臺灣博碩士論文加值系統

本研究目的是藉由自編的「平均數問題工作單」進行調查研究，以探討某高職271位學生在平均數問題的解題類型和所佔的比例，並分析其中3位個案學生的解題表現。研究結果如下：
高職學生面對熟悉的問題的解題類型，大多以「總和/個數」求解(75%)；錯誤類型，則多數未考慮權值(9%)與欠缺程序性知識(11%)。面對不熟悉的平均速率問題，多數以假設兩地距離方式求解(4%)；在解「有相同增量」的問題，多數以「調整前、後總分」求解(3%)。面對等差數列的平均數問題，可能因欠缺等差數列的平均數即是中位數的先備知識，導致解題發生困難。面對「由各數與最小數差的總和求資料個數」的平均數問題，多以「直觀」列式求解。
個案學生的解題表現：小維解熟悉且抽象問題時，以嘗試錯誤法解題，但因未「控制」所代入數字，以致解題失敗。小琪解不熟悉且具體的問題時，因欠缺對答案的「控制」，誤以為答案需再加1，以致解題失敗。小政解各資料均以相同增量調整的不熟悉且抽象問題時，欠缺多餘訊息的解題經驗，即「資源」不足，而解題失敗。

This study aimed to investigate 271 vocational high school students’ ability of solving average problem and the percentage of the types of problem solving by using a self-designed “Average Problem Worksheet”. According to the analysis of resources, heuristics, and control of 3 case students’ performance of problem solving, the research findings could be found.
Problem solving could be divided into three types when the students dealt with the familiar average problems: 75% students chose to use “total sum divided by the number”; students of using error type were 9% due to the lack of considering “weighted”; 11% students did not have procedural knowledge. When facing unfamiliar questions of average speed, most students solved the problems by the assumption of the distance in between two locations (4%); when facing “restoration average problems”, most students used the method of “work backward” (3%). In terms of the average problems of arithmetic progression, students had difficult solving it possibly because they did not have prior knowledge of arithmetic progression’s median. Referring to the averaging problems of “seeking the total sum of each number and the minimum number of the sum of the difference”, the students tended to use ‘intuitive method’.
Regarding the performance of the case students who failed to solve the average problems, Hsiao-Wei solved familiar and abstract problems by using the method of error type. The results were close or far from the average problems because he did not control the used numbers. Next, Hsiao-Chi solved unfamiliar and concrete problems due to the lack of control towards the answer. Finally, Hsiao-Cheng solved unfamiliar and abstract problems such as “restoration average” because he had insufficient resources and did not identify the extra message.

中文摘要 i
英文摘要 ⅱ
目次 ⅲ
表次 ⅴ
圖次 ⅵ
第一章 緒論
第一節 研究背景與動機 1
第二節 研究目的與研究問題 3
第三節 名詞釋義 4
第四節 研究範圍與研究限制 5
第二章 文獻探討
第一節 問題解決 7
第二節 數學解題 12
第三節 平均數問題 22
第四節 學生在平均數問題的認知概念與解題表現 32
第三章 研究方法
第一節 研究方法與研究架構 39
第二節 研究對象 42
第三節 研究工具 44
第四節 研究實施 49
第五節 資料蒐集、整理與分析 50
第四章 研究結果與討論
第一節 高職一年級學生解平均數問題的解題類型和所占的比例 52
第二節 個案學生在平均數問題的解題表現 79
第五章 結論與建議
第一節 結論 95
第二節 建議 100
參考文獻
中文部分 102
英文部分 104
附錄
附錄一 平均數問題工作單 108

中文部分
Skemp, R. R. (1987) 。數學學習心理學(陳澤民譯)。台北市：九章。
林碧珍 (2001) 。教師如何培養學生形成數學問題的能力。國教世紀，198，5-14。
邱皓政 (2000) 。量化研究與統計分析。台北市：五南圖書出版公司。
張少同 (2003) 。青少年數學概念學習研究－子計畫七：青少年統計概念學習研究。行政院國家科學委員會專題研究計畫成果報告(報告編號：NSC 91-2521-S-003-007-)。台北市：國科會。
張少同 (2005) 。青少年數學概念學習與教學之研究－子計畫三：青少年統計概念學習與教學之研究(1/3)。行政院國家科學委員會專題研究計畫期中進度報告(報告編號：NSC 93-2521-S-003-014-)。台北市：國科會。
張平東 (1989) 。國小數學教材教法新論。台北市：五南圖書出版公司。
張春興 (1994) 。教育心理學。台北市：東華書局。
張英傑 (2009) 。國民小學數學第十二冊(六下)。台南市：南一書局。
張鴻林 (2001) 。澳洲數學競試(AMC)試題解析。台北：九章出版社。.
教育部 (2003) 。國民中小學九年一貫課程綱要數學學習領域。台北市：教育部。
教育部 (2009) 。職業學校群科課程綱要暨設備基準－一般科目。台北市：教育部。
教育部 (2010) 。國民中小學九年一貫課程綱要數學學習領域。台北市：教育部。
黃敏晃 (1991) 。淺談數學解題。教與學，23，2-15。
黃寶園 (2009) 。心理與教育研究法。臺北市：華立圖書。
楊世瑩 (2010) 。SPSS統計分析實務。台北市：旗標出版股份有限公司。
部編版 (2010) 。小六下(第十二冊)習作。台北市：。
陳吳煜 (2011) 。數學CⅢ。新北市：泰宇出版。
陳冒海 (2010) 。國中數學第六冊。台南市：南一書局。
郭生玉 (2005) 。心理與教育研究法。台北市：精華。
喻平 (2002) 。論數學解題教學的現代理論基礎。數學傳播，26(4)，60-68。
莊純芬 (2011) 。個案資優生對平均數問題解題表現之研究。未出版之碩士論文，國立嘉義大學數學教育研究所。
劉秋木 (1996) 。國小數學科教學研究。台北市：五南圖書出版公司。
劉錫麒 (1993) 。數學思考教學研究。台北市：師大書苑。
蔡子雲、劉祥通(2007)。資優生在想什麼？－速率篇。資優教育研究，7(1)，29-47。
歐用生 (1995) 。國民小學新課程標準的精神與特色。台北市：台灣省國民學校教師研習會。
陳冒海 (2010) 。國中數學第六冊。台南市：南一書局。
 
外文部分
Arthur, E. N. (2009) 。心理與教育統計學(黃瓊蓉、蘇文賢與江吟梓合譯)。台北市：學富文化.
Barba, R.H. (1990). Problem solving pointer. Science Teacher, 57(7), 32-35.
Beisel, J. D. (2001). An experiment in three approaches to teaching average to elementary school children. School Science and Mathematics, 101(1), 23-31.
Cai, J. (1995). Beyond the computational algorithm: students' understanding of the arithemetic average concept. Paper presented at the Proceedings of the 19th Psychology of mathematics education conference, Sao Paulo, Brazil.
Cai, J. (1998). Exploring students' conceptual understanding of the averaging algorithm. School Science and Mathematics, 98(2), 93-98.
Cai, J. (2002). Intended Treatments of arithmetic average in U.S. and Asian school mathematics textbooks. School Science and Mathematics, 102(8), 391-404.
Campbell, P. F., &; Bamberger, H. J. (1990). The vision of problem solving in the standards. Arithmetic Teacher, 38(9), 14-17.
Carpenter, T.P. et al. (1981). Results from the Second Mathematics Assessment of the National Assessment of Academic Progress. Reston, VA: National Council of Teachers of Mathematics .
Gagné, R. M. (1985). The conditions of learning and theory of instruction (4th ed.). Holt, NY: Rinehart and Winston.
Gagné R. M. &; Driscoll, M. P. (1988). Essentials of learning for instruction (2nd ed.). Englewood Cliffs, NJ: Prentice-Hall.
Gal, I. (1995). Statistical tools and statistical literacy: the case of the average. Teaching Stastistics, 17, 97-99.
Gick, M. L. (1986). Problem-solving strategies. Educational Psychologist, 21, 99-120.
Goodchild, S. (1983). School pupils' understanding of average. Teaching Stastistics, 10, 77-81.
Hiebert, J. , Lefevre, P. (1986). Conceptual and procedural knowledge in mathematics：An introductory analysis, In J. Hiebert (Ed.), Conceptual and procedural knowledge：The case of mathematics, 1-27, Hillsdale, NJ: Lawrence Erlbaum Associates.
Kelley, T. L. (1939). The selection of upper and lower groups for the validation of test item, Educational Psychology, 30, 17-24。
Kilpatrick, J. (1985). A retrospective account of the past 25 years of research on teaching mathematical problem solving. In E.A.Silver(Ed.), Teaching and learning mathematical problem solving：Multiple research Perspectives,1-15. Hillsdale, NJ：Erlbaum .
Klausmeier, H. J., Ghatala, E. S., &; Frayer, D. A. (1974). Conceptual learning and development. New York: Academic.
Lester, F. K. (1980). Research on mathematical problem solving. In R. J. Shumway(Ed.) . Research in Mathematics Education. 286-323, Reston, VA: National Council of Teachers of Mathematics .
Liu, S. T. (1993). Effects of teaching calculator use and problem solving strategies on mathematics performance and attitude of fifth grade Taiwannese male and female students. A Dissertation of the Doctor of Education Degree of University of Memphis, USA.
Makros, J., &; Rusell, S. J. (1995). Children 's concepts of average and representativeness. Journal for Research in Mathematics Education , 26(1), 20-39.
Mayer, R. E. (2003). Thinking, Problem solving, Cognition (2nd ed.). NY: Freeman.
Mevarech (1983). A deep structure model of students' statistical misconceptions. Educational Studies in Mathematics. 14(4). 415-429.
Morris, J. (1983). How to develop problem solving using a calculator. Reston, VA: National Council of Teachers of Mathematics.
National Council of Supervisors of Mathematics. (1977). Position paper on basic mathematical skills. Arithmetic Teacher, 25, 19-22.
National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
National Council Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. (1980). An Agenda for Action. Washington, DC: National Council of Teachers of Mathematics.
Polya, G. (1981). Mathematical Discovery. New York: John Wiley.
Ploya, G. (2006). 怎樣解題(蔡坤憲譯).。台北:天下文化。(原著出版於1945)。
Pollatsek, L., Lima, S. &; Well, A D. (1981). Concept or computation: students' understanding of the mean. Educational Studies in Mathematics, 12, 191-204.
Reed, S. K. (1999). Word Problems. San Diego: Lawrence Erlbaum Associates.
Rosser, R. (1994). Cognitive development: Psychological and biological perspectives. Boston, Massachusetts: Allyn and Bacon.
Russell S. J. (1999). Mathematical reasoning in elementary grade. In L. V. Stiff, &; F. R.Curcio (Eds.), Developing mathematical reasoning in grades K-12, 1-12. Reston, VA: National Council of Teachers of Mathematics.
Schoenfeld, A. H. (1985). Mathematocal problem solving. San Diego, CA: Academic press.
Skemp, R. R. (1976). Relational Understanding and Instrumental Understanding. Mathematics Teaching, 77, 20-26.
Stanic, G., &; Kilpatrick, J. (1988). Historical Perspectives on Problem Solving in the Mathematics Curriculum. In R. I. Charles &; E. A. Silver (Eds.), The teaching and assessing of mathematical problem solving (pp. 1-22). Reston, VA: National Council of Teachers of Mathematics.
Strauss, S., &; Bichler, E.(1988). The Development of Children's Concepts of the Arithmetic Average, Journal for Research in Mathematics Education, 19(1),64-80.
Watson, J. M., &; Moritz, J. B. (1999). The development of concepts of average. Focus on Learning Problems in Mathematics, 21(4), 15-39.
Watson, J. M., &; Moritz, J. B. (2000). The longitudinal development of understanding of average. Mathematical Thinking and Learning, 2(1&;2), 11-50.
Zawojewski. (1986). Putting Research Into Practice in the Elementary Grades: Readings from Journals of the National Council of Teachers of Mathematics. In D. L. Chambers (Ed.), Teaching Statistics: Mean, median, and mode (pp. 238-240). Reston, VA: National Council of Teachers of Mathematics.