[[alternative]]A study into the ability of seventh grade mathematically gifted students on isomorphic transformation and the effect of external representation on problem solving

[[abstract]]In resent years, problem solving has become a key concern in many disciplines, especially in mathematics education. Since problem representation is an important component of problem solving, and the mastery of the use of external representation can be of much help in problem solving, so problem representation is a worthy area of research and discussion. The reason why representation can help in problem solving is because representation involved some form of isomorphism. Moreover, the relationship between different representations also depends on isomorphism to a certain extent. The purpose of this study is to explore and understand about how isomorphic transformation would influence on problem solving. In addition, this study investigated into how a problem solver would choose among various representations, and also into the relationship among different problem representations. The subjects of this study included 24 mathematically gifted students, most of them being seventh graders. The duration of this study was about one semester with 12 sessions, each lasting for 3 hours. Most of the time instruction were conducted in a small group discussion setting. The subjects were divided into 5 groups. The first ten class materials were uniformly used across all groups, with each group having different emphasis. The last two sessions were conducted in a one to one interview format, so as to find out more about how students would approach solving problem. Because of the small sample size, data analysis were performed mostly by means of descripitive statistics, and qualitative case study method. According to the data of this study, isomorphic transformation had positive influences on problem solving, but its use by students was not spontaneous. Moreover, students did not easily make a habit of using it, nor transfer its use on similar setting. However, The ability to use isomorphic transformation could be trained and fostered by means of solving isomorphic problems in a sequential way. The data also revealed that the factors that affected students' choice of representation included personal characteristic, experience of problem solving, and the formats of representation. For the last factor mentioned above, it was found that students tended to consider the deep structure of problem first, and then the format of the representation. Yet regardless of how the three factors would influence the choice of representation, it all depended on the students' ability of isomorphic transformation. It was found that the order of choice of the representational format was mental computation, drawing, and then physical manipulation. As regards mental computation, it was found that the better the ability of the student to think abstractly, the better his/her problem solving ability. As for drawing, if the drawn figure matched the feature of the problem and had good quality, then it would facilitate the solving of the problem. As regards physical manipulation, it would not not help if either the problem solver was not handy in terms of manipulation, or the problem was not difficult enough, or if it was not of the appropriate problem type. Based on the results of the study, it is suggested that teachers should try to utilize isomorphic transformation in their teaching. Moreover, they should foster the ability of students to perform isomorphic transformation, but yet notice the limits of its use. As regards the use of drawing and physical manipulation as the means of external representation, teachers should pay attention to how effective problem solving depended on the problem solver and the problem itself. Teachers should also try to provide a convenient and easy environment for students to solve problems by physical manipulation. Furthermore, teachers who design real-life problems should notice the extent to which students are familiar with the mathematical concept behind the problems as well as the way in which the problems are worded. Otherwise, it may cause opposite effect. Future study may consider developing of programs to teach internal and external representation, so as to enhance students' ability to identify both the deep structure and the key points of the problems. Besides, more work should be devoted to study the relationship among the types of problems, their representations, and the problem solvers while solving problem.