拖曳對國小生理解四邊形包含關係之研究

A study of Dragging in Dynamic Geometry Environment in Developing Elementary School Students' Understanding of Inclusive Relationships among Quadrilaterals

10.6278/tjme.202004_7(1).003

蔡淑君(Shu-Chun Tsai)；許慧玉(Hui-Yu Hsu)；鄭英豪(Ying-Hao Cheng)；陳建誠(Jian-Cheng Chen)

四邊形 ； 包含關係 ； 幾何性質 ； 動態幾何環境 ； quadrilateral ； inclusive relationship ； geometric properties ； Dynamic Geometry Environment (DGE)

臺灣數學教育期刊

7卷1期（2020 / 04 / 01）

27 - 54

繁體中文

本研究旨在探討動態幾何環境之拖曳行為如何影響學生理解不同四邊形間的包含關係。從三位五年級學生個案資料分析得知：第一，即使學生能在紙筆測驗正確回答四邊形定義，動態幾何環境圖形的呈現方式仍會改變他們對於定義的想法。第二，從分析學生在動態幾何環境拖曳，區別出六種不同認知行為和思維類別：（1）認為拖曳無法窮盡各種圖形；（2）關注圖形外觀；（3）微幅調度；（4）以特定性質為目標；（5）結合表格分析策略；（6）從一般例到特例。不同拖曳行會影響學生後續是否成功建構四邊形的包含關係。第三，研究提出學生不同判定四邊形關係的思維模式，包括分類思維、交集思維和包含思維。依據研究結果，本文進一步提出動態幾何環境拖曳和包含關係的研究和教學相關建議。

This study investigated how dragging in a Dynamic geometry environment (DGE) influences fifth graders in understanding inclusive relationships among quadrilaterals. We selected students who correctly recalled quadrilateral definitions but did not understand inclusive relationships. An analysis revealed that the diagrams in the DGE influenced students' responses to quadrilateral definitions, indicating that their concept definitions of the quadrilaterals are not stable. Six types of student cognitive behaviors associated with dragging in the DGE were identified. First, students could not understand that dragging can help them find all diagram examples constructed based on certain geometric properties. Second, students may only focus on diagram outlooks when dragging. Third, students may slightly drag vertices or segments to avoid big diagram changes. Fourth, students may focus on what diagrams they intended to drag without noticing the dragging process. Fifth, students may only use data shown in table to evaluate inclusive relationships. Sixth, students can notice generalization and specification in various dynamic examples. Additionally, the types of student evaluations on relationships between quadrilaterals were recognized. They were evaluations based on classification, intersection, and inclusive relationship. Implications and suggestions for research and practices about the use of dragging to learn inclusive relationships are proposed.

1. 鄭英豪, Y. H.,陳建誠, J. C.,許慧玉, H. Y.(2017)。國中生在動態幾何軟體輔助下臆測幾何性質之研究。臺灣數學教育期刊，4(1)，1-34。
   連結：
2. Arzarello, F.,Olivero, F.,Paola, D.,Robutti, O.(2002).A cognitive analysis of dragging practises in Cabri environments.ZDM Mathematics Education,34(3),66-72.
3. Baccaglini-Frank, A.,Mariotti, M. A.(2010).Generating conjectures in dynamic geometry: The maintaining dragging model.International Journal of Computers for Mathematical Learning,15(3),225-253.
4. Bartolini Bussi, M. G.,Baccaglini-Frank, A.(2015).Geometry in early years: Sowing seeds for a mathematical definition of squares and rectangles.ZDM Mathematics Education,47(3),391-405.
5. Birks, M.,Mills, J.(2011).Grounded theory: A practical guide.Los Angeles, CA:Sage.
6. de Villiers, M.(1986).The role of axiomatization in mathematics and mathematics teaching.Stellenbosch, South Africa:University of Stellenbosch.
7. de Villiers, M.(1994).The role and function of a hierarchical classification of quadrilaterals.For the Learning of Mathematics,14(1),11-18.
8. Fischbein, E.(1993).The theory of figural concepts.Educational Studies in Mathematics,24(2),139-162.
9. Fischbein, E.,Nachlieli, T.(1998).Concepts and figures in geometrical reasoning.International Journal of Science Education,20(10),1193-1211.
10. Fujita, T.(2012).Learners’ level of understanding of the inclusion relations of quadrilaterals and prototype phenomenon.The Journal of Mathematical Behavior,31(1),60-72.
11. Fujita, T.,Jones, K.(2007).Learners' understanding of the definitions and hierarchical classification of quadrilaterals: Towards a theoretical framing.Research in Mathematics Education,9(1),3-20.
12. Hershkowitz, R.(1989).Visualization in geometry: Two sides of the coin.Focus on learning problems in mathematics,11(1-2),61-76.
13. Kaur, H.(2015).Two aspects of young children's thinking about different types of dynamic triangles: Prototypicality and inclusion.ZDM Mathematics Education,47(3),407-420.
14. Kilpatrick, J.(Ed.),Hoyles, C.(Ed.),Skovsmose, O.(Ed.),Valero, P.(Ed.)(2005).Meaning in mathematics education.New York, NY:Springer.
15. LeCompte, M. D.(Ed.),Millroy, W. L.(Ed.),Preissle, J.(Ed.)(1992).The handbook of qualitative research in education.New York, NY:Academic Press.
16. Leung, I. K. C.(2008).Teaching and learning of inclusive and transitive properties among quadrilaterals by deductive reasoning with the aid of SmartBoard.ZDM Mathematics Education,40(6),1007-1021.
17. Lin, F.-L.,Yang, K.-L.(2002).Defining a rectangle under a social and practical setting by two seventh graders.ZDM Mathematics Education,34(1),17-28.
18. Okazaki, M.,Fujita, T.(2007).Prototype phenomena and common cognitive paths in the understanding of the inclusion relations between quadrilaterals in Japan and Scotland.Proceedings of the 31st International Group for the Psychology of Mathematics Education Conference,Seoul, Korea:
19. Piaget, J.(1971).Science of education and the psychology of the child.New York, NY:Viking Press.
20. Tall, D.,Vinner, S.(1981).Concept image and concept definition in mathematics with particular reference to limits and continuity.Educational Studies in Mathematics,12(2),151-169.
21. Walcott, C.,Mohr, D.,Kastberg, S. E.(2009).Making sense of shape: An analysis of children's written responses.The Journal of Mathematical Behavior,28(1),30-40.
22. Winer, G. A.(1980).Class-inclusion reasoning in children: A review of the empirical literature.Child Development,51(2),309-328.
23. 李源順（編）, Yuan-Shun (Ed.)(2019).國民小學數學.臺南=Tainan, Taiwan:南一=Nani.
24. 范力分, Li-Fen(2017)。新竹市=Hsinchu，國立清華大學=National Tsing Hua University。
25. 教育部（2018）。十二年國民基本教育課程綱要國民中小學暨普通型高級中等學校：數學領域。臺北：作者。【Ministry of Education. (2018). Curriculum guidelines of 12-year basic education curriculum national primary and secondary schools and ordinary senior secondary schools: Mathematics. Taipei: Author. (in Chinese)】
26. 楊瑞智（編）, Jui-Chih(Ed.)(2019).國民小學數學.新北=New Taipei, Taiwan:康軒=Kang Hsuan.
27. 蔡淑君, Shu-Ghun(2017)。新竹市=Hsinchu，國立清華大學=National Tsing Hua University。