本研究透過Schoenfeld數學解題歷程，分析國小一般智能資優生數學解題歷程、解題策略、及情意特質，再提供資優班與普通班教師具體的教學建議。六位參與者為高雄市某國小三年級表達能力較佳的資優生。以專家效度方式篩選出四題非例行性數學問題，再讓學生以放聲思考的方式進行數學解題，並採專家信度進行原案分析。
研究結果發現，第一，資優生數學解題大都符合解題歷程六階段，其中四位呈現較多歷程階段比較會解題；更有一位在各題解題過程皆無驗證階段，答對二題；還有一位則較少出現分析、計畫、探討、與驗證階段，僅答對一題。第二，資優生的解題策略具多元性且靈活，資優生在各題解題過程中靈活運用抽象表徵、繪圖表徵、逆推、替代、及嘗試錯誤等策略，來輔助瞭解及探索題意完成解題。第三，資優生的情意特質均具正向態度。六位解題者均具毅力及堅持態度；另外，有四位對數學感到有興趣；有四位對自己具有信心。這些正向態度的情意特質符合Sternberg的創造力特質與Krutetskii的數學天才特質。
教學建議方面，雖然六位學生皆為資優生，但並未所有的一般智能資優生其探討、計畫、與驗證等後設認知都具備。所以，建議資優班教師，可利用數學來訓練學生探討、計畫與驗證等後設認知的能力；善用團體討論的方式，讓學生獲得更多的解題策略；普通班教師可利用資優生的解題思考特質，塑造同儕相互學習的機會與空間。

The purpose of this research is to use Schoenfeld’s mathematics problem-solving model to analyze processes, strategies, and affective characteristics of children in a gifted primary program, and then, to propose concrete suggestions for gifted class and general class teachers. Participants were six third-grade gifted children who were great in articulation, and enrolled in one primary school in Kaohsiung. The investigator analyzed think-aloud protocols of them who solved four non-routine problems selected by several expert teachers.

The findings of this study were three. First, all six gifted students' thought processes mostly conformed to Schoenfeld’s problem-solving model, though with various differences by individuals, and by problems. One of them provided two correct answers, having no verification stage in all problems. And one only provided one correct answer, had less analysis, exploration, design, and verification stage in solving all problems. Second, children exhibited diversified and flexible strategies. They used representing, drawing figures, working backward, introducing auxiliary element, and attempting mistakes to solve four non-routine mathematical problems. Last, the affective characteristics of students were positive. They were patient and perseverant and showed personal mathematics curiosity, excitement, and confidence, which were given as creative characteristics by Sternberg, and as mathematical talent or characteristics by Krutetskii.

The investigator concluded that not all gifted students possessed meta-cognition ability: including exploration, design, and verification. The gifted class teachers could use non-routine mathematics problems to discipline students' meta-cognitive ability, including exploration, design, and verification, and encourage them to generate more solving strategies by group discussion in class. Finally, the general class teachers could adopt problem-solving characteristics of gifted students as materials for gifted students and general students to learn together in class.

第一章 緒論……………………………………… 1
第一節 研究背景與動機……………………………………… 1
第二節 研究目的與待答問題………………………………… 8
第三節 名詞釋意……………………………………………… 9
第二章 文獻探討………………………………… 11
第一節 數學解題的意義……………………………………… 12
第二節 解題歷程與相關研究………………………………… 15
第三節 解題策略與相關研究………………………………… 34
第四節 資優生解題特質與相關研究………………………… 38
第三章 研究方法與設計………………………… 43
第一節 研究方法……………………………………………… 43
第二節 研究架構……………………………………………… 45
第三節 研究對象……………………………………………… 48
第四節 研究工具……………………………………………… 49
第五節 資料蒐集與分析……………………………………… 55
第六節 研究步驟與流程……………………………………… 58


第四章 研究結果與分析………………………… 60
第一節 一般智能資優生數學解題歷程分析………………… 60
第二節 一般智能資優生數學解題策略之分析……………… 102
第三節 一般智能資優生數學解題之情意特質分析………… 108
第五章 結論與建議……………………………… 113
第一節 有關資優生解題的發現……………………………… 113
第二節 資優生教學與研究方向建議………………………… 115
參考書目…………………………………………… 118
一、中文部分…………………………………………………… 118
二、英文部分…………………………………………………… 121
附錄………………………………………………… 127
附錄一 專家篩選非例行性數學題目題本…………………… 127
附錄二 正式研究之情境佈題題本…………………………… 129
附錄三 逐字稿………………………………………………… 131

中文部分：
毛連塭（2001）。如何實施資優教育。台北：心理出版社。
吳武典(2003)。三十年來的台灣資優教育。資優教育季刊，88，1-5。
吳武雄(2001)。資優學生的情緒特質與輔導。資優教育簡訊，3，1-4。
李伯黍等譯（1983）（Krutetskii著）。中小學生數學能力心理學。上海：上海教育出版社。（原著出版年：1976年）。
汪榮才（1990）。增進智力的教學：Sternberg智力三層面論的應用。台南師院學報，23，29-42。
香港特別行政區教育統籌局(2003)。校本資優課程教師培訓教材。2003年9月2日，取自：http://prod1.e1.com.hk/education5/Maths_TTP/pii.htm。
孫達剛（1992）。雄中、雄女學生數學解題之研究。國立高雄師範大學數學教育研究所碩士論文，未出版，高雄。

徐則洲、陳潔雲、李金生、李濟元(2000a)。小學奧林匹克讀本(二年級)。新竹市：凡異出版社。
徐則洲、陳潔雲、李金生、李濟元(2000b)。小學奧林匹克讀本(三年級)。新竹市：凡異出版社。
中華民國特殊教育法（1984）。
中華民國特殊教育法（1997）。
高雄市特殊教育資訊網(2003)。特教班統計概況。2003年9月5日，取自：http://www.spec.kh.edu.tw。
教育部(2000)。國民中小學九年一貫課程暫行綱要。台北：教育部。
黃光雄等譯(2001)。Bogdan R.C. & Biklen S.K.著。質性教育研究理論與方法。嘉義：濤石文化。（原著出版年：2001年）
黃敏晃（1991）。淺談數學解題。教與學，23，2-15。
黃敏晃(2000)。規律的尋求。台北：心理出版社。
黃瑞琴（1994）。質的教育研究方法。台北：心理出版社。
劉貞宜(2001)。數學資優生的解題歷程分析—以建中三位不同能力的數學資優生為例。資優教育研究，2(1)，97-120。
謝淡宜(1998)。小學五年級數學資優生與普通生數學解題時思考歷程之比較。臺南師院學報，31,225-268。
謝淡宜(1999)。國小數學資優生及普通生「數學解題」歷程之比較(四年級)。臺南師院學報，32,297-367。

英文部分：
Cohen, S.A., & Stover, G. (1981). Effects of teaching sixth grade students to modify format variables of math word problems. Reading Research Quarterly, 16, 175-199.
Gallagher J.J. (1985). Teaching the Gifted (3rd). Newton, MA:Allpn and Bacon Inc.
Goldin G.A. (1982). Department of Mathematical Sciences Northern Illinois University. In F.K. Lester, & J. Garofalo, Mathematical Problem Solving Issues in Research（pp.87-101）. Philadelphia, Pa.: Franklin Institute Press.
Greenes, C. (1981). Identifying the gifted student in mathematics. Arithmetic Teacher, 28, 14-18.

Gruber, H.E.（1986）. The self-construction of the extraordinary. In R.J. Sternberg and J.E. Davidson (Eds.), Conceptions of giftedness. Cambridge University Press.
Johnson, H. A. (1980). The nature of student monitoring process in mathematical problem solving tasks. (Doctoral Dissertation, Cloumbia University.) University Microfilms International.
Kilpatrick, J. (1967). Analyzing the solution of word problems in mathematics: An exploratory study. Dissertation Abstracts International, 28(11), 4380A.
Kilpatrick, J. (1985). A retrospective account of the past twenty-five years of research on teaching mathematical problem solving. In E.A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp.1-15). Hilsdale, N.J.: Erlbaum.
Krutetskii, M.A. (1963). Some characteristics of the thinking of pupils with little capacity for mathematics. In B. Simon, & J. Simon (Eds.), Educational psychology in the USSR. Stanford, Calif.: Stanford University Press.
Krutetskii, V.A. (1969). An analysis of the individual structure of mathematical abilities in school children. In J. Kilpatrick, & I. Wirszup (Eds), Soviet studies in the psychology of learning and teaching mathematics (Vol.2). Stanford, Calif.: School Mathematics Study Group.
Krutetskii, V.A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago: University of Chicago Press.
Lester, F.K.（1980）. Problem Solving: Is it a problem? In M.M. Lindquist (Eds.), Selected issues in mathematics education (pp.29-45). Berkeley, Calif: McCutchan Pub. Co.
Lester, F.K.（1994）. Musings about mathematical problem-solving research: 1970-1994. Journal for Research in Mathematics Education, 25(6), 660-675.
Mayer, R.E. (1982). The Psychology of Mathematical Problem Solving. In F.K. Lester, & J. Garofalo, Mathematical Problem Solving Issues in Research. Philadelphia, Pa.: Franklin Institute Press.
Mayer, R.E. (1985). Implications of cognitive psychology for instruction in mathematical problem. In E.A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp.123-138). Hilsdale, N.J.: Erlbaum.
National Council of Supervisors of Mathematics. (1977). Position paper on basic mathematical skill. Arithmetic Teacher, 25, 19-22.
National Council of Teachers of Mathematics (NCTM) (2000). Principles and standards for school mathematics. Retrieved November 20, 2003 from http://standards.nctm.org/document/prepost/cover.htm.
Polya, G.（1945）. How to solve it. New York: Doubleday.
Reid, C., & Romanoff, B. (1997). Using multiple intelligence theory to identify gifted children. Educational Leadership, 55(1), 71-74.
Renzulli, J. S. (1978). What makes giftedness? Re-examining a definition. PhiDelta Kappan, 60, 180-184.
Schoenfeld, A.H.（1985）.Mathematical problem solving. Orlando, Fla.: Academic Press.
Schoenfeld, A.H.(1989). Explorations of students’mathematical beliefs and behavior. Journal for Research in Mathematics Education, 20(4), 338-355.
Sriraman B. (2003). Mathematical Giftedness, Problem Solving, and the Ability to Formulate Generalizations: The Problem-Solving Experiences of four Gifted Students. The Journal of Secondary Gifted Education, 3(14), 151-165.
Sternberg, R.J.（1987）. Teaching intelligence: The application of cognitive psychology to the improvement of intellectual skills. In Baron and R.J. Sternberg (Eds.), Teaching thinking skills: Theory and practice. New York: W.H. Freeman and Company.
Sternberg, R.J. (2000). Identifying and developing creative giftedness. Roeper Review, 23(2), 60-84.
Sullivan, P. (1997). Open-ended maths activities: Using good questions to enhance learning. New York: Oxford University Press.
Vernon, P.E.（1970）. The structure of human abilities（2nd ed）. Landon: Methuen.
Vlahovic-Stetic V., Vidovic V.V., Arambasic L. (1999). Motivational Characteristics in Mathematical Achievement: a study of gifted high-achieving, gifted underachieving and non-gifted pupils. High Ability Studies, 10(1), 37-49.
Webb, Norman. L. (1975). An Exploration of Mathematical Problem Solving Processes. (ERIC Document Reproduction Service No. ED106148)