Accumulation, Integrals, and the Fundamental Theorem of Calculus in the Calculus I Pathway
Several scholars have described consequential ways of understanding the fundamental theorem of calculus and how students might apply definite integrals in modeling problems. Multiple studies (Orton, 1983; Orton, 1984; Serhan, 2015; Rasslan & Tall, 2002) suggest that students might not hold quantitative meanings for the components of an integral despite being proficient with integral calculations. Compounding this potential lack of meaning are various studies documenting the challenges students experience when attempting to apply definite integrals to contexts in physics or engineering (Sealey, 2014; Meredith & Marrongellle, 2008; Jones 2013; Jones 2015; Simmons & Oehrtman, 2017; Chhetri & Oehrtman, 2015; Bajracharya & Thompson, 2014). Other research has documented students’ difficulties with coordinating the product structure f(xi)Δx of an accumulated quantity (e.g., Sealey, 2014). Mathematics educators have responded to these difficulties by demonstrating the effectiveness of engaging students in tasks that require them to consider how to approximate the accumulation of a quantity (or to construct a function that represents the value of an accumulated quantity) by assuming that a varying quantity (a rate, a force, etc.) is constant over some interval of its variation, and then to approximate the change in the accumulated quantity over each successive interval by computing the product of the (assumed) constant quantity and the change in the independent variable. Generally, an important instructional goal is to help students conceptualize the product of the integrand and the change in the function’s independent variable as an approximation of the change of the accumulated quantity.
Faculty at the MIP Calculus I Initiation Workshop in August, 2019 recommended that instructional resources developed by CoRDs and ARCs addressing Accumulation, Integrals, and the Fundamental Theorem of Calculus for success in the Oklahoma Calculus I Pathway should:
1. Interpret Riemann sums quantitatively by conceiving individual terms in the sum as approximations of “bits of change” of some quantity and the sum itself as an approximate change in the value of a quantity over a particular interval of the independent quantity’s variation.
2. Interpret Riemann sums geometrically as approximations of bounded areas in the Cartesian plane.
3. Interpret definite integrals in context as the exact change in the value of a quantity over a particular interval of the independent quantity’s variation.
4. Interpret definite integrals geometrically as exact values of bounded areas in the Cartesian plane.
5. Leverage the idea of local linearity to approximate the accumulation of a quantity by assuming that it varies at a constant rate over small (possibly infinitesimal) intervals of the independent variable.
6. Generalize their “adding up pieces” strategy of approximation in kinematic contexts to reason about quantifying accumulation in less intuitive contexts.
Addressing Components of Inquiry
Participants of the MIP Workshop on Calculus I suggested the following ways resources about Accumulation, Integrals, and the Fundamental Theorem of Calculus could address the three MIP components of mathematical inquiry:
Active Learning: By computing successively refined approximations to the accumulation of a quantity, students may engage in a process reflecting the structure of an integral as a limit of Riemann sums. Each approximation will require students to attend to the multiplicative structure f(xi)⋅∆x as an estimate of a portion of the accumulated quantity (based on an assumption of small variation in f across each subinterval). They must attend to the Riemann sum as the estimated accumulation over an interval, and refining their approximations experience the limiting process.
Meaningful Applications: Focusing on quantitative interpretations of all components of a Riemann sum helps students i) understand why the integral is defined as it is, ii) understand how to interpret the meaning of of an integral in context, and iii) understand how to develop an integral to model a quantity in an appropriate situation. Without a focus on quantitative reasoning, students are likely to think that an integral just adds up values of f(x) or only be able to interpret integrals as “area under a curve. Furthermore, asking students to reason about a variety of contexts that involve accumulation (not just area) provides the opportunity for them to generalize their reasoning. As a result, they are better positioned to both interpret an integral in terms of its generalized, abstract mathematical structure and to create or interpret integrals in a broad range of novel contexts.
Academic Success Skills: Students often interpret the algebraic computations involved in applying the fundamental theorem of calculus as the “real math” that replaces ideas about limits of Riemann sums that they dismiss as the “hard way” of working with integrals. Students need experiences seeing that these underlying meanings enable powerful use of ideas about definite integrals.
References
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. In N. Patemen (Ed.), Proceedings of the 2003 Meeting of the International Group for the Psychology of Mathematics Education–North America (Vol. 2, pp. 165–172). Honolulu, HI: University of Hawaii.
Chhetri, K., & Oehrtman, M., (2015) The equation has particles! How calculus students construct definite integral models. Proceedings of the 18th Annual Conference on Research in Undergraduate Mathematics Education, Pittsburgh, Pennsylvania.
Ferrini-Mundy, J., & Graham, K. (1994). Research in Calculus Learning: Understanding of Limits, Derivatives, and Integrals. In J. J. Kaput & E. Dubinsky (Eds.), MAA Notes Number 33 (pp. 31–45). Mathematical Association of America.
Jones, S. (2013). Understanding the integral: Students’ symbolic forms. Journal of Mathematical Behavior, 32, 122–141.
Jones, S. R. (2015). Areas, anti-derivatives, and adding up pieces: Definite integrals in pure mathematics and applied science contexts. The Journal of Mathematical Behavior, 38, 9-28.
Oehrtman, M. & Simmons, C. (2023). Emergent quantitative models for definite integrals. International Journal of Research in Undergraduate Mathematics Education, 9(1), 36-61.
Rasslan S, Tall D. (2002). Definitions and images for the definite integral concept. In: A. Cockburn & E. Nardi (Eds.), Proceedings of the 26th Conference of the International Group for the Psychology of Mathematics Education. Norwich, UK.
Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33, 230-245.
Serhan, D. (2015). Students’ understanding of the definite integral concept. International Journal of Research in Education and Science, 1(1), 84-88.
Simmons, C. & Oehrtman, M. (2017). Beyond the product structure for definite integrals. Proceedings of the 20th Annual Conference on Research in Undergraduate Mathematics Education, San Diego, CA, pp. 912-919.
Thompson, P. W. (1994). Images of rate and operational understanding of the Fundamental Theorem of Calculus. Educational Studies in Mathematics, 26(2-3), 229-274.