Case Study on the Learning Process of Gifted Students in the GSP Environment

Abstract

It is widely known that gifted students put more emphasis on compressed reasoning than ordinary students, having a stronger desire for justification. This research is designed to find out whether such characteristics in thinking are observed in the GSP environment, and which role the GSP environment plays in the learning process of gifted students. Krutetskii, V. A.(1976) observed and analyzed the problem-solving process, behaviors, and personalities of gifted students in mathematics over a long period of time, before reaching the following conclusion. The gifted students tend to identify rules by generalizing mathematical facts. Furthermore, their thinking process is so flexible that their idea is quickly transferred from one structure to another. They even have reversibility in thinking. Trying to come up with easy, clear, and economical solutions, they often formalize special mathematical concepts that are easily found in our daily lives - though this process is carried out at a beginner level. Polya, G(1965) explains that mathematical research is composed of demonstrative reasoning and probable reasoning. The former is safe, non-controversial, and final, while the latter is adventurous, controversial, and tentative. Polya emphasized that the process of hypothesizing mathematically, researching and confirming the validity of the hypothesis is very important in mathematical activities. With regard to courses for gifted students in mathematics, such a perspective has significant implications for identifying and developing their potential. This research will identify the requirements of teaching materials for gifted students by reviewing prior research and find out how the GSP environment can contribute to meeting the requirements. In reality, we will develop teaching materials for gifted students in mathematics that can satisfy the requirements, comparing their reasoning to the original goal of the materials, for analysis. In particular, the research will study how the gifted identify the format and structure of problems, whether logical thinking, compression in thinking, flexibility, and reversibility are observed, and whether they are trying to generalize mathematical concepts and develop mathematical symbols. The ultimate purpose of this research is to identify what potential the GSP environment can create for education for gifted students in mathematics, through case analysis.