[[alternative]]A Study of the Assessment of Mathematical Problem Solving Ability from a Schema Perspective on a Group of Sixth and Seventh Grade Mathematically Gifted Students

[[abstract]]The purpose of this study is to explore the possibility of a new approach to assess mathematical problem solving ability from a schema perspective on a group of sixth and seventh grade mathematically gifted students. This is done by developing three specially designed problems. The basic concepts behind these problems included ratio, area, length, time, etc., that are all covered in the mathematics curriculum of the elementary school. A total of twenty-four subjects were involved in this study, and most of them had represented Taiwan to participate in international mathematical competitions. All subjects were required to solve the first two problem sets within an hour. As for the last problem, it was solved by five specially chosen students, which was then followed by a semi-structured interview. The whole process was videotaped and last for one and a half hours in order to further understand their reasoning process. The first two problem sets were so designed as to evaluate the extent of related schema knowledge of the subjects, as well as their ability to identify the given data in the problems, ability to guess what question will be asked based on part of the information of the problem, ability to decompose the problems into a set of small problems, as well as their problem solving ability. These two problem sets were required to be solved individually and their solution times were recorded for each subproblem. The purpose of the last problem is to understand the role of related schema knowledge regarding whether it will enhance or disturb students’ problem solving performance. Both quantitative and qualitative procedures were adopted for analysis of the first two problem sets. In particular, students were classified into three levels of schema knowledge as well as three levels of problem solving performance. A detailed analysis was then performed to compare if there was qualitative differences among students of difference schema levels with respect to the quality of their solutions. As for the last problem, after assessing the abilities of reverse thinking of the five students, a detailed analysis was then performed to compare their problem solving strategies by visualizing their solution process. For the first problem sets, it was found that the students’ ability to identify the given data in the problems was highly correlated to their problem solving performance. As for the second problem sets, the extent of their related schema knowledge and their ability to decompose the problem into smaller problems were found to be highly correlated to their problem solving performance. In general, it was found that for those students with weaker connections among concepts, their schema knowledge were also more choppy, and that they were weaker in terms of deriving extra information from the given data in the problems. Meanwhile, they tended to be less careful with respect to breaking down the problems into smaller problems, as well as less able to implement their plans nor to solve the problems. As for those with medium schema knowledge, they were more able in extracting extra information from the given information, as well as representing the problems in another formats. Yet, such treatments of the problems were usually not very much in depth, and the success rate was not very high. Finally, for those students with high schema knowledge, they exhibited very good connections among concepts, and could utilize, to a better extent, their schema knowledge together with the given information of the problem to generate useful information to solve the problems. Further more, they could break down the problems into subproblems in a more extensive way. Finally, they could implement their plan better and achieved better rate of success. The major finding with respect to the last problem was that strong schema knowledge could sometimes hinder the process of problem solving, by channeling the solution into a wrong direction. The major suggestions of this study were that teachers should consider providing more accurate explanations of mathematical concepts, provide multiple representations of the problem, and consider using a comprehensive way to assess students’ mathematical cognitive knowledge, parallel to the examples used in this study. Topics for future research were also suggested at the end of this report.