Understanding the integral: Students’ symbolic forms

Areas, anti-derivatives, and adding up pieces: Definite integrals in pure mathematics and applied science contexts

“…Sealey and Engelke (2012) suggest that the area conception alone is not sufficient for robust understanding of integrals, and (Jones, 2013) supports this conclusion with an example of a student who struggled to interpret a physics integral through the area lens. Furthermore, Jones describes how a shift to an adding up pieces conception helped this student make sense of the integral he was struggling with.…”

“…This paper builds on a previous study (Jones, 2013) that details several conceptualizations of the definite integral held by calculus students. In this section, the reader is briefly acquainted with three of the conceptualizations described in that study, which deal with how students cognitively hold the familiar notions of area under a curve, anti-derivatives, and Riemann sums in connection with integrals.…”

“…In the last decade there has been increased attention given to researching students' understanding and use of the calculus topic of the definite integral (e.g., Bajracharya & Thompson, 2014;Jones, 2013;Kouropatov & Dreyfus, 2013;Rasslan & Tall, 2002;Sealey, 2014;Thompson & Silverman, 2008;Wemyss, Bajracharya, Thompson, & Wagner, 2011). Integration is a key topic that deserves our attention for several reasons.…”

“…While we are beginning to learn how students create and hold knowledge about the definite integral (see Grundmeier, Hansen, & Sousa, 2006;Jones, 2013;Rasslan & Tall, 2002), there is scant evidence about how these cognitive constructs actually play out for students in making sense of the expressions and formulas in which integrals are present. Sealey and Engelke (2012) suggest that the area conception alone is not sufficient for robust understanding of integrals, and (Jones, 2013) supports this conclusion with an example of a student who struggled to interpret a physics integral through the area lens.…”

“…The study examines, in both of these contexts, student conceptualizations of the definite integral that are related to three common interpretations of the definite integral: (a) as the area under a curve, (b) as the values of an anti-derivative, and (c) as the limit of Riemann sums (see Salas et al, 2006;Stewart, 2012;Thomas et al, 2009). Note that even though the anti-derivative notion is often thought of as the province of indefinite integrals, students show a tendency to interpret definite integrals through the anti-derivative lens as well (Jones, 2013). Each of these three conceptualizations of the integral is evaluated for how helpful or useful it is for making sense of definite integral expressions and equations.…”