ERG SES C 07, Mathematics and Education
Calculus is accepted as one of the main courses in mathematics and a basis for learning concepts in more advanced mathematics (Mahir, 2009; Sevimli, 2013). The importance of concepts covered in calculus can also be seen through the applications of these concepts in different disciplines such as science, engineering and economics (Firouzian, 2014; Nasari, 2008). According to the literature review, studies regarding various concepts of calculus have been conducted by researchers such as limit and continuity (e.g., Aydos, 2015; Bezuidenhout, 2001), derivative (e.g., Kula, 2013; Orton, 1983), definite integral (e.g, Oberg, 2000; Rasslan & Tall, 2002), and indefinite integral (e.g., Metaxas, 2007; Swidan & Yerushalmy, 2014). This study aims to contribute to the literature by investigating integral.
Integral is not only a central concept of calculus in university level (Mahir, 2009; Rasslan & Tall, 2002; Rosasco, 2013), but also a fundamental component of high school mathematics curriculum in Turkey (Yazlık & Erdoğan, 2015). In other words, according to the objectives of the 12th grade mathematics curriculum in Turkey stated by the Ministry of National Education (MoNE, 2013), some concepts of calculus such as limit, derivative and integral are introduced to students at high school level. The studies undertaken regarding integral showed that students at every level have difficulties in integral (Oberg, 2000; Rasslan & Tall, 2002; Yazlık & Erdoğan, 2015). To develop both high school and undergraduate students’ understanding of integral and to address the difficulties they have, students’ level of conceptual and procedural knowledge and the deficiencies they have related to integral should be determined as an important step (Mahir, 2009). Then, teacher educators might also improve their further teaching experiences based on this step.
According to Ross (1996, as cited in Goerdt, 2007), to foster students’ conceptual understanding, they are expected to interpret and use numerical, graphical, symbolic and verbal representations in concepts of calculus accurately. Similarly, to be able to move between various representations in concepts of calculus is among the objectives of calculus course (Goerdt, 2007). Moreover, undergraduate students have some difficulties in graphical interpretations of integral and area relations (Grundmeier, Hansen, & Sousa, 2006; Sağlam, 2011) and they have tendency to use algebraic methods instead of geometrical or graphical methods in calculation of integral (Oberg, 2000). Since calculus is a basis for the following mathematics courses in mathematics teacher education program (Nasari, 2008), prospective middle school mathematics teachers should have necessary content knowledge about each concept of the calculus and apply them properly (Mahir, 2009). In this respect, the purpose of the study is to investigate prospective middle school mathematics teachers’ interpretations of graphs related to integral in terms of year level. Based on this purpose, research questions were stated as follows:
1. To what extent can prospective middle school mathematics teachers interpret graphs related to integral?
2. Does year of enrollment in teacher education program affect their interpretations?
Firouzian, S. S. (2014). Correlations between students’ multiple ways of thinking about the derivative and their abilities to solve applied derivative problems. (Doctoral dissertation). The University of Maine. Fraenkel, J. R., & Wallen, N. E. (2005). How to design and evaluate research in education (6th ed.). Boston: McGraw Hill. Goerdt, L. S. (2007). The effect of emphasizing multiple representations on calculus students’ understanding of the derivative concept. (Doctoral dissertation). The University of Minnesota. 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. Mahir, N. (2009). Conceptual and procedural performance of undergraduate students integration. International Journal of Mathematical Education in Science and Technology, 40(2), 201-211. Ministry of National Education [MoNE] (2013). Ortaöğretim Matematik Dersi 9–12 Sınıflar Öğretim Programı. Retrieved in May 16 from http://ttkb.meb.gov.tr/www/ogretim-programlari/icerik/72. Nasari, Y. G. (2008). The effect of graphıng calculator embedded materıals on college students’ conceptual understandıng and achıevement ın a calculus I course. (Doctoral dissertation). Available from ProQuest Dissertations and Theses Database. (UMI No. 3296875). Oberg, R. (2000). An investigation of under graudate calculus students understanding of the definite integral. (Doctoral dissertation). Available from ProQuest Dissertations and Theses Database. (UMI No. 9993971). Orton, A. (1983). Student’s understanding of Integration. Educational Studies in Mathematics, 14(1), 1-18. Rasslan, S., & Tall, D. (2002). Definitions and images for the definite integral concept. Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education, 4, 89-96, Norwich: England. Rosasco, M. E. (2013). Factors associated with success in college calculus II. (Doctoral dissertation). Available from ProQuest Dissertations and Theses Database. (UMI No. 3572564). Sağlam, Y. (2011). Üniversite öğrencilerinin integral konusunda görsel ve analitik stratejileri. (Doctoral dissertation), Hacettepe University. Sevimli, E. (2013). Bilgisayar cebiri sistemi destekli öğretimin farklı düşünme yapısındaki öğrencilerin integral konusundaki temsil dönüşüm süreçlerine etkisi. (Doctoral dissertation), Marmara University. Stewart, J. (2001). Calculus:Concepts and Contexts. (2nd edition). Brooks/Cole, Thomson Learning. Swidan, O., & Yerushalmy, M. (2014). Learning the indefinite integral in a dynamic and interactive technological environment. ZDM-The International Journal on Mathematics Education, 46(4), 517–531. Thomas, G. B., Weir, M. D., Hass, J., & Giordano, F. R. (2010). Thomas’ Calculus. (12th edition). Boston: Pearson Education. Yazlık, D.Ö., Erdoğan, A. (2015). İntegralde alan uygulamaları konusunda Flash programı ile geliştirilen öğretim materyalinin değerlendirilmesi. Hacı Bektaş Veli University Journal of Social Sciences, 4(2), 144-159.
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