Supporting Argumentation on the Mathematical Connections Between the Graphs of f and f' by Rational Questioning

Year 2024, Volume: 22 Issue: 3, 1919 - 1953, 31.12.2024

Selin Urhan

https://doi.org/10.37217/tebd.1514191

Abstract

The aim of this study is to investigate how teachers’ rational questioning during argumentation influences students’ reasoning about the mathematical connections between the graphs of f and f'. The study was conducted with 13 senior students in a mathematics education program at a university in Türkiye. Students were requested to individually work on a task involving the graph of f' and subsequently asked to derive the graph of f, followed by argumentation on their individual works. The teacher engaged in argumentation by asking rational questions, encouraging the students to behave rationally, and monitoring their performance within the context of rationality. The structure of argumentation was analyzed using Toulmin’s model, the teacher’s questioning by the Teacher Rational Questioning Framework, and students’ behaviors using Habermas’ Construct of Rationality. Through the teacher’s rational questioning, it was observed that students were able to reason more rationally in constructing mathematical connections between the graphs of f and f'. The teacher’s rational manner during argumentation directed students towards rational questioning of each other, supporting them to perform more rationally. The findings suggest that designing learning environments supported by rational questioning could foster a culture of rational behavior in mathematics classrooms.

Keywords

Argumentation , Mathematical reasoning , Rationality theory , Rational questioning

f ve f' Grafikleri Arasındaki Matematiksel Bağlantılar Üzerine Argümantasyonun Akılcı Sorgulama ile Desteklenmesi

Abstract

Bu çalışmanın amacı, öğretmenin argümantasyon sürecini akılcı sorgulama destekli yürütmesinin öğrencilerin f ile f' grafikleri arasındaki matematiksel bağlantılar üzerine akıl yürütmesini nasıl etkilediğini araştırmaktır. Çalışma, Türkiye’de bir üniversitede matematik eğitimi programında son sınıfta öğrenim görmekte olan 13 öğretmen adayı ile yürütülmüştür. Öğrencilerden f' grafiğinin sunulduğu ve buradan f grafiğinin oluşturulmasının istendiği bir görev üzerinde bireysel çalışmaları ve ardından bireysel çalışma ürünleri üzerine tartışmaları istenmiştir. Dersi veren öğretim elemanı, argümantasyona akılcı sorgulama yaparak dahil olmuş ve öğrencilere akıl yürütme sürecindeki performanslarını akılcılık bağlamında denetleyici ve akılcı davranmaya teşvik edici sorular sormuştur. Argümantasyonun yapısı Toulmin modeli, öğretmenin öğrencileri sorgulaması süreci Öğretmenin Akılcı Sorgulama Çerçevesi ve öğrencilerin öğretmenin soruları karşısındaki söylemleri ve davranışları Habermas Akılcı Davranış Teorisi ile analiz edilmiştir. Öğretmenin akılcı sorgulaması sayesinde öğrencilerin f ve f' grafikleri arasındaki matematiksel bağlantıları kurmada daha akılcı davranabildiği görülmüştür. Öğretmenin akılcı sorgulama davranışı, öğrencileri birbirlerini akılcı sorgulamaya ve akılcı davranma konusunda desteklemeye yöneltmiştir. Elde edilen sonuçlar, öğretmenin matematik sınıflarında akılcı sorgulama destekli öğrenme ortamları oluşturmasının akılcı davranma kültürü oluşturabileceğine işaret etmektedir.

Keywords

Argümantasyon , Matematiksel akıl yürütme , Akılcılık teorisi , Akılcı sorgulama

References

• Boero, P. (2006). Habermas' theory of rationality as a comprehensive frame for conjecturing and proving in school. J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Ed.), Proceedings of the 30th conference of the International Group for the Psychology of Mathematics Education içinde (c. 2, s. 185-192). Prague, Czech Republic: PME. https://files.eric.ed.gov/fulltext/ED496932.pdf sayfasından erişilmiştir.
• Boero, P., Douek, N., Morselli, F., & Pedemonte, B. (2010). Argumentation and proof: A contribution to theoretical perspectives and their classroom implementation. M. M. F. Pinto & T. F. Kawasaki (Ed.), Proceedings of the 34th conference of the International Group for the Psychology of Mathematics Education içinde (c. 1, s. 179-205). Belo Horizonte: PME. https://www.igpme.org/publications/current-proceedings/ sayfasından erişilmiştir.
• Boero, P. & Planas, N. (2014). Habermas’ construct of rational behavior in mathematics education: New advances and research questions. P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Ed.), Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education içinde (c. 1, s. 205-235). Vancouver, Canada: PME. https://www.igpme.org/publications/current-proceedings/ sayfasından erişilmiştir.
• Boero, P. & Morselli, F. (2009). The use of algebraic language in mathematical modelling and proving in the perspektive of Habermas’ theory of rationality. V. Durand-Guerrier, S. Soury-Lavergne, & F. Arzarello (Ed.) Proceedings of the Sixth Congress of the European Society for Research in Mathematics Education içinde (s. 964-973). Lyon, France. http://erme.site/wp-content/uploads/2021/06/cerme6_proceedings.pdf sayfasından erişilmiştir.
• Cai, J. & Ding, M. (2015). On mathematical understanding: Perspectives of experienced Chinese mathematics teachers. Journal of Mathematics Teacher Education, 18(5), 1-25.
• Cohen, L., Manion, L., & Morrison, K. (2018). Research methods in education. Routledge.
• Conner, A. M., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). Identifying kinds of reasoning in collective argumentation. Mathematical Thinking and Learning, 16(3), 181-200.
• Douek, N. (2014). Pragmatic potential and critical issues. P. Liljedahl, C. Nicol, S. Oesterle, & D. Allan (Ed.), Proceedings of the 38th Conference of the International Group for the Psychology of Mathematics Education and the 36th Conference of the North American Chapter of the Psychology of Mathematics Education içinde (c. 1, s. 209-213). Vancouver, Canada: PME. https://www.igpme.org/publications/current-proceedings/ sayfasından erişilmiştir.
• Dubinsky, E., Weller, K., Mcdonald, M. A., & Brown, A. (2005). Some historical issues and paradoxes regarding the concept of infinity: An apos-based analysis: Part 1. Educational Studies in Mathematics, 58, 335–359.
• Duval, R. (2006). A cognitive analysis of problems of comprehension in a learning of mathematics. Educational Studies in Mathematics, 61(1-2), 103-131.
• Duval, R. (2017). Understanding the mathematical way of thinking – the registers of semiotic representations. Springer.
• Francisco, J. M. (2013). The mathematical beliefs and behavior of high school students: Insights from a longitudinal study. Journal of Mathematical Behavior, 32, 481-493.
• García-García, J. & Dolores-Flores, C. (2018). Intra-mathematical connections made by high school students in performing calculus tasks. International Journal of Mathematical Education in Science and Technology, 49(2), 227-252.
• García-García, J. & Dolores-Flores, C. (2021). Pre-university students’ mathematical connections when sketching the graph of derivative and antiderivative functions. Mathematics Education Research Journal, 33, 1-22.
• Habermas, J. (1998). On the pragmatics of communication. The MIT.
• Hufferd-Ackles, K., Fuson, K. C., & Sherin, M. G. (2004). Describing levels and components of a math-talk learning community. Journal for Research in Mathematics Education, 35(2), 81-116.
• Inglis, M., Mejia-Ramos, J. P., & Simpson, A. (2007). Modeling mathematical argumentation: The importance of qualification. Educational Studies in Mathematics, 66(1), 3-21.
• Jeannotte, D. & Kieran, C. (2017). A conceptual model of mathematical reasoning for school mathematics. Educational Studies in Mathematics, 96(1), 1-16.
• Kastberg, S. E. (2002). Understanding mathematical concepts: the case of the logarithmic function. Unpublished dissertation. University of Georgia. United States of America.
• Krummheuer, G. (1995). The ethnography of argumentation. P. Cobb & H. Bauersfeld (Ed.), The emergence of mathematical meaning: Interaction in classroom cultures içinde (s. 229–269). Erlbaum.
• McCarthy, P., Sithole, A., McCarthy, P., Cho, J. P., & Gyan, E. (2016). Teacher questioning strategies in mathematical classroom discourse: A case study of two grade eight teachers in Tennessee, USA. Journal of Education and Practice, 7(21), 80–89.
• McDonald, M. A., Mathews, D., & Strobel, K. (2000). Understanding sequences: A tale of two objects. Research in Collegiate Mathematics Education, 4(8), 77-102.
• Morselli, F. & Boero, P. (2009). Habermas' construct of rational behaviour as a comprehensive frame for research on the teaching and learning of proof. F.-L. Lin, F.-J. Hsieh, G. Hanna, & M. de Villiers (Ed.), Proceedings of the ICMI Study 19 Conference: Proof and proving in mathematics education içinde (c. 2, s. 100-105). Taipei: Normal University. https://link.springer.com/book/10.1007/978-94-007-2129-6 sayfasından erişilmiştir.
• Morselli, F. & Boero, P. (2011). Using Habermas’ theory of rationality to gain insight into students’ understanding ofalgebraic language. J. Cai & E. Knuth (Ed.), Early algebraization: A global dialogue from multiple perspectives içinde (s. 453–481). Springer.
• Rodriguez‑Nieto, C. A., Rodriguez‑Vasquez, F. M., & Font-Moll, V. (2023). Combined use of the extended theory of connections and the onto‑semiotic approach to analyze mathematical connections by relating the graphs of f and f’. Educational Studies in Mathematics, 114, 63-88.
• Rokeach, M. (1968). Beliefs, attitudes and values: a theory of organization and change. Jossey-Bass.
• Sahin, A. & Kulm, G. (2008). Sixth grade mathematics teachers’ intentions and use of probing, guiding, and factual questions. Journal of Mathematics Teacher Education, 11(3), 221-241.
• Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22, 1–36.
• Toulmin, S., Rieke, R., & Janik, A. (1984). An introduction to reasoning. Macmillan.
• Urhan, S. & Bülbül, A. (2022). The analysis of the algebraic proving process based on Habermas' construct of rationality. Hacettepe Üniversitesi Eğitim Fakültesi Dergisi, 37(3), 1154-1175.
• Urhan, S. & Bülbül, A. (2023a). Habermas’ construct of rationality in the analysis of the mathematical problem-solving process. Educational Studies in Mathematics, 112(7), 175-197.
• Urhan, S. & Bülbül, A. (2023b) Analysis of mathematical proving in geometry based on Habermas’ construct of rationality. Mathematics Education Research Journal, 35(6), 929–959.
• Urhan, S. & Yüksel, N. S. (2019). The analysis of problem solving process of pre-service physics teachers by Habermas’ theory of rationality. B. Akkus, R. B. Cakiril-Mutlu, E. Gudekli, B. Kinaci, F. Guzelcimen, G. Susoy-Dogan, F. Ozturk, & A. Ertoprak (Ed.), Proceedings of ATP Conference içinde (2178(1), s. 030065). AIP Publishing LLC. https://doi.org/10.1063/1.5135463 sayfasından erişilmiştir.
• Zhuang, Y. & Conner, A. (2022). Teachers’ use of rational questioning strategies to promote student participation in collective argumentation. Educational Studies in Mathematics, 111(3), 345-365.