The notion of metaphor has been increasingly popular in research of mathematics education. In particular, metaphor becomes a useful unit for analysis to provide a profound insight into mathematical reasoning and problem solving. In this context, this paper takes metaphor as an analytic unit to examine the relationship between objectivity and subjectivity in mathematical reasoning. Specifically, the discourse analysis focuses on the code switching between literal language and metaphor in mathematical discourse. It is shown that the linguistic code switching is parallel with the switching between two different kinds of mathematical knowledge, that is, factual knowledge and mathematical imagination, which constitute objectivity and subjectivity in mathematical reasoning. Furthermore, the pattern of the linguistic code switching reveals the dialectical relationship between the two poles of mathematical reasoning. Based on the understanding of the dialectical relationship, this paper provides some educational implications. First, the code-switching highlights diverse aspects of mathematics learning. Learning mathematics is concerned with developing not only technicality but also mathematical creativity. Second, the dialectical relationship between objectivity and subjectivity suggests that teaching and teaming mathematics is socioculturally constructed. Indeed, it is shown that not all metaphors are mathematically appropriated. They should be consistent with the cultural model of a mathematical concept under discussion. In general, this sociocultural perspective on mathematical metaphor highlights the sociocultural organization of teaching and loaming mathematics and provides a theoretical viewpoint to understand epistemological diversities in mathematics classroom.

This study investigated the new revised Mathematics curriculums of elementary schools in Korea and Japan at the end of the 20th century. The comparison was made especially with revising direction, purposes, and contents of elementary school mathematics curriculum in both countries. I began by comparing and confirming the ratio of instruction hours of Mathematics to the total instruction hours of all the subjects at as whole. This comparison was done of the elementary and middle school mathematics. The next part of the study was to compare in detail the purposes of revised mathematics in elementary and middle schools of both countries. Particular attentions was paid to the important revised points of Japanese elementary school Mathematics. Finally, I concluded by comparing the contents of elementary school Mathematics of the two countries. New mathematic text books in both countries having been published by revised curriculum, puts the future task in comparing, in detail, the concrete contents of each textbook.

Given that the culture of the mathematics classroom has been perceived as an important topic in mathematics education research, this paper deals with the construct of sociomathematical norms which can be used as an analytical tool in understanding classroom mathematical culture. This paper first reviews the theoretical foundations of the construct such as symbolic interactionism and ethnomethodology, and describes the actual classroom contexts in which social and sociomathematical norms were originally identified. This paper then provides a critical analysis of the previous studies with regard to sociomathematical norms. Whereas such studies analyze how sociomathematical norms become constituted and stabilized in the specific classroom contexts, they tend to briefly document sociomathematical norms mainly as a precursor to the detailed analysis of classroom mathematical practice. This paper reveals that the trend stems from the following two facts. First, the construct of sociomathematical norms evolved out of a classroom teaching experiment in which Cobb and his colleagues attempted to account for students' conceptual loaming as it occurred in the social context of an inquiry mathematics classroom. Second, the researchers' main role was to design instructional devices and sequences of specific mathematical content and to support the classroom teacher to foster students' mathematical learning using those sequences Given the limitations in terms of the utility of sociomathematical norms, this paper suggests the possibility of positioning the sociomathematical norms construct as more centrally reflecting the quality of students' mathematical engagement in collective classroom processes and predicting their conceptual teaming opportunities. This notion reflects the fact that the construct of sociomathematical norms is intended to capture the essence of the mathematical microculture established in a classroom community rather than its general social structure. The notion also allows us to see a teacher as promoting sociomathematical norms to the extent that she or he attends to concordance between the social processes of the classroom, and the characteristically mathematical ways of engaging. In this way, the construct of sociomathematical norms include, but in no ways needs to be limited to, teacher's mediation of mathematics discussions.

The major purpose of this study is to show the insufficiency of traditional epistemology and consructivism as epistemology of mathematics education. Traditional epistemology such as empiricism, rationalism, Kant's epistemology, and Piaget's genetic epistemology is not sufficient to explain episteme in educational situation because it regards that epsteme is the phenomenon occurs between the abstract individual subject and the object world. Modern epistemology like constructivism recognize the public or social character of epsteme. So it is more appreciate than traditional epistemology to explain episteme in math educational situation. But constructivist pedagogy derived from constructivist learning theory has the following important shortcoming: The lack of clear criteria by which instructional effectiveness might be evaluated from a constructivist perspective.

Traditionally, there are many inherent restrictions in highschool mathematics textbooks. They are restricted in its contents and inevitably resorted to reader's ability of intuition. So they are usually lacked logical precisions and have various differences in expressions. We are mainly concerned with the definite integral and its applications in current highschool mathematics II textbooks according to 6th curriculum. We choose 6 of them arbitrarily and survey by comparison to deduce some controversial topics among them as follows. 1) absurd metaphors in formula process 2) confusions in important notations and too much choices in terms and statements. 3) lack of precisions in - teaching hierarchy (between some contents of Physics and the applications of definite integral) - introducing a proof of theorem (fundamental theorem of Calculus I) - introducing the methods (integral substitutions 1, ll) 4) adopting small topics such as - mean value theorem of integral - integrals with variable limits. In coming 7th curriculum, highschool students in Korea are supposed to choose calculus as a whole, independent course. So we hope that the suggested controversial topics are to be referred by authors to improve the preceding Mathematics ll textbooks and for teachers to use them for better mathematics education.

The researcher compared the curricula, the textbooks, the methods of didactic transposition, teacher practices of elementary and secondary mathematics education. The study reviewed several mathematics education theories in the aspect of the differences between the two levels. Findings from this study show a little difference between elementary and secondary mathematics education. There are special difficulties in didactic transposition in elementary mathematics education since it is so far from scientific mathematics. From this study we can conclude that further research needs to be done in the area of development of perspectives on elementary mathematics education.

Infinity is one of the important concept in mathematics, science, philosophy etc. In history of mathematics, potential infinity concept conflicts with actual infinity concept. Reason that mathematicians refuse actual infinity concept during long period is because that actual infinity concept causes difficulty in our perceptions. This phenomenon is called epistemological obstacle by Brousseau. Potential infinity concept causes difficulty like history of development of infinity concept in mathematics learning. Even though students team about actual infinity concept, they use potential infinity concept in problem solving process. Therefore, we must make clear epistemological obstacles of infinity concept and must overcome them in learning of infinity concept. For this, it is useful to experience visualization about infinity concept. Also, it is to develop meta-cognition ability that students analyze and control their problem solving process. Conclusively, students must adjust potential infinity concept, and understand actual infinity concept that is defined in formal mathematics system.

Piaget's genetic epistemology has been known as the basis of the 'New Math' and as the opposite point of view to the historico-genetic principle. But these days Piaget's theory is considered to support the historico-genetic principle so that it influences many studies. This study shows the reason of the difference of interpretations of Piaget's theory.

The purpose of this thesis is, through analysing the characteristics of the definitions in Korean school mathematics textbooks, to explore the levels of them and to make suggestions for definition - teaching as a mathematising activity, Definitions used in academic mathematics are rigorous. But they should be transformed into various types, which are presented in school mathematics textbooks, with didactical purposes. In this thesis we investigated such types of transformation. With the result of this investigation we tried to identify the levels of the definitions in school mathematics textbooks. And in school mathematics textbooks there are definitions which carry out special functions in mathematical contexts or situations. We can say that we understand those definitions, only if we also understand the functions of definitions in those contexts or situations. In this thesis we investigated the cases in school mathematics textbooks, when such functions of definition are accompanied. With the result of this investigation we tried to make suggestions for definition-teaching as an intellectual activity. To begin with we considered definition from two aspects, methods of definition and functions of definition. We tried to construct, with consideration about methods of definition, frame for analysing the types of the definitions in school mathematics and search for a method for definition-teaching through mathematization. Methods of definition are classified as connotative method, denotative method, and synonymous method. Especially we identified that connotative method contains logical definition, genetic definition, relational definition, operational definition, and axiomatic definition. Functions of definition are classified as, description-function, stipulation-function, discrimination-function, analysis-function, demonstration-function, improvement-function. With these analyses we made a frame for investigating the characteristics of the definitions in school mathematics textbooks. With this frame we identified concrete types of transformations of methods of definition. We tried to analyse this result with van Hieles' theory about levels of geometry learning and the mathematical language levels described by Freudenthal, and identify the levels of definitions in school mathematics. We showed the levels of definitions in the geometry area of the Korean school mathematics. And as a result of analysing functions of definition we found that functions of definition appear more often in geometry than in algebra or analysis and that improvement-function, demonstration-function appear regularly after demonstrative geometry while other functions appear before demonstrative geometry. Also, we found that generally speaking, the functions of definition are not explained adequately in school mathematics textbooks. So it is required that the textbook authors should be careful not to miss an opportunity for the functional understanding. And the mathematics teachers should be aware of the functions of definitions. As mentioned above, in this thesis we analysed definitions in school mathematics, identified various types of didactical transformations of definitions, and presented a basis for future researches on definition teaching in school mathematics.

The purpose of this study is to conceptualize 'proof' school mathematics. We based on the assumption the following. (a) There are several different roles of 'proof' : verification, explanation, systematization, discovery, communication (b) Accepted criteria for the validity and rigor of a mathematical 'proof' is decided by negotiation of school mathematics community. (c) There are dynamic relations between mathematical proof and empirical theory. We need to rethink the nature of mathematical proof and give appropriate consideration to the different types of proof related to the cognitive development of the notion of proof. 'proof' in school mathematics should be conceptualized in the broader, psychological sense of justification rather than in the narrow sense of deductive, formal proof 'proof' has not been taught in elementary mathematics, traditionally, Most students have had little exposure to the ideas of proof before the geometry. However, 'proof' cannot simply be taught in a single unit. Rather, proof must be a consistent part of students' mathematical experience in all grades, in all mathematics.

The axiomatic geometry unit of the space figure in Mathematics II in the expository book of high school math curriculum (published by Ministry of Education, June 20, 1995) suggests some teaching points to bear in mind, so as not to make use of the system of axiom. However, it doesn't take the axiom about the space geometry as a starting point of argument, and so many textbooks can be found, in which intuitively true propositions are proved acceptable by the logical ambiguous statements. Thus, this study analyzes the contents of axiomatic geometry in high school math II textbooks and draws their problems. As an alternative improvement, 3 kinds of axiom on the space geometry and some important propositions, which are basic to proofs of proposition, will be presented here in this paper.