing countries, and some developed countries with multicultural background as well. In order to connect it to school mathematics as an educational endeavour, the complementary relation between critical mathematics education and ethnomathematics has been deliberated in both way. After this both way of consideration, the importance of objectization of mathematical activities is pointed out and the foundational framework has been proposed to fortify ethnomathematics as a research. (1) to reflect critically mathematics education through ethnomathematics a) mathematical implication b) social implication (2) to reflect critically ethnomathematics by means of critical mathematics education c) mathematical implication d) social implication

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The aim of this paper is to adopt the cultural approach to mathematics curriculum, referring to "Mathematical Enculturation" written by A. J. Bishop. He has described three components of the curriculum, which labeled the symbolic, societal, and cultural components. Those have two complementary values. He presents a curriculum structure which allows 'rationalism' to be stressed more than 'objectism', where 'progress' can be emphasised more than 'control' and where 'openness' can be more significant than 'mystery'. The societal and cultural component are necessary to adopt exemplifying to historical development of knowledge, which offer an individualising aspect of teaching. On the otherhand, the symbolic component generates concepts of mathematics through activities. I interpreted the significant on the cultural approach from a Vygotskian perspective in the following way. The symbolic component generates mainly an intermental category, and then the societal and cultural components generate an intramental category. Finally, I suggested as the cultural approach to the mathematics curriculum an alternative to the 'technique' curriculum.

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The study of creativity and creative thinking in educational/psychological research has been highly fashionable since the 1950s. Recently, in Japan, it is emphasized that mathematics education should contribute to help students develop their creativity and creative thinking in school mathematics for the 21st century. We can find some important suggestions for fostering students' creativity out from the results of research done within the subject of school mathematics, though we do not have enough information about such mathematical problems that could be used in teaching and learning of mathematics suited for students to develop their creativity and creative thinking. The purpose of this research is to identify and make clear such mathematical problems that could be used in teaching and learning of school mathematics in order to foster students' creativity and creative thinking from the view point of mathematics education. To do it, we first summarize main results of the research on creativity and creative thinking in the area of psychology. Second we think about our conception of students' creativity that should be fostered in mathematics education. Finally, basing on these considerations, we identify five different types of mathematical problem as follows and show two illuminating sample problems to each type with the key-question in each type of mathematical problem. Type 1: Mathematical problem of encouraging students to find as many mathematical patterns/relations as possible in a given mathematical situation Type 2: Mathematical problem of encouraging students to think as many ways as possible to a given mathematical task Type 3: Mathematical problem of encouraging students to have and represent as many images as possible in a given mathematical situation Type 4: Mathematical problem of encouraging students to understand different ideas represented by the given different mathematical representations Type 5: Mathematical problem of encouraging students to pose new and extended mathematical problems by changing some attributes of a given mathematical problem All types are related to divergent thinking in general, but each type has its own characteristic feature from the mathematical view point. The framework of these types could be useful and helpful for us in mathematics education.

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Similarity recognition is familiar in everyday life and everyone has the ability to recognize the similarity. But we don't make use of it in mathematical lessons. In this paper, we stated the relation of similarity recognition with understanding of mathematical knowledge and fostering of problem solving ability and mathematical thinking in the first place. According to the psychological research, commonalities and differences affect the recognition of similarity and there are two types of differences: those connected to the commonalities (called alignable differences) and those not connected to the commonalities (called nonaliginable differences). We indicated that it is important for mathematics learning to recognize the alignable differences as similarity. Under such consideration, we proposed comparison lesson for making children construct the mathematical concepts. It is the lesson which children construct the mathematical concepts by recognizing the similarity in comparison pairs. We practiced some comparison lessons that were formed by 1-pair, 2-pairs and 3-pairs comparison. It has been shown by these comparison lessons that 2-pairs comparison is more effective for the construction of mathematical concepts. But it has been shown also that it is not so easy for children to recognize the similarity in 2-pairs comparison. So, we considered some reasons for it.

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The Purpose of this study is to make Vygotsky's Theory clear. Especially Zone of Proximal Development. And to investigate the adaptation of Vygotsky's Theory for mathematics education. In this paper, I examine about how to understand Vygotsky's Theory, and understand the learning guidance in mathematics course, in accordance with the basical study of Vygotsky's Theory (Yoshida, 1997a; Yoshida, 1998), the comparative study between Vygotsky's Theory and the succession Approaches (Yoshida, 1997b). I construct a process of learning on mathematical concepts based on Vygotsky's Theory. The results are following; 1. A child has the cognition of an object through his/her experience. 2. He/She comes across a new mathematical concept. 3. He/She has the cognition of the gap between spontaneous and scientific cencepts. 4. He/She exchanges the concept with society, for example, adults, more expert children, books and so on, by mathematical languages. 5. He/She exchanges the concept with his/herself by mathematical languages. 6. He/She has get the mathematical languages and/or mathematical concepts.

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In this article, the conception and the roles of the variable "OG/NOG (Ongoing/Not Ongoing)" which is the component of the framework of this study are examined. Firstly, the notion of "the change of goal", which is an important viewpoint when we think of this variable, is examined making relation to the preceding studies. Then, the variable "OG/NOG" is elaborated to "Previous Goal/Current Goal/Concurrent (PG/CG/CCR)". "Previous Goal Self-Referential Activity (Previous Goal SRA)" is such a self-referential activity that the solver refers to the processes or products in problem-solving phases based on previous goal. "Current Goal Self-Referential Activity (Current Goal SRA)" is such a self-referential activity that the solver refers to the processes or products in problem-solving phases based on current goal except for referring to them just concurrently. "Concurrent Self-Referential Activity (Concurrent SRA)" is such a self-referential activity that the solver refers to the emerging processes or products just concurrently. Finally, it is examined about the roles and significance of the three types of self-referential activity which are prescribed by the elabrated variable. For example, based on an analysis of a problem solving process, it is suggested that the Previous Goal SRA plays a role "possibility of recovery" in problem solving process.

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The research focus of this study is to explore the ways in which teachers can make their everyday teaching more fruitful from a metaknowledge perspective. For the study, qualitative, hermeneutical and micro-ethnographical case studies are utilized. Two teaching episodes (episode I and episode II) are analyzed from an epistemological point of view. The two episodes are taken from an 8th grade mathematics classroom; in both episodes, the context is the teaching of basic triangle congruence theorems. Episode I is a transcript from a standard classroom setting; episode II is a transcript from a similar setting but in a co-operative teaching experiment, which seems more fruitful from a metaknowledge perspective. An epistemological triangle is used as a theoretical perspective to analyze the two episodes. It is not fixed but is kept flexible to enable further improvement of the analysis. That is, there is an interactive relationship between the developing epistemological triangle as a theoretical perspective and the interpretation of the two episodes, especially the differences between them. Analysis of the two episodes from the perspective of the epistemological triangle reveals that in episode II there are clear interactions between developing the students' reference context and developing the students' symbol system. However, there are no such interactions in episode I. It becomes more clearly from a metaknowledge perspective how do the differences between them influence the students' mathematics learning and is episode II more fruitful. It is observed that the teacher's instructional approach in episode II is a complete contrast to the teacher's approach in episode I. The teacher in episode I attempted to teach the new symbol system directly through language without developing the students' reference context. The teacher in episode II attempted to create a productive 'tension' between the symbol system and a structural reference context and to develop and maintain this tension (Steinbring, H. 1997, p.79). To achieve these goals, the teacher in episode II attempted: (1) to try to understand the extent of the student's reference context and work on it by presenting geometrical constructions, which would put the students' reference context off its balance (2) to encourage the students to formulate their own conceptions of the reference context structure (3) to introduce a 'free element' into the subject matter, where the word "free' means the teacher is no "obligation for action', which is typical of institutionalized educational processes (Bauersfeld, H. 1988, p.37). These three actions are significant elements in fostering the interactions between developing the students' reference context and developing their symbol system.

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The purpose of this study is to explore experimentally the role of metacognition in mathematical problem solving. The present paper aims to investigate the metacognition which forth and fifth graders could use during problem solving. For this purpose, the author proposed four types of metacognitive activities; 『device』, 『check』, 『correction』, 『attention』. The metacognitive activity of 『device』 is to determine what to do in order to solve the problem very well. The metacognitive activity of 『check』 is to look at his/her activities. The metacognitive activity of 『correction』 is to reflect his/her own activity, to give up it, and to have a new that. The metacognitive activity of 『attention』 is to be on the lookout for his/her own activity. The method of this investigation is that all pupils individually have to solve the four problems on the work-sheet and answer the metacognitive questionnaire. The way of analysis is to mark at pupil's work-sheet and to identify his/her metacognitive activities. The main findings of this investigation are the followings: (1) As the pupil became older, the number of his/her metacognitive activities (in particular, 『device』 『correction』 『attention』) increased. (2) The number of pupil's metacognitive activity and marks at his/her work-sheet were positive correlation. (3) Pupils who marked high scores at their work-sheets, the number of their metacognitive activities the of four problems were almost positive correlation. Pupils who marked middle scores at their work-sheets, the number of their metacognitive activities the of four problems were almost negative correlation.

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The purpose of this study is to indicate how student self-assessment and self-evaluation are effective methods for students in mathematical problem solving. The study outlines the differences between the functions of self-assessment and self-evaluation. This research concludes that there is a different degree of effectiveness between self-assessment and self-evaluation, in which self-evaluation is more effective method for students in problem solving. In this paper, I define "reflective self-evaluation" as: (1) Students reflect on their own work after problem solving, and (2) Students assess and evaluate their own problem solving work. This study presents two examples of the "reflective self-evaluation". One example consists of students comparing their own solution with other solutions. The second example, the students assess and score their own problem solving work. This research indicates that "reflective self-evaluation" is more effective method for students in mathematical problem solving.

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The purpose of this paper is to examine some factors that influence access to knowledge during mathematical problem solving. In the preceding study, three major factors of mathematical knowledge-access difficulties were identified. Especially, it is important to improve management of the problem solving process such as planning, rereading and monitoring for accessing to weakly linked knowledge. I indicated that it is important to make students be aware of accessing to knowledge. By trying this, teacher could be able to make students understand the importance of reading a problem well and planning.

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The small group learning is taken as peer collaborative learning that bring out the various verbal interactions that result from students' active discourse. According to Artzt & Armour-Thomas (1996, p.118), "Small problem-solving groups serve as natural settings for discussion in which the interpersonal monitorings and regulating of members' goal-directed behaviors occur." The purpose of this article examine the cognitive processes of mathematical problem solving in small group learning in connection with those metacognitive behaviors, using the researches of Artzt & Armour-Thomas (1992, 1996, 1997). Firstly, a time-line representation of protocol was produced by using the protocols of their research (1996). It is indicated that this time-line representation of protocol is similar to the one that Schoenfeld had provided as expert problem solver's. Secondly, it was illustrated that the monitoring and control of metacognitive skills serve as interpersonal function in group settings, and they contribute to the successful mathematical problem-solving. The findings from the analyses is that the cognitive process of mathematical problem-solving in small group learning has the following features: 1) Many interpersonal monitorings and controls of peer group member's occur through the dialogue space that makes verbal interactions possible. 2) The various heuristic episodes continuously occur in group learning, so that mathematical problem is solved successfully.

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　　In this paper, I analyze a process of knowing a theorem through some kinds of concrete operations, like action proofs. For this sake, I intend to use the theory of excluding the third term, invented by Imamura, Hitoshi, a Japanese philosopher.

[fig. 1]

[fig. 2]

　　Consequently, this process consists of four phases.
　　We can display each phase by a kind of diagrams. In this diagram, O(a), O(b), etc mean operations on concrete objects a,b,etc. And O'(a), O'(b), etc mean how to manipulate concrete objects. Then, the fig.1 mean that O(b) represents O(a).
　　The first phase can be pictured like fig.2.
　　This diagram means, if O(b) represents O(a) then the manipulating way of the object b is represented by O(b) being able to represent O(a). It seem natural that we transform the fig.2 to fig.3. These two figures will display the first phase. And the second phase, pictured by the fig.4,

[fig. 3]

[fig. 4]

where only two manipulations O(a) and O(b) are pictured. In this diagram, there are no distinctions between O(a) and O(b). And all objects are lacking distinction.
　　We can draw fig.5 for the third phase.
　　In this phase, one object "a" is picked up in order to represent the other manipulates O(b), O(c), O(d), etc. And the same time, each manipulate can represent O'(a).
　　Finally, the fourth phase, we can picture like fig.6. In the diagram, O(*) mean the form of O(a), O(b) etc that is the manipulating way itself. Usually, we give expression to this O(*) through a kind of algebraic symbols etc.
　　And then, we can obtain a kind of formal proofs.

[fig. 5]

[fig. 6]

　　In these phases, there exist at least two problems. The first, whether it is natural to transform to fig.3 from fig.2 or not. The second, what is the condition to be able to transform to fig.6 from fig.5.
　　We have to do further studies on these problems.

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Mathematics teachers want to educate their student's logical thinking abilities. However there are many obstacles to the aim because of some characters concerning to mathematics teaching: teaching method, textbooks and curriculum. I clarified the existence of teaching obstacles and its reasons. Also I suggested how to deal with teaching obstacles. Especially from this study I proposed the following. *Statistics should be taught in another subject than mathematics. *We should make a mathematics curriculum which contains educating logical abilities as a practical aim.

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I would like to focus attention on linguistic representation of the concept of geometry. And the purpose of this research is to clarify the meaning of linguistic representation from the viewpoint of epistemology. For this purpose I considered two issues here. 1. What are the characteristics of linguistic representation used in the teaching of geometry? 2. What kinds of problems occur for the reason of the characteristics? There are five types of representation of the concept of geometry: realistic, operational, figural, linguistic and symbolic represantation. And the linguistic representation is a representation which is used natural language. One of the representaional style is "term", that is to say the names of the concept of geometry and the other style is "sentence", that is to say definitions, characteristics and propositions. Because of the rules of natural language an interpretation of geometrical representation is influenced frequently. Characteristics of terms: ・Naming the concept of geometry is not only labeling the concept but also identifying the object. ・In addition the name of the concept is not a lexicon definition but should be grasped as a characteristic list. Characteristics of sentences: ・Sometimes lack of rigidity of a mathematical sentence may be revealed due to the daily interpretation of natural language. ・The representations of definitions reflect the hierarchy of the concepts of geometry. ・To interpret the meaning of sentences we have to understand lexicon understanding about the terms, and syntactic and semantic understanding of the sentences. As a final point, I should emphasize that it is necessary to make children become aware of the rigid aspects of mathematical representation even though it is described by natural language.

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There have been a little epistemological interests in the phenomenon of inconsistencies in the concept of geometry. So the purpose of this research is to clarify the characteristics of inconsistencies in the concept of geometry. For this purpose I have to consider here three main issues. 1. What are inconsistencies in the concept of geometry? 2. What are the factors of the inconsistencies? 3. What are the functions of the inconsistencies in the teaching and learning of geometry? The term "inconsistence" is used as the situation that if p is a proposition, then both p and〜p hold simultaneously. And inconsistencies in the concept of geometry can be classified into two kinds of types. One kind of type is from a viewpoint of obuject of inconsistencies. External inconsistencies: inconsistencies between the student's concept of geometry and mathematical concept of geometry. Internal inconsistencies: inconsistencies within the student's concept of geometry. The other kind of type is from a viewpoint of student's awareness. Explicit (cognitive) inconsistencies: student is aware of inconsistencies. Inplicit inconsistencies: student is not aware of inconsistencies. External inconsistencies have been called "conflict" or "misconception" and explicit inconsistencies have been called "cognitive conflict" or "disequillibrium". External inconsistencies are easily recognized by observers because mathematical varidity of the student's concept of geometry is judged on the basis of abstract mathematical concept of geometry. According to the representational model of concept of geometry, the inconsistencies can be interpreted. For example the definition of rectangle can not be written because of the lack of meaning of linguistic representation about rectangle and rectangle is recognized as long shape because of the effect of figural representation about rectangle. On the other hand internal inconsistencies are not always recognized by both observers and students. According to the representational model, internal inconsistencies are considered as the difference between linguistic representation and imaginary representation. Furthermore according to the aspect model of understanding on individual concept of geometry, internal inconsistencies are described as the inconsistencies between the aspects of understanding. For example definition of rectangle is written by use of linguistic representation but rectangle shape is identified by use of imaginary representation. There are many factors that interpret inconsistencies: natural language, meaning of definition, linguistic interpretation, prototype judgement and generalization of figure for external inconsistencies and the difference of cognitive operation, the awareness of definition and the context-bound nature for internal inconsistencies. What is significant in the teaching and learning of geometry in the light of inconsistencies is to make student become aware of the inconsistencies, that is to exchange from implicit inconsistencies to explicit inconsistencies. Then on the basis of the facters of inconsistencies they can dissolve the inconsistencies. It is the conscious of teachers on the student's individual concept of geometry that needs to be reformed.

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　　The purpose of this study was to analyze the structure of childrens' own concept. Children in this study included 50 fifth graders who had received instruction about trapezoids during the previous year. One group, A, could not write the definition and the second group, B, could do this. Two methods of analysis, a conviction degree rating scaling method and a protocol, were applied to the data. The analysis of the data from group A (i.e. childrens' own concept) was as follows.
　　1. There exists 'a set of trapezoids', that was constructed with a continuous change distribution based upon the probability representation.
　　2. Every trapezoid was arranged according to a conviction degree rating in order.
　　3. When any instance was presented, it was neither recognized as a snap shot nor discriminated by definitive representation. It was recognized as a continuous change distribution by probability representation.
　　4. When children discriminated presented trapezoids, they retrieved the reference that was 'a set of trapezoids' that consisted of well-experienced instance, focused instance studied at the first teaching, and geometrical figure which was similar to trapezoids. Therefore, they may have mistaken the relevant instances.
　　In contrast to the concept of group A, the analysis of the data from group B indicated that whether or not a presented instance was perceived as a trapezoid was based upon definitive representation.

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The purpose of this study is to establish a teaching principle based on the duality on the recognition of mathematical concepts. Inoue (1997) theoretically considered the principle. This paper aims to verify the validity and the application possibility of this principle by means of teaching experiment of function. As a result, it is suggested that the teaching in line with the principle promotes the structural conception of students in learning of function. Through the teaching experiment, the following are found as noteworthy point in applying the principle to one mathematical concept; (1) Making mathematical concept clear which is needed for students to acquire. (2) Certifying the operational foundation of students at the stage of objectification. (3) Taking learning for "operational-structural" repetition.

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In learning of mathematics, writing mathematics and speaking mathematics are very important in the process of solving the problems. Because students ought to comprehend of the learning contents through activities of writing mathematics and speaking mathematics. This study deals with the following: 1) The development of a new index to measure the change between the power of writing and the power of speaking in the process of solving the mathematics problems. 2) The relation between the power of writing and the power of speaking in the process of solving mathematics problems. We found a new index which was divided the value of the slope of a regression line by using the cumulative marks of the tests by allotment of marks of the tests. We call this new index the coefficient of variation of the cumulative marks. We experimented on the power of writing mathematics and the power of speaking mathematics for the junior high shcool students grade 2. As a result, the following became clear: 1) The value of the correlation coefficient between the marks of writing mathematics and the marks of speaking mathematics is very large. 2) Generally, the power of writing mathematics is higher than the power of speaking mathematics.

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This is the third report of the series of studies, which are based on potential test, topic tests, and four kinds of questionnaire developed by KassEx Project. The purpose of this research is to make an investigation into pupils' progress of mathematical ability at secondary school level, to appreciate the factors that enhance or hinder mathematics teaching, and to make recommendations for improvement of mathematics teaching curriculum. In this paper, we review the results of Year 1 and Year 2 potential test done by the same pupils in Japan for the cross-sectional comparison between European countries and Japan, and the longitudinal comparison between Year 1 and Year 2 in Japan. The main results are as follows: In the longitudinal comparison, 1) the progress of potential ability was statistically significant. Learning mathematics during one year after the Year 1 test could cause the development of it. 2) But the reliability of potential test was confirmed by cross table concerning the ratio of correct answer to each question. And in the cross-sectional comparison, 3) the potential ability of Japanese pupils is higher as a whole than that of British and German pupils. The results of Japanese pupils on three questions out of 26 ones, however, are worse significantly than that of British and German pupils. Answers of those questions could be got by try and error. Pupils in Japan might be inferior to pupils in Britain and Germany in intellectual toughness.

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The first aim of this study is to explor the professionalism of teachers in mathematics teaching and, based on it, to refer the ideal mathematics teachers. And the second one is to make some proposals on preservice and inservice teacher education programs. To explor the professionalism of teachers in mathematics teaching, I considered both social and individual aspect of mathematics teachers. In sosial aspect, I searched some studies and I considered six characteristics of teaching profession. The professionalism of teaching profession is that teachers perform their duty through being conscious of these six characteristics. In individual aspect, I assumed that the individual professionalism of teachers would appear in style of thinking and performing. And I compared expert teachers with novice teachers by searching some referentional studies so that I could extract the individual professionalism of teachers through drawing out the characteristics of expert teachers. Experts are proficient in complex cognitive skill and improvisational performance. They are sperior in reading a classroom and making decision than novices. The ability on reading a classroom and making decision is based on their knowledge that is relevant to subject matter, pupil and student, and teaching and management. Experts are excellent in all these respects. Hence the ability on reading a classroom and making decision is a significant component of the individual professionalism of teachers in mathematics teaching. The individual professionalism of teachers in mathematics teaching is proportion to a wealth of expertise. Therefore it is very important to pursue knowledge and to improve teacher's ability on reading a classroom and making decision in preservice and inservice teacher education programs, because a teacher who has a wealth of expertise and who is sperior in reading a classroom and making decision is one of the ideal mathematics teachers. And a teacher who is autonomous is also one of the ideal mathematics teachers. To be autonomous is important for professional development. Summarizing them, I maked some proposals on preservice and inservice teacher education programs.

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The purpose of this paper is to assert the importance of mathematical thinking on orientation of geometrical objects. Our main contents are summarized as follows. H. Freudenthal once pointed out, with various reasons, that orientation of spaces and figures is significant in mathematical thinking, in his study of didactical phenomenology. We assert that the current curriculum has few materials which instruct students the efficiency of the positive use of orientation, although it contains many materials which are related to orientation. We investigate, through several questions to some university students, how they are conscious of mathematical thinking when they face some issues concerning orientation. As a result, we can admit some tendency that many students are unconscious of mathematical methods on orientation. Our next assertion is that mathematical thinking of orientation enables students to recognize polarities, as negative direction for instance. We offer some mathematical problems which are related to some 2 color's property of diagrams and might be difficult to be solved without any idea of orientation. We imposed those problems on the students in the above questions. As a result, we could only find some partial answers, but no complete solutions. It implies that mathematical thinking of orientaion gives students a good opportunity to experience the efficiency of polarities.

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The purpose of this paper is to consider the structure of number system in school mathematics and the possibility of constructive teaching, from a viewpoint of instructional contents in mathematics education. In the algebraic structure, the real number field is a extension field of raitional number field, and the complex number field is a extension field of real number field. But, when it thinks about the extension of field, real number field can't be finitely generated from the rational number field, though complex number field can be finitely generated from the real number field. This difference suggests that we must be careful in teaching of number system in school mathematics. Moreover, we must pay attention so that a student can compose mathematical knowledge when it thinks from the actual teaching. I thought about these points, and showed one idea about the teaching of the complex number at the high school. This idea is based on the finitely generation of a field, and it thinks to be effective in the constructive teaching of school mathematics.

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