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RESEARCH PAPER
Learning trajectory of geometry proof construction: Studying the emerging understanding of the structure of Euclidean proof
1
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Negeri Malang, Malang, INDONESIA
2
Institute for Science Education and Communication, University of Groningen, Groningen, NETHERLANDS
3
Department of Mathematics and Applied Mathematics, University of Crete, Crete, GREECE
Online publication date: 2023-04-06
Publication date: 2023-05-01
EURASIA J. Math., Sci Tech. Ed 2023;19(5):em2266
KEYWORDS
ABSTRACT
This paper presents a learning trajectory of geometry proof (LTGP) for Indonesian prospective
mathematics teachers (PMTs) during their first year of studies at an Indonesian university. The
trajectory aims at PMTs’ progression of their understanding of the structure of proof and their
proof construction abilities. We designed and implemented teaching materials with geometry
problems based on the use of the flow-chart proof format and the model of understanding of
proof structure from Miyazaki et al. (2017). We present an analysis of data from pre- and posttests of proof construction problems, written answers to proof problems during intervention with
60 PMTs, and individual interviews with eight PMTs. We found that the intervention supports
PMTs’ understanding of the structure of proof and their proof construction abilities. Our findings
contribute to knowledge about teaching strategies to support students in their understanding
and construction of a proof. From our findings, we suggest the use of the flow-chart proof format
together with other more formal proof formats in creating, reading, and rewriting proof of
geometric propositions and the use of open problems to encourage students to think forward
and backwards interactively to help students plan for proof construction.
REFERENCES (28)
1.
Antonini, S., & Mariotti, M. A. (2010). Abduction and the explanation of anomalies: The case of proof by contradiction. In V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Eds.), Proceedings of the 6th Conference of European Research in Mathematics Education (pp. 322-331). PME.
2.
Anwar, L., Mali, A., & Goedhart, M. (2022). Formulating a conjecture through an identification of robust invariants with a dynamic geometry system. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/002073....
3.
Anwar, L., Mali, A., & Goedhart, M. J. (2021). The effect of proof format on reading comprehension of geometry proof: The case of Indonesian prospective mathematics teachers. EURASIA Journal of Mathematics, Science and Technology Education, 17(4), 1-15. https://doi.org/10.29333/EJMST....
4.
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. https://doi.org/10.1007/s10758....
5.
Bakker, A. (2018). Design research in education: A practical guide for early career researchers. Routledge. https://doi.org/10.4324/978020....
6.
Cirillo, M., & Herbst, P. G. (2012). Moving toward more authentic proof practices in geometry. The Mathematics Educator, 21(2), 11-33.
7.
Durand-Guerrier, V., Boero, P., Douek, N., Epp, S. S., & Tanguay, D. (2012). Examining the role of logic in teaching proof. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education: New ICMI study series (pp. 369-389). Springer. https://doi.org/10.1007/978-94....
8.
Heinze, A., Cheng, Y. H., Ufer, S., Lin, F. L., & Reiss, K. M. (2008). Strategies to foster students’ competencies in constructing multi-steps geometric proofs: Teaching experiments in Taiwan and Germany. ZDM-International Journal on Mathematics Education (A), 40(3), 443-453. https://doi.org/10.1007/s11858....
9.
Inagaki, K., Hatano, G., & Morita, E. (1998). Construction of mathematical knowledge through whole-class discussion. Learning and Instruction, 8(6), 503-526. https://doi.org/10.1016/S0959-....
10.
Ivars, P., Fernández, C., Llinares, S., & Choy, B. H. (2018). Enhancing noticing: Using a hypothetical learning trajectory to improve pre-service primary teachers’ professional discourse. EURASIA Journal of Mathematics, Science and Technology Education, 14(11), em1599. https://doi.org/10.29333/ejmst....
11.
Mchugh, M. L. (2012). Interrater reliability: The kappa statistic. Biochemia Medica, 22(3), 276-282. https://doi.org/10.11613/BM.20....
12.
Mckee, K., Savic, M., Selden, J., & Selden, A. (2010). Making actions in the proving process explicit, visible, and “reflectable.” In S. Brown (Ed.), Proceedings of the 13th Annual Conference on Research in Undergraduate Mathematics Education.
13.
Mckenney, S., & Reeves, T. C. (2012). Conducting educational design research. Routledge. https://doi.org/10.4324/978020....
14.
Mills, G. E. (2014). Action research: A guide for the teacher researcher. Prentice Hall Columbus.
15.
Miyazaki, M., Fujita, T., & Jones, K. (2012). Introducing the structure of proof in lower secondary school geometry: A learning progression based on flow-chart proving. In S. J. Cho (Ed.), Proceedings of the 12th International Congress on Mathematical Education (pp. 2858-2867). ICMI.
16.
Miyazaki, M., Fujita, T., & Jones, K. (2015). Flow-chart proofs with open problems as scaffolds for learning about geometrical proofs. ZDM-Mathematics Education, 47(7), 1211-1224. https://doi.org/10.1007/s11858....
17.
Miyazaki, M., Fujita, T., & Jones, K. (2017). Students’ understanding of the structure of deductive proof. Educational Studies in Mathematics, 94(2), 223-239. https://doi.org/10.1007/s10649....
18.
Plomp, T. (2013). Educational design research: An introduction. In T. Plomp & N. Nieveen (Eds.), Educational design research (p. 204). SLO. https://doi.org/10.1007/978-1-....
19.
Prediger, S., Gravemeijer, K., & Confrey, J. (2015). Design research with a focus on learning processes: An overview on achievements and challenges. ZDM-Mathematics Education, 47(6), 877-891. https://doi.org/10.1007/s11858....
20.
Selden, A. (2012). Transitions and proof and proving at tertiary level. In G. Hanna & M. de Villiers (Eds.), Proof and proving in mathematics education: New ICMI study series (pp. 391-420). Springer. https://doi.org/10.1007/978-94....
21.
Selden, A., & Selden, J. (2017). A comparison of proof comprehension, proof construction, proof validation and proof evaluation. In R. Göller, R. Biehler, R. Hochmuth, & H. G. Rück (Eds.), Proceedings of the Conference on Didactics of Mathematics in Higher Education as a Scientific Discipline (pp. 339-345). KHDM.
22.
Selden, A., Selden, J., & Benkhalti, A. (2018). Proof frameworks: A way to get started. PRIMUS, 28(1), 31-45. https://doi.org/10.1080/105119....
23.
Stavrou, G. S. (2014). Common errors and misconceptions in mathematical proving by education undergraduates. IUMPST: The Journal, 1, 1-8.
24.
Stylianides, G. J., Stylianides, A. J., & Weber, K. (2017). Research on the teaching and learning of proof: Taking stock and moving forward. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 237-266). National Council of Teachers of Mathematics.
25.
van Engen, H. (1970). Strategies of proof in secondary mathematics. The Mathematics Teacher, 63(8), 637-645. https://doi.org/10.5951/MT.63.....
26.
Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101-119. https://doi.org/10.1023/A:1015....
27.
Weber, K. (2004). A framework for describing the processes that undergraduates use to construct proofs. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th Annual Meeting of the International Group for the Psychology of Mathematics Education (pp. 425-432). PME.
28.
Yang, K. L., & Lin, F. L. (2008). A model of reading comprehension of geometry proof. Educational Studies in Mathematics, 67(1), 59-76. https://doi.org/10.1007/s10649....
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| eISSN: | 1305-8223 |
| ISSN: | 1305-8215 |
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