The Emphasis on Generalization Strategies in Teaching Integral: Calculus Lesson Plans

Authors

  • Nourooz Hashemi Department of Mathematics Education, Farhangian University,Tehran, Iran
  • Hamidreza Kashefi Department of Mathematics Education, Farhangian University,Tehran, Iran http://orcid.org/0000-0001-5463-4453
  • Mohd Salleh Abu Department of Science, Mathematics and Creative Multimedia Education, School of Education, Faculty of Social Sciences and Humanities, Universiti Teknologi Malaysia, 81310, UTM Johor Bahru, Johor, Malaysia

DOI:

https://doi.org/10.11113/sh.v12n3.1687

Keywords:

Generalization, Lesson Plans, Undergraduate, Integral

Abstract

Although integral is one of the important concepts in mathematics, most students have problems learning it. A well-designed lesson plan consisting generalization and its related activities can play an important role to overcome the students’ difficulties in the learning process. In this study, generalization means going from particular to something general by looking at common things and expanding their domains of validity. The main goal of this study is to investigate the rate of using generalization by mathematics instructors in teaching of integral concepts based on lesson plans. Thus, the instructors were interviewed on what generalization means to them and to find the reasons of emphasis on generalization or lack of it in their lesson plans. Findings revealed that most instructors did not emphasise the teaching of generalization in their lesson plans. Although most of the instructors expressed that generalization can be used to extend the theorems, the interview clarified that majority of the instructors did not know about the important applications of generalization in the teaching of integral. The results demonstrated that instructors have oppressed generalization in the integral teaching process.

References

Brijlall D., Ndlaz, N. J. (2019). Analysing engineering students’ understanding of integration to propose a genetic decomposition. The Journal of Mathematical Behavior, 55, 1-12. https://doi.org/10.1016/j.jmathb.2019.01.006.

Ding, M., & Carlson, M. A. (2013). Elementary teachers’ learning to construct high quality mathematics lesson plans. The Elementary School Journal, 113(3), 359-385.

Dorko, A., Weber, E. (2014). Generalising calculus ideas from two dimensions to three: how multivariable calculus students think about domain and range. Research in Mathematics Education, 16(3), 269-287.

Fernandez, C.; Cannon, J.; Chokshi, S. (2003). A US–Japan lesson study collaboration reveals critical lenses for examining practice. Teaching and Teacher Education , 19, 171–185.

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. (2008). Role of Generalization in High School Mathematics Teaching. Unpublished Master Thesis, FSRIA University. Iran: Shiraz.

Hashemi, N.; Mohdsalleh, A.; Kashefi, H.; Rahimi, K. (2013a). What are difficulties of learning derivation and integral among undergraduate students? Proceeding of 4th International Graduate Conference on Engineering Science & Humanity 2013 (IGCESH2013), 1056-1063.

Hashemi, N.; Mohdsalleh, A.; Kashefi, H.; Rahimi, K. (2013b). Generalization in the Learning of Mathematics. Proceeding of 2nd International Seminar on Quality and Affordable Education (ISQAE 2013), 1, 208-215.

Hashemi, N., Mohd Salleh Abu., and Kashefi, H. (2019a). Undergraduate Students’ Difficulties in Solving Derivative and Integral Mathematical Problems, Sains Humanika, 11(2), 65-74.

Hashemi, N., Mohd Salleh Abu., and Kashefi, H. (2019b). Generalization Strategies in the Problem Solving of Derivative and Integral, International Journal on Emerging Mathematics Education, 3(1), 77-92.

Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. The Journal of Mathematical Behavior, 32(20), 122-141.

Jones, S. R., Lim, Y., Chandler, K. R. (2017). Teaching Integration: How Certain Instructional Moves May Undermine the Potential Conceptual Value of the Riemann Sum and the Riemann Integral. International Journal of Science and Mathematics Education, 15, 1075-1095.

Karamzadeh, S. O. (2000). Generalization, Incredible Results in Mathematics. University of Shahid Chamran Press, Ahvaz, Iran.

Li, V. L., Julaihi, N. H., Eng, T. H. (2017). Misconceptions and Errors in Learning Integral Calculus, Asian Journal of University Education, 13(2), 17-39.

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.Springer, New York, 23 - 47.

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.Seoul: PME, 3, 265 - 272.

Miles, M. B.; Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook. Sage.

Mitchelmore, M. C. (2002). The Role of Abstraction and Generalisation in the Development of Mathematical Knowledge, Paper presented at the East Asia Regional Conference on Mathematics Education (EARCOME)(2nd) and the Southeast Asian Conference on Mathematics Education (SEACME), (9th, Singapore, May 27-31, 2002).

Moru, E. K., & Qhobela, M. (2019). Social science students’ concept images and concept definitions of anti-derivatives. Pythagoras-Journal of the Association for Mathematics Education of South Africa, 40(1),1-12.

Nursyahidah, F., Albab, I. U. (2017). Investigating student difficulties on integral calculus based on critical thinking aspects, Jurnal Riset Pendidikan Matematika, 4(2), 211-218.

Orton, A. (1983). Students' Understanding of Integration. Educational Studies in Mathematics, 14, 1-18.

Rettig, M. D.; Mccullough, L. L.; Santos, K. (2003). A blueprint for increasing student achievement. Educational Leadership, 61(3), 71-76.

Roselainy, A. R. (2009). Changing My Own and My Students’ Attitudes to Calculus Through Working on Mathematical Thinking.Unpublished Ph. D. Thesis.Open University. UK.

Saepuzaman, D., Zulfikar, A., Girsang, D. Y. (2017). Correlation between Students' Understanding on Derivative and Integral Calculus with Thermodynamics, Proceedings of the 2016 International Conference on Mathematics and Science Education.

Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33, 230-245.

Sri Mentari, S Saragih, Mulyono. (2018). The Development of Mathematic Lesson Plan to Increase Mathematic Communication Ability Students Through the Model of Problem Based Learning Contextually on Java Culture, Proceedings of the 3rd Annual International Seminar on Transformative Education and Educational Leadership (AISTEEL 2018).

Sriraman, B. (2004). Reflective Abstraction Uniframes and the Formulation of Generalization. Journal of Mathematical Behavior, 23, 205-222.

Stacey, K. (2006). What Is Mathematical Thinking And Why Is It Important? University of Melbourne, Australia.

Tall, D. (1993). Students' Difficulties in Calculus. Paper Presented At The Working Group 3 On Students' Difficulties In Calculus, Quebec, Canada.

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

Tall, D. (2004). Introducing Three Worlds of Mathematics. For the Learning of Mathematics, 23(3), 29–33.

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

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

Tatar, E., Zengin, Y. (2016). Conceptual Understanding of Definite Integral with GeoGebra, Interdisciplinary Journal of Practice. Theory, and Applied Research, 33(2), 120-132.

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.

Wagner, J. F. (2018). Students’ Obstacles to Using Riemann Sum Interpretations of the Definite Integral. International Journal of Research in Undergraduate Mathematics Education, 4, 327-356.

West, L.; Staub, F. C. (2003). Content-Focused Coaching: Transforming Mathematics Lessons. Portsmouth, NH: Heinemann.

Wild, M. (2000). Designing and evaluating an educational performance support system. British Educational Communications and Technology Agency, 31(1), 5-20.

Zazkis, R., Liljedahl, P., & Sinclair, N. (2009). Lesson plays: Planning teaching versus teaching planning. For the Learning of Mathematics, 29(1), 40-47.

Downloads

Published

2020-08-27

How to Cite

Hashemi, N., Kashefi, H., & Abu, M. S. (2020). The Emphasis on Generalization Strategies in Teaching Integral: Calculus Lesson Plans. Sains Humanika, 12(3). https://doi.org/10.11113/sh.v12n3.1687

Issue

Section

Articles