Designing Learning Strategy to
Improve Undergraduate Students’ Problem Solving in Derivatives and Integrals: A Conceptual Framework

Nourooz Hashemi; Mohd Salleh Abu; Hamidreza Kashefi; Mahani Mokhtar; Khadijeh Rahimi

doi:10.12973/eurasia.2015.1318a

Designing Learning Strategy to Improve Undergraduate Students’ Problem Solving in Derivatives and Integrals: A Conceptual Framework

Full Text (PDF)

Nourooz Hashemi ¹ ^* , Mohd Salleh Abu ¹, Hamidreza Kashefi ¹, Mahani Mokhtar ¹, Khadijeh Rahimi ²

More Detail

¹ Universiti Teknologi Malaysia, MALAYSIA² Center for Mathematics Education, IRAN^* Corresponding Author

Abstract

Derivatives and integrals are two important concepts of calculus which are precondition topics for most of mathematics courses and other courses in different fields of studies. A majority of students at the undergraduate level have to master derivatives and integrals if they want to be successful in their studies However, students encounter difficulties in the learning of derivatives and integrals. Most of these difficulties arise from the students’ weakness in problem solving. This paper presents a learning strategy which has been designed to overcome these difficulties based on mathematical thinking and generalization strategies with prompts and questions.

Keywords

learning difficulties
problem solving
learning strategy
mathematical thinking
generalization
derivatives and integrals

License

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Article Type: Research Article

EURASIA J Math Sci Tech Ed, 2015, Volume 11, Issue 2, 227-238

https://doi.org/10.12973/eurasia.2015.1318a

Publication date: 02 Apr 2015

Article Views: 2281

Article Downloads: 1514

Open Access References How to cite this article

References

Aghaee, M. (2007). Investigation of Controlling Abilities for Solving Problem in Infinite Integral. Unpublished Master Thesis. Shahid Bahonar University, Iran.
Azarang, Y. (2012). Quality of Leaning Calculus in Iran. Roshd Mathematics Education Journal, 27(1): 24- 30.
Cruz, J. A. G., Martinon, A. (1998). Levels of Generalization in Linear Patterns. 36th Conference of the International Group for the Psychology of Mathematics Education. 2: 329-336.
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. (2012). Drwing Figures as a Problem Solving Method. Roshd Mathematics Education Journal, 27(1): 38- 41.
Gray, E. and Tall, D. (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.
Harel, G., Tall, D. (1991). The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics, 11(1), 38-42.
Javadi, M. (2008). Perception of Concepts and Definition of Concept for Calculus. Roshd Mathematics Education Journal. 27(2): 23-27.
Kabael, T. (2011). Generalizing Single Variable Functions to Two-variable Functions, Function Machine and APOS. Educational Sciences: Theory and Practice, 11(1), 484-499.
Karamzadeh, S., O. (2000). Generalization. Incredible Results in Mathematics. University of Shahid Chamran Press. Ahvaz, Iran.
Kashefi, H., Ismail, Z., Yusof, Y. M. (2013). Learning Functions of Two Variables Based on Mathematical Thinking Approach. Jurnal Teknologi, 63(2), 59-69.
Kirkley, J. (2003). Principles for teaching problem solving. PLATO Learning Inc. USA.
Larsen, L. C. (1999). Problem-Solving Through Problems, Springer.
Lithold, L. (1968). Calculus and Linear Algebra. Amazon, 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., Stacey, K. and Burton, L. (2010). Thinking Mathematically (2th edition), Edinburgh: Pearson.
Mejia-Ramos, J. P. (2006, November). The long-term cognitive development of different types of reasoning and proof. In Conference on Explanation and Proof in Mathematics: Philosophical and Educational Perspectives, Essen, Germany
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.
Orton, A. (1983). Students' Understanding of Integration. Educational Studies in Mathematics. 14: 1-18.
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 TopicsPhysics 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.
Silverman, R. (1988) . Modern Calculus and Analytic Geometry. Amazon, USA.
Sriraman, B. (2004). Reflective Abstraction, Uniframes and the Formulation of Generalizations. The Journal of Mathematical Behavior. 23(2): 205–222.
Stacey, K. (2006). What Is Mathematical Thinking and Why Is It Important? University of Melbourne, Australia.
Stewart, S. (2008). Understanding Linear Algebra Concepts Through the Embodied, Symbolic and Formal Worlds of Mathematical Thinking. Unpublished Ph.D. thesis. University of Auckland, New Zealand.
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,ICME7, Québec, Canada: 13–28.
Tall, D. (1995). Mathematical Growth in Elementary and Advanced Mathematical Thinking, (Plenary Address). In Luciano Meira and David Carraher (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. (2002b). Differing Modes of Proof and Belief in Mathematics, International Conference on Mathematics: Understanding Proving and Proving to Understand, 91–107. National Taiwan Normal 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. (2005). The Transition from Embodied Thought Experiment and Symbolic Manipulation to Formal Proof. In M. Bulmer, H. MacGillivray and C. Varsavsky(Eds.), Proceedings of Kingfisher Delta’05: Fifth Southern Hemisphere Symposium on Undergraduate Mathematics and Statistics Teaching and Learning. (pp. 23– 35).Fraser Island, Australia.
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., Yudariah Mohd Yusof. (1995). Professors’ Perceptions Of Students’ Mathematical Thinking: Do They Get What They Prefer or What They Expect? In L. Meira, D. Carraher, (Eds.), Proceedings of PME 19, Recife, Brazil, II: 170–177.
Tarmizi, R., A. (2010). Visualizing Students’ Difficulties in Learning Calculus. Procedia Social and Behavioral Science. 8: 377- 383.
Thomas, w., J. (2009). Calculus and Linear Algebra. Amazon, USA.
Villiers, M. D. and Garner, M. (2008). Problem Solving and Proving via Generalization, Journal of Learning and Teaching Mathematics. 5: 19-25.
Watson, A. (2000). Going Across The Grain: Mathematical Generalization in a Group of Low Attainers. Nordisk Matematikk Didaktikk (Nordic Studies in Mathematics Education).8(1): 7–2.
Watson, A. (2002). Embodied Action, Effect, and Symbol in Mathematical Growth. In A. Cockburn, E. Nardi (Eds.). Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education. Norwich: UK. 4: 369- 376.
Watson, A. and Mason, J. (1998). Questions and Prompts for Mathematical Thinking. ATM, Derby.
Watson, A. and Mason, J. (2006). Seeing an Exercise as a Single Mathematical Object: Using Variation to Structure Sense-Making. Mathematical Thinking and Learning. 8(2): 91-111.
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. Yudariah Mohamad.
Yusof. (1997). Undergraduate Mathematics Education:Teaching Mathematical Thinking Or Product Of Mathematical Thought? Jurnal Teknologi. 26(June): 23 – 40. Yudariah Mohd.
Yusof., Tall, D. (1999). Changing Attitudes to University Mathematics through Problem-solving. Educational Studies in Mathematics. 37: 67-82.
Yudariah Mohammad Yusof. and Roselainy Abd. Rahman. (2004). Teaching Engineering Students to Think Mathematically. Paper presented at the Conference on Engineering Education, Kuala Lumpur, 14- 15. December.

How to cite this article

APA

Hashemi, N., Abu, M. S., Kashefi, H., Mokhtar, M., & Rahimi, K. (2015). Designing Learning Strategy to Improve Undergraduate Students’ Problem Solving in Derivatives and Integrals: A Conceptual Framework. Eurasia Journal of Mathematics, Science and Technology Education, 11(2), 227-238. https://doi.org/10.12973/eurasia.2015.1318a

Vancouver

Hashemi N, Abu MS, Kashefi H, Mokhtar M, Rahimi K. Designing Learning Strategy to Improve Undergraduate Students’ Problem Solving in Derivatives and Integrals: A Conceptual Framework. EURASIA J Math Sci Tech Ed. 2015;11(2):227-38. https://doi.org/10.12973/eurasia.2015.1318a

AMA

Hashemi N, Abu MS, Kashefi H, Mokhtar M, Rahimi K. Designing Learning Strategy to Improve Undergraduate Students’ Problem Solving in Derivatives and Integrals: A Conceptual Framework. EURASIA J Math Sci Tech Ed. 2015;11(2), 227-238. https://doi.org/10.12973/eurasia.2015.1318a

Chicago

Hashemi, Nourooz, Mohd Salleh Abu, Hamidreza Kashefi, Mahani Mokhtar, and Khadijeh Rahimi. "Designing Learning Strategy to Improve Undergraduate Students’ Problem Solving in Derivatives and Integrals: A Conceptual Framework". Eurasia Journal of Mathematics, Science and Technology Education 2015 11 no. 2 (2015): 227-238. https://doi.org/10.12973/eurasia.2015.1318a

Harvard

Hashemi, N., Abu, M. S., Kashefi, H., Mokhtar, M., and Rahimi, K. (2015). Designing Learning Strategy to Improve Undergraduate Students’ Problem Solving in Derivatives and Integrals: A Conceptual Framework. Eurasia Journal of Mathematics, Science and Technology Education, 11(2), pp. 227-238. https://doi.org/10.12973/eurasia.2015.1318a

MLA

Hashemi, Nourooz et al. "Designing Learning Strategy to Improve Undergraduate Students’ Problem Solving in Derivatives and Integrals: A Conceptual Framework". Eurasia Journal of Mathematics, Science and Technology Education, vol. 11, no. 2, 2015, pp. 227-238. https://doi.org/10.12973/eurasia.2015.1318a