Components of collective argumentation in geometric construction tasks

Year 2023, , 50 - 71, 31.01.2023

Esra Demiray , Mine Işıksal , Elif Saygı

https://doi.org/10.19128/turje.1176981

Cited By: 1

Abstract

This study aims to examine the components of collective argumentation of pre-service middle school mathematics teachers during geometric construction activities. To scrutinize this issue, case study research was utilized. The participants were 14 pre-service middle school mathematics teachers who worked collectively by forming four groups. During the data collection process, the groups worked on four geometric construction tasks by using compass-straightedge or GeoGebra. The findings presented that the collective argumentation processes were depicted by means of eleven components. In more detail, the six components of Toulmin’s argument model which are data, warrant, claim, backing, rebuttal, and qualifier were insufficient to represent collective argumentation. Instead of claim, the term conclusion was used in this study since the associated data and warrant were provided in the argumentation. The collective argumentation processes of the groups involved not only the mentioned six components but also the five additional components, which were named conclusion/data, target conclusion, guidance, challenger, and objection. The new components might be used while investigating the argumentation process in other disciplines.

Keywords

Argumentation components, Collective argumentation, Geometric construction

Supporting Institution

Scientific and Technological Research Council of Türkiye

Project Number

Grant 2211-A

Thanks

This work was supported by the Scientific and Technological Research Council of Türkiye (TÜBİTAK) under Grant 2211-A.

Geometrik inşa etkinliklerinde ortaklaşa argümantasyonun bileşenleri

Abstract

Bu çalışma, ortaokul matematik öğretmen adaylarının geometrik inşa etkinlikleri sürecindeki ortaklaşa argümantasyon bileşenlerini incelemeyi amaçlamaktadır. Bu konuyu araştırmak için durum çalışmasından yararlanılmıştır. Katılımcılar, dört grup oluşturarak ortaklaşa çalışan 14 ortaokul matematik öğretmen adayı olarak belirlenmiştir. Veri toplama sürecinde, gruplardan dört geometrik inşa etkinliği sırasında araç olarak pergel-çizgeç veya GeoGebra kullanarak çalışmaları istenmiştir. Çalışmanın bulguları, ortaklaşa argümantasyon sürecinin on bir bileşen aracılığıyla betimlenebildiğini ortaya koymuştur. Daha ayrıntılı ifade etmek gerekirse, Toulmin'in argüman modelinde veri, gerekçe, iddia, destek, çürütücü ve niteleyen olarak isimlendirilen altı bileşenin ortaklaşa argümantasyonu temsil etmekte yetersiz kaldığı görülmüştür. Ayrıca, bu çalışmada yer alan argümantasyon süreçlerinde katılımcılar birbiriyle bağlantılı veriler ve gerekçeler ortaya koymuştur. Bu nedenle, Toumin’in argümantasyon modelindeki iddia terimi yerine sonuç terimi kullanılmıştır. Grupların ortaklaşa argümantasyon süreçleri, sadece bahsedilen altı bileşeni değil, aynı zamanda sonuç/veri, hedef sonuç, rehber, meydan okuma ve itiraz olarak adlandırılan beş ek bileşeni de içermektedir. Yeni bileşenlerin, diğer disiplinlerdeki argümantasyon süreçleri araştırılırken kullanılabilmesi beklenmektedir.

Keywords

Argümantasyon bileşenleri, Geometrik inşa, Ortaklaşa argümantasyon

Project Number

Grant 2211-A

References

• Balacheff, N. (1999). Is argumentation an obstacle? Invitation to a debate. International Newsletter on the Teaching and Learning of Mathematical Proof. http://eric.ed.gov/PDFS/ED435644.pdf
• Barabash, M. (2019). Dragging as a geometric construction tool: Continuity considerations inspired by students’ attempts. Digital Experiences in Mathematics Education, 5(2), 124-144. https://doi.org/10.1007/s40751-019-0050-2
• Bench-Capon, T. J. M. (1998). Specification and implementation of Toulmin dialogue game. In J. C. Hage, T. Bench-Capon, A. Koers, C. de Vey Mestdagh, & C. Grutters (Eds.), Foundation for Legal Knowledge Based Systems (pp.5-20). Gerard Noodt Institut
• 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 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 179-204). Belo Horizonte
• Brinkerhoff, J. A. (2007). Applying Toulmin's argumentation framework to explanations in a reform oriented mathematics class (Unpublished doctoral dissertation). Brigham Young University.
• 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
• Carrascal, B. (2015). Proofs, mathematical practice and argumentation. Argumentation, 29(3), 305-324. https://doi.org/10.1007/s10503-014-9344-0
• Cervantes-Barraza, J. A., Hernandez Moreno, A., & Rumsey, C. (2020). Promoting mathematical proof from collective argumentation in primary school. School Science and Mathematics, 120(1), 4-14. https://doi.org/10.1111/ssm.12379
• Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014a). Teacher support for collective argumentation: A framework for examining how teachers support students’ engagement in mathematical activities. Educational Studies in Mathematics, 86(3), 401-429. https://doi.org/10.1007/s10649-014-9532-8
• Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014b). Identifying kinds of reasoning in collective argumentation. Mathematical Thinking and Learning, 16(3), 181-200. https://doi.org/10.1080/10986065.2014.921131
• De Villiers, M. (2014). A variation of Miquel's theorem and its generalisation. The Mathematical Gazette, 98(542), 334-339. https://doi.org/10.1017/S002555720000142X
• Freeman, J. B. (2005). Systematizing Toulmin’s warrants: An epistemic approach. Argumentation, 19(3), 331-346. https://doi.org/10.1007/s10503-005-4420-0
• Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modelling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(1), 3-21. https://doi.org/10.1007/s10649-006-9059-8
• Janičić, P. (2010). Geometry constructions language. Journal of Automated Reasoning, 44(1-2), 3-24. https://doi.org/10.1007/s10817-009-9135-8
• Knipping, C. (2008). A method for revealing structures of argumentations in classroom proving processes. ZDM – Mathematics Education, 40(3), 427-441. https://doi.org/10.1007/s11858-008-0095-y
• Knipping, C., & Reid, D. (2013). Revealing structures of argumentations in classroom proving processes. In A. Aberdein & I. J. Dove (Eds.), The argument of mathematics (pp 119-146). Springer.
• Knipping, C., & Reid, D. (2015). Reconstructing argumentation structures: A perspective on proving processes in secondary mathematics classroom interactions. In A. Bikner-Ahsbahs, C. Knipping, & N. Presmeg (Eds.), Approaches to Qualitative Research in Mathematics Education (pp. 75-101). Springer.
• Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), Emergence of mathematical meaning (pp. 229-269). Lawrence Erlbaum.
• Kuzle, A. (2013). Construction with various tools in two geometry courses in the United States and Germany. In B. Ubuz, C. Haser, M. A. Mariotti (Eds.), Proceedings of the 8th Congress of the European Society for Research in Mathematics Education (pp. 675-685). Ankara, Türkiye: Middle East Technical University and ERME.
• Lim, S. K. (1997). Compass constructions: a vehicle for promoting relational understanding and higher order thinking skills. The Mathematics Educator, 2(2), 138-147.
• Lin, P. J. (2018). The development of students’ mathematical argumentation in a primary classroom. Educação & Realidade, 43(3), 1171-1192. https://doi.org/10.1590/2175-623676887
• Mariotti, M. A., Bartolini-Bussi, M., Boero, P., Ferri, F., & Garuti, R. (1997). Approaching geometry theorems in contexts: From history and epistemology to cognition. In E. Pehkonen (Eds.), Proceedings of the 21st conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 180-195). Lahti, Finland: PME.
• Mariotti, M. A., Durand-Guerrier, V., & Stylianides, G. J. (2018). Argumentation and proof. In T. Dreyfus, M. Artigue, D. Potari, S. Prediger, & K. Ruthven (Eds.), Developing research in mathematics education: Twenty years of communication, Cooperation and collaboration in Europe (1st ed., pp. 75–89). Routledge.
• Metaxas, N., Potari, D., & Zachariades, T. (2016). Analysis of a teacher’s pedagogical arguments using Toulmin’s model and argumentation schemes. Educational Studies in Mathematics, 93(3), 383-397. https://doi.org/10.1007/s10649-016-9701-z
• Nardi, E., Biza, E., & Zachariades, T. (2012). ‘Warrant’ revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation. Educational Studies in Mathematics, 79(12), 157-173. https://doi.org/10.1007/s10649-011-9345-y
• O’Donnell, A. M. (2006). Introduction: Learning with technology. In A. M. O’Donnell, C. E. Hmelo-Silver & G. Erkens (Eds.), Collaborative learning, reasoning, and technology (pp. 1-15). Erlbaum.
• Pandiscio, E. A. (2002). Exploring the link between pre-service teachers’ conception of proof and the use of dynamic geometry software. School Science & Mathematics, 102(5), 212–221.
• Pedemonte, B., & Balacheff, N. (2016). Establishing links between conceptions, argumentation and proof through the ck¢-enriched Toulmin model. The Journal of Mathematical Behavior, 41, 104-122. https://doi.org/10.1016/j.jmathb.2015.10.008
• Sanders, C. V. (1998). Geometric constructions: Visualizing and understanding geometry. The Mathematics Teacher,91(7), 554-556.
• Schreck, P., Mathis, P., & Narboux, J. (2012). Geometric construction problem solving in computer-aided learning. 24th IEEE International Conference on Tools with Artificial Intelligence (Vol. 1, pp. 1139-1144). Athens, Greece: IEEE.
• Smith, R. C. (2010). A comparison of middle school students’ mathematical arguments in technological and non-technological environments (Unpublished doctoral dissertation). North Carolina State University
• Stephan, M., & Rasmussen, C. (2002). Classroom mathematical practices in differential equations. The Journal of Mathematical Behavior, 21(4), 459-490. https://doi.org/10.1016/S0732-3123(02)00145-1
• Stupel, M., & Ben-Chaim, D. (2013). A fascinating application of Steiner's theorem for trapezium: geometric constructions using straightedge alone. Australian Senior Mathematics Journal, 27(2), 6-24.
• Stupel, M., Sigler, A., & Tal, I. (2018). Investigative activity for discovering hidden geometric properties. Electronic Journal of Mathematics & Technology,12(1), 247-260.
• Toulmin, S. (1958). The uses of argument. Cambridge University Press.
• Toulmin, S. (2003). The uses of argument. Cambridge University Press.
• Van Ness, C. K., & Maher, C. A. (2018). Analysis of the argumentation of nine-year-olds engaged in discourse about comparing fraction models. The Journal of Mathematical Behavior, 53, 13-41. https://doi.org/10.1016/j.jmathb.2018.04.004
• Verheij, B. (2005). Evaluating arguments based on Toulmin’s scheme. Argumentation, 19(3), 347-371. https://doi.org/10.1007/s10503-005-4421-z
• Verheij, B. (2009). The Toulmin argument model in artificial intelligence. In I. Rahwan & G. Simari (Eds.), Argumentation in artificial intelligence (pp. 219-238). Springer.
• Voss, J. F. (2005). Toulmin's model and the solving of ill-structured problems. Argumentation, 19, 321–329. https://doi.org/10.1007/s10503-005-4419-6
• Walton, D. (2006). Fundamentals of critical argumentation. Cambridge University Press.
• Yackel, E. (2001). Explanation, justification and argumentation in mathematics classrooms. In M. van den Heuvel-Panhuizen (Eds.), Proceedings of the 25th Annual Conference of the International Group for the Psychology of Mathematics Education, (Vol. 1, p. 9-24). University of Utrecht.
• Yackel, E. (2002). What we can learn from analyzing the teacher’s role in collective argumentation. Journal of Mathematical Behavior, 21(4), 423-440. https://doi.org/10.1016/S0732-3123(02)00143-8
• Yackel, E., & Cobb, P. (1996). Sociomathematical norms, argumentation, and autonomy in mathematics. Journal for Research in Mathematics Education, 22, 390-408. https://doi.org/10.5951/jresematheduc.27.4.0458
• Yin. R. K. (2014). Case study research: Design and methods (5th ed.). Sage Publications.
• Yu, S., & Zenker, F. (2020). Schemes, critical questions, and complete argument evaluation. Argumentation, 34(4), 469-498. https://doi.org/10.1007/s10503-020-09512-4