Understanding the definite integral with the help of Riemann sums

Year 2022, Volume: 9 Issue: 3, 445 - 465, 01.05.2022

Özkan Ergene , Ahmet Şükrü Özdemir

https://doi.org/10.17275/per.22.75.9.3

Cited By: 1

Abstract

Students encounter difficulties in understanding integral because the concept of integral requires the usage of theorems, formulas, daily life practices, and interdisciplinary approaches. From this point of view, in this study we examine the effects of a teaching process consisting of modelling activities on understanding the definite integral with the help of Riemann sums. The research is designed according to the case study based on a qualitative research method. Participants consist of 28 pre-service mathematics teachers who have limited understanding of integral although they have completed a Calculus course. The modelling activities were prepared in accordance with the emergent modelling approach. Data were collected through integral test and semi-structured interviews conducted before and after the teaching process. Before the teaching process consisting of modelling activities, pre-service mathematics teachers’ knowledge about the definite integral was included in the area under a curve, inverse of derivative and integral with known boundaries. In addition, participants did not refer to Riemann sums or cumulative sums. After the teaching process consisting of modelling activities, it was seen that almost all participants could explain the following equality (lim)┬(n→∞)⁡〖∑_(k=1)^n▒f(c_k ) ∆x_k 〗=∫_a^b▒f(x)dx. At the end of the teaching process consisting of modelling activities, this process has enabled pre-service teachers to establish the relationship between Riemann sum and definite integral. Findings revealed that teaching process of the definite integral enhanced the understanding the definite integral.

Keywords

definite integral, Riemann sums, modelling activities, teaching process

Thanks

The manuscript was produced from the doctoral dissertation the first author completed under the supervision of the second author.

References

• Adams, R., & Essex, C. (2010). Calculus a complete course. Toronto, Ontario: Pearson.
• Artigue, M. (1991). Analysis, in Advanced Mathematical thinking, edited by D. Tall. Kluwer, Boston, pp. 167–198
• Bajracharya, R., & Thompson, J. (2014). Student understanding of the fundamental theorem of calculus at the mathematics-physics interface. Proceedings of the 17th special interest group of the Mathematical Association of America on research in undergraduate mathematics education. Denver (CO).
• Berry, J. S., & Nyman, M. A. (2003). Promoting students’ graphical understanding of the calculus. The Journal of Mathematical Behavior, 22(4), 479-495.
• Carlson, M. P., Smith, N., & Persson, J. (2003). Developing and Connecting Calculus Students' Notions of Rate-of Change and Accumulation: The Fundamental Theorem of Calculus. International Group for the Psychology of Mathematics Education, 2, 165-172.
• Chapell, K. K., & Kilpartrick, K. (2003). Effects of concept-based instruction on students’ Conceptual understanding and procedural knowledge of calculus. PRIMUS: problems, resources, and issues in mathematics undergraduate studies, 13(1), 17-37.
• Chhetri, K., & Oehrtman, M. (2015). The Equation Has Particles! How Calculus Students Construct Definite Integral Models. In Proceedings of the 18th Annual Conference on Research in Undergraduate Mathematics Education (pp. 19-25).
• Cohen, L., Manion, L., & Morrisson, K. (2000). Research methods in education (5th Ed.). London: Routlenge Falmer.
• Creswell, J. W. (2003). Research design: Qualitative, quantitative, and mixed methods approaches (2nd ed.). Thousand Oaks, CA: Sage
• Darvishzadeh, M., Shahvarani-Semnani, A., Alamolhodaei, H., & Behzadi, M. (2018). Analysis of student´s challenges and performances in solving integral´s problems. Journal for Educators, Teachers and Trainers, 9(1), 164 – 177.
• De Souza, C. (2012). The Greek method of exhaustion: Leading the way to modern integration. Columbus, Ohio: The Ohio State University.
• Doorman, L. M., & Gravemeijer, K. P. E. (2009). Emergent modeling: discrete graphs to support the understanding of change and velocity. ZDM Mathematics Education, 41(1), 199-211.
• Erbaş, A. K., Çetinkaya, B., Alacacı, C., Çakıroğlu, E., Aydoğan Yenmez, A., Şen et all. (2016). Lise matematik konuları için günlük hayattan modelleme soruları [Real life modelling questions for high school mathematics topics]. Ankara: Türkiye Bilimler Akademisi.
• Ferrini-Mundi, J., & Graham, K. (1994) Research in calculus learning: Understanding of limits, derivatives and integrals. In J. J. Kaput & E. Dubinsky (eds.), Research Issues in Undergraduate Mathematics Learning, MAA (Notes 33, pp.31-45). Washington DC: MAA.
• Girard, N. R. (2002). Students' representational approaches to solving calculus problems: Examining the role of graphing calculators. Unpublished EdD, Pittsburg: Universityof Pittsburg.
• González-Martín, A. S., & Camacho, M. (2004). What is first-year Mathematics students' actual knowledge about improper integrals?. International journal of mathematical education in science and technology, 35(1), 73-89.
• Gravemeijer, K. (1999). How emergent models may foster the constitution of formal mathematics. Mathematical thinking and learning, 1(2), 155-177.
• Gravemeijer, K. (2007). Emergent modeling as precursor to mathematical modelling. In W. Blum, P. Galbraith, W. Henn ve M. Niss Modelling and applications in mathematics education. The 14th ICMI Study (pp. 137-144). New York: NY: Springer.
• Gravemeijer, K., & Stephan, M. (2002). Emergent models as an instructional design heuristic. In K. Gravemeijer, R. Lehrer, B. Oers ve L. Verschaffel, Symbolizing, modeling and tool use in mathematics education (pp. 145-169). Dordrecht, the Netherlands: Kluwer Academic Publishers.
• Grundmeier, T. A., Hansen, J., & Sousa, E. (2006). An exploration of definition and procedural fluency in integral calculus. Problems, Resources, and Issues in Mathematics Undergraduate Studies, 16(2), 178-191.
• Heath, T. (1912). The method of Archimedes. Cambridge: Cambridge University Press.
• Jones, S. R. (2015a). Areas, anti-derivatives, and adding up pieces: Definite integrals in pure mathematics and applied science contexts. The Journal of Mathematical Behavior. 38, 9–28.
• Jones, S. R. (2015b). The prevalence of area-under-a-curve and anti-derivative conceptions over Riemann-sum based conceptions in students' explanations of definite integrals. International Journal of Mathematics Education in Science and Technology, 46(5), 721–736.
• Jones, S. R. (2018). Prototype images in mathematics education: The case of the graphical representation of the defnite integral. Educational Studies in Mathematics, 97(3), 215–234.
• 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(6), 1075-1095.
• Knuth, E. J. (2002). Secondary school mathematics teachers’ conceptions of proof. Journal for research in mathematics education, 33(5), 379-405. Koçak, Ş., & Erdoğan, N. K. (2013). Matematik II [Mathematics II]. Anadolu Üniversitesi.
• Mahir N. (2009) Conceptual and procedural performance of undergraduate students in integration, International Journal of Mathematical Education in Science and Technology. 40(2), 201-211.
• McDermott, L. E., Rosenquist, M.L., & van Zee, E. H. (1987). Student difficulties in connecting graphs and physics: Examples from kinematics. American Journal of Physics. 55. 503-513.
• Meredith, D. C., & Marrongelle, K. A. (2008). How students use mathematical resources in an electrostatics context. American Journal of Physics, 76(6), 570-578.
• Nguyen, D. H., & Rebello, N. S. (2011). Students’ understanding and application of the area under the curve concept in physics problems. Physical Review Special Topics- Physics Education Research. 7, 010112.
• Orton, A. (1983). Students' understanding of differentiation. Educational studies in mathematics, 14(3), 235-250.
• Orton, A. (1984). Understanding rate of change. Mathematics in school, 13(5), 23-26.
• Ostebee, A., & Zorn, P. (1997). Calculus from graphical, numerical and symbolic points of view. Fort Worth, TX: Saunder College Publishing.
• Rasslan, S., & Tall, D. (2002). Definitions and images for the definite integral concept. In Cockburn A.; Nardi, E. (eds.) Proceedings of the 26th PME, 4, 89-96.
• Sealey, V. (2008). Calculus students' assimilation of the Riemann integral into a previously established limit structure. Unpublished doctoral dissertation, Arizona: Arizona State University.
• Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33(1), 230–245.
• Serhan, D. (2015). Students’ Understanding of the Definite Integral Concept. International Journal of Research in Education and Science, 1(1), 84-88.
• Sevimli, E. (2013). Bilgisayar cebiri sistemi destekli öğretimin farklı düşünme yapısındaki öğrencilerin integral konusundaki temsil dönüşüm süreçlerine etkisi [The effect of a computer algebra system supported teaching on processes of representational transition in integral topics of students with different types of thinking] Unpublished doctoral dissertation. Marmara University.
• Shekutkovski, N. (2013). Definition of the definite integral. The Teaching of Mathematics, 16(1), 1, 29–34.
• Simmons, C., & Oehrtman, M. (2017). Beyond the product structure for definite integrals Proceedings of the 20th special interest group of the Mathematical Association of America on research in undergraduate mathematics education. SIGMAA on RUME.
• Stein, S.K., & Barcellos, A. (1992) Calculus and Analytic Geometry. 5th Edition, McGraw-Hill, Inc., New York.
• Thomas, G., & Finney, R. (1998). Calculus and analytic geometry (9th Edition). Wokingham, England: Addison-wesley Publishing Company.
• Thomas, G., Weir, M., Hass, J., & Heil, C. (2014). Thomas’ Calculus. (13th Edition) Boston: Pearson Education.
• Thompson, P. W. (1994). The development of the concept of speed and its relationship to concepts of rate. In G. Harel, & J. Confrey. The development of multiplicative reasoning in the learning of mathematics (pp. 181-234). Albany: NY: SUNY Press
• Thompson, P. W., & Silverman, J. (2007). The Concept of accumulation in calculus. In M. Carlson ve C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics (pp. 117-131).
• Wagner, J. F. (2018). Students’ obstacles to using Riemann sum interpretations of the definite integral. International Journal of Research in Undergraduate Mathematics Education, 4(3), 327-356.
• Yin, R. (2003). Case study research: Design and methods (3th edition). Thousand Oaks, Sage Publications.