Mathematization is cognitive process of empirical phenomena into mathematics. The article shows that mathematization is an fundamental element in the process of westernization and the difference between the East and tile West is due to the existence of mathematics.

This paper aims to investigate how Chinese and Korean evaluate $\pi$ and measure tile area of circle by reviewing the problems in the old mathematical books. The books are Gu-Jang-San-Sul(The nine chapters on tile mathematical art) for China and Gu-Il-Jib for Chosun Dynasty. The result shows that our ancestors used the different values of ${\pi}$ in relation to the accuracy and the various methods for measuring the area of circle.

The Pythagorean equation $x^2{＋}y^2{=}z^2$ and Pythagorean triple had appeared in the Babylonian clay tablet made between 1900 and 1600 B. C. Another quadratic equation called Pell equation was implicit in an Archimedes' letter to Eratosthenes, so called ‘cattle problem’. Though elliptic equation were contained in Diophantos’ Arithmetica, a substantial progress for the solution of cubic equations was made by Bachet only in 1621 when he found infinitely many rational solutions of the equation $y^2{=}x^3{-}2$. The equation $y^2{=}x^3{+}c$ is the simplest of all elliptic equations, even of all Diophantine equations degree greater than 2. It is due to Bachet, Dirichlet, Lebesque and Mordell that the equation in better understood.

In this paper, we deal with the ‘dearithmetization’ of algebra. Historically, the origin of algebra comes from arithmetic. Also school algebra is related to arithmetic in general. However, we have many difficulties in teaching school algebra, and there is many problems for students to learn algebra from elementary arithmetic knowledges. This paper supposed that the solution of these problem may be founded in the ‘dearithmetization’ of algebra. And we supposed that the ‘dearithmetization’ of algebra may be developed by three historical achievements - the completion of symbolic algebra, the principle of permanence of form, and the expansion of number concepts. In order to justify these supposition, we investigate Peacock's ideas, i.e. ‘symbolic algebra’, ‘the principle of permanence of form’, and consider how the integer is introduced in modern mathematics. And we analyze various textbooks, and investigate the ‘dearithmetization’ of school algebra which has been progressed in the three fields - symbolic algebra, the principle of permanence of form, and the expansion of number concepts.

The goals of a major reform effort are to enable us to educate informed citizens who are better able to function in our increasingly technological society. Discrete mathematics is an exciting and appropriate vehicle for working toward and achieving these goals. it is an excellent tool for improving reasoning and problem solving skills. Discrete mathematics has many practical applications that are useful for solving some of the problems of our society and that are meaningful to our students. Its problems make mathematics come alive for students, and help them see the relevance of mathematics th the real word. To build up the role of Discrete mathematics in the school, this study is to investigate various theories and curricula related to discrete mathematics, and to collect a great deal of valuable material that will help teachers introduce discrete mathematics in their classrooms. In conclusion. mathematics teachers will find the need and importance of why and how discrete mathematics can be introduced into their curricula by this study.

This paper is a sequel to [11]. We introduce $\sigma$-coherent frames, stably countably approximating frames and strongly Lindelof frames, and show that a stably countably approximating frame is a strongly Lindelof frame. We also show that a complete chain in a Lindelof frame if and only if it is a strongly Lindelof frame by using the concept of strong convergence of filters. Finally, using the concepts of super compact frames and filter compact frames, we introduce an example of a strongly Lindelof frame which is not a stably countably approximating frame.

Observing that if f: $Y{\leftrightarro}$Χ is a covering map and Χ is a countably locally weakly Lindelof space, then Y is countably locally weakly Lindelof and that every dense countably weakly Lindelof subspace of a basically disconnected space is basically disconnected, we show that for a countably weakly Lindelof space Χ, its minimal basically disconnected cover ${\bigwedge}$Χ is given by the filter space of fixed ${\sigma}Ζ(Χ)^#$- ultrafilters.

We define the second Gausssian curvature $K_{II}$ by using Brioschi's formula and shall disscuss the helicoidal surfaces satisfying $K_{II}$=Η.