İlköğretim matematik öğretmeni adaylarının matematiksel tümevarımın versiyonlarına yönelik beceri ve güçlükleri

Year 2025, Issue: 74, 667 - 691, 30.04.2025

Fikret Cihan , Muhammet Doruk

https://doi.org/10.21764/maeuefd.1456545

Abstract

Bu çalışmada ilköğretim matematik öğretmeni adaylarının matematiksel tümevarımın farklı versiyonlarına yönelik beceri ve güçlüklerinin ortaya çıkarılması amaçlanmıştır. Tümevarımın; zayıf, genişletilmiş zayıf, güçlü ve genişletilmiş güçlü tümevarım yöntemi olmak üzere dört versiyonuna odaklanılmıştır. Nitel araştırma yaklaşımı ile tasarlanan çalışma bir durum çalışması örneğidir. Araştırmanın çalışma grubunu 2023-2024 eğitim-öğretim yılı güz döneminde Türkiye’deki bir devlet üniversitesinin ilköğretim matematik öğretmenliği bölümünde öğrenimlerine devam eden ikinci sınıf öğrencileri oluşturmaktadır. Veriler araştırmacılar tarafından geliştirilen Matematiksel Tümevarım Versiyonları Beceri Formu yardımıyla toplanmıştır. Formdan elde edilen veriler betimsel analizle çözümlenmiştir. Çalışma sonucunda öğretmen adaylarının çoğunun zayıf tümevarım dışındaki tümevarım versiyonlarında başarısız oldukları, bu başarısızlığın zayıf tümevarımdan genişletilmiş güçlü tümevarıma doğru gittikçe arttığı tespit edilmiştir. Araştırmanın diğer ilgi çekici bir sonucu da öğretmen adayları zayıf ve genişletilmiş zayıf tümevarım yönteminde çoğunlukla tümevarım adımını oluşturmakta güçlük yaşarken güçlü ve genişletilmiş güçlü tümevarım yönteminde ise daha bu adıma geçemeden tümevarım hipotezlerini yazma aşamasında güçlük yaşadıkları ve bu yüzden ispatları tamamlayamadıkları saptanmıştır. Sonuçlar öğretmen yetiştirme bağlamında tartışılmış ve önerilerde bulunulmuştur.

Keywords

İspat , matematik eğitimi , öğretmen yetiştirme , tümevarım , tümevarım versiyonları

Prospective primary school mathematics teachers’ skills and difficulties towards versions of mathematical induction

Abstract

This study aimed to reveal the skills and difficulties of prospective primary school mathematics teachers towards different versions of mathematical induction. We focus on four induction versions: weak, extended weak, strong, and extended strong induction. The study designed with a qualitative research approach is a case study. The study group consisted of second-year students studying in the Department of Elementary School Mathematics Teacher Education at a state university in Türkiye in the fall semester of the 2023-2024 academic year. The data of the study were collected with the help of the Mathematical Induction Versions Skill Form developed by the researchers. The data obtained from the form were analyzed by descriptive analysis. As a result of the study, it was found that most of the pre-service teachers were unsuccessful in induction versions other than weak induction, and this failure increased from weak induction to extended strong induction. In weak and extended weak induction, the students mostly had difficulty in forming the induction step, whereas in strong and extended strong induction, the students had difficulty in writing induction hypotheses. The results were discussed in the context of teacher training and suggestions were made.

Keywords

Proof , mathematics education , teacher training , induction , versions of induction

References

• Arnesen, K. K., & Skartsæterhagen, Ø. I. (2025). Mathematical induction in education research: A systematic review. Educational Studies in Mathematics. https://doi.org/10.1007/s10649-024-10373-x
• Avital, S., & Libeskind, S. (1978). Mathematical induction in the classroom: Didactical and mathematical issues. Educational Studies in Mathematics, 9(4), 429-438. https://doi.org/10.1007/BF00410588
• Baker, J. D. (1996, April 8–12). Students’ difficulties with proof by mathematical induction [paper presentation]. Annual Meeting of the American Educational Research Association, New York, NY, United States.
• Baxandall, P. R., Brown, W. S., Rose, G. St. C., & Watson, F. R. (1978). Proof in mathematics. Institute of Education, University of Keele.
• Beck, M., & Geoghegan, R. (2010). The art of proof: Basic training for deeper mathematics. Springer Science + Business Media.
• Bloch, E. D. (2011). Proofs and fundamentals: A first course in abstract mathematics (2nd ed.). Springer Science + Business Media.
• Brown, S. (2003). The evolution of students’ understanding of mathematical induction: A teaching experiment (Order No. 3090458) [Doctoral Dissertation. University of California] ProQuest Dissertations & Theses Global. (305355390). https://www.proquest.com/dissertations-theses/evolution-students-understanding-mathematical/docview/305355390/se-2
• Campbell, C. M. (2012). Introduction to advanced mathematics: A guide to understanding proofs. Brooks/Cole, Cengage Learning.
• Chartrand, G., Polimeni, A. D., & Zhang, P. (2013). Mathematical proofs: A transition to advanced mathematics (3rd ed.). Pearson Education, Inc. Cihan, F. (2019). Matematik öğretmen adaylarının ispatla ilgili alan ve pedagojik alan bilgilerini geliştirmeye yönelik bir ders tasarımı (Tez No. 570220) [Doktora tezi, Marmara Üniversitesi]. Yükseköğretim Kurulu Başkanlığı Tez Merkezi. https://tez.yok.gov.tr/UlusalTezMerkezi/
• Cihan, F., & Akkoç, H. (2023). An intervention study for improving pre-service mathematics teachers’ proof schemes. Mathematics Teaching-Research Journal, 15(2), 56-80. https://eric.ed.gov/?id=EJ1394137
• Cornu, B. (1991). Limits. In D. Tall (Ed.), Advanced mathematical thinking (pp. 153-166). Kluwer.
• Creswell, J. W. (2014). Araştırma deseni: Nitel, nicel ve karma yöntem yaklaşımları. (S. B. Demir, Çev.). (4.baskıdan çeviri). Eğiten Kitap.
• Cunningham, D. W. (2012). A logical introduction to proof. Springer.
• Cusi, A., & Malara, N. A. (2009). Improving awareness about the meaning of the principle of mathematical induction. PNA, 4(1), 15-22. https://doi.org/10.30827/pna.v4i1.6170
• Daymon, C., & Holloway, I. (2011). Qualitative research methods in public relations and marketing communications (2nd ed.). Routledge. https://doi.org/10.4324/9780203846544
• Doğan-Dunlap, H., Özdemir-Erdoğan, E., & Kılıç, Ç. (2013). Matematiksel tümevarım: Karşılaşılan kavram yanılgıları ve öğrenme güçlükleri. M. F. Özmantar, E. Bingölbali ve H. Akkoç (Ed.), Matematiksel kavram yanılgıları ve çözüm önerileri (3. Baskı) içinde (s. 291-328). PegemA.
• Dubinsky, E. (1986). Teaching mathematical induction I. Journal of Mathematical Behavior, 5, 305-317.
• Dubinsky, E. (1990). Teaching mathematical induction II. Journal of Mathematical Behavior, 8, 285-304.
• Epp, S. S. (2011). Discrete mathematics: An introduction to mathematical reasoning (Brief ed.). Brooks/Cole, Cengage Learning.
• Ernest, P. (1984). Mathematical induction: A pedagogical discussion. Educational Studies in Mathematics, 15(2), 173-189. https://dx.doi.org/10.1007/BF00305895
• Feil, T., & Krone, J. (2003). Essential discrete mathematics for computer science. Upper Saddle Pearson Education, Inc.
• Fischbein E., & Engel, I. (1989, July). Psychological difficulties in understanding the principle of mathematical induction. In G. Vergnaud, J. Rogalski, & M. Artigue (Eds.), Proceedings of the 13th International Conference for the Psychology of Mathematics Education (Vol. I, pp. 276-282). Paris, France: CNRS.
• García-Martínez, I., & Parraguez, M. (2017). The basis step in the construction of the principle of mathematical induction based on APOS theory. The Journal of Mathematical Behavior, 46, 128-143.https://doi.org/10.1016/j.jmathb.2017.04.001
• Garnier, R., & Taylor, J. (2009). Discrete mathematics: Proofs, structures, and applications (3rd ed.). CRC Press, Taylor & Francis Group, LLC.
• Gerstein, L. J. (2012). Introduction to mathematical structures and proofs (2nd ed.). Springer.
• Gossett, E. (2009). Discrete mathematics with proof (2nd ed.). John Wiley & Sons.
• Guba, E. G., & Lincoln, Y. S. (1981). Effective evaluation: Improving the usefulness of evaluation results through responsive and naturalistic approaches. Jossey-Bass.
• Güler, G., & Ekmekci, S. (2016). Matematik öğretmeni adaylarının ispat değerlendirme becerilerinin incelenmesi: Ardışık tek sayıların toplamı örneği. Bayburt Eğitim Fakültesi Dergisi, 11(1), 59-83. http://dergipark.gov.tr/befdergi/issue/23129/247062
• Güler, G., Özdemir, E., & Dikici, R. (2012). Öğretmen adaylarının matematiksel tümevarım yoluyla ispat becerileri ve matematiksel ispat hakkındaki görüşleri. Kastamonu Eğitim Dergisi, 20(1), 219-236. https://dergipark.org.tr/tr/pub/kefdergi/issue/48696/619520
• Hair, J. F., Black, W. C., Babin, B. J., & Anderson, R. E. (2014). Multivariate Data Analysis (7th ed.). Pearson Education, Inc.
• Hammack, R. (2013). Book of proof (3rd ed.). Publisher: Author. https://www.people.vcu.edu/~rhammack/BookOfProof/
• Harel, G. (2001). The development of mathematical induction as a proof scheme: A model for DNR-based instruction. In S. Campbell, & R. Zaskis (Eds.), Learning and teaching number theory (pp.185-212). Ablex Publishing Corporation.
• Knapp, J. (2005). Learning to prove in order to prove to learn. http://mathpost.asu.edu/~sjgm/issues/2005_spring/SJGM_knapp.pdf
• Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for Research in Mathematics Education, 33(5), 379-405. https://www.jstor.org/stable/4149959
• Lestyanto, L. M., Sudirman, S., Susiswo, S., & Hidayanto, E. (2020, April). The level of students’ reading comprehension on proof by mathematical induction. In AIP Conference Proceedings (Vol. 2215, No. 1, 060014). AIP Publishing. https://doi.org/10.1063/5.0000564
• Lomonaco, S. J. (t.y.). Class notes on mathematical induction. Publisher: Author.
• Marrades, R., & Gutiérrez, A. (2000). Proofs produced by secondary school students learning geometry in a dynamic computer environment. Educational Studies in Mathematics, 44(1-2), 87-125. https://dx.doi.org/10.1023/A:1012785106627
• Michaelson, M. T. (2008). A literature review of pedagogical research on mathematical induction. Australian Senior Mathematics Journal, 22(2), 57-62. https://eric.ed.gov/?id=EJ819415
• Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook (2nd ed.). Sage Publications, Inc.
• Miyazaki, M. (2000). Levels of proof in lower secondary school mathematics. Educational Studies in Mathematics, 41(1), 47-68. https://dx.doi.org/10.1023/A:1003956532587
• Movshovitz-Hadar, N. (1993a). Mathematical induction: A focus on the conceptual framework. School Science and Mathematics, 93(8), 408-417. https://doi.org/10.1111/j.1949-8594.1993.tb12271.x
• Movshovitz-Hadar, N. (1993b). The false coin problem, mathematical induction and knowledge fragility. Journal of Mathematical Behavior, 12(3), 253-268. https://eric.ed.gov/?id=EJ484115
• Norton, A., Arnold, R., Kokushkin, V., & Tiraphatna, M. (2023). Addressing the cognitive gap in mathematical induction. International Journal of Research in Undergraduate Mathematics Education, 9(4), 295-321. https://doi.org/10.1007/s40753-022-00163-2
• Palla, M., Potari, D., & Spyrou, P. (2012). Secondary school students’ understanding of mathematical induction: structural characteristics and the process of proof construction. International Journal of Science and Mathematics Education, 10(5), 1023-1045. https://doi.org/10.1007/s10763-011-9311-2
• Patton, M. Q. (2014). Nitel araştırma ve değerlendirme yöntemleri (3. baskıdan çeviri) (M. Bütün & S. B. Demir, Çev. Haz.). Pegem Akademi.
• Reid, D. (1992). Mathematical induction: An epistemological study with consequences for teaching (Order No. MM80943) [Master thesis, Concordia University]. ProQuest Dissertations & Theses Global. (304027786). https://www.proquest.com/dissertations-theses/mathematical-induction-epistemological-study-with/docview/304027786/se-2
• Ron, G., & Dreyfus, T. (2004, July). The use of models in teaching proof by mathematical induction. In M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 113-120). Bergen, Norway: Bergen University College.
• Rossi, R. J. (2006). Theorems, corollaries, lemmas, and methods of proof. John Wiley & Sons, Inc.
• Rotman, J. J. (2007). Journey into mathematics: An introduction to proofs. Dover Publications, Inc.
• Stylianides, G. J., Sandefur, J., & Watson, A. (2016). Conditions for proving by mathematical induction to be explanatory. The Journal of Mathematical Behavior, 43, 20-34. https://doi.org/10.1016/j.jmathb.2016.04.002
• Stylianides, G. J., Stylianides, A. J., & Philippou, G. N. (2007). Preservice teachers’ knowledge of proof by mathematical induction. Journal of Mathematics Teacher Education, 10(3), 145-166. http://dx.doi.org/10.1007/s10857-007-9034-z
• Sundstrom, T. (2014). Mathematical reasoning: Writing and proof. Pearson Education, Inc. Şahin, Ç. (2014). Verilerin analizi. R. Y. Kıncal (Ed.), Bilimsel araştırma yöntemleri (3. Baskı) içinde (183-219). Nobel Yayın Dağıtım.
• Telloni, A. I., & Malara, N. A. (2021). A constructive and metacognitive teaching path at university level on the principle of mathematical induction: Focus on the students’ behaviours, productions and awareness. Teaching Mathematics and Computer Science, 19(1), 133-161. https://doi.org/10.5485/TMCS.2021.0525
• Yıldırım, A., & Şimşek, H. (2016). Sosyal bilimlerde nitel araştırma yöntemleri (10. baskı). Seçkin Yayınevi.