[[alternative]]The influence of problem presentation formats and other relevant variables on the performances of mathematically gifted students in solving Euclidean geometry problems

[[abstract]]The main purposes of this study are threefold. The first one is to find out if there are differences between the way students solve Euclidean geometry problems that are given either in a story context or in terms of the conventional symbolic format. The second one is to identify if there are relevant variables that may affect students' performance in solving geometry problems. The third one is to find out what attitudes the students had regarding the presentation formats of the problems they have to solve. Our research sample are twenty one seventh to ninth graders who are gifted in mathematics. Most of them have participated in various international mathematics competition. This research focuses mainly on geometric concepts that are relevant to the curriculum of the junior high level. Various topics that involve computational, derivational and constructional skills are covered in the study. We design two types of problems that are isomorphic to each other. Of which, one is presented in the form of mathematical symbols and figures only, while the other is by way of stories with or without figures. The story-type problems are further divided into the straight forward descriptive and the dramatic types. The purpose behind this design is to investigate into the relationship between students' performance in relation to the context which the problems are presented in. The result will shed some light on the role of situated learning so far as junior high Euclidean geometry is concerned. In addition, a pretest is used to measure their prior knowledge in geometry. Nevertheless, due to the small sample size, data analysis is mainly by way of descriptive statistics and qualitative analysis. The result shows that our students perform relatively better on the problems presented in mathematical symbols. However, there are not many students who can first solve the mathematical symbol problems, and then also solve the corresponding isomorphic problems that are presented with a context. On the contrary, there is a high percentage of students who can first solved the problems in a contextualized format and then succeed in solving their isomorphic counterparts. Furthermore, it is noticed that not many students are able to tell that the problems presented in two different formats are actually isomorphic to each other. With respect to the second purpose, it is found that the prior knowledge of students, the presentation formats of problem, and the accuracy of the figures are some relevant variables that may affect students' performance. More specifically, students with better mathematical skill perform equally well on problems in either presentation formats. However, students with weaker skill perform relatively well on problems presented in mathematical symbols only. Although on the whole, students are better in solving problems presented in mathematical symbols, most of them prefer to solve problems with a storyline. They indicate that problems with a storyline are more interesting and enable them to understand the problems better. Nevertheless, it is found that some students may consider extraneous variables in real life which they may bring in to solve problems with a context. This reveals that students can make connection between problem solving and real life situation. Based on this study, it is quite acceptable to present geometry problems in more real-life format. It is suggested that teachers should discuss with their students various variables that can be identified in real life. This will enable the students to transform the story situation into a more realistic mathematical model. Besides, teacher should pay attention to the figures they provide in problems, especially with respect to the length of lines and the ratio between lines. This will prevent the students from mistaking special cases as a general principle. In the future, other variables, such as students' comprehensive ability on reading, students' gender, teachers' teaching methods and the length of presentation formats can be included for further study. A large sample size as well as extending the study to more general classrooms should also be considered.