### Concept of Triangle: Examples of Mathematical Abstraction in Two Different Contexts

#### Abstract

Geometry has abstract notions to be learnt so that all those notions cannot be just transferred into students’ mind like a bunch of information that should be memorized. Students need to construct those concepts during their learning process. This process of knowledge construction can be considered as an abstraction process. This study aimed to qualitatively compare abstraction process of students who learned the topic of triangle in conventional method and in van Hiele model of teaching aided by Geometers’ sketchpad. Subjects of this study were junior high school students in grade 7. This is a qualitative study with grounded theory design. Data were collected through classroom observation, test, and task-based interview. Results of the study show that theoretical abstraction processes tend to dominate classrom with conventional method of teaching while classroom with van Hiele model of teaching aided by Geometers’ sketchpad accommodated empirical abstraction process of the students.

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Alwasilah, C. (2003). Pokoknya Kualitatif. Bandung: PT Kiblat Buku Utama.

Battista, M. T. (2007). The Development of Geometric and Spatial Thinking. In F. K. Lester (Ed.), Second Handbook of Research on Mathematics Teaching and Learning (pp. 843-908). Charlotte, NC: Information Age.

Chapko, M. A., & Buchko, M. (2004). Math Instruction for Inquiring Minds. Principal (Reston,Va.), 84, 30-33.

Chazan, D., & Yerushalmy, M. (1998). Charting a course for secondary geometry. In R. Lehrer & D. Chazan (Eds.), Designing learning environment for developing understanding of geometry and space (pp.67-90). Mahwah, Nj: Erlbaum.

Choi-koh, S. (2000). The Activities Based on van Hiele Model Using Computer as a Tool. Journal of the Korea Society of Mathematics education series D: research in Mathematics Education vol. 4, No. 2. November 2000, 63-67. Seoul.

Crowley, L Mary. (1987). “The Van Hiele Model of the Development of Geometric Thought.” Dalam Learning and Teaching Geometry, K-12. National of Teachers of Mathematics (NCTM). Reston Virginia: NCTM, Inc.

Dreyfus, T. & Gray, E. (2002). Research Forum “Abstraction: Theories about the emergence of knowledge structures”. In Cockburn, A. and Nardi, E (Eds). Proceedings of the 26th Annual Conference of PME, Norwich: University of East Anglia, pp. 113-133.

Dreyfus, T. (1991). On the Status of Visual Reasoning in Mathematics and Mathematics Education, In F. Finghetti (Ed.), Proceeding of PME 15 vol. 1, pp. 32-48. Assisi, Italy.

Dreyfus, T. (2002). The Nature of Advanced Mathematical Thinking, in David, T., (Ed). Advanced Mathematical thnking. (Vol. 11. Pp 25-41). New York: Kluwer Academic Publisher.

Dubinsky, E. (2002). Reflective Abstraction in Advanced Mathematical Thinking, in David, T. (Ed). Advanced Mathematical Thinking. (Vol 11, pp 95-123). New York: Kluwer Academic Publishing.

Ferrari, P.L. (2003). Abstraction in mathematics. Philosophical Transactions of the Royal Society of London. Series B: Biological Sciences, 358(1435), 1225–1230.

Goldenberg, E. P., & Cuoco, A. A. (1998). What is Dynamic Geometry? In R. Lehrer & D. Chazan (Eds), Designing Learning Environments for Developing Understanding of Geometry and Space. (pp. 351-368). Hillsdale, New Jersey: Lawrence Erlbaum Associates.

Goodson, T., & Espy. (2005). Why Reflective Abstraction Remains Relevant In Mathematics Education Research. Proceedings of the 27th Annual Meeting of PME, NA, Virginia Tech, October 2005 2.

Gray, E. & Tall, D. (2001.) Relationships between embodied objects and symbolic procepts: an explanatory theory of success and failure in mathematics. In Heuvel-Panhuizen, M.(Ed.) Proceedings of the 25th conference of the international group for the psychology of mathematics education. Utrecht, Netherland: Freudenthal Institute.

Hazzan, O. (2003). How students attempt to reduce abstraction in the learning of mathematics and in the learning of computer science,’ Computer Science Education 13(2), 95–122.

Hershkowitz, R., Schwarz, B. B., & Dreyfus, T. (2001). Abstraction in context: Epistemic actions. Journal for Research in Mathematics Education 32, 195-222.

Hong, J. Y., & Kim, M.K. ( 2015). Mathematical abstraction in the solving of ill-structured problems by elementary school students in Korea. Eurasia J. Math. Sci. & Tech. Ed., 12(2), 267-281.

Hoyles C & Healy L 1997. Unfolding Meanings for Reflective Symmetry. International Journal of Computers for Mathematical Learning, 2:27-59.

Laborde, C., Chronis, K., Hollebrands, K., Stasser, R. (2006). Teaching and Learning Geometry with Technology. Dalam Handbook of Research on the Psychology of Mathematics Education. Past, present and Future. London: Sense Publisher.

Mitchelmore, M.C., & White, P. (2000). Development of Angle concepts by Progressive Abstraction and Generalization. Educational Studies in Mathematics, 41, 209-238.

Mitchelmore, M & White, P. (2004). Abstraction in Mathematics and Mathematics Learning. Proceeding of the 28th Conference of the International Group for the Psychology of Mathematics Education, 3, 329-336.

Mitchelmore, M. C., & White, P. (2007). Abstraction in Mathematics Learning. Mathematics Education Research Journal, 19(2), 1-9.

Nurhasanah, F. (2004). Proses Berpikir Siswa Sekolah Menengah Tingkat Pertama Dalam Belajar Geometri Pada Pokok Bahasan Jajargenjang, Belahketupat, Layang-Layang Dan Trapesium. Unpublished Thesis. FKIP UNS Surakarta.

Nurhasanah, F. (2010). Abstraksi Siswa SMP dalam Belajar Geometri Melalui Penerapan Model van Hiele dan Geometers’ Sketchpad. Unpublished Thesis. Universitas Pendidikan Indonesia: Bandung.

Nurhasanah, F. (2011). Junior High School Students’ Abstraction in Learning Geometry. Proceeding of International Seminar and the Fourth National Conference on Mathematics Education 2011, Department of Mathematics Education, Yogyakarta State University, Yogyakarta, July 21-23 2011.

Nurhasanah, F., Sabandar, J., & Kusumah, Y. S. (2013). Abstraction Processes in Learning Geometry Using GSP. Proceeding of 6th East Asia Regional Conference on Mathematics Education (Earcome6) 17-22 March 2013. Phuket: Prince Songkla University.

Olkun, S., Snoplu, N.B., & Deryakulu, D. (2002). Geometric explorations with Dynamic Geometry Applications based on van Hiele Levels. International Journal for Mathematics teaching and Learning, 1-12.

Piaget, J. (1970). Science of Education and Psychology of the Child. New York: Viking.

Raine, D., & Colett, J. (2003). Problem-based learning in Astrophysics. European Journal of Physics, 24(2), 1-9.

Ross, B. H. (1996). Category representation and the effects of interacting with instances. Journal of Experimental Child Psychology: Learning, Memory, and Cognition, 22(5), 1249-1265.

Saads, S. & Davis, G. (1997). Spatial abilities, van Hiele levels, and language use in three dimensional geometry. In Pehkonen, E. (Ed.), Proceedings of the 21st Conference of the International Group for the Psychology of Mathematics Education, 4, 104-111. Lahti, Finland.

Saitta, L., & Zucker, J.-D. (2013). Abstraction in Different Disciplines, in Saitta, L., & Jean, -D., Z (Ed.), Abstraction in Artificial Intellegent and Complex System. New York: Springer.

Skemp, R. (1986). The Psychology of Learning Mathematics. London: Penguin.

White, P., & Mitchelmore, M. C. (2010). Teaching for abstraction: A model. Mathematical Thinking and Learning, 12, 205-226.

William, G. (2007). Abstracting in the context of spontanious learning. Mathematics Education Research Journal, 19(2), 69-88.

DOI: http://dx.doi.org/10.12928/ijeme.v1i1.5782

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