Student Understanding of the Sign of Negative Definite Integrals in Mathematics and Physics

In a study of student understanding of negative definite integrals at two institutions, we administered a written survey and follow-up clinical interviews at one institution and found that “backward integrals”, where the integral was taken from right to left on the x -axis, were the most difficult for students to interpret. We then conducted additional interviews focused on backward integrals at a second institution. Our analysis uses the concept image framework and a recent categorical framework for mathematical sense making. We found that students were most successful using the Fundamental Theorem of Calculus to determine the sign of an integral when a symbolic expression was provided. When considering a definite integral in a graphical context, students often had difficulty if they viewed the integral as a spatial area, stating that area must always be positive. Some students were able to conceptualize Δ x or dx as a difference or change and thus a signed quantity; when these students were able to view the area as a sum of smaller pieces, they were more successful in justifying a negative backward integral. Students who used amounts or similar images often had difficulty making sense of the negative sign. Even more progress was made when students either invoked or were asked explicitly about a specific physical context to be represented by the backward integral, other than spatial area. The context provided a meaning to the difference represented by Δ x or dx and thus to the sign of that difference and the definite integral.


Motivation
The learning of physics concepts often requires the ability to interpret and reason with underlying mathematics concepts. In the context of physics, mathematics not only serves as a representational tool for physics concepts (e.g., equations, graphs, etc.), but also provides logical paths to solve complex physics problems. Thus, students are expected to possess a good understanding of basic mathematics to learn physics. We also believe that physics contexts can help students gain a better understanding of mathematical quantities and structures.
One mathematical concept that is used across a broad spectrum of physics contexts is the definite integral. An in-depth knowledge of definite integrals is extremely important to understanding various physical phenomena, since many of those phenomena are expressed using definite integrals. For example, the displacement of an object described by a velocity function in kinematics is expressed as Δx = ∫ t 2 t 1 v(t)dt , and the potential difference between two points in (one-dimensional) space in electrostatics is expressed as ΔV = − ∫ x 2 x 1 E(x)dx . In these contexts, the symbolic definite integral expression provides a "shorthand" representation of the physical relationship between quantities. For example, the displacement of an object is the accumulated sum of the product of instantaneous velocity and an infinitesimally small time interval. In fact, for many introductory calculus-based physics courses, differential calculus is only corequisite, so students learn about the calculus expressions and meaning through their physics classes well before the concepts are taught in a calculus course. Thus, in many cases relating the symbolic expression to the physics helps students understand the mathematical symbols (Hu & Rebello, 2013a, b).
Student understanding of signs of physical quantities are important because they are not just the outcomes of some mathematical operations, but also the outcomes of deep underlying physical meaning. The signs of integrals in physics often carry physical meaning, the interpretation of which requires understanding of both physical and mathematical aspects of the signs. This inspired us to investigate how students reason about the signs of definite integrals. One type of integral that results in a negative sign that is common in physics classes is the integral in the backward direction. Thus, we later focused on the backward integral.
In physics the differential has multiple meanings, but we believe it is typically treated as a "physically infinitesimal" quantity, i.e., much smaller than any relevant scale in the context but still finite (Courant & John, 1999). Thus, the properties and meaning of dx are the same as those of Δx . This is consistent with Roundy et al.'s (2015) view of a differential in a "thick derivative" in their measurement-focused extension of Zandieh's (2000) derivative framework. Thus dx is not merely the indicator of the integration variable. In physical contexts, dx has dimensions so that the product of f (x) and dx has the dimensions of the quantity represented by the antiderivative. One could also view ∫ b a f (x)dx as a shorthand for lim Δx→0 ∑ n k=1 f (x * k )Δx . In this case, it is sufficient to consider the sign of Δx , where Δx = b−a n . Note that in the case of a "backward" integral where b < a, this would make Δx < 0 , and thus, dx would also be negative. As with the differential (or Δx ), the quantity represented by

Relevant Literature
The existing literature on definite integrals tends to support ways that encourage students to develop an understanding of definite integrals as the sum of infinitely small products, i.e., Riemann sums (Jones, 2013(Jones, , 2015aJones & Ely, 2023;Meredith & Marrongelle, 2008;Sealey, 2014). Jones (2015b) found that a conceptualization of an integral as area under a curve was the most prominent representation among students, and Orton's (1983) work showed that students had little difficulty when asked to solve problems that related integrals with area. We note that it is possible to think about area as a whole, "macroscopic", quantity (i.e., the area under the curve) or as the sum of the areas of many "microscopic" rectangles (i.e., a Riemann sum perspective). It has been argued that reasoning about a definite integral as area under the curve, while among the most common conceptualization in calculus texts (Meredith & Marrongelle, 2008) may limit students' ability to apply the integral concept to quantities other than area (Norman & Prichard, 1994;Sealey, 2006;Thompson & Silverman, 2008). For example, Sealey (2006) pointed out that viewing area as a macroscopic quantity was not helpful to students when they were attempting to set up an integral that represented energy, and not simply area. Instead, the students needed to connect the area of the rectangles to the quantities that multiply to form the quantity of energy, thus emphasizing a Riemann sum perspective. Recent work shows that area-oriented problems are least prevalent in common physics texts, with the adding up pieces model (Jones, 2015a;Sealey, 2014) being most common (Pina & Loverude, 2019). Previous studies have found that students lacking an understanding of definite integrals have difficulties in physics, in the contexts of kinematics graphs (Beichner, 1994;McDermott et al., 1987), thermodynamics (Pollock et al., 2007), and electrostatics (Nguyen & Rebello, 2011). The results of these studies suggest that some specific difficulties when solving problems in particular physics contexts and/or with particular representations may have some connection to difficulties with definite integrals, such as connecting definite integrals to the area under a curve.

3
Additionally, various ways of understanding the meaning and role of the differential in an integral can lead to student difficulties. Students could view the differential simply as an indicator of the variable of integration rather than a fundamental element of the product in integration of both single-and multivariable functions (Artigue et al., 1990;Hu & Rebello, 2013a;Jones, 2013;Nguyen & Rebello, 2011). This could stem from, or lead to, a failure to understand the product layer of the integral (Sealey, 2014;Von Korff & Rebello, 2012). Students could treat dx as a small amount or quantity of whatever the x-axis represents (Artigue et al., 1990;Hu & Rebello, 2013a;Jones, 2015a;Roundy et al., 2015), or an infinitesimal difference, and thus change in x (Sealey & Thompson, 2016;Von Korff & Rebello, 2012).
McCarty and Sealey (2019) have classified themes in mathematics experts' conceptualizations of the differential. They found that mathematicians had various ways of reasoning about the dx. Some thought about it as an infinitesimal quantity, sometimes even thinking about it as a tiny, but finite quantity. Other experts described it as notation that was a non-mathematical object, and still others defined it as dx = lim Δx→0 Δx . Of course, this limit would always be 0, which would cause significant issues in many contexts involving dx (e.g., Oehrtman's (2009) collapse metaphor, also seen in Nilsen & Knutsen (2023)), but experts still claimed this conceptualization was useful at times.
Researchers have documented student difficulties with the signs of definite integrals in both physics and calculus (Grundmeier et al., 2006;Mahir, 2009;McDermott et al., 1987;Orton, 1983;Stevens & Jones, 2023). Definite integrals that have a negative result are of particular difficulty geometrically. Students often do not treat the area as a signed quantity, in this case as a negative quantity when the integrand is negative, i.e., below the x-axis, effectively treating it as a spatial area (also known as the area of a surface (Weisstein, 2003)) rather than either a signed area or the physical quantity represented by f (x)dx (Bezuidenhout & Olivier, 2000;Lobato, 2006;Orton, 1983;Rasslan & Tall, 2002). Recently, Kontorovich (2023) focused on the converse of this situation in more detail, dealing with tasks asking for the spatial area under a curve or between two curves ("enclosed area") for an integrand whose sign may change between the limits of integration. He found that students often determined the signed area with a single integral over the full region rather than breaking up the integral into sub-regions and adding up the spatial area of each region to get the total area. While this prior work deals with many aspects of definite integrals that are relevant to mathematicians, it has not addressed some of the aspects and features of definite integrals that are most applicable to physics contexts. One such aspect is a definite integral taken in the backward direction, i.e., ∫ b a f (x)dx where a > b, and thus dx (or Δx ) is negative. Several physically meaningful quantities are obtained from these "backward" definite integrals, most having to do with energy changes or transfers, including the work done on a thermodynamic system (e.g., an ideal gas) during a compression process: the volume of the system is decreasing during the process, so the integral is taken from a larger initial volume to a smaller final volume. While some quantities are genuinely line integrals rooted in vector quantities, initial instruction in one dimension reduces them to standard integrals.
The differential, not surprisingly, is discussed quite a bit in standard calculus texts. For example, when Stewart (2015) introduces a linear approximation, dx is defined as being equal to Δx , and it "can be given the value of any real number" (p. 254). However, within the context of a definite integral, it would not make sense for dx to be a large finite number. Stewart (2015) introduces the symbolic expression for a definite integral and states that, "For now, [emphasis added] the symbol dx has no meaning by itself…The dx simply indicates that the independent variable is x" (p. 379). Thus, the definition and meaning of dx is context dependent.

Theoretical Perspectives
We apply two specific frameworks to the studies reported here, the primary one being from the mathematics education literature, but we also use the physics education research (PER) framework of mathematical sensemaking to further frame our work.
The broad framework we apply is Tall and Vinner's (1981) concept image, which refers to the overall cognitive structure in one's mind associated with a given concept, including all the mental pictures, properties, and processes associated with the concept. Evoked concept images are provided in response to tasks and may have conflicting elements across contexts. A concept definition consists of the words one uses to define a concept and it should be noted that one's personal concept definition may or may not be aligned with the formal concept definition, which is "a concept definition which is accepted by the mathematical community at large" (p. 152). Over time, as one learns more about a given concept, their concept image tends to reflect the formal concept definition. As noted in the previous section, we argue that while there is a formal concept definition of definite integral, there is not a single formal concept definition of dx. Thus, we cannot discuss how closely a student's concept image of differentials aligns with a concept definition of differential. Instead, we will discuss the ways in which their concept image enabled them to make sense of a physical context problem.
Another relevant perspective is sensemaking, in particular mathematical sensemaking. Sensemaking is a commonly articulated goal in science and mathematics instruction. Not surprisingly, there are multiple definitions in the literature. Recently, Odden and Russ (2018) defined sensemaking as noticing a gap or inconsistency in one's knowledge and generating an explanation to resolve that gap or inconsistency. In physics, mathematical sensemaking (MSM) is crucial, given the centrality of mathematical modeling and reasoning in physics problem solving (Uhden et al., 2012). Gifford and Finkelstein (2020) state that MSM is "a subset of sense making activities that privilege the use of mathematical formalisms (as either tools or objects) in generating an explanation" (p. 3). They have created a categorical framework for MSM, using mediated cognition and activity theory, that focuses on scientific sensemaking modes in which mathematics or physics serves as the tool employed in and/or the object of the sensemaking (see Fig. 1). Gifford and Finkelstein consider mathematics to be focused on "the use and manipulation of symbolic and graphical representations in a self-consistent fashion" and note that any mathematical representation illustrates a "specific relationship between abstract quantities", while physics focuses on "emphasizing conceptual relationships and principles" between quantities that are "anchored in the real world -or specific models of the world" (pp. 3-4). We use these descriptors here as well.
There are four combinations of modes, labeled with shorthand to distinguish the domain of tool (first letter) and object (last letter) as mathematics (M) or physics (P): Msm-M, Msm-P, Psm-M, and Psm-P. Thus Psm-M is using a physics concept or principle (P) as a tool to understand a mathematical object (M). Students can "translate" between these modes, and reasoning can be "chained" in a multi-step sequence, in which the object of one mode becomes the tool in the next mode. Two different ways of reasoning can be "coordinated" to provide two ways to make sense of the same idea (object). In Gifford and Finkelstein's work, coordination strengthens sensemaking by providing complementary lines of reasoning. As we show below, coordination can also lead to conflict if the two reasoning lines lead to different conclusions, e.g., if one approach is flawed.
Because the tasks discussed here ask for students to generate an explanation for the negative sign of a backward integral beyond invocation of a recalled definition, and since we expect explanations to involve both mathematical formalism and physical context, the MSM framework should provide insight into the tools, and the flavor of tools -mathematical or physical -that are used in student reasoning.

Methods
We conducted our studies at two public US universities. Both involve clinical, thinkaloud interviews of individual students. The first study, at a medium-sized university, consisted of follow-up interviews based on student responses on written surveys focused on determining the signs of graphically represented definite integrals. Although the initial written survey questions involved only positive integrals, several students identified them as negative integrals by misusing various calculus concepts, such as the Fundamental Theorem of Calculus (FTC) and the Riemann sum (Bajracharya et al., 2012). This led us to conduct follow-up clinical interviews with a small number of those students to explore students' understanding of the signs of definite integrals that are relevant in physics contexts, particularly with graphical representations. During the interviews, we found that students struggled with one  Gifford and Finkelstein (2020). The student uses the tool as a mediated pathway to make sense of an object Student Tool Object particular type of integral, the backward definite integral. Because this type of integral is common in many physical contexts, we conducted a second related study at a second, larger university, focusing particularly on student understanding of the relationship between forward and backward definite integrals. In this second study, we conducted clinical interviews of students with a wider range of mathematical backgrounds. All students in our study had used a version of Stewart's calculus text (e.g., Stewart (2015)) either during their time as a calculus student or instructor.

First Study -Individual Interviews Focusing on Negative Integrals
Based on the student responses to the written surveys, we developed follow-up interview questions by varying representational features -e.g., integration direction and integrand sign -of the survey graph. We also varied some notations and features in the new graphs. In this paper, we focus on data from a graph involving a backward integral.
In the first study, we asked for the signs of integrals of the functions f (y) and g(y) from a to b using the graphs shown in Fig. 2. To minimize rule-based reasoning such as ∫ b a f (y)dy = − ∫ a b f (y)dy and to promote qualitative reasoning, the limits were swapped between the graphs instead of in the integral symbol, and the axes were labeled z and y. These graphs were designed to probe the robustness of the area reasoning provided by the students.
Seven students from the survey population volunteered to participate in the interviews (Table 1, Study 1). Four students were in a second-semester physics course; the other three were in Calculus III (Multivariable Calculus) and had completed the physics course in the previous semester. Each interview was videorecorded and lasted 45-60 min. Our interview protocol included asking students to provide reasoning to their responses and elaborate on them further. For example, whenever students used the term "negative area," they were asked to elaborate on their meaning of the term.

Second Study -Individual Interviews Comparing Forward and Backward Integrals
After analyzing the data from the first study, we found that the backward integrals were most problematic for students. As such, we designed a study focusing on how or if students could justify why  Table 1). The interview subjects were volunteers and received a small monetary incentive at the conclusion of the interview.
The interview protocol questions were asked in a specific order, starting with open-ended general expressions and concluding with a physical example (see Fig. 3). In each case we gave the forward integral first, then asked about the backward integral of the same expression. Interviews were semi-structured, following the same order of topics for each of the five interviews, but allowing for off-script questions to clarify our understanding of students' responses. For example, with the general expressions, we first presented the students with a written expression of the integral ∫ b a f (x)dx and asked them how they would read that expression. Next, we asked them to, "Explain what you know and understand about that expression." We asked follow-up questions to clarify our understanding of their responses. In the third section of the interview, students were asked about the work done on a spring. Our initial task provided the integral expression and the expression for the force on a spring and a figure to demonstrate the scenario. Students were asked to interpret this integral for an extension of the spring (i.e., where x 1 < x 2 ), and then to interpret the integral with the limits reversed, rather than explicitly asking about a compression of the spring. We also allowed for time at the end of each interview to go off script and ask additional questions. In each task, students typically needed to explain how they knew that the backward integral was negative, either from a direct prompt by the interviewers or through their own desire to understand the relationship. Both sets of interviews were videorecorded and transcribed for analysis. At least two authors viewed each video multiple times and made notes regarding the ways students explained the sign of the integral in both the forward and backward directions. In subsequent analysis, we paid particular attention to how (or whether) the students could describe definite integrals as a sum of products and/or as area under a curve. We noted which of the differential concept images each student held and which images were evoked during each interview question along with the integral image(s), and also attempted to identify other concept images that emerged from the data. We also noted the ways in which students engaged in mathematical sensemaking, identifying modes, tools, and objects in students' reasoning.
While choosing interview excerpts, the best attempt was made to choose those that were representative of the relevant issues. The interview data were analyzed using a thematic analysis approach (Braun & Clarke, 2006;Guest et al., 2012). The approach was used to identify, analyze, and interpret the concept images invoked by the students while dealing with the integrals. As prescribed by Braun and Clark, our analysis involves reading and re-reading the transcripts, generating initial codes and labels to represent important features of the data relevant to the research question, identifying concepts contained within the data, observing patterns to construct meaningful themes and comparing them with existing literature, and putting together the logical descriptions to convey the findings.

Results
While some of the students' concept images corroborate the findings of previous researchers, others are novel. This section describes the concept images that students invoked to deal with negative integrals and categorizes the mathematical sensemaking modes relating to each concept image. We present the results from both studies and discuss the extent to which our results are consistent with other literature. Table 2 depicts the concept images manifested by students during the first set of interviews. The principal concept images invoked in the interviews are based on FTC, area, and Riemann sums. Students also used physical contexts to make sense of the sign. We compared the student concept images with their formal concept definitions from standard textbooks (Pina & Loverude, 2019;Stewart, 2015). Whether a student invoked a concept image that is aligned (A) or not aligned (N) with the concept definition is indicated by the symbol 'A' or 'N'.
In the second study, we asked students questions to elicit their understanding of why the backward integral was negative in all contexts (symbolic, graphical, and within the context of a physical scenario). Thus, all students in Study 2 reasoned about backward integrals in all contexts. However, within some contexts, students were productive in explaining why the backward integral was negative, and in other contexts they were not productive. Table 3 summarizes the productivity of students' initial attempts at justifying why the backward integral was negative. The sections   that follow describe students' justifications and show how students' understanding developed throughout the interview.

The Fundamental Theorem of Calculus Concept Image
Students in both studies invoked the FTC when determining the sign of an integral.
In the first study, all but one student inappropriately invoked the FTC at some point, while in the second study, all implemented the FTC correctly. In the latter study, all students initially used the FTC to justify why ∫ 3 1 2x dx = − ∫ 1 3 2x dx , by subtracting the values of x 2 at the two limits in each case. They were able to generalize the rela- , which would have the opposite sign. Alex, however, stated that area should always be positive, and decided to drop the negative that resulted in the second integral.
In the first study, Ben, James, and Abbie successfully determined the sign of the backward integral using a graphical interpretation of the FTC, by interpreting F(b) as the area from the origin to b, F(a) as the area from the origin to a, and then evaluating the sign of F(b) − F(a) . The other four used inappropriate expressions for the FTC. Although we did not deeply probe their concept images of the FTC, the expression they used and the justifications they provided verbally were not adequate descriptions of why an integral was negative or positive. For example, Freddie and Mary wrote that the integral was equal to b − a and F(b) − a respectively instead of without providing a satisfactory explanation.
In his interview, Ben used incorrect notation to express the FTC, but once he was probed deeper, he was able to explain each term in the expression, appropriately showing an adequate concept image of the FTC.
Ben: … once you integrate it, it out, you'd basically, you, end up having [writes | b a on the board] b, a, um, and then you're subtracting them. So, you have b − a and these are, like, once they have been integrated …and since you know b is smaller, once you do … once you subtract larger value, it's gonna become negative. Interviewer: Where? Where do you plug the number? Ben: In, uh, to the function like this…when I'm, like, what it is supposed to look like is basically, Ben started with a response that indicated he was considering the limits of integration, but when the interviewer asked Ben a follow-up question, he changed his response to ∫ f (b) − ∫ f (a) (Fig. 4), and specified that ∫ f (b) is the antiderivative of the function. In this case, we believe that Ben understood the FTC graphically, but did not express it with conventional notation.
In Ben's case, although it appears that he used the FTC as a quick procedure to find the sign of integrals, his painting of the area under the curve to represent the antiderivative of the function indicated that it was more than just a superficial implementation of calculations. This is consistent with Rasslan and Tall's (2002) categorization of students' definition of definite integrals, in which students often provide the "procedure of calculation" when asked to provide the definition.
Six students initially evoked the FTC concept image in response to the backward integral question without using the notion of area, consistent with Grundmeier et al.'s (2006) findings. Particularly in written surveys, we documented that students mischaracterized the FTC as either ∫ to justify their response for the signs of integrals, which has not been reported in other studies (Bajracharya et al., 2012). Our interview results indicate that the students' misapplication of the FTC might be due to the lack of concept images of the antiderivative function and of its rate of change (Bezuidenhout & Olivier, 2000;Thompson, 1994). We also consider that the graphical nature of the questions in this study may have encouraged the reasoning that evoked the FTC concept images that were misaligned with concept definitions.
In terms of mathematical sensemaking (MSM) modes, these data demonstrate Msm-M, using mathematical information to make sense of other mathematical formalism (Fig. 5). Here the students were attempting to use the graphs to determine values for antiderivative terms to put in the FTC expression. That outcome is then chained to be the tool to understand the sign of the definite integral. These students generally used the function or limit values rather than the antiderivative values, leading to an incorrect result.

Using Area under the Curve Concept Image to Justify the Negative Backward Integral
The area concept image was dominantly manifested in both studies. In this section, we discuss instances where students conceived of an integral as the entire area under a curve (a macroscopic perspective) as well as instances where they discussed area as the sum of the areas of many small quadrilaterals, typically rectangles (a microscopic perspective, consistent with a Riemann sum image).
For the purposes of this work, we classify spatial area as a mathematical tool (or object) rather than a physical one. As shown below, reasoning about definite integrals using area under the curve can lead to cognitive conflict when the integral represents a physical quantity that could be negative.

Macroscopic Spatial Area
Many students perceived the integral as a spatial area, which according to them is always positive. Here we provide examples of students not considering area from a Riemann sum perspective, treating area as a macroscopic ("non-differential") quantity.
For example, Mary conceptualized the definite integral as an area that is always positive and concluded that the signs of the backward integral and the integral of a negative function were both positive.

Mary:
Calculating something is one thing… I know that I'm calculating an area. But for me, why would the area be negative? Why would it be any different? Like, that's not something that I grasp.
The fact that this concept image of area as spatial area did not help with determining the sign is consistent with earlier work (Bezuidenhout & Olivier, 2000;Lobato, 2006;Orton, 1983;Rasslan & Tall, 2002).
Other students seemed to hold two parallel images for the definite integral, treating it differently as a symbolic computation than as the area under the curve. Freddie noted that the sign of an integral had different outcomes based on the purpose:

Freddie:
It depends on what you're doing. If you're wanting to find the area, then I would say the area is always going to be positive, regardless, but when you do out the math, this looks like, it'd give you a negative number.
For Freddie, the sign of the integral is context dependent; it is positive if one uses it to find area under the curve or it can be positive or negative depending on the outcome of a mathematical procedure (such as the FTC). Similarly, Erica believed that although area could be negative graphically, it could not be negative physically, in the sense of spatial area.
In the second set of interviews, some students also initially held parallel images for the sign of the backward integral but knew that they should be able to make sense of why the integral was negative, even though area was positive. Matt, a junior math major, noted that the backward integral represented the same area as the forward integral, but the backward integral would have to be negative since the limits were reversed "because I already know that, like as a fact, that it's a negative if you want to flip the bounds." He did state that he believed there should be a graphical justification, but he did not know what one would be. While his reasoning about area was correct, it was not productive in making sense of the sign of the backward integral.
Matt's, Freddie's, and Erica's reasoning can be represented in the MSM diagrams shown in Fig. 6. They believed the integral was negative from the FTC, but positive if it were represented as area, leading to a conflict when attempting to coordinate the two.  Sara, a first-year mathematics and physics double major in the second set of interviews, evaluated ∫ 3 1 2x dx by finding the area of the large triangle (Fig. 7a) and subtracting the area of the small triangle (Fig. 7b) to obtain the desired area (Fig. 7c). She noticed that these calculations corresponded to the values she obtained when applying the FTC to the same problem. Then, when computing ∫ 1 3 2x dx , she reversed the order of her subtraction, subtracting the area of the large triangle (Fig. 7a) from the area of the small triangle (Fig. 7b), and said, "But I'm not sure why that order is. I mean I know why for the integral [symbolically] because it's written that way, but if you were to solve this geometrically, I don't know why you would change the order of the subtraction".
However, after a short time of thinking, Sara said, "Oh, that makes sense, ok. So, by this saying the integral from 1 to 3 [points to integral from 3 to 1], it would be like saying subtracting the area of the function up to 1 minus of the area of the function up to 3." After further questioning, it became clear that Sara had started dx and the backward integral as the reverse of that. Nick's reasoning was similar to Sara's, and he specifically mentioned this symbolic expression. Both Sara and Nick interpreted the antiderivatives as areas under the curve from the origin to the limit value to make sense of the FTC in this setting. The subtraction of the antiderivatives as the subtraction of areas under the curve was productive in helping them make sense of the negative sign of the backward integral (Fig. 8). There was no discussion of whether the area is negative, just that the integral is negative.
Matt, however, treated the area between the limits as a fixed amount regardless of the direction of integration, which conflicted with use of the limit reversal equation to justify a sign reversal. He ultimately trusted the symbolic expression more and could not make sense of the negative sign graphically. All three of these participants used area-based solutions that sidestep the need for thinking about area in terms of a Riemann sum.

Riemann Sum-based Area
Almost every student in both sets of interviews seemed comfortable discussing the forward integral as the sum of small rectangles (or other quadrilaterals). We call Fig. 8 Sara's (and Nick's) MSM diagrams for reasoning using graphical area and the FTC this a Riemann sum concept image of area, or a microscopic view, in contrast to thinking about the entire spatial area as large chunks of area (as Matt and Sara did in the previous section). Students who exhibited evidence of a concept image of area as a Riemann sum were able to draw rectangular strips under the curves, labeling the width ∆x or dx, the heights as f (x) , and stating that the total area under the curve is the limit as we increase the number of rectangles or decrease the width of the rectangles. This concept image is similar to a category of student "conversion between definite integrals and areas" labeled "sums of areas of bars involving infinitesimals" seen in Norwegian first-year engineering students in a calculus course (Nilsen & Knutsen, 2023). Although most of the students recognized the Riemann method of generating rectangles and adding them up (Sealey's product and summation layers (Sealey, 2014;Von Korff & Rebello, 2012) and Jones's adding-up-pieces conceptualization (Jones, 2015a)), some treated the Riemann rectangles as always positive, either because the area was inherently a positive quantity or because they were thinking of ∆x (or dx) as a (sign-less) width rather than a difference. Thus, despite their familiarity with the Riemann sum, these students were unable to make sense of why the backward integral was negative.
Earlier, Simon invoked the FTC concept image in response to the backward integral. When asked whether he could determine the sign without using the FTC, Simon invoked the Riemann sum concept image, drawing a series of rectangles under the curve, as an alternative to the FTC. However, as he drew the rectangles (Fig. 9), he changed his response for the sign from negative to positive: Simon used an adding up pieces model (Jones, 2015a), but the pieces -the rectangles he drew -were not signed. Simon then invoked the commutative property of addition -although he called it "commutalative" -to justify that adding in either order yields the same result, and thus the signs of the total area did not change depending on the direction he added them. Simon's MSM (Fig. 10) shows a chain of sensemaking from spatial area (implicitly, Psm-M) to graphical area to determine the sign of a Riemann sum to be positive and then that the sum of those rectangles is also positive. He then used the commutative property to make sense of the backward integral also being positive, and coordinated these two sets.
In analyzing the second set of interviews, we coded the ways in which students were making sense of ∆x and dx. Students mentioned that dx indicates the variable of integration, and most also were able to describe ∆x, but not dx, as the width of individual rectangles under a curve. These are similar categories to those described by Nilsen and Knutsen (2023) when studying student interpretations of differentials. Of particular interest to us is whether the students could conceive of ∆x and/or dx as signed quantities, as either a negative width, or as a negative value obtained from x 2 − x 1 . According to our analysis, most of the students attributed a sign to ∆x at some point during the interview. None of the students considered dx as a signed quantity on their own accord, but with prompting from the interviewers, most were able to do so (see Table 4). We show two examples of how students made sense of the notion of area below.
Anna, a senior math major, was one of the few students who thought about ∆x as a negative quantity without intervention from the interviewers. When considering a backward integral near the beginning of the interview, she said, "You're going to have that negative width times a positive value [height of rectangle], which is going  to give you a negative number, so you're going to get the addition of a bunch of negative numbers." While she had no trouble thinking about ∆x as a negative width, she did not seem comfortable thinking about dx as being positive or negative. Much later in the interview, one of the interviewers asked Anna if it was possible for dx to be positive or negative, and Anna responded, "I've actually never thought of that. So, I'm not sure. I mean I guess it could, but I just always viewed the dx as the indication of what term to integrate to. So, I'm not actually sure, I guess." It seems that Anna was reluctant to change her image of dx from an integration variable indicator to an infinitesimally small, signed quantity. Throughout his interview, Nick focused his explanation as to why the backward integral was negative on the direction of integration. He seemed to be thinking about the variable x representing time, and mentioned more than once that the backward integral would be like playing a movie in reverse (a macroscopic view). On another note, Nick spent a great deal of time during the interview talking about the two terms that made up the product in the definite integral, namely the 2x and the dx in ∫ 1 3 2x dx . He knew that when multiplying two quantities to obtain a negative result, exactly one of the terms multiplied must be negative. He debated whether the x or the dx turned negative. He "voted" for the dx to be negative, saying that the dx represents a change, and that change implies motion. But this answer did not sit well with him: "I've just, in my head for so long, always assumed dx to be a positive, you know, change in x." It seems as if Nick had a change concept image of the differential, which could then be signed; however, Nick still did not consider the possibility of a negative dx until "confronted" with the issue.
In the MSM framework (Fig. 11), Nick used the fact that the backward integral is negative to make sense of dx being negative, an Msm-M move. He supported this by reconsidering dx as change and how that applies to a backward integral, using motion as a means of sensemaking the negative sign, represented as a Psm-M move.

Sensemaking with Physical Contexts
Another method used to justify the signs of definite integrals during the interviews in both studies was mentioning physics contexts (other than area), demonstrating instances of using physics ideas as tools to understand a mathematical object, or Psm-M. This was expected in the second study, since one of the interview tasks explicitly occurred in the context of a spring extension. In the first study, in which tasks were without any physical context, students spontaneously invoked a context in their explanations. For most, but not all, the use of physics contexts helped determine the correct signs of integrals. We start with examples from Ben and Abbie, students in the first study, before discussing responses from Matt, Anna, and Alex in the second study.
Although Ben was able to reconcile the negative sign of the backward integral using the definition of dx, he was still uncomfortable with the notion of negative area perceiving it as a spatial area. Later, Ben made sense of the negative integral by describing dx as b−a n . When the interviewer pressed him to think about situations where one can have negative areas, he immediately responded: Ben: … if I'm speaking space, there is no negative area, but in physics, depending on what you're doing, you could have a negative value for a lot of different things. Like…, we're talking about the electrical currents, specifically, ac current. All the functions follow, like, a sine wave kind of [draws a sinusoidal curve (Fig. 12) When the interviewer asked if he was referring to the signs of the integral, pointing at the signs, Ben responded: "Yeah. The area under that would be positive, the area under that would be negative, …" Although Ben drew the sinusoidal curve referring to the context of alternating current, he did not clearly indicate what physical quantity the integral represented. However, the physics context helped him to make sense of the significance of negative area beyond spatial area.
Like Ben, Abbie also initially stated that area cannot be negative, invoking spatial area reasoning, but later used a physics context to make sense of the signs of the integrals. When Abbie was asked to clarify her reasoning about the sign of the integral in her written response, she drew rectangular strips under the curve, calling the process "slicing." The interviewer asked if she could use this notion to determine the sign of the integral in the graph with the backward integral. She seemed to be uncertain until she brought in the concept of thermodynamic work: Abbie: I know by my intuition that compressing gas, work is done, like, is not done by the, this system…of the gas, the gas is working against that to be happened … so it's working this way [lifting her hands upward]. So, it's negative work done by the gas, whereas the opposite is true on the system. Abbie reasoned with thermodynamic work, a physical quantity that could be represented by the graph, to make sense of the sign of the integral (Fig. 13). She used the principle that work done on an object by a system is the opposite sign of the work done by the object on the system, which follows from Newton's Third Law.
Ben's and Abbie's uses of physics contexts to make sense of negative integrals are consistent with the findings of Marrongelle (2004). In her study, two participants (Rob and Terry) used physics concepts, to different extents, to solve calculus tasks. However, those students had taken a novel integrated, co-taught physics-calculus course; her study did not show to what extent the participants' use of physics concepts to deal with calculus tasks was due to their unique instructional experience.
Our study suggests that students' use of physical contexts to answer calculus problems is not limited to these kinds of integrated physics-calculus courses; students in traditional calculus courses can also demonstrate this ability, when given the opportunity to engage in similar reasoning.
In the second study, the spring task seemed to evoke concept images of dx as a small but finite amount, change, or difference in the position of the end of the spring -in students who already were engaging with the idea of the differential.
Matt had indicated a concept image of dx as a variable indicator, but during the spring extension discussion, he specified it using a physics context as: Matt: … the change in the distance from each, like, individual from like your x 1 to x 2 and it's going to be every, like, small, small distance.
Upon prompting from an interviewer about the change in his view of dx, Matt was able to make sense of dx as a small negative displacement of the spring, given the physical meaning:

Matt:
When you think about just, like, the pure math problems, that's all you really think about -just the fact that dx is just telling you […] what variable to use […] But […] here, it represents, it represents something… Similarly, Anna pointed out that before the interview, she thought of dx as "the term you're supposed to integrate to." She also had a clear sense of the product layer and of the adding up pieces conceptualization of definite integrals, multiplying f (x) by Δx "to get the area limits" for a Riemann rectangle (Sealey, 2014;Von Korff & Rebello, 2012). She immediately realized that Δx is negative for the backward integral in the first task, and how that leads to the negative sign on the result. While discussing the spring problem, she stated that dx is "kind of representative of that infinitely small piece which would still be able to have a negative change as we are heading smaller, because it's directional so because it's getting smaller it's negative." Matt's and Anna's MSM diagrams for these excerpts are shown in Fig. 14. However, we also saw a case where context did not promote recognition of dx as an infinitesimal signed quantity. In the math contexts, Alex displayed reasonable area under the curve and antiderivative images of definite integrals, although he showed some confusion graphically with the integrand representing the area. He never mentioned dx in his discussion (except when reading definite integrals aloud), instead focusing on more macroscopic differences, either Δx or the whole interval ( b − a ). His image of Δx was primarily as a distance on the x-axis, but this could be a signed quantity, since reversing the limits of the integral "would make your width negative." But Alex had a strong aversion to the idea of negative area; for him the negative sign was added when limits were reversed "to negate the negative distance" because otherwise you would have a negative area, "which isn't really a thing." Similarly, he did not connect the sign of the backward integral (for a positive function) to the sign of Δx.
With the spring task, Alex considered that the forward work integral represented the energy put into the spring and that the backward integral was energy lost. He discussed the meaning of this: "It is totally okay for there to be negative work, but that's usually, that's defined upon reference frames. Whether work was done on a system or by a system." As Alex continued, his discomfort with a negative integral emerged: Alex: Whether you start at x 1 or you start at x 2 , it's still the same amount of force over the same amount of distance so there's no reason why the flipped bound, this second, x 2 to x 1 , there's no reason why that shouldn't be able to have a positive answer.
It seems as if Alex used the idea of "reference frames," by which he means changing what is labelled as the system in an interaction that involves mechanical work being done, to make sense of the arbitrary positive sign of the integral. (There is positive work done by an object in each direction, but it is done by a different object in each case.) This line of reasoning avoided the need to consider the sign of the differential displacement. Alex's excerpt instantiates a common difficulty of considering the spring displacement as a distance, which is an unsigned and thus effectively positive quantity. Even though he used physical reasoning, and is correct that the magnitude of the work would be the same in either direction, his incomplete understanding of the context led to an incorrect conclusion (Fig. 15).
The inclusion of the physical context of the spring, as well as Nick's invoking motion or time contexts for the integrals, seemed to help subjects make sense of the differential itself and of the sign thereof. This supports our reasoning for including the spring task in the second study and is consistent with students in the first study invoking physical contexts on their own to interpret a negative integral.
On the other hand, it is possible that any reasoning on the spring task about the differential in backward integrals may have arisen due to priming of this issue in the previous two tasks of the interview. While Alex could be considered a counterexample of this, to be certain, additional data would need to be collected with a rearranged or otherwise varied protocol. As mentioned earlier, invoking a physical context is consistent with the Psm-M mode of MSM. In Gifford and Finkelstein's (2020) study, Psm-M was productive in interpreting graphical or symbolic representations of quantum mechanical phenomena. In our first study, some students spontaneously invoked a context to make sense of the sign of the integral; the second study contained a physical context of work on a spring to probe the effectiveness of the context on sense making, effectively cueing Psm-M. (See Fig. 16). It is worth noting that using spatial area to make sense of a signed graphical area was not productive in determining the integral sign. This

Psm-M
Physical context (voltage, gas compression, spring) is consistent with Jones's (2015a) finding that the area interpretation of the integral was not helpful in making sense of different concepts. While technically Psm-M, the fact that the same term can be used for both mathematical (graphical) and physical (spatial) contexts suggests that contexts need to be considered carefully to be productive and must match the mathematical model. The user of the context needs to have a solid understanding of the concepts as they relate to the problem. This pairs with the findings of Kontorovich (2023), where students were explicitly asked to find the (geometrical-spatial) "enclosed areas" of regions of a function that include sub-region(s) with a negative integrand.

Area Concept Image Revisited
In the second set of interviews, we explicitly asked some of the students to revisit the integral as area after they had completed the spring problem, to see if their concept image had changed in any way. For some students (but not all), the spring problem seemed to help them to make sense of Δx or dx as a signed quantity to modify their evoked image of area. When thinking about the spring problem, Matt realized that the Δx in the backward integral represented a negative quantity and described why Δx would be positive if we stretched the spring from x 1 to x 2 , but negative if we released the spring from x 2 to x 1 . He also nicely described Δx in a way that showed he was thinking about it as a signed finite number (typically small), and that dx was what the Δx became once you applied the limit to the Riemann sum. After many attempts from Matt, the interviewer initiated the following conversation.
Interviewer: So, you said Δx over here was negative and Δx over here was positive. Could dx be negative or positive? Matt: That's probably the hidden spot that I couldn't figure out before. Yeah, I would say that this dx would be negative [pointing to a backward integral] and this one would be positive [pointing to a forward integral] because it's approaching 0 so this [dx in forward integral] would still stay positive … and this one [dx in backward integral] would stay under [negative].
Next, the interviewer asked Matt to reconsider forward and backward integrals as area and to explain graphically why one integral would be positive and one negative. Eventually, Matt summarized his reasoning as follows.
Matt: When you're going from b to a instead of a to b, we were talking about how to say graphically, and like I was saying here, this would be when if you were drawing rectangles here. From b to a … you'd be going back this way … So, all of these Δx 's are negative because you're going backwards, you're going in the negative direction. So, then your dx is negative, not a hundred percent sure why, but I know that your Δx is basically like turned into your dx when you're changing it from the Riemann sum writing of it … so I'd imagine that if the Δx were negative then the dx would be negative.
Matt recognized the displacement, ∆x, as a signed quantity and used that to make sense of the sign change with the direction change. By also connecting dx to ∆x to write the integral form, he imbues dx with the same meaning, and thus the same signed properties. He coordinates these two lines of reasoning to reconcile that dx is signed and is negative for a backward integral (see Fig. 17).
Sara, however, was never willing to think about dx as a negative quantity, even after completing the spring problem. Sara was comfortable with f (x) being negative or positive, depending on if it was above or below the x-axis, but when she was directly asked if dx could have a sign, said, "Well no, I don't think dx would ever be negative because it's just a distance, it's not like an actual value." She continued her explanation, pointing to the negative part of the x-axis and saying, "Like, even if it was over here, dx wouldn't be negative because, yeah, it's distance." This is consistent with McCarty and Sealey's (2019) theme of the differential as unquantifiable, but also with multiple other findings of identifying dx as a small amount of a quantity and thus unsigned. It is in exactly these cases where this image becomes problematic.
Interestingly, Sara and Alex never mentioned the term Δx during their interviews, but did discuss the width of rectangles and/or the distance along the x-axis. Sara focused her explanation of the negative backward integral on the application of the FTC and was unable to see why the backward integral would be negative when thinking about area as the sum of smaller pieces, while Alex argued that both integrals should be positive since they represented the same area, and the width of the rectangles would be the same regardless of the direction of integration.

Discussion & Conclusion
In this study we explored two research questions related to signs of definite integrals, especially negative integrals: What concept images do students invoke while reasoning about the sign of negative definite integrals, in particular backward integrals? What role do students' evoked concept images play in making sense of the sign of these integrals? Written responses led to follow-up interviews within that population and a second set of interviews with a different population and different tasks. In the interviews, we found four prevalent concept images for integrals that led to specific ways to make sense of the signs of integrals: invoking the Fundamental Theorem of Calculus, using macroscopic area under the curve -spatial or graphical -considering the Riemann sum (microscopic area, both types), and using a physical context. The concept image of the differential and of ∆x affected student reasoning as well. While some students invoked concept images that align with formal concept definitions, others manifested images that do not align with the concept definitions. Similarly, most student sensemaking was appropriate, using different tools and often chaining reasoning. However, even when combined with various forms of correct MSM, students who invoked unaligned images tended to reach an incorrect conclusion or arrived at a conceptual conflict.
The most common method for justifying the sign of the backward integral was invoking the FTC and the property of reversing the sign of the antiderivative difference when the limits are reversed. For the second set of interviews, the symbolic expressions led to procedural or rote recall justifications, specifically that the forward integral's value was 8 and thus the backward integral must be -8. This was generally productive when used symbolically. Several students seemed to use this knowledge to make sense of the negative backward integral sign graphically. However, some students were still uneasy about considering the area to be negative, so this task seems insufficient on its own for resolving any cognitive conflict.
Invoking the FTC and/or having an image of dx as a change or difference -with an adding up pieces image of definite integrals -involves explicit acknowledgement of the operation of subtraction in determining the result F(b) − F(a) or the sum of the products of f (x) and a negative dx. Thinking of dx as a width or amount eliminates any sign cues, which becomes problematic for a backward integral. This latter concept image is complementary to, and even corroborates, the idea of area as spatial area.
While the use of "direction" on the horizontal axis can cue the negative sign for an integral, this seems similar to, but less sophisticated than, assigning the sign of dx as a difference or change. Occasionally in physics the positive direction on an axis may be to the left; we did not check the robustness of student thinking to this variation. So, while direction should typically provide the correct sign, more research is needed to determine whether this line of reasoning is correct.
An overemphasis on the area under the curve representation to reason about definite integrals is well known (Bezuidenhout & Olivier, 2000;Orton, 1983). As with other literature, we found that the use of area reasoning does not necessarily imply a complete understanding of definite integrals (Grundmeier et al., 2006;Jones, 2015a;Orton, 1983;Sealey, 2006). Our interview results suggest that students have difficulties using area under the curve to interpret the sign of a backward integral.
Students' failure to deal with negative integrals due to their overgeneralization of area as spatial area corroborates with what Norman and Prichard (1994) called cognitive obstacles. In their study, the area under the curve notion induced cognitive obstacles to students' computational skills, whereas in this study, it led to obstacles with their graphical reasoning. While other work demonstrated area overgeneralization for forward integrals with negative integrands, we documented this when students determined the sign of backward integrals; students who used spatial area language did not treat a backward integral as negative either. In fact, in some cases computation led to a conflicting outcome for the sign: students computed the integral sign to be negative but struggled to reconcile this with their spatial area reasoning, in which all area is positive. While this latter line of reasoning is consistent with some textbook approaches (e.g., Kontorovich, 2023;Stewart, 2015), it is unproductive when the integral represents a physical quantity that could be negative.
Overall, our findings indicate that students have difficulties with negative definite integrals, particularly with backward integrals, which have physical significance. While students generally recognized the role of direction of integration in determining the signs of integrals, they did not recognize that the change in direction causes a change in sign of dx or ∆x and thus in the sign of the integral.
We found that students are better able to interpret negative integrals, including backward integrals, when they are able to link the mathematical concepts of integrals to some physical meaning, beyond spatial area, whether self-generated or prompted. Contexts such as the spring problem have potential to cue the direction and thus the sign of ∆x or dx, but are not necessarily a consistent cue on their own. Our results suggest that the pairing of opportunities to emphasize dx and ∆x as differences and applications of the notion of a negative differential beyond a mathematical procedure to relevant physical situations should result in a deeper understanding of the differential and the meaning of definite integrals, no matter the sign. This is similar to and consistent with the findings and suggestions of Oehrtman and Simmons (2023). Gifford and Finkelstein (2020) suggest that their categorical sense making modes can be cued in instructional interventions. While they have suggested this for physics instruction and interpreting graphical and symbolic representations of physical phenomena (Gifford & Finkelstein, 2021), this could also apply to sense making of mathematical objects, such as definite integrals and differentials, by invoking physical contexts as a tool for sense making, tying the mathematical representations to real-world phenomena. A nod to applications of calculus, especially unique situations such as backward integrals, would give students more experience with this expert-like skill of tying mathematical formalism to physical phenomena, a key component of mathematical modeling (e.g., Czocher, 2017) and a desired outcome for problem solving in the sciences and engineering (e.g., Uhden et al., 2012).
Moreover, physics instructors may overestimate students' abilities to apply their understanding of (negative) integrals from mathematics to physics despite the substantial differences in notations and representations of the concept across the two fields (Jones, 2015a). Coupled with the difficulties documented here with negative integrals in particular, this may lead to additional difficulties with other related concepts, such as work done on a spring. Our findings suggest that not assigning a sign to graphical areas can cause difficulties in considering the sign of an integral quantity, which could be related to the sign of the integrand or the sign of the differential. Having two sign conventions -area is always positive on a graph, but the product of the two quantities represented by the area could be negative -seems to lead to conflict.