本研究旨在探討九年級資優生與一班生的數學解題歷程，分析資優生與一般生在解題過程中所採用的策略之差異，進而提供具有參考價值的教學建議給資優生及一般生的數學教師。本研究採個案研究法，挑選兩位九年級資優班學生及兩位九年級普通班學生作為研究對象。研究者篩選十題非例行性的數學問題，對學生施測以觀察他們在解題歷程中的表現，並透過分析訪談資料深入了解學生解題歷程，施測與訪談皆以錄影的方式記錄。
研究結果發現，第一，資優生的答對率高於一般生，兩類學生大都符合解題歷程四階段，四名學生中只有一名在完成解題後進行驗算與回顧的步驟，其中三名學生則會在計算過程中出現檢驗的動作。第二，資優生的解題策略更具多元性且能夠靈活運用，在解題歷程中能夠巧妙的運用許多不同的先備知識以及策略來完成解題。第三，資優生及一般生具有以下特點：1、資優生能夠全面且迅速的組織題目中的資訊來擬定解題計畫，而一般生時常擷取到片面的資訊。2、一般生容易錯誤使用題目中所給予的數字來進行解題。3、遇到解題困難時，資優生能夠迅速的改變解題策略。4、資優生能夠監控解題結果的合理性。5、一般生在可能會猜測作答或被以前習得的數學知識混淆。6、資優生能夠利用較困難的先備知識進行解題。7、資優生的解題歷程比一般生更為跳躍，會避免多餘的計算。8、資優生在解題歷程中具有更多元的解題策略。9、資優生解題經驗相較於一般生更為豐富。10、資優生具有迅速敏捷的推理能力。
本研究除了對研究結果進行討論之外，也與其他研究進行比較，最後提出對兩類學生的教學建議以及未來的研究建議。在教學方面，資優生與一般生的所使用的解題策略以及先備知識有許多的不同，建議教師可透過解題融入教學來使兩類學生互相學習彼此的優缺點；在未來研究方面，將來可擴大研究範圍及將解題融入教學作為研究主題。

The purpose of this study is to explore problem solving process of gifted and regular students. By analyzing the difference of mathematical problem-solving strategies between gifted and regular junior high school students, the researcher proposed teaching suggestions which have reference value for both mathematics teachers. The participants of this case study were two gifted junior high school students and two regular junior high school students, studying in the ninth grade. To gain insights into mathematical problem solving process, the researcher gave the participants a test based on the chosen ten non-routine math problems. Additionally, we analyzed the data collected from the interview processes were all collected.
The major findings of this study are listed as follows. First, The correct answer rate of the gifted students is higher than that of the regular students. Both of them mostly conformed to Polya's 4-step problem solving process. Only one of the four students reflected and looked back at what he had done after solving the problems, and three of them checked calculations during the process. Second, the gifted students can utilize diverse problem solving strategies, and can flexibly use lots of different advanced concepts and strategies to solve problems. Third, the gifted students and the regular students respectively have the following characteristics: (1) The gifted students are able to intergrate the information in the question stem to devise a plan, but the regular students usually rely on partial information to solve a problem. (2) The regular students are prone to use the wrong quantity given in the question stem to solve a problem. (3) When encountering difficulties in problem solving process, the gifted students can immediately shift problem solving strategies. (4) As for gifted students, they are able to check for the rationaility of their answers. (5) The regular students may answer the questions by guessing or have some misconceptions on the mathematics knowledge which they have acquired. (6) The gifted students have the ability to use difficult prior knowledge. (7) The problem solving process of the gifted students is non-linear, and they can avoid many redundant calculations. (8) The gifted students can utilize more diverse problem solving strategies. (9) The problem solving experience of the gifted students are more profound than the regular students.(10) The gifted students have better reasoning ability.
This study also includes discussions of the results, compared with other research. Some suggestions are proposed for mathematic teaching of the gifted and regular students, and some suggestions are provided for future research. In terms of teaching, the gifted and regular students use different problem solving strategies and advanced concepts, so the suggestions is that teachers can integrate problem solving into mathematical teaching, and make them learn from each other's advantages and disadvantages. In future researches, the areas of the research can be expanded, and the integration of problem solving into mathematical teaching can be studied.

摘要
目次
第一章 緒論---------------------1
第一節 研究背景與研究動機--------1
第二節 研究目的與待答問題--------3
第三節 名詞釋義-----------------4
第四節 研究範圍與限制------------6
第二章 文獻探討------------------8
第一節 數學解題------------------8
第二節 數學解題歷程--------------10
第三節 資優生與一般生的解題特質---21
第四節 實徵研究------------------27
第三章 研究方法------------------31
第一節 個案研究法----------------31
第二節 研究流程------------------32
第三節 研究情境與對象-------------35
第四節 研究工具------------------36
第五節 資料蒐集與分析-------------39
第四章 研究結果-------------------45
第一節 資優生與一般生的數學解題歷程-45
第二節 資優生和一般生的解題歷程差異-106
第五章 結論、討論與建議------------115
第一節 結論-----------------------115
第二節 討論-----------------------117
第三節 建議-----------------------118
參考文獻---------------------------121
中文文獻---------------------------121
英文文獻---------------------------123
附錄一：數學預試測驗試題-------------125
附錄二：數學測驗試題----------------127
附錄三：學生家長同意書--------------131
附錄四：逐字稿----------------------132
附錄五：試卷施測訪談逐字稿-----------133

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