The concept of accumulation in calculus

Patrick W. Thompson, Jason Silverman

Research output: Chapter in Book/Report/Conference proceedingChapter

43 Scopus citations

Abstract

The concept of accumulation is central to the idea of integration, and therefore is at the core of understanding many ideas and applications in calculus. On one hand, the idea of accumulation is trivial. You accumulate a quantity by getting more of it. We accumulate injuries as we exercise. We accumulate junk as we grow older. We accumulate wealth by gaining more of it. There are some details to consider, such as whether it makes sense to think of accumulating a negative amount of a quantity, but the main idea is straightforward. On the other hand, the idea of accumulation is anything but straightforward. First, students find it hard to think of something accumulating when they cannot conceptualize the “bits” that accumulate. To understand the idea of accomplished work, for example, as accruing incrementally means that one must think of each momentary total amount of work as the sum of past increments, and of every additional incremental bit of work as being composed of a force applied through a distance. Second, the mathematical idea of an accumulation function, represented as, involves so many moving parts that it is understandable that students have difficulty understanding and employing it. Readers already sophisticated in reasoning about accumulations may find it surprising that many students are challenged to think mathematically about them. The ways in which it is difficult, though, are instructive for a larger set of issues in calculus.

Original languageEnglish (US)
Title of host publicationMaking the Connection
Subtitle of host publicationResearch and Teaching in Undergraduate Mathematics Education
PublisherMathematical Association of America
Pages43-52
Number of pages10
ISBN (Electronic)9780883859759
ISBN (Print)9780883851838
DOIs
StatePublished - Jan 1 2008

ASJC Scopus subject areas

  • Mathematics(all)

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