| Summary: | 博士 === 國立中正大學 === 課程研究所 === 100 === The purpose of this study was to investigate the performance on solving proportional problems from two sixth gifted students. The case study approach with purposive sampling was adopted as the methodology of this study. The structured task-based interviews were applied here as well. Besides, the researcher designed the worksheets based on the non-routine proportional problems within four models, including well-chunked measures, part-part-whole, associated sets and stretchers & shrinkers problems, and made participants solve the easier problems on a higher priority basis. What’s more, the researcher went a step further to analyze the participants’ performance on solving problems and used the analysis as the foundation of the following interviews or scaffolding teaching. After the interviews or the scaffolding teaching, the researcher used the analysis adopted from Mayer, who summarized the different stages into four main steps, as the basic framework and further discussed the five knowledge models involved in the solving process.
To be brief, the conclusion of this study could be listed as following seven points. First of all, from the perspective of levels of difficulties, the well-chunked problems were the easiest ones but the part-part-whole measures were just on the opposite. However, the two participants both considered the latter as the most challenging model of problems. Secondly, in the solving process, basically these two gifted students could both demonstrate their four steps of solving process. The stage of problem integration as well as the stage of solution planning and monitoring could easily hinder them from proceeding. Thirdly, in the aspect of knowledge models, basically these two students were able to utilize the five knowledge models to solve the problems. However, the knowledge they could not grasp well was the schematic and strategic knowledge. Fourthly, as for the solving process, the most challenging aspect for the two participants was the stage of integration in the part-part-whole problems; the stage of solution planning and monitoring in the associated sets and stretchers & shrinkers problems was secondary. Fifthly, in the aspect of knowledge models, the schematic knowledge required in the stage of problem integration was the most difficult part for both of the participants, and so was the strategic knowledge required in the stage of solution planning and monitoring. Moreover, in the stage of solution planning and monitoring, it was possible to use the semantic knowledge required in processing the strategic knowledge. Sixthly, when it comes to the knowledge models, the most difficult part for the gifted was the schematic knowledge in the part-part-whole problems, and the strategic knowledge in the associated sets and stretchers & shrinkers problems was posterior. At last, in the scaffolding teaching, asking questions could help the gifted to strengthen the knowledge models unavailable in students’ solving process and then help them to solve the problems.
Discussion is made on the basis of these findings and some suggestions on future instructions and research are proposed. First, utilizing challenging non-routine proportional problems can inspire the gifted students’ potential and learning motivation. Second, when it comes to the models of problems, solving process and models of knowledge which are hard to grasp well for the gifted, by means of asking questions, the scaffolding teaching can timely help a lot. Third, we can offer the enriched or the accelerated curriculum on the basis of the gifted students’ performance. At last, the researchers can use various research materials and research methods to study how gifted students solve the proportional problems.
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