教學及研究建議

在文檔中 高一學生視覺化轉化為幾何推理之過程及其特徵 (頁 160-176)

(一)教學方面

目前有關幾何推理的教學多受考題形式影響,傾向強調對幾何圖形相關性質的應用,致 使學生容易重視結果而忽略推理的過程,更可能無法辨識如何能在最少的條件下合理地作論 述,這些都是從訪談對象身上看到的普遍現象。因此,強化學生對定義、性質及推理的理解,

以及對幾何概念的表述都是重要的教學目標。比較本研究學生填寫的紙本問卷與訪談過程中 所處理的推理情形看來,視覺化的情況在文字表述一般不易察覺,也因以文字表達推理過程 通常需呈現較完整的邏輯結構及結果,對沒有把握的內容,學生容易放棄作答,若只以書寫 的結果來看,將不易看到學生在推理過程的困難。因此建議教學中除了需要強化學生局部的 視覺化能力外,也要啟發他們對幾何物件之間的視覺結構,尤其著重引導學生在多個子圖的 複雜圖像中,應如何適當地選取子圖進行推理。

有關幾何推理任務的設計方面,建議考慮兼顧推理過程中的各個階段,即包括提出猜想、

注意限制、提出論述策略、尋找不同的不變量作論述到提出證明等機會,這些都是幾何推理 中的重要過程,這樣的證明發展脈絡是提供學生理解證明意涵的機會以及可能可以提高他們 對學習證明的意願。與此同時,任務的設計應考慮循序漸進地培養不同視覺化轉化幾何推理 的階層,尤其是能發展階層 4 以邏輯關係為主導的任務,而學生能夠處理這些任務必須要有 穩固的圖像組態模型,換句話說,教學中應同時強化學生對各幾何概念的知識結構、圖像基 模和證明途徑,而不只是理解相關概念的性質及應用。不同視覺化轉化幾何推理的階層提供 任務的設計依據,可成為發展幾何推理能力的參考,也是逐步銜接視覺化到幾何推理之間的 可能途徑。

動態幾何軟體環境雖然有很多幫助幾何推理的功能及優勢,然而這樣的工具在教學現場 中並未被普遍使用,在動態幾何軟體環境下的推理任務的設計(參考表 2-1),需要有別於傳 統給定命題及圖像,只注重提出論述的結果及依據的呈現方式,而是需要提供機會及引導以 激發學習者的想像能力。需要特別注意的是,由於物件可藉由拖曳隨意改變,部分圖形原來 的狀態會隨拖曳而消失,若相關狀態與後續推理有關,對推理能力較弱的學生來說,不容易 察覺圖形改變前後之間的關係,因此在環境設計時將依據需要考慮保留重要元素的痕跡,以 產生有效的資訊以幫助學習者進行推理。

(二)研究方面

從研究結果發現,學生所具備的幾何知識對他們形成臆測並沒有證據顯示有密切的關係。

換句話說,除了學生的先備知識外,還存在其他因素會影響學生進行合適的猜想,例如空間 推理能力(Clements & Battistia, 1992; Pittalis & Christou, 2010),未來研究可進一步探討影響 學生進行猜想的因素。本研究所使用之分析架構,主要整合了 Toulmin 論證模型以及在動態 幾何環境下進行論證的重要階段,並探討幾何物件與幾何知識之間的連結,然而在各個階段 中,視覺化的角色還需要作進一步的探討,以分析除了幾何知識外,其他影響視覺化的可能 因素。另外,分析學生在不同推理階段中的困難及特徵,亦是理解他們的幾何推理發展的必 要工作,以提供更詳細的資訊以發展幾何推理之教學及工具設計之參考。

不同視覺化轉化幾何推理的階層除了說明學生的推理層次外,也可以用作分析學生回答 推理任務時的層次分類,或分析當前教科書中任務所達的層次,從而提供發展培養不同階層 推理的任務及教學環境的參考。而在這些不同的階層中可能有影響推理表現的不同因素,將 有待後續研究作進一步的討論及分析。

動態幾何環境(如:GeoGebra)雖能提供使用者更多探索幾何物件的機會,並提供標示 邊角相等的不同記號、手寫筆等功能,但這些記號的操作並不方便,對學生的探索無法提供 足夠的支持,未來研究可針對在動態幾何軟體環境中保留圖形的痕跡、提供不同的簡便標記 功能等設計的開發,為幫助幾何推理學習提供更有效的展示平台或探索工具。

參考文獻

吳志揚、陳文豪(2004)。幾何學發展史簡介。數學傳播,28(1),24-33。

呂柏彥(2017)。台北市高中生數學作業態度和學習表現之研究(未出版之碩士論文)。國立 臺灣師範大學,臺北市。

林志能、洪振方(2008)。論證模式分析及其評量要素。科學教育月刊,312,2-18。

翁立衛(2008)。圖在幾何解題中所扮演的角色。科學教育月刊,308,7-15。

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

Aleven, V. A. W. M. M., & Koedinger, K. R. (2002). An effective metacognitive strategy: Learning by doing and explaining with a computer-based Cognitive Tutor. Cognitive Science, 26, 147-179.

Ali, I., Bhagawati, S., & Sarmah, J. (2014). Performance of geometry among the secondary school students of Bhurbandha CD Block of Morigaon District, Assam, India. International Journal of Innovative Research & Development, 3(11), 73-77.

Arcavi, A. (2003). The role of visual representations in the learning of mathematics. Educational Studies in Mathematics, 52(3), 215-241.

Arici, S., & Aslan-Tutak, F. (2013). The effect of origami-based instruction on spatial visualization, geometry achievement, and geometric reasoning. International Journal of Science and

Mathematics Education, 13(1), 179-200.

Arzarello, F., Olivero, F., Paola, D., & Robutti, O. (2002). A cognitive analysis of dragging

practises in Cabri environments. ZDM – The International Journal on Mathematics Education, 34(3), 66-72.

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.

Baccaglini-Frank, A., Antonini, S., Leung, A., & Mariotti, M. A. (2017). Designing

non-constructability tasks in a dynamic geometry environment. In A. Leung & A. Baccaglini-Frank (Eds.), Digital technologies in designing mathematics education tasks: Potential and pitfalls (pp. 99-120). Cham, Switzerland: Springer.

Battista, M. T. (2007). The development of geometric and spatial thinking. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 843-908). Charlotte, NC: Information Age Publishing.

Boero, P. (1999). Argumentation and mathematical proof: A complex, productive, unavoidable relationship in mathematics and mathematics education. International newsletter on the teaching and learning of mathematical proof, 7/8.

Boero, P., Douek, N., Morselli, F., & Pedemonte, B. (2010). Argumentation and proof: A

contribution to theoretical perspectives and their classroom implementation. In M. M. F. Pinto

& T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 179-209). Belo Horizonte, Brazil: PME.

Boz, N. (2005). Dynamic visualization and software environments. The Turkish Online Journal of Educational Technology, 4(1), 26-32.

Brockriede, W., & Ehninger, D. (1978). Toulmin on argument: An interpretation and application.

Quarterly Journal of Speech, 46(1), 44-53.

Bussi, M. G. B., & Mariotti, M. A. (2008). Semiotic mediation in the mathematics classroom:

Artifacts and signs after a Vygotskian perspective. In L. D. English (Ed.), Handbook of international research in mathematics education (pp. 746-783). New York, NY: Routledge.

Casey, E. S. (2000). Imagining: A phenomenological study (2nd ed.). Bloomington, IN: Indiana University Press.

Chinnappan, M. (1998). Schemas and mental models in geometry problem solving. Educational Studies in Mathematics, 36(3), 201-217.

Chinnappan, M., Ekanayake, M. B., & Brown, C. (2012). Knowledge use in the construction of geometry proof by Sri Lankan students. International Journal of Science and Mathematics Education, 10(4), 865-887.

Choi-Koh, S. S. (1999). A student’s learning of geometry using the computer. The Journal of Educational Research, 92(5), 301-311.

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. 420-464). New York, NY:

Macmillan.

Common Core State Standards Initiative. (2010). Common core state standards for mathematics.

Retrieved from http://www.corestandards.org/wp-content/uploads/Math_Standards1.pdf Crowley, M. L. (1987). The van Hiele model of the development of geometric thought. In M. M.

Lindquist (Ed.), Learning and teaching geometry, K-12: 1987 Yearbook of the National Council of teachers of mathematics (pp. 1-16). Reston, VA: National Council of Teachers of Mathematics.

de Guzmán, M. (2002, July). The role of visualization in the teaching and learning of mathematics analysis. Paper presented at the 2nd International Conference on the Teaching of Mathematics (at the Undergraduate Level), Hersonissos, Greece.

de Villiers, M. (1990). The role and function of proof in mathematics. Pythagoras, 24, 7-24.

de Villiers, M. D. (1995). An alternative introduction to proof in dynamic geometry. MicroMath, 11(12), 14-19.

Dreyfus, T. (1991). On the status of visual reasoning in mathematics and mathematics education. In F. Furinghetti (Ed.), Proceedings of the 15th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 33-48). Assisi, Italy: PME.

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). Berlin, Germany: Springer.

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, The Netherlands: Kluwer Academic Publishers.

Fischbein, E. (1993). The theory of figural concepts. Educational Studies in Mathematics, 24(2), 139-162.

Gal, H., & Linchevski, L. (2010). To see or not to see: Analyzing difficulties in geometry from the perspective of visual perception. Educational Studies in Mathematics, 74(2), 163-183.

Gawlick, T. (2005). Connect arguments to actions – Dynamic geometry as means for the attainment of higher van Hiele levels. ZDM – The International Journal on Mathematics Education, 37(5), 361-370.

Gegenfurtner, A., Lehtinen, E., & Säljö, R. (2011). Expertise differences in the comprehension of visualizations: A meta-analysis of eye-tracking research in professional domains. Educational Psychology Review, 23(4), 523-552.

Guven, B. (2008). Using dynamic geometry software to gain insight into a proof. International Journal of Computers for Mathematical Learning, 13(3), 251-262.

Habre, S. (2001). Visualization enhanced by technology in the learning of multivariable calculus.

International Journal for Technology in Mathematics Education, 8(2), 115-130.

Hadas, N., Hershkowitz, R., & Schwarz, B. B. (2000). The role of contradiction and uncertainty in promoting the need to prove in dynamic geometry environments. Educational Studies in Mathematics, 44(1-3), 127-150.

Hadas, N., Hershkowitz, R., & Schwarz, B. B. (2002). Analyses of activity design in geometry in the light of student actions. Canadian Journal of Science, Mathematics and Technology Education, 2(4), 529-552.

Hanna, G., & de Villiers, M. (2008). ICMI study 19: Proof and proving in mathematics education.

ZDM Mathematics Education, 40(2), 329-336.

Hanna, G., & de Villiers, M. (Eds.) (2012). Proof and proving in mathematics education: The 19th ICMI study. Dordrecht, The Netherlands: Springer.

Harel, G., & Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. CBMS Issues in Mathematics Education, 7, 234-283.

Hattermann, M. (2010). The drag-mode in three dimensional dynamic geometry environments – Two studies. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (pp. 786-795). Lyon, France: Institut National de Recherche Pédagogique.

Hauptman, H., & Cohen, A. (2011). The synergetic effect of learning styles on the interaction between virtual environments and the enhancement of spatial thinking. Computers &

Education, 57(3), 2106-2117.

Healy, L. (2000). Identifying and explaining geometrical relationship: Interactions with robust and soft Cabri constructions. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 103-117). Hiroshima, Japan: Hiroshima University.

Hegarty, M. (2004). Dynamic visualizations and learning: Getting to the difficult questions.

Learning and Instruction, 14(3), 343-351.

Hemmi, K., & Löfwall, C. (2010). Why do we need proof. In V. Durand-Guerrier, S.

Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education (pp. 201-210). Lyon, France: Institut National de

Recherche Pédagogique.

Herbst, P. G. (2002). Establishing a custom of proving in American school geometry: Evolution of the two-column proof in the early twentieth century. Educational Studies in Mathematics, 49, 283-312.

Hohenwarter, M., & Fuchs, K. (2004, July). Combination of dynamic geometry, algebra and calculus in the software system GeoGebra. Paper presented at Computer Algebra Systems and Dynamic Geometry Systems in Mathematics Teaching Conference, Pécs, Hungary.

Hollebrands, K. F. (2007). The role of a dynamic software program for geometry in the strategies high school mathematics students employ. Journal for Research in Mathematics Education, 38(2), 164-192.

Hölzl, R. (1996). How does 'dragging' affect the learning of geometry. International Journal of Computers for Mathematical Learning, 1(2), 169-187.

Houdement, C., & Kuzniak, A. (2004). Elementary geometry split into different geometrical

paradigms. In M. Mariotti (Ed.), Proceedings of the Third Conference of the European Society for Research in Mathematics Education. Bellaria, Italy: University of Pisa and ERME.

Retrieved from

http://www.dm.unipi.it/~didattica/CERME3/proceedings/Groups/TG7/TG7_Houdement_cerm e3.pdf

Hoyles, C., & Jones, K. (1998). Proof in dynamic geometry contexts. In C. Mammana & V. Villani (Eds.), Perspectives on the teaching of geometry for the 21st century (pp. 121-128). Dordrecht, The Netherlands: Kluwer Academic Publishers.

Jackiw, N. (2001). The Geometer's Sketchpad (Version 4.0) [Computer software]. Emeryville, CA:

Key Curriculum Press.

Johnson, G. J., Thompson, D. R., & Senk, S. L. (2010). Proof-related reasoning in high school textbooks. Mathematics Teacher, 103(6), 410-417.

Keşan, C., & Çalişkan, S. (2013). The effect of learning geometry topics of 7th grade in primary education with dynamic geometer’s sketchpad geometry software to success and retention. The Turkish Online Journal of Educational Technology, 12(1), 131-138.

Kilpatrick, J., Swafford, J., & Findell, B. (Eds.). (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.

Koedinger, K. R., & Anderson, J. R. (1990). Abstract planning and perceptual chunks: Elements of expertise in geometry. Cognitive Science, 14(4), 511-550.

Komatsu, K., & Jones, K. (2019). Task design principles for heuristic refutation in dynamic geometry environments. International Journal of Science and Mathematics Education, 17(4), 801-824.

Kordaki, M., & Balomenou, A. (2006). Challenging students to view the concept of area in triangles in a broad context: Exploiting the features of Cabri-II. International Journal of Computers for Mathematical Learning, 11(1), 99-135.

Kösa, T., & Karakuş, F. (2010). Using dynamic geometry software Cabri 3D for teaching analytic geometry. Procedia Social and Behavioral Sciences, 2(2), 1385-1389.

Kospentaris, G., Spyrou, P., & Lappas, D. (2011). Exploring students’ strategies in area conservation geometrical tasks. Educational Studies in Mathematics 77(1), 105-127.

Kramarski, B., & Gutman, M. (2006). How can self-regulated learning be supported in

mathematical E-learning environments? Journal of Computer Assisted Learning, 22(1), 24-33.

Kramarski, B., & Hirsch, C. (2003). Using computer algebra systems in mathematics classrooms.

Journal of Computer Assisted Learning, 19(1), 35-45.

Kramarski, B., & Ritkof, R. (2002). The effects of metacognition and email interactions on learning graphing. Journal of Computer Assisted Learning, 18(1), 33-43.

Kutluca, T. (2013). The effect of geometry instruction with dynamic geometry software; GeoGebra on Van Hiele geometry understanding levels of students. Educational Research and Reviews, 8(17), 1509-1518.

Laborde, C. (1993). The computer as a part of the learning environment: The case of geometry. In C. Keitel & K. Ruthven (Eds.), Learning from computers: Mathematics education and technology (pp. 48-67). Berlin, Germany: Springer-Verlag.

Laborde, C. (2000). Dynamic geometry environments as a source of rich learning contexts for the complex activity of proving. Educational Studies in Mathematics, 44(1/2), 151-156.

Laborde, C. (2005a). The hidden role of diagrams in students' construction of meaning in geometry.

In J. Kilpatrick, C. Hoyles, O. Skovsmose, & P. Valero (Eds.), Meaning in mathematics education (pp. 159-179). New York, NY: Springer.

Laborde, C. (2005b). Robust and soft constructions: Two sides of the use of dynamic geometry environments. In S. C. Chu, H. C. Lew, & W. C. Yang (Eds.), Proceedings of the 10th Asian Technology Conference in Mathematics (pp. 22-35). Cheongju, South Korea: Korea National University of Education.

Lanzing, J. W. A., & Stanchev, I. (1994). Visual aspects of courseware engineering. Journal of Computer Assisted Learning, 10(2), 69-80.

Larkin, J. H., & Simon, H. A. (1987). Why a diagram is (sometimes) worth ten thousand words.

Cognitive Science, 11(1), 65-99.

Leung, A. (2015). Discernment and reasoning in dynamic geometry environments. In S. J. Cho (Ed.), Selected regular lectures from the 12th International Congress on Mathematical

Leung, A. (2015). Discernment and reasoning in dynamic geometry environments. In S. J. Cho (Ed.), Selected regular lectures from the 12th International Congress on Mathematical

在文檔中 高一學生視覺化轉化為幾何推理之過程及其特徵 (頁 160-176)