課程內容：
<br>
<br>主題一：幾何思考的發展
<br>
<br>教學說明：認識幾何思考、能力發展的兩大理論基礎：van Hiele 與Piaget 幾何發展的理論。從認
<br>識這兩大理論，瞭解國中小學生的幾何發展的階層以及學習的可能面臨的問題。
<br>
<br>說明van Hiele 理論裡幾何發展的五個階層 (Burger & Shaughnessy, 1986; Clements, 
<br>Battista, & Sarama, 2001; Fuys, Geddes, & Tischler, 1988)。
<br>摘要論述Piaget兒童的幾何概念與空間概念兩本書中，與國中學生幾何課程發展相關的重點。 
<br>(Piaget & Inhelder, 1967; Piaget, Inhelder, & Szeminska, 1960)。
<br>
<br>主題二：視覺推理
<br>
<br>教學說明：幾何學習除了需要一般的邏輯推理之外，視覺推理能力也是成功幾何學習的重要能力之
<br>一。幾何的視覺推理與幾何圖形息息相關。而視覺推理的目的則是幫助學生找到有用的幾何性質，進
<br>而使用這些幾何性質進行解題活動。
<br>
<br>視覺推理和幾何推理的關係 (Duval, 1995, 1998)
<br>學生的視覺推理與圖形中隱含的幾何性質的關係為何 (Blerk, Christiansen, & Anderson, 
<br>2008)
<br>視覺推理如何造成幾何論證學習的障礙 (Aleven, Koedinger, Sinclair, & Snyder, 1998)
<br>學生幾何圖形的辨識與幾何性質及定義的關係研究(Fischbein & Nachlieli, 1998)
<br>學生的在特殊幾何圖形上的推理表現 (K&#252;chemann & Hoyles, 2006)
<br>
<br>主題三：幾何推理
<br>
<br>教學說明：什麼叫做幾何推理？幾何推理與幾何論證學習的關係為何？從Duval研究開始瞭解幾何推
<br>理的定義。然後進一步探討如何增進學生的推理能力。
<br>
<br>幾何推理的定義與幾何論證學習的關係 (Duval, 1998)
<br>另一種的推理 transformational reasoning (Simon, 1996)
<br>如何增進學生的幾何推理能力 (Mistretta, 2000)
<br>臆測與其涵蓋的幾何推理 (Canadas, Deulofeu, Figueiras, Reid, & Yevdokimov, 2007; 
<br>Lee & Sriraman, 2011)
<br>
<br>主題四：論證的有效性研究
<br>
<br>教學說明：論證的有效性(validity)是成功幾何學習的關鍵之一。從目前的文獻看來，學生對於論
<br>證的有效性瞭解相當的不足。如何幫助學生瞭解論證的有效性，進一步提升學生的幾何論證學習是本
<br>主題探討的重點。
<br>
<br>台灣學生論證對有效性理解的研究報告 (Yu, 2001; Yu, Chin, & Lin, 2004)
<br>英國高成就學生對論證有效性理解的研究報告 (Hoyles & K&#252;chemann, 2002)
<br>大學生辨識論證與數學理論之間的有效關係(Selden & Selden, 2003)
<br>大學生如何validate證明的結果(Alcock & Weber, 2005)
<br>
<br>主題五：探討國中、小幾何推論與論證的銜接與學習困難
<br>
<br>教學說明：不同國家對幾何推論與論證持有不同的觀點。不同國家的論證課程之下，學生產生的迷思
<br>概念與學習困難也不盡相同。
<br>
<br>歐洲實行論證課程教學的觀點 (Mariotti, 2006)
<br>台灣與德國學生論證表現 (Heinze, Cheng, & Yang, 2004)
<br>台灣與英國學生論證表現 (Hoyles & Kuchemann, 2002)
<br>歐洲實行論證課程教學的觀點 (Mariotti, 2006)
<br>英國學生論證表現 (Hoyles & Kuchemann, 2002)
<br>
<br>教學型式：
<br>重視理論與實務的結合。由文獻探討與實例學生學習與教學分析來建構學生該科的專業知識。並能由
<br>實務分析中，轉化理論知識到實際教學層面。
<br>
<br>教學評量方式：
<br>教學評量主要有三面向：（1）出席率（30％）。（2）表現評量(30%)：包括期末報告（20％）、課
<br>堂簡短報告(10%).（3）上課參與(50%)：包括上課參與討論、出缺席、及課前閱讀準備等
<br>
<br>參考書目
<br>Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and 
<br>checking warrants. Journal of Mathematical Behavior, 24, 125-134.
<br>
<br>Aleven, V., Koedinger, K. R., Sinclair, H. C., & Snyder, J. (1998). Combatting 
<br>shallow learning in a tutor for geometry problem solving. Paper presented at the 
<br>Intelligent tutoring systems: Fourth International Conference, ITS ’98, Berlin.
<br>
<br>Blerk, A. v., Christiansen, I. M., & Anderson, T. R. (2008). Learners’ visual 
<br>recognition of geometry theorems. Paper presented at the The International 
<br>Congress on Mathematical Education, Monterrey, Mexico.
<br>
<br>Burger, W. F., & Shaughnessy, M. (1986). Characterizing the van Hiele levels of 
<br>development in geometry. Journal for Research in Mathematics Education, 17(1), 
<br>31-48.
<br>
<br>Canadas, M., Deulofeu, J., Figueiras, L., Reid, D., & Yevdokimov, O. (2007). The 
<br>conjecturing process: Perspectives in theory and implications in practice. 
<br>Journal of Teaching and Learning, 5(1), 55-72.
<br>
<br>Clements, D. H., Battista, M., & Sarama, J. (2001). Logo and geometry. Reston: 
<br>The National Council of Teachers of Mathematics.
<br>
<br>Duval, R. (1995). Geometrical pictures: Kinds of representation and specific 
<br>processings. In R. Sutherland & J. Mason (Eds.), Exploiting mental imagery with 
<br>computers in mathematics education (pp. 142-157). Berlin: Springer.
<br>
<br>Duval, R. (1998). Geometry from cognitive point of view. In C. Mammana & V. 
<br>Villani (Eds.), Perspectives on the teaching of geometry for 21st century (pp. 
<br>37-52). Dordrecht: Kluwer.
<br>
<br>Fischbein, E., & Nachlieli, T. (1998). Concepts and figures in geometrical 
<br>reasoning. International Journal of Science Education, 20(10), 1193-1211.
<br>
<br>Fuys, D., Geddes, D., & Tischler, R. (1988). The van Hiele model of thinking in 
<br>geometry among adolescents (Vol. 3). Reston, VA: NCTM.
<br>
<br>Heinze, A., Cheng, Y.-H., & Yang, K.-L. (2004). Students’ performance in 
<br>reasoning and proof in Taiwan and Germany: Results, paradoxes and open 
<br>questions. ZDM-The International Journal on Mathematics Education, 36(5), 162-
<br>171.
<br>
<br>Hoyles, C., & K&#252;chemann, D. (2002). Students’ understanding of logical 
<br>implication. Educational Studies in Mathematics, 51, 193-223.
<br>
<br>Hoyles, C., & Kuchemann, D. (2002). Students’ explanations in geometry: Insights 
<br>from a large-scale longitudinal survey. Paper presented at the 2002 
<br>International Conference on Mathematics: Understanding Proving and Proving to 
<br>Understand, Taipei, Taiwan.
<br>
<br>K&#252;chemann, D., & Hoyles, C. (2006). Influences on students’ mathematical 
<br>reasoning and patterns in its development: Insights from a longitudinal study 
<br>with particular reference to geometry. International Journal of Science and 
<br>Mathematics Education, 4, 581-608.
<br>
<br>Lee, K.-H., & Sriraman, B. (2011). Conjecturing via reconceived classical 
<br>analogy. Educational Studies in Mathematics, 76(2), 123-140.
<br>
<br>Mariotti, M. A. (2006). Proof and proving in mathematics education. In &#193;. 
<br>Guti&#233;rrez & P. Boero (Eds.), Handbook of research on the psychology of 
<br>mathematics education (pp. 173-204). Rotterdam: Sense Publishers.
<br>
<br>Mistretta, R. M. (2000). Enhancing geometric reasoning. Adolescence, 35(138), 
<br>365-379.
<br>
<br>Piaget, J., & Inhelder, B. (1967). The child’s conception of space. New York: 
<br>The Norton Library.
<br>
<br>Piaget, J., Inhelder, B., & Szeminska, A. (1960). The child’s conception of 
<br>geometry (E. A. Lunzer, Trans.). New York: W.W. Norton & Company.
<br>
<br>Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can 
<br>undergraduates tell whether an argument proves a theorem? Journal for Research 
<br>in Mathematics Education, 34(1), 4-14.
<br>
<br>Simon, M. (1996). Beyond inductive and deductive reasoning: The search for a 
<br>sense of knowing. Educational Studies in Mathematics, 30, 197-210.
<br>
<br>Yu, J.-Y. W. (2001). Reasoning difficulty related to validity of argument and 
<br>truth of assertion in mathematics. (doctoral thesis), Pittsburgh University, 
<br>Pittsburgh, PA.
<br>
<br>Yu, J.-Y. W., Chin, E.-C., & Lin, C.-J. (2004). The understanding of Taiwanese 
<br>junior high school students about the validity of conditional statement. 
<br>International Journal of Science and Mathematics Education, 2(2), 257-285.