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研究生: 袁珮倫
論文名稱: 探討數學系學生對函數極限概念之抽象性的感知
Investigating the Perceptions of Abstractness with Respect to the Limit of Function in Calculus by Mathematics Students
指導教授: 譚克平
學位類別: 碩士
Master
系所名稱: 科學教育研究所
Graduate Institute of Science Education
論文出版年: 2014
畢業學年度: 102
語文別: 中文
論文頁數: 218
中文關鍵詞: 微積分函數的極限ε - δ 定義成對比較法反思抽象
英文關鍵詞: calculus, the limit of function, ε – δ definition, pair comparison, reflective abstract
論文種類: 學術論文
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  • 本研究欲探討數學系學生在學習基礎微積分時,對於函數極限的形式化定義是否會感到抽象?他們感受到的抽象性是有何內涵?這種抽象觀念從何而來?他們通常會如何處理及面對自己感到抽象的概念?學生們心中的抽象直覺與文獻中的抽象意涵是否具有相關性?或者要如何界定?

    本研究採用自基礎微積分教科書上所節錄從口語化到形式化逐步調整的五項函數極限定義,以成對比較法作為問卷設計的主軸,讓受試者進行五項定義之間抽象程度的比較。量化分析的部分利用了多元尺度法的知覺圖瞭解可能影響受試者判斷抽象程度的重要因素;並利用Bradley-Terry模型探討細部不同群組之間的差異。最後再從有效問卷樣本的受試者中抽樣進行深入晤談,晤談時以五項定義之間的差異性比較的方式進行,為瞭解影響受試者判斷抽象程度的真正原因。結果發現,學生對於函數極限的形式化定義感到抽象的型態可簡單歸納為三種類型,第一類的學生將抽象視為難懂,但第二類與第三類的學生則與其將問卷中的五項函數極限定義轉譯為其心目中最能表達函數極限定義的不同方式有關;雖然學生心中的抽象直覺與文獻中的抽象意涵有所不同,但在學生進行成對比較與定義之間的差異性比較時,可以觀察到學生如何以文獻中的抽象思考方式,對五項函數極限定義產生數學性質的知識建構。

    The main purpose of this study is to investigate the abstract perceptions of the students who major in Mathematics when they study the concepts of limit of the function in their calculus class.

    We adopt five different descriptions of the definition of limit of the function from Spivak’s famous seminal textbook on Calculus. The five descriptions of definition are displayed from daily speaking language to formative mathematics language. The pair comparisons of these five descriptions of definition are designed in our questionnaire for the quantitative data analysis. We use the perceptual map of multi-dimensional scaling to analyze the possible abstract perception of students, and utilize Bradley-Terry model to cluster the different groups of students’ preference behaviors of abstract definitions. We also interview some students to look for the where their abstract perceptions come from.

    At last, we conclude three types of students with different abstract perceptions, and found out their construction of reflective abstract as described in our literature review when students compare the difference between the five definitions of limit of function.

    第壹章 緒論 1 第一節 研究背景與動機 1 第二節 研究目的與問題 4 第三節 名詞解釋 6 第四節 研究範圍與限制 7 第五節 研究貢獻 9 第貳章 文獻探討 11 第一節 抽象的意涵 11 第二節 函數極限的學習困難與迷思模式 26 第參章 研究方法 29 第一節 研究方法設計 29 第二節 研究對象 32 第三節 量化分析方法 34 第四節 量化研究工具的內容、設計與修正 39 第五節 研究的步驟與流程 67 第肆章 量化資料分析結果 71 第一節 背景資料與學習情形分析 71 第二節 多元尺度法分析 83 第三節 Bradley-Terry模型分析 87 第伍章 質性訪談分析結果 95 第一節 定義之間的差異性比較 96 第二節 五項定義由抽象到具體的排序與排序的改變 136 第三節 五項定義形成學生抽象感的可能原因 145 第陸章 討論、結論與建議 154 第一節 討論 154 第二節 結論 162 第三節 建議 164 參考文獻 166 附錄 174 附錄A-1 函數極限定義抽象感調查的問卷版本一 174 附錄A-2 函數極限定義抽象感調查的問卷版本二 180 附錄A-3 函數極限定義抽象感調查的問卷版本三 201

    中文部分

    詹志禹(1996)。認識與知識:建構論 V.S. 接受觀。教育研究雙月刊,49。

    伍振鷟、林逢祺、黃坤錦、蘇永明(1999)。教育哲學。臺北市:五南。

    林翠屏(2010)。以學生認知發展為主軸的微積分教學模式:教與學的特徵 (未出版之碩士論文)。國立彰化師範大學數學研究所,彰化縣。

    蔡森任(2011)。成對比較資料中個別差異模型結構之分類(未出版之碩士論文)。國立臺灣師範大學數學研究所,台北市。

    西文部分

    Abstract (2004). Merriam-Webster's collegiate dictionary (11th ed.). Springfield, MA:Merriam-Webster.

    Abstract (2009). Oxford English Dictionary (2nd ed., version 4.0). New York: Oxford University Press.

    Barnhart, R. K. (Ed.) (1988). Barnhart dictionary of etymology. Bronx, NY: H.W. Wilson Co.

    Bartlett, F. C. (1932). When smart groups fail. The Journal of Learning Science, 12, 307-359.

    Beth, E. W., & Piaget, J. (1966). Mathematical epistemology and psychology (W. Mays, trans.). Dordrecht, the Netherlands: Reidel.

    Bradley, R. A., & Terry, M. E. (1952). Rank analysis of incomplete block designs: I. The methods of paired comparisons. Biometrika, 39, 324-345. 168

    Brook, A. (1997). Approaches to abstraction: A commentary. International Journal of Educational Research, 27, 77-88.

    Brown, J. S., Collins, A., & Duguid, P. (1989). Situated cognition and the culture of learning. Educational Researcher, 18(1), 32-42.

    Cornu, B. (1981). Apprentissage la notion de limite: Modèles spontanés et modèles propres. In Proceedings of the 5th annual conference of the International Group for the Psychology of Mathematics Education, (pp.322-326). Grenoble, France: PME.

    Cornu, B. (1983). Apprentissage de la notion de limite: Conceptions et obstacles. Thèse de doctorat de troisième cycle, L’Université Scientifique et Médicale de Grenoble, France.
    Cornu, B. (1991). Limit. In D. Tall, (Ed.), Advanced mathematical thinking (pp. 153-166). Dordrecht, the Netherlands: Kluwer Academic Press.

    Cuevas, G. (1984). Mathematics learning in English as a second language. Journal of Research in Mathematics Education, 15, 134-144.

    Davis, R., & Vinners, S. (1986). The notion of limit: Some seemingly unavoidable misconception stages. Journal of Mathematical Behavior, 5, 281-303.

    Davydov, V. V. [1972] (1990). Types of generalization in instruction: Logical and psychological problems in the structuring of school curricula (J. Kilpatrick, Ed., & J. Teller, trans.). Reston, VA: National Council of Teachers of Mathematics.

    Dubinsky, E. (1991). Reflective abstraction in advanced mathematical thinking. In D. Tall, (Ed.), Advanced mathematical thinking (pp. 95-123). Dordrecht, The Netherlands: Kluwer Academic Press.

    Gorman, A. M. (1961). Recognition memory for nouns as a function of abstractness and frequency. Journal of Expirimental Psychology, 61(1), 23-29.

    Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education, 32, 195-222.

    Herzog, O. (1919). Der abstrakte expessionismus. Der Sturm, 10, 29.

    Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws (Ed.). Handbook of research on mathematics education (pp. 195-222). New York: Macmillan.

    Hiebert, J., & LeFevre, P. (Eds.). (1986). Conceptual and procedural knowledge in mathematics: An introductory analysis. In J. Hiebert, (Ed.), Conceptual and procedural knowledge: The case of mathematics (pp. 1-27). Hillsdale, NJ: Lawrence Erlbaum.

    Hume, D. [1758] (2005). Inquiry concerning human understanding. Stilwell, KS: Digireads.

    Hume, D. [1739] (2007). A treatise of human nature. In D. F. Norton and M. J. Norton (Eds.), David Hume: A treatise of human nature. New York: Oxford University Press.

    Jampolsky, M. (1950). Êtude de quelques épreuves de Reconnaissance. Annee Psychology, 49, 63-97.

    Kant, I. [1781] (1998). Critique of pure reason. Cambridge: Cambridge University Press.

    Keil, F. C. (1989). Concept, kinds, and cognitive development. Cambridge, MA: MIT.

    Kiehl, K. A., Liddle, P. F., Smith, A. M., Mendrek, A., Forster, B. B., & Hare, R. D. (1999). Neural pathways involved in the processing of concrete and abstract words. Human Brain Mapping, 7, 225-233.

    Koczkodaj, W. W. (1998). Testing the accuracy enhancement of pairwise comparisons by a monte carlo experiment. Journal of Statistical Planning and Inference, 69, 21-31.

    Kruskal, J. B., & Wish, M. (1978). Multidimensional scaling. Beverly Hills: Sage Publications.

    Lave, J. (1988). Cognition in practice: Mind, mathematics and culture in everyday life. Cambridge: Cambridge University Press.

    Li, L., & Tall, D. (1992). Constructing different concept images of sequence and limits by programming. In I. Hirabayashi, N. Nohda, K. Shigematsu, & F. Lin (Eds.). Proceedings of the 17th annual conference of the International Group for the Psychology of Mathematics Education, (Vol.2, pp.41-48). Tsukuba, Japan: PME.

    Locke, J. [1689] (1979). An essay concerning human understanding (Clarendon edition of the works of John Locke). New York: Oxford University Press.

    Larger, C. A. (2006). Types of mathematics-language reading interactions that unnecessarily hinder algebra learning and assessment. Reading Psychology, 27, 192.
    Mitchelmore, M. (2007). Abstraction in mathematics learning. Mathematics Education Research Journal, 19(2), 1-9.

    Monaghan, J. (1991). Problems with the language of limits. For the Learning of Mathematics, 11(3), 20-24.

    Monaghan, J., Sun, S., & Tall, D. O. (1994). Construction of the limit concept with a computer algebra system. In J. P. da Ponte, & J. F. Matos (Eds.), Proceedings of the 17th annual conference of the International Group for the Psychology of Mathematics Education, (Vol.3, pp.279-286). Lisbon, Portugal: PME.

    Noss, R., & Hoyle, C. (1996). Windows on mathematics meaning. Dordrecht, The Netherlands: Kluwer Academic Press.

    Ozmantar, M. F., & Monaghan, J. (2007). A dialectical approach to the formation of mathematical abstractions. Mathematics Education Research Journal, 19(2), 89-112.

    Piaget, J. (1971). Biology and knowledge (B. Walsh, trans.). Chicago: University of Chicago Press. 172

    Piaget, J. (1972). The principles of genetic epistemology (W. May, trans.). London: Routledge & Kegan Paul.

    Piaget, J. (1975). Piaget's theory (G. Cellerier & J. Langer, trans.). In P. B. Neubauer (Ed.), The process of child development (pp. 164-212). New York: Jason Aronson.

    Piaget, J. (1978). Success and understanding (A. J. Pomerans, trans.). Cambridge, MA: Harvard University Press.

    Piaget, J. (1980). Adaptation and intelligence (S. Eames, trans.). Chicago: University of Chicago Press.

    Piaget, J. (1985). The equilibration of cognitive structures (T. Brown & K. J. Thampy, trans.). Cambridge, MA: Harvard University Press.

    Piaget, J., & Garcia, R. (1983). Psychogenèse et histoire des sciences. Paris: Flammarion.

    Schwarz, B. B., Dreyfus, T., & Hershkowitz, R. (2009). The nested epistemic actions model for abstraction in context. In B. B. Schwarz, T. Dreyfus, & R. Hershkowitz (Eds.). Transformation of knowledge through classroom interaction (pp. 11-40). New York: Routledge.

    Schwarzenberger, R. L. E., & Tall, D. O. (1978). Conflicts in the learning of real number and limits. Mathematics Teaching, 82, 44-49.

    Spivak, M. (1967). Calculus. New York: W.A. Benjamin, Inc..

    Stoke, S. M. (1929). Memory for onomatopes. The Journal of Genetic Psychology, 36, 594-596.

    Thorndike, E. L., & Lorge, I. (1944). The teacher’s word book of 30,000 words. New York: Bureau of Publications, Teachers College, Columbia University.

    Toglia, M. P., & Battig, W. F. (1978). Handbook of semantic word norms. Hillsdale, NJ: Lawrence Erlbaum.

    Van Oers, B. (1998). The fallacy of decontextualization. Mind, Culture, and Activity, 5, 135-142.

    Van Oers, B. (2001). Contextualization for abstraction. Cognitive Science Quarterly, 1, 279-305.

    Vygotsky, L. S. [1934] (1987). Thinking and speech. In R. W. Rieber & A. S. Carton (Eds.) & N. Minick (trans.). The collected works of L. S. Vygotsky: Vol. 1 Problems of general psychology. New York: Plenum.

    Williams, S. (1991). Models of limit held by college calculus students. Journal for Research in Mathematics Education, 22(3), 219-236.

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