摘 要

本研究主要為探索個體於幾何解題活動時之動態心像的類型與運作機制，以及動態幾何環境對建構動態心像的影響。據此目的，選擇台北市內湖區某公立國中二年級之兩個班級作為研究對象，並將其分實驗與控制兩組；全程參與之學生共計63名，並從實驗組分層隨機抽樣選取14名個案，以經審訂之問題進行個案訪談，依此探究個體原生型之動態心像的種類與運作機制，而後以動態的觀點整合GSP環境設計實驗教學活動。於活動中期及結束時各進行一次訪談，並於末了進行紙筆測驗，以瞭解學生的變化及評估實驗組相對於控制組的學習成效。

研究方法採質與量並重的方式進行，以質的詮釋性研究探究個體心像操作的機制，而以紙筆測驗之結果進行統計分析，由此獲得學習成效評估及各試題作答情形之關連性。依所得資料之類型，以紙筆測驗、個案動態心像類型、個案解題認知歷程分析與個案縱向分析等分別描述與討論本研究之發現。

針對研究目的，本研究之主要的研究結果如下：
1. 本研究發現14名個案皆會產生動態心像，且可將其引入解題活動中，然而不同的學生操作心像的頻率並不相同，與層次無關，唯高層次學生運用動態心像解題時較有系統。
2. 在以動態及靜態語意所佈置之問題情境中，台法兩地學生的表現恰好相反，其中台灣的學生較偏好動態語意情境，其原因為學生可因此引進動態心像，擬定自己所認為之較佳的解題策略。
3. 本研究所發現之動態心像類型有割補、變換、拓樸與動態模擬等四種主類型；除拓樸型之外，其餘三類可作為具威力之解題方法，而拓樸型雖不具此能力，然而仍對解題具有輔助的功能。
4. 個體於解題活動中所運用之動態心像類型通常與問題情境有關，不過某些學生易傾向某種類型之心像操作方式。

依據研究之結果，教師在進行教學時，應特別注意動態與靜態語意對學生理解方式與解題方法的影響，並且應以較為有系統的方式，配合動態幾何工具的使用，培養學生運用動態心像以增強學習的成效。未來則可以較多的樣本數、專家與生手的對比，以及幾何、分析與調和等類型的個體是否傾向使用某種特定類型之動態心像等方向進行相關的研究。

關鍵字：動態幾何軟體、動態幾何環境、心像、心像操作、動態心像

參考文獻
一、中文部分
1. Kline, M. (1995)。 西方文化中的數學（張祖貴 譯）。台北市：九章出版社。
2. Papert, S. (1997)。 WWW.新家庭：開創網路時代的親子學習文化（李鐏龍、賴慈芸、周文萍 譯）。台北市：大塊文化出版股份有限公司。
3. Patton, M. Q. (1995)。 質的評鑑與研究（吳芝儀、李奉儒 譯）。台北縣：桂冠圖書股份有限公司。
4. 王文科 (2001)。 教育研究法（六版）。台北市：五南圖書出版有限公司。
5. 左台益 (1998)。網路科技對高中教育的影響之研究—網路科技對高中數學教學之影響與效能(1/2)。台北市：行政院國家科學委員會專題研究計畫進度報告，NSC 87-2514-S-003-006-N。
6. 左台益、王惠中 (2000)。動態幾何實驗設計。中華民國第十六屆科學教育學術研討會短篇論文彙編 (pp. 337-345)。台北市：國立台灣師範大學科學教育研究所。
7. 林福來、陳美芳、吳毓瑩、金鈐、江南青、李源順、鄭英豪、林佳蓉、朱綺鴻、陳英娥、吳慧真、陳儀君、黃凡玲 (1997)。 教學思維的發展：整合數學教學知識的教材教法（1/3）。台北市：行政院國家科學委員會專題研究計畫進度報告，NSC 86-2511-S-003-025。
8. 張春興 (1999)。 教育心理學：三化取向的理論與實踐（修訂版）。台北市：台灣東華書局。
9. 蔡志仁 (2000)。 動態連結多重表徵視窗環境下橢圓學習之研究。國立台灣師範大學數學研究所碩士論文，未出版，台北市。
10. 劉錫麒 (1997)。 數學思考教學研究。台北市：師大書苑。
二、英文部分
1. Anderson, J. R. (1983). The architecture of cognition. Cambridge, MA: Harvard University Press.
2. Anderson, J. R. (1990). Cognitive psychology and its implications, (3rd ed.). New York: W. H. Freeman and Company.
3. Balacheef, N., & Kaput, J. J. (1996). Computer-Based Learning Environments in Mathematics. In A. J. Bishop, & K. Clements, C. Keitel, J. Kilpatick, & C. Laborde (Eds.), International Handbook of Mathematics Education (pp. 469-501). Dordrecht: Kluwer Academic Publishers.
4. Bishop, A. J. (1989). Review of research on visualization in mathematics education. Focus on Learning Problems in Mathematics, 11(1), 7-16.
5. Brown, D. L., & Presmeg, N. C. (1993). Types of imagery used by elementary and secondary school students in mathematical reasoning. Proceedings of the 17th Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. II, pp. 137-144), Tsukuba, Japan.
6. Clements, D. H., & Battista, M.T. （1992）. Geometry and spatial reasoning. In D. A. Grouws（Ed.）, Handbook of research on mathematics teaching and learning （pp. 437-442）. New York: Macmillan.
7. Cooper, L. A., & Shepard, A. N. (1973). Chronometric studies of the rotation of mental images. In W. G. Chase (Ed.), Visual information processing (pp. 75-176). New York, NY: Academic Press.
8. Cuoco, A., & Goldenberg, P. (1997). Dynamic Geometry as a Bridge from Euclidean Geometry to Analysis. In J. King & D. Schattschneider (Eds.), Geometry Turned On (pp. 33-46). Washington, DC: The Mathematical Association of America.
9. Dixon, J. K. (1997). Computer use and visualization in students’ construction of reflection and rotation concepts. School Science and Mathematics, 97(7), 352-358.
10. Duval, R. (1995). Geometrical pictures: Kinds of Representation and Specific Processings. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education (pp. 142-157). New York, NY: Springer-Verlag.
11. Duval, R. (1998). Geometry from a Cognitive Point of View. In C. Mammana & V. Villani (Eds.), Perspectives on the Teaching of Geometry for the 21st Century (pp. 37-52). Dordrecht: Kluwer Academic Publishers.
12. Fischbin, E. (1987). Intuition in science and mathematics. Dordrecht, The Netherlands: D. Reidel.
13. Fomenko, A. T. (1990). Mathematical Impressions. Providence, RI: American Mathematical Society.
14. Gagn?, E. D. (1985). The cognitive psychology of school learning. Boston: Little, Brown.
15. Gagn?, E. D., Yekovich, C. W., & Yekovich, F. R. (1993). The cognitive psychology of school learning (2nd ed.). New York: HarperCollins.
16. Gagn?, R. M. (1985). The conditions of learning and theory of instruction, (4th ed.). New York, NY: Holt, Rinehart and Winston.
17. Goldenberg, E. P. (1995). Ruminations about Dynamic Imagery. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education (pp. 202-224). New York, NY: Springer-Verlag.
18. Hadamard, J. (1945). The psychology of Invention in the Mathematical Field. Princeton, NJ: Princeton University Press.
19. Harada, K., Gallou-Dumiel, E., & Nohda, N. (2000). The role of figures in geometrical, proof-problem solving – students’ cognitions of geometrical figures in France and Japan. Proceedings of the 24th Annual Meeting of the International Group for the Psychology of Mathematics Education (Vol. 3, pp. 25-32), Hiroshima, Japan.
20. Hershkowitz, R., Parzysz, B., & van Dormolen, J. (1996). Space and Shape. In A. J. Bishop, K. Clements, C. Keitel, J. Kilpatick, & C. Laborde (Eds.), International Handbook of Mathematics Education (pp. 161-204). Dordrecht: Kluwer Academic Publishers.
21. Hoyles, C., & Healy, L. (1997). Unfolding meanings for reflective symmetry. International Journal of Computers for Mathematical Learning, 2, 27-59.
22. Jackiw, N. (1997). The Geometer’s sketchpad (Version 3.10) [Computer Software]. Berkeley, CA: Key Curriculum Press.
23. Jackiw, N. (2001). The Geometer’s sketchpad (Version 4.00) [Computer Software]. Emeryville, CA: Key Curriculum Press.
24. Jonassen, D. H. (1996). Computers in the classroom: Mindtools for critical thinking. Englewood Cliffs, NJ: Prentice Hall.
25. Kappraff, J. (2000). A Secret on Ancient Geometry. In C. A. Gorini (Ed.), Geometry at Work: A Collection of Papers Showing Applications of Geometry (pp. 26-36). Washington, DC: The Mathematical Association of America.
26. Kaput, J. J. (1995). Overcoming Physicality and the Eternal Present: Cybernetic Manipulatives. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education (pp. 161-177). New York, NY: Springer-Verlag.
27. King, J. R. (1996). Geometry Through the Circle with the Geometer’s Sketchpad. Berkeley, CA: Key Curriculum Press.
28. Krutetskii, V. A. (1976). The psychology of mathematical abilities in schoolchildren. Chicago, IL: University of Chicago Press.
29. Lean, G., & Clements, M. A. (1981). Spatial ability, visual imagery, and mathematical performance. Educational Studies in Mathematics, 12, 267-299.
30. Moreno A., L. E., & Sacrist?n R., A. I. (1995). On Visual and Symbolic Representations. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education (pp. 178-189). New York, NY: Springer-Verlag.
31. Morrow, J. (1997). Dynamic Visualization from Middle School through College. In J. King & D. Schattschneider (Eds.), Geometry Turned On: Dynamic software in learning, teaching, and research (pp. 47-54). Washington, DC: The Mathematical Association of America.
32. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
33. Noss, R., & Hoyles, C. (1995). The Dark Side of the Moon. In R. Sutherland & J. Mason (Eds.), Exploiting Mental Imagery with Computers in Mathematics Education (pp. 191-201). New York, NY: Springer-Verlag.
34. Osta, I. (1998). Computer Technology and the Teaching of Geometry. In C. Mammana & V. Villani (Eds.), Perspectives on the Teaching of Geometry for the 21st Century (pp. 109-112). Dordrecht: Kluwer Academic Publishers.
35. Papert, S. (1980). Mindstoms: Children, computer, and powerful ideas. New York: Basic Books.
36. Paivio, A. (1983). The Empirical Case for Dual Coding. In J. C. Yuille (Ed.), Imagery, memory, and cognition: essays in honor of Allan Paivio (pp. 307-332). Hillsdale, NJ: Lawrence Erlbaum Associates.
37. Paivio, A. (1986). Mental Representations: A dual coding approach (pp. 53-83). Oxford: Oxford University Press.
38. Paivio, A., & Clark, J. M. (1991). Static versus dynamic imagery. In C. Cornoldi & M. A. McDaniel (Eds.), Imagery and Cognition (pp. 221-245). New York: Springer-Verlag.
39. Presmeg, N. C. (1985). The role of visually mediated processes in high school mathematics: A classroom investigation. Unpublished doctoral dissertation. University of Cambridge, Cambridge, England.
40. Presmeg, N. C. (1997). Generalization Using Imagery in Mathematics. In L. D. English (Ed.), Mathematical Reasoning: Analogies, Metaphors, and Images (pp. 299-312). Mahwah, NJ: Lawrence Erlbaum Associates.
41. Reusser, K. (1996). From cognitive Modeling to the design of pedagogical tools. In S. Vosniadou, E. de Corte, R. Glaser, & H. Mandl (Eds.), International Perspectives on the Design of Technology-Supported Learning Environments (pp. 81-103). Mahwah, NJ: Lawrence Erlbaum Associates.
42. Richardson, J. T. E. (1991). Imagery and the Brain. In C. Cornoldi & M. A. McDaniel (Eds.), Imagery and Cognition (pp. 1-45). New York: Springer-Verlag.
43. Tall, D. （1992）. The transition to advanced mathematical thinking: functions, limits, infinity, and proof. In D. A. Grouws（Ed.）, Handbook of research on mathematics teaching and learning （pp. 495-511）. New York: Macmillan.
44. Thomas, N. J. T. (2001). Mental Imagery. Retrieved March 29, 2002, from the World Wide Web: http://plato.stanford.edu/entries/mental-imagery/
45. Thomas, N. J. T. (2002). Mental Imagery, Philosophical Issues About. Retrieved March 29, 2002, from the World Wide Web:
http://www.calstatela.edu/faculty/nthomas/mipia.htm
46. Toffler, A. (1980). The Third Wave. New York: Morrow.
47. Usiskin, Z. (1982). Van Hiele levels and achievement in secondary school geometry (Final report of the Cognitive Development and Achievement in Secondary School Geometry Project). Chicago, IL: University of Chicago, Department of Education.
48. Vinner, S. (1983). Concept definition, concept image and notion of function. International Journal of Mathematical Education in Science and Technology, 14(3), 293-305
49. Walter, M. (2001). Looking at a Painting with a Mathematical Eye. For the Learning of Mathematics, 21(2), 26-30.
50. Wheatley, G. H. (1997). Reasoning with images in Mathematical activity. In L. D. English (Ed.), Mathematical Reasoning: Analogies, Metaphors, and Images (pp. 281-297). Mahwah, NJ: Lawrence Erlbaum Associates.
51. Wu, D. B. (1994). A study of the use if the van Hiele model in the teaching of non-Euclidean geometry to prospective elementary school teachers in Taiwan, the Republic of China. Unpublished doctoral dissertation. University of Northern Colorado, Greeley, Colorado.
52. Zimmermann, W., & Cunningham, S. (1991). What is Mathematical visualization? In W. Zimmermann & S. Cunningham (Eds.), Visualization in teaching and learning Mathematics. Washington, DC: The Mathematical Association of America.