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Abstract
This study aims to broadly examine how commonly various conceptualizations of the definite integral are drawn on by students as they attempt to explain the meaning of integral expressions. Previous studies have shown that certain conceptualizations, such as the area under a curve or the values of an anti-derivative, may be less productive in making sense of contextualized integrals. On the other hand, interpreting the integral using Riemann sum-based conceptions proves much more productive for understanding contextualized integrals. This study investigates how frequently students from a US calculus population drew on these three conceptualizations (as well as others) to interpret the meaning of definite integrals. The results were achieved by asking a large sample of students from two US colleges (n = 150) four open-ended questions regarding the underlying meaning of definite integrals. Data from the student responses show a high prevalence of area and anti-derivative ideas and a relatively low occurrence of multiplicatively based summation ideas for interpreting these integrals. Possible reasons for and implications of the results are discussed.
Disclosure statement
No potential conflict of interest was reported by the author.
Notes
1. The bulk of the sample (127 students) came from one institution and the remainder (30 students) came from the second institution.
2. The final sample contained 120 students from one institution and 30 from the other. The results from the students at the two institutions were quite similar and χ2-tests performed between them for each of the four survey items (see section 3.2) showed no significant difference at the α = 0.05 level. The p-values for each survey item were as follows: Item 1, p = 0.382; Item 2, p = 0.499; Item 3, p = 0.634; Item 4, p = 0.879.