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In this paper we analyze various mathematical theorems and problems related with a perpendicular of triangle, classify the these as followings: perpendicular's properties related with its length, triangle's area, and triangle's angle. We suggest proof process of these properties in detail, and discuss didactical meaning of the proof and property.

The concept of function as an integrated idea is not only a rule that uniquely associates elements of one set with elements of another set but also a common topic which is found in various areas of mathematics such as algebra, geometry, and probability. Moreover, a function is a mathematical representation describing a relationship between input and output which can easily be found in our daily lives. Functional thinking pervades all of mathematics and is one of mathematical literacy required to get involved in the future society. The purposes of the study were to investigate the difference between Korea's middle grade textbooks and MathThematics textbooks in the unit of function and to seek ways to solve middle school students' problems or difficulties in learning functions.

The purpose of this study is to examine what kinds of comments Korean language arts teachers make about the students' persuasive essay writings[Nonsul] and to analyze how the teachers perceive their own comments. The data were obtained from 100 essay writings that 50 middle school teachers of Korean language arts read and commented through a government-funded learning support website. Surveys were also conducted to identify the teachers' perceptions about the comments they offered to students. Standards for data analysis were adapted from Peterson & Kennndy(2006) and reformed for this study. This study found out that the Korean language arts teachers perceived that they made comments on students' writings mainly focusing on the meaning and the structure of the essay, students' thought, and their background knowledge. However, the teachers' actual comments were overwhelmingly related to students' spelling mistakes, the word spacing, and orthography. The results show there are differences and gaps between teachers' perceptions about their comments on students' writings and their actual contents of the comments.

The purpose of the study was to investigate the difference between an experimental group and a control group. An experimental group was exposed to small group cooperative learning environments using MIC textbook and students in a control group studied with traditional expository lessons. As a result, students in an experimental group performed better than the ones in a control group in a mathematics test implemented after the treatment. Also students in an experimental group performed better in mathematics disposition test except one item. In other words, small group cooperative learning method using MIC textbook not only influence mathematics performance, but also mathematics dispositions of the students.

This study is going to analyze mathematical communication in internet network according to trend of families that are having a computer each of them as wide support of internet. In this view, there have been many study which are performed in teacher centered network system. But in this study, I performed a study in the system of students centered network encouraging participation of students by using a blog at the popular website. For this study, I will design a blog which is one of social networkings that are familiar with teachers as well as mid and high school students to post, cooperate and debate mathematically various categories of problems so it is designed for the students as guinea pigs to take part in this study and learn aggressively. To achieve this goal, I set subjects of inquiry on the following below to analyze the way and ability of communication after classification some inquiry coming up on the blogs into some categories and describing of student's personality taking part in this study. 1. Cooperative learning tasks through blogs 1) how are the ways of communication based on inquiry typology? 2) how are the abilities of communication according to students personality? 2. Do the cooperative learning through blogs affect on the level of interesting toward mathematics of students? To find the out inquries as above, I processed experiment for 2months in the environment which I made for doing cooperative learning with selected 9 students by personalities out of first year students, middle school which is located in Cheongweon-gun.

This study is intended to provide a method using multimedia in a history class and analyze the perception of the teachers and the students about utilizing multimedia for a class. The teachers think that multimedia for teaching can encourage students' interests, and if enough materials are offered, they are willing to use them for a class. The teachers pointed the problem that multimedia in a class can distract the students and requires a lot of time for a teacher to prepare, despite the help to enhance the students' achievement in a history class based on multimedia. After experiencing a multimedia class, students showed a change in situational interest; the students felt bored less about the contents of textbooks. In addition, some information difficult to be explained became easier to understand through multimedia materials and students could relate the historical knowledge learned in a class to the real life. Especially, through multimedia for teaching, historical thinking skills and self-directed learning have improved, and the students showed many interests in new data and a variety of multimedia materials presented.

The purpose of the writing education is to make students as a skilful writer. Skilful writer has a writing strategies and writing abilities, and writing motivation can make to work writing strategies and writing abilities well. Therefore, it's important to search writing motivation aspect. In this research, 10 middle school students of the first grade were interviewed to search writing motivation aspect of writing motivation factor. In cooperactive interaction factor, the results were as follows. The students of high level and middle level writing motivation expected a compliment and approval through cooperative interaction and they wanted to write better after cooperative interaction with others. But low level students tended to avoid a cooperative interaction because they don't have a confidence about their writing and a positive experience in cooperative interaction. In writing efficacy factor, the writing motivation aspects were the most various. And the students of high level writing motivation related writing efficacy factor had a compliment and approval of parents and teachers. In competitive effort factor, most ot students revised their writing for assessment but the students of high level writing motivation revised their writing that they write spontaneously. And the more students get higher level writing motivation, the more they had revision. In challenging factor, the students of high level writing motivation made an effort to solve when they were in a writing difficult but the low level students just tried to end.

Despite much attention to the changes in teaching-learning environment caused by electronic boards and much necessity for them, it is almost impossible to supply all classrooms with electronic boards considering our educational budget. To solve this cost problem, I designed and constructed an extremely low-cost electronic board which is even very simple and convenient to install, and easy to maintain and repair. We improved and complemented the existing software, based on the open source made by Johnny Lee, a graduate student at Carnegie Mellon University, in 2007. By using an infrared LED pen and a Wiimote, the controller for Nintendo's Wii home video game console, I implemented an electronic board perfectly with only 30,000 won (28,000 won for Wimote and 2,000 won for an IR LED pen), and made a web site for teachers to share our softwares and tips. I expect that this low cost electronic board will be very useful for teaching in math as well as many other subjects.