# 教學及研究建議

（一）教學方面

（二）研究方面

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。

參考文獻

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。

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.

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