The aim of this study is to analyze the relationships and propose the desirable relationships between mathematics curriculum and curriculum-introduction. Under the purpose of this study, we investigate the coherence of mathematics curriculum and curriculum-introductions which have been revised through the seven curriculum-revisions to the present. And we analyze the change in the education. On the basis on this analysis, we propose mathematics textbooks which can be said that have a strong impact on the practice of mathematics the important points that should be considered in the mathematics curriculum-revision. First, we need to reconsider the order of the mathematics curriculum-revisions which has been going on from curriculum-introduction to mathematics curriculum. Second, we should take more efforts in developing the mathematics textbooks and reflect the voice of the mathematics textbooks and reflect the voice of athematics leachers in the more positive way.

This study began with an epistemological question about the nature of mathematical cognition in relation to the learner's activity. Therefore, by examining Piaget's 'reflective abstraction' theory which can be an answer to the question, we tried to get suggestions which can be given to the mathematical education in practice. 'Reflective abstraction' is formed through the coordination of the epistmmic subject's action while 'empirical abstraction' is formed by the characters of observable concrete object. The reason Piaget distinguished these two kinds of abstraction is that the foundation for the peculiar objectivity and inevitability can be taken from the coordination of the action which is shared by all the epistemic subjects. Moreover, because the mechanism of reflective abstraction, unlike empirical abstraction, does not construct a new operation by simply changing the result of the previous construction, but is forming re-construction which includes the structure previously constructed as a special case, the system which is developed by this mechanism is able to have reasonability constantly. The mechanism of the re-construction of the intellectual system through the reflective abstraction can be explained as continuous spiral alternance between the two complementary processes, 'reflechissement' and 'reflexion'; reflechissement is that the action moves to the higher level through the process of 'int riorisation' and 'thematisation'; reflexion is a process of 'equilibration'between the assimilation and the accomodation of the unbalance caused by the movement of the level. The operational learning principle of the theorists like Aebli who intended to embody Piaget's operational constructivism, attempts to explain the construction of the operation through 'internalization' of the action, but does not sufficiently emphasize the integration of the structure through the 'coordination' of the action and the ensuing discontinuous evolvement of learning level. Thus, based on the examination on the essential characteristic of the reflective abstraction and the mechanism, this study presents the principles of teaching and learning as following; $\circled1$ the principle of the operational interpretation of knowledge, $\circled2$ the principle of the structural interpretation of the operation, $\circled3$ the principle of int riorisation, $\circled4$ the principle of th matisation, $\circled5$ the principle of coordination, reflexion, and integration, $\circled6$ the principle of the discontinuous evolvement of learning level.

In this paper, we tried to establish the concept of 'Mathematics for Elementary Teachers(MET)'. There are 4 kinds of Mathematics for Teachers(MT). MET is one of tried to establish the concept in contradistinction to school mathematics(SM), mathematics, and Teaching Materials(TM). We suggested the outline of MET by suggesting the parts to which SM, MTs. The concept of MET is established variously according to various views. Here, we mathematics, and TM can not approach.

This paper explored the impact of dynamic geometry software such as CabriII, GSP on student's understanding deductive justification, on the assumption that proof in school mathematics should be used in the broader, psychological sense of justification rather than in the narrow sense of deductive, formal proof. The following results have been drawn: Dynamic geometry provided positive impact on interacting between empirical justification and deductive justification, especially on understanding the necessity of deductive justification. And teacher in the computer environment played crucial role in reducing on difficulties in connecting empirical justification to deductive justification. At the beginning of the research, however, it was not the case. However, once students got intocul-de-sac in empirical justification and understood the need of deductive justification, they tried to justify deductively. Compared with current paper-and-pencil environment that many students fail to learn the basic knowledge on proof, dynamic geometry software will give more positive ffect for learning. Dynamic geometry software may promote interaction between empirical justification and edeductive justification and give a feedback to students about results of their own actions. At present, there is some very helpful computer software. However the presence of good dynamic geometry software can not be the solution in itself. Since learning on proof is a function of various factors such as curriculum organization, evaluation method, the role of teacher and student. Most of all, the meaning of proof need to be reconceptualized in the future research.

In this paper, we tried to find out what makes the students feel anxious to mathematics and the ways to decrease their anxiety by comparing two learning types, self-directed-learning and learning in a body, by means of continuous observation and interviews. To perform this study, two classes of self-directed-learning and other two classes of learning-in a body were chosen from the third year-students in Wando Middle School in Chollanamdo. In this study, we obtained the following results: 1. In high group in math grade, students in self-directed-learning are less anxious than students in learning-in a body. 2. In average group in math grade, students in self-directed-learning are much more anxious than students in learning-in a body. 3. In low group in math grade, both students in self-directed-learning and students in teaming-in a body feel anxious about math and there is no difference between them. 4. Anxiety about math hove positive influence on high group in math grade. 5. Anxiety about math have negative influence on average and low groups in math grade. Especially, low group students had no interests about mathematics because of their math anxiety. We observed that some students got over the math anxiety to some meaningful extent by means of interviews or appropriate advices, and became to have confidence and interests in mathematics.

This study is to develop a mathematics assessment framework based on the mathematics assessment framework and content strands suggested by KEDI, NCTM, NAEP, TIMSS, Oregon State, New Zealand. According to the literature review, there has been more emphasis that students themselves 'communicate' what they 'understood' and how they 'thought' during the situation of 'solving problems'. As a result, communication ability is considered one of the most important factors in assessment situation, which always accompany the abilities of understanding, thinking, problem-solving, etc. In conclusion, the framework related to mathematical knowledge consists of content and behavior domains. The content domain is categorized into 6 content areas of the 7th mathematics curriculum, and the behavior domain is divided into computation, understanding, inference, problem-solving, and communication.

The learning of mathematics happens in some situations. It is natural that students should learn mathematics in more appropriate situations. But, so far It has been hardly studied about concrete situation and milieu where math can be successfully taught. In today's math education, the situation of education as a external circumstance become realized more and more importantly with influence of open education. But they don't embody situation as an internal circumstance where the intrinsic concept of mathematics can be obtained. We started this thesis from this tried to answer it on the basis of Brousseau's question, have theory about the didactical situations. One of the purpose of this study is to understand the theory of didactical situations, which focuses on how we can elaborate situations which really make a mathematical notion function. In this study, It is attempted clarify some concepts of the theory of didactical situations. The other is to discuss about what the theory of didactical situations suggests us in math zeducation. The method of math teaching and learning and the teacher's role were discussed in the viewpoint of Brousseau's theory. Finally, We elaborated and presented some didactical situations which make the notion of the area of rectangle.