Conceptions of Area: In Students and in History

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

“…I wish to note here that students may hold other conceptualizations of the definite integral, beyond what the students in this study exhibited, such as the "method of indivisibles" described in Section 2.1 from Czarnocha et al (2001). The results of this paper may help provide a way for determining whether a given conceptualization, such as "indivisibles," might have productive power for helping students make sense of contextualized definite integrals.…”

“…However, there is an important distinction between "indivisibles" as described by Czarnocha et al (2001) and how the students in Jones (2013) seemed to talk about the integral. That is, the students typically did not think of the rectangles with width dx as having "collapsed" in dimension to straight lines.…”

“…In Jones (2013), it was demonstrated that, within this conceptualization, students tended to think of the x from the Riemann sum as having shrunk to an "infinitesimal" size, or size "zero," before the addition takes place. As a result, this conception is related to what Czarnocha, Dubinsky, Loch, Prabhu, and Vidakovic (2001) call the "method of indivisibles," which considers area under a curve to be the summation of straight lines that fill in the space underneath it. As an example of what I mean, consider Chris' explanation:…”

Geometrical representations in the learning of two-variable functions

“…One can use the theory to study students' constructions after they have finished a course or a series of courses related to the topic under study (Czarnocha, Dubinsky, Loch, Prabhu, & Vidakovic, 2001;Dubinsky, Weller, McDonald, & Brown, 2005;Trigueros, 2000). One can also use the theory to design activities to teach the topic and then analyze which constructions were made by students and which were not (Brown, DeVries, Dubinsky, & Thomas, 1998;Dubinsky & Yiparaki, 2000;McDonald, Mathews, & Strobel, 2000).…”

Technology as a Tool for Teaching Undergraduate Mathematics

“…Other studies into student conceptions of Riemann sums and integration include those by Czarnocha, Dubinsky, Loch, Prabhu, and Vidakovic (2002) and Czarnocha, Loch, Prabhu, and Vidakovic (2001). The first paper concentrates on the relationship between Riemann sums and area and describes different students' intuitive ideas including that of indivisibles.…”