Generalization Strategies in the Problem Solving of Derivative and Integral

Nourooz Hashemi, Hamidreza Kashefi, Mohd Salleh Abu


This study proposes a learning strategy of derivatives and integrals (LSDI) based on specialized forms of generalization strategies to improve undergraduate students’ problem solving ofderivative and integral. The main goal of this study is to evaluate the effects of LSDI on students’ problem solvingofderivative and integral. The samples of this study were 63 undergraduate students who took Calculus at Islamic Azad University of Gachsaran, Iran. The students were divided into two classes based on their marks in the pre- test of derivative and integral. The results indicated that there was a significant difference between the achievements of students in experimental and control groupsafter treatment. Thus, the findings reveal that using generalization strategies improves students’ achievements in solving problems of derivative and integral.


derivative; generalization strategies; integral; problem solving; undergraduate studentsderivative; generalization strategies; integral; problem solving; undergraduate students

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Aghaee, M. (2007). Investigation of Controlling Abilities for Solving Problem in Infinite Integral, Unpublished Master Thesis. Shahid Bahonar University, Kerman, Iran.

Azarang, Y. (2008). Learning of Calculus With Concepts of Limit and Symbolic. Roshd Mathematics Education Journal . 27(1), 4-10.

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. Proceedings of the 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. (2010). Looking to Change of Calculus in Iran. Roshd 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.M., and Tall, D. (2001). Relationships between embodied objects and symbolic procepts: anexplanatory theory of success and failure in mathematics. Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education, 3, 65-72, Utrecht, The Netherlands.

Harel, G., Tall, D. (1991). The general, the abstract, and the generic in advanced mathematics. For the Learning of Mathematics,11(1), 38-42.

Hashemi, N., MohdSalleh, A., Kashefi, H., Rahimi, K. (2013). What are Difficulties of Learning Derivative and Integral among Undergraduate Students? Proceeding of 4th International Graduate Conference on Engineering Science & Humanity 2013 (IGCESH2013), 1056- 1063.

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.

Kashefi, H., Zaleha Ismail., and Yudariah Mohd Yusof. (2012). Overcoming Students Obstacles in Multivariable Calculus through Blended Learning: A Mathematical Thinking Approach, Procedia - Social and Behavioral Sciences, 56, 579-586.

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.

Kirkpatrick, L.A, Feeney, B.C. (2012). A Simple Guide to IBM SPSS Statistics. Wadsworth Cenage Learning, USA.

Larsen, L.C. (1999). Problem-Solving Through Problems, Springer.

Mason, J. (2002). Generalisation and Algebra: Exploiting Children's Powers. In L. Haggerty (Ed.) Aspects of Teaching Secondary Mathematics: perspectives on practice. (pp. : 105-120). London: RoutledgeFalmer.

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. (pp. 23-47). Springer, New York.

Mason, J., Stacey, K., 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. (pp. 265-272). Seoul: PME.

Pallant, J. (2010). SPSS survival manual: A step by step guide to data analysis using SPSS. Open University Press.

Parhizgar, B. (2008). Conceptual Understanding of Function. Unpublished Master Thesis. University of Shahid Beheshti, Tehran, Iran

Pepper, R., Chasteen, E.S.V., Pollock, S.J., Perkins, K.K. (2012). Observations on Student Difficulties with Mathematics in Upper-Division Electricity and Magnetism. Physical Review Special Topics- Physics Education Research. (pp. 1-15).

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, Kerman, 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 of 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.

Scruggs, T.E., Mastropieri, M.A. (1993). Special education for the twenty-first century: Integrating learning strategies and thinking skills. Journal of Learning Disabilities, 26, 392–398.

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.

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, In Luciano Meira and David Carraher (Eds.), Proceedings of PME, 19(1), 61–75, Recife, Brazil.

Tall, D. (2001). Cognitive Development in Advanced Mathematics Using Technology, Mathematics Education Research Journal,12(3), 196–218.

Tall, D. (2002a). Differing Modes of Proof and Belief in Mathematics, Proceedings of International Conference on Mathematics: Understanding Proving and Proving to Understand, 91–107., National TaiwanNormal University, Taipei, Taiwan.

Tall, D. (2002b). Advanced Mathematical Thinking (11 Ed.). London: Kluwer academic publisher.

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, Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education, 4, 281-288, Bergen: Bergen University College, Norway.

Tall, D. (2008). The Transition to Formal Thinking in Mathematics. Mathematics Education Research Journal. 20(2), 5-24.

Tall, D. (2010). 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. (2011). Looking for the Bigger Picture. For the Learning of Mathematics. 31 (2), 17-18.

Tall, D., Yudariah Mahommad 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.

Villiers, M. D., Garner, M. (2008). Problem Solving and Proving via Generalization, Journal of Learning and Teaching Mathematics, 5, 19-25.

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. (2006). Seeing an Exercise as a Single Mathematical Object: Using Variation to Structure Sense-Making. Mathematical Thinking and Learning, 8(2), 91-111.

Watson, A., Mason, J. (1998). Questions and Prompts for Mathematical Thinking. ATM, Derby.

Willcox, K., Bounova, G. (2004). Mathematics in Engineering: Identifying, Enhancing and Linking the Implicit Mathematics Curriculum. 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, Kerman, Iran.

Yudariah Mohammad Yusof., Roselainy Abd. Rahman. (2004). Teaching Engineering Students to Think Mathematically. Proceedings of the Conference on Engineering Education, Kuala Lumpur, 14- 15, December.

Yudariah Mohd. Yusof., Tall, D. (1999). Changing Attitudes to University Mathematics through Problem-solving. Educational Studies in Mathematics, 37, 67-82.


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