# Students’ Obstacles to Using Riemann Sum Interpretations of the Definite Integral

Article

## Abstract

Students use a variety of resources to make sense of integration, and interpreting the definite integral as a sum of infinitesimal products (rooted in the concept of a Riemann sum) is particularly useful in many physical contexts. This study of beginning and upper-level undergraduate physics students examines some obstacles students encounter when trying to make sense of integration, as well as some discomfort and skepticism some students maintain even after constructing useful conceptions of the integral. In particular, many students attempt to explain what integration does by trying to use algebraic sense-making to interpret the symbolic manipulations involved in using the Fundamental Theorem of Calculus. Consequently, students demonstrate a reluctance to use their understanding of “what a Riemann sum does” to interpret “what an integral does.” This research suggests an absence of instructional attention to subtle differences between sense-making in algebra and sense-making in calculus, perhaps inhibiting efforts to promote Riemann sum interpretations of the integral during calculus instruction.

## Keywords

Calculus instruction Definite integral Riemann sums Physics

## References

1. Bezuidenhout, J., & Olivier, A. (2000). Students’ conceptions of the integral. In T. Nakahara & M. Koyama (Eds.), Proceedings of the 24th Conference of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 73–80). Hiroshima, Japan.Google Scholar
2. Doughty, L., McLoughlin, E., & van Kampen, P. (2014). What integration cues, and what cues integration in intermediate electromagnetism. American Journal of Physics, 82, 1093–1103.
3. Dray, T., & Manogue, C. A. (2010). Putting differentials back into calculus. The College Mathematics Journal, 41(2), 90–100.
4. Dray, T., Edwards, B., & Manogue, C. A. (2008). Bridging the gap between mathematics and physics. Retrieved from http://tsg.icme11.org/document/get/659.
5. Engelke, N., & Sealey, V. (2009). The great gorilla jump: A Riemann sum investigation. Paper presented at the 12th Annual Conference on Research in Undergraduate Mathematics Education, Raleigh, NC. Retrived from http://sigmaa.maa.org/rume/crume2009/proceedings.html.
6. Ferrini-Mundy, J., & Graham, K. (1994). Research in calculus learning: Understanding of limits, derivatives, and integrals. In J. Kaput & E. Dubinsky (Eds.), Research Issues in Undergraduate Mathematics Learning, MAA notes #33 (pp. 31–46). Washington, DC: Mathematical Association of America.Google Scholar
7. 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.Google Scholar
8. Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. The Journal of Mathematical Behavior, 32(2), 122–141.
9. Jones, S. R. (2015a). Areas, anti-derivatives, and adding up pieces: Integrals in pure mathematics and applied contexts. The Journal of Mathematical Behavior, 38(1), 9–28.
10. 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.
11. Kieran, C. (2007). Learning and teaching algebra at the middle school through college levels: Building meaning for symbols and their manipulation. In F. K. Lester Jr. (Ed.), Second handbook of research on mathematics teaching and learning (pp. 707–762). Greenwich: Information Age Publishing.Google Scholar
12. Kouropatov, A., & Dreyfus, T. (2014). Learning the integral concept by constructing knowledge about accumulation. ZDM, 46(4), 533–548.
13. 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.
14. Meredith, D. C., & Marrongelle, K. A. (2008). How students use mathematical resources in an electrostatics context. American Journal of Physics, 76(6), 570–578.
15. Nguyen, D. H., & Rebello, N. S. (2011a). Students’ understanding and application of the area under the curve concept in physics problems. Physical Review Special Topics - Physics Education Research, 7(1), 010112.
16. Nguyen, D. H., & Rebello, N. S. (2011b). Students’ difficulties with integration in electricity. Physical Review Special Topics - Physics Education Research, 7(1), 010113.
17. Oehrtman, M. (2009). Collapsing dimensions, physical limitations, and other students metaphors for limit concepts. Journal for Research in Mathematics Education, 40(4), 396–426.Google Scholar
18. Rasslan, S. & Tall, D. (2002). Definitions and images for the definite integral concept. In a. D. Cockburn & E. Nardi (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education (Vol. 4, pp. 89–96). Norwich, UK. Google Scholar
19. Sealey, V. (2006). Definite integrals, Riemann sums, and area under a curve: What is necessary and sufficient. In S. Alatorre, J. L. Cortina, M. Sáiz & A. Méndez (Eds.) Proceedings of the 28th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (Vol. 2, pp. 46-53). Mérida: Universidad Pedagógica Nacional.Google Scholar
20. Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33, 230–245.
21. Sherin, B. L. (2001). How students understand physics equations. Cognition and Instruction, 19(4), 479–541.
22. Thompson, P. W., & Silverman, J. (2008). The concept of accumulation in calculus. In M. P. Carlson & C. Rasmussen (Eds.), Making the connection: Research and teaching in undergraduate mathematics, MAA notes #73 (pp. 43–52). Washington, DC: Mathematical Association of America.Google Scholar
23. Thompson, P. W., Byerly, C., & Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools, 30, 124–147.
24. Wagner, J. F. (2016). Analyzing students’ interpretations of the definite integral as concept projections. In (Eds.) T. Fukawa-Connelly, N. Infante, M. Wawro, and S. Brown, Proceedings of the 19th Annual Conference on Research in Undergraduate Mathematics Education (pp. 1385-1392). Pittsburgh, PA. Retrived from http://sigmaa.maa.org/rume/RUME19v3.pdf.
25. Yeatts, F. R., & Hundhausen, J. R. (1992). Calculus and physics: Challenges at the interface. American Journal of Physics, 60(8), 716–721.