Öz

Bu çalışma, öğretmenlerin, salt sanal ib sayısının özellikle Kartezyen formunu anlamalarını incelemiştir. Bu çalışma, beş öğretmenin karmaşık sayıların çeşitli formlarına ilişkin anlayışlarını keşfetmeyi amaçlayan bir mesleki gelişim (MG) programının geliştirilmesini içeren kapsamlı bir tasarım-tabanlı araştırma çalışmasının bir parçasıdır. Veri toplama süreci, MG öncesi ve sonrası yazılı oturumlar ile MG sonrasında gerçekleştirilen yarı yapılandırılmış görüşmeleri içermektedir. Veriler, öğretmenlerin i'yi kavramsallaştırmalarındaki değişime işaret etmektedir ve başlangıçta bazı öğretmenlerin i hakkında cebirsel veya geometrik olarak muhakeme yapamadıklarını göstermiştir. Ancak, MG'nin tamamlanmasının ardından, tüm katılımcılar i'yi x2+1=0 ikinci dereceden denkleminin köklerinden biri olarak tanımlamış ve geometrik olarak karmaşık düzlemde (0,1) noktası olarak temsil edebilmiştir. Ayrıca, tüm katılımcılar i'nin 90 derecelik bir dönme olarak operatör yorumunu farketmiştir. Bir katılımcı ayrıca b reel sayısı i ile çarpıldığında b'nin genişletme anlamına dikkat çekmiş ve bir başka katılımcı da tekrarlı toplama anlamı üzerinde durmuştur. Sonuçlar ayrıca öğretmenlerin salt sanal sayıyı kavramsallaştırırken karşılaştıkları belirli zorlukları da vurgulamıştır. Sonuçlar toplu olarak, öğretmen eğitiminde karmaşık sayıların Kartezyen formunun salt sanal kısmının ve karmaşık sayıların operatör anlamlarının ele alınmasının öneminin altını çizmektedir. Ayrıca bu sonuçlar, nicel muhakemenin, i birimi de dahil olmak üzere karmaşık sayıları anlamlandırmak için temel bir düşünme biçimi olarak hizmet edebileceğini göstermektedir.

Anahtar Kelimeler: Karmaşık sayıların Kartezyen formu, Öğretmen bilgisi, Nicel muhakeme

Kaynakça

  1. Anevska, K., Gogovska, V., & Malcheski, R. (2015). The role of complex numbers in interdisciplinary integration in mathematics teaching. Procedia-Social and Behavioral Sciences, 191, 2573-2577. https://doi.org/10.1016/j.sbspro.2015.04.553
  2. Atmaca, U. İ., Uygun, N. G., Çağlar, M. F., & Çünür, I. (2014, August). Haberleşme mühendisliği için mobil öğrenme uygulaması. Paper presented at the VII. URSI-Türkiye Bilimsel Kongresi, Elazığ. https://www.researchgate.net/profile/Mehmet-Caglar/publication/280732542_Haberlesme_Muhendisligi_icin_Mobil_Ogrenme_Uygulamasi/links/55c3aa1a08aeb97567401a6f/Haberlesme-Muehendisligi-icin-Mobil-Oegrenme-Uygulamasi.pdf
  3. Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special?. Journal of Teacher Education, 59(5), 389-407. https://doi.org/10.1177/0022487108324554
  4. Benítez, J., Giménez, M. H., Hueso, J. L., Martínez, E., & Riera, J. (2013). Design and use of a learning object for finding complex polynomial roots. International Journal of Mathematical Education in Science and Technology, 44(3), 365-376. https://doi.org/10.1080/0020739X.2012.729678
  5. Caglayan, G. (2016). Mathematics teachers' visualization of complex number multiplication and the roots of unity in a dynamic geometry environment. Computers in the Schools, 33(3), 187-209. https://doi.org/10.1080/07380569.2016.1218217
  6. Clement, J. (2000). Analysis of clinical interviews: Foundations and model viability. In A. E. Kelly & R. A. Lesh (Eds.), Research design in mathematics and science education (pp. 547-589). Lawrence Erlbaum Associates, Inc.
  7. Common Core State Standards Initiative. (2010). Common Core State Standards for mathematics (Report No. ED522005). https://learning.ccsso.org/wp-content/uploads/2022/11/ADA-Compliant-Math-Standards.pdf
  8. Conference Board of the Mathematical Sciences. (2012). The mathematical education of teachers II. American Mathematical Society. https://doi.org/10.1090/cbmath/017
  9. Conner, E., Rasmussen, C., Zandieh, M., & Smith, M. (2007). Student understanding of complex numbers. Proceedings of the 10th Annual Conference on Research in Undergraduate Mathematics Education. http://sigmaa.maa.org/rume/crume2007/papers/conner-rasmussen-zandieh-smith.pdf
  10. Çelik, A., & Özdemir, M. F. (2011). Ortaöğretimde Kompleks sayılarla ilgili kavram yanılgılarının belirlenmesi ve çözüm önerileri. Dokuz Eylül Üniversitesi Buca Eğitim Fakültesi Dergisi, (29), 203-229.
  11. Edwards, T., Özgün-Koca, S. A., & Chelst, K. (2021). Visualizing complex roots of a quadratic equation. Mathematics Teacher: Learning and Teaching PK-12, 114(3), 238-243. https://doi.org/10.5951/MTLT.2020.0028
  12. Fauconnier, G., & Turner, M. (2002). The way we think: Conceptual blending and the mind’s hidden complexities. Basic Books.
  13. Fischbein, E., Deri, M., Nello, M. S., & Marino, M. S. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education, 16(1), 3-17. https://doi.org/10.5951/jresematheduc.16.1.0003
  14. Gaona, J., López, S. S., & Montoya-Delgadillo, E. (2022). Prospective mathematics teachers learning complex numbers using technology. International Journal of Mathematical Education in Science and Technology, 55(9), 2219-2248. https://doi.org/10.1080/0020739X.2022.2133021
  15. Glas, E. (1998). Fallibilism and the use of history in mathematics education. Science & Education, (7), 361-379. https://doi.org/10.1023/A:1008695214877
  16. Harding, A., & Engelbrecht, J. (2007). Sibling curves and complex roots 1: Looking back. International Journal of Mathematical Education in Science and Technology, 38(7), 963-973. https://doi.org/10.1080/00207390701564680
  17. Harel, G. (2013). DNR-based curricula: The case of complex numbers. Journal of Humanistic Mathematics, 3(2), 2-61. https://doi.org/10.5642/jhummath.201302.03
  18. Hedden, C. B., & Langbauer, D. (2003). Balancing problem-solving skills with symbolic manipulation skills. In H. L. Schoen & R. I. Charles (Eds.), Teaching mathematics through problem solving: Grades 6-12 (pp. 155-159). National Council of Teachers of Mathematics.
  19. Hoch, M., & Dreyfus, T. (2004). Equations - a structural approach. In I. M. J. Høines & A. B. Fuglestad (Eds.), Proceedings of the 28th Conference of The International Group for the Psychology of Mathematics Education (Vol. 1, pp. 152-155). Bergen University College.
  20. Karagöz Akar, G., Belin, M., Arabacı N., İmamoglu, Y., & Akoğlu, K. (2023a). Knowledge of different forms of complex numbers through quantitative reasoning: The case of teachers. In P. Drijvers, C. Csapodi, H. Palmér, K. Gosztonyi, & E. Kónya (Eds.), Proceedings of the Thirteenth Congress of the European Society for Research in Mathematics Education (CERME13) (pp. 3785-2792). Alfréd Rényi Institute of Mathematics and ERME.
  21. Karagöz Akar, G., Belin, M., Arabacı, N., İmamoğlu, Y., & Akoğlu, K. (2023b). Different meanings of pure imaginary numbers. In M. Shelley, O. T. Ozturk, & M. L. Ciddi (Eds.), Proceedings of ICEMST 2023-International Conference on Education in Mathematics, Science and Technology (pp. 259-277). ISTES.
  22. Karagöz Akar, G., Belin, M., Arabacı, N., İmamoğlu, Y., & Akoğlu, K. (2024). Teachers’ knowledge of different forms of complex numbers through quantitative reasoning. Mathematical Thinking and Learning, 1-25. https://doi.org/10.1080/10986065.2024.2378910
  23. Karagöz Akar, G., Sarac, M. & Belin, M. (2023). Exploring prospective teachers’ development of the Cartesian form of complex numbers. Mathematics Teacher Educator 12(1), 49-69. https://doi.org/10.5951/MTE.2022.0034
  24. Karagöz Akar, G., Watanabe, T., & Turan, N. (2022). Quantitative Reasoning as a Framework to Analyze Mathematics Textbooks. In G. Karagöz Akar, İ. Ö. Zembat, S. Arslan, & P. W. Thompson (Eds.), Quantitative reasoning in mathematics and science education (pp. 107-132). Springer, Cham. https://doi.org/10.1007/978-3-031-14553-7_5
  25. Karagöz Akar, G., Zembat, İ. Ö., Arslan, S., & Belin, M. (2022). Revisiting geometric transformations through quantitative reasoning. In G. Karagöz Akar, İ. Ö. Zembat, S. Arslan, & P. W. Thompson (Eds.), Quantitative reasoning in mathematics and science education (pp. 199-219). Springer, Cham. https://doi.org/10.1007/978-3-031-14553-7_8
  26. Karakok, G., Soto-Johnson, H., & Dyben, S. A. (2015). Secondary teachers’ conception of various forms of complex numbers. Journal of Mathematics Teacher Education, 18(4), 327-351. https://doi.org/10.1007/s10857-014-9288-1
  27. Karam, R. (2020). Why are complex numbers needed in quantum mechanics? Some answers for the introductory level. American Journal of Physics, 88(1), 39-45. https://doi.org/10.1119/10.0000258
  28. Karyağdı, B. (2022). Investigating preservice teachers’ content knowledge on multiplication and division (Thesis No. 764493) [Master’s thesis, Boğaziçi University]. Council of Higher Education National Thesis Center.
  29. Kontorovich, I. (2018a). Undergraduate’s images of the root concept in R and in C. The Journal of Mathematical Behavior, (49), 184-193. https://doi.org/10.1016/j.jmathb.2017.12.002
  30. Kontorovich, I. (2018b). Roots in real and complex numbers: A case of unacceptable discrepancy. For the Learning of Mathematics, 38(1), 17-19. https://www.jstor.org/stable/26548478
  31. Kontorovich, I., Zazkis, R., & Mason, J. (2021). The metaphor of transition for introducing learners to new sets of numbers. In Y. H. Leong, B. Kaur, B. H. Choy, J. B. W. Yeo, & S. L. Chin (Eds.), Excellence in mathematics education: foundations and pathways (pp. 259-264). MERGA.
  32. Melliger, C. (2007). How to graphically interpret the complex roots of a quadratic equation (University of Nebraska–Lincoln MAT Exam Expository Papers No. 35). University of Nebraska–Lincoln. https://digitalcommons.unl.edu/mathmidexppap/35
  33. Moore, K. C., Carlson, M. P. & Oehrtman, M. (2009). The role of quantitative reasoning in solving applied precalculus problems. Paper presented at the Twelfth Annual Conference on Research in Undergraduate Mathematics Education, Atlanta, GA.
  34. Murray, N. T. (2018). Making imaginary roots real. The Mathematics Teacher, 112(1), 28-33. https://doi.org/10.5951/mathteacher.112.1.0028
  35. Nachlieli, T., & Elbaum-Cohen, A. (2021). Teaching practices aimed at promoting meta-level learning: The case of complex numbers. The Journal of Mathematical Behavior, (62), 1-13. https://doi.org/10.1016/j.jmathb.2021.100872
  36. Nahin, P. J. (2010). An imaginary tale: The story of √-1. Princeton University Press. https://doi.org/10.1515/9781400833894
  37. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Author.
  38. Nemirovsky, R., Rasmussen, C., Sweeney, G., & Wawro, M. (2012). When the classroom floor becomes the complex plane: Addition and multiplication as ways of bodily navigation. Journal of the Learning Sciences, 21(2), 287-323. https://doi.org/10.1080/10508406.2011.611445
  39. Nordlander, M. C., & Nordlander, E. (2012). On the concept image of complex numbers. International Journal of Mathematical Education in Science and Technology, 43(5), 627-641. https://doi.org/10.1080/0020739X.2011.633629
  40. Novotná, J., Stehlíková, N., & Hoch, M. (2006). Structure sense for university algebra. In J. Novotná, H. Moraová, M. Krátká, & N. Stehlíková (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 249-256). Charles University in Prague.
  41. Panaoura, A., Elia, I., Gagatsis, A., & Giatilis, G. P. (2006). Geometric and algebraic approaches in the concept of complex numbers. International Journal of Mathematical Education in Science and Technology, 37(6), 681-706. https://doi.org/10.1080/00207390600712281
  42. Saldanha, L., & Thompson, P. W. (1998). Re-thinking co-variation from a quantitative perspective: Simultaneous continuous variation. In S. B. Berenson & W. N. Coulombe (Eds.), Proceedings of the Annual Meeting of the Psychology of Mathematics Education - North America (Vol 1, pp. 298-304). North Carolina State University.
  43. Saraç, M. (2016). A prospective secondary mathematics teacher's development of the meaning of the cartesian form of complex numbers (Thesis No. 459451) [Master’s thesis, Boğaziçi University]. Council of Higher Education National Thesis Center.
  44. Saraç, M., & Karagöz Akar, G. (2017). A prospective secondary mathematics teacher’s development of the meaning of complex numbers through quantitative reasoning. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 267-271). Hoosier Association of Mathematics Teacher Educator.
  45. Seloane, P. M., Ramaila, S., & Ndlovu, M. (2023). Developing undergraduate engineering mathematics students' conceptual and procedural knowledge of complex numbers using GeoGebra. Pythagoras, 44(1), 1-14. https://doi.org/10.4102/pythagoras.v44i1.763
  46. Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational Studies in Mathematics, 22(1), 1-36. https://doi.org/10.1007/BF00302715
  47. Simon, M. (2000). Research on mathematics teacher development: The teacher development experiment. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education (pp. 335-359). Routlege.
  48. Smith III, J. P., & Thompson, P. W. (2007). Quantitative reasoning and the development of algebraic reasoning. In J. J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95-132). Erlbaum.
  49. Soto-Johnson, H., & Troup, J. (2014). Reasoning on the complex plane via inscriptions and gesture. The Journal of Mathematical Behavior, (36), 109-125. https://doi.org/10.1016/j.jmathb.2014.09.004
  50. Stevens, I. E. (2019). Pre-service teachers' constructions of formulas through covariational reasoning with dynamic objects (Thesis No. 14794) [Doctoral dissertation, University of Georgia]. https://openscholar.uga.edu/record/14794?v=pdf
  51. Stevens, I. E., & Moore, K. C. (2017). The intersection between quantification and an allencompassing meaning for a graph. In E. Galindo & J. Newton (Eds.), Proceedings of the 39th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 709-716). Hoosier Association of Mathematics Teacher Educators.
  52. Tekin, D. (2019). A model for developing the multiplication of complex numbers (Thesis No. 597960) [Master’s thesis, Boğaziçi University]. Council of Higher Education National Thesis Center.
  53. Thompson, P. W. (1990, March). A theoretical model of quantity-based reasoning in arithmetic and algebra. Paper presented at the Annual Meeting of the American Educational Research Association, San Francisco, CA. https://pat-thompson.net/PDFversions/1990TheoryQuant.pdf
  54. Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Vergnaud, G. Harel, & J. Coufrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 179-234). SUNY Press.
  55. Thompson, P. W. (2011). Quantitative reasoning and mathematical modeling. In L. L. Hatfield, S. Chamberlain, & S. Belbase (Eds.), New perspectives and directions for collaborative research in mathematics education, WISDOMe Monographs (Vol. 1, pp. 33-57). University of Wyoming.
  56. Thompson, P. W. (2016). Researching mathematical meanings for teaching. In L. English & D. Kirshner, (Eds.), Handbook of international research in mathematics education (pp. 435-461). Taylor and Francis.
  57. Thompson, P. W., & Carlson, M. P. (2017). Variation, covariation, and functions: Foundational ways of thinking mathematically. In J. Cai (Ed.), Compendium for research in mathematics education (pp. 421-456). National Council of Teachers of Mathematics.
  58. Thompson, P. W., Hatfield, N. J., Yoon, H., Joshua, S., & Byerley, C. (2017). Covariational reasoning among US and South Korean secondary mathematics teachers. The Journal of Mathematical Behavior, 48, 95-111. https://doi.org/10.1016/j.jmathb.2017.08.001
  59. Thompson, P. W., & Saldanha, L. A. (2003). Fractions and multiplicative reasoning. In Research companion to the principles and standards for school mathematics (pp. 95-113). National Council of Teachers of Mathematics.
  60. Usiskin, Z., Peressini, A. L., Marchisotto, E. A., & Stanley, D. (2003). Mathematics for high school teachers: An advanced perspective. Prentice Hall.
  61. Warren, E., Trigueros, M., & Ursini, S. (2016). Research on the learning and teaching of algebra. In Á. Gutiérrez, G. Leder, & P. Boero (Eds.), The second handbook of research on the psychology of mathematics education (pp. 73-108). Brill. https://doi.org/10.1007/978-94-6300-561-6_3
  62. Yin, R. K. (2009). Case study research: Design and methods (5th ed.). Sage.

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

Karagöz Akar, G., Belin, M., Arabacı, N., İmamoğlu Y., & Akoğlu, K. (2025). Öğretmenlerin salt sanal sayıların farklı anlamlarına ilişkin kavramsallaştırmaları. Eğitim Ve Bilim, 50(223), 199-222. https://doi.org/10.15390/ES.2025.2570