###### Abstract

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Problem solving that enables the experience of the power and usefulness of mathematics is a field of mathematics education that is an important part of school mathematics, and is actively studied and becomes an object of the overall practical curriculum. In particular, the 2009 Revised Curriculum co...

Problem solving that enables the experience of the power and usefulness of mathematics is a field of mathematics education that is an important part of school mathematics, and is actively studied and becomes an object of the overall practical curriculum. In particular, the 2009 Revised Curriculum consistently instructs problem solving in the whole areas of each semester of all grades. However, in spite of this great deal of research and effort, the student does not have confidence in their problem solving but feel difficulty. This reality becomes more serious as they move up to higher grades, and in the end it is the factor to make them lose their interest in mathematics. Accordingly, offering the chances to the student that is the subject of learning to actively participate in the course of problem solving, and to have self-directed learning is required to make them feel the true meaning of independent problem solving. In this connection, this study is attempted to suggest efficient the problem solving methods and new directions of mathematics teaching through discussions for the improvement of the problem solving ability and teaching and learning methods of mathematics for the elementary student that is at the starting point and in the course to strengthen the foundation of mathematics education. In order to achieve these purposes, it established below research questions. 1. Development of a student-oriented problem solving mechanism model to improve the mathematical thinking and problem solving ability 2. Verification of the efficiency of the developed student-oriented problem solving mechanism model 3. Analysis of the didactic situations in the problem solving process that applies the student-oriented problem solving mechanism model In order to solve the research question 1, the domestic and overseas preceding studies on the problem solving process were compared and analyzed, and the transitions of our elementary school mathematics education course and various strategies and teaching methods of problem solving were examined. In addition, as for Poly's problem solving characteristics and limits, it analyzed the conversations that were provided as examples by Polya and the problem solving process of 5th graders. Based on this literature review and case study, it developed a student-oriented ‘problem solving mechanism’ model that schematizes a series of courses that structurize mathematical thinking and solve problems with the premise of basic communication and learner's ideas from the beginning to the termination of problem solving. As for the solution of the research question 2, the methodological aspects of research such as research subject, method, procedure, test tool and data analysis were reviewed, and experimental research on two classes of elementary school was performed to verify the efficacy of the developed student-oriented problem solving mechanism model. The efficacy of the problem solving mechanism was statistically compared and analyzed each other according to group, sex and level factors about the whole academic performance, problem type and problem area. As a result, the experimental class showed more significance than the comparative group and this means that the application of problem solving mechanism was efficient. In order to solve the research question 3, the theoretical content about didactic situations was examined, and didactic situations shown in the problem solving process were analyzed. The analysis of didactic situations was reviewed through the data such as observations of student's problem solving process, results and interviews about interaction and individual difference in the problem solving situations of three students by level. Based on student-oriented problem solving mechanism, after it was applied to the experimental class, the didactic situations shown in the problem solving process of three students were analyzed. As a result, the students all succeeded in problem solving but they showed differences in the problem solving methods or technical aspects to recognize problem situations and establish, perform and review their ideas. From the characteristics or facts found in the course to develop and apply the mechanism model of problem solving, several didactic suggestions to improve mathematics class could be acquired. First, student-oriented problem solving mechanism provides a practical plan on how the student should think to solve problems and how the teacher should prepare for class to teach the student. Second, establishing an idea and practicing it as it is, is the basis of problem solving. The habit of maintaining and reviewing the basis of problem solving can enable creative problem solving in various situations. Third, the problem solving worksheet provided for this study played a guiding role for students with insufficient problem solving methods or ability. Therefore, it is necessary to review the utilization of the problem solving worksheet in school class. Fourth, in order to verify the efficacy of the problem solving mechanism, both of experimental study and case study were utilized. The combination of the above two studies is required to be considered to have qualitative improvement of this study type. This study was progressed in the aspect to suggest a plan for efficient problem solving and a new direction of teaching and learning by verifying the efficacy of the student-oriented problem solving mechanism that was developed to improve the student's problem solving ability and teaching and learning method and the didactic situations that are shown in the problem solving process that applies the mechanism. We expect that along with the above discussion, various follow-up studies will continue to deal with problem solving mechanism such as the effects of the problem solving mechanism on the positive aspects and relevance of the evaluation method that were not studied by this paper.