臺灣博碩士論文加值系統

1980年起，擬題活動逐漸受到國際間的重視。想出一個數學題目出來，即為數學擬題，而其中自由擬題的出題有著強烈的生活連結。針對近年來台灣學生數學學習成就高而興致低迷的情況，擬題活動可能作為改善學生態度的一種教學方法。另一方面，機率作為處理不確定性事件的數學基礎，也特別容易與生活連結產生。

本研究目的在以自由擬題為學習活動，提升學生學習數學的興趣，並了解機率概念的發展。因此，本研究將探討：1 )學生所擬的題目有何進展？2 )機率思維有何發展？3)面對超過國中範圍之問題，解題的能力為何？

本研究對台北市一所國中，兩個九年級班級共62名學生，進行自由擬題學習活動。在為期3個月，共10節課的活動歷程裡，讓學生進行機率方向的擬題。蒐集資料的工具有機率裸測試題、學生擬題單及擂台賽答題單三種。本研究方法採用內容分析法，以「機率自由擬題評量規準」為研究工具，量化學生擬題各向度表現，並使用成對樣本t檢定來分析擬題版本之間的變化。

根據研究結果顯示，第一，學生的擬題品質在可解性、可讀性、精緻性與機率概念層次有所進展，但生活性可能從一開始就具備良好的品質，所以沒有明顯的進步數據。第二，學生的機率思維，也可藉由一題多磨的擬題歷程得到提升與發展。第三，學生有能力解決超過國中課程的問題。最後再根據研究結果於自由擬題活動、未來機率課程提出若干建議。

Since 1980, the problem-posing activities have gradually received international attention. Think of a math problem, which is the mathematical problem-posing, and particularly the free problem-posing has strong life link. In recent years, Taiwan students have high achievements and low attitudes for mathematics learning, the problem-posing activities may serve as a teaching method to improve students' attitudes. On the other hand, Probability as the mathematical basis for dealing with uncertainty events. It is especially easy to link with life.

The purpose of this thesis in the free problem-posing, enhance students' interest in learning mathematics, and understand the development of the concept of probability. Therefore, the two research questions are: 1) To what extend do student’s problem-posing improve? 2) What is the development of probability thinking? 3) When facing the problems exceeds the junior high school level, what is the ability to solve the problem?

The research is conducted to 62 ninth grade students from two classes of junior high school of Taipei City. In a period of 3 months, a total of 10 lessons in the course of activities, so that students in the direction of the probaility problem-posing. The following data are collected : the probability of naked test questions, problems between versions and answers of the probability challenge. The results were analyzed in a content analysis, evaluate the quality of the problems by posers, And use the paired sample t test to analyze the problem between versions.

The results show that: 1) The student-posed problems made significant progress throughout the three-version process of “solvability”, “readability”, “Exquisite”, “Probability level”. 2) The probability of students thinking can enhance and develop by the process of problem-posing of “one problem, many revised versions”. 3) Students have the ability to solve problems beyond the course of the junior high school.

Finally, the researcher offers some suggestions related to the free problem-posing, probability course in the future.

第一章 緒論 1
第一節 研究動機與背景 1
第二節 研究目的 3
第三節 研究問題 4
第四節 名詞釋意 4
第二章 文獻探討 7
第一節 數學擬題 7
第二節 機率概念研究 14
第三章 研究方法 21
第一節 研究對象 21
第二節 研究活動設計 21
第三節 研究工具 26
第四節 擬題題目範例解說 30
第五節 資料收集 35
第六節 資料分析 38
第四章 研究結果 41
第一節 機率擬題品質的進展 41
第二節 有修改題幹者擬題品質分析 61
第三節 機率概念層次 67
第四節 結果討論 80
第五章 結論與建議 85
第一節 結論 85
第二節 建議 86
參考文獻 89
一、中文部分 89
二、英文部分 90
附錄一、單維彰老師主編---機率擬題學習單20151021 93

參考文獻
一、中文部分

丁村成 (2008)。走一趟高中機率教學之旅。數學傳播，32(4)，33-50。
江家瑋 (2014)。數學擬題活動的合作效果─五年級學童之經驗。(未出版之
碩士論文)。取自http://ir.lib.ncu.edu.tw/handle/987654321/65895#.
WV5K1IiGPIU
呂溪木 (1986)。變遷時代中我國數學課程的發展。臺灣省國民學校教師研習
會三十年紀念專刊。新北市：台灣省國民學校教師研習會。
呂溪木 (2007)。民國 75年之前我國數學課程演變。論文發表於「吳大猷先生
百歲冥誕科學教育學術研討會～我國近五十年之科學教育發展」研討會，
臺灣師範大科學教育所，臺北市。
周祝瑛 (2003)。台灣教育改革之研究。論文發表於「民辦教育」研討會。上
海華東師範大學。取自http://www3.nccu.edu.tw/~iaezcpc/
C-%20The%20research%20of%20taiwan%20education%20revolution%201.htm
陳斐卿、江家瑋、張鐵懷、黃佩岑、單維彰 (2015)。數學自由擬題之設計
與評量—一個合作的取徑。科學教育學刊，23(2)，185-211。
陳玟樺 (2017)。民國五十至八十年代（1961-2000年）數學課程改革之探究。教
育部普通高級中學課程 數學學科中心 (臺北市立建國高級中學)。
翁秉仁 (2016)。小朋友適合學機率嗎？科學人雜誌，170。取自http://sa.ylib.com/
MagArticle.aspx?Unit=columns&id=3058
教育部 (1964)。高級中學生物、化學、物理教材編輯大綱及數學教材大綱。臺
北市：正中書局。
教育部 (1971)。高中數學課程標準。臺北市：正中書局。
教育部 (1975)。國民小學課程標準。臺北市：正中書局。
教育部 (2008)。國民中小學九年一貫課程綱要數習領域。取自http://
teach.eje.edu.tw/data/files/class_rules/math.pdf
教育部 (2014)。十二年國民基本教育課程綱要總綱。取自http://www.naer.
edu.tw/ezfiles/0/1000/attach/87/pta_5320_2729842_56626.pdf
梁淑坤 (1994)。「擬題」的研究及其在課程的角色。國民小學數學科新課程概說
(低年級)。新北市：台灣省國民學校教師研習會。
許哲毓、單維彰、劉柏伸 (2016)。樹狀圖在機率教學的應用-臺灣與英國教科書
之比較。第四屆師資培育國際學術研討會，國立臺灣大學
單維彰、陳斐卿、許哲毓 (2017)。以裸測與擬題探討九年級學生的自發性機率
概念。科技部專題研究成果報告(編號：NSC-104-2511-S-008-002-MY2)，
未出版。
甯平獻 (2010)。數學教材教法。臺北市：五南。
劉秋木 (1996)。 國小數學科教學研究。臺北市：五南。
鄭章華、單維彰 (2015)。素養導向之數學教材初探。邁向十二年國教新課綱的
第一哩路：從課綱轉化到學校教育的系統性變革學術研討會，國家教育研究
院。

二、英文部分

Bognar & Nemetz(1977). On the teaching of probability at secondary level.
Educational Studies in Mathematics, 8 , 399-404.
Bush, W., & Fiala, A. (1986). Problem Stories: New Twist on Problem Posing. The
Arithmetic Teacher, 34(4),6-9.
Brown , S. I. & Walter, M. I. (1993). Problem Posing in Mathematics Education. In
S. I. Brown & M. I. Walter (Eds.) Problem Posing: Reflection and
Applications 16-27, Hillsdale, New Jersey: Lawrence Erlbaum Associates.
Charles,R . I., & Silver, E. A. (Eds.). (1988). Thet eachinga nd assessing of
mathematical problem solving.Reston, VA: National Council of Teachers of
Mathematics.
Cunningham, R. (2004). Problem posing: an opportunity for increasing student
responsibility, Mathematics and Computer Education, 38(1), 83-89.
Cifarelli, V. & C. Sheet,(2009). Problem Posing and Problem Solving: A Dynamic
Connection. Sch. Sc. Math, 109(5): 245-246.
Ellerton, N. F. (1986). Children's made-up mathematics problems: a new perspective
on talented mathematicians. Educational Studies in Mathematics, 17(3),
261-271.
English. L. D. (1996). Children’s problem posing and problem solving preferences, In
J. Mulligan & M. Mitchelmore (Eds.), Research in Early Number Learning
,227-242. Australian Association of Mathematics Teachers.
English, L. D. (1997). The development of fifth-grade children’s problem-posing
abilities. Educational Studies in Mathematics, 34(3), 183–217
Hart, K. (Ed.). (1981). Children's understanding of mathematics: 11-16.
London: John Murray.
Hashimotto, Y. (1987). Classroom parctice of problem solving in Japanese elementary
schools. In J. P. Becker & T. Miwa (Eds), Proceedings of the U. S.-Japan
seminar on mathematical problem solving (pp. 94-119). Carbon dale, IL:
Southern Illinois University.
Jones, G. A., Thornton, C. A., Langrall, C. W., & Tarr, J. E. (1999). Understanding
students' probabilistic reasoning. In L. V. Stiff, & F. R. Curcio (Eds.),
Developing mathematical reasoning in Grades K-12: 1999 Yearbook (pp.146-
155). Reston, VA: National Council of Teachers of Mathematics.
Li, J. (2000)., Chinese students understanding of probability, Unpublished doctoral
dissertation,National Institute of Education. Nanyang Technological University,
Singapore, 2000.
Lavy, I. & Shriki, A. (2007). Problem posing as a means for developing mathematical
knowledge of prospective teachers. In W. Jeong-Ho, P. Kyo-Sik, L. Hee-Chan
& S. Dong-Yeop (Eds.), Proceedings of the 31th international conference on
the Psychology of Mathematics Education (PME) (pp. 129–136). Seoul, South
Korea, III.
NCTM (1989). Curriculum and Evaluation Standards for School Mathematics
National Council of Teachers of Mathematics Commission on Standards for
School Mathematics. Reston, VA: NCTM
NCTM (2000). National Council of Teachers of Mathematics (2000). Principles and
standards for school mathematics. Reston, VA: NCTM
Piaget, J., & Inhelder, B. (1975). The origin of the idea of chance in children (L. J.
Leake & P. D. Burrell & H. D. Fischbein, Trans.). London: Routledge & Kegan
Paul. (Original work published 1951)
Reitman, W.(1965). Cognition and thought. New York：Wiley.
Shaughnessy, J. M. (1992). Research in probability and statistics: Reflections and
directions. In D. A. Grouws (Ed.), Handbook of research on mathematics
teaching and learning (pp. 465-494). New York: National Council of Teachers
of Mathematics and MacMillan.
Silver, E. (1993). On mathematical problem posing. In I. Hirabayashi, N. Nohda, K.
Shigematsu, & F.L. Lin (Eds.). Proceedings of the Seventeenth International
Conference for the Psychology of Mathematics Education (Vol. 1, pp. 66-85).
Tsukuba (Japan): International Group for the Psychology in Mathematics
Education.
Silver, E. A. (1994). On mathematical problem posing, For the Learning of
Mathematics, 14(1), 19-28
Silver, E. A (1995). The nature and use of open problems in mathematics education:
Mathematical and pedagogical perspectives.International Reviews on
Mathematical Educqtion, 2, 67-72.
Silver, E. A, & Cai, J. (1996). An analysis of arithmetic problem posing by middle
school students. Journal for Research in Mathematics Education, 27, 521-539.
Silver, E. A., Mamona-Downs J., Leung, S. S. & Kenney P. A. (1996). Posing
Mathematical Problems: An Exploratory Study Journal for Research in
Mathematics Education, 27 (3), 293-309.
Stoyanova, E., & Ellerton, N. F. (1996). A framework for research into students’
problem posng in school mathematics. In P. C. Clarkson (Ed.), Technology in
mathematics education (pp. 518-525). Mathematics Education Research Group
of Australasia. The University of Melbourne.
Mullis, I. V. S., Martin, M. O., Foy, P., & Hooper, M. (2016). TIMSS 2015 International Results in Mathematics. Retrieved from Boston College,
TIMSS & PIRLS International Study Center website:
http://timssandpirls.bc.edu/timss2015/international-results/
Writz, R. W. & Kahn, E. (1982). Another look at application in elementary school
mathematics. Arithmetic Teacher, 30,21-25.