Prospective Mathematics Teachers’ Strategies for Evaluating the Accuracy of Proofs in the Field of Analysis

Year 2018, Volume: 47 Issue: 2, 623 - 666, 01.10.2018

Muhammet Doruk , Abdullah Kaplan

Abstract

The purpose of this study is to reveal how
prospective mathematics teachers evaluate proofs that are proved by others in
the field of analysis. In this regard, skills of prospective teachers to
evaluate the accuracy of arguments are presented in various ways, and
strategies they use during the evaluation process are examined. This study, in
which the qualitative approach is adopted, is a case study. The sample
consisted of eight prospective teachers who were studying primary school
mathematics teaching in their third year at a state university in Turkey. The
data were collected with the help of task-based clinical interviews on subjects
of functions, sequences, limit and derivatives. In the study, it was found that
prospective teachers were successful at choosing valid proofs, whereas they had
difficulties in identifying invalid proofs. It was determined that especially
some prospective teachers were not able to distinguish proving methods, they
were not aware of the power of counterexample, and they considered inductive
arguments and, even if they were not correct, they accepted deductive arguments
as valid proofs. It was found that prospective teachers used three strategies
while evaluating proofs. These were structural examination, argument
examination and authoritarian examination. 

Keywords

Mathematical proof, Proof evaluation, Prospective mathematics teachers, Analysis

References

• Akkaş, S., Hacısalihoğlu, H. H., Özel, Z., & Sabuncuoğlu, A. (1998). Soyut matematik. Ankara: Gazi Üniversitesi Yayınları.
• Alcock, L., & Weber, K. (2005). Proof validation in real analysis: Inferring and evaluating warrants. Journal of Mathematical Behavior, 24(2), 125-134.
• Almeida, D. (2003). Engendering proof attitudes: Can the genesis of mathematical knowledge teach us anything?. International Journal of Mathematical Education in Science and Technology, 34(4), 479-488.
• Altun, M. (2013). Ortaokullarda (6, 7 ve 8. sınıflarda) matematik öğretimi. (9. Baskı). Bursa: Aktüel Alfa Akademi.
• Altun, M. (2014). Eğitim fakülteleri ve matematik öğretmenleri için liselerde matematik öğretimi. (5. Baskı). Bursa: Aktüel Alfa Akademi.
• Barkai, R., Tsamir, P., Tirosh, D., & Dreyfus, T. (2002). Proving or refuting arithmetic claims: The case of elementary school teachers. Paper presented at the annual meeting of the International Group of the Psychology of Mathematics Education, Norwich, England.
• Boero, P., Douek, N., Morselli, F., & Pedemonte, B. (2010). Argumentation and proof: A contribution to theoretical perspectives and their classroom implementation. In Proceedings of the 34th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, pp. 179-204). Belo-Horizonte, Brazil: PME. Cambridge University (2013). Cambridge Advenced Learner’s Dictionary. (4th edition). McIntosh, C. (Ed.). UK: Cambridge University Press.
• Cusi, A., & Malara, N. (2007). Proofs problems in elementary number theory: Analysis of trainee teachers' productions. In D. Pitta-Pantazi, & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp. 591-600). Cyprus, Larnaca.
• Doruk, M., & Kaplan, A. (2013b). İlköğretim Matematik Öğretmeni Adaylarının Dizilerin Yakınsaklığı Kavramı Üzerine İspat Değerlendirme Becerileri. Eğitim ve Öğretim Araştırmaları Dergisi, 2(1), 241-252.
• Doruk, M., & Kaplan, A. (2015b). Prospective mathematics teachers’ difficulties in doing proofs and causes of their struggle with proofs. Bayburt Üniversitesi Eğitim Fakültesi Dergisi, 10(2), 315-328.
• Douek, N. (1998). Some remarks about argumentation and mathematical proof and their educational implications. In European Research in Mathematics Education 1.1, Proceedings of the First Conference of the European Society for Research in Mathematics Education (pp. 125-139).
• Galbraith, P.L. (1981). Aspects of proving: A clinical investigation of process. Educational Studies in Mathematics, 12(1), 1-28.
• Gholamazad, S., Liljedahl, P. & Zazkis, R. (2004). What counts as proof? Investigation of pre-service elementary teachers’ evaluation of presented ‘proofs’. In D. E.
• McDougall & J. O. Ross (Eds.), Proceedings of the Twenty-sixth Annual Meeting of the North American Chapter of the International Group for the Psychology of
• Mathematics Education (vol. 2, pp. 639–646), University of Toronto, Toronto
• Gibson, D. (1998). Students' use of diagrams to develop proofs in an introductory analysis course. InA. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education. Ill (pp. 284-307). Providence, RI: American Mathematical Society.
• Goetting, M. (1995). The college students' understanding of mathematical proof (Unpublished doctoral dissertation). Available from ProQuest Dissertations and Theses database. (UMI No. 9539653).
• Güler, G. (2013). Matematik öğretmeni adaylarının cebir öğrenme alanındaki ispat süreçlerinin incelenmesi. Yayımlanmamış doktora tezi. Atatürk Üniversitesi Eğitim Bilimleri Enstitüsü, Erzurum.
• Güler, G., & Ekmekçi, S. (2016). Matematik öğretmeni adaylarının ispat değerlendirme becerilerinin incelenmesi: Ardışık tek sayıların toplamı örneği. Bayburt Eğitim Fakültesi Dergisi, 11(1), 59-83.
• Harel, G, & Sowder, L. (1998). Students' proof schemes: Results from exploratory studies. InA. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234-283). Providence, R.I.: American Mathematical Society.
• Irmak, H. (2008). Soyut matematik. Ankara: Pegem Akademi.
• Kaleli Yılmaz, G. (2015). Durum çalışması. Mustafa Metin (Ed.). Kuramdan uygulamaya eğitimde bilimsel araştırma yöntemleri içinde (s. 261-285). Ankara: Pegem Akademi.
• Knuth, E. (2002). Secondary school mathematics teachers' conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405.
• Ko, Y.Y. (2010). Proofs and Counterexamples: Undergraduate Students' Strategies for Validating Arguments, Evaluating Statements, and Constructing Productions. (Unpublished doctoral dissertation). Available from ProQuest Dissertations and Theses database. (UMI No. 3437186)
• Ko, Y.Y., & Knuth, E. (2009). Undergraduate mathematics majors’ writing performance producing proofs and counterexamples about continuous functions. The Journal of Mathematical Behavior, 28(1), 68-77.
• Lakatos, I. (1976). Proofs and refutations: The logic of mathematical discovery. Cambridge, UK: Cambridge University Press.
• Lampert, M. (1990). When the problem is not the question and the solution is not the answer: Mathematical knowing and teaching. American Educational Research Journal, 27(1), 29-63.
• Martin, G., & Harel, G. (1989). Proof frames of preservice elementary teachers. Journal for Research in Mathematics Education, 20, 41-51.
• Mejia-Ramos, J.P., & Inglis, M. (2009). What are the argumentative activities associated with proof?. Research in Mathematics Education, 11(1), 77-78.
• Merriam, S.B. (2013). Nitel araştırma desen ve uygulama için bir rehber. (Çev. Ed. S. Turan). Ankara: Nobel Akademik Yayıncılık.
• Moore, R.C. (1994). Making the transition to formal proof. Educational Studies in Mathematics. 27, 249-266.
• Morris, A.K. (2002). Mathematical Reasoning: Adults' ability to make the inductive- eductive distinction. Cognition and Instruction, 20(1), 79-118.
• Oxford University. (2010). Advenced Learner’s Dictionary (International students’ edition).(8th edition). New York: Oxford University Press
• Patton, M.Q. (2014). Nitel araştırma ve değerlendirme yöntemleri. (Çev. Ed. M. Bütün ve S. B. Demir). Ankara: Pegem Akademi.
• Pedemonte, B. (2007). How can the relationship between argumentation and proof be analysed? Educational Studies in Mathematics, 66, 23–41.
• Pedemonte, B. (2008). Argumentation and algebraic proof. ZDM Mathematics Education, 40(3), 385–400.
• Raman, M. (2003). Key ideas: What are they and how can they help us understand how people view proof?. Educational Studies in Mathematics, 52(3), 319-325.
• Riley, K.J. (2003). An investigate of prospective secondary mathematics teachers' conceptions of proof and refutations (Doctoral dissertation). Available from ProQuest Dissertation and Theses database. (UMI No. 3083484)
• Ross, K.A. (1998). Doing and proving: The place of algorithms and proofs in school mathematics. The American Mathematical Monthly, 105(3), 252-255.
• Rumsey, C. W. (2012). Advancing fourth-grade students' understanding of arithmetic properties with instruction that promotes mathematical argumentation (Unpublished doctoral dissertation). Available from ProQuest Dissertations and Theses database. (UMI No. 3520912)
• Sarı, M., Altun, A., & Aşkar, P. (2007). Üniversite öğrencilerinin analiz dersi kapsamında matematiksel kanıtlama süreçleri: örnek olay çalışması. Ankara Üniversitesi Eğitim Bilimleri Fakültesi Dergisi, 40(2), 295–319.
• Segal, J. (2000). Learning about mathematical proof: Conviction and validity. Journal of Mathematical Behavior, 18(2), 191-210.
• Selden, A., & Selden, J. (2003). Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? Journal for Research in Mathematics Education, 34(1), 4-36.
• Stylianides, A.J., & Stylianides, G.J. (2009). Proof constructions and evaluations. Educational Studies in Mathematics, 72(2), 237-253.
• Stylianou, D., Chae, N., & Blanton, M. (2006). Students' proof schemes: A closer look at what characterizes students' proof conceptions. In Alatorre, S. Cortina, J. and Mendez A.(Eds, 2006). Proceedings of the 28th annual meeting of the North American Chapters of the International Group of the Psychology of Mathematics Education. Merida, Mexico.
• Toulmin, S.E. (2003). The uses of argument. Cambridge: Cambridge University Press.
• Uygan, C., Tanışlı, D., & Köse, N. Y. (2014). İlköğretim Matematik Öğretmeni Adaylarının Kanıt Bağlamındaki İnançlarının, Kanıtlama Süreçlerinin ve Örnek Kanıtları Değerlendirme Süreçlerinin İncelenmesi. Turkish Journal of Computer and Mathematics Education, 5(2), 137-157.
• Weber, K. (2001). Student difficulty in constructing proofs: The need for strategic knowledge. Educational Studies in Mathematics, 48(1), 101-1 19.
• Weber, K. (2005). Problem solving, proving and learning: the relationship between problem solving processes and learning opportunities in the activity of proof construction. Journal of Mathematical Behaviour, 24, 351-360.
• Weber, K. (2008). How mathematicians determine if an argument is a valid proof. Journal for Research in Mathematics Education, 39, 431-459.
• Weber, K. (2009). How syntactic reasoners can develop understanding, evaluate conjectures, and generate counterexamples in advanced mathematics. Journal of Mathematical Behavior, 25(2-3), 200-208.
• Whiteley, W. (2009). Refutations: the role of counter-examples in developing proof. In F. L. Lin, F. J. Hsieh, G. Hanna & M. Villiers (Eds), Proceedings of the ICMI Study 19 conference: Proof and Proving in Mathematics Education, vol. 2 Taipei, Taiwan (pp. 257-262).
• Williams, E. (1979). An investigation of senior high school students ' understanding of the nature of mathematical proof. Unpublished doctoral dissertation, University of Alberta, Edmonton.
• Yasuhiro, S. (1991). An investigation on proofs and refutations in the mathematics classroom. Unpublished doctoral dissertation, University of Georgia, Atlanta.
• Yıldırım, A. ve Şimşek, H. (2011). Sosyal bilimlerde nitel araştırma yöntemleri. (8. baskı). Ankara: Seçkin Yayıncılık.
• Yıldırım, C. (2014). Matematiksel düşünme. (10. Baskı). İstanbul: Remzi Kitapevi.
• Zaslavsky, O., & Peled, I. (1996). Inhibiting factors in generating examples by mathematics teachers and student teachers: The case of binary operation. Journal for Research in Mathematics Education, 67-78.
• Zaslavsky, O., & Ron, G. (1998). Students' understanding of the role of counter-examples. In Olivier A. & Newstead K. (Eds.), Proceedings ofthe Twenty-second Annual Meeting of the International Groupfor the Psychology of Mathematics Education (Vol. 4, pp. 225-232). Stellenbosch, South Africa.