Matematik Öğretmeni Adaylarının Tümdengelimsel Akıl Yürütme Yoluyla İspat Anlayışları

Yıl 2024, Sayı: 62, 362 - 404, 23.09.2024

Emine Gaye Çontay

https://doi.org/10.9779/pauefd.1359924

Öz

Bu çalışmanın amacı matematik öğretmeni adaylarının tümdengelimsel akıl yürütme yoluyla ispat yapma becerilerini, ispat yöntemlerine ilişkin bilgilerini ve ispat yöntemlerini kullanma becerilerini ispat şemalarıyla ilişkilendirerek “ispat anlayışları” bağlamında ortaya koymaktır. Çalışma bir devlet üniversitesinde son sınıfta öğrenim gören 44 öğretmen adayı ile gerçekleştirilmiştir. Çalışmada öğretmen adaylarının ispat anlayışlarını ortaya koyması amacıyla iki farklı ölçme aracı geliştirilmiş ve uygulanmıştır. Çalışmanın bulguları öğretmen adaylarının büyük bölümünün ispat becerilerinin zayıf olduğunu, ispat yaparken kullandıkları yöntemlerden habersiz biçimde önermede yer alan değişkene değerler vererek ispatı yapılandırdıklarını, çoğunlukla deneysel ispat şeması göstergeleri ile hareket ettiklerini ortaya koymuştur. Öğretmen adayları ispat yöntemlerine ilişkin bilgileri açısından değerlendirildiklerinde, en çok aksine örnek verme ile ispat yöntemine ilişkin bilgiye sahip oldukları görülmüştür. Öğretmen adaylarının analitik dönüşümsel ispat şemasına ilişkin göstergelerle hareket ettikleri tek ispat sorusu ise durumlarla ispat yöntemi ile çözülebilecek ispat sorusudur. Analitik ispat şeması göstergeleriyle ispatlarını yapılandıran öğretmen adaylarından hiçbirinin, kullandıkları ispat yöntemini bilmedikleri belirlenmiştir. Dolayısıyla bu çalışmaya katılan matematik öğretmeni adaylarının ispat anlayışları yetersiz bulunmuştur.

Anahtar Kelimeler

matematik öğretmeni adayı, ispat becerisi, ispat öğretimi, ispat şeması, ispat yöntem bilgisi

Etik Beyan

Bu araştırma, Pamukkale Üniversitesi Sosyal ve Beşeri Bilimleri Araştırma ve Yayın Etiği Kurulunun 0909//2020 tarihli 68282350/2018/G07 sayılı kararı ile alınan izinle yürütülmüştür.

Destekleyen Kurum

Pamukkale Üniversitesi BAP Birimi

Proje Numarası

2019BSP017

Preservice Mathematics Teachers’ Understanding of Proof Through Deductive Reasoning

Öz

The aim of the study is to investigate preservice mathematics teachers' skills in proving through deductive reasoning, their knowledge of proof methods, and their ability to use proof methods in the context of their "understanding of proof" by associating them with proof schemes. The study was conducted with 44 preservice teachers studying in their final year at a state university. Two different assessment instruments were developed and applied. The findings of the study revealed that the majority of preservice teachers had weak proof skills that they structured the proof by giving values to the and that they frequently acted with experimental proof scheme indicators. When preservice teachers were evaluated in terms of their knowledge of proof methods, they had the knowledge about the method of counterexample. The only proof question in which preservice teachers acted with indicators related to the analytical transformational proof scheme was that could be solved by using the method of “proof cases.” It was identified that none of the preservice teachers who structured their proofs with analytical proof scheme indicators knew the proof method they used. Consequently, the proof understanding of the preservice mathematics teachers who participated in this study was found insufficient.

Anahtar Kelimeler

Preservice mathematics teachers, Proof ability, Proof education, Proof scheme, Knowledge of the proof method

Proje Numarası

2019BSP017

Kaynakça

• Almeida, D. (2003). Engendering proof attitudes: Can the genesis of mathematical knowledge teach us anyting. International Journal of Mathematical Education in Science and Technology, 34(4), 479-488. https://doi.org/10.1080/0020739031000108574.
• Barak, B. (2018). Investigation of pre-service middle school mathematics teachers’ proving processes [Doctoral Dissertation, Anadolu University]. National Dissertation Center.
• Brown, J., & Stillman, G. (2009). Preservice secondary teachers’ competencies in proof. Proceedings of the ICMI Study 19 conference: Proof and Proving in Mathematics Education Mathematics Education, 1, 1-94.
• Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics (CCSSM). Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. http://www.corestandards.org/Math/
• Çontay, E. G. & Duatepe Paksu, A. (2019). The Proof Schemes of Preservice Middle School Mathematics Teachers and Investigating the Expressions Revealing These Schemes. Turkish Journal of Computer and Mathematics Education, 10(1), 59-100. https://doi.org/10.16949/turkbilmat.397109
• Demircioğlu, H. (2022). Investıgatıon of preservıce mathematıcs teachers’ ,mathematıcs teachers’ and academıcıan's proof skılls, The Journal of Academic Social Science Studies, 73, 493-508. http://dx.doi.org/10.9761/JASSS7906.
• Doruk, M. (2019). preservice mathematics teachers' determination skills of the proof techniques: The case of ıntegers. International Journal of Education in Mathematics, Science and Technology, 7(4), 335-348.
• Doruk, M., & Kaplan, A. (2017). The characterıstıcs of proofs produced by preservıce prımary mathematıcs teachers ın calculus. Mehmet Akif Ersoy Journal of Education Faculty, (44), 467-498. https://doi.org/10.21764/maeuefd.305605.
• Doruk, M., & Kaplan, A. (2013). Prospectıve prımary mathematıcs teachers’ proof evaluatıon abıltıes on convergence of sequence concept. Journal of Research in Education and Teaching, 2(1), 241-252.
• Gholamazad, S., Liljedahl, P., & Zazkis, R. (2004, October 21-24). What counts as proof? Investigation of preservice elementary teachers' evaluation of presented 'Proofs’, Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Toronto, Canada. https://www.pmena.org/pmenaproceedings/PMENA%2026%202004%20Proceedings%20Vol%201.pdf.
• Grabiner, J. V. (2012). Why proof? A historian’s perspective. G. Hanna, & M. De Villers (Ed.), Proof and Proving in Mathematics Education. Springer.
• Güler, G. (2013). Investigating Pre-Service Teachers’ Knowledge for Teaching Mathematics: The Sample of Algebra, [Doctoral Dissertation, Atatürk University]. National Dissertation Center.
• Güler, G., & Ekmekci, S. (2016). Examination of the proof evaluation skills of the prospective mathematics teachers: The example of sum of successive odd numbers. Bayburt Journal of Faculty of Education, 11(1), 59-83.
• Güler, G., Özdemir, E., & Dikici, R (2012). Pre-servıce teachers’ provıng skılls usıng mathematıcal ınductıon and theır vıews on mathematıcal proving, Kastamonu Education Journal, 20(1), 219-236.
• Güner, S. (2012). Examınıng the ways of understandıng and thınkıng of mathematıcs teacher candıdates accordıng to dnr based teachıng ın theır proof process [Master’s Thesis, Marmara University]. National Thesis Center.
• Hanna, G. (1990). Some pedagogical aspects of proof. Interchange, 21(1), 6-13.
• Harel, G., & Sowder, L. (1998). Students proof schemes: Results from exploratory studies, CBMS Issues in Mathematics Education, 7, 234-283.
• Hersch, R. (1993). Proving is convincing and explaining. Educational Studies in Mathematics, 24, 389-399. https://doi.org/10.1007/BF01273372.
• Imamoglu, Y., & Togrol, A. Y. (2015). Proof construction and evaluation practices of prospective mathematics educators. European Journal of Science and Mathematics Education, 3(2), 130-144.
• İskenderoğlu, T. (2010). Proof schemes used by prerservıce mathematıcs teachers and their ıdeas about proof [Doctoral Diisertation, Karadeniz Technic University]. National Thesis Center.
• İskenderoğlu, T., Baki, A., & 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.
• Karunakaran, S., Freeburn, B., Konuk, N., & Arbaugh, F. (2014). Improving preservice secondary mathematics teachers' capability with generic example proofs. Mathematics Teacher Educator, 2(2), 158-170. https://doi.org/10.5951/mathteaceduc.2.2.0158.
• Kleiner, I. (1991). Rigor and proof in mathematics: A historical perspective. Mathematics Magazine. 64(5), 291-314. https://doi.org/10.1080/0025570X.1991.11977625.
• Knuth, E.J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405. https://doi.org/10.2307/4149959.
• Köğce, D. (2013). Primary mathematics pre-service teachers’ opinions about the contribution of doing proof on learning mathematics and their levels of doing mathematical proof.. Electronic Turkish Studies, 8(12).
• Mariotti, M.A. (2006). Proof and proving in mathematics education. A. Gutierrez, & P. Boero (Ed.), Handbook of research on the psychology of mathematics education. Past, present and future (pp. 173-204). Sense Publishers.
• Mason, J, Burton, L., ve Stacey, K. (2010). Thinking mathematically (2. Ed.). Pearson.
• National Council of Teachers of Mathematics (NCTM) (2000). Principles and Standards for School Mathematics, Reston.VA: NCTM.
• Noto, M.S., Priatna, N., & Dahlan, J.A. (2019). Mathematical proof: Learning obstacles pre-service teachers on transformation geometry. Journal on Mathematics Education, 10(1), 117-126.
• Oflaz, G., Bulut, N., & 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.
• Öztürk, M., & Kaplan, A. (2022). Secondary Mathematics Teacher Candidates’ Geometric Proof Process: A Case Study. Eurasian Journal of Teacher Educaiton. 3(1), 39-54.
• Pala, O., & Narlı, S. (2018). Examining Proof Schemes of Prospective Mathematics Teachers Towards Countability Concept. Necatibey Faculty of Education Electronic Journal of Science & Mathematics Education, 12(2).
• Pekşen Sağır, P. (2013). Analysıng the process of prospectıvemath teachers’ doıng mathematıcal proof [Master’s Thesis, Marmara University]. National Thesis Center.
• Reid, D.A., & Knipping, C. (2010). Proof in mathematics education. Research, Learning and Teaching. Sense Publishers.
• Sarı, M., Altun, A., & Aşkar, P. (2007). Undergraduate students’ mathematical proof processes in a calculus course: case study, Journal of Faculty of Educational Sciences, 40(2), 295-319.
• Sowder, L., & Harel, G. (1998). Types of students’ justifications, The Mathematics Teacher, 91(8), 670-675. https://doi.org/10.5951/MT.91.8.0670.
• Sears, R., Mueller, E., ve Karadeniz, I. (2013, November 6-8). Preservice teachers perception of their preparation program to cultivate their ability to teach proof. I Congreso de Education Matematica de America Central y El Caribe. http://funes.uniandes.edu.co/4271/1/SearsPreserviceCemacyc2013.pdf.
• Sears, R. (2019). Proof schemes of pre-service middle and secondary mathematics teachers. Investigations in Mathematics Learning, 11(4), 258-274. https://doi.org/10.1080/19477503.2018.1467106.
• Stylinou, D., Chae, N., & Blanton, M. (2006, Kasım 9-12). Students’ proof schemes: A closer look at what characterizes students’ proof conceptions, Proceedings of the annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mexico. https://www.pmena.org/pmenaproceedings/PMENA%2028%202006%20Proceedings.pdf.
• Şengül, S., & Güner, P. (2013). Investigation of preservice mathematics teachers’ proof schemes according to DNR based instruction. International Journal of Social Science, 6(2), 869-878. http://dx.doi.org/10.9761/JASSS_401
• Şen ve Şen, C.., & Güler, G.(2022). Examining Proof-Writing Skills of Pre-Service Mathematics Teachers' in Geometric Proofs: van Hiele Model. Ahi Evran universityi Kırşehir Journal of Faculty of Education, 23, 128-176. https://doi.org/10.29299/kefad.997311
• Stylianides,G.J., Stylianides,A.J., & Philippou, G.N.(2007). Preservice teachers' knowledge of proof by mathematical induction. Journal of Mathematics Teacher Education, 10, 145-166. https://doi.org/10.1007/s10857-007-9034-z.
• Uygan, C., Tanışlı, D., & Köse, N.Y. (2014). Research of Pre-Service Elementary Mathematics Teachers’ Beliefs in Proof, Proving Processes and Proof Evaluation Processes, Turkish Journal of Computer and Mathematics Education, 5(2), 137-157. https://doi.org/10.16949/turcomat.33155
• Uğurel, I., Morali, H., Yiğit Koyunkaya, M., & Karahan, O. (2016). Pre-Service secondary mathematics teachers' behaviors in the proving process. Eurasıa Journal Of Mathematıcs Scıence And Technology Education, 12(2), 203-231. 10.12973/eurasia.2016.1523a.
• Uygur Kabael, T. (2020). Methods of Prooving. I. Uğurel (Ed.) Mathematical Proof and teaching (227-242). Anı Publishing
• Varghese, T. (2007). Student teachers' conceptions of mathematical proof, Faculty of Graduate Studies and Research. [Master’s Thesis, University of Alberta]. Admonton. https://era.library.ualberta.ca/items/e2c86876-2f0a-4982-9812-ff314d023fcd.
• Yıldırım, A. & Şimşek, H. (2021). Methods of qualitative research. Seçkin Publishing.
• Yin, R. K. (2003). Case study research: Design and methods. Thousand Oaks, CA: Sage.
• Yoo, S. (2008). Effects of Traditional and Problem-Based Instruction on Conceptions of Proof and Pedagogy in Undergraduates and Prospective Mathematics Teachers [Doctoral Dissertation, The University of Texas, Austin, USA]. https://www.proquest.com/docview/304473805?pq-origsite=gscholar&fromopenview=true&sourcetype=Dissertations%20&%20Theses.
• Zaimoğlu, Ş. (2012). Geometrıcal proof processes and tendencıes of 8th grade students. [Master’s Thesis, Kastamonu University]. National Thesis Center.