動態幾何環境下大學生幾何探索之研究

Analyzing College Students' Geometric Investigation within the Dynamic Geometry Environment

10.6278/tjme.20140307.001

許舜淵(Shun-Yuan Xu)；胡政德(Cheng-Te Hu)

動態表徵 ； 動態幾何環境 ； 幾何探索 ； dynamic representation ； dynamic geometry environment ； geometric investigation

臺灣數學教育期刊

1卷1期（2014 / 04 / 01）

49 - 77

繁體中文

本研究目的在探索動態幾何環境下大學生幾何探索之思考運作模式。透過個案研究來進行探究並以質性分析來詮釋資料。研究結果顯示：（1）當學生觀察動態幾何軟體所產生動態表徵時，通常透過幾何思考後再做適當的拖曳行動；（2）動態表徵其外顯的行為和內在的數學性質會激發個體產生猜測，並在心智中模擬操作數學物件以及分析可能的動態行為來驗證猜測，進而產生宣告；（3）學生會依據模擬操作的複雜程度，再決定是否使用DGS具體操作以驗證幾何思考過程中的想法；（4）學生在動態幾何環境下進行幾何實驗並與幾何思考不斷地交互作用下探索幾何性質。

The aim of this study is to explore, in a dynamic geometry environment (DGE), the operation models of the geometric thinking of college students. To achieve this, we conducted a case study. We recorded the process of geometry exploration activities by math-major college students, interviewed them, and interpreted their operation models of thinking through a qualitative analysis. The results are summarized as follows: (1) When students observe the dynamic representations generated by dynamic geometry software (DGS), they seldom react immediately, and instead engage in geometric thinking before they carry out appropriate dragging. (2) The apparent actions and intrinsic mathematical properties of dynamic representations tend to inspire students' conjectures. Students then mentally manipulate mathematical objects and analyze possible dynamic behaviors to confirm their conjectures. Finally, they are able to produce a declaration. This process is a basic model for geometric thinking. (3) Students manipulate mathematical objects mentally based on the complexity of operation, and then decide whether to use a DGS-specific claim or conjecture in geometric thinking. (4) Students explore geometry properties in DGE under the constant interactions between geometry experiments and geometric thinking.

1. Arzarello, F.,Olivero, F.,Paola, D.,Robutti, O.(2002).A cognitive analysis of dragging practises in Cabri environments.ZDM,34(3),66-72.
2. Fischbein, E.(1993).The theory of figural concepts.Educational Studies in Mathematics,24(2),139-162.
3. Furinghetti, F.,Paola, D.(2003).To produce conjectures and to prove them within a dynamic geometry environment: A case study.Proceedings of the 27th International Group for the Psychology of Mathematics Education Conference Held Jointly with the 25th PME-NA Conference,Honolulu, HI:
4. Hazzan, O.,Goldenberg, E. P.(1996).Students' understanding of the notion of function in dynamic geometry environments.International Journal of Computers for Mathematical Learning,1(3),263-291.
5. Hölzl, R.(1996).How does 'dragging' affect the learning of geometry.International Journal of Computers for Mathematical Learning,1(2),169-187.
6. Laborde, C.(1993).The computer as part of the learning environment: The case of geometry.Learning from computers: Mathematics education and technology,Berlin, Germany:
7. Lakatos, I.(1976).Proofs and refutations: The logic of mathematical discovery.Cambridge, UK:Cambridge University Press.
8. Lester, F. K.(Ed.)(2007).Second handbook of research on mathematics teaching and learning.Charlotte, NC:Information Age.
9. Lopez-Real, F.,Leung, A.(2006).Dragging as a conceptual tool in dynamic geometry environments.International Journal of Mathematical Education in Science and Technology,37(6),665-679.
10. Mammana, C.(Ed.),Villani, V.(Ed.)(1998).Perspectives on the Teaching of Geometry for the 21st Century.Dordrecht, The Netherlands:Kluwer Academic.
11. Mariotti, M. A.(2000).Introduction to proof: The mediation of a dynamic software environment.Educational Studies in Mathematics,44(1-3),25-53.
12. Mariotti, M. A.,Laborde, C.,Falcade, R.(2003).Function and graph in DGS environment.Proceedings of the 27th International Group for the Psychology of Mathematics Education Conference Held Jointly with the 25th PME-NA Conference,Honolulu, HI:
13. Pinto, M.,Tall, D.(2002).Building formal mathematics on visual imagery: A case study and a theory.For the Learning of Mathematics,22(1),2-10.
14. Sorensen, R. A.(1992).Thought experiments.New York, NY:Oxford University Press.
15. Sutherland, R.(Ed.),Mason, J.(Ed.)(1995).Exploiting mental imagery with computers in mathematics education.New York, NY:Springer-Verlag.
16. Tall, D.(1998).Information technology and mathematics education: Enthusiasms, Possibilities & Realities.Proceedings of the 8th International Congress on Mathematical Education,Sevilla, Spain:
17. Tall, D.(1999).The chasm between thought experiment and mathematical proof.Mathematische bildung und neue technologien,Leipzig, Germany:
18. Talmon, V.,Yerushalmy, M.(2004).Understanding dynamic behavior: Parent-child relations in dynamic geometry environments.Educational Studies in Mathematics,57(1),91-119.
19. Vygotsky, L. S.,Cole, M.(Ed.Trans.),John-Steiner, V.(Ed.Trans.),Scribner, S.(Ed.Trans.),Souberman, E.(Ed.Trans.)(1978).Mind in society: The development of higher mental psychological processes.Cambridge, MA:Harvard University Press.
20. 陳英娥、林福來(1998)。數學臆測的思維模式。科學教育學刊，6(2)，191-218。

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