This paper discusses and searches for mathematical analysis and efficient algorithm for types of wallpapers. We study some previous classification methods, develop a systematic process, and present some examples of determining types of wallpaper through our algorithm. Through this approach, we expect to introduce a mathematical perspective on relation between real life and mathematics.

The effort to find the volume of pyramids has been done by mathematicians for a long time, and many trial-and-error calculations and proofs give various perspectives and educational material. In the early days, finding the volume of pyramids was mainly studied by calculating the volume of triangular pyramids or quadrangular pyramids by cutting and the relationship between pyramids. Thereafter, methods based on infinite, infinitesimal, limit, etc. appeared, but the research topic was still about them. The purpose of this study is to examine the four themes appearing the mathematics history in terms of methodology, and to think about its implications from the viewpoint of improving the professionalism of the teachers.

The purpose of this study is to investigate and analyze the understandings of elementary pre-service teachers about 'estimation' in the elementary mathematics. Together with this analysis, we identify elementary pre-service teacher's Mathematical Pedagogical Contents Knowledge(MPCK), especially focusing to Subject Matter Knowledge(SMK). In order to this goals, we investigate contents relating to 'estimation' from $1^{st}$ curriculum to 2009 revised curriculum and compare 'rounding up', 'rounding off', 'rounding' in the elementary mathematics textbooks. As results of investigations, 'estimation' has been teaching at the 'Measurement' domain from $3^{rd}$ curriculum, but contexts of measuring weaken from $7^{th}$ curriculum. 'Rounding up(off)' is defined three types in the textbooks from $1^{st}$ to 2009 revised curriculum. And we examine elementary pre-service teachers through the questions on these 'estimation' contents. On the analysis of pre-service teacher's understanding relating MPCK, four themes is summarized as followings; the understanding of '0' in the 'rounding up', the cognitive gap between 'rounding up' and 'rounding off', the difference of percentage of correct answers according to types of question in the 'rounding up', and the difference between the definition of 'rounding up' and the definition of 'rounding'.

A rational number as operator is eventually that it is considered a mapping. Depending on how selecting domain (the target of operation by rational number) and codomain (including the results of operations by rational number), it is possible to see the rational in two aspects. First, rational numbers can be deal with functions if we choose the target of operation by rational number as a number field containing rationals. On the other hand, if we choose the target of operation by rational number as integral domain $\mathbb{Z}$, then rational numbers can be regarded as partial functions on $\mathbb{Z}$. In this paper, we regard the rational numbers with a view of partial functions, we investigate the theoretical background of the relationship between the multiplication of rational numbers and the composition of rational numbers as operators.

This study expects that Team Based Project Learning can act positive role to improve pre-service elementary teachers' teaching competence from previous studies. So designing and executing Team Based Project Learning program, I can provide some suggestions to the teaching method for teacher training education. In this study, Team Based Project Learning focuses on the use of history of mathematics in school mathematics. Also from organizing the team to evaluation, I conduct team activity systematically both inside and outside the classroom. The result of this study shows that pre-service elementary teachers' teaching competence has improved and I could identify the positive response about the value of using the history of mathematics and effect of Team Based Project Learning in mathematics learning.

This study focuses on the teaching of fraction related to curriculum, introducing time of fraction, the meaning of fractions in textbook, material of teaching of fraction concept, teaching model of introducing time of fraction concept, special cases of teaching fraction and common points of representation of fraction among Korea, New Zealand and Singapore. For this study, Korea's mathematics textbooks(3-1, 3-2, 4-1, 5-1, 6-1) and New National Curriculum Mathematics(3, 4, 5. 6. 7)of New Zealand and New Syllabus Primary Mathematics(2B, 3B, 4A, 4B, 5A, 6A)of Singapore were selected for comparison and analysis. As a results we will suggest a reference to the development of mathematical curriculum, teaching fraction and improving the quality of the textbook through a method of comparative analysis of Korea, New Zealand and Singapore.