Varied ways to teach the definite integral concept

Iiris Attorps; Kjell Björk; Mirko Radic; Timo Tossavainen

doi:10.29333/iejme/275

Full Text (PDF)

Iiris Attorps, Kjell Björk, Mirko Radic, Timo Tossavainen

More Detail

Abstract

In this paper, we report on a collaborative teaching experiment based on the Learning Study model (LS model) which grounds on the Variation Theory. Until today, most of such studies have focused on the teaching and learning of elementary school mathematics; ours was carried out in undergraduate mathematics education. In the following, we discuss how we managed to promote students’ conceptual learning by varying the treatment of the object of learning (the concept of definite integral and the Fundamental Theorem of Calculus) during three lectures on an introductory course in calculus. We also discuss the challenges and possibilities of the LS model and the Variation Theory in the development of the teaching of tertiary mathematics in general. The experiment was carried out at a Swedish university. The data of the study consists of the documents of the observation of three lectures and the students’ answers to the pre- and post-tests of each lesson. The analysis of learning results revealed some critical aspects of the definite integral concept and patterns of variations that seem to be effective to a significant degree. For example, we found several possibilities to use GeoGebra to enrich students’ learning opportunities.

Keywords

definite integral
GeoGebra
learning study
tertiary education
variation

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

INT ELECT J MATH ED, 2013, Volume 8, Issue 2-3, 81-99

https://doi.org/10.29333/iejme/275

Publication date: 10 Nov 2013

Article Views: 3457

Article Downloads: 5232

Open Access References How to cite this article

References

Adams, R. A. (2006). Calculus a complete course. Toronto: Addison Wesley.
Artigue, M. (2001). What can we learn from educational research at the university level? In D. Holton (Ed.), The teaching and learning of mathematics at university level: An ICMI study (pp.207-220). Dordrecht: Kluwer Academic Publishers.
Attorps, I. (2006). Mathematics teachers’ conceptions about equations. Doctoral dissertation, Research Report 266. Department of Applied Sciences of Education, University of Helsinki, Finland.
Attorps, I., Björk, K., Radic, M., & Tossavainen, T. (2010). The learning study model and the teaching of the definite integral concept. In M. Asikainen, P. E. Hirvonen, & K. Sormunen (Eds.), Reports and Studies in Education, Humanities, and Theology, 77-86. University of Eastern Finland, Joensuu, Finland.
Barton, B., & Thomas, M. (2010). Editorial. CULMS Newsletter, 1.
Burton, L. (1989). Mathematics as a cultural experience: Whose experience? In C. Keitel, P. Damerow, A. Bishop & P. Gerdes (Eds.), Mathematics education and society - Science and Technologhy Education (Document Series 35, pp.16-19). Paris: Unesco.
Blum, W. (2000). Perspektiven für den Analysisunterricht. Der Mathematikunterricht, 46(4/5), 5-17.
Booth, S. A. (1992). Learning to program: A pehenomenographic perspective. Göteborg Studies in Educational Sciences, 89. Göteborg: Acta Universitatis Gothoburgensis.
Brown, A. L. (1992). Design experiments: Theoretical and methodological challenges in creating complex interventions in classroom settings. The Journal of the Learning Sciences, 2(2), 141-178.
Calvo, C. (1997). Bases para una propuesta didáctica sorbe integrales. Unpublished Master Thesis.
Camacho, M., Depool, R. & Santos-Trigo, M. (2010). Student’s use Derive software in comprehending and making sense of definite integral and area concepts. CBMS Issues in Mathematics Education, 16, 29-62.
Cobb, P., Confrey, J., diSessa, A., Lehrer, R., & Scauble, L. (2003). Design experiments in educational recearch. Educational Researcher, 32(1), 9-13.
Collins, A.(1992). Towards a design science of education. In E. Scandlon & T. D. Shea (Eds.), New directions in educational technology. Berlin: Springer.
Emanuelsson, J. (2001). En fråga om frågor. Hur lärarens frågor i klassrummet gör det möjligt att få reda på elevernas sätt att förstå det som undervisningen behandlar i matematik och naturvetenskap. Göteborg Studies of Educational Sciences. Göteborg: Acta Universitatis Gothoburgensis.
Gustavsson, L. (2008). Att bli bättre lärare. Hur undervisningsinnehållets behandling blir till samtalsämne lärare emellan. Högskolan i Kristianstad, sektion för lärarutbildningen.
Häggström, J. (2008). Teaching systems of linear equations in Sweden and China: What is made possible to learn? Göteborg Studies of Educational Sciences. Göteborg: Acta Universitatis Gothoburgensis.
Leung, A. (2003). Dynamic geometry and the theory of variation. In N. Pateman, B. J. Doughherty, & J. Zillox, Proceedings of the 27th International Conference for the Psychology of Mathematics Education (Vol. 3, pp.197-204). Honolulu: University of Hawaii.
Lewis, C. (2002). Lesson study: A handbook of teacher-led instructional change. Philadelphia: Research for Better Schools Inc.
Machín, M. C., & Rivero, R. D. (2003). Using DERIVE to understand the concept of definite integral. International Journal for Mathematics Teaching and Learning, 4, 1-16.
Marton, F. (1993). Phenomenography. In T. Husén & T. N. Postlethwaite (Eds.). The International Encyclopaedia of Education (2nd Ed) (pp.4424-4429). Oxford: Pergamon.
Marton, F., & Booth, S. (1997). Learning and awareness. Mahwah, N.J.: Law Earlbaum.
Marton, F., & Morris, P. (Eds.) (2001). What matters? Discovering critical conditions of classroom learning. Göteborg Studies of Educational Sciences. Göteborg: Acta Universitatis Gothoburgensis.
Marton, F., Runesson, U., & Tsui, A. (2004). The space of learning. In F. Marton and A. Tsui (Eds.), Classroom discourse and the space of learning (pp. 3-40). New Jersey: Lawrence Erlbaum Associates, INC Publishers.
Marton, F., & Tsui, A. (2004). Classroom discourse and the space of learning. Mahwah, NJ: Lawrence Erlbaum.
Orton, A. (1983). Student’s understanding of integration. Educational Studies in Mathematics, 14(1), 1-18.
Pang, M. F., & Marton, F. (2003). Beyond “lesson study”: Comparing two ways of facilitating the gasp of some economic concepts. Instructional Science, 31, 175-195.
Rasslan, S., & Tall, D. (2002). Definitions and images for the definite integral concept. In A. Cockburn & E. Nardi (Eds.), Proceedings of the 26th International Conference for the Psychology of Mathematics Education (Vol.4, pp.89-96). Norwich, UK.
Rovio-Johansson, A., & Johansson, I-L. (2005). Lärandets kontextualitet: Hur utvecklas ekonomstudenters förståelse av ett grundläggande ekonomiskt begrepp under utbildningen? Didaktiskt Tidskrift, 16(2-3), 61-92.
Runesson, U. (1999). Variationens pedagogik. Skilda sätt att behandla ett matematiskt innehåll. Göteborg Studies of Educational Sciences. Göteborg: Acta Universitatis Gothoburgensis.
Rösken, B., & Rolka, K. (2007). Integrating intuition: The role of concept image and concept definition for students’ learning of integral calculus. The Montana Mathematics Enthusiast, 3, 181-204.
Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coins. Educational Studies in Mathematics, 22, 1-36.
Stigler, J. W., & Hiebert, J. (1999). The teaching gap. New York: The Free Press.
Tall, D. (1994). The transition to advanced mathematical thinking. Functions, limits, infinity and proof. In D. A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 390-419). New York: Macmillan.
Tall, D., & Vinner, S. (1981). Concept image and concept definition with particular reference to limits and continuity. Educational Studies in Mathematics, 12, 151-169.
Tossavainen, T., Haukkanen, P., & Pesonen, M. (2013). Different aspects of monotonicity. International Journal of Mathematical Education in Science and Technology. doi:10.1080/0020739X.2013.770088.
Touval, A. (1997). Investigating a definite integral – from graphing calculator to rigorous proof. Mathematics Teacher, 90(3), 230-232.
Vinner, S. (1991). The role of definitions in teaching and learning. In D. Tall (Ed.). Advanced mathematical thinking (Chapter 5, pp.65-81). Dordrecht: Kluwer Academic Publishers.
Viirman, O., Attorps, I., & Tossavainen, T. (2011). Different views – Some Swedish mathematics students’ concept images of the function concept. Nordic Studies in Mathematics Education, 15(4), 5-24.

How to cite this article

APA

Attorps, I., Björk, K., Radic, M., & Tossavainen, T. (2013). Varied ways to teach the definite integral concept. International Electronic Journal of Mathematics Education, 8(2-3), 81-99. https://doi.org/10.29333/iejme/275

Vancouver

Attorps I, Björk K, Radic M, Tossavainen T. Varied ways to teach the definite integral concept. INT ELECT J MATH ED. 2013;8(2-3):81-99. https://doi.org/10.29333/iejme/275

AMA

Attorps I, Björk K, Radic M, Tossavainen T. Varied ways to teach the definite integral concept. INT ELECT J MATH ED. 2013;8(2-3), 81-99. https://doi.org/10.29333/iejme/275

Chicago

Attorps, Iiris, Kjell Björk, Mirko Radic, and Timo Tossavainen. "Varied ways to teach the definite integral concept". International Electronic Journal of Mathematics Education 2013 8 no. 2-3 (2013): 81-99. https://doi.org/10.29333/iejme/275

Harvard

Attorps, I., Björk, K., Radic, M., and Tossavainen, T. (2013). Varied ways to teach the definite integral concept. International Electronic Journal of Mathematics Education, 8(2-3), pp. 81-99. https://doi.org/10.29333/iejme/275

MLA

Attorps, Iiris et al. "Varied ways to teach the definite integral concept". International Electronic Journal of Mathematics Education, vol. 8, no. 2-3, 2013, pp. 81-99. https://doi.org/10.29333/iejme/275