Öz

Bu çalışmanın amacı, üstün yetenekli öğrencilerin bir matematiksel ifadeye dair hem doğrulayan hem de çelişen örneklerle karşılaştıklarında matematiksel akıl yürütme biçimlerini ele almaktır. Bu amaçla çalışma, öğrencilerin doğrulayan ve çelişen örnekler oluşturabilecekleri matematiksel ifadeler ile karşılaştıktan sonra geliştirdikleri örnek, genelleme ve argüman türlerini araştırmayı hedeflemektedir. Bilim ve Sanat Merkezi'nde eğitim gören sekiz öğrenci, yarı yapılandırılmış bireysel görüşmelere katılmak üzere gönüllü olmuştur. Çalışmanın bulguları, önerilen örnek türlerinin ve önerilme amaçlarının öğrenciler arasında farklılık gösterdiğini ortaya koymuştur. Öğrenci akıl yürütmesini araştıran çalışmalar, öğrencilerin kanıt şemalarının öğrencilerin matematiksel bir genellemenin geçerliliği için yeterli kabul ettiği örnek uzaya dair mevcut görüşlerini yansıttığını önermektedir. Bu çalışma ise öğrenciler tarafından matematiksel ifadelerin doğruluğunu araştırmak için oluşturulan örnek türlerinin, aynı zamanda öğrencilerin oluşturduğu genelleme ve argüman türleri hakkında bilgi verici olduğunu göstermiştir.

Anahtar Kelimeler: Genelleme, Gerekçelendirme, Matematiksel akıl yürütme, Örnek, Özel yetenekli öğrenciler

Kaynakça

  1. Akkanat, H. (2004). Üstün veya özel yetenekliler. In M. R. Şirin, A. Kulaksızoğlu, & A. E. Bilgili (Eds.), Seçilmiş makaleler kitabı (pp. 167-193). İstanbul: Çocuk Vakfı.
  2. Alcock, L. J. (2004). Uses of example objects in proving. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (pp. 17-24). Bergen, PME 28.
  3. Alcock, L., & Inglis, M. (2008). Doctoral students’ use of examples in evaluating and proving conjectures. Educational Studies in Mathematics, 69(2), 111-129. doi:10.1007/s10649-008-9149-x
  4. Alcock, L., & Weber, K. (2010). Undergraduates’ example use in proof construction: Purposes and effectiveness. Investigations in Mathematics Learning, 3(1), 1-22. doi:10.1080/24727466.2010.11790
  5. Balacheff, N. (1988). Aspects of proof in pupils' practice of school mathematics. In D. Pimm (Ed.), Mathematics, teachers and children (pp. 216-235). London: Hodder & Stoughton.
  6. Balacheff, N. (1991). Treatment of refutations: Aspects of the complexity of a constructivist approach to mathematics learning. In E. von Glaserfeld (Ed.), Radical constructivism in mathematics education (pp. 89-110). Hingham, MA: Kluwer.
  7. Ball, D. L., Hoyles, C., Jahnke, H. N., & Movshovitz-Hadar, N. (2002). The teaching of proof. In L. I. Tatsien (Ed.), Proceedings of the International Congress of Mathematicians (pp. 907-920). Beijing: Higher Education Press.
  8. Ben-Zeev, T., & Star, J. (2001). Intuitive mathematics: Theoretical and educational implications. In B. Torff & R. J. Sternberg (Eds.), Understanding and teaching the intuitive mind: Student and teacher learning (pp. 29-56). Mahwah, NJ: Lawrence Erlbaum Associates Publishers.
  9. Berg, D. H., & McDonald, P. A. (2018). Differences in mathematical reasoning between typically achieving and gifted children. Journal of Cognitive Psychology, 30(3), 281-291. doi:10.1080/20445911.2018.1457034
  10. Blanton, M. L., Levi, L., Crites, T., & Dougherty, B. J. (2011). Developing essential understandings of algebraic thinking for teaching mathematics in grades 3-5. Reston, VA: National Council of Teachers of Mathematics.
  11. Buchbinder, O., & Zaslavsky, O. (2009). A framework for understanding the status of examples in establishing the validity of mathematical statements. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education (pp. 225-232). Thessaloniki: Aristotle University of Thessaloniki.
  12. Budak, I. (2012). Mathematical profiles and problem solving abilities of mathematically promising students. Educational Research and Reviews, 7(16), 344-350. doi:10.5897/ERR12.009
  13. Common Core State Standards for Mathematics. (2010). Common core state standards mathematics. VA: Author.
  14. Dörfler, W. (1991). Forms and means of generalization in mathematics. In A. J. Bishop, S. Mellin-Olsen, & J. Dormolen (Eds.), Mathematical knowledge: Its growth through teaching (pp. 61-85). Dordrecht: Springer Netherlands.
  15. Ellis, A. B., Ozgur, Z., Vinsonhaler, R., Dogan, M. F., Carolan, T., Lockwood, E., … Zaslavsky, O. (2019). Student thinking with examples: the criteria-affordances-purposes strategies framework. The Journal of Mathematical Behavior, 53, 263-283. doi:10.1016/j.jmathb.2017.06.003
  16. English, L., & Warren, E. (1995). General reasoning processes and elementary algebraic understanding: Implications for instruction. Focus on Learning Problems in Mathematics, 77(4), 1-19.
  17. Epp, S. S. (2003). The role of logic in teaching proof. The American Mathematical Monthly, 110(10), 886-899. doi:10.1080/00029890.2003.11920029
  18. Gal, H. (2019). When the use of cognitive conflict is ineffective-problematic learning situations in geometry. Educational Studies in Mathematics, 102, 239-256. doi:10.1007/s10649-019-09904-8
  19. Giannakoulias, E., Mastorides, E., Potari, D., & Zachariades, T. (2010). Studying teachers’ mathematical argumentation in the context of refuting students’ invalid claims. The Journal of Mathematical Behavior, 29, 160-168. doi:10.1016/j.jmathb.2010.07.001
  20. Glaser, B. G., & Strauss, A. L. (1967). The discovery of grounded theory: Strategies for qualitative research. New Brunswick, NJ: Aldine Transaction.
  21. Goldenberg, P., & Mason, J. (2008). Shedding light on and with example spaces. Educational Studies in Mathematics, 69(2), 183-194. doi:10.1007/s10649-008-9143-3
  22. Gorodetsky, M., & Klavirb, R. (2003). What can we learn from how gifted/average pupils describe their processes of problem solving?. Learning and Instruction, 13(3), 305-325. doi:10.1016/S0959-4752(02)00005-1
  23. Hadamard, J. W. (1945). Essay on the psychology of invention in the mathematical field. Princeton, NJ: Princeton University Press.
  24. Harel, G. (1998). Two dual assertions: The first on learning and the second on teaching (or vice versa). The American Mathematical Monthly, 105(6), 497-507. doi:10.1080/00029890.1998.12004918
  25. 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). Norwood, NJ: Ablex Publishing Corporation.
  26. Harel, G., & Sowder, L. (1998). Students' proof schemes: Results from exploratory studies. In A. H. Schoenfeld, J. Kaput, & E. Dubinsky (Eds.), Research in collegiate mathematics education III (pp. 234-283). Providence, RI: American Mathematical Society.
  27. Healy, L., & Hoyles, C. (2000). A study of proof conceptions in algebra. Journal for Research in Mathematics Education, 31(4), 396-428. doi:10.2307/749651
  28. Hong, E., & Aqui, Y. (2004). Cognitive and motivational characteristics of adolescents gifted in mathematics: Comparisons among students with different types of giftedness. Gifted Child Quarterly, 48(3), 191-201. doi:10.1177/001698620404800304
  29. Komatsu, K. (2010). Counter-examples for refinement of conjectures and proofs in primary school mathematics. Journal of Mathematical Behavior, 29(1), 1-10. doi:10.1016/j.jmathb.2010.01.003
  30. Leikin, R. (2021). When practice needs more research: The nature and nurture of mathematical giftedness. ZDM-Mathematics Education, 53, 1579-1589.
  31. Ma, L. (1999). Knowing and teaching elementary mathematics. Teachers' understanding of fundamental mathematics in China and the United States. Erlbaum: Mahwah.
  32. Marton, F. (1986). Phenomenography - a research approach to investigating different understandings of reality. Journal of Thought, 21(3), 28-49. Retrieved from https://www.jstor.org/stable/42589189
  33. Mason, J., Burton, L., & Stacey, K. (2010). Thinking mathematically (2nd ed.). Harlow: Pearson.
  34. Mata-Pereira, J., & da Ponte, J. P. (2017). Enhancing students’ mathematical reasoning in the classroom: teacher actions facilitating generalization and justification. Educational Studies in Mathematics, 96(2), 169-186. doi:10.1007/s10649-017-9773-4
  35. Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook. Thousand Oaks, CA: SAGE.
  36. Ministry of National Education. (2018). Ortaokul matematik dersi (1, 2, 3, 4, 5, 6, 7 ve 8. sınıflar) öğretim programı. Ankara: MEB.
  37. Ministry of National Education. (2022). Science and art centers directive. Retrieved from https://orgm.meb.gov.tr/meb_iys_dosyalar/2022_12/06214921_BIYLIYM_VE_SANAT_MERKEZLERIY_YOYNERGESIY.pdf
  38. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. VA: Author.
  39. Patton, M. Q. (2002). Qualitative research and evaluation methods (3rd ed.). Thousand Oaks, CA: SAGE.
  40. Pedemonte, B., & Buchbinder, O. (2011). Examining the role of examples in proving processes through a cognitive lens: The case of triangular numbers. ZDM Mathematics Education, 43(2), 257-267. doi:10.1007/s11858-011-0311-z
  41. Péter-Szarka, S. (2012). Creative climate as a means to promote creativity in the classroom. Electronic Journal of Research in Education Psychology, 10(28), 1011-1034. doi:10.25115/ejrep.v10i28.1547
  42. Piaget, J. (1975). The child’s conception of the world. Totowa, NJ: Littlefield, Adams.
  43. Sriraman, B. (2004). Gifted ninth graders’ notions of proof: Investigating parallels in approaches of mathematically gifted students and professional mathematicians. Journal for the Education of the Gifted, 27(4), 267-292.
  44. Stavy, R., & Tirosh, D. (1996). Intuitive rules in science and mathematics: The case of ‘more of A - more of B’. International Journal of Science Education, 18(6), 653-667. doi:10.1080/0950069960180602
  45. Stylianides, A. (2007). Proof and proving in school mathematics. Journal for Research in Mathematics Education, 38(3), 289-321. doi:10.2307/30034869
  46. Stylianides, A. J., & Ball, D. L. (2008). Understanding and describing mathematical knowledge for teaching: Knowledge about proof for engaging students in the activity of proving. Journal of Mathematics Teacher Education, 11, 307-332. doi:10.1007/s10857-008-9077-9
  47. Stylianides, A. J., & Stylianides, G. J. (2022). Introducing students and prospective teachers to the notion of proof in mathematics. Journal of Mathematical Behavior, 66, 100957. doi:10.1016/j.jmathb.2022.100957
  48. Stylianides, G. J., & Stylianides, A. J. (2009). Facilitating the transition from empirical arguments to proof. Journal for Research in Mathematics Education, 40(3), 314-352. doi:10.5951/jresematheduc.40.3.0314
  49. Trigwell, K. (2006). Phenomenography: An approach to research into geography education. Journal of Geography in Higher Education, 30(2), 367-372. doi:10.1080/03098260600717489
  50. Yopp, D. A. (2015). Prospective elementary teachers’ claiming in responses to false generalization. The Journal of Mathematical Behavior, 39, 79-99. doi:10.1016/j.jmathb.2015.06.003
  51. Yopp, D. A., Ely, R., Adams, A. E., Nielsen, A. W., & Corwine, E. C. (2020). Eliminating counterexamples: A case study intervention for improving adolescents’ ability to critique direct arguments. The Journal of Mathematical Behavior, 57, 100751. doi:10.1016/j.jmathb.2019.100751
  52. Watson, A., & Mason, J. (2005). Mathematics as a constructive activity: Learners generating examples. Mahwah, NJ: Lawrence Erlbaum Associates.
  53. Wechsler, D. (1975). Intelligence defined and undefined: A relativistic appraisal. American Psychologist, 30(2), 135-139. doi:10.1037/h0076868
  54. Wihlborg, M. (2004). Student nurses' conceptions of internationalisation in general and as an essential part of Swedish nurses' education. Higher Education Research & Development, 23(4), 433-453. doi:10.1080/0729436042000276459
  55. Yıldırım, A., & Şimşek, H. (2006). Sosyal bilimlerde nitel araştirma yöntemleri (6th ed). Ankara: Seçkin Yayıncılık.
  56. Zaskis, R., & Chernoff, E. J. (2008). What makes a counterexample exemplary?. Educational Studies in Mathematics, 68(3), 195-208. doi:10.1007/s10649-007-9110-4
  57. Zeybek Şimşek, Z. (2021). “Is it valid or not?”: Pre-service teachers judge the validity of mathematical statements and student arguments. European Journal of Science and Mathematics Education, 9(2), 26-42. doi:10.30935/scimath/10772

Telif hakkı ve lisans

Telif Hakkı © 2025 Yazar(lar). Açık erişimli bu makale, orijinal çalışmaya uygun şekilde atıfta bulunulması koşuluyla, herhangi bir ortamda veya formatta sınırsız kullanım, dağıtım ve çoğaltmaya izin veren Creative Commons Atıf Lisansı (CC BY) altında dağıtılmıştır.

Nasıl atıf yapılır

Zeybek Şimşek, Z., & Kılıçoğlu, E. (2025). Öğrencilerin Doğrulayan ve Çelişen Örnekler İçeren Matematiksel bir İfade Karşısındaki Akıl Yürütmelerinin Araştırılması. Eğitim Ve Bilim, 50(222), 45-66. https://doi.org/10.15390/EB.2025.12962