Assessing Conceptual Understanding 
In Mathematics: Using Derivative Function To Solve Connected Problems

Year 2013, Volume: 14 Issue: 3, 138 - 151, 01.09.2013

Nevin Orhun

Abstract

Open and distance education plays an important role in the actualization of cultural goals as well as in societal developments. This is an independent teaching and learning method for mathematics which forms the dynamic of scientific thinking. Distance education is an important alternative to traditional teaching applications. These contributions brought by technology enable students to participate actively in having access to information and questioning it. Such an application increases students’ motivation and teaches how mathematics can be used in daily life. Derivative is a mathematical concept which can be used in many areas of daily life. The aim of this study is to enable the concept of derivatives to be understood well by using the derivative function in the solution of various problems. It also aims at interpreting difficulties theoretically in the solution of problems and determining mistakes in terms of teaching methods. In this study, how various aspects of derivatives are understood is emphasized. These aspects concern the explanation of concepts and process, and also their application to certain concepts in physics. Students’ depth of understanding of derivatives was analyzed based on two aspects of understanding; theoretical analysis and contextual application. Follow-up interviews were conducted with five students. The results show that the students preferred to apply an algebraic symbolic aspect instead of using logical meanings of function and its derivative. In addition, in relation to how the graph of the derivative function affects the aspect of function, it was determined that the students displayed low performance.

Keywords

Mathematics Education, Learning, Teaching, Derivative, Solving Problems.

References

• Amit, M., and Vinner, S., (1990). Some misconceptions in calculus: Proceedings of the 14th International Conference for the Psychology of Mathematics Education, 1, 3-10.
• Amoah, V., and Laridon, P. (2004). Using multiple representations to assess students’ understanding of the derivative concept. Proceeding of the British Society for Research into Learning maths. 24(1), 1-6.
• Asiala, M., Cottrill, J., and Dubinsky, E. (1997). The Development of Students’ Graphical
• Understanding of the Derivative. Journal of Mathematical Behavior 16 (4) 399-431. Aspinwall L., Shaw, K., and Presmeg, N. (1997), Uncontrollable images:Graphical connections between a function and its derivative. 78th Annual Meeting of the Mathematical Association of
• America and the 101st Annual Meeting of the American Mathematics Society, San Francisco, California. Breidenbach, D., Dubinsky, E., Hawkes, J., and Nichols, D. (1992). Development of the process conception of function. Educational Studies in Mathematics, 23,247-285.
• Berry, J.S., and Nyman, M. N. (2003), Promoting students’ graphical understanding of the calculus. Journal of Mathematical Behavior, 22,481-497.
• Eisenberg,T. (1992). On the development of a sense for functions. In Guershon Harel and ED Dubinsky, The concept of function: Aspects of epistemology and pedagogy,
• MAA notes 25, 153-174. Washington, DC: Mathematical Association of America. Eisenberg ,T., and Dreyfus,T.(1991). On the reluctance to visualize in mathematics in W.
• Zimmermann and S.Cunningham (eds), Visualization in Teaching and Learning Mathematics, Mathematical Association of America, Washington,DC, pp 25-37. Ferrini-Mundy, J., and Lauten, D. (1994). ”Learning About Calculus Learning”. The Mathematics Teacher, 87, 2 February.
• Hallet, D. H, (1991). Visualization and calculus reform in W.Zimmermann and S.Cunningham
• (eds),Visualization in Teaching and Learning Mathematics, Mathematical Association of America, Washington, DC, pp.121-126. Heid, M. K. (1988). Resequencing skills and concepts in applied calculus using the computer as a tool. Journal for Research in Mathematics Education, 19, 3-25.
• Kendal, M., and Stacey, K. (2000). Teachers in transition: Moving towards CAS- supported classrooms.Zentralblatt für Didaktik der Mathematik, 34(5), 196-203.
• Kieran, C. (1994). The learning and teaching of school algebra. In Grouws, D. A. (ed.)
• Handbook of Research on Mathematics Teaching and Learning. New York:Macmillan. 390-419. Ofsted. Klymchuk, S., Zverkova, T. Gruenwald, and N. Sauerbier, G. University. (2010). Students’ in
• Solving Application problems in Calculus: Student Perspectives. Mathematical Education Research Journal, 22(1), 81-91. Lauten, A., D, Graham, K., and Ferini-Mundy, J. (1994). Students understanding of basic calculus concepts. Journal of Mathematical Behavior, 13, 225-237.
• Leinhardt,G., Zaslavsky, O., and Stein, M. (1990). Functions, graphs, and graphing:
• Tasks, learning, and teaching. Review of Educational Research, 60, 1-64. Lesh, R. (2000). What mathematical abilities are most needs for success beyond school in technology based age of information? Proceedings of Time 2000an International
• Conference on the Technology in Mathematics Education 73-83. Jong, O. D., and Brinkman, F. (1997). Teacher Thinking and Conceptual Change in
• Science and Mathematics Education. European Journal of Teacher Education 20(2), 121- Orton, A. (1983). Students understanding of differentiation. Educational Studies in Mathematics, 14,235-250.
• Radatz, H. (1979). Error analysis in mathematics education, Journal for Research in
• Mathematics Education 10 (3). Schwalbach, M. E., and Dosemagen, M. D. (2000). Developing Students Understanding:
• Contextualizing Calculus Concepts. School Science and Mathematics 100(2) 90-98. Sigel, I. E. (1999). Development of mental representation: Theories and applications.
• New Jersey: Lawrence Erlbaum Associates. Tall, D. (1991). Visualization in Teaching and Learning Mathematics, Mathematical
• Association of America, Washington, DC, 105-119. Tall, D., and Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics, 12 151- 1
• Tucker, T. (1981). Priming the Calculus Pump, Innovations and Resources
• Mathematical Association of America, Washington, DC, 17. Vinner, S. (1983). Concept definition, concept image and the notion of function.
• International Journal of Mathematics Education in Science and Technology, 14 (3), 293-305. Vinner, S. (1989). The avoidance of visual considerations in calculus students. Focus on
• Learning Problems in Mathematics, 11, 149-156. Vinner, S., & Dreyfus, T. (1989). Images and definitions for the concept of function.
• Journal for Research in Mathematics Education, 20, 356-366. Zandieh, M. J. (2000). A theoretical framework for analyzing student understanding of the concept of derivative. CBMS Issues in Mathematics Education, 8, 103-127.
• Zandieh, M. J., and Knapp, J. (2006). Exploring the role of metonymy in mathematical understanding and reasoning: The concept of derivative as an example. Journal of
• Mathematical Behavior, 25, 1-17. Zimmermann, W. (1991). Visual thinking in mathematics. In W. Zimmermann and S.
• Cunningham (Eds), Visualization in teaching and learning mathematics (127-137). Washington, DC: Mathematical Association of America.