How argumentation relates to formal proof process in geometry

Esra Demiray; Mine Işıksal-Bostan; Elif Saygı

doi:10.29333/iejme/13214

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Esra Demiray ¹ ^* , Mine Işıksal-Bostan ² , Elif Saygı ¹

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¹ Faculty of Education, Mathematics and Science Education, Hacettepe University, Ankara, TÜRKİYE² Faculty of Education, Mathematics and Science Education, Middle East Technical University, Ankara, TÜRKİYE^* Corresponding Author

Abstract

This study aims to examine how being involved in an argumentation process relates to the formal proof process in geometry. Prospective mathematics teachers were involved in an argumentation process while producing conjectures before engaging in formal proof of the recently produced conjectures. To collect data, four geometry-proof tasks that involve two sections were employed. The first section of the tasks demands the production of conjectures, which stands for the term argumentation. The second section asks for the formal proof of one of the recently produced conjectures. Based on the data analysis, the affordances of being involved in argumentation before engaging in the formal proof process were listed as positive affective occasions, arrangement of knowledge related to the content of the task, visual aspect, and the veracity of the statement. Negative affective occasions and confusion related to the difference between conjecturing and proving were coded as constraints of being involved in argumentation before formal proof.

Keywords

argumentation
proof
geometry

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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

INT ELECT J MATH ED, Volume 18, Issue 3, August 2023, Article No: em0741

https://doi.org/10.29333/iejme/13214

Publication date: 01 Jul 2023

Online publication date: 19 Apr 2023

Article Views: 1369

Article Downloads: 1000

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References

Alexander, D. C., & Koeberlein G. M. (2011). Elementary geometry for college students. Brooks/Cole, Cengage Learning.
Antonini, S., & Mariotti, M. A. (2008). Indirect proof: What is specific to this way of proving? ZDM, 40(3), 401-412. https://doi.org/10.1007/s11858-008-0091-2
Baccaglini-Frank, A. (2010). Conjecturing in dynamic geometry: A model for conjecture-generation through maintaining dragging [Doctoral dissertation, University of New Hampshire].
Boero, P. (2017). Cognitive unity of theorems, theories and related rationalities. In T. Dooley, & G. Gueudet (Eds.), Proceedings of the 10^th Congress of the European Society for Research in Mathematics Education (pp. 99-106). DCU Institute of Education and ERME.
Boero, P., Chiappini, G., Garuti, R., & Sibilla, A. (1995). Towards statements and proofs in elementary arithmetic: An explanatory study about the role of the teachers and the behaviour of students. In L. Meira, & D. Carraher (Eds.), Proceedings of the 19^th Conference of the International Group for the Psychology of Mathematics Education (pp. 129-136). PME.
Boero, P., Douek, N., Morselli, F., & Pedemonte, B. (2010). Argumentation and proof: A contribution to theoretical perspectives and their classroom implementation. In M. F. Pinto, & T. F. Kawasaki (Eds.), Proceedings of the 34^th Conference of the International Group for the Psychology of Mathematics Education (pp. 179-204). PME.
Bogdan, R. C., & Biklen, S. K. (2007). Qualitative research for education: A introduction to theory and methods. Allyn and Bacon.
Brown, R. A. J. (2017). Using collective argumentation to engage students in a primary mathematics classroom. Mathematics Education Research Journal, 29(2), 183-199. https://doi.org/10.1007/s13394-017-0198-2
Duval, R. (1991). Structure du raisonnement déductif et apprentissage de la demonstration [Structure of deductive reasoning and demonstration learning]. Educational Studies in Mathematics, 22(3), 233–261. https://doi.org/10.1007/BF00368340
Duval, R. (1995). Sémiosis et pensée humaine [Semiosis and human thought]. Peter Lang.
Ellis, A. B., Bieda, K., & Knuth, E. J. (2012). Developing essential understanding of proof and proving for teaching mathematics in grades 9-12. National Council of Teachers of Mathematics.
Elo, S., & Kyngäs, H. (2008). The qualitative content analysis process. Journal of Advanced Nursing, 62(1), 107-115. https://doi.org/10.1111/j.1365-2648.2007.04569.x
Fan, L., Qi, C., Liu, X., Wang, Y., & Lin, M. (2017). Does a transformation approach improve students’ ability in constructing auxiliary lines for solving geometric problems? An intervention-based study with two Chinese classrooms. Educational Studies in Mathematics, 96(2), 229-248. https://doi.org/10.1007/s10649-017-9772-5
Fiallo, J., & Gutiérrez, A. (2017). Analysis of the cognitive unity or rupture between conjecture and proof when learning to prove on a grade 10 trigonometry course. Educational Studies in Mathematics, 96(2), 145-167. https://doi.org/10.1007/s10649-017-9755-6
Fujita, T., Jones, K., & Kunimune, S. (2010). Students’ geometrical constructions and proving activities: A case of cognitive unity. In M. F. Pinto, & T. F. Kawasaki (Eds.), Proceedings of the 34^th Conference of the International Group for the Psychology of Mathematics Education (pp. 9-16). PME.
Furinghetti, F., & Morselli, F. (2009). Every unsuccessful problem solver is unsuccessful in his or her own way: Affective and cognitive factors in proving. Educational Studies in Mathematics, 70(1), 71-90. https://doi.org/10.1007/s10649-008-9134-4
Garuti, R., Boero, P., & Lemut, E. (1998). Cognitive unity of theorems and difficulty of proof. In A. Olivier, & K. Newstead (Eds.), Proceedings of the 22^nd Conference of the International Group for the Psychology of Mathematics Education (pp. 345-352). PME.
Garuti, R., Boero, P., Lemut, E., & Mariotti, M. A. (1996). Challenging the traditional school approach to theorems: A hypothesis about the cognitive unity of theorems. In L. Puig, & A. Guierrez (Eds.), Proceedings of the 20^th Conference of the International Group for the Psychology of Mathematics Education (pp. 113-120). PME.
Gillis, J. (2005). An investigation of student conjectures in static and dynamic geometry environments [Doctoral dissertation, Auburn University].
Golafshani, N. (2003). Understanding reliability and validity in qualitative research. The Qualitative Report, 8(4), 597-606.
Guba, E. G. (1981). Criteria for assessing the trustworthiness of naturalistic inquiries. Educational Communication and Technology Journal, 29, 75-91. https://doi.org/10.1007/BF02766777
Gutenmacher, V., & Vasilyev, N. B. (2004). Lines and curves: A practical geometry handbook. Springer Science & Business Media. https://doi.org/10.1007/978-1-4757-3809-4
Harel, G., & Sowder, L. (2009). College instructors’ views of students vis-à-vis proof. In M. Blanton, D. Stylianou, & E. Knuth (Eds.), Teaching proof across the grades: A K-12 perspective (pp. 275-289). Routledge. https://doi.org/10.4324/9780203882009-16
Jahnke, H. N. (2007). Proofs and hypotheses. ZDM, 39(1-2), 79-86. https://doi.org/10.1007/s11858-006-0006-z
Jones, K. (2002). Issues in the teaching and learning of geometry. In L. Haggarty (Eds.), Aspects of teaching secondary mathematics (pp 121-139). Routledge.
Lincoln, Y. S., & Guba, E. G. (1985). Naturalistic inquiry. SAGE. https://doi.org/10.1016/0147-1767(85)90062-8
Marrades, R., & Gutiérrez, A. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44, 87-125. https://doi.org/10.1023/A:1012785106627
Mejia-Ramos, J. P., & Inglis, M. (2009). Argumentative and proving activities in mathematics education research. In F. L. Lin, F. J. Hsieh, G. Hanna, & M. de Villiers (Eds.), Proceedings of the ICMI Study 19 Conference: Proof and Proving in Mathematics Education (pp. 88-93).
Merriam, S. B. (2009). Qualitative research: A guide to design and implementation. John Wiley & Sons.
Moore, R. C. (1994). Making the transition to formal proof. Educational Studies in Mathematics, 27(3), 249-266. https://doi.org/10.1007/BF01273731
Morris, D. W., & Morris, J. (2009). Proofs and concepts: The fundamentals of abstract mathematics. University of Lethbridge.
NCTM. (2000). Principles and standards for school mathematics. National Council of Teachers of Mathematics.
Patton, M. Q. (2002). Qualitative research and evaluation methods. SAGE.
Pedemonte, B. (2002). Relation between argumentation and proof in mathematics: Cognitive unity or break? In J. Novotna (Ed.), Proceedings of the 2^nd Conference of the European Society for Research in Mathematics Education (pp. 70-80). ERME.
Pedemonte, B. (2005). Quelques outils pour l’analyse cognitive du rapport entre argumentation et demonstration [Some tools for the cognitive analysis of the relationship between argumentation and demonstration]. Recherches en Didactique des Mathématiques, 25(3), 313-348.
Pedemonte, B. (2007). How can the relationship between argumentation and proof be analyzed? Educational Studies in Mathematics, 66(1), 23-41. https://doi.org/10.1007/s10649-006-9057-x
Pedemonte, B. (2008). Argumentation and algebraic proof. Zentralblatt fur Didaktik der Mathematik [Central Journal for Didactics of Mathematics], 40(3), 385-400. https://doi.org/10.1007/s11858-008-0085-0
Reiss, K. M., Heinze, A., Renkl, A., & Groß, C. (2008). Reasoning and proof in geometry: Effects of a learning environment based on heuristic worked-out examples. ZDM, 40(3), 455-467. https://doi.org/10.1007/s11858-008-0105-0
Selden, A., & Selden, J. (2013). Proof and problem solving at university level. The Mathematics Enthusiast, 10(1), 303-334. https://doi.org/10.54870/1551-3440.1269
Sinclair, N., Pimm, D., Skelin, M., & Zbiek, R. M. (2012). Developing essential understanding of geometry for teaching mathematics in grades 6-8. National Council of Teachers of Mathematics.
Stylianides, A. J., & Stylianides, G. J. (2018). Addressing key and persistent problems of students’ learning: The case of proof. In A. J. Stylianides, & G. Harel (Eds.), Advances in mathematics education research on proof and proving (pp. 99-113). Springer. https://doi.org/10.1007/978-3-319-70996-3_7
Stylianides, A. J., Bieda, K. N., & Morselli, F. (2016). Proof and argumentation in mathematics education research. In A. Gutiérrez, G. C. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 315-351). Sense Publishers. https://doi.org/10.1007/978-94-6300-561-6_9
Stylianides, G. J., & Stylianides, A. J. (2017). Research-based interventions in the area of proof: The past, the present, and the future. Educational Studies in Mathematics, 96(2), 119-127. https://doi.org/10.1007/s10649-017-9782-3
Toulmin, S. E. (2003). The uses of argument. Cambridge University Press. https://doi.org/10.1017/CBO9780511840005
Velleman, D. J. (2006). How to prove it. Cambridge Universtiy Press. https://doi.org/10.1017/CBO9780511808234

How to cite this article

APA

Demiray, E., Işıksal-Bostan, M., & Saygı, E. (2023). How argumentation relates to formal proof process in geometry. International Electronic Journal of Mathematics Education, 18(3), em0741. https://doi.org/10.29333/iejme/13214

Vancouver

Demiray E, Işıksal-Bostan M, Saygı E. How argumentation relates to formal proof process in geometry. INT ELECT J MATH ED. 2023;18(3):em0741. https://doi.org/10.29333/iejme/13214

AMA

Demiray E, Işıksal-Bostan M, Saygı E. How argumentation relates to formal proof process in geometry. INT ELECT J MATH ED. 2023;18(3), em0741. https://doi.org/10.29333/iejme/13214

Chicago

Demiray, Esra, Mine Işıksal-Bostan, and Elif Saygı. "How argumentation relates to formal proof process in geometry". International Electronic Journal of Mathematics Education 2023 18 no. 3 (2023): em0741. https://doi.org/10.29333/iejme/13214

Harvard

Demiray, E., Işıksal-Bostan, M., and Saygı, E. (2023). How argumentation relates to formal proof process in geometry. International Electronic Journal of Mathematics Education, 18(3), em0741. https://doi.org/10.29333/iejme/13214

MLA

Demiray, Esra et al. "How argumentation relates to formal proof process in geometry". International Electronic Journal of Mathematics Education, vol. 18, no. 3, 2023, em0741. https://doi.org/10.29333/iejme/13214