Undergraduate Students’ Difficulties in Solving Derivative and Integral Mathematical Problems
DOI:
https://doi.org/10.11113/sh.v11n2.1477Keywords:
Derivative, difficulties, mathematical thinking, problem solving, integral, undergraduateAbstract
Undergraduate students often experienced difficulty in solving problem related to derivative and integral topics. The main goal of this study was to investigate the reasons of students’ difficulties in solving derivative and integral problems based on mathematical thinking approach. The participants of the study consisted of 63 undergraduate students. A test contained derivative and integral problems was given to students and the results was analyzed by quantitative and qualitative methods. Results revealed that the reasons of students’ difficulties in solving problems were inability to use suitable problem solving framework, weakness in recalling previous knowledge in entry and attack steps of specialization and generalization, weakness in making connection between embodied and symbolic worlds of mathematical thinking and using symbolic world rather than embodied world.
References
Aghaee, M. (2007). Investigation of Controlling Abilities for Solving Problem in Infinite Integral. Unpublished Master Thesis. Shahid Bahonar University, Iran.
Azarang, Y. (2008). Learning of Calculus With Concepts of Limit and Symbolic. Iranian Mathematics Education Journal . 27(1): 4-10.
Azarang, Y. (2012). Quality of Leaning Calculus in Iran. Roshd Mathematics Education Journal, 27(1): 24- 30.
Cline, K., Parker, M., Zullo, H., Stewart, A. (2012). Addressing Common Student Errors With Classroom Voting in Multivariable Calculus. PRIMUS, 23(1), 60-75.
Dubinsky, E. (1991). Reflective Abstraction in Advanced Mathematical Thinking. In D. O. Tall (Ed.), Advanced Mathematical Thinking (pp. 95-123). Dordrecht: Kluwer Academic Publishers.
Ghanbari, G. (2010). Looking to Change of Calculus in Iran. Mathematics Education Journal, 27(1): 59- 61.
Ghanbari, G. (2012). Drwing Figures as a Problem Solving Method. Roshd Mathematics Education Journal, 27(1): 38- 41.
Gray, E.and Tall, D. O. (2001). Relationships Between Embodied Objects and Symbolic Procepts: An Explanatory Theory of Success and Failure in Mathematics. Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, Utrecht, The Netherlands. 3: 65-72.
Haber, S.and Abboud, M. (2006). Students’ Conceptual Understanding of a Function and Its Derivative in an Experimental Calculus Course. Journal of Mathematical Behavior. 25: 57–72.
Hashemi, N., Mohd Salleh, A., Kashefi, H., Rahimi, K. (2013). What are Difficulties of Learning Derivative and Integral among Undergraduate Students? Proceeding of4th International Graduate Conference on Engineering Science and Humanity (IGCESH2013).1: 1056- 1063.
Hirst, K. (2002). Classifying students’ mistakes in Calculus. In Proceedings of the 2nd International Conference on the Teaching of Mathematics (at the undergraduate level).
Javadi, M. (2008). Perception of Concepts and Definition of Concept for Calculus. Roshd Mathematics Education Journal. 27(2): 23-27.
Kashefi, H. (2012). Mathematical Thinking in Multivariable Calculus Through Blended Learning. Unpublished Ph.D Thesis. Universiti Teknologi Malaysia (UTM). Malaysia
Kiat, S. E. (2005). Analysis of Students’ Difficulties in Solving Integration Problems. The Mathematics Educator, 9(1), 39-59.
Kirkley, J. (2003). Principles for teaching problem solving. PLATO Learning Inc.USA.
Mason, J. (2002). Generalisation and Algebra: Exploiting Children's Powers. In L. Haggerty (Ed.) Aspects of Teaching Secondary Mathematics: perspectives on practice. London: RoutledgeFalmer: 105-120.
Mason, J. (2010). Attention and Intention in Learning About Teaching Through Teaching. In R. Leikin and R. Zazkis (Eds.) Learning Through Teaching Mathematics: Development of Teachers' Knowledge and Expertise In Practice. p23-47. Springer, New York.
Mason, J. (2012). A Phenomenal Approach to Mathematics. Retrieved on March, 2013 from http://www.xtec.cat/centres/a8005072/articles/phenomenal_approach.pdf.
Mason, J., Stacey, K.and Burton, L. (2010).Thinking Mathematically (2th edition), Edinburgh: Pearson.
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. 3: 265-272. Seoul: PME.
National Assessment of Educational Progress (NAEP). (2003). What Does the NAEP Mathematics Assessment Measure? Retrieved on March, 2013 in http//:www.nces.ed.gov/nationsreportcard/mathematics/abilities.asp.
Orton, A. (1983a). Students' Understanding of Integration. Educational Studies in Mathematics. 14: 1-18.
Orton. A. (1983b).Students' Understanding of Differentiation. Educational Studies in Mathematics. 14: 235-250.
Parhizgar, B. (2008). Conceptual Understanding of Function. Unpublished Master Thesis. University of Shahid Beheshti. Iran.
Pepper, R., E. Stephanie V. Chasteen, Steven J. Pollock and Katherine K. Perkins. (2012). Observations on Student Difficulties with Mathematics in Upper-Division Electricity and Magnetism. Physical Review Special Topics- Physics Education Research. 8(010111): 1- 15.
Polya, G. (1982). Mathematical Discovery: on Understanding, Learning and Teaching Problem Solving (Combined Ed.). New York, NY: Wiley.
Polya, G. (1988). How to Solve It. USA: Princeton University Press.
Roknabadi, H. A. (2007). Varieties of Conceptual Understanding: Different Theories. Unpublished Master Thesis. Shahid Bahonar University, Iran.
Roselainy Abdolhamid. (2008). Changing My Own and My Students’ Attitudes to Calculus Through Working on Mathematical Thinking. Unpublished Ph. D. Thesis. Open University. UK.
Rubio, B. S.and Chacón, Gómez-I.M. (2011).Challenges with Visualization. The Concept of Integral with Undergraduate Students. Proceeding the Seventh Congress of European Society for Research in Mathematics Education (CERME-7). 9th and 13th Feb, University of Rezeszow, Poland.
Schoenfeld, A. H. (1992). Learning to Think Mathematically: Problem Solving, Metacognition, and Sense-making Mathematics. Grouws, D. (Ed). Research on Mathematics Teaching and Learning: 334–370. Macmillan, New York. USA.
Shahshahani, S. (2012). Why Calculus in Iran? Retrieved from: http://matheducation.blogfa.com/post-6.aspx.
Stacey, K. (2006). What Is Mathematical Thinking and Why Is It Important? University of Melbourne, Australia.
Tall, D. (1986).Building and Testing a Cognitive Approach to the Calculus Using Interactive Computer Graphics, Ph.D Thesis, the University of Warwick.
Tall, D. (1992). Conceptual Foundations of the Calculus. Proceedings of the Fourth International Conference on College Mathematics Teaching: 73- 88.
Tall, D. (1993). Students’ Difficulties in Calculus. Proceedings of Working Group 3 on Students’ Difficulties in Calculus,ICME-7, Québec, Canada: 13–28.
Tall, D. (1995).Mathematical Growth in Elementary and Advanced Mathematical Thinking, (Plenary Address). In Luciano Meiraand DavidCarraher (Eds.), Proceedings of PME 19(1): 61–75. Recife, Brazil.
Tall, D. (1997). Functions and Calculus. Retrieved from: http//: http://www.davidtall.com/.
Tall, D. (2001). Cognitive Development in Advanced Mathematics Using Technology, Mathematics Education Research Journal. 12 (3): 196–218.
Tall, D. (2002a). Advanced Mathematical Thinking (11 Ed.). London: Kluwer academic publisher.
Tall, D. O. (2002b). Differing Modes of Proof and Belief in Mathematics, International Conference on Mathematics: Understanding Proving and Proving to Understand, 91–107. National TaiwanNormal University, Taipei, Taiwan.
Tall, D. (2004a). Introducing Three Worlds of Mathematics. For the Learning of Mathematics. 23 (3): 29–33.
Tall, D. (2004b).Thinking through three worlds of mathematics.InProceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education (4: 281-288). Bergen: Bergen University College, Norway.
Tall, D. (2007).Developing a theory of mathematical growth. ZDM. 39(1-2): 145-154.
Tall, D. (2008).The Transition to Formal Thinking in Mathematics. Mathematics Education Research Journal. 20(2): 5-24.
Tall, D. (2010a). Perceptions, Operations and Proof in Undergraduate Mathematics, CULMS Newsletter (Community for Undergraduate Learning in the Mathematical Sciences). University of Auckland, New Zealand, 2, November 2010: 21-28.
Tall, D. (2010b). A Sensible Approach to the Calculus. (Plenary at The National and International Meeting on the Teaching of Calculus. 23–25th September 2010, Puebla, Mexico).
Tall, D. (2011). Looking for the Bigger Picture. For the Learning of Mathematics. 31 (2): 17-18.
Tall, D. (2012). Making Sense of Mathematical Reasoning and Proof. Plenary at Mathematics and Mathematics Education: Searching for Common Ground: A Symposium in Honor of Ted Eisenberg. April 29-May 3, 2012, Ben-Gurion University of the Negev, Beer Sheva, Israel.
Tall, D.and Mohammad Yusof, Y. (2009). Changing Attitudes to University Mathematics Through Problem Solving. Educational Studies in Mathematics.Colección Digital Eudoxus, 1(3).
Tarmizi, R., A. (2010). Visualizing Students’ Difficulties in Learning Calculus. Procedia Social and Behavioral Science. 8: 377- 383.
Villiers, M. D.and Garner, M. (2008). Problem Solving and Proving via Generalization, Journal of Learning and Teaching Mathematics. 5: 19-25.
Watson, A.and Mason, J. (1998).Questions and Prompts for Mathematical Thinking. ATM, Derby.
Willcox, K. and Bounova, G. (2004). Mathematics in Engineering: Identifying, Enhancing and Linking the Implicit Mathematics Curriculum. In Proceedings of the 2004 American Society for Engineering Education Annual Conference and Exposition.USA.
Yazdanfar, M. (2006). Investigation of Studying Skills in Calculus for Undergraduate. Unpublished Master Thesis. Shahid Bahonar University. Iran.
Yee, N. K., Lam, T. T. (2008). Pre-University Students’ Errors in Integration of Rational Functions and Implications for Classroom Teaching. Journal of Science and Mathematics Education in Southeast Asia, 31(2), 100-116.
Yudariah Mohamad Yusof. (1995). Thinking Mathematically: A Framework for Developing Positive Attitudes Amongst Undergraduates. Unpublished Ph.D. thesis. University of Warwick. UK.
YudariahMohd. Yusof.and Tall, D. (1999). Changing Attitudes to University Mathematics through Problem-solving.Educational Studies in Mathematics. 37: 67-82.
