Dönel Cisimlerin Yüzey Alanının Hesaplanması Sürecindeki Argümanların İncelenmesi

Ayşe Tekin Dede; Ali Özgün Özer; Esra Bukova Güzel

doi:10.30703/cije.1072163

EN TR

Examination of the Arguments in the Process of Calculating the Surface Area of the Solids of Revolution

Abstract

In the study, it is aimed to examine the collective arguments of students in the process of constructing a model to calculate the surface area of the solids of revolution in the Calculus I course. The participants of the study, which was designed as a case study, were forty-two students enrolled in Calculus I course in the elementary mathematics teaching department. The data consisted of the video recordings of the argumentation process and the observation notes of the researcher during the construction of the surface area model of the solids of revolution. By analysing the data, Toulmin argumentation schemes containing the components of the argumentation process were created. The findings of the study revealed that the students actively participated in the argumentation process by justifying and refuting each other's claims. In this active participation, the actions of the researcher who conducted the course to support the argumentation process, the students' pre-learning and the norms in the classroom were effective. Data, claim, warrant and rebuttal components emerged from the components of the argumentation process. Particularly, it was a remarkable finding that the students put forward rebuttals by listening to each other's explanations carefully and that these rebuttals were justifications for the subsequent arguments. Another task of the rebuttals was to invalidate not only the claim but also, in some cases, the sub-arguments containing data, claim and justification. In addition, it was understood that the students expressed their justifications with the researcher's questioning in the transition from the data to the claim. In addition to all these, it was seen that the participants did not express backings or qualifiers in the argumentation process.

Keywords

Dönel Cisimlerin Yüzey Alanının Hesaplanması Sürecindeki Argümanların İncelenmesi

Öz

Çalışmada Analiz I dersinde dönel cisimlerin yüzey alanını hesaplamayı sağlayacak modelin oluşturulması sürecinde ilköğretim matematik öğretmenliği öğrencilerinin ortaklaşa argümanlarının incelenmesi amaçlanmaktadır. Durum çalışması olarak desenlenen çalışmanın katılımcıları Analiz I dersine kayıtlı kırk iki ilköğretim matematik öğretmenliği öğrencileridir. Veriler dönel cisimlerin yüzey alanı modelinin oluşturulması esnasındaki argümantasyon sürecinin video kayıtları ile araştırmacının gözlem notlarından oluşmaktadır. Veriler analiz edilerek argümantasyon sürecinin bileşenlerini içeren Toulmin argümantasyon şemaları oluşturulmuştur. Çalışmanın bulguları öğrencilerin birbirlerinin iddialarına gerekçe sunarak ve bu iddiaları çürüterek argümantasyon sürecine aktif katılım sağladıklarını ortaya koymaktadır. Bu aktif katılımda dersi yürüten araştırmacının argümantasyon sürecini destekleyici eylemleri, öğrencilerin ön öğrenmeleri ve sınıf içindeki normlar etkili olmuştur. Argümantasyon sürecinin bileşenlerinden veri, iddia, gerekçe ve çürütücü bileşenleri ortaya çıkmıştır. Özellikle öğrencilerin birbirlerinin açıklamalarını dikkatle dinleyerek çürütücüler öne sürdükleri ve bu çürütücülerin kendinden sonraki gelen argümanların gerekçeleri olması dikkat çekici bir bulgu olmuştur. Çürütücülerin başka bir görevi de sadece iddiayı değil bazı durumlarda veri, iddia ve gerekçeyi içeren alt argümanların da geçerliğini yok etmek olmuştur. Ayrıca verilerden iddiaya geçişte araştırmacının sorgulatmasıyla birlikte öğrencilerin gerekçelerini ifade ettikleri anlaşılmaktadır. Tüm bunlarla birlikte katılımcıların argümantasyon sürecinde destekleyici ya da niteleyicileri ifade etmedikleri de görülmektedir.

Anahtar Kelimeler

Supporting Institution

Dokuz Eylül Üniversitesi Bilimsel Araştırma Projeleri Birimi

Project Number

2018.KB.EGT.008

References

Anthony, G., & Walshaw, M. (2009). Characteristics of effective teaching of mathematics: a view from the west. Journal of Mathematics Education, 2(2), 147-164. https://doi.org/10.12691/education-6-1-1
Ayalon, M., & Hershkowitz, R. (2019). Mathematics teachers' attention to potential classroom situations of argumentation. The Journal of Mathematical Behavior, 49, 163-173. https://doi.org/10.1016/j.jmathb.2017.11.010
Aydın Güç, F., & Kuleyin, H. (2021). Argümantasyon kalitesinin matematiksel modelleme sürecine yansıması. Uludağ Üniversitesi Eğitim Fakültesi Dergisi, 34(1), 222-262. https://doi.org/10.19171/uefad.850230
Berry, J. S., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22(4), 479-495. https://doi.org/10.1016/j.jmathb.2003.09.006
Bülbül, A. & Urhan, S. (2016). Argümantasyon ve matematiksel kanıt süreçleri arasındaki ilişkiler. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi , 10(1) , 351-373. https://doi.org/10.17522/nefefmed.00387
Caglayan, G. (2016). Teaching ideas and activities for classroom: integrating technology into the pedagogy of integral calculus and the approximation of definite integrals. International Journal of Mathematical Education in Science and Technology, 47(8), 1261-1279. https://doi.org/10.1080/0020739X.2016.1176261
Cobb, P., Wood, T., Yackel, E., & McNeal, B. (1992). Characteristics of classroom mathematics traditions: An interactional analysis. American Educational Research Journal, 29(3), 573-604. https://doi.org/10.3102/00028312029003573
Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics. Retrieved from http://www.corestandards.org/assets/CCSSI_Math%20Standards.pdf

Conner, A. (2008). Expanded Toulmin diagrams: A tool for investigating complex activity in classrooms. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano& A. Sepulveda (Eds.), Proceedings of the Joint Meeting of PME 32 and PME-NA XXX (Vol. 2, pp. 361–368). Morelia, Mexico: Cinvestav-UMSNH.
Conner, A., Singletary, L. M., Smith, R. C., Wagner, P. A., & Francisco, R. T. (2014). 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
Çetin, İ., & Dev, Ş. (2021): Pre-service elementary mathematics teachers’ methods when solving integral-volume problems and the rationale behind their selections. International Journal of Mathematical Education in Science and Technology. https://doi.org/10.1080/0020739X.2021.1951859
Delice, A., & Sevimli, E. (2010). Mathematics teacher candidates’ multiple representation and conceptual-procedural knowledge level in definite integral. Gaziantep Üniversity Journal of Social Sciences, 9(3), 581–605.
Dogruer, S.S., & Akyuz, D. (2020). Mathematical practices of eighth graders about 3d shapes in an argumentation, technology, and design-based classroom environment. International Journal of Science and Mathematics Education, 18, 1485–1505. https://doi.org/10.1007/s10763-019-10028-x
Doruk, M. (2016). İlköğretim matematik öğretmeni adaylarının analiz alanındaki argümantasyon ve ispat süreçlerinin incelenmesi. Yayımlanmamış Doktora Tezi, Atatürk Üniversitesi Eğitim Bilimleri Enstitüsü, Erzurum, Türkiye.
Doruk, M., Duran, M., & Kaplan, A. (2018). Argümantasyon tabanlı olasılık öğretiminin ortaokul öğrencilerinin matematiksel üstbiliş farkındalıklarına ve olasılıksal muhakeme becerilerine etkisinin incelenmesi. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 12(1), 83-121. https://doi.org/10.17522/balikesirnef.437714
Erkek, Ö., & Işıksal Bostan, M. (2019). Prospective middle school mathematics teachers’ global argumentation structures. International Journal of Science and Mathematics Education, 17(3), 613-633. https://doi.org/10.1007/s10763-018-9884-0
Forman, E. A. (2003). A sociocultural approach to mathematics reform: Speaking, inscribing, and doing mathematics within communities of practice. In J. Kilpatrick, W. G. Martin, & D. Schifter (Eds.), A Research Companion to Principles and Standards for School Mathematics (pp. 333-352). Reston, VA: NationalCouncil of Teachers of Mathematics. Forman, E. A., Larreamendy-Joerns, J., Stein, M. K., & Brown, C. A. (1998). “You're going to want to find out which and prove it”: Collective argumentation in a mathematics classroom. Learning and Instruction, 8(6), 527-548. https://doi.org/10.1016/S0959-4752(98)00033-4
Goos, M. (2004). Learning mathematics in a classroom community of inquiry. Journal for Research in Mathematics Education, 35(4), 258–291. https://doi.org/30034810
Hathaway, RA. (2008). Simple acronym for doing calculus: CAL. PRIMUS, 18. 542–545. https://doi.org/10.1080/10511970701604014
Herbel-Eisenmann, B. A., Steele, M. D., & Cirillo, M. (2013). (Developing) Teacher discourse moves: a framework for professional development. Mathematics Teacher Educator, 1(2), 181–196. https://doi.org/10.5951/mathteaceduc.1.2.0181
Hiebert, J., & Carpenter, T. P. (1992). Learning and teaching with understanding. In D. A. Grouws, (Ed.), Handbook of research on mathematics teaching and learning: (A project of the national council of teachers of mathematics) (pp. 65–97). Macmillan.
Hollebrands, K. F., Conner, A., & Smith, R. C. (2010). The nature of arguments provided by college geometry students with access to technology while solving problems. Journal for Research in Mathematics Education, 41(4), 324-350.
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. https://doi.org/30034933
Hunter, R. (2007). Can you convince me: Learning to use mathematical argumentation. In J. H. Woo, H. C. Lew, K. S. Park & D. Y. Seo (Eds.), Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol 3, pp. 81-88). Seoul: PME.
Hunter, R., & Anthony, G. (2011). Learning to “friendly argue” in a community of mathematical inquiry (Teaching and Learning Research Initiative Report). Wellington: New Zealand Educational Research Council.
Knipping, C. (2004). Argumentations in proving discourses in mathematics classrooms. In E. Cohors-Fresenborg, H. Maier, K. Reiss, G. Toerner & H. G. Weigand (Eds.), Selected Papers from the Annual Conference on Didactics of Mathematics (pp. 73–84). Hildesheim: Franzbecker Verlag.
Knipping, C. (2008). A method for revealing structures of argumentation in classroom proving processes. Zentralblatt für Didaktik der Mathematik-ZDM, 40(3), 427–441. https://doi.org/10.1007/s11858-008-0095-y
Knipping, C., & Reid, D. (2013). Revealing structures of argumentation in classroom proving processes. In A. Aberdein & I. J. Dove (Eds.), The Argument of Mathematics (pp. 181– 208). Dordrecht, Springer. https://doi.org/10.1007/978-94-007-6534-4_8
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: Dordrecht. https://doi.org/10.1007/978-94-017-9181-6_4
Kosko, K. W., Rougee, A., & Herbst, P. (2014). What actions do teachers envision when asked to facilitate mathematical argumentation in the classroom? Mathematics Education Research Journal, 26(3), 459–476. https://doi.org/10.1007/s13394-013-0116-1
Krummheuer, G. (1995). The ethnography of argumentation. In P. Cobb & H. Bauersfeld (Eds.), Emergence of Mathematical Meaning (pp. 229-269). Hillsdale, NJ: Lawrence Erlbaum.
Krummheuer, G. (2000). Studies of argumentation in primary mathematics education. Zentralblatt für Didaktik der Mathematik-ZDM, 32(5), 155-161.
Krummheuer, G. (2007). Argumentation and participation in the primary mathematics classroom: Two episodes and related theoretical abductions. The Journal of Mathematical Behavior, 26(1), 60-82.
Lampert, M., & Cobb, P. (2003). Communication and language. In J. Kilpatrick, W. G. Martin & D. Schifter (Eds.) A Research Companion to Principles and Standards for School Mathematics (pp. 237-249). Reston, VA:NCTM.
Lee, T. N. (2015). Developing a theoretical framework to assess taiwanese primary students' geometric argumentation. In M. Marshman, V. Geiger, & A. Bennison (Eds.). Mathematics education in the margins (Proceedings of the 38th annual conference of the Mathematics Education Research Group of Australasia) (pp. 365–372). Sunshine Coast: MERGA.
Lerman, S. (2000). The social turn in mathematics education research. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 19–44). Westport, CT: Ablex Publishing.
Lerman, S. (2001). Cultural, discursive psychology: A sociocultural approach to studying the teaching and learning of mathematics. Educational Studies in Mathematics, 46, 87–113. https://doi.org/10.1023/A:1014031004832
Manouchehri, A., & Enderson, M. C. (1999). Promoting mathematical discourse: Learning from classroom examples. Mathematics Teaching in the Middle School, 4(4), 216-222. https://doi.org/10.5951/MTMS.4.4.0216
Martino, A. M., & Maher, C. A. (1999). Teacher questioning to promote justification and generalization in mathematics: What research practice has taught us. The Journal of Mathematical Behavior, 18(1), 53–78. https://doi.org/10.1016/S0732-3123(99)00017-6
Merriam, S. B. (2013). Nitel araştırma: desen ve uygulama için bir rehber. S. Turan (Çev.) Ankara: Nobel.
Milovanović, M., Takači, Đ, & Milajić, A. (2011). Multimedia approach in teaching mathematics – Example of lesson about the definite integral application for determining an area. International Journal of Mathematical Education in Science and Technology, 42(2), 175–187. https://doi.org/10.1080/0020739X.2010.519800
National Council of Teachers of Mathematics [NCTM], (2000). Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.
Öztürk, M., Akkan, Y., & Kaplan, A. (2019). Sınıf öğretmenliği öğrencilerinin temel matematik ispatlarını yapma sürecindeki bilişsel yapılar ve argümanları. Cumhuriyet Uluslararası Eğitim Dergisi, 8(2), 429-452. http://dx.doi.org/10.30703/cije.490887
Pedemonte, B. (2002). Relation between argumentation and proof in mathematics: Cognitive unity or break? In J. Novotna´ (Ed.), Proceedings of the 2nd Conference of the European Society for Research in Mathematics Education (pp. 70–80). Marienbad: ERME.
Pedemonte, B. (2008). Argumentation and algebraic proof. Zentralblatt für Didaktik der Mathematik-ZDM, 40(3), 385-400. https://doi.org/10.1007/s11858-008-0085-0
Planas, N., & Morera, L. (2011). Revoicing in processes of collective mathematical argumentation among students. In M. Pytlak, E. Swoboda & T. Rowland (Eds.), Proceedings of the VII Congress of the European Society of Research in Mathematics Education (pp. 1356–1365). Reszów, Poland: ERME.
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. https://doi.org/10.1007/s10857-008-9071-2
Schwarz, B. B. (2009). Argumentation and learning. In N. Muller Mirza & A. N. Perret-Clermont (Eds.), Argumentation and Education: Theoretical Foundations and Practices (pp. 91-126). Boston, MA: Springer. https://doi.org/10.1007/978-0-387-98125-3_4
Sfard, A. (2001). There is more to discourse than meets the ears: Looking at thinking as communicating to learn more about mathematical learning. Educational Studies in Mathematics, 46(1-3), 13-57. https://doi.org/10.1023/A:1014097416157
Solar, G., Ortiz, A., Deulofeu, J., & Ulloa, R. (2021). Teacher support for argumentation and the incorporation of contingencies in mathematics classrooms. International Journal of Mathematical Education in Science and Technology, 52(7), 977-1005. https://doi.org/10.1080/0020739X.2020.1733686
Stein, M. K., Engle, R. A., Smith, M. S., & Hughes, E. K. (2008). Orchestrating productive mathematical discussions: Five practices for helping teachers move beyond show and tell. Mathematical Thinking and Learning, 10(4), 313-340. https://doi.org/10.1080/10986060802229675
Stephan, M., Cobb, C., & Gravemeijer, K. (2003). Coordinating social and individual analyses: Learning as participation in mathematical practices. Journal for Research in Mathematics Education. Monograph, 12, 67-102.
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
Tatar, E., & Zengin, Y. (2016) Conceptual understanding of definite ıntegral with geogebra, computers in the schools, 33(2), 120-132. https://doi.org/10.1080/07380569.2016.1177480
Tekin Dede, A. (2019). Arguments constructed within the mathematical modelling cycle. International Journal of Mathematical Education in Science and Technology, 50(2), 292-314. https://doi.org/10.1080/0020739X.2018.1501825
Topuz, F., & Gunhan, B.C. (2021). Türkiye’de matematik eğitimindeki argümantasyon çalışmalarının eğilimi nasıldır?. Akdeniz Eğitim Araştırmaları Dergisi, 15(36), 55-80. https://doi.org/10.29329/mjer.2020.367.4
Toulmin, S. E. (2003). The Uses of Argument (updated ed.). New York, NY: Cambridge University Press. (Original work published 1958).
Van de Walle, J.E. (1989). Elementary School Mathematics. Virjinia Commonwealth University.
Weber, K., Maher, C., Powell, A., & Lee, H. S. (2008). Learning opportunities from group discussions: warrants become the objects of debate. Educational Studies in Mathematics, 68, 247–261. https://doi.org/10.1007/s10649-008-9114-8
Wood, T. (1998). Funneling or focusing? Alternative patterns of communication in mathematics class. In H. Steinbring, M. g. Bartolini- Bussi, & A. Sierpinska (eds.), Language and communication in the mathematics classroom (pp. 167–178). Reston, VA: National Council of Teachers of Mathematics.
Wood, T., Williams, G., & McNeal, B. (2006). Children's mathematical thinking in different classroom cultures. Journal for Research in Mathematics Education, 37(3), 222-255. https://doi.org/30035059
Yackel, E. (2004). Theoretical perspectives for analyzing explanation, justification and argumentation in mathematics classrooms. Communications of Mathematical Education, 18(1), 1-18. 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, 27(4), 458-477.
Yin R. K. (2018). Case study research and applications: Design and methods, 6 th edition. London: Sage. https://doi.org/10.2307/749877

Details

Primary Language

Turkish

Subjects

-

Journal Section

Research Article

Authors

Ayşe Tekin Dede ^*
0000-0002-8971-1970
Türkiye

Ali Özgün Özer
0000-0002-4204-9115
Türkiye

Esra Bukova Güzel
0000-0001-7571-1374
Türkiye

Publication Date

December 30, 2022

Submission Date

February 11, 2022

Acceptance Date

November 3, 2022

Published in Issue

Year 2022 Volume: 11 Number: 4

DOI

https://doi.org/10.30703/cije.1072163

IZ

https://izlik.org/JA99LD42ST

Cite

RIS / Bibtex

APA

Tekin Dede, A., Özer, A. Ö., & Bukova Güzel, E. (2022). Dönel Cisimlerin Yüzey Alanının Hesaplanması Sürecindeki Argümanların İncelenmesi. Cumhuriyet Uluslararası Eğitim Dergisi, 11(4), 587-603. https://doi.org/10.30703/cije.1072163

Examination of the Arguments in the Process of Calculating the Surface Area of the Solids of Revolution

Abstract

Keywords

Dönel Cisimlerin Yüzey Alanının Hesaplanması Sürecindeki Argümanların İncelenmesi

Öz

Anahtar Kelimeler

Supporting Institution

Project Number

References

Details

Primary Language

Subjects

Journal Section

Authors

Publication Date

Submission Date

Acceptance Date

Published in Issue

DOI

IZ

Cite

© Cumhuriyet University, Faculty of Education