MATEMATİK ÖĞRETMEN ADAYLARININ GERÇEL SAYILARIN TAMLIK ÖZELLİĞİNE İLİŞKİN KAVRAYIŞLARININ İNCELENMESİ

Öz

Son yıllarda kavram öğretimi ve kavramsal anlama vurgusu matematik eğitimi müfredatlarında merkezi rol oynamaktadır. Sayı sistemlerinin kavramsallaştırılması matematiğin tüm alanlarının anlaşılması bakımından önemlidir. Gerçel sayıların kavramsallaştırılmasında yaşanan güçlüklere bakıldığında, öğrencilerin gerçel sayıların diğer sayı sistemleriyle olan ilişkilerini anlamakta zorlandıkları görülmektedir. Bu çalışmanın amacı matematik öğretmen adaylarının gerçel sayı kümesini diğer sayı sistemlerinden ayıran en temel unsur olan tamlık özelliğine ilişkin kavrayışlarını incelemektir. Bu amaç doğrultusunda bireyin bir kavrama ilişkin zihinsel yapılarını ve mekanizmalarını ortaya koyan APOS teorisi kullanılmıştır. Çalışmanın verileri üç öğretmen adayıyla yarı-yapılandırılmış görüşmeler yapılarak elde edilmiştir. Veriler betimsel analiz yöntemiyle incelenmiş ve analiz sonucunda bulgular belirlenen iki ana temaya göre, gerçel sayıların tamlık özelliği kavramının epistemolojisi ve gerçel sayıların tamlık özelliğine ilişkin şemalarda yer alan zihinsel yapılar olarak sunulmuştur. Bulgulara bakıldığında öğretmen adaylarının gerçel sayıların tamlık özelliğine ilişkin kavrayışlarının çoğunlukla eylem düzeyinde olduğu görülmüştür. Buna bağlı olarak gerçel sayıların tamlık özelliğine ilişkin APOS teorisi bağlamında oluşturulacak genetik ayrışımda yer alması gereken rasyonel sayıların bir doğru üzerinde temsil edilmesi gibi zihinsel yapılar belirlenmiştir.

Anahtar Kelimeler

• Tamlık özelliği
• kavramsal anlama
• kavrayış
• APOS teorisi

INVESTIGATING PRE-SERVICE MATHEMATICS TEACHERS' CONCEPTIONS OF THE PROPERTY OF COMPLETENESS OF REAL NUMBERS

Öz

In recent years, the emphasis on concept teaching and conceptual understanding has played a central role in mathematics education curricula. The conceptualization of number systems is important to understand all areas of mathematics. Considering the difficulties experienced in the conceptualization of real numbers, it is seen that students have difficulty in understanding the relationship between real numbers and other number systems. This study aims to investigate pre-service mathematics teachers' conceptions of the property of completeness, which is the most basic element that distinguishes a real number set from other number systems. To this end, the APOS theory, which reveals the mental structures and mechanisms of the individual regarding a concept, was used. Data were collected via semi-structured interviews with three pre-service teachers. The data were analyzed using the descriptive analysis method, and the findings were presented under two themes, which are the epistemology of the concept of the property of completeness of real numbers and the mental structures in the schemas related to the property of completeness of real numbers. The findings revealed that the pre-service teachers' conceptions of the property of completeness of real numbers was mostly at the action level. The study found some mental structures such as the representation of rational numbers on a line, which should be included in the genetic decomposition to be created in the context of the APOS theory regarding the completeness of real numbers.

Anahtar Kelimeler

• Completeness axiom
• conceptual understanding
• conception
• APOS theory

Kaynakça

1. Arnon, I., Cottrill, J., Dubinsky, E., Oktaç, A., Roa Fuentes, S., Trigueros, M., Weller, K. (2014). APOS Theory: A framework for research and curriculum development in mathematics education. New York: Springer.
2. Asiala, M., Brown, A., DeVries, D., Dubinsky, E., Mathews, D., & Thomas, K. (1996). A framework for research and curriculum development in undergraduate mathematics education. In Research in Collegiate mathematics education II. CBMS issues in mathematics education (Vol. 6, pp. 1–32). Providence, RI: American Mathematical Society.
3. Awodey, S., & Reck, E. H. (2002). Completeness and categoricity, part I: 19th century axiomatics to 20th century metalogic. History and Philosophy of Logic, 23, 1–30.
4. Berge, A. (2008). The completeness property of the set of real numbers in the transition from calculus to analysis, Educational Studies in Mathematics. 67, pp. 217–235.
5. Bergé, A. (2010). Students’ perceptions of the completeness property of the set of real numbers. International Journal of Mathematical Education in Science and Technology, 41(2), 217–227.
6. Bosch, M., Gascon, J., & Trigueros, M. (2017). Dialogue between theories interpreted as research praxeologies: the case of APOS and the ATD. Educational Studies in Mathematics, 95, 39–52.
7. Dubinsky, E. (Eds.) (1991). Reflective abstraction in advanced mathematical thinking, Advanced mathematical thinking (pp. 95-123). Dordrecht. The Netherlands: Kluwer.
8. Dubinsky, E., Weller, K., Mcdonald, M.A., Brown, A. (2005). Some Historical Issues and Paradoxes Regarding the Concept of Infinity: An Apos-Based Analysis: Part 1. Educational Studies in Mathematics, 58, 335–359.

9. Durand-Guerrier, V., Montoya Delgadillo, E., & Vivier, L. (2019). Real exponential in discreteness-density-completeness contexts. Calculus in upper secondary and beginning university mathematics, University of Agder, Kristiansand, Norway, August 6-9, 2019.
10. Fischbein, E., Jehiam, R., & Cohen, D. (1995). The concept of irrational number in high-school students and prospective teachers, Educational Studies in Mathematics. 9, pp. 29–44.
11. Font, V., Trigueros, M., Badillo, E., & Rubio, N. (2016). Mathematical objects through the lens of two different theoretical perspectives: APOS and OSA. Educational Studies in Mathematics, 91(1), 107–122.
12. Gray, E., & Tall, D. (1994). Duality, ambiguity and flexibility: A proceptual view of simple arithmetic. Journal for Research in Mathematics Education, 26(2), 115–141.
13. Malara, N. (2001). From fractions to rational numbers in their structure: Outlines for an innovative didactical strategy and the question of density. In J. Novotná (Ed.), Proceedings of the 2nd Conference of the European Society for Research Mathematics Education (pp. 35–46). Praga: Univerzita Karlova v Praze, Pedagogická Faculta.
14. Maschietto, M. (2002). L’enseignement de l’analyse au lycée: les débuts du jeu global/local dans l’environment de calculatrices. Thèse doctorale, Université Paris VII.
15. McDonald, M. A., Mathews, D., & Strobel, K. (2000). Understanding sequences: A tale of two objects. Research in collegiate mathematics education IV, 8, 77-102.
16. Merenluoto, K., & Lehtinen, E. (2002). Conceptual change in mathematics: Understanding the real numbers. In M. Limón & L. Mason (Eds.), Reconsidering conceptual change: Issues in theory and practice (pp. 233–257). Dordrecht: Kluwer.
17. Ministry of National Education [MoNE]. (2009). İlköğretim matematik dersi 6–8. sınıflar öğretim programı. Ankara: Milli Eğitim Basımevi.
18. Ministry of National Education [MoNE]. (2013). Ortaöğretim matematik dersi (9, 10, 11 ve 12. sınıflar) öğretim programı. Ankara.
19. Ministry of National Education [MoNE]. (2018). Matematik dersi öğretim programı. Ankara: Milli Eğitim Bakanlığı Talim ve Terbiye Kurulu Başkanlığı, Ankara.
20. National Council of Teachers of Mathematics [NCTM]. (2000). Principles and standards for school mathematics. Reston, VA: NCTM.
21. National Council of Teachers of Mathematics [NCTM]. (2006). Curriculum focal points. Reston, VA: Author.
22. Neumann, R. (1998). Students’ ideas on the density of fractions. In H. G. Weigand, A. Peter Koop, N. Neil, K. Reiss, G. Törner, & B. Wollring (Eds.), Proceedings of the Annual Meeting of the Gesellschaft fur Didaktik der Mathematik on Didactics of Mathematics (pp. 97–104). Munich: Gesellschaft fur Didaktik der Mathematik.
23. Pantziara, M. & Philippou, G. N. (2012). Levels of students’ “conception” of fractions. Educational Studies in Mathematics, 79(1), 61–83.
24. Patton, M. K. (1987). How to use qualitative methods in evaluation. Newbury Park: SAGE publications.
25. 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, 1–36.
26. Stenger, C., Weller, K., Arnon, I., Dubinsky, E., & Vidakovic, D. (2008). A search for a constructivist approach for understanding the uncountable set P(N). Revista Latinoamericana De Investigacion En Matematica Educativa-Relime, 11(1), 93-125.
27. Thomas, G. B., Finney, R. L., Weir, M. D., & Giordano, F. R. (2003). Thomas’ calculus. USA: Addison Wesley.
28. Uzun Erdem Ö. & Dost Ş. (2023). Content Analysis of Qualitative Studies on Irrational Numbers in Turkey: A Meta-Synthesis Study. Mehmet Akif Ersoy Üniversitesi Eğitim Fakültesi Dergisi, (65), 1-11.
29. Vamvakoussi, X. & Vosniadou, S. (2007). How Many Numbers are there in a Rational Numbers Interval? Constrains, Synthetic Models and the Effect of the Number Line. In S. Vosniadou, A. Baltas, & X. Vamvakoussi (Eds.) Reframing the Conceptual Change Approach in Learning and Instruction (pp. 265-282). The Netherlands: Elsevier.
30. Yıldırım, A., Şimşek, H. (2013). Sosyal bilimlerde nitel araştırma yöntemleri (9. Baskı). Ankara: Seçkin Yayıncılık.
31. Yin, R. K. (2003). Case Study Research: Design and Methods (3rd edition). Sage Publications, Thousand Oaks, CA.