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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
There are, of course, entirely valid ways of extracting conceptual sense of the definite integral by interpreting antiderivatives of the integrand. These do not, however, make use of Riemann sum reasoning, nor do they shed light on the symbolic manipulations involved in finding the antiderivatives.
One beginning physics student, B1, showed sporadic use of Riemann sum-based reasoning, but did not use it consistently and he could not decide on the units for The Integral of Position Problem (see below). Primarily for reasons of space, this case will not be examined here, but I will use the language that “all beginning students” failed to use this reasoning strategy for ease of presentation.
A few beginning students occasionally made mention of sums or rectangles while discussing the definite integral, however none of them could successfully use them to explain what the integral did or what its units were.
The area chosen by U5 lends itself to a more general geometric analysis that reveals algebraic sense behind the antiderivative area calculation as the difference of areas of triangles. This geometric argument does not hold for polynomials of higher degree, however, nor does it shed any light on why the algebraic representation should correspond to the antiderivative of the bounding function.
Both of his interpretations, with some careful tweaking, were quite reasonable, but the details are outside the scope of this paper.
References
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.
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.
Dray, T., & Manogue, C. A. (2010). Putting differentials back into calculus. The College Mathematics Journal, 41(2), 90–100.
Dray, T., Edwards, B., & Manogue, C. A. (2008). Bridging the gap between mathematics and physics. Retrieved from http://tsg.icme11.org/document/get/659.
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.
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.
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.
Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. The Journal of Mathematical Behavior, 32(2), 122–141.
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.
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.
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.
Kouropatov, A., & Dreyfus, T. (2014). Learning the integral concept by constructing knowledge about accumulation. ZDM, 46(4), 533–548.
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.
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. (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.
Nguyen, D. H., & Rebello, N. S. (2011b). Students’ difficulties with integration in electricity. Physical Review Special Topics - Physics Education Research, 7(1), 010113.
Oehrtman, M. (2009). Collapsing dimensions, physical limitations, and other students metaphors for limit concepts. Journal for Research in Mathematics Education, 40(4), 396–426.
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.
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.
Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33, 230–245.
Sherin, B. L. (2001). How students understand physics equations. Cognition and Instruction, 19(4), 479–541.
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.
Thompson, P. W., Byerly, C., & Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools, 30, 124–147.
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.
Yeatts, F. R., & Hundhausen, J. R. (1992). Calculus and physics: Challenges at the interface. American Journal of Physics, 60(8), 716–721.
Acknowledgements
This research was supported in part by the National Science Foundation under grant No. PHY-1405616, and in part by a faculty development leave from Xavier University. Any opinions, findings, conclusions, or recommendations expressed in this paper are those of the author and do not necessarily reflect the views of the National Science Foundation or Xavier University. I would like to thank Corinne Manogue, Tevian Dray, John Thompson, and Mike Loverude for their thoughtful conversations and feedback throughout the course this research.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wagner, J.F. Students’ Obstacles to Using Riemann Sum Interpretations of the Definite Integral. Int. J. Res. Undergrad. Math. Ed. 4, 327–356 (2018). https://doi.org/10.1007/s40753-017-0060-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40753-017-0060-7