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.

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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 some other ‘original function,’ which has become the integrand through a derivative. The purpose of the integral is to ‘get back’ to this original function. A key feature of this form is that the differential, ‘d[],’ indicates what the variable of the function is. Adding up pieces (i.e. multiplicatively based summation). This form casts the differential, ‘d[],’ as being an infinitesimally tiny piece of the domain, wherein the quantities represented by the integrand and differential are multiplied together to create a small amount of the resultant quantity. The integral symbol dictates an ‘infinite’ summation of this resultant quantity that ranges over the domain. Adding up the integrand. This conceptualization is a problematic deviant of the Riemann sum. The difference is that within each tiny piece, only the quantity represented by the integrand is added up. For example, in the integral of density with respect to volume, mass = S ρ dV,itisdensity that is added up within each piece to get the ‘total density’. This resulting total from the integrand is then multiplied by the entire domain (length, area, or volume) to get the resulting quantity, or in this example ‘total density × total volume = total mass’.

2.2. Manifold view of knowledge Symbolic forms can be considered a subset of ‘cognitive resources’,[23] which are any piece of cognition that can be drawn on and employed as a unit, whether large or small. The main idea from cognitive resources that is used for this paper is the push away from a ‘unitary view’ of concepts to a ‘manifold view’ of knowledge.[24] The theory of cognitive resources argues that a ‘concept’ such as the integral comprises many small and large elements – like rectangles, graphs, functions, ideas about summation, areas, limits, anti-derivative rules, and so forth – that are too complex to be considered a single entity. Thus, this study assumes that students can think about the integral by drawing on certain aspects of their integral knowledge while other aspects either remain dormant or are cognitively ‘set aside’. This process is known as ‘activating resources’.[23,25]For example, a student may look at an integral and immediately think ‘area under a curve’ without ever thinking about Riemann sums. Or they might, in fact, initially think ‘Riemann sums,’ but decide they would rather think about anti-derivatives instead and base their subsequent interpretation on anti-derivatives. This suggests that students may possess a 724 S.R. Jones given conception in their cognition, but choose to draw on a different one to make sense of an integral expression. It may be that the conceptualization the student draws on is more familiar or readily accessible, which might make them prefer it over others. The results of the study are interpreted through this lens. That is, this paper is not claiming that the responses to this study’s survey constitute an exhaustive probing of a student’s entire understanding of integrals. Instead, this study is attempting to relate the frequency with which students from a calculus population might draw on a given conceptualization in their attempts to explain the meaning of deﬁnite integral expressions.

2.3. Framing Since students may hold several different notions of the integral in their cognition, a tacit ‘choice’ must be made about which conceptualization to draw on for a given task. This choice is made through framing,[24,26] which is how an individual goes about interpreting a given situation in order to determine how to think or act. Thus, a student’s expectations regarding the speciﬁc survey items they are responding to and what counts as a ‘good answer’ affect which symbolic forms the student might employ. Framing played a key role in the design of this study as a justiﬁcation to include both ‘pure mathematics’ and ‘applied physics’ items in the survey. This design allowed for increased opportunities for students to activate different aspects of their integral knowledge.

3. Methods 3.1. Student recruitment In order to investigate the prevalence of the area, anti-derivative, summation, and other conceptions of the definite integral, 157 students at two large colleges1 in the United States who had successfully completed first-semester calculus were recruited to participate in a four-question survey that asked them open-ended questions about definite integrals. The choice to use successful first-semester students is based on two reasons. First, many fundamental aspects of the integral are explored during the first semester at these schools: areas under curves, the Riemann integral definition, the FTC, the Net Change Theorem (NCT), velocity/position applications, and anti-derivative techniques (including u-substitution). Second, there are many courses in mathematics, physics, and engineering that make use of the definite integral, but have only first-semester calculus as a prerequisite (see Section 3.2 for greater detail on this point). Thus, it is important to understand how students attempt to interpret the definite integrals in these courses with only a first-semester calculus background. To recruit students, several second-semester calculus courses were visited within the first two days of the semester in order to administer the four-question survey. Students who had already taken second semester calculus (or the equivalent calculus BC in high school) were asked not to complete the survey, in order to keep the focus of the study only on students who had finished first-semester calculus. Nine sections of calculus (roughly 30–35 students each) in total were visited, yielding an average of about 17–18 participants from each section. Of those not participating, the majority had already taken the equivalent of second semester calculus, meaning that the actual response rate of those eligible to participate was quite high. The students in these courses came from a variety of backgrounds, including having taken first-semester calculus at a four-year university, at a community college, or in high school (calculus AB). International Journal of Mathematical Education in Science and Technology 725

The students who participated were given 15 minutes to complete the four-question survey. Only surveys that had responses to at least two of the four items were included in the data. Consequently, seven of the surveys were excluded since those students did not answer either three or all four of the questions. The exclusion of these students yielded a ﬁnal sample size of n = 150 students.2 Given the limited time frame, it is obvious that the results of this study cannot account for all of these students’ possible conceptions of the deﬁnite integral. However, this is not the point of the study. Rather, this study investigates how commonly students draw on various interpretations of the integral as they initially try to make sense of integral expressions.

3.2. Survey items The four survey items were split between two ‘pure mathematics’ problems and two ‘applied physics’ problems. The questions alternated between math and physics items, so that items 1 and 3 were pure mathematics and items 2 and 4 were applied physics. This was done purposefully to mix together the two contexts, hopefully providing students more opportunities to draw on their ideas about definite integrals (see Section 2.3). The four survey items are listed in Table 1. The first survey item asked the students to describe as much as they could about the integral, and this instruction was reiterated to the students when the surveys were administered. The intent of this first item was to let students take whatever direction they wanted in their interpretations, without the item implying any particular conceptualization of the integral. This allows for an examination of which conceptualizations of definite integrals the students tend to use to describe the meaning of a generic, decontextualized integral expression. A note should be made here that in a study by Rasslan and Tall [9], a very similar question to item 1 was asked. However, there are important differences between their survey question and this item. The questionnaire used by Rasslan and Tall included six items, the last of which was an open-ended question that asked the students to state what b they thought a f (x) dx was. Yet, the first five items were all calculational in nature and all dealt with either the area under graphs (items 2, 3, and 5) or anti-derivatives (items 1, 3, 4, and 5) – including a special case involving improper integrals. Thus, the students

Table 1. Survey items presented to the students. b f x dx Item 1 (math) Explain in detail what a ( ) means. If you think of more than one way to describe it, please describe it in multiple ways. Please use words, or draw pictures, or write formulas, or anything else you want to explain what it means. Item 2 (physics) From physics, force = pressure × area. If the pressure on a surface, S,is not the same at every point, we use an integral to calculate it: F = PdA S . Why does this integral actually calculate the total force on the surface? That is, what is the integral actually doing? If you don’t know, please make your best guess as to why. Item 3 (math) Why does an integral need a ‘dx’ on it? For example, why can’t it just be 1 x3 1 x3 dx 0 instead of 0 ? Explain in as much detail as you can. Item 4 (physics) From physics, mass = density × volume. If an object, S,hasvarying density (i.e. it’s not constant throughout the object), we use an integral to M = DdV calculate it: S . Why does this integral actually calculate the total mass of the object? That is, what is the integral actually doing? If you don’t know, please make your best guess as to why. 726 S.R. Jones in that study would have been heavily ‘primed’ to be thinking of integrals as areas and anti-derivatives by the time they reached item 6. This priming is reflected in the responses, given that many students described the integral as an area under a curve or a calculational technique and no students used any Riemann sum-type conception. Consequently, the study by Rasslan and Tall, while offering other useful results, does not adequately address the questions posed by this research study. This study is meant to leave the interpretation open to the students, to see which conceptualizations they naturally tend to activate as they attempt to make sense of definite integral expressions. The second and fourth survey items asked the students to think of applied formulas where definite integrals are used, which gave the students opportunities to make interpretations of definite integral expressions in other contexts. A potential criticism of these particular items should be addressed here. One might observe that these integrals use functions of several variables, which the students would not have seen in first-semester calculus. This is absolutely true. However, many students may have to deal with these types of integral expressions in their physics or engineering courses before they ever take multivariate calculus (see [12,13,27,28]). The integral of pressure with respect to area is discussed in both Bedford and Fowler’s and Hibbeler’s ‘Engineering Statics’ textbooks. Yet the requirements for engineering statics typically include only calculus courses that do not deal with multivariate integration (e.g. [29–31]). Similarly, the integral of pressure with respect to area and the integral of density with respect to volume both appear in Serway and Jewett’s and Tipler and Mosca’s calculus-based physics textbooks, which may be covered in a first-semester physics course. The requirements listed for typical first-semester physics courses usually only involve the completion or concurrent enrolment with pre-multivariate calculus (e.g. [32–34]). Therefore, it is important to include these integrals in a survey intended for successful first-semester calculus students, since many students may confront them early in their studies, before reaching multivariate calculus. It is necessary to know what sort of interpretations students with only a first-semester background make of these kinds of integrals, in order to know how they might think about them in their entry-level, calculus-based physics and engineering courses. Furthermore, the multiplicative relationship between pressure and area and density and volume was provided to the students, meaning that no prior physics knowledge was required. The multiplicative relationship is sufficient for describing what the integral is ‘doing’. Finally, the purpose of the third survey item was to give the students a second mathematics context to discuss their ideas about integrals. It required the students to break apart the integral symbol template to discuss the integral expression in more detail. The way in which the students explained the existence of ‘dx’ was compared to the symbolic forms of the integral (see Section 2.1) to help determine what conceptualizations of the integral the students were drawing on.

3.3. Coding student responses The students’ responses for each interview item were coded through what Knuth [35] calls an ‘analytical-inductive’ method. First, the various symbolic forms of the integral, described in Section 2.1, provided an ‘external,’ or in other words ‘researcher-generated,’ set of codes,[35] which were established prior to the data analysis. Responses were coded into these four categories based on their explicit relationship to one of these symbolic forms. Then, while the data were examined, other categories were inductively created, which are considered the ‘internal’ or ‘data-grounded’ codes.[35] These codes were created International Journal of Mathematical Education in Science and Technology 727 out of need for student responses that did not fit into one of the external codes. Once the full coding scheme, consisting of both external and internal codes, was created, the student responses were re-coded a final time to ensure that each response was properly categorized. The initial set of external codes consisted of four basic categories: ‘area,’ ‘anti- derivative,’ ‘summation,’ and ‘sum integrand,’ based off of the symbolic forms perimeter and area, function matching, adding up pieces, and adding up the integrand described in Section 2.1. The internal codes later refined the summation category into ‘summation’ and ‘weak summation’ to capture responses that hinted at a summation notion, but were not articulated enough to be conclusive. A new category was also created for a previously undocumented conceptualization: ‘average’. For examples of responses for each category, see Section 4.1. The internal codes also yielded an ‘other/blank’ category, which was used for responses that were indecipherable (e.g. ‘When graphed within the range (S, density) the mass is calculated’), too vague for classification (e.g. ‘It can give you an estimate’), blank, or that consisted of a statement such as ‘Because my professor said so’. Since the focus of this paper deals with various conceptualizations of the definite integral, the other/blank category receives little attention and is mentioned here simply to show how responses of these types were dealt with. While a potential criticism of this action could be that these responses essentially constitute a ‘no conception’ category, I contend that most of the responses coded into other/blank were an attempt to communicate something by the student, yet these responses simply resisted any kind of meaningful classification. Thus, this code does not represent a ‘no conception’ category, but rather a ‘non-code-able’ category. Following the coding of responses into these categories, the frequencies of each category was recorded, per survey item. Many individual responses were coded into multiple categories since students often expressed more than one idea in their response to a particular survey item. As a result, the total frequencies for each survey item add to more than the sample size, n = 150. After the frequencies were tabulated, 95%-level confidence intervals were used to estimate the percentages of the overall calculus student population that might respond similarly to each survey item.[36] By using these descriptive statistics, a picture of the general calculus student population emerges from the data.

4. Results 4.1. Sample responses for each category In Table 2, I present examples of student responses that fit into each of the six categories listed in Section 3.3. The purpose of doing so is to allow the reader the opportunity to more fully understand each coding category and to see how students articulated these conceptualizations of the definite integral. Note that minor editing was conducted on the students’ responses for ease of readability. For example, spelling errors were corrected and first letters of sentences were capitalized. In any case where there was concern that an editing change might alter the content of the response, no editing was made.

4.2. The student population using descriptive statistics In order to describe the calculus student population, Table 3 displays the frequencies of responses to each survey item for each of the six categories. Note that since many students provided more than a single response to a given survey item, the frequencies add up to more than the sample size. An additional category, ‘S + WS + SI,’ has been included to capture 728 S.R. Jones

Table 2. Examples of student responses coded into each category.

Category Example student responses (and the survey items they came from)

Area Item 1: ‘Essentially, the amount of area underneath the curve of the function’. Item 2:‘P would be your function and thus finding the area would give you your force’. Item 3: ‘Since the function ranges from 0 to 1...the dx allows the total area under the function to be found’. Anti-derivative Item 1: ‘The integral is a tool that is used to undo the derivative process’. Item 2:‘P changes, so you can view it as a function. For this to work, force must be the derivative of pressure, or F ’ = P’. Item 3: ‘The “dx” stands as the derivative notation, the “d”istakingthe derivative of “x”’. Summation Item 1: ‘It’s the sum of f(x)ࢫx from a to b and x is so small that it’s close to zero’. Item 2: ‘The integral is actually calculating the pressure × area for every point on the surface, then adding all of those points together’. Item 3: ‘The “dx” supplies the “width” of each rectangle’. Weak summation Item 1:‘Sumofparts’. Item 3: ‘To know that you are taking an infinite amount of increments between 0 and 1’. Item 4: ‘I think it is because the integral considers an infinite number of masses based on the volume’. Sum integrand Item 1: ‘It accumulates the values of f(x) from x = a to x = b’. Item 2: ‘It is adding up the pressure at each point to find the total pressure’. Item 4: ‘The integral is the addition of all densities throughout the object and multiplying it by the volume’. Average Item 2: ‘The integral calculates the total force across the entire surface because it takes the average force on a point and spreads that across the entire surface’. Item 4: ‘The integral is taking the avg. density and finding the mass from that’. Item 4: ‘The integral takes the average mass at any point and spreads it throughout the whole object’.

the proportion of students that made any reference to any kind of summation idea for that item, whether mathematically correct, incorrect, articulated, or vague. The frequency for this category is calculated by adding together the frequencies for the summation, weak summation, and sum integrand categories. Along with the numeric frequencies, 95%-level confidence intervals have been calculated for each category within each survey item. Assuming that this sample is at least reasonably representative of the overall US calculus student population, these confidence intervals can extrapolate from this specific data set to what might be true for the overall student population. That is, they provide an estimation range for the proportion of all successful first-semester calculus students that might respond similarly to each survey item. While this sample’s representativeness of the overall population is debatable given that the students all came from only two colleges, the confidence intervals are still provided as a suggestion for what could possibly be true of the overall population. For more details on confidence intervals, the reader is referred to [36]. International Journal of Mathematical Education in Science and Technology 729

Table 3. Frequency of responses (n = 150), with 95%-conﬁdence intervals providing an estimated range of the per cent of the overall student population that might respond similarly.

Responses from Responses from Conceptualization item 1 (math) item 2 (physics)

Area 131 (82.0% < p < 92.7%) 51 (26.4% < p < 41.6%) Anti-derivative 60 (32.2% < p < 47.8%) 22 (9.0% < p < 20.3%) Summation 10 (2.7% < p < 10.7%) 25 (10.7% < p < 22.6%) Weak summation 5 (0.5% < p < 6.2%) 19 (7.3% < p < 18.0%) Sum integrand 2 (n/a low frequency) 17 (6.3% < p < 16.4%) S + WS + SI 17 (6.3% < p < 16.4%) 61 (32.8% < p < 48.5%) Average 0 (n/a low frequency) 15 (5.2% < p < 14.8%)

Responses from Responses from Conceptualization item 3 (math) item 4 (physics)

Area 7 (1.3% < p < 8.0%) 37 (17.8% < p < 31.6%) Anti-derivative 114 (69.2% < p < 82.8%) 15 (5.2% < p < 14.8%) Summation 11 (3.2% < p < 11.5%) 27 (11.9% < p < 24.1%) Weak summation 2 (n/a low frequency) 17 (6.3% < p < 16.4%) Sum integrand 0 (n/a low frequency) 16 (5.7% < p < 15.6%) S + WS + SI 13 (4.2% < p < 13.2%) 60 (32.2% < p < 47.8%) Average 0 (n/a low frequency) 15 (5.2% < p < 14.8%)

4.3. High frequencies for the area and anti-derivative conceptions In order to answer how commonly students from a calculus population draw on each of the conceptualizations listed above to explain the meaning of the various definite integral expressions contained in the survey items, I now discuss the relative frequencies of the conceptualizations invoked by the students. The data demonstrate that the area and anti- derivative notions come to dominate students’ explanations of integrals in pure mathematics contexts. Perhaps it is no surprise that the area conception is so prevalent because it is often used as the primary motivator for the study of integrals (see [37–39]). Again, this is not bad, since areas under curves can be a useful idea to work with in both mathematics and science settings. However, what is striking is the high prevalence of anti-derivative conceptions for the definite integral, when anti-derivatives do not actually compose the underlying meaning of the definite integral. It is simply a tool used for calculational purposes a` la FTC. Yet, 60 out of 150 students in this study (40%) interpret the basic meaning of the definite integral (according to their responses to item 1) as being an anti-derivative. In addition, 13 of the students (8.7%) invoked only the anti-derivative conception when providing a response to item 1. Furthermore, when asked to justify the existence of the ‘dx’ in the integral (item 3), a large majority of the students used an anti-derivative perspective to explain it. In fact, roughly three-fourths (76%) appealed to anti-derivatives to make sense of the differential. Many of the responses included statements like the ‘dx signifies that x3 is in a derivative form,’ or the ‘d is taking the derivative of x,’ or the ‘dx is like a reverse version of derivative d/dx’. It is clear that for many students, the ‘dx’ does not readily call up the ‘x’fromthe Riemann sum, but rather is more closely tied to the notion of derivatives. Even the applied physics items (2 and 4) had a relatively high rate of responses related to areas and anti-derivatives. Roughly one-third of students applied an area conceptualization 730 S.R. Jones to these two items. Unlike the pure mathematics context, the area conceptualization is not always as productive in making sense of these types of contextualized integrals.[19,21]In fact, many of the responses for items 2 and 4 that invoked ‘area’ seemed overly general and indistinct, suggesting that the students did not have a robust idea for how the area notion might be applied to those physics integrals. For example, the statement, ‘The integral is finding the area under the curve of pressure,’ is too vague to address how the integral operates in these contexts. Therefore, it might be a problem that so many students initially attempt to analyse these types of physics integrals through the area lens. Additionally, around one-eighth of students tried to make sense of items 2 and 4 through the anti-derivative lens. They made statements to the effect that pressure is the derivative of force and density is the derivative of mass, so that these derivatives could be ‘undone’ to get back to force and mass. While it is possible, in a very roundabout way, to construe pressure as the rate at which more force is accumulated as one enlarges the area over which the force is applied (and hence might be considered an unorthodox ‘derivative’), this is typically not the way that pressure is defined nor discussed in physics and engineering textbooks.[12,13,27,28] The same holds true for density and mass. Thus, this is a problematic way to interpret these integrals and generally does not provide the meaning that is intended in physics and engineering.

4.4. Low frequencies for the summation conception The data clearly demonstrate a low frequency of responses that make use of any type of summation idea for the two mathematics items. Even taking ‘summation,’ ‘weak summation,’ and ‘sum integrand’ together, only 17 out of 150 students (11.3%) made any kind of statement dealing with summations on item 1, which asked them to explain what an integral is at its essence. Furthermore, 117 out of 150 students (78%) made no mention of anything related to summations whatsoever on either mathematics item, 1 or 3. This result implies that roughly three-fourths of successful first-semester calculus students may not appeal to any summation conception to explain the meaning of integrals in pure mathematics contexts. This may complicate the learning of volumes of revolution, arc length, and other topics requiring an immediate use of the Riemann sum lens in order to construct meaning beyond memorized formulas. I note here that these students certainly may have Riemann sum-based ideas in their cognitions. However, they largely did not activate and use them when interpreting these integral expressions. If students are not actively drawing on a multiplicatively based summation conception in these contexts, it implies the need for additional scaffolding by the instructor to link the Riemann sum structure to the formulas that are being developed. In the responses to the applied physics items (2 and 4), we see somewhat of a shift toward the summation conceptions (cf. [1]). However, despite this shift the picture is still fairly unsatisfactory. Only 25 students (16.7%) used an explanation that included a multiplicatively based summation conception for item 2, and only 27 (18%) used one for item 4. This is problematic considering that many of these students proceed directly to physics and engineering courses that make use of integrals similar to those presented on this survey (see [12,13,27,28]). When an instructor or a textbook displays an integral expression or equation, it is likely that most of the students will attempt to make sense of it with something other than a well-constructed summation conception. Again, this implies the need for instructors to highlight the multiplicative relationship between integrand and differential, and the accumulation of the resultant quantity throughout the domain. Instructors will need International Journal of Mathematical Education in Science and Technology 731 to explicitly draw out the Riemann sum idea in order to help students make sense of the integrals they see in physics and engineering. Additionally, a count was conducted of all students who made no mention of any summation idea anywhere on all four survey items (including summation, weak summation, and sum integrand). The number of students who fell into this category was 71 out of 150 (47.3%), yielding a confidence interval of 39.3% < p < 55.3%. It is therefore likely that around half of all successful first-semester calculus students might not draw on any type of summation conception as a basis for explaining the meaning of these types of definite integral. Again, this is not to say that these students have no summation conception in their cognition; most of these students probably do cognitively possess the Riemann sum idea. Yet, given the important nature of the multiplicatively based summation conception, it might not be sufficient simply to have it. It may be necessary for it to be readily available for activation by the students in order to make sense of applied integrals. It is striking that half of the students did not activate this conception at all throughout four different questions that asked them to explain the underlying meaning of the definite integral in a variety of contexts. Furthermore, the count in the previous paragraph even includes vague or incorrect statements from students about summations, meaning that the reality of the picture may be worse. For example, adding up the integrand makes use of a mathematically incorrect summation process, which is incompatible with physics and engineering integrals. In fact, only 38 out of the 150 students (25.3%) provided a response to at least one of the four survey items that could be counted in the ‘summation’ category instead of ‘weak summation’ or ‘sum integrand’. Thus, instructors of subsequent classes must realize that the large majority of their students may not be interpreting (at least initially) integral expressions in a way that identifies the multiplicative relationship of the integrand and the differential as well as the summation/accumulation of that quantity over the domain prescribed by the bounds of integration.

4.5. The ‘average’ conception Finally, I describe here a previously undocumented conceptualization of the definite integral. Many of the students surveyed expressed an ‘average’ conception for the integral that they used to answer the physics items (2 and 4). The data suggests that roughly one-tenth of the overall calculus student population may draw on this notion, making it worth understanding. The students depicted the integral as a process that takes a non-uniform function and ‘smooths’ it out over the domain to make it as though it were uniform. For example, students made statements such as, ‘It takes the average force on a point and spreads that across the entire surface,’ and ‘The integral is taking the average density and finding the mass from that’. That is, after the integral ‘spreads out’ the non-uniform function to make it uniform, the scenario is then reduced to the equations for uniform pressure or density: F = P × A,orM = D × V. It is interesting that many students made a connection between integrals and average values in a way that made the integral personally meaningful to them. However, it is not clear from this data alone what the individual students used as the basis for this conception. It is possible to think of an ‘averaging’ process as a convenient way to compute summations, such as adding ‘5 + 6 + 7 + 8 + 9’ by thinking ‘the average is 7, so let me just add 7 five times’. On the other hand, it is possible that students are conflating an application of the integral, namely finding the average value of a function through the mean value 732 S.R. Jones f¯ = 1 b f x dx theorem for integrals, b−a a ( ) , with the actual meaning of the integral. If the latter is the case, it results in interesting circular reasoning, since the integral is determined through an average value, but yet is also the process for finding that average value. Thus, while the average conception is an interesting way to understand the resulting value of the integral, it is possible that it might lead to a problematic notion of the underlying meaning of the integral. It should be noted, though, that these data alone do not provide a conclusive answer as to whether this is the case.

5. Discussion 5.1. Activation of knowledge is required in addition to simply possessing it If calculus textbooks say anything about how calculus instructors may typically present integration, then we can assume that students are taught about Riemann sums and that the definite integral is defined as the limit of Riemann sums (see [37–39]). However, the manifold view of knowledge implies that simply having this piece of knowledge may not be sufficient. In addition to gaining knowledge about Riemann sums, students need to construct this knowledge in a way in which they may actually activate it in situations where it is useful.[23,25] It is likely that most students, having learned about Riemann sums in their calculus courses, have some type of summation conception in their cognition. However, the results of this study indicate that regardless of whether they possess it or not, many students are often not drawing on it in their attempts to explain the meaning definite integral expressions. Consequently, many students might not be deriving the sense-making benefits that the summation conception affords for contextualized integrals – especially those in physics and engineering (see [21]). I believe we would ideally want 100% of the students to flexibly draw on both summation and area conceptions when thinking about integrals, and then to also recognize the usefulness of the anti-derivative calculational technique. The results of this study show that interpreting the integral as an area is certainly not the problem, but rather the underuti- lization of the multiplicatively based summation conception. This resonates with Sealey’s [40] finding that ‘area under a curve is not sufficient for understanding the definite integral’ (p. 52, emphasis in original). Many applications, both in pure mathematics and in physics and engineering, use the Riemann integral definition to create integral formulas. This is true of volumes of revolution, arc-lengths of curves, and line integrals, as well as myriad formulas in science such as the integral of pressure over area, density over volume, force over distance, position over mass, force over time, flow over area, and power over time. Sealey and Oehrtman [41] use this as a reason to promote deeper understanding of the Riemann sum structure in calculus courses. Thus, it might be ideal for a student, first, to make sense of the integral through a summation conception to know what it is calculating and why it is calculating it and then, second, to apply an area or anti-derivative lens onto it for calculational or analytic purposes.

5.2. Why students might rely so much on area and anti-derivatives The conclusion that so few students draw on the summation conception in their attempt to interpret integral expressions may seem odd since textbooks typically explore the idea of the Riemann sum and subsequently use the Riemann integral definition for definite integrals (e.g. [37–39]). Why, then, are students not readily and actively drawing on this International Journal of Mathematical Education in Science and Technology 733 conceptualization to make sense of integrals? Perhaps the typical methods used in intro- ducing integration rely too heavily on ‘areas under curves’. The three calculus textbooks cited here all use the ‘area’ problem as the primary motivator for integration. It seems that the Riemann sum is mostly used as a tool for getting at these complex, irregular areas. If students come to see the Riemann sum simply as a calculational method, then when anti-derivatives (through the FTC) offer a more convenient calculational technique that is easier to use, students may view the Riemann sum as an outdated instrument, much like an abacus compared to a computer. Thus, the Riemann sum may take a ‘cognitive backseat’ to the more ‘efficient’ anti-derivative. As a result, the summation conception may not be regularly activated in making sense of integral expressions. The problem is that the Riemann sum is not simply a ‘tool’. It is the heart of meaning for definite integrals. Even when a student could reasonably use an anti-derivative technique to calculate an integral, anti-derivatives fall short of providing coherence to the integral expression and a reason for why it calculates what it does.[21] Many mathematics and science problems can be tackled by invoking Riemann sums, allowing the problem to be set up as an integral. It is certainly nice that anti-derivatives (or numeric methods) can be used to get solutions quickly, but only because the Riemann integral definition applies to the problem, letting it to be turned into an integral in the first place. Thus, it is critical that we examine how to support not only the creation, but also the activation of a robust summation conception in students’ cognitions if we hope to assist students in having a deep understanding of the main ideas of first-year calculus.

5.3. Possible ways to address the issue The data from this study offer a place to begin this examination, by looking at the relationship that exists between summation ideas in a mathematics and physics context. It appears that there is a strong dependence between the tendency to draw on a summation idea in one context and the tendency to do so in the other. The following table illustrates the frequencies of responses to either items 1 or 3 (pure mathematics) and items 2 or 4 (applied physics) that include any element of summation in them, including the summation, weak summation, and sum integrand categories. Table 4 demonstrates clearly that students who referred to summation ideas in the mathematics items (1 or 3) were much more likely to draw on summation ideas in the physics items (2 or 4), by a margin of 26 to 1. On the other hand, students who did not invoke any summation conception for items 1 or 3 were much less likely to draw on a summation conception for items 2 or 4, by a margin of 46 to 71. A χ 2-test for independence yields a p-value << 0.001 for the relationship between drawing on summation ideas for items 1 or 3 and items 2 or 4. This result may hint at the idea that scaffolding a robust summation conception for students in mathematics courses might drastically improve the likelihood that that conception will be activated during physics and engineering learning

Table 4. Summation ideas present in pure mathematics items versus applied physics items.

Any summation idea present No summation idea present (items 2 or 4) (items 2 or 4)

Any summation idea present 26 1 (items 1 or 3) No summation idea present 46 71 (items 1 or 3) 734 S.R. Jones as well. If this is the case, then the issues raised in this study are not simply problems to pass over to science educators; calculus instructors might have a powerful impact on their students’ understanding of integration in subsequent courses. If we consider Table 4 in the reverse direction, another possible interpretation emerges – that increased experience with applied, science-based integral expressions in calculus courses may help strengthen students’ activation of the multiplicatively based summation conception in pure mathematics contexts. That is, employing in mathematics courses definite integrals that come from physics, engineering, or other science contexts might help develop this critical conception, which may, in turn, help students see how the Riemann sum is used to create integration formulas for volumes of revolution, arc length, or line integrals. In general, the correlation indicated by Table 4 cannot state for sure which way the causation (if any) runs, but it is clear that the more a student draws on a summation conception of integration in one context the more likely they are to do so in the other context. Actively drawing on the notion that an integral is inherently a summation (or accumulation) of a quantity over many tiny pieces may help students overcome some of the documented difficulties in working with integrals. They might better understand how the definition of the definite integral and the FTC fit together (see [9,17]), since they would be in a better position to see how the accumulation notion of the integral (see [4]) leads to an anti-derivative. This connection would help students see how the common anti-derivative procedure for calculating integrals, and hence the FTC itself, emerges from Riemann sums and areas under curves (see [5]). Students may also be in a better position to apply their mathematical knowledge to physics and engineering (see [6,15,19,21]), since many of the integrals found in these areas of study make use of Riemann integrals to develop formulas.[12,13]

6. Future directions In order to investigate this problem further, educators are calling for a closer examination of the instructional content provided during the introduction of integration in first-semester calculus.[19,21,40,41] Perhaps in doing so we could shed light on the reasons for the high prevalence of area and anti-derivative conceptualizations and the low frequency of the summation conceptualization when students attempt to explain definite integral expressions, as seen in this paper. It may also show what can be done to improve the proportion of successful first-semester calculus students that develop a robust, readily accessible summation conception in regards to integration. To explore these issues the author is currently involved in the second phase of a larger study, of which this paper is a part. The second phase consists of observing first-semester calculus courses in order to correlate what happens during instruction with the types of responses students give to these survey items. Additionally, a design experiment is underway to look at the effects of using Riemann sums (not area under a curve) as the primary motivator of integration, which is based on recommendations found in the literature that this may be beneficial for students (see [19,21,40,41]). This experiment will help answer whether emphasizing the multiplicatively based summation conception primarily and the area under a curve conception secondarily–the opposite of the current typical model provided in many calculus textbooks (as in [37–39]) – can assist students in developing a robust summation conception of the integral. The hope is that such presentations and discussions will enable students to not only construct, but also have readily accessible at the forefront of their International Journal of Mathematical Education in Science and Technology 735 cognition, so to speak, the important multiplicatively based summation conception of definite integrals.

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 χ 2-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.

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