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I have long held the belief (Bressoud, 2011) that we should revert to the original name, the Fundamental Theorem of Integral Calculus (FTIC), for what in the 1960s came to be known as the Fundamental Theorem of Calculus (FTC). The reason is that the real importance of this theorem is not that integration and differentiation are inverse processes—for most students that is the working definition of integration—but that we have two very distinct ways of viewing integration, as limits of Riemann sums and in terms of anti-differentiation, and that for all practical purposes they are equivalent.
Figure 1. Students working on integral as accumulator, reproduced from the homepage of Coherent Labs to Enhance Accessible and Rigorous Calculus Instruction (CLEAR Calculus)
A recent paper by Joseph Wagner (2017) is an insightful study of the confusion experienced by most students about the nature of integration. As he points out, this is not about student deficits, but about common misconceptions that can be traced to the way we teach integration.
Previous work by Sealey (2006, 2014) and Jones (2013, 2015a, 2015b) has shown that
there are three ways in which students describe the meaning of the definite integral,
Nevertheless, when confronted with a problem in physics that requires integration, the interpretation in terms of a summation is more common. Jones (2015b), after reminding second term calculus students that force is pressure times area, asked why
calculates the total force. Of 150 students, 61 (41%) produced an argument that involved summation, although only 25 of them (17%) indicated that any product was involved.Following up on this insight, Wagner explored the understanding of definite integrals by physics students. He interviewed eight students in an introductory calculus-based physics course focused on classical mechanics and seven third-year physics majors. Of the students in the introductory course, five had completed both single and multi-variable calculus, two were currently enrolled in multi-variable calculus, and one was still in single variable calculus. All were in majors that required this physics course.
When students in the introductory course were asked what Riemann sums have to do with definite integrals, they split evenly between two types of answers: either as something that accomplishes the same task as an integral (usually finding areas) or as a means of approximating definite integrals. As we shall see, the connection between integration as a limit of Riemann sums and in terms of antiderivatives was hazy at best and not recognized as significant. As Wagner reports, several were mystified why they had to learn about Riemann sums, “Because like when they were teaching this, they were kind of like oh, like you’ll do this for the first test, and then you get rid of it and never have to do it again.”
On the other hand, the third-year physics students were much more inclined to explain the meaning of the definite integral in terms of a summation. They were conversant with how to convert an accumulation problem into a definite integral. As Wagner suggested privately, this appears to be the result of repeated exposure to problems from physics in which definite integrals arise from “slice and add” procedures.
But Wagner uncovered an intriguing gap in their understanding. All fifteen students were asked to make up a simple area problem and then solve it. All of them did so correctly, using a polynomial function and antidifferentiation. As an example the area under the graph of y=x^3 from 0 to 2 was calculated as follows,
He then pushed each of these students to explain why this sequence of calculations produced the area. Only one of the fifteen, a third-year physics student, indicated that this was a consequence of FTC. Several of the others struggled to make sense of how the symbols in the definite integral led to the functional transformation implied by the first equality. Wagner argues that many students are looking for algebraic sense-making in that first equality. With two of the third-year students, he documented their growing sense of frustration as they realized that they could not explain why it works. Quoting the first student:
"Yeah, I do it. I don’t–. I’m not proud of it, but I hope there is some way to justify it. […] When I think about integration as a sum of differentials, quantities–. When I think about that, I go, OK, that makes intuitive sense, and it works. Great. But then I wonder, you know, what is, in terms of more modernized math that I’m doing. Because I usually feel like what I’m doing is kind of a trick. And it works. I don’t feel great about doing this, like, intuitively I feel fine."
From the second student:
"So math gives us these sort of weird tools, and they behave differently than any, like, the physical tools we know of, and it doesn’t really make sense to ask why they work or how they work, because they work mathematically, not physically. So this mathematical tool called the integral allows us to change functions, to apply this operation that changes functions into other functions."
Wagner concludes this article with a thoughtful discussion of the distinction between the algebraic equivalence of two expressions, a notion of equivalence with which students are familiar, and the transformational equivalence that is enabled by FTC. As he laments, “Nothing, however, in the standard calculus curriculum prepares students for the sudden transition from making sense of the symbolic processes of algebra to making sense of the symbolic processes of calculus.” He points out that a great deal of attention has been devoted to a Riemann-sum based understanding of the definite integral, but virtually none to helping students understand the transformational aspects of calculus that are so central.
I believe that a shift from FTC to FTIC can help. As Thompson with others (2008, 2013, 2016) has shown, and I have discussed in earlier columns (Re-imagining the Calculus Curriculum, I, and Re-imagining the Calculus Curriculum, II), it makes sense to first develop the definite integral as an accumulator, making it very clear that Riemann sums are neither an introduction to a subject that eventually will be about antiderivatives nor just a tool for finding approximations, but the very essence of what a definite integral is and how it is used. Then, we bring in FTIC to show that there is another—entirely distinct because it is transformational—expression for this same integral and that this equivalent expression facilitates calculation. Wagner’s third-year physics students were struggling because they failed to realize that integration has these two very different manifestations. It is a very big deal that it does.
References
Bressoud, D. (2011). Historical reflections on teaching the Fundamental Theorem of Integral Calculus. American Mathematical Monthly. 118:99–115. http://www.jstor.org/stable/10.4169/amer.math.monthly.118.02.099?seq=1 - page_scan_tab_contents
Jones, S. R. (2013). Understanding the integral: Students’ symbolic forms. The Journal of Mathematical Behavior, 32(2), 122–141. https://www.sciencedirect.com/science/article/pii/S0732312312000612
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. https://www.sciencedirect.com/science/article/pii/S0732312315000024
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. http://www.tandfonline.com/doi/abs/10.1080/0020739X.2014.1001454
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. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.500.3209&rep=rep1&type=pdf
Sealey, V. (2014). A framework for characterizing student understanding of Riemann sums and definite integrals. The Journal of Mathematical Behavior, 33, 230–245. https://www.sciencedirect.com/science/article/pii/S0732312313001065
Thompson, P.W., and 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 Vol. 73, pp. 43–52). Washington, DC: Mathematical Association of America. https://doi.org/10.5948/UPO9780883859759.005
Thompson, P.W., Byerley, C. and Hatfield, N. (2013). A conceptual approach to calculus made possible by technology. Computers in the Schools. 30:124–147. http://pat-thompson.net/PDFversions/2013CalcTech.pdf
Thompson, P.W., and Dreyfus, T. (2016). A coherent approach to the Fundamental Theorem of Calculus using differentials. In R. Göller. R. Biehler & R. Hochsmuth (Eds.), Proceedings of the Conference on Didactics of Mathematics in Higher Education as a Scientific Discipline (pp. 355–359 ) Hannover, Germany: KHDM. https://www.researchgate.net/publication/306108323_A_Coherent_Approach_to_the_Fundamental_Theorem_of_Calculus_Using_Differentials
Wagner, J.F. (2017). Students’ obstacles to using Riemann sum interpretations of the definite integral. International Journal of Research in Undergraduate Mathematics Education. https://doi.org/10.1007/s40753-017-0060-7
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