Ortaokul Matematik Öğretmen Adaylarının Belirli İntegral ve Belirsiz İntegral Tanımlamaları

Abstract

Araştırmacılar analizin matematik, matematik eğitimi ve diğer disiplinlerdeki önemini ve bu dersin temel kavramlarından biri olan integrale ilişkin öğrencilerin kavramsal anlamalarının geliştirilmesi gerektiğini vurgulamaktadır. Bu nedenle, bu çalışmanın amacı, ortaokul matematik öğretmen adaylarının belirli integrali ve belirsiz integrali nasıl tanımladıklarını ve ayrıca belirli integral ve belirsiz integral arasındaki ilişkiyi ne derece ifade edebildiklerini araştırmaktır. Bu amaçla, 173 ortaokul matematik öğretmen adayından üç soruyu cevaplamaları istenmiştir. Elde edilen bulgulara göre, ortaokul matematik öğretmen adayları hem belirli hem de belirsiz integrali tanımlarken benzer kavramlara değinmektedir. Bu kavramlar sınır, notasyon, matematiksel formül, örnek verme, alan, hacim, ters türev, hesaplama süreci, sonucun yapısı (sayı, fonksiyon, cebirsel ifade veya bilinmeyen) ve c sabiti olarak sınıflandırılmıştır. Ayrıca her bir integral türü için en çok bahsedilen kavramın sınır olduğu görülmüştür. Ancak, katılımcıların azınlığının cevaplarında belirli integral ve belirsiz integral arasındaki ilişkiye dair ifadeler bulunmuştur.

Keywords

• Belirli integral
• Belirsiz integral
• Analiz Ortaokul matematik öğretmeni adayı

Pre-service Middle School Mathematics Teachers’ Descriptions of Definite Integral and Indefinite Integral

Abstract

The importance of calculus in mathematics, mathematics education, and other disciplines and the necessity of developing students’ conceptual understanding regarding integral, which is one of the major concepts in calculus course, are among the issues emphasized by researchers. Thus, the purposes of this study were to examine how pre-service middle school mathematics teachers describe definite integral and indefinite integral and also to what extent they can see the relation between definite integral and indefinite integral. For these purposes, 173 pre-service middle school mathematics teachers were asked to answer three questions. According to the findings, the concepts pre-service middle school mathematics teachers mentioned while describing both definite and indefinite integral are similar which are bound, notation, mathematical formula, example, area, volume, antiderivative, calculation process, the form of result (number, function, algebraic expression or unknown), and the constant c. It was also seen that mostly mentioned concept is bound for each type of integral. However, the minority of them presented evidence regarding the relation between definite integral and indefinite integral in their responses.

Keywords

• Definite integral
• Indefinite integral
• Calculus
• Pre-service middle school mathematics teachers

References

1. Adams, R. A., & Essex, C. (2010). Calculus-A complete course (7th ed.). Toronto, Pearson.
2. Aspinwell, L., & Miller, D. (1997). Students’ positive reliance on writing as a process to learn first semester calculus. Journal of Instructional Psychology, 24(4), 253-261.
3. Attorps, I., Björk, K., & Radic, M. (2013). Varied ways to teach the definite integral concept. International Electronic Journal of Mathematics Education, 8(2-3), 81-99.
4. Berry, J., & Nyman, M. (2003). Promoting students’ graphical understanding of the calculus. Journal of Mathematical Behavior, 22(4), 481-497.
5. Bezuidenhout, J. (1998). First-year university students' understanding of rate of change. International Journal of Mathematical Education in Science and Technology, 29(3), 389-399.
6. Bezuidenhout, J. (2001). Limits and continuity: some conceptions of first-year students. International Journal of Mathematical Education in Science and Technology, 32(4), 487-500.
7. Bezuidenhout, J., & Olivier, A. (2000). Students’ conceptions of the integral. In T. Nakahara, & M. Koyama (Eds.), Proceeding of the 24th conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 73–80). Hiroshima, Japan.
8. Braun, V., & Clarke, V. (2006). Using thematic analysis in psychology. Qualitative Research in Psychology, 3(2), 77-101.

9. Ferrini-Mundy, J., & Graham, K.G (1991). An overview of the Calculus curriculum reform effort: Issues for learning, teaching and curriculum development. American Mathematical Monthly, 98(7), 627-635.
10. Goerdt, L. S. (2007). The effect of emphasizing multiple representations on calculus students’ understanding of the derivative concept. (Doctoral dissertation). The University of Minnesota.
11. Grundmeier, T. A., Hansen, J., & Sousa, E. (2006). An exploration of definition and procedural fluency in integral calculus. Problem, Resources and Issues in Mathematics Undergraduate Studies, 16(2), 178-191.
12. Hall, W. L. (2010). Language and area: influences on student understanding of integration. (Master's thesis). University of Maine.
13. Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. Journal of Mathematical Behavior, 32(2), 122-141.
14. Jones, S. R. (2015). Areas, anti-derivatives, and adding up pieces: Integrals in pure mathematics and applied contexts. The Journal of Mathematical Behavior, 38, 9-28.
15. Jones, S. R., Lim, Y., & Chandler, K.R. (2017). Teaching integration: How certain instructional moves may undermine the potential conceptual value of the Riemann sum and the Riemann integral. International Journal of Science and Mathematics Education, 15(6), 1075-1095.
16. Liu, P. H. (2009). History as a platform for developing college students’ epistemological beliefs on mathematics. International Journal of Science and Mathematics Education, 7(3), 473-499.
17. Mahir, N. (2009). Conceptual and procedural performance of undergraduate students in integration. International Journal of Mathematical Education in Science and Technology, 40(2), 201-211.
18. McGee, D., & Martinez-Planell, R. (2013). A study of effective application of semiotic registers in the development of the definite integral of functions of two and three variables. International Journal of Science and Mathematics Education, 12(4), 883-916.
19. Metaxas, N. (2007). Difficulties on Understanding the Indefinite Integral. In Woo, J.H., Lew, H.C., Park, K.S., Seo, D.Y. (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education (Vol. 3, pp.265-272). Seoul: PME.
20. Ministry of National Education [MoNE] (2013). Ortaöğretim matematik dersi 9-12 sınıflar öğretim programı [Secondary school mathematics curriculum grades 9 to 12]. Retrieved on February 15, 2018 from http://mufredat.meb.gov.tr/ProgramDetay.aspx?PID=343.
21. Nasari, Y. G. (2008). The effect of graphing calculator embedded materials on college students’ conceptual understanding and achievement in a calculus I course. (Doctoral dissertation). Wayne State University.
22. Oberg, R. (2000). An investigation of undergraduate calculus students understanding of the definite integral. (Doctoral dissertation). University of Montana.
23. Orton, A. (1983). Students' understanding of integration. Educational Studies in Mathematics, 14(1), 1-18.
24. Özdemir, Ç. (2017). The development of an inquiry-based teaching unit for Turkish high school mathematics teachers on integral calculus: the case of definite integral. (Master's thesis). Bilkent University.
25. Park, J. (2015). Is the derivative a function? If so, how do we teach it? Educational Studies in Mathematics, 89(2), 233-250.
26. Park, J. (2016). Communicational approach to study textbook discourse on the derivative. Educational Studies in Mathematics, 91(3), 395-421.
27. Pyzdrowski, L. J., Sun, Y., Curtis, R., Miller, D., Winn, G., & Hensel, R. A. (2013). Readiness and attitudes as indicators for success in college calculus. International Journal of Science and Mathematics Education, 11(3), 529-554.
28. Rasslan, S., & Tall, D. O. (2002). Definitions and images for the definite integral concept. In A. D. Cockburn & E. Nardi (Eds.) Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 89-96). Norwich, UK.
29. Sağlam, Y. (2011). Üniversite öğrencilerinin integral konusunda görsel ve analitik stratejileri [Undergraduate students’ visual and analytic strategies in integral topic]. (Doctoral dissertation). Hacettepe University.
30. Skemp, R. (1986). The psychology of mathematics learning. Suffolk: Penguin Books.
31. Stewart, J. (2010). Calculus: Concepts and contexts. (4th edition). Brooks/Cole, Thomson Learning.
32. Swidan, O., & Naftaliev, E. (2019). The role of the design of interactive diagrams in teaching–learning the indefinite integral concept. International Journal of Mathematical Education in Science and Technology, 50(3), 464-485.
33. Swidan, O., & Yerushalmy, M. (2014). Learning the indefinite integral in a dynamic and interactive technological environment. ZDM, 46(4), 517-531.
34. Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12(2), 151-169.
35. Thomas, G. B., Weir, M. D., Hass, J., & Giordano, F. (2005). Thomas’ calculus (11th Edition). Pearson Education: Addison-Wesley.
36. Vinner, S. (1983). Concept definition, concept image and the notion of function. International Journal of Mathematical Education in Science & Technology, 14(3), 239–305.
37. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of a function. Journal for Research in Mathematics Education, 20(4), 356-366.
38. Yerushalmy, M., & Swidan, O. (2012). Signifying the accumulation graph in a dynamic and multi- representation environment. Educational Studies in Mathematics, 80(3), 287-306.