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研究生:李聿晟
研究生(外文):LEE,YU-CHENG
論文名稱:極限與連續解題策略與診斷之研究
論文名稱(外文):Problem-Solving Strategies and Error Diagnosis for Limits and Continuity
指導教授:曾勵新
指導教授(外文):TSENG,LI-HSIN
口試委員:李英豪張桂芳
口試委員(外文):LEE,IN-HAOCHANG,KUEI-FANG
口試日期:2017-06-22
學位類別:碩士
校院名稱:逢甲大學
系所名稱:應用數學系
學門:數學及統計學門
學類:數學學類
論文種類:學術論文
論文出版年:2017
畢業學年度:105
語文別:中文
論文頁數:33
中文關鍵詞:建構反應題錯誤類型適性診斷學習專家知識結構函數定義域函數極限函數連續
外文關鍵詞:constructed-response testserroneous categoriesaptitude diagnosis learning,expert knowledge structurefunction domainfunction limitsfunction continuity
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本研究源自科技部 104 年度計畫 “微積分多重解題策略診斷與補救教學系統” 之成果報告,原計畫是由張桂芳教授主持,研究生助理為陳建樺、李聿晟,及博士生楊期壹。
本研究旨在編製大學微積分當中的極限與連續兩大主題之建構反應題型測驗試題、施測收集作答資料、分析建構反應題策略及錯誤類型類別與自動化分析反應題型機制之建立。期望有助於授課教師能夠據此發現、瞭解、及消除學習者在學習中的困難及迷思,而且將可以用來建立補救教學系統,或建構線上的適性診斷學習軟體。
本研究成果為:建立極限和連續的子技能表、專家知識結構、15 題極限和連續的建構反應題。並已收集約 200 份逢甲大學大學部學生施測結果,分析極限和連續的建構反應題策略及錯誤類型。依建構反應題策略及錯誤類型建置相關之命題卡。

The thesis focuses on improving the efficiency of teaching the limits and continuity in college Calculus, by setting up constructed-response test questions, collecting response data thereof, and analyzing strategies and erroneous categories. In this way, not only the instructors can discover/understand/ease students’ learning difficulty/misconceptions, but also a remedial teaching system or on-line software for aptitude diagnosis learning can be thus implemented.
The results are as follows. A sub-skill table and expert knowledge structure for limits and continuity, and fifteen constructed-response test questions are established. Test results (around 200 copies) from Feng-Chia undergraduate students have been analyzed for strategies and erroneous categories, and corresponding question cards have been created.
Being one of the program research assistants for Professor Kuei-Fang Chang’s Ministry of Science and Technology Projects, part of the thesis appears also in the project’s research report (本研究源自科技部 104 年度計畫 “微積分多重解題策略診斷與補救教學系統” 之成果報告,原計畫是由張桂芳教授主持,研究生助理除了本人外,另有陳建樺,及博士生楊期壹).

第一章 動機與背景 1
1.1 背景 1
1.2 文獻探討 2

第二章 子技能與知識結構 8
2.1 極限子技能與連續子技能 8
2.2 極限知識結構與連續知識結構 9

第三章 命題卡與實測分析 12
3.1 建置建構反應題 12
3.2 分析建構反應題策略及錯誤類型類別 13
3.3 極限與連續的錯誤類型與迷失概念 22

第四章 結果與討論 24

參考文獻 25

一、中文文獻:
[1] 呂溪木(1983)。從國際科展看我國今後科學教育的發展方向。科學教育月刊,64,13-19。
[2] 邱佳寧(2001)。國小數學學習障礙學生解題策略之研究(未出版之碩士論文)。國立彰化大學,彰化。
[3] 李隆生 (2000) 大學生對數學和微積分的認知 。數學傳播,24,3。
[4] 林清山、張景媛(1994)。國中生代數應用題教學策略效果之評估。國立台灣師範大學教育心理與輔導學系教育心理學報,27,35-62。
[5] 莊峰魁、王文卿、劉育隆、郭伯臣 (2010)。「光」單元之電腦化建構反應試題與診斷模式開發初探。2010電腦與網路科技在教育上的應用研討會。國立新竹教育大學。新竹市。
[6] 黃雅琪 (2006)。高三學生矩陣基本運算及應用錯誤類型之分析研究。雄工學報 7 ,175-188。
[7] 陳淑琳(2010)。國小二年級學童乘法文字題解題歷程之研究-以屏東市一所國小為例(未出版之博士論文)。國立屏東教育大學,屏東縣。
[8] 彭子怡(2007)。二維形狀之美-幼兒型式積木之幾何解題歷程(未出版之碩士論文)。國立臺南大學,臺南市。
二、外文文獻:
[9] Bagni, C. (2005). Historical roots of limit notion, developments of its representation registers and cognitive development. Canadian Journal of Science, Mathematics and Technology Education, pages 453-468. 1, 2.1, 2.3, 2.7, 4.1.4
[10] Bergsten, C. (2008). How do theories inuence the research on teaching andlearning limits of functions? ZDM Mathematics Education, 40:189-199.2.3
[11] Bezuidenhout, J. (1998). First-year University Students’ Understanding of Rate of Change in International Journal of Mathematical Education in Science and Technology 29(3): 389-399
[12] Chi, M. T. H., & Roscoe, R. D. (2002). The process and challenges of conceptual change. In M. Limon & L. Mason (Eds.), "Reconsidering conceptual change: Issues in theory and practice" (pages. 3-27). Dordrecht: Kluwer
[13] Cornu, B. (1991). Limits in Advanced Mathematical Thinking, edited by Tall, D. ME Library: Kluwer Academic Press: 153-166
[14] Cottrill, J; Dubinsky, E; Nichols, D; Schwingendorf, K; Thomas, K & Vidakovic, D. (1996). Understanding the Limit Concept: Beginning with a Coordinated Process Scheme in Journal of Mathematical Behaviour 15:167-192
[15] Davis, R.B. & Vinner, S. (1986). The Notion of Limit: Some Seemingly Unavoidable Misconception Stages in Journal of Mathematical Behaviour 5: 281-303
[16] Jordaan, T. (2005). Misconceptions of the Limit Concept in a Mathematics Course for Engineering Students. Master’s Thesis, University of South Africa
[17] Ketterlin-Geller, L. R. & Yovanoff, P. (2009). Diagnostic Assessments in Mathematics to Support Instructional Decision Making. Practical Assessment, Research & Evaluation, 14(16). Available online: http://pareonline.net/getvn.asp?v=14&n=16
[18] Knuth, D. (1968). The Art of Computer Programming, Vol 1. Addison-Wesley.
[19] Laridon, PE. (1992). Teaching Calculus in Moodley, M, Njisane, RA & Presmeg, NC (eds) Mathematics education for Inservice and Preservice Teachers. Pietermaritzburg: Shuter & Shooter: 381-401
[20] Markle, S.M. & Tiemann, P. (1970). Really understanding concepts: Or, in frumious pursuit of the jabberwock: Instructional manual. Champaign, IL: Stipes Publishing Company
[21] Markovits, Z., Eylon B., & Bruckheimer M. (1986). Functions Today and Yesterday, Forthe Learning of Mathematics 6, 18-24
[22] Ochieng, P. A. (2009), An Analysis of the Strengths and Limitation of Qualitative and Quantitative Research Paradigms, Problems of Education in the 21st Century [PEC] (13), 13-18
[23] Olivier, A. (1989). Handling Pupils’ Misconceptions in Presidential address delivered at the Thirteenth National Convention on Mathematics, Physical Science and Biology Education, 3-7 July: 1-12
[24] Özkan, E. M., & Ünal, H. (2009). Misconception in Calculus-I: Engineering students’ misconceptions in the process of finding domain of functions. Procedia-Social and Behavioral Sciences, 1(1), 1792-1796
[25] Peters, P. (1982). Even Honors Students Have Conceptual Difficulties With Physics. Am. J. Phys., 50, 501-508
[26] Pólya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press
[27] Sarvestani, A. (2011). Contemplating Problems Taken From the History of Limits as a Way to Improve Students' Understanding of the Limit Concept. Thesis. Universiteit van Amsterdam
[28] Skemp, R. R. (1971). The Psychology of Learning Mathematics (1st ed.). Harmondsworth: Penguin
[29] Tall, D. (1992). Students’ Difficulties in Calculus. Plenary presentation in Working Group 3, ICME, Québec, august 1992. Mathematics Education Research Centre. University of Warwick
[30] Tall, D. & Vinner, S. (1981). Concept Image and Concept Definition in Mathematics with Particular Reference to Limits and Continuity. Educational Studies in Mathematics 12: 151-169
[31] Thabane, JL. (1998). Students’ Understanding of the Limit Concept in a First- year Calculus Course. MSc Dissertation: Wits University
[32] Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematics Education in Science and Technology, 14,293-305
[33] Westling, D. L., & Fox, L. (2000). Teaching students with severe disabilities (2nd ed.). Upper Saddle River, NJ: Merrill
[34] Williams, SR. (1991). Models of Limit held by College Calculus Students in Journal for Research in Mathematics Education 22(3):219-236

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