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Derivative Conceptions Over Riemann Sum-Based Conceptions in Students’ Explanations of Definite Integrals

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Derivative Conceptions Over Riemann Sum-Based Conceptions in Students’ Explanations of Definite Integrals International Journal of Mathematical Education in Science and Technology ISSN: 0020-739X (Print) 1464-5211 (Online) Journal homepage: http://www.tandfonline.com/loi/tmes20 The prevalence of area-under-a-curve and anti- derivative conceptions over Riemann sum-based conceptions in students’ explanations of definite integrals Steven R. Jones To cite this article: Steven R. Jones (2015) 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 Mathematical Education in Science and Technology, 46:5, 721-736, DOI: 10.1080/0020739X.2014.1001454 To link to this article: https://doi.org/10.1080/0020739X.2014.1001454 Published online: 29 Jan 2015. Submit your article to this journal Article views: 193 View related articles View Crossmark data Citing articles: 6 View citing articles Full Terms & Conditions of access and use can be found at http://www.tandfonline.com/action/journalInformation?journalCode=tmes20 International Journal of Mathematical Education in Science and Technology, 2015 Vol. 46, No. 5, 721–736, http://dx.doi.org/10.1080/0020739X.2014.1001454 The prevalence of area-under-a-curve and anti-derivative conceptions over Riemann sum-based conceptions in students’ explanations of definite integrals Steven R. Jones∗ Department of Mathematics Education, Brigham Young University, Provo, UT, USA (Received 29 May 2014) 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. Keywords: calculus; definite integral; Riemann sum; area; anti-derivative 1. Introduction The first-year calculus concept of the definite integral has become a topic of significant interest in the undergraduate mathematics education community in the last decade (e.g. [1–9]). Perhaps this interest is due to the fact that the integral is used extensively in both advanced mathematics (see [10,11]) and applied sciences (see [12,13]). Since the definite integral is a foundational concept, helping students understand and work with it should be an important educational goal for calculus instruction. Unfortunately, however, several studies demonstrate that students are struggling to use their knowledge of integration effectively in both mathematics and science courses. For example, students appear to have difficulties using graphic representations to work with integrals,[14–16] and students often fail to grasp the meaning of the definition of the definite integral, or how the definition relates to the Fundamental Theorem of Calculus (FTC).[5,9,17,18] These problems may leave students unprepared for subsequent course- work that requires an understanding of definite integrals (see [6,15]). These findings have led some researchers to begin to examine why students are having this difficulty. Sealey and Engelke [19] have suggested that the ‘area under a curve’ notion by itself is not sufficient for understanding definite integrals and Bressoud [20] has proposed ∗Email: [email protected] C 2015 Taylor & Francis 722 S.R. Jones that students’ over-reliance on anti-derivatives may be at fault. On the other hand, Jones [1] analysed an example of a student who productively used a multiplication-summation conception to make good sense of a physics integral. Yet these three papers, individually or collectively, do not make up a deep analysis of whether these three conceptualizations are productive or not. Jones [21] subsequently analysed the anti-derivative, area under a curve, and Riemann sum-based interpretations of the definite integral in both mathematics and science contexts. The results demonstrate that a multiplicatively based summation conception is highly productive for understanding definite integrals that are either situated in a larger context or that contain variables representing physical quantities. For this paper a ‘multiplicatively-based summation conception’ is one that incorporates both (1) the multiplicative relationship between the integrand and the differential to produce a ‘resultant quantity’ and (2) the summation or accumulation of this resulting quantity throughout the domain 560 prescribed by the bounds of integration (see [21]). By contrast, the study confirms that the ‘area under a curve’ and ‘values of an anti-derivative’ conceptions are less productive in making sense of these types of contextualized definite integrals. While the findings do not imply that the area and anti-derivative ideas are not important (nor that they should not be learned) they do suggest that it is critical for students to both have and actively draw on a robust and accessible summation conception of integration. Based on these results, it seems important to know what types of conceptions the general calculus student population tends to draw on for making sense of integrals in mathematics and physics contexts. For example, if I am a second-semester calculus instructor, or an introductory-level, calculus-based physics instructor, what can I expect from my students who walk through the door? Would the students, in general, draw on a Riemann sum conception to interpret a definite integral I write on the board, or would they typically rely on area or anti-derivative notions? No study to date has attempted to survey a large sample of calculus students in order to see how commonly the students draw on each of these three conceptualizations to explain the meaning of definite integrals. Consequently, two important questions emerge that this paper aims to address: (1) How prevalently do students draw on the (a) area under a curve, (b) anti-derivative, or (c) multiplicatively based summation conceptualizations of the definite integral to explain (in writing) the meaning of integral expressions? (2) Do these three conceptualizations tend to categorize the vast majority of students’ (written) interpretations of definite integrals, or are there other student conceptualizations of the integral that can be documented and understood from the students’ written responses? The answer to the first question will enable instructors of calculus-based courses to understand the make-up of a typical group of their students, in regards to how the students attempt to interpret integral expressions presented in their classes. The answer to the second question will provide instructors and researchers with a baseline of the various ways in which students tend to interpret definite integral expressions. If the results show that students are not typically drawing on summation conceptions to interpret definite integrals, then instruction pertaining to the definite integral needs to be re-evaluated to determine how best to support not only the creation, but also the activation of this important idea in mathematics, science, and engineering courses. 2. Theoretical perspective 2.1. Students’ conceptualizations of the integral For this study, the manner in which students hold their knowledge of the integral is char- acterized through the lens of symbolic forms.[22] In brief, a symbolic form is a blend International Journal of Mathematical Education in Science and Technology 723 between a symbol template and a conceptual schema. The symbol template refers to the [] d d arrangement of the symbols in an equation or expression, such as [] [] [] or [] [] [], where each ‘box’ can be filled in with symbols. The conceptual schema, on the other hand, is the underlying meaning ascribed to the symbols in the template and their relationships to each other. Jones [1] documented several students’ symbolic forms of the definite integral, including ones that are associated with the standard notions of area under a curve, values of an anti-derivative, and Riemann sums, as well as a ‘deviant’ of the typical Riemann sum conception. These four symbolic forms aided the analysis of the student data by helping determine when students were drawing on each of the area, anti-derivative, or summation conceptualizations of the integral. A brief description of these symbolic forms is provided here (for a more detailed account, see [1]). Perimeter and area. Each ‘box’ in the symbol template corresponds to one part of the perimeter of a shape in the (x–y) plane, whose area is the value of the integral. An important component of this symbolic form is that the differential, ‘d[],’ dictates the variable that resides on the horizontal axis, which represents the ‘bottom’ of the shape. Function matching. This conceptualization views the integrand as having come from
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