The paper is analyzed the crisis of mathematics for paradoxes in mathematics, also its process of conquest in the history of mathematics. We have found that various views of the infinity and infinitesimal in the history of mathematics. This study be tried to find out the process development of mathematics on passing through a crisis conquest of mathematics for contradictions in the hi story of mathematics.

The first part of this thesis gives some brief explanation of the theory and history of imaginary elements in analytic geometry in the 19th century. The second part of this thesis discusses the theory of imaginary elements of synthetic geometry in the first half of the 19th century. Then the next part mentions the theory of imaginary elements of geometry in the second half of that same century. Particularly Christian von Staudt's and Felix Klein's theories are handled in this part. Von Staudt, who has completed the system of the synthetic projective geometry, used ‘involution’ in order to introduce a new concept ‘imaginary elements’- imaginary points, imaginary lines and imaginary plane-in synthetic geometry. Klein applied von Staudt's theory as he convey the result of the research in algebraic geometry in a picture. Von Staudt's and Klein's research may be regarded as the top of the effort to investigate possible relationship between real and imaginary structures.

In this article we study various proofs of Euler's theorem(the number of faces of any polyhedron, together with the number of vertices, is two more than the number of edges), from these proofs extract some mathematical ideas. In this paper we in detail introduce eight different proofs from various articles and textbooks.

One of the characteristics of modem mathematics is to use algebra in every fields of mathematics. But we don't have the exact definition of algebra, and we can't clearly define algebraic thinking. In order to solve this problem, this paper investigate the history of algebra. First, we describe some of the features of proportional Babylonian thinking by analysing some problems. In chapter 4, we consider Greek's analytical method and proportional theory. And in chapter 5, we deal with Diophantus' algebraic method by giving an overview of Arithmetica. Finally we investigate Viete's thinking of algebra through his ‘the analytical art’. By investigating these history of algebra, we reach the following conclusions. 1. The origin of algebra comes from problem solving(various equations). 2. The origin of algebraic thinking is the proportional thinking and the analytical thinking. 3. The thing that plays an important role in transition from arithmetical thinking to algebraic thinking is Babylonian ‘the false value’ idea and Diophantus’ ‘arithmos’ concept.

Platonist asserts the existence of abstract entities. Social constructivism views mathematics as a social construction. Platonism seems to be opposed to social constructivism. But this paper discusses the compatibility of Platonism and social constructivism.

This paper is concerned with the mathematical concept displayed in Buddhism, which is reasonable enough to consider as a philosophy and encompasses the concept of infinity as scientific as that of mathematics. The purpose of this paper is to examine the changing process of the Buddhism concept of infinity on the basis of time sequence and to combine this with that of mathematics.

In this paper we investigate the esthetic elements and the beautiful pieces of mathematics. And we also survey the role of mathematical beauty in the work of mathematicians and in teaching mathematics.

Mathematics and music play very different roles in our society. However, they are closely related to each other In the time of the ancient Greeks, they were strongly connected. This paper identifies such connection and concludes that music has mathematical characteristics.

Gu-Jang-San-Sul is a book of Chinese ancient mathematics and has had an impact on Korean mathematics. The book is organized into nine chapters and each chapter is composed of problems, answers, and their computation algorithms. The contents reflect the practicality of Chinese mathematics. Especially the first chapter covers the computation of fractions for land surveying. This paper suggests how the computation methodology is used in teaching fractions for primary school students. Five strategies for fractions related to the reduction, addition, subtraction, multiplication, and division are followed by: 1) developing the ability to apply rules to problems by practicing the computation process according to the given algorithm; 2) developing the communication skill by comparing the differences of various computation algorithms; 3) setting computation problems; 4) understanding the characteristics of terminology in mathematics; and 5) being exposed to new ideas in mathematics.

The main purpose of this study is to classify types of class using computer in the school mathematics classroom. For this purpose, we will first survey the Tyle's theory relate to the curriculum model and its details. Then we will investigate the crucial points using computer in the school mathematics classroom in the viewpoint of Glaser's teaching model. From this, we will device several types of mathematics class using computer.

The purpose of this paper is to analyze the the effect on the academic improvement, applying various teaching methods throughout cooperative teaming in small groups to inspire slow-learners'interest on mathematics and impove capability to solve problems and achievement degree of studies. The subject of the study is as follows: 1. What is the difference of students'interest on mathematics between cooperative learning in small groups and large group learning\ulcorner 2. Is there any difference of slow-learners' achievement degree of studies between cooperative learning in small groups and large group learning?

In this paper, we explain independence and mutually exclusiveness of events. Also we explain independence and uncorrelation of random variables. Finally we compare with differences as examples.

We investigate the history of the research of the existence of periodic solutions of a nonlinear wave equation with jumping nonlinearity, suggested by Mckenna and Lazer (cf. [15]). We also investigate the recent research of it; a relation between multiplicity of solutions and source terms of the equation when the nonlinearity -($bu^+$-$au^-$) crosses eigenvalues and the source term f is generated by eigenfuntions.

We show that if the Stone-Cech compactification of $\textit{AX}$ and the minimal basically disconnected cove. of $\beta$Χ we homeomorphic and every real $\sigma$$Z(X)^#$-ultrafilter on X has the countable intersection property, then there is a covering map from $\nu$(ΛΧ) to $\nu$Χ and every real $\sigma$$Z(X)^#$-ultrafilter on Χ has the countable intersection property if and only if there is a homeomorphism from the Hewitt realcompactification of ΛΧ to the minimal basically disconnected space of $\nu$Χ.

In this paper we prove that tile property of product irreducible characters is, if $\Phi$$\in$$\textit{Irr(H)}$ and $\Theta$$\in$$\textit{Irr(K)}$ are faithful, then $\Phi$$\times\theta$ is faithful if and only if │$\textit{Z(H)}$│ and │$\textit{Z(K)}$│ are relative primes where G=$\textit{H}\times\textit{K}$.

Since Noether has consolidated the structural approach to the study of algebra, the lattice theory has reemerged as a tool for the structural study for algebra and its own right as well in 1930s. We investigate the process which the mathematical structures made their foundations in Mathematics through the lattice theory in the period.