Currently, our country operates gifted education only as a special curriculum, which results in many problems, e.g., there are few beneficiaries of gifted education, considerable time and effort are required to gifted students, and gifted students' educational needs are ignored during the operation of regular curriculum. In order to solve these problems, the present study formulates the following research questions, finding it advisable to conduct gifted education in elementary regular classrooms within the scope of the regular curriculum. A. To devise a teaching plan for the gifted students on mathematics in the elementary school regular classroom. B. To develop a learning program for the gifted students in the elementary school regular classroom. C. To apply an in-depth learning program to gifted students in mathematics and analyze the effectiveness of the program. In order to answer these questions, a teaching plan was provided for the gifted students in mathematics using a differentiating instruction type. This type was developed by researching literature reviews. Primarily, those on characteristics of gifted students in mathematics and teaching-learning models for gifted education. In order to instruct the gifted students on mathematics in the regular classrooms, an in-depth learning program was developed. The gifted students were selected through teachers' recommendation and an advanced placement test. Furthermore, the effectiveness of the gifted education in mathematics and the possibility of the differentiating teaching type in the regular classrooms were determined. The analysis was applied through an in-depth learning program of selected gifted students in mathematics. To this end, an in-depth learning program developed in the present study was applied to 6 gifted students in mathematics in one first grade class of D Elementary School located in Nowon-gu, Seoul through a 10-period instruction. Thereafter, learning outputs, math diaries, teacher's checklist, interviews, video tape recordings the instruction were collected and analyzed. Based on instruction research and data analysis stated above, the following results were obtained. First, it was possible to implement the gifted education in mathematics using a differentiating instruction type in the regular classrooms, without incurring any significant difficulty to the teachers, the gifted students, and the non-gifted students. Specifically, this instruction was effective for the gifted students in mathematics. Since the gifted students have self-directed learning capability, the teacher can teach lessons to the gifted students individually or in a group, while teaching lessons to the non-gifted students. The teacher can take time to check the learning state of the gifted students and advise them, while the non-gifted students are solving their problems. Second, an in-depth learning program connected with the regular curriculum, was developed for the gifted students, and greatly effective to their development of mathematical thinking skills and creativity. The in-depth learning program held the interest of the gifted students and stimulated their mathematical thinking. It led to the creative learning results, and positively changed their attitude toward mathematics. Third, the gifted students with the most favorable results who took both teacher's recommendation and advanced placement test were more self-directed capable and task committed. They also showed favorable results of the in-depth learning program. Based on the foregoing study results, the conclusions are as follows: First, gifted education using a differentiating instruction type can be conducted for gifted students on mathematics in the elementary regular classrooms. This type of instruction conforms to the characteristics of the gifted students in mathematics and is greatly effective. Since the gifted students in mathematics have self-directed learning capabilities and task-commitment, their mathematical thinking skills and creativity were enhanced during individual exploration and learning through an in-depth learning program in a differentiating instruction. Second, when a differentiating instruction type is implemented, beneficiaries of gifted education will be enhanced. Gifted students and their parents' satisfaction with what their children are learning at school will increase. Teachers will have a better understanding of gifted education. Third, an in-depth learning program for gifted students on mathematics in the regular classrooms, should conform with an instructing and learning model for gifted education. This program should include various and creative contents by deepening the regular curriculum. Fourth, if an in-depth learning program is applied to the gifted students on mathematics in the regular classrooms, it can enhance their gifted abilities, change their attitude toward mathematics positively, and increase their creativity.