MATEMATİK ÖĞRETMENİ ADAYLARININ İRRASYONEL SAYILARLA İLGİLİ ANLAYIŞLARI

Öz

Bu çalışmanın amacı matematik öğretmeni adaylarının, literatürde var olan ancak ders kitaplarında Calculus, Analiz, Cebir, vb… pek yer verilmeyen ve bir sayının irrasyonel olup olmadığını göstermek için kullanılan farklı çözüm yolları ile ilgili anlayışlarını ve yaklaşımlarını belirlemektir. Bu bağlamda çalışmanın katılımcılarını 40 matematik öğretmeni adayı the fourth-year students oluşturmaktadır. Çalışma betimsel niteliktedir. Çalışmanın verileri açık uçlu sorulardan oluşan bir testten elde edilmiştir. Elde edilen sonuçlardan matematik öğretmeni adaylarına ilk etapta uygulanan testin analiziyle, bir sayının irrasyonelliğinin gösterimi noktasında sınırlı sayıda yolun kullanıldığı ortaya çıkmıştır. İkinci testin sonuçlarından ise katılımcıların irrasyonel sayıların gösterimi ile ilgili farklı çözüm yollarını tercih ettikleri görülmüştür.

Anahtar Kelimeler

• İrrasyonel Sayılar,Matematik Öğretmeni Adayı,Farklı Çözüm Yolları

Pre-service Mathematics Teachers’ Understandings of Irrationality Numbers

Abstract

The aim of this study is to determine pre-service mathematics teachers’ understandings and approaches of different methods of solution which exist in literature but are not featured much in textbooks Calculus, Analysis, Algebra, etc. , and which are used to show whether or not a number is irrational. In this regard, the participants of the study are composed of 40 pre-service mathematics teachers. The study is of descriptive quality. The data of the study were obtained via a test composing of open-ended questions. In view of the obtained data and via analysis of the test that was administered to the pre-service mathematics teachers in the first step, it was found that a limited number of methods were used to represent the irrationality of a number. In view of the results of the second test, it was observed that the participants preferred different methods of solution regarding the representation of irrational numbers

Keywords

• Irrational numbers,Pre-service mathematics teacher,Different methods of solution

References

1. Arcavi, A., Bruckheimer, M. and Ben-Zvi, R. (1987). History of mathematics for teachers: The case of irrational numbers. For the Learning of Mathematics, 7(2), 18–23.
2. Baki, A. (2008). Kuramdan uygulamaya matematik eğitimi. Ankara: Harf Yayınları.
3. Berge, A. (2008). The completeness property of the set of real numbers in the transition from calculus to analysis. Educational Studies in Mathematics, 67, 217 –235.
4. Fischbein, E., Jehiam, R. & Cohen, D. (1995). The concept of irrasyonel numbers in high
5. school students and prospective teachers. Educational Studies in Mathematics, 29(1), 29-44.
6. Giannakoulias, E., Souyoul, A. & Zachariades, T. (2007). Students’ thinking about fundamental real numbers properties. In D. Pitta-Pantazi & G. Philippou (Eds.), Proceedings of the Fifth Congress of the European Society for Research in Mathematics Education (pp.416-425), Cyprus: CERME, Department of Education, University of Cyprus.
7. Gözen, Ş. (2006). Matematik ve öğretimi, İstanbul: Evrim Yayınevi.
8. Güven, B., Çekmez, E. & Karataş, I. (2011). Examining preservice elementary mathematics teachers’ understandings about irrational numbers. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies, 21(5), 401-416.

9. Lee, Y. R. (2006). A case study on the introducing methods of the irrational numbers based on the Freudenthal's Mathematising Instruction. In Novotná, J., Moraová, H., Krátká, M. & Stehlíková, N. (Eds). Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, (1), (p. 277), July 16-21, Prague, Czech Republic.
10. Merenluoto, K. & Lehtinen, E. (2004). Number concept and conceptual change towards systemic model of the processes of change. Learning and Instruction, 14, 519- 539.
11. Moseley, B. (2005). Students’ early Mathematical Representation knowledge: The effects of emphasizing single of multiple perspectives of the rational number domain in problem solving, Educational Studies in Mathematics, 60(1), 37-69.
12. Peled, I. & Hershkovitz, S. (1999). Difficulties in knowledge integration: Revisiting Zeno’s paradox with irrational numbers. International Journal of Mathematical Education in Science and Technology, 30(1), 39–46.
13. Sertöz, S. (2002). Matematiğin aydınlık dünyası, TÜBİTAK popüler bilim kitapları 36, Ankara: Semih ofset.
14. Shinno, Y. (2007). On the teaching situation of conceptual change: Epistemological considerations of irrational numbers. In J. H. Woo, K. S. Park, & D. Y. Seo (Eds.), Proceedings of the 31st. Annual General Meeting of the International Group for the Psychology of Mathematics Education (PME31), 4, 185-192.
15. Sirotic, N. & Zazkis, R. (2007a). İrrational numbers: The gap between formal and intuitive knowledge. Educational Studies in Mathematics, 65, 49-76.
16. Sirotic, N. & Zazkis, R. (2007b). İrrational numbers on the number line – where are they? International Journal of Mathematical Education in Science and Tecnology, 38(4), 477-468.
17. Soylu, Y., Akgün, L., Dündar, S. & İşleyen, T. (2011). Bir sayının irrasyonel sayı olduğunu farklı bir metotla gösterme. Procedia Social and Behavioral Sciences 15, 3277–3280, http://dx.doi.org/10.1016/j.sbspro.2011.04.285
18. Tall, D. O. (2001). Natural and formal infinities. Educational Studies in Mathematics, 48(2-3), 199-238.
19. Tall, D. O. & Schwarzenberger, R. L. E. (1978). Conflicts in the learning of real numbers and limits. Mathematics Teaching, 82, 44–49.
20. Tiketar, V. G. (2007). Seven different proofs of the irrationality of . Resonance, 12(12), 31-39.
21. Tirosh, D., Fischbein, E., Graeber, A. & Wilson, J. (1998). Prospective elementary teachers’ conceptions of rational numbers. Retrieved March 5th, 2012 from the World http://jwilson.coe.uga.edu/Texts.Folder/Tirosh/Pros.El.Tchrs.html. Wide Web
22. Tirosh, D. & Tsamir, P.(2006). Conceptual change in mathematics learning: The case of infinite sets. Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education, 1, p.159, July 16-21, Prague, Czech Republic.
23. Vamvakoussi, X. & Vosniadou, S .(2004). Understanding the structure of the set of rational numbers: a conceptual change approach. Learning and Instruction, 14, 453–467.
24. 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, V. Baltas, & X. Vamvakoussi, (Eds), Re-framing the conceptual change approach in learning and instruction (pp. 265-282). Amsterdam: Elsevier.
25. Zachariades, T., Christou, C. & Pitta-Pantazi, D. (2013). Reflective, systemic and analytic thinking in real numbers. Educational Studies in Mathematics, 82, 5-22.
26. Zazkis, R. (2005). Representing numbers: Prime and irrational. International Journal of Mathematical Education in Science and Technology, 36(2–3), 207–218.
27. Zazkis, R. & Sirotic, N. (2004). Making sense of irrational numbers: Focusing on representation. In M.J. Hoines, A.B. Fuglestad (eds.), Proceedings of 28th International Conference for Psychology of Mathematics Education, 4, 497–505, Bergen, Norway.