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The purpose of this study is to examine that high school students recognize mathematical situation when they are requested for changing same mathematical situation into different situation, and to find the cause of nonrecognition if they cannot identify it. The results of the study are followings. First, percentage of correct answers to the questions of turning equal mathematical situation into function is higher than the one of turning equal mathematical situation into equation and inequality. And percentage of correct answers to the questions of relation between linear equation, linear inequality and function is lower than the one of relation between quadratic equation, quadratic inequality and function. As a result of individual interview for comprehensibility of the students on these relations , it is found that if degree goes up and there is different expressions of questionaries although mathematical situation is identical, it affects comprehensibility of the subjects. Second, the students who are acquainted with mechanical calculation have difficulty in solving subjective questions such as drawing graphs. It is found that they cannot understand identical mathematical situation because they have trouble in drawing graph or applying to get the answer while many students understand a point of intersection on the graph as correct answer. Third, as a result of individual interview for comprehensibility of the students on relation between equation, inequality and function the students, it is found that they manage to get correct answer even without perfect comprehensibility on this relation. And there are the students who comprehend these relations after they solved the problem in which they made errors in the process of calculating or they failed to completing graph.

The purpose of this study, middle school students as self-directed learning ability, which affect problem solving in mathematics. Study in the Gyeonggi and M 2nd grade of middle school students from eight classes of 200 patients were randomly selected, the learner did not properly study of 26 patients, were excluded. Results of this study are as follows: First, high-grade math learners, self-directed learning ability is very high. Second, learner self-directed learning skills, high math scores. Third, of the three sub-factors of self-directed learning skills, cognitive factors and behavioral factors have a direct impact on mathematical problem solving ability, motivation factors indirectly affect. Fourth, the details of the factors of the three sub-factors of self-directed learning skills, all different factors affecting each step of the mathematical problem solving ability. Step1 (understand the problem) 'help save' and 'learning strategy', Step2 (planning) 'learning strategies' and' information processing, Step3 (execution plan) 'time management' and 'learning strategy', the last step4 (Results and reflection) 'academic motivation' and 'learning strategies' influence was. In other words, there are significant differences depending on the ability of self-directed learning mathematical problem solving ability of middle school students. Learners identify factors that habitual use of self-directed learning skills and, therefore, if we apply the teaching of mathematics scarce factors that can help build students' mathematical problem solving Ability to be able to promote.

This study examined self-regulation activity in draft writing according to writers’ level. First, I classified student writers into proficient writers and unexperienced writers, and analyzed their self-regulated writing activity by using a reconstituted framework based on preceding research. As a result, there were similarities and differences on writers’ internal level. Both proficient writers and unexperienced writers realized, regulated, or judged their background knowledges, experiences or emotions. Also, they commonly had writing block phenomenon during writing. However, there were differences in several aspects : responses when writing block happened, performance and control of writing strategies such as choosing ways of expressions or genres by considering readers, judgement of background knowledge and information application, and assessment of their writings, and concreteness of assessment of fellow students’ writings. On the external level, both proficient writers and unexperienced writers showed differences in their behaviors or control of environment.

The purpose of this study was to develop mathematics instructional model and its application based on Ausubel's meaningful reception learning. The process was First, to select instructional elements based on Ausubel's theory, second, to develop instructional model reflecting these elements, and then to analysis three classes applying the new instructional model in H-girls' high school. Students who learned mathematics with new instructional model responded that 'it was easy to study mathematics with this model', 'I was much interested in mathematics after studying mathematics with this model', 'I preferred teacher to explain new knowledge', 'I was sure that I learned mathematics by my math's level' and 'It was easy to remember new knowledge after studying mathematics with this model'. Teacher who teach students with new instructional model was satisfied at 'to select suitable teaching method for student's level', ‘to prepare Advanced Organizer for students', 'to give students learning tasks by hierarchical order', at the end of this model 'to use principle of integrative reconciliation' and 'to incite a critical approach' ect. Therefore, we were abel to be sure that mathematics instructional model based on Ausubel's meaningful reception learning was suitable for teacher centered teaching method.

The purpose of the study was to analyze the elementary mathematics gifted program developed by Seoul school district as a first step to evaluate mathematics gifted program developed for elementary children in general. Fourteen mathematics gifted program for elementary children developed in 2009 were analyzed for the study. The instrument for the study was a standard for gifted education program developed by Kim S. W and Huh M in 2005 . It includes the purpose of the program, the contents of the program and instructional methods. The results of the study reveals that the program was appropriate for gifted elementary children in general, even though some area such as future occupational guidance and cultivation of personality needed more attention in the future development of gifted program.