重讀幾何思維的階層論：載體理論的啟示

Reconsidering the Hierarchical View in Geometric Thinking from the Perspective of Genre Theory

10.6173/CJSE.2009.1702.04

吳毓瑩(Yuh-Yin Wu)；呂金燮(Chin-Hsieh Lu)；吳昭容(Chao-Jung Wu)

情境認知 ； 幾何思維 ； 階層論 ； 概念發展 ； 載體理論 ； Hierarchical View in Conceptual Development ； Genre Theory ； Geometric Thinking ； Situated Cognition

科學教育學刊

17卷2期（2009 / 04 / 01）

157 - 177

繁體中文

本文以獲取知識的「媒介」，亦即載體，為理論核心，重新解讀幾何思維的階層論。我們首先提問幾何思維的發展除了階層論外，是否另有可能？由此提問再思概念發展的不同歷程，闡釋載體理論所彰顯的特性，包括社會文化動態結構、仲介思考的符號工具、以及知識與載體無法分離的性質。從載體之觀點，研究者綜合出幾種可能性：1.幾何思維的層次亦可能是性質相異之載體的不同運作；2.個體對性質相異載體的掌握可以同時並進；3.載體所導引的內隱規則會促成不同學習策略；4.幾何教學可因不同載體的利用而各有安排。最後研究者結語對於部份無法歸納在階層順序下的學習者，如果從載體理論的觀點，或可接受幾何思維具有多元與非線性的特質。

This article aimed to revise geometric thinking from the perspective of genre theory. We first criticize the following properties of geometric thinking through recent research findings of conceptual development: hierarchy, concrete-abstract formation, and generalization. Then we discuss genre theory and explore its three main points: social cultural emphasis, language/instrument system, and knowledge being mediated by genre. We then conclude some possibilities about geometric thinking under genre theory: 1. The levels of hierarchical structure might be learners' operation on different genres, 2. Learners might grasp operations on different genres simultaneously, 3. Different genres produce different implicit rules and might lead different geometric learning strategies, and 4. Geometric teaching strategies might have changed according to different genres. Genre theory may serve as another perspective to approach geometric thinking.

1. Abravanel, E.(1977).The figural simplicity of parallel lines.Child Development,48,708-710.
2. As cited in P. M.,van Hiele(1986).Structure and insight.New York:Academic Press.
3. Ausbel, D. P.(1968).Educational psychology: A cognitive view.NY:Holt, Rinehart, & Winston.
4. Barsalou, L. W. ,S. Vosniadou (Eds.),A. Ortony (Eds.)(1989).Similarity and analogical reasoning.NY:Cambridge University Press.
5. Beilin, H.(1965).Learning and operational convergence in logical thought development.Journal of Experimental Child Psychology,2,317-339.
6. Berkenkotter, C.,Huckin, T.(1995).Genre knowledge in disciplinary communication: Cognition/culture/power.Hillsdale, NJ:Erlbaum.
7. Brainerd, C. J.(1978).The stage question in cognitive-developmental theory.Behavioral and Brain Sciences,2,173-213.
8. Brainerd, C. J.(1977).Cognitive development and concept learning: An interpretative review.Psychological Bulletin,84,919-939.
9. Brown, J. S.,Collins, A.,Duguid, P.(1989).Situated cognition and the culture of learning.Educational Researcher,18(1),32-41.
10. Brown, J. S.,Duguid, P.,H. McLellan (Ed.)(1996).Situated learning perspectives.Englewood Cliffs, NJ:Educational Technology.
11. Chirwa, A. S.(1997).Knowledge acquisition in computer-based learning environments among elementary children.Journal of Educational Technology Systems,26(1),19-25.
12. Clements, D. H.,Battista, M. T.,D. A. Grouws (Ed.)(1992).Handbook of research on mathematics teaching and learning.New York:Macmillan.
13. Cole, M.(1995).Culture, communication, and cognition.Cambridge:Cambridge University Press.
14. Duval, R.,C. Mammana (Eds),V. Villani (Eds)(1998).Perspective on the teaching of geometry for the 21st century.Dordrecht:Klu-wer.
15. Duval, R.,R. Sutherland (Eds.),J. Mason (Eds.)(1995).Exploiting mental imagery with computers in mathematics education.New York:Springer.
16. Feldman, C. F.,K. Nelson (Ed.)(1989).Narrative from the crib.Cambridge, MA:Harvard University Press.
17. Geertz, C.(1983).Local knowledge: Further essays in interpretive anthropology.New York:Basic Books.
18. Gigerenzer, G.,Hug, K.(1992).Domain-specific reasoning: Social contracts, cheating, and perspective change.Cognition,43,127-171.
19. Gopnik, A.,Meltzoff, A., Kuhl, P.(1999).The scientist in the crib: Mind, brain, and how children learn.NY:William Marrow & Company, Inc.
20. Greenfield, P. M.(1999).Cultural change and human development.New Directions in Child and Adolescent Development,83,37-59.
21. Greeno, J. G.,Collins, A. M.,Resnick, L. B.,D. C. Berliner (Eds.),R. C. Calfee (Eds.)(1996).Handbook of educational psychology.New York:Simon & Schuster Macmillan.
22. Gutierrez, A.,Jaime, A.,Fortuny, J. M.(1991).An alternative paradigm to evaluate the acquisition of the van Hiele levels.Journal for Research in Mathematics Education,22(3),237-251.
23. Hadamard, J.(1945).An essay on the psychology of invention in the mathematical field.Princeton, NJ:Princeton University Press.
24. Hershkowitz, R.,C. Mammana (Eds.),V. Villani (Eds.)(1998).Perspective on the teaching of geometry for the 21st century.Dordrecht:Kluwer.
25. Inhelder, B.,Sinclair, H.,Bovet, M.(1974).Apprentissage et structures de la connaissance.Paris:Presses Universitaires de France.
26. Jackiw, N.(2006).Geometer's sketchpad 4.07. (Computer software).Berkeley, CA:Key Curriculum Press.
27. Jones, S. S.,Smith, L. B.(1993).The place of perception in children's concepts.Cognitive Development,8,113-139.
28. Keil, F. C.,D. Kuhn (Eds.),R. S. Siegler (Eds.)(2006).Handbook of child psychology Vol. 2: Cognition, perception, and language.New York:John Wiley & Sons.
29. Keil, F. C.,K. A. Franklin (Ed.)(1998).Handbook of child psychology, fifth edition.New York:John Wiley & Sons.
30. Koning, J.(1948).Enige problemen uit de didactiek der natuurwetenschappen in het bijzonder van de scheikunde.Dissertation, Utrecht. Dordrecht:Petel & Felkers.
31. Lave, J.(1988).Cognition in practice: Mind, mathematics and culture in everyday life.Cambridge, UK:Cambridge University Press.
32. Lave, J.,Wenger, E.(1991).Situated learning: Legitimate peripheral performance.Cambridge:Cambridge University Press.
33. Mark, W.(1990).Measuring a van Hiele geometry sequence: A reanalysis.Journal for Research in Mathematics Education,21(3),230-237.
34. Mason, M. M.(1997).The van Hiele model of geometric understanding and mathematically talented students.Journal for the Education of the Gifted,21(1),38-53.
35. Miller, C.,A. Freedman (Eds.),P. Medway (Eds.)(1994).Genre and the new rhetoric.London:Taylor & Francis.
36. Mistretta, R. M.,Wagner, C.(2000).Enhancing geometric reasoning.Adolescence,35(138),365-378.
37. Murphy, G. L.,Medin, D. L.(1985).The role of theories in conceptual coherence.Psychological Review,92,289-316.
38. Nespor, J.(1994).Knowledge in motion.London:Falmer.
39. P. M. van Hiele(1986).Structure and insight.New York:Academic Press.
40. Papert, S.,Logo Computer Systems (Ed.)(1999).Logo philosophy and implementation.Montreal, Canada:Logo Computer Systems Inc.
41. Piaget, J.(1927).La Causalite physique chez l'enfant.Neuchatel:Delachaux & Niestle S. A.
42. Piaget, J.(1968).Le structuralisme.Presses Universitaire de France.
43. Phardrus
44. Aping Einstein
45. Rosch, E.,E. Rosch (Eds.),B. B. Lloyd (Eds.)(1978).Cognition and categorization.Hillsdale, NJ:Erlbaum Association.
46. Rosser, R.(1994).Cognitive development: Psychological and biological perspectives.Boston:Allyn & Bacon.
47. Ryan, M. L.(1981).On the why, what, and how of generic taxonomy.Poetics,10,109-126.
48. Sharp, J. M.(2001).Distance education: A powerful medium for developing teachers' geometric thinking.Journal of Computers in Mathematics & Science Teaching,20(2),199-219.
49. Siegler, R. S.(1998).Children's thinking.Upper Saddle River, NJ:Prentice Hall.
50. Simons, D. J.,Keil, F. C.(1995).An abstract to concrete shirt in the development of biological thought: The insides story.Cognition.
51. Simons, D. S.(1996).In sight, out of mind: When object representations fail.Psychological Science,7,301-305.
52. Simons, D. S.,Keil, F. C.(1995).An abstracts to concrete shift in cognitive development: The inside story.Cognition,56,129-163.
53. Smith, L. B.,Heise, D.,B. Burns (Ed.)(1992).Percepts, concepts and categories.Amsterdam:Elsevier Science.
54. Smith, L. B.,Jones, S. S.(1993).Cognition without concepts.Cognitive Development,8,181-188.
55. Strauss, S.,Langer, J.(1970).Operational thought inducement.Child Development,41,163-175.
56. Teppo, A.(1991).van Hiele levels of geometric thought revisited.Mathematics Teacher,84(3),210-221.
57. van Hiele, P. M.(1986).Structure and insight: A theory of mathematics education.New York:Academic Press.
58. van Hiele, P. M.(1959).Development and learning process: A study of some aspects of Piaget's psychology in relation with the didactics of mathematics.Groningen, Holland:J. B. Wolters.
59. van Hiele, P. M.(1999).Developing geometric thinking through activities that begin with play.Teaching Children Mathematics,5(6),310-316.
60. van Hiele, P. M.(1957).De problematiek van het inzicht, gedemonstreerd aan het inzicht van schoolkinderen in meetkunde-leerstof.Groningen, Holland:J. B. Wolters.
61. van Hiele, P. M. ,D. Fuys (Eds.),D. Geddes (Eds.),R. Tischler (Eds.)(1984).English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele.Brooklyn, NY:Brooklyn College, C.U.N.Y.
62. van Hiele, P. M.,D. Fuys (Eds.),D. Geddes (Eds.),R. Tischler (Eds.)(1984).English translation of selected writings of Dina van Hiele-Geldof and Pierre M. van Hiele.Brooklyn, NY:Brooklyn College, C.U.N.Y.
63. van Hiele, P. M.,van Hiele-Geldof, D.,H. Freudenthal (Ed.)(1958).Report on methods of initiation into geometry.Groningen, Holland:J. B. Wolters.
64. Vaughan, M. W.,Dillon, A.(1998).The role of genre in shaping our understanding of digital documents.Proceedings of the ASIS annual meeting
65. Vygotsky, L. S.(1978).Mind in society: The development of higher psychological processes.Cambridge, MA:Harvard University Press.
66. Wilson, P. S.(1990).Inconsistent ideas related to definitions and examples.Focus on Learning Problems in Mathematics,12(3-4),31-47.
67. Wu, C. J.,Wong, W. K.,Cheng, Y. H.,Lien, Y. W.,J. Novotná(Eds.),J. Moraová(Eds.),M. Krátká(Eds.),N. Stehliková(Eds.)(2006).Generating and evaluating geo metry conjectures with self-directed experiments.Proceedings of the 30th conference of the international group for the psychology of mathematics education,Prague:
68. Yates, J. A.,Orlikowski, W. J.(1992).Genres of organizational communication: A structurational approach.Academy of Management Review,17,299-326.
69. 卡密羅史密斯、謬小春譯(2001)。超越模組性：認知科學的發展觀。上海:華東師範大學出版社。
70. 左台益(2002)。van Hiele模式之國中幾何教材設計。中等教育，53(3)，44-53。
71. 左台益、梁勇能(2001)。國二學生空間能力與van Hiele幾何思考層次相關性研究。師大學報：科學教育類，46(1-2)，1-20。
72. 何森豪(2001)。van Hiele幾何發展水準之量化模式－以國小中高年級學生在四邊形概念之表現為例。測驗統計年刊，9，81-129。
73. 吳德邦、謝翠玲(1998)。根據van Hiele理論來探討國小數學實驗課程之幾何教材。中師數理學報，2(1)，22-62。
74. 徐于婷(2005)。碩士論文(碩士論文)。台北市，國立台北師範學院數學教育研究所。
75. 張英傑(2001)。兒童幾何形體概念之初步探究。國立臺北師範學院學報，14，491-527。
76. 陳于倩、姚如芬(2002)。Van Hiele幾何思考層次與SOLO分類法之比較。屏師科學教育，16，18-26。
77. 劉宜芳、吳昭容(2008)。兒童在數位式數學實驗情境下的發現學習。師大學報：科學教育類，53(1)，87-111。
78. 盧銘法(1999)。國小學生四邊形幾何概念之分析。中師數理學報，3(1)，1-37。