This paper is a sequel to our paper [3]. Although polynomials in the tianyuanshu induce perfectly the algebraic structure of polynomials, the tianyuan(天元) is always chosen by a specific unknown in a given problem, it can't carry out the role of the indeterminate in ordinary polynomials. Further, taking the indeterminate as a variable, one can study mathematical structures of polynomials via those of polynomial functions. Thus the theory of polynomials in East Asian mathematics could not be completely materialized. In the previous paper [3], we show that Jeong Yag-yong disclosed in his Gugo Wonlyu(勾股源流) the mathematical structures of Pythagorean polynomials, namely polynomials p(a, b, c) where a, b, c are the three sides gou(勾), gu(股), xian(弦) of a right triangle, respectively. In this paper, we show that Jeong obtained his results through his recognizing Pythagorean polynomials as polynomial functions of three variables a, b, c.

Mathematics is a kind of language, and even a tool of cognition for human beings. Mathematics has been used to communicate and to develop the civilizations through the history. So mathematics is one of the most important subjects for human to teach and learn. Especially, developed countries believe that mathematics will play very important roles in the developments of future industries and so future society. In this article, we clarify that combinatorics which is mainly represented by counting is an important ingredient of future mathematics education. To do so, we investigate the characteristics of combinatorics from the educational and cognitive perspectives.

In this study, Bumwoo KIM Chi Young's (1916.12.24~1995.4.22) papers, essays and books on mathematics education were collected and classified. Bumwoo KIM Chi Young was the most important person on the New Math Movement (Modernization for Mathematics Education), and in 1973, he led the Third Reformation of National School Mathematics Curriculum in Korea. Bumwoo emphasized mathematical structure in mathematics education, and he was a strong advocate of importance of set theory, creativity and the type of spiral learning.

In tropical geometry, the sum of two numbers is defined as the minimum, and the multiplication as the sum. We learned that dynamic programming in tropical algebraic geometry can be used to find the shortest path in graphs. We have also learned about the Bezout's Theorem, which is a theorem concerning the intersections of tropical plane curves, and the stable intersection principle.