Research Article
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Examination of Mathematics Teacher Candidates' Arguments and Proof Schemes in the Process of Collective Argumentation

Year 2024, Volume: 13 Issue: 1, 102 - 119, 27.03.2024

Buse Gizem Yitmez , Süha Yılmaz

https://doi.org/10.30703/cije.1302410

Abstract

The study, aims to examine the arguments and proof schemes created by the mathematics teacher candidates in the process of collective argumentation and to interpret the arguments in question by taking into account the proof schemes. The participants of this research, which is designed as one group case study, are four mathematics-teaching undergraduate students. The data collection tools of this research are the video recordings, observation notes, and the students' solution papers. Data analysis was carried out in two stages in this study. Firstly, the video recordings were transcribed and the data obtained were analyzed using Toulmin's argumentation schema, and then the proof schemes used in this process were determined. The results of the research show that four different sub-arguments emerged in the argumentation process in which teacher candidates actively participated. It is especially noteworthy that the teacher candidates presented rebuttals against each other's explanations and some of them were refuted by others. In addition, it was determined that external and experimental proof schemes were used in the sub-arguments formed in the argumentation process, but all of these sub-arguments were refuted. The data of the study show that analytical proof schemes were predominantly used in the main claim on which the teacher candidates reached a consensus. This study is limited to one group and one proof question, and it is recommended to conduct more in-depth research in the future.

Keywords

Argumentation proof schemes mathematics teacher candidates case study

References

  • Aydoğdu-İskenderoğlu, T. (2016). Kanıt ve kanıt şemaları. E. Bingölbali, S. Arslan. ve İ. Ö. Zembat (Eds.), Matematik eğitiminde teoriler içinde (ss. 101-114). Pegem Akademi.
  • Bingölbali, E. (2015) Türev kavramına ilişkin öğrenme zorlukları ve kavramsal anlama için öneriler. M. F. Özmantar, E. Bingölbali ve H. Akkoç (Eds.), Matematiksel kavram yanılgıları ve çözüm önerileri içinde (4. Baskı, ss. 223-252). Pegem Akademi.
  • Boero, P., Douek, N., Morselli, F., and Pedemonte, B. (2010). Argumentation and proof: a contribution to theoretical perspectives and their classroom implementation. In M. M. F. Pinto and T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol 1, pp. 179-209). Brasile: PME.
  • Brown, R. A. J., and Renshaw, P. D. (2000). Collective argumentation: A sociocultural approach to reframing classroom teaching and learning. In H. Cowie and G. Van Der Aalsvoort (Eds.), Social interaction in learning and instruction: The meaning of discourse for the construction of knowledge (pp. 52–66). Pergamon: Elsevier Science Inc.
  • Brown, R. (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
  • Bülbül, A., ve 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
  • Canbazoğlu-Bilici, S. (2019). Örnekleme yöntemleri. H. Özmen ve O. Karamustafaoğlu (Eds.), Eğitimde araştırma yöntemleri içinde (s. 56-78). Ankara: Pegem Akademi
  • Conner, A. (2008). Expanded Toulmin diagrams: A tool for investigating complex activity in classrooms. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano and 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., and 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., and 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
  • Doruk, M. (2016). İlköğretim matematik öğretmeni adaylarının analiz alanındaki argümantasyon ve ispat süreçlerinin incelenmesi. [Yayımlanmamış Doktora Tezi]. Atatürk Üniversitesi.
  • Flores, A. (2006). How do students know what they learn in middle school mathematics is true?. School Science and Mathematics, 106(3), 124-132. https://doi.org/10.1111/j.1949-8594.2006.tb18169.x
  • Harel, G., and Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. American Mathematical Society, 7, 234-283.
  • Harel, G., and Sowder, L. (2007). Toward a comprehensive perspective on proof. In F. Lester (Ed.), Handbook of Research on Teaching and Learning Mathematics (Vol 2, pp. 805-842). NCTM.
  • Housman, D., and Porter, M. (2003). Proof schemes and learning strategies of above-avarage mathematics students, Educational Studies in Mathematics, 53(2), 139-158. https://doi.org/10.1023/A:1025541416693
  • Inglis, M., Mejia-Ramos, J.P., and 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
  • İskenderoğlu, T., Baki, A., and İskenderoğlu, M.(2010). Proof schemes used by first grade of preservice mathematics teachers about function topic. Procedia Social and Behavioral Sciences, 9, 531-536. https://doi.org/10.1016/j.sbspro.2010.12.192
  • Knipping, C., Reid, D. (2015). Reconstructing Argumentation Structures: A Perspective on Proving Processes in Secondary Mathematics Classroom Interactions. In Bikner-Ahsbahs, A., Knipping, C., and Presmeg, N. (Eds) Approaches to Qualitative Research in Mathematics Education, (pp. 75-101). Dordrecht: Springer. https://doi.org/10.1007/978-94-017-9181-6_4
  • Knuth, E., Choppin, J., and Bieda, K. (2009). Middle school students’ productions of mathematical justification. In M. Blanton, D. Stylianou, and E. Knuth (Eds.), Teaching and Learning Proof Across the Grades: A K-16 Perspective (pp. 153–212). NY: Routledge.
  • Krummheuer G. (1995) The ethnography of argumentation. In Cobb P., and Bauersfeld H. (Eds), Emergence of Mathematical Meaning (pp. 229–269). NJ: Routledge
  • 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. https://doi.org/10.1016/j.jmathb.2007.02.001
  • Liu, Y., and Manouchehri, A. (2013). Middle school children’s mathematical reasoning and proving schemes. Investigations in Mathematics Learning, 6(1), 18-40. https://doi.org/10.1080/24727466.2013.11790328
  • Merriam, S. B. (2018). Nitel araştırma desen ve uygulama için bir rehber. (Çev. S. Turan). Ankara: Nobel Yayıncılık.
  • Oflaz, G., Bulut, N., and Akcakin, V. (2016). Pre-service classroom teachers’ proof schemes in geometry: a case study of three pre-service teachers. Eurasian Journal of Educational Research, 63, 133-152. http://dx.doi.org/ 10.14689/ejer.2016.63.8
  • Pala, O. ve Narlı, S. (2018). Matematik öğretmen adaylarının sayılabilirlik kavramına yönelik ispat şemalarının incelenmesi. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 12(2), 136-166. https://doi.org/10.17522/balikesirnef.506425
  • Pedemonte, B. (2007a). How can the relationship between argumentation and proof be analysed?, Educational Studies in Mathematics, 66, 23-41. https://doi.org/10.1007/s10649-006-9057-x
  • Pedemonte, B. (2007b). Structural relationships between argumentation and proof in solving open problems in algebra. In Proceedings of the V Congress of the European Society for Research in Mathematics Education CERME 5, (pp. 643–652). Larnaca, Cyprus.
  • Pedemonte, B. (2008). Argumentation and algebraic proof. Zentralblatt für Didaktik der Mathematik, 40, 385-400. https://doi.org/10.1007/s11858-008-0085-0
  • Planas, N., and Morera, L. (2011). Revoicing in processes of collective mathematical argumentation among students. In M. Pytlak, T. Rowland, and E. Swobod (Eds.), Proceedings of the 7th Congress of the European Society for Research İn Mathematics Education (pp. 1356-1365). Poland: CERME.
  • Schabel, C. (2005). An Instructional Model for Teaching proof Writing in the Number Theory Classroom. Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 15(1). 45-59. https://doi.org/10.1080/10511970508984105
  • Stephan, M., and 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
  • Şahin, A., and 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
  • Şen, C., and Güler, G. (2015). Examination of secondary school seventh graders’ proof skills and proof schemes. Universal Journal of Educational Research, 3(9), 617-631. https://doi.org/10.13189/ujer.2015.030906
  • Şengül, S., ve Güner, P. (2013). DNR tabanlı öğretime göre matematik öğretmen adaylarının ispat şemalarının incelenmesi. International Journal of Social Science, 6(2), 869-878.
  • Tall, D. (1995). Cognitive development, representations and proof. Justifying and Proving in School Mathematics, 27-38.
  • Tekin-Dede, A., Özer, A. Ö., ve Bukova-Güzel, E. (2022). Dönel cisimlerin yüzey alanının hesaplanması sürecindeki argümanların incelenmesi. Cumhuriyet International Journal of Education, 11(4): 587-603. https://doi.org/10.30703/cije.1072163
  • 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
  • Vygotsky L. (1978). Interaction between learning and development. In M. Gauvain and M. Cole (Eds), Readings on the Development of Children (pp. 34-40). NY: Scientific American Books.

Matematik Öğretmeni Adaylarının Ortaklaşa Argümantasyon Sürecindeki Argümanlarının ve İspat Şemalarının İncelenmesi

Year 2024, Volume: 13 Issue: 1, 102 - 119, 27.03.2024

Buse Gizem Yitmez , Süha Yılmaz

https://doi.org/10.30703/cije.1302410

Abstract

Bu araştırmada, matematik öğretmeni adaylarının ortaklaşa argümantasyon sürecinde oluşturdukları argümanların ve ispat şemalarının incelenmesi ve söz konusu argümanların ispat şemaları dikkate alınarak yorumlanması amaçlanmaktadır. Tekli durum çalışması olarak desenlenen araştırmanın katılımcılarını dört matematik öğretmenliği lisans öğrencisi oluşturmaktadır. Araştırmanın veri toplama araçlarını ortaklaşa argümantasyon sürecinin video kayıtları, araştırmacının gözlem notları ve öğretmen adaylarının çözüm kağıtları oluşturmakta olup veri analizi iki aşamada gerçekleştirilmiştir. İlk olarak video kayıtlar transkript edilerek elde edilen veriler Toulmin’in argümantasyon şeması kullanılarak analiz edilmiş, daha sonra bu süreçte kullanılan ispat şemaları belirlenmiştir. Araştırma sonuçları öğretmen adaylarının aktif olarak katıldıkları argümantasyon sürecinde dört farklı alt argümanın ortaya çıktığını göstermektedir. Argümantasyon sürecinde veri, iddia, gerekçe, çürütücü ve destekleyici bileşenleri ortaya çıkarken, niteleyici bileşeninin kullanılmadığı görülmüştür. Özellikle öğretmen adaylarının birbirlerinin açıklamalarına karşılık çürütücüler sunmaları ve bunlardan bazılarının diğerleri tarafından çürütülmesi dikkat çekmektedir. Ayrıca argümantasyon sürecinde oluşan alt argümanlarda dışsal ve deneysel ispat şemalarının kullanıldığı, ancak bu alt argümanların tamamının çürütüldüğü belirlenmiştir. Araştırmanın verileri öğretmen adaylarının üzerinde fikir birliği sağladıkları ana iddiada ağırlıklı olarak analitik ispat şemalarının kullanıldığını göstermektedir. Bu araştırma tek grup ve bir ispat sorusu ile sınırlı olup gelecekte daha geniş katılımcı grupları ve farklı ispat soruları ile daha derinlemesine araştırmalar yapılması önerilmektedir.

Keywords

Argümantasyon ispat şemaları matematik öğretmen adayları durum çalışması

References

  • Aydoğdu-İskenderoğlu, T. (2016). Kanıt ve kanıt şemaları. E. Bingölbali, S. Arslan. ve İ. Ö. Zembat (Eds.), Matematik eğitiminde teoriler içinde (ss. 101-114). Pegem Akademi.
  • Bingölbali, E. (2015) Türev kavramına ilişkin öğrenme zorlukları ve kavramsal anlama için öneriler. M. F. Özmantar, E. Bingölbali ve H. Akkoç (Eds.), Matematiksel kavram yanılgıları ve çözüm önerileri içinde (4. Baskı, ss. 223-252). Pegem Akademi.
  • Boero, P., Douek, N., Morselli, F., and Pedemonte, B. (2010). Argumentation and proof: a contribution to theoretical perspectives and their classroom implementation. In M. M. F. Pinto and T. F. Kawasaki (Eds.), Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol 1, pp. 179-209). Brasile: PME.
  • Brown, R. A. J., and Renshaw, P. D. (2000). Collective argumentation: A sociocultural approach to reframing classroom teaching and learning. In H. Cowie and G. Van Der Aalsvoort (Eds.), Social interaction in learning and instruction: The meaning of discourse for the construction of knowledge (pp. 52–66). Pergamon: Elsevier Science Inc.
  • Brown, R. (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
  • Bülbül, A., ve 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
  • Canbazoğlu-Bilici, S. (2019). Örnekleme yöntemleri. H. Özmen ve O. Karamustafaoğlu (Eds.), Eğitimde araştırma yöntemleri içinde (s. 56-78). Ankara: Pegem Akademi
  • Conner, A. (2008). Expanded Toulmin diagrams: A tool for investigating complex activity in classrooms. In O. Figueras, J. L. Cortina, S. Alatorre, T. Rojano and 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., and 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., and 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
  • Doruk, M. (2016). İlköğretim matematik öğretmeni adaylarının analiz alanındaki argümantasyon ve ispat süreçlerinin incelenmesi. [Yayımlanmamış Doktora Tezi]. Atatürk Üniversitesi.
  • Flores, A. (2006). How do students know what they learn in middle school mathematics is true?. School Science and Mathematics, 106(3), 124-132. https://doi.org/10.1111/j.1949-8594.2006.tb18169.x
  • Harel, G., and Sowder, L. (1998). Students’ proof schemes: Results from exploratory studies. American Mathematical Society, 7, 234-283.
  • Harel, G., and Sowder, L. (2007). Toward a comprehensive perspective on proof. In F. Lester (Ed.), Handbook of Research on Teaching and Learning Mathematics (Vol 2, pp. 805-842). NCTM.
  • Housman, D., and Porter, M. (2003). Proof schemes and learning strategies of above-avarage mathematics students, Educational Studies in Mathematics, 53(2), 139-158. https://doi.org/10.1023/A:1025541416693
  • Inglis, M., Mejia-Ramos, J.P., and 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
  • İskenderoğlu, T., Baki, A., and İskenderoğlu, M.(2010). Proof schemes used by first grade of preservice mathematics teachers about function topic. Procedia Social and Behavioral Sciences, 9, 531-536. https://doi.org/10.1016/j.sbspro.2010.12.192
  • Knipping, C., Reid, D. (2015). Reconstructing Argumentation Structures: A Perspective on Proving Processes in Secondary Mathematics Classroom Interactions. In Bikner-Ahsbahs, A., Knipping, C., and Presmeg, N. (Eds) Approaches to Qualitative Research in Mathematics Education, (pp. 75-101). Dordrecht: Springer. https://doi.org/10.1007/978-94-017-9181-6_4
  • Knuth, E., Choppin, J., and Bieda, K. (2009). Middle school students’ productions of mathematical justification. In M. Blanton, D. Stylianou, and E. Knuth (Eds.), Teaching and Learning Proof Across the Grades: A K-16 Perspective (pp. 153–212). NY: Routledge.
  • Krummheuer G. (1995) The ethnography of argumentation. In Cobb P., and Bauersfeld H. (Eds), Emergence of Mathematical Meaning (pp. 229–269). NJ: Routledge
  • 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. https://doi.org/10.1016/j.jmathb.2007.02.001
  • Liu, Y., and Manouchehri, A. (2013). Middle school children’s mathematical reasoning and proving schemes. Investigations in Mathematics Learning, 6(1), 18-40. https://doi.org/10.1080/24727466.2013.11790328
  • Merriam, S. B. (2018). Nitel araştırma desen ve uygulama için bir rehber. (Çev. S. Turan). Ankara: Nobel Yayıncılık.
  • Oflaz, G., Bulut, N., and Akcakin, V. (2016). Pre-service classroom teachers’ proof schemes in geometry: a case study of three pre-service teachers. Eurasian Journal of Educational Research, 63, 133-152. http://dx.doi.org/ 10.14689/ejer.2016.63.8
  • Pala, O. ve Narlı, S. (2018). Matematik öğretmen adaylarının sayılabilirlik kavramına yönelik ispat şemalarının incelenmesi. Necatibey Eğitim Fakültesi Elektronik Fen ve Matematik Eğitimi Dergisi, 12(2), 136-166. https://doi.org/10.17522/balikesirnef.506425
  • Pedemonte, B. (2007a). How can the relationship between argumentation and proof be analysed?, Educational Studies in Mathematics, 66, 23-41. https://doi.org/10.1007/s10649-006-9057-x
  • Pedemonte, B. (2007b). Structural relationships between argumentation and proof in solving open problems in algebra. In Proceedings of the V Congress of the European Society for Research in Mathematics Education CERME 5, (pp. 643–652). Larnaca, Cyprus.
  • Pedemonte, B. (2008). Argumentation and algebraic proof. Zentralblatt für Didaktik der Mathematik, 40, 385-400. https://doi.org/10.1007/s11858-008-0085-0
  • Planas, N., and Morera, L. (2011). Revoicing in processes of collective mathematical argumentation among students. In M. Pytlak, T. Rowland, and E. Swobod (Eds.), Proceedings of the 7th Congress of the European Society for Research İn Mathematics Education (pp. 1356-1365). Poland: CERME.
  • Schabel, C. (2005). An Instructional Model for Teaching proof Writing in the Number Theory Classroom. Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 15(1). 45-59. https://doi.org/10.1080/10511970508984105
  • Stephan, M., and 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
  • Şahin, A., and 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
  • Şen, C., and Güler, G. (2015). Examination of secondary school seventh graders’ proof skills and proof schemes. Universal Journal of Educational Research, 3(9), 617-631. https://doi.org/10.13189/ujer.2015.030906
  • Şengül, S., ve Güner, P. (2013). DNR tabanlı öğretime göre matematik öğretmen adaylarının ispat şemalarının incelenmesi. International Journal of Social Science, 6(2), 869-878.
  • Tall, D. (1995). Cognitive development, representations and proof. Justifying and Proving in School Mathematics, 27-38.
  • Tekin-Dede, A., Özer, A. Ö., ve Bukova-Güzel, E. (2022). Dönel cisimlerin yüzey alanının hesaplanması sürecindeki argümanların incelenmesi. Cumhuriyet International Journal of Education, 11(4): 587-603. https://doi.org/10.30703/cije.1072163
  • 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
  • Vygotsky L. (1978). Interaction between learning and development. In M. Gauvain and M. Cole (Eds), Readings on the Development of Children (pp. 34-40). NY: Scientific American Books.
There are 38 citations in total.

Details

Primary Language Turkish
Subjects Mathematics Education
Journal Section Research Article
Authors

Buse Gizem Yitmez 0000-0002-4163-489X

Süha Yılmaz 0000-0002-8330-9403

Publication Date March 27, 2024
Published in Issue Year 2024Volume: 13 Issue: 1

Cite

APA Yitmez, B. G., & Yılmaz, S. (2024). Matematik Öğretmeni Adaylarının Ortaklaşa Argümantasyon Sürecindeki Argümanlarının ve İspat Şemalarının İncelenmesi. Cumhuriyet Uluslararası Eğitim Dergisi, 13(1), 102-119. https://doi.org/10.30703/cije.1302410
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