國小六年級資優生在比例問題之解題表現

The Performance on Proportional Problem Solving: A Case Study in the Gifted Sixth Graders

翁若溱(JO-CHEN WENG)

指導教授 ： 蔡清田

國立中正大學/教育學院/課程研究所/博士(2012年)

摘要

本研究旨在探討國小資優生在比例問題的解題表現，運用立意抽樣，選取二位六年級學生為對象進行個案研究，並運用結構式工作單為基礎的方法進行訪談。研究者以比例的非例行性問題為主軸，設計關聯集合、合成量數、部分整體，以及放大縮小等四種問題類型作為題材，依據難易順序讓個案解題，並針對其解題表現進行初步分析，以作為後續訪談或鷹架教學的基礎。在訪談或鷹架教學告一段落後，研究者依據Mayer(1992)提及的四種解題歷程作為資優生解題表現之分析架構，並探討解題歷程中涉及的五種知識類型。
本研究之結論共有七點：第一、從問題類型的難易度來看，合成量數問題最簡單，部分整體問題最困難，然而後者是兩位資優生認為最具挑戰性的問題類型；第二、在解題歷程中，基本上兩位資優生都能呈現四種解題階段，而較容易產生困難的是問題整合階段與解答計劃和監控階段；第三、在知識類型方面，基本上兩位資優生都能運用五種知識類型解題，但較無法掌握的是基模知識和策略知識；第四、在綜合問題類型與解題歷程方面，兩位資優生感到最困難的是部份整體的問題整合階段，其次是關聯集合與放大縮小等兩種問題類型的解答計劃與監控階段；第五、在綜合解題歷程與知識類型方面，兩位資優生較容易產生困難的是問題整合階段所需的基模知識，以及解答計劃與監控階段所需的策略知識，而解答計劃與監控階段，可能還需運用「執行策略知識所需的語意知識」；第六、在綜合問題類型與知識類型方面，兩位資優生最感到困難的是部分整體問題所需的基模知識，其次是關聯集合與放大縮小等兩類問題所需的策略知識；第七、透過提問之鷹架教學來加強解題歷程中所缺乏的知識類型，似乎能幫助資優生解題。
本研究之建議共有四點：第一、運用具挑戰性的非例行比例問題，激發資優生的潛能與學習動機；第二、針對資優生解比例問題時較容易產生困難的問題類型、解題歷程或知識類型，適時運用提問之鷹架教學協助；第三、依據資優生的解題表現，提供充實或加速的課程；第四、運用不同研究題材與研究方法，進行資優生解比例問題之研究。

關鍵字

知識類型 ； 解題歷程 ； 問題類型 ； 解題表現 ； 比例問題 ； 資優生

並列摘要

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.

並列關鍵字

solving process ； models of problems ； performance on problem solving ； proportional problems ； gifted students ； knowledge models

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