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The WSU Libraries' goal is to provide excellent customer service. Let us know how we are doing by responding to this short survey: https://libraries.wsu.edu/access_services_survey Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 DOI 10.1007/s40753-017-0060-7

Students’ ObstaclestoUsingRiemannSum Interpretations of the Definite Integral

Joseph F. Wagner1

Published online: 22 September 2017 # Springer International Publishing AG 2017

Abstract Students use a variety of resources to make sense of integration, and interpreting the definite integral as a sum of infinitesimal products (rooted in the concept of a Riemann sum) is particularly useful in many physical contexts. This study of beginning and upper-level undergraduate physics students examines some obstacles students encounter when trying to make sense of integration, as well as some discom- fort and skepticism some students maintain even after constructing useful conceptions of the integral. In particular, many students attempt to explain what integration does by trying to use algebraic sense-making to interpret the symbolic manipulations involved in using the Fundamental Theorem of Calculus. Consequently, students demonstrate a reluctance to use their understanding of Bwhat a Riemann sum does^ to interpret Bwhat an integral does.^ This research suggests an absence of instructional attention to subtle differences between sense-making in algebra and sense-making in calculus, perhaps inhibiting efforts to promote Riemann sum interpretations of the integral during calculus instruction.

Keywords Calculus instruction . Definite integral . Riemann sums . Physics

Introduction

Researchers have argued that the Riemann sum interpretation of the definite integral is perhaps the most valuable interpretation for making sense of integration in applied contexts, particularly in physics (e.g., Jones 2015a, b;Sealey2006). The term Riemann sum inter- pretation (or Riemann sum reasoning) is used here to include conceptions of the definite integral as a sum of Bvery small pieces^ or infinitesimals, such that the infinitesimal pieces are imagined to be a product of the value of some function and an infinitesimal (Bvery

* Joseph F. Wagner [email protected]

1 Department of Mathematics, Xavier University, 3800 Victory Parkway, Cincinnati, OH 45207, USA 328 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 small^) change in the independent variable of the function across a selected interval of integration. Such an interpretation can be suggested by defining the integral as the limit of a Riemann sum, such that, under standard definitions of the embedded symbols,

n ∫b ðÞ ¼ ∑ ðÞΔ ; a fxdx lim fxi x i¼1 n→∞ though not all uses of Riemann sum interpretations are equivalent to such a formal definition. Indeed, the point is that at a conceptual, interpretive level, a certain blurring of the meaning of the symbols takes place, such that the values of f(x)anddx in the definite integral are imagined to be the height and the width, respectively, of each of the rectangles in a Riemann sum area approximation. The notation of the integral, of course, is intended to highlight just such an association. Unless otherwise noted, in the context of this paper, the function f is presumed to be a continuous, real-valued function of a single variable. Despite the utility of Riemann sum-based interpretations of the definite integral, research has shown that many students do not develop such interpretations in their calculus courses (Jones 2015b; Sealey 2014). This is true despite the fact that the same students have learned about both Riemann sums and area-under-curve interpretations. Some have suggested that this is due to the fact that Riemann sum interpretations are not emphasized or well taught in traditional calculus curricula (Dray et al. 2008; Yeatts and Hundhausen 1992). The analysis presented in this paper will reveal, however, that there are additional factors at work in students that may discourage the development and use of Riemann sum reasoning, and that these factors may well be heightened by pedagogical factors. Using data taken from interviews with beginning and upper-level undergraduate students studying physics, I will argue that because a Riemann sum interpretation does not readily map onto the symbolic process for finding the value of a definite integral using the Fundamental Theorem of Calculus (FTC), students see no reason to use it. Furthermore, students may expect that the symbolic processes involved in antidifferentiation should be able to be subjected to the same or similar sense-making processes that they have learned to understand and interpret the symbol manipulations of algebra. This results in students’ attempts to find or impose algebraic meaning where none in fact exists. Finally, even among students who do learn to adopt Riemann sum reasoning, some of them remain skeptical of its validity. These obstacles to adopting Riemann sum reasoning have not been previously observed in the literature, yet they suggest the need for instructional attention to some subtle but fundamental differences between sense-making in algebra and sense-making in calculus.

Background

The definite integral is a central topic in the study of calculus, and it lends itself to widespread applications in other disciplines, particularly physics. Because of its importance, students’ understanding of integration has attracted considerable at- tention in mathematics education research. A good deal of this research has Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 329 focused on revealing the usually limited and tenuous nature of students’ concep- tual knowledge of integration (Ferrini-Mundy and Graham 1994;Grundmeier et al. 2006;Mahir2009; Rasslan and Tall 2002). What typical students do know about integration is often confined to the procedural knowledge necessary for evaluating certain integrals symbolically, while their conceptual understanding is limited to interpreting the definite integral as an Barea under a curve^ (Bezuidenhout and Olivier 2000;Jones2015b). These results are not surprising in the United States where the calculation of areas through symbolic procedures have usually been emphasized in traditional calculus textbooks and curricula. The definite integral lends itself to a variety of interpretations, however. Apart from procedural and area interpretations, the definite integral can be understood as a measure of accumulated or total change in a quantity whose rate of change is known, for example, or as a sum of Bvery small^ or Binfinitesimal^ products such that the integrand and the differential of an integral are interpreted through the lens of a Riemann sum (by which the definite integral is often formally defined). This latter way of thinking, of course, requires an ability to imagine a limiting process that transforms a discrete Riemann sum into its continuous equivalent. Some students have developed additional conceptions as well, such as an inter- pretation of the definite integral as the area bounded by the average value of the integrand over the interval of integration (Jones 2015b; Wagner 2016). The expert has the ability to understand the fundamental equivalence of all these interpretations. Students, however, need to acquire them cumulatively over time, and develop an appreciation for their equivalence over even more time. Jones (2015b) found that the majority of students with an introductory background in integral calculus tended to use an area interpretation as the central meaning of the integral, while the next largest number interpreted the integral’s meaning as an antiderivative or a tool used to reverse the process of differentiation (a notable conflation of the definite integral as a value and the symbolic process used to compute it). Summation interpretations were much less common. When asked to interpret problems within a physics context, summation interpretations increased modestly, but most of the students offering such interpretations did not do so in a way that reflected a correct or normative understanding. Wagner (2016) found similar results among students with a slightly more advanced background in integral calculus. In contrast to the evidence that area interpretations of the definite integral are dominant in most students’ understandings, research has suggested that area interpre- tations are often not the most useful for using the integral in applied contexts (Sealey 2006;Jones2013, 2015a). Furthermore, although many students do not have a well- developed understanding of the definite integral as a sum, several researchers have argued that a Riemann sum interpretation is perhaps the most useful to support students in making sense of many applications of integration (Doughty et al. 2014;Jones2013, 2015a; Meredith and Marrongelle 2008; Nguyen and Rebello 2011a, b; Sealey 2006, 2014; Thompson and Silverman 2008). I note, however, that this situation exists even though most traditional calculus textbooks and curricula in the United States include presentations of Riemann sums and their limits, as well as numerical methods for approximating areas based on a Riemann sum understanding. In short, even though 330 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 students are familiar with areas, Riemann sums, and methods of integration, a large number of students complete their study of calculus without being able to interpret the definite integral as a sum, or with only a tenuous familiarity with such an understanding (Jones 2015b). Some have suggested that there is a mismatch between what is emphasized in traditional calculus curricula and what is actually most useful to students in applications to disciplines outside of the mathematics classroom (Dray et al. 2008;Meredithand Marrongelle 2008; Yeatts and Hundhausen 1992). I think that this is likely to be true. I argue in this paper, however, that learning to interpret the definite integral using a Riemann sum conception offers particular challenges to students that may, in fact, be exacerbated by traditional pedagogical approaches. More specifically, I claim that the problem is also one rooted in students’ attempts to reconcile a Riemann sum interpre- tation with the symbolic manipulations involved in computing the value of a definite integral using the FTC. Despite students’ introduction to Riemann sums in most calculus curricula, the vast majority of their subsequent time studying integral calculus is spent using the FTC to evaluate integrals. It may well be the case that this emphasis on finding and using antiderivatives may lead students to seek algebraic and conceptual meaning beneath the definite integral not in Riemann sums but in the antidifferentiation processes used in integral computations. I would like to emphasize that this paper does not address the mathematical history and controversies that have surrounded the notions of differentials and infinitesimals. Rather, I take as a starting point the fact that throughout the physical sciences the interpretation of differentials as arbitrarily small or Binfinitesimal^ changes in given quantities is a widely used and beneficial heuristic tool for thinking about and interpreting derivatives and integrals. As such, it is to our students’ benefit to include such an interpretation among their conceptual resources for understanding calculus. (For a more thorough mathematical discussion of this matter, see Dray and Manogue 2010.)

Conceptual Framework

The focus of this paper is to examine the challenges students encounter that may limit their use of Riemann sum reasoning when thinking about integration. In this section, I offer a clarifying example of what is meant by a Riemann sum interpretation of an integral, and I describe what I argue to be the primary conceptual challenges that students encounter to discourage its usage. In the data analysis sections to follow, I offer evidence for these claims.

Riemann Sum Reasoning in Use

As an example of what other researchers (noted above) and I classify as a Riemann sum interpretation of the definite integral, consider this example of a third-year physics major (identified as U3) who was asked simply to describe his interpretation of the following symbols: ∫b ðÞ a fxdx Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 331

As part of his interpretation, he drew the sketch shown in Fig. 1. Asked to interpret what the integral means, he explained as follows:

U3: So this would get us essentially the area under this curve, this block here.

JFW: And can you pull apart exactly why that is?

U3: So, this is our y [indicates height of first rectangle], if you will, and this is our small change in x [indicates width of rectangle, which he has labeled dx]. So that's this times this amount gives us the area of this rectangle. And then we take another little step, which gives us another rectangle that we're adding up [draws second rectangle to the right of the first]. And so basically, this is an approxima- tion of this curve. If you make these, like we say in physics, infinitesimally small, you get closer and closer to this curve as you add up all of these little blocks, and so since you're taking the area of each of these little ones and adding them up, eventually you have this entire area.

This is a clear example of a Riemann sum interpretation of a definite integral. The differential dx serves to describe both the width of a drawn rectangle as well as the width of an Binfinitesimally small^ imagined rectangle, and both the sketch and the student’s language suggest that the infinitesimals themselves are added up in sequence from left to right. This interpretive frame serves well in making sense of actual physical processes. For example, if one integrates a velocity function v(t)overanintervalof time, a ≤ t ≤ b, one can imagine that one is breaking the interval of time into infinites- imally small increments. By multiplying the velocity at each instant by the infinitesimal time increment, dt, the product v(t) dt may be interpreted as the distance traveled over each time increment. The sum of all such Binfinitesimal distances^ gives the total distance over the entire interval from a to b. Note that when U3 introduced the language of infinitesimals, he prefaced it with the caveat, Blike we say in physics.^ At the very least, this suggests an expectation on his part that the interpretation he used may not be universally acceptable, and/or that it may be something he learned in his physics courses but not in his mathematics courses. I now turn to describing what I believe to be a considerable obstacle to the use of Riemann sum reasoning encountered by many students.

A Challenge to Riemann Sum Reasoning

Although students and experts alike commonly use an Barea under a curve^ interpretation of the definite integral, it is important to note that this is really a means of interpreting what the value of the integral might represent. Riemann sums, however, can be used as part of a process through which that value is found. Of interest to this research is how students understand the process of using Riemann sums, and how they understand the computational process of evaluating a definite integral. That is, from a student’s perspective, what do these two processes do and what do they have to do with each other? 332 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356

Fig. 1 A physics student's graphical representation of the area found by a definite integral. (The additional tangent and secant notations were added during later discussion.)

Under standard definitions of the embedded symbols, the definite integral can be expressed (or defined) as a Riemann sum:

n lim ∑ fxðÞi Δx i¼1 n→∞

As a mathematical, algebraic statement, the Riemann sum representation of an area as a combination of rectangular areas has explanatory sense built into it, in that a meaning- ful algebraic and geometric process for finding area can be mapped onto the symbols of the expression, and the process can be directly modeled and investigated. The area of each rectangle is computed by multiplying its height and width, and is represented by f(xi)Δx. The summation indicates that all the rectangles are to be added together and combined into a total area. The limiting process mathematizes the notion of letting the width of the intervals get Bvery small^ (or, equivalently, letting the number of rectan- gles increase without bound). Furthermore, the fact that the limiting process may be imagined to have a completion point, the limit itself, supports the notion of the rectangles themselves becoming Binfinitesimally thin.^ In short, from the students’ perspective, the Riemann sum process does algebraically what it says it does. There is a clear way in which the familiar algebraic and geometric meaning of area can be mapped onto the algebraic syntax. The situation with regard to the definite integral, however, is quite different fromtheperspectiveofastudentwhoconsidersadefiniteintegraltobethemeans of calculating an area using antidifferentiation. Although the symbols used in expressing a definite integral itself lend themselves to being interpreted as the sum of the products of lengths and widths of rectangles, the actual process of computing the definite integral using the FTC is entirely different. Consider a simple example: Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 333

2 1 2 16 ∫ x3dx ¼ x4 ¼ −0 ¼ 4 0 4 0 4

In this case, what the integral Bdoes^ is transform the integrand, x3, into one of its 1 4 antiderivatives, 4 x , through a process that cannot be subjected to algebraic sense- making. The power rule for finding such an antiderivative can be readily proven, of course, but the computation takes advantage of a known pattern associated with antiderivatives of polynomials, and the symbolic manipulation is immune to any sort of geometric explanation or metaphorical algebraic interpretation, except perhaps in the simplest cases. In addition, no actual Bsummation^ of any sort takes place, nor does anything Bget very small,^ and the differential dx appears simply to be extraneous and to evaporate in the solution process. Riemann sum reasoning may fit the syntax of the original integral expression, but it cannot be used to explain or extract meaning from the computational process resulting from use of the FTC. It may help to pause at this point and consider an expert’s understanding of the relationships among these various processes. The expert understands that the FTC demonstrates that the definite integral (typically defined as the limit of a Riemann sum) can be evaluated by using a process that transforms the integrand into one of its antiderivatives. The accumulation of a variety of Bmethods of integration^ permits one to recognize symbolic patterns that relate families of functions and their (anti)derivatives. An expert is capable of relying on the definition of the definite integral as the limit of a Riemann sum to interpret the conceptual meaning of the integral, while independently recognizing the implications of the FTC for computing its value. This involves a rather sophisticated and subtle appreciation for how and why the conceptual meaning of the integral is found in its Riemann sum definition, but not through the computational process provided by the FTC.1 Furthermore, the expert understands that symbolic patterns and processes of antidifferentiation, arising from a function transformation, cannot be subjected to the conceptual sense-making that underlies basic algebra. The central thesis of this paper is that many, if not most, undergraduate students complete several courses in calculus without much exposure (if any) to the subtleties of the expert understanding described above. While students’ under- standing of basic algebra permits them to see why Riemann sums can be used to find areas, that same algebraic understanding cannot be used to shed light on the function transformations they routinely learn in order to apply the FTC. The algebraic solution process for findinganareaasalimitofaRiemannsumis inherently different from the solution process for finding an area through the computation of an antiderivative, and this difference can cause varying levels of confusion and puzzlement to students. At the very least, nothing would suggest that importing a Riemann sum-based explanation into the process of integration via antidifferentiation ought to be automatic or natural for students, at least, not for students who are accustomed to trying to make sense of their mathematical activities. I am not aware of any other circumstances in typical mathematics

1 There are, of course, entirely valid ways of extracting conceptual sense of the definite integral by interpreting antiderivatives of the integrand. These do not, however, make use of Riemann sum reasoning, nor do they shed light on the symbolic manipulations involved in finding the antiderivatives. 334 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 curricula prior to the study of calculus that ask students to use the semantics of one mathematical process to interpret the syntax of another. The purpose of this research is to document how this conflict between syntax and meaning is mani- fested in the reasoning used by undergraduate physics students. This suggests that a few words of caution and clarification are in order before proceeding. It may be easy to interpret this paper as yet another revelation of Bdeficits^ in our students’ mathematical skills and understanding. My point, to contrary, is that the challenges that these students face are quite likely indicative of deficits in the experi- ences, instructional practices, and curricula to which they have been exposed.

Methods

Student Participants

Students who took part in this study were enrolled in a large public university using a quarter system of eight-week terms. Volunteers were selected from two groups: Eight beginning students were enrolled in an introductory calculus-based physics course focusing primarily on classical mechanics. All but one of the beginning students had completed differential and integral calculus, and had already completed (five students) or were concurrently enrolled in (two students) multivariable calculus. (The exception was concurrently enrolled in integral calculus for a second time.) The beginning students had a variety of majors and minors, at least one of which required them to take physics. Seven upper-level students were all third-year physics majors who had already completed two terms of multivariable (vector) calculus, and at least one additional course in advanced mathematics, such as differential equations or linear algebra. Except for an attempt to balance male and female participants, all students were selected randomly from pools of volunteers. Nearly all participants self-reported as typically earning mathematics grades in the A-B range, except beginning student B1 who reported occasional Cs and upper-level student U3 who reported occasional Ds. Beginning students were offered a modest stipend for each interview, and the advanced students were offered no motivating compensation but were given a modest gift certificate at the end of the study. For ease of presentation, students will be referred to by a letter and number combination, with beginning students identified as B1-B8 and upper-level students as U1-U7.

Interviews

Students were interviewed individually by me, the author, every other week during the course of an eight-week term, with each interview typically lasting 45–60 min. Partic- ipating students knew me only as an educational researcher, and I had no instructional or evaluative role in the students’ academic performance. All students completed at least two interviews, and most completed four. Questions and problems involved conceptual and procedural aspects of integration, differentiation, and other aspects of calculus, some in purely abstract mathematical form, and others in applied contexts. Physical devices and manipulatives were sometimes used. The questions varied from highly open-ended queries to focused, textbook-like problems. The specific questions Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 335 and problems used that are relevant to this paper will be presented as needed below. Each segment of an interview typically began with a written question or problem that the student was asked to read aloud, and the student was then asked to respond, thinking aloud as much as possible and explaining his or her thinking as clearly as possible. With the written question as a starting point, further questioning was open- ended and free-flowing, though some questions were addressed to all students. Except in rare circumstances, questions were always posed to elicit students’ explanations and understanding, and the interviewer avoided taking on an intentionally instructive role. In general, I asked probing questions until I believed, in the moment, that I understood the answers and explanations offered by the student, without offering judgment on them. All students were permitted and encouraged to write anything they found helpful on paper provided to them. A calculator was offered if requested. The interviews were audiotaped and videotaped using two cameras and an additional audio recorder. One camera was focused downward on the desktop to capture written work, and a second was focused on the student to capture facial expressions, gestures, etc. All written work was saved and digitized, and I transcribed many of the interviews for further analysis.

Method of Analysis

Data used in this paper were analyzed using qualitative methods. At the top level, I reviewed interviews in their entirety, multiple times, breaking them into episodes in which a student was using or interpreting a definite integral. For each episode, and as part of a larger research program, the students’ responses and explanations were analyzed with an eye toward a variety of forms of reasoning about the integral, including Riemann sum reasoning. The phenomena of interest here, however, were neither hypothesized nor predicted prior to data collection, so none of the tasks or interview methods were intentionally designed to capture the students’ reasoning and behavior described here. Rather, as I attempted to interpret carefully each student’s ways of under- standing and making sense of the definite integral, I noticed that certain patterns of reasoning that emerged all seemed to suggest a common underlying source, namely the apparent discontinuity described above between students’ knowledge of Riemann sums and their knowledge of the processes involved in solving integrals using the FTC. I emphasize this point to highlight the Baccidental^ and purely observational nature of this work: It did not arise from prior research questions or research design. In the work that follows, I present a collection of students’ responses through which I make my argument. In a sense, I am using the data as a type of existence proof for the phenomena of interest, and I argue that the observations made here represent a matter of concern to educational researchers that has not been previously raised.

Data Analysis

The data resulting from this research will be presented in three sections. First, I will examine evidence that the beginning physics students entering the study had not yet 336 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 learned to use Riemann sum-based reasoning to interpret the definite integral, although the upper-level students had. I briefly discuss additional evidence for students’ perceptions of integration and Riemann sums as distinct processes whose relationship is, to them, ambiguous. In the second section, I will consider a variety of ways in which students attempted to impose algebra-based reasoning onto their integration processes. Finally, in the third section, we will consider upper-level students’ ongoing discomfort with and limited understanding of the antidifferentiation process, even though they had adopted Riemann sum-based reasoning to interpret integration.

The Occurrence of Riemann Sum-Based Reasoning

Evidence for the Absence of Riemann Sum-Based Reasoning

All of the upper-level students, but none of the beginning physics students, entered this study using Riemann sum-based reasoning to interpret integration.2 In the case of the upper-level students, each of them was able to give descriptions of Riemann sum-based reasoning similar to U3’s description presented earlier. (Further evidence of this will not be presented here, for the present concerns are related to students who did not use such reasoning.) Demonstrating the absence of a particular reasoning strategy is challenging, of course, for the absence of evidence should not be taken as evidence of absence. In this case, I make the assertion of absence based largely on students’ responses to two problems used in the study. These problems are presented in Fig. 2. The key piece of evidence used to determine if a student was using Riemann sum-based reasoning was the student’s interpretation of the units associated with the integral. A student’s failure to determine the correct units was taken as strong evidence that the student did not perceive a product in the expressions R(t) dt and x(t) dt,and,hence,wasnot using Riemann sum-based reasoning. Students’ responses to The Integral of Position Problem are particularly compelling because none of them (by their own observation) had ever been exposed to the question before. The integral of a position function with respect to time has no widely acknowledged physical or otherwise meaningful interpretation, nor are its units of meter-seconds imbued with any familiar meaning to most individuals. All but one (B6) of the beginning students failed to identify the correct units of one or both of the problems in Fig. 2. In trying to make sense of the problem, students often offered explanations that suggested that they did not have any tools with which to analyze the units, but tried to rely on contextual clues instead:

B4: I'm guessing the engineers would learn, like, through this crazy like variation, whatever, that they learn like, per minute over a span of time, the revolutions per minute, or how many revolutions are there in each minute at that-. I don't know the variable of the y-axis. I'm going to guess it. Wouldn't-, I guess it would be distance? Because it's a car? But, that's what I think. I don't know if that answers your question [posed by the RPM Problem].

2 One beginning physics student, B1, showed sporadic use of Riemann sum-based reasoning, but did not use it consistently and he could not decide on the units for The Integral of Position Problem (see below). Primarily for reasons of space, this case will not be examined here, but I will use the language that Ball beginning students^ failed to use this reasoning strategy for ease of presentation. Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 337

The beginning students who correctly identified units did so using alternative forms of reasoning, often entirely legitimate:

B2: R(t) would be the change in revolutions, in change of time, if I were to integrate that. Like that's a form-, it's like a derivative of some function, and if I were to integrate that, it would just become a function that was the revolutions rather than the change of revolutions, in-, per minute, for example. Revolutions per minute indicates that-, like a ratio of revolutions and minutes. So whatever the integral of this is, it's going to be just an equation that gives you revolutions.

B2 went on to explain correctly that the integral found the total number of revolutions over the 600-min time interval. He interpreted the integrand as a derivative, and so he deduced the units of its antiderivative using his knowledge of functions and their rates of change. His reasoning is entirely legitimate, but it does not make use of Riemann sum-based reasoning. When faced with The Integral of Position Problem, whose integrand could not easily be interpreted as a rate of change, B2 was stumped:

JFW: Is there anything you can tell me about the units it would give?

B2: Well, hmm. [...] Seconds, no that-, maybe, probably not. [...] Well, if position is meters or some-, something like that. Meters or inches or something. And when I take the derivative of that and get velocity, it's meters per second. If I take-, I get rid of the seconds when I do the integral. So if I do the integral of me-, I don't know, maybe seconds over meters. That doesn't really sound right.

B2 again tried to consider rates of change by recalling the derivative of position, considering that he would Bget rid of the seconds when I do the integral^ of velocity, but this did not help him make sense of the integral of position. One student, B6, had developed a method for analyzing the units of an integral by considering the units of area on a graph of the function:

The RPM Problem The durability of a car engine is being tested. The engineers run the engine at varying levels of “revolutions per minute” for a period of time. Denote the number of revolutions per minute at time t by R(t). Interpret the following: 600 R(t)dt 0

The Integral of Position Problem The position of a particle traveling along a line is given by the function x(t), where t is measured in seconds. Interpret the following: 5 x(t)dt 0

Fig. 2 The RPM Problem and The Integral of Position Problem. The RPM Problem was adapted from Jones (2013). The statement of each of these problems inadvertently omitted some units. In all cases, students were informed that t was measured in minutes for The RPM Problem, and that x(t) was measured in meters for The Integral of Position Problem 338 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356

B6: OK. So the y-axis is revolutions per minute, which is the same as revolutions over minutes. And then the x-axis is minutes, which-. So given any curve, or any area you're trying to find, which in this case would be 0 to 600 of whatever curve is there, even if it was just a little square, its-. To find any area would be revolutions per minute times minutes. Because length times width, or whatever curve times width would be revolutions per minute of that unit, times the unit here, which is minutes, which again, it's just a simple multiplication problem for units, would be revolutions per minute times minutes, which cancels out the minutes, leaving only revolutions

JFW: So, you're, you just said it's a multiplication problem, and you're doing multiplication. Is there, is that multiplication in this? [points to original integral]

B6: No. That multiplication is only used to derive the correct units. Because that is simply denoting what the area is under that curve, unit-wise. In this [points to original integral], this is finding out what the actual area is….

B6 reasoned that units of area on a Cartesian graph would be the product of the units of the vertical and the horizontal axes (BBecause that [multiplication] is simply denoting what the area is under that curve, unit-wise.^). When asked explicitly, however, if he perceived the multiplication process as part of the actual integral, he denied it, indicating that his multiplication method was Bonly used to derive the correct units.^ The integral Bis finding out what the actual area is,^ suggesting that he saw it as a means of deriving the numerical value of the area, but not the units. His Cartesian analysis allowed him to determine the area Bunit-wise.^ When he later considered The Integral of Position Problem, he analyzed units again by drawing a graph, and pointing to the axes of the graph as he suggested that the units were meter-seconds. Finding this answer unappealing, however, he further argued that multiplication by seconds Bwould just give you more meters,^ suggesting that the final units were simply meters. Evidence of students’ failure to identify units is quite compelling, I think, in supporting my claim that none of the beginning students had learned Riemann sum- based reasoning to interpret the definite integral.3 In all but one case (B5), however, additional triangulating evidence also existed, such as B6’s explicit denial of his perception of a product in the integrand. Additional evidence included students’ failure to explain the meaning or purpose of the differential, or if it could be represented on area graphs. Finally, all beginning students in this study took part in an activity during their final interview designed to introduce them to Riemann sum-based reasoning. Some of them expressed notable surprise that the integral could be interpreted this way, and this was also taken as evidence that they were not able to use such reasoning earlier.

The Relationship between the Definite Integral and a Riemann Sum

The beginning physics students varied in their explanations for how integrals and Riemann sums were related. When asked explicitly what Riemann sums have to do

3 A few beginning students occasionally made mention of sums or rectangles while discussing the definite integral, however none of them could successfully use them to explain what the integral did or what its units were. Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 339 with the definite integral, four of the eight students indicated, directly or indirectly, that integrals and Riemann sums were two different processes for finding the same thing:

B1: They both find area under a curve.

B3: So I guess the integration is a simplified version of that [Riemann sums], as presented as an equation.

B5: I mean it's a different process [Riemann sums], but, yes, it finds the same thing.

B7: That’s [Riemann sums] basically what the integral’sdoing.

The other four interpreted Riemann sums and integrals as somehow similar, but not quite the same, sometimes emphasizing Riemann sums as a means of approx- imating a definite integral:

B2: It's [Riemann sums] related, sort of like a similar idea [to the definite integral]. You could do it-. I know that this is more, this is, like, if you can integrate, and you can solve, then that's better. But in some equations like, you can't solve the integral, so you can do this method, sort of, and take the sum of all these, and get kinda close to the answer.

B4: You're just finding the area underneath the curve of that function, but like with a great amount uncertainty in rectangles. I didn't like them!

B6: A Riemann sum is basically the easy way of finding a definite integral that isn't particularly exact.

B8: I don't remember the specifics, but finding this area [by integration] is kind of like this [Riemann sums], but in a more fluid approach. I don't know.

Both B2 and B6 also indicated, however, that the approximation given by a Riemann sum would be exact in the limit, appearing to treat that as theoretically true, but not of practical use. Several students explicitly noted that they did not understand why learning Riemann sums was important, and that they did not find them useful in understanding integra- tion. Some further noted that that they were told explicitly during instruction that after learning the FTC, they did not need to bother using Riemann sums again. (B5: BBecause like when they were teaching this, they were kind of like, oh, like you'll do this for the first test, and then you can get rid of it and never have to do it again.^)

Summary

At best, it would seem that the beginning physics students perceived some vague relationships between the definite integral and Riemann sums, perhaps because these topics were taught sequentially. In general, however, they interpreted them either as two 340 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 different methods of finding the same thing (usually an area), or they assumed that Riemann sums could be useful for approximating a definite integral, particularly if the integral was symbolically intractable. In the end, their understanding of these concepts gave them no reason to think it necessary or helpful to talk about Riemann sums when asked to talk about or explain integration. It is interesting to recall that, when learning about derivatives, students are typically first introduced to the method of finding a derivative using its definition as a limit. Soon after, they learn general rules that render the limit definition obsolete in practice, even if it appears occasionally theoretically. Perhaps it should not be surprising that students treat the relationship between definite integrals and Riemann sums in the same way.

The Search for Meaning in the Antidifferentiation Process

All of the student participants in this study, both beginners and upper-level, were asked to make up a simple area problem and to solve it. All of the students chose to find the area beneath some polynomial function, and all of them did so correctly using antidifferentiation. They were then asked why the antiderivative was used to find area. None of the beginning students was able to answer the question, responding directly or indirectly with some version of BI don’t know.^ Most attempted no further explanation, sometimes noting that it was simply what they had been taught, but that they could not explain it. Of the seven upper-level students, five of them also indicated that they did not know how to answer the question. Only one student, U4, indicated that it was a consequence of the FTC. The one remaining student, U2, responded that either Newton or Leibniz Bcreated or discovered^ what he described as Balistofrules^ to transform a function into another function that gives the area under the first function. Some students sug- gested that they probably once knew how to answer the question, but had since forgotten, while others said that they had never really thought to ask the question in the first place. Because the majority of the students were not able to explain why area could be found using antidifferentiation, they typically made no attempt to find meaning in their solution process. A few, however, showed evidence that they believed that the antidifferentiation process ought to be explained through algebraic or geomet- ric interpretation, even if they did not know how to explain it themselves. I offer here several examples of such students.

The Search for Meaning in the Symbolic Manipulations

During the first three interviews, beginning student B2 frequently noted that he did not understand why the differential was included in the notation for the integral:

B2: And again, I don’t really know what, what’s the deal with this differential. I don’t know why that’s there. I mean, it just goes away.

The observation that the differential Bjust goes away^ was made by several other students in the study, as well. It directly suggests that these students interpreted the integral as a process for finding something, and that, by just Bgoing away,^ the Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 341 differential plays no role in the solution process. The implication, perhaps, is that the algebraic symbols ought to be doing something, serving an algebraic role. B2 was very explicit about his expectations for the existence of algebraic meaning in the antidifferentiation process during his fourth interview. The issue arose here as he tried to explain why integrating Brevolutions per minute^ with respect to time gives a result with units of Brevolutions^:

B2: I don't know what the integral does specifically. Like, I can solve it, if I see this, I can solve it. I don't know why, like, bringing up a constant in the exponent, or whatever you have to do to solve it, is reversing-. I mean I know that it's exactly reversing the derivative, but I don't know why that means that it's now revolutions instead of revolutions per minute if I was integrating revolutions per minute. I don't know what-, like, where the minutes go.

Note the active nature of his description, focusing on explaining the symbolic manip- ulation process. In trying to explain the change in units, he referred to the symbolic process of Bbringing up a constant in the exponent^ but failing to find an explanation for Bwhere the minutes go^ in this process. Later in the interview, although he indicated his belief that the integral would find the total number of revolutions, he exhibited continued confusion about why the process gave that result: BI still don’tunderstand how integration does this.^ During the rest of the fourth interview, B2 worked through an activity (based on Engelke and Sealey (2009)) that was designed to help highlight a Riemann sum-based interpretation of the definite integral as a sum of infinitesimal products. By the end of it, we discussed explicitly how, in general, the symbols f(x) dx could be interpreted as a product:

B2: Yeah. I just-, OK, so, I follow that, and I get that this is-. So this is actually multiplication. That's what I thought. I don't get why solving this is like-. I don't understand why, like, how I solve this is the same thing-, or. I don't know. Like, when I bring exponents up, or do like integration by parts, or whatever it is I'm doing to solve this problem, I get another equation that gives me positions-.

B2’s response suggests a direct confrontation or conflict between a Riemann sum-based interpretation of a definite integral, and the calculation of an integral through an antidifferentiation process that has no connection to that interpretation. Despite accepting the product interpretation, B2 still expressed his inability to understand Bwhy solving this^ should be subject to that interpretation. By Bsolving this,^ he made it clear that he referred to Bwhen I bring exponents up, or do like integration by parts, or whatever it is I’m doing to solve this problem.^ Even after he was able to use the Riemann sum-based interpretation, B2 expressed continued confusion because the explanation did nothing to inform or explain the process he had been taught. He could not find meaning in the symbolic manipulations that corresponded to the interpretation I was encouraging him to use. At that point, I engaged B2 in a general discussion about the FTC, explaining among other things, that the power and significance of the theorem is found, in part, in its unanticipated result, and that the rules of integration that he has learned result from patterns that can be demonstrated, but that aren’t subject to algebraic or geometric 342 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 interpretations in quite the way he was used to. After we talked for a while, I asked him if my responses helped to clarify things for him:

B2: Yeah, it does a lot. So there's just not-, it's not just like simple as to why I do the methods I do to solve an integral, like integration by parts or anything?

His final response confirmed, again, that searching for meaning in symbolic methods of integration had been his central concern and obstacle. In the end, he seemed satisfied (and perhaps a bit relieved) to learn that the meaning he had been seeking in the integration process was not, in fact, implicit in the symbolic procedures themselves. At least one other student suggested a search for meaning in the symbolic manip- ulations of the power rule. In this case, an upper-level student, U7, who was accus- tomed to using a Riemann sum-based interpretation of the integral, was asked to explain the role of the antidifferentiation process when integrating the function f(x)=x:

JFW: OK, so when you solve this, you-, and in both cases you went from an x and you did a procedure that led you to one-half x-squared. Why does this give you area?

U7: It's similar to, it's right here [circles his rectangle drawing from earlier], I'm guessing. Where you have [...] you have the height, which is f(x), in this case, f(x) equals x. And then you have your width, which is your infinitesimally small change in x. Which would be right here [indicates width of a Riemann rectangle].

JFW: But I guess I'm not seeing what that has to do with changing this x into a one-half-x-squared.

U7: Oh! Um ... something that has been drilled into me so much that I forgot the reason why. [Both laugh.]

JFW: Specifically, the reason why behind what?

U7: Well, I know this, what is this, the power rule? I know it's the power rule, but I guess they never showed me why behind the power rule, or like the visual, a connection between the graph and-. It's just kind of something that, it's because that's the way it is [laughing]. Which is a terrible answer to anything, and I'm probably going to look this up when I get home, because now I'm curious.

JFW: Well, because it's not just the power rule, if you integrated cosine, you would-.

U7: It would be-, yeah, there's a bunch of other stuff going on. But, I mean, it is, it's the area under the curve, but we also have to understand that there's a rate of change involved, and that, these are like tiny little slices. But now I, now I want to go over and to find out the concept behind it. That would make it so much easier to explain to my parents [laughing]. Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 343

JFW: OK, so, just to be clear, so at this point, you really don't have an explanation for why this, this algebraic manipulation leads to something that gives you this area.

U7: Yeah, I mean, I could just give you, like I said, I could just quote the power rule, or just that. But I wouldn't be able to tell you the [garbled] concepts behind it. So, I don't know! [laughs]

As noted above, U7 was among the five upper-level students who could not explain the role of the antiderivative in finding the area under a curve. When I first asked the question, he replied with a Riemann sum-based interpretation. After I restated my question concerning the role of the antiderivative, he immediately indicated that he could not remember, but he went further in identifying the power rule, specifically, as a procedure he could not explain. His observation that Bthey never showed me why behind the power rule, or like the visual, a connection between the graph and-,^ indicates an expectation that the power rule should be subject to some of Bvisual^ explanation or Bconnection^ to the graph. Perhaps the student expected there to be a geometric interpretation of some sort, but it is clear that he believed that the symbolic manipulation process ought to be subject to some kind of Bvisual^ explanatory interpretation.

The Search for Meaning in the Substitution Procedures

Some students found significance in the substitution procedures implicit in the FTC, particularly as a means of explaining the change in units between the initial integrand and the value of the integral. While discussing the RPM Problem, beginning student B8 tried to explain why integrating Brevolutions per minute^ with respect to time yields an answer in units of Brevolutions^:

B8: Ok, so if this is in revolutions per minute. Oops, I did it backwards. If this is in revolutions per minute, and then-, when we evaluate it we evaluate it from this minute to this minute. Then it would be revolutions times 600 minutes-, yeah, over minutes. [Writes Br(600m)/m^] And then these could cancel and then you would get revolutions.

JFW: I'm not sure where the times came from here [points to the multiplication by 600].

B8: If you-, after you-, like after you antidifferentiate, then you evaluate at the-, at the-. Oh, shoot. What are they called? At the limits, let's call them.

JFW: So, let's give an example. Suppose I give you 0 to 600 and we'll just say it's 2 ∫600 2 t .[Iwrite 0 t dt.] Why don't you evaluate it?

B8: OK. So, this would become-.

JFW: Actually, I am going to need a little more-. Why don't we make it something like t-squared plus 1. [I adjust the integrand to t2 +1.] 344 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356

1 3 þ B8: OK. So then this would be 3 t t. It would-, and then you would have to evaluate this at 0 and 600. So then it'd be 1/3, 600, times 600 cubed, plus 600. [...] Minus 1/3, 0 cubed plus 0. So this would-, this would go to 0. And then this would give you revolutions between time 0 and 600.

JFW: What I want to ask you about is here [previous written work] you were talking about these things being multiplied. And I'm wondering where you multiplied in there.

1 3 B8: Well, here is t, so you substitut-, you-. [...] This [points to 3 t ] is-. This is-. 1 3 þ [sighs] [...] Maybe I misspoke. [...] Well this [points to 3 t t] is still a function of revolutions per minute. Or-, no. This [glides pen under "t2 +1dt"] is a function 1 3 þ of revolutions per minute. This [points to 3 t t] is giving you the number of revolutions, and-, for time t.

JFW: And I guess that's my question. OK, so that changes my question. How do you know that? Or what makes you think that?

B8: Because essentially you're just plugging in a time point, and then that's giving you how many revolutions there were at that time point.

JFW: OK, but I'm still trying to figure out how you-, why it's revolutions.

B8: Because you're adding in the time component. You're substituting in.

Throughout her explanation, B8 attributed the change in units from Brevolutions per minute^ to Brevolutions^ to the substitution process in which units of Bminutes^ are introduced into the function. At first, without an explicit function to integrate, she appeared to attribute it to a multiplication process, as Brevolutions per minute^ are multiplied by B600 min^ in substituting the endpoints of integration. Because I thought she might be interpreting the function R(t) as a constant, I made up an explicit function that made it clear that she was not simply multiplying by the endpoints. At this point, she adjusted her reasoning, admitting that multiplication was not the source of the unit change (BMaybe I misspoke.^), but she continued to attribute the change in units to the substitution process, explaining, Byou’re adding in the time component. You’re substituting in.^ I note first of all that, although we would like students to see multiplication in the expression R(t) dt, this is not what student B8 was seeing. Her first suggestion of multiplication occurred after the integration process: Bwhen we evaluate it, we evaluate it from this minute to this minute. Then it would be revolutions times 600 minutes.^ Second, B8 repeatedly attributed the change in units via integration to take place through the substitution process. That is, the time sequence in all her explanations made it clear that she believed that the unit change did not occur until the substitution process took place. At first, she explicitly noted that the antiderivative Bis still a function of revolutions per minute^ (emphasis added), suggesting that the units did not change through antidifferentiation. She immedi- ately appeared to question this, however, deciding that, in contrast to the Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 345 integrand, the antiderivative Bis giving you the number of revolutions, and-, for time t.^ The addition of the phrase Bfor time t,^ intentionally added as an apparent clarification to a sentence that she was about to continue (with Band-^), may indicate the importance she placed on the substitution of the time component. She affirmed this interpretation after my further questioning about why the unit change occurred: BBecause you’re adding in the time component. You’re substitut- ing in.^ Evidence for this perception of changing units only through the substitu- tion process was seen occasionally in other students as well, and offers a variation of students’ likely search for meaning in integration through their usual algebraic interpretations.

The Search for Meaning in the Geometry of the Antiderivative

I noted earlier that advanced student U7 expressed his belief that the power rule for antidifferentiation of a polynomial might be amenable to some sort of graphical or geometric explanation. Another advanced student, U5, went on a search of her own for a geometric interpretation of the antiderivative when asked why it was needed to compute an area. I asked student U5 to make up a simple integral problem and solve it. She chose to integrate the function f(x)=3+5x over the interval 6 ≤ x ≤ 10. She did it correctly, identifying her process as Bantideriving.^ (See Fig. 3.) I then asked her why Bantideriving^ was used to find area, and she went silent for an extended period of time, thinking, writing, and sketching. When I finally interrupted her to ask what she was doing, she offered her initial response:

U5: I'm trying to relate my x-squared value and my evaluated integral, and relating it to x and y coordinates, but I'm having a hard time doing it. [...] [Softly:] I don't remember why you do that.

She described herself as attempting to relate the antiderivative she found [Bmy x- squared value^] and her final answer [Bmy evaluated integral^] to a graphical repre- sentation of the area that she found, described verbally as Bx and y coordinates,^ but visually apparent as the sketch that she drew. (See Fig. 4.) After a little discussion, she continued to go about her search until I interrupted her again:

JFW: Can you say a little bit what you're thinking about?

U5: I'm trying to break it down into like different parts to describe it.

JFW: What's the Bit ^ you're trying to break down?

U5: This part, right here [points to f(10)-f(6) calculations], so that I can maybe work backwards. So I could say that 3, 10, [mumbles while calculating] is the area that's right here. But I still have this part left over, that I don't know how to explain. 346 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356

Fig. 3 Student U5’s integral computation At this point, it was clear that she was taking the values that arose in her computation and attempting to map them onto the picture that she had drawn. Her written work made it clear that her first attempt involved searching for significance behind the 3x term in her antiderivative. She wrote 3(10) − 3(6)=, and (mumbling) identified it as 12. She then Bshaded^ in the lower portion of her area of integration, the rectangle of height 3 and width extending from x =6tox = 10, using slanted lines drawn from lower right to upper left, and identified the value 12 as Bthe area that’s right here.^ In fact, her association of her calculation with the area of the rectangle she identified was correct, however, it was clear from the behavior that she was searching for correspondences and stumbled upon it, initially without explanation. As she indicated, she did not Bknow how to explain^ the remaining area. In the minutes that followed, she did, in fact, come up with a correct correspondence between her calculated values and the area of the region she found. She seemed both excited and satisfied to have found this connection, though a bit deflated that she could not offer a geometric explanation for why those correspondences held or explain why the antiderivative produced this result. At that point, she went back to the drawing board, wrote down the general expression for the power rule, thought for a bit, and concluded BIdon’t know.^ The point of this analysis, of course, is that, like other students, U5 also searched for meaning in the algebraic symbols of the of antiderivative, in this case, geometric meaning. The correspondences she found were correct, but did not offer any explan- atory power, nor could they if the area had been a more complicated one.4 In her search, she lost track of the original question concerning why antidifferentiation was useful in finding area at all. When she returned to the question, she again turned attention briefly to the power rule for the antiderivative as if to seek an answer there, but could not offer any further explanation.

Summary

Each of the cases above highlights a different way in which students can make thoughtful efforts to interpret and make sense of the mathematical procedures that they have been taught. In each case, as well, those students brought to bear what they likely

4 The area chosen by U5 lends itself to a more general geometric analysis that reveals algebraic sense behind the antiderivative area calculation as the difference of areas of triangles. This geometric argument does not hold for polynomials of higher degree, however, nor does it shed any light on why the algebraic representation should correspond to the antiderivative of the bounding function. Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 347

Fig. 4 Student U5’s sketch corresponding to her evaluated integral. She identified the area found by the integral to be the region Bshaded^ by the slanted lines drawn from upper left to lower right expected to be the most obvious and useful tools for doing so: the same tools they use for making good sense of algebra and geometry. The difficulty, of course, is that antidifferentiation is not subject to ordinary algebraic interpretation; it represents a function transformation—a notion given little or no explicit attention in traditional calculus curricula. As the next section will reveal, however, even students who do have awareness of antidifferentiation as a function transformation can still be at a loss to explain the process conceptually, leaving them reluctant to interpret it through the lens of Riemann sums.

Advanced Students’ Discomfort with Riemann Sum-Based Reasoning

The cases of U5 and U7 above reveal that even some upper-level students who demonstrated an ability to use Riemann sum-based reasoning were still subject to searching for other levels of meaning in the process of antidifferentiation. In this section, I show how two additional upper-level students, equally competent in using Riemann sum-based reasoning, nevertheless showed discomfort or dissatisfaction with the use of such reasoning, because it did not appear to them to be justified by what was actually taking place mathematically. Both of these students reported consistently receiving Bmostly As^ in their mathematics courses.

The Case of U1: BIdon’t feel great about doing this.^

∫b ðÞ When first asked to explain the meaning of the symbols a fxdx, upper-level student U1 immediately offered a Riemann sum-based interpretation: 348 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356

U1: So when I think of an integral sign, I usually have this kind of analog model of, OK, what you're going to be doing is essentially chopping it up and adding it. And "chopping it up" refers to the function.

By referring to his explanation as an Banalog model,^ it is clear that U1 understood himself to be applying an interpretation that serves only analogously; that is, it does not directly represent what the process of integration actually does. Although he only spoke of Bchopping^ and Badding,^ further conversation revealed that he interpreted f(x) dx as a product representing the area of a Bsmall rectangle,^ as well as explain what happens as the number of rectangles approaches infinity. He referred to the process by name as Riemann integration. Later in the interview, I asked U1 to make up a real-world problem that could be solved using a definite integral. He chose to model the potential energy of a horizontal mass- spring system that required integrating a force F over a distance, with the position of the mass given by x. He explained that the work done on the block by the spring could be represented as a sum of products, Fdx, geometrically represented by rectangles, conclud- ing, BEach little rectangle would be like a little contribution of work done on the block.^ I asked if he could directly connect his explanation with the symbols in the integral:

U1: I guess you could say, if you have F at a point x, dx, that would be some contribution, because you have, say F, the force at this point is some value, and then you're gonna push it some, you know, dx, which is some infinitesimal quantity, which, ehh, you know [spoken as if to express skepticism]. And so that would- this would be, kind of a little-, this would actually be, you know. [Writes dw = F(x) dx.] Yeah, I do it. I don't-. I'm not proud of it, but I hope that there's some way to justify it. So this [dw] would be some contri-, little contribution of work. Right? And, so, essentially, you can say, at least, if you're a physicist you can say, that from a to b, you integrate all the d-, all the differential work, right?

The student gave a standard Riemann sum-based explanation for the integral, yet his explanation was peppered with dissatisfaction, apparently because of the use of infinitesimal language. Immediately after introducing the language of an infinitesimal, he indicated skepticism through his tone of voice and qualifier, BEhh. You know.^ After writing down a differential expression, he followed up with a clear statement of skepticism, BYeah, I do it. I don’t-. I’m not proud of it, but I hope there is some way to justify it.^ He further noted his discomfort by suggesting that, Bat least, if you’re a physicist, you can say^ that the process integrates Ball the differential work,^ suggest- ing a belief that his interpretation is used by physicists, but is mathematically suspect. Later in the interview, I asked U1 if he usually interpreted integration as a sum of differentials:

U1: When I think about integration as a sum of differentials, quantities-. When I think about that, I go, OK, that makes intuitive sense, and it works. Great. But then I wonder, you know, what is, in terms of more modernized math that I'm doing. Because I usually feel like what I'm doing is kind of a trick. And it works. I don't feel great about doing this, like, intuitively I feel fine." Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 349

I asked him to say more about why he thought it was a trick:

U1: I don't know, I guess I just-. So, some combination of hearing, hearing that this isn't very rigorous, or at least that the way that it's done-. I guess I just-. I don't feel that confident that this is something that you can do, in a sense….

In his third interview, U1 raised the issue again. While discussing the meaning of a line integral, he referred to the explanations he has heard for using and interpreting differentials as Bhokey.^ I think the transcript makes it clear that U1 understood that the reasoning he was using was an Banalog model,^ indicating a disconnect between the computational process and the interpretation of the result. It is worth noting, however, that even though U1 could offer a fine explanation for finding an area as a limit of a Riemann sum, he never appeared to use that reasoning as a justification or explanation for why the Banalog model^ he used for interpreting the definite integral might be legitimate. His discomfort, it seems, was further heightened by Bhearing that this isn’t very rigorous,^ a reference, perhaps, to mathematical controversies over the use and inter- pretation of differentials and differential notation.

The Case of U2: BIt’s impossible to explain this integral conceptually.^

Upper-level student U2 was, in a sense, the exception that proves the rule. On the one hand, he varied between two different conceptual interpretations of the definite integral: a Riemann sum-based interpretation, and a second interpretation that relied on an imagined weighted average of the values of the integrand, with the differentials serving as weights.5 Simultaneously, he exhibited one of the most sophisticated understandings of the antidifferentiation process, explaining it as a function transformation. On the other hand, although U2 did not attempt to place interpretive meaning on the symbolic manipulations of antidifferentiation, he interpreted the process as a deeply mysterious and inexplicable one, and he shared U1’s mistrust of reasoning about infinitesimals. In an early discussion of integration, U2 described the Riemann process of summing areas of rectangles, and he used h as the width of each rectangle:

U2: And so the smaller we make our h, the more accurate we get. And so if we made h go to zero, or as close to zero as we can get, we actually, this ceases to be an approximation and becomes the actual area. Because you can quantify the error, the difference between these is an expression that involves h.Soitwillbe something multiplied by h. So the bigger h is, the bigger the error is. When h equals zero, the error is actually zero. And it's hard to conceptualize that, conceptualize taking the area of a rectangle where one of the sides, one of the dimensions of the rectangle is zero. That doesn't make any conceptual sense, but it makes mathematical sense, and it works, and it gives us numbers.

5 Both of his interpretations, with some careful tweaking, were quite reasonable, but the details are outside the scope of this paper. 350 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356

This was the first indication of U2’s belief that although the infinitesimal interpretation of the integral Bmakes mathematical sense,^ it Bdoesn’t make any conceptual sense.^ On the one hand, U2 used what Oehrtman (2009)identifiedasaBcollapse metaphor for definite integrals,^ suggesting that Bwhen h equals zero,^ the two-dimensional rectan- gle would actually collapse into a one-dimensional object: Bone of the dimensions of the rectangle is zero.^ On the other hand, rather than rely on this metaphor as part of his explanation, U2 immediately concluded that Bthat doesn’t make any conceptual sense.^ This lack of conceptual sense made him reluctant to place geometric interpretations on the individual symbols of the definite integral, suggesting some progress beyond dependence on the sorts of metaphors Oehrtman observed. Here U2 interpreted the symbols of the integral of a function g(x):

U2: Well, in order to find the area, we're just multiply-, you just multiply the height by the width to find the area of a tall column like that, and so dx is the width and g(x) is the height. And so d of x and delta x,deltax is a way of kind of talking about d of x, but in this case when, when delta x gets infinitesimally small, it becomes d of x. And I mean, you can't actually do math with d of x because d of x is an infinitesimal number, so you can't do arithmetic with it. So this expression really doesn't mean anything without this. This whole expression is a symbol, and it's not like you can really do anything to the particular things inside the expression. You have to deal with it as a whole.

It is clear from the context and written work that U2 transitioned, apparently inadver- tently, into pronouncing the differential dx as Bd of x.^ His point, however, was that, because he interpreted the antidifferentiation process as a transformation of the inte- grand, he insisted that the entire integral expression had to be dealt with Bas a whole.^ Throughout his explanations, he juggled his interpretation of antidifferentiation as a transformation with his Riemann sum-based reasoning. The point of conflict, it seems, occurred whenever his reasoning led him to arguments about infinitesimals. On the one hand, he first interpreted the dx as Ban infinitesimal number,^ but on the other hand, he denied that you could Bdo anything to the particular things inside the [integral] expression.^ That is, the dx was not meaningful in itself, but only as part of Bawhole.^ I noted above that U2 demonstrated perhaps the most sophisticated understanding of the definite integral. He gave every indication that he understood that Riemann sum- based reasoning was useful, but that the antidifferentiation process was a process of function transformation that did not, in itself, involve the algebraic processes of multiplication and summation. The surprising aspect of U2’s understanding, however, and as his quotation above first suggested, was that the absence of algebraic sense- making in the transformation process led him to conclude that there was really no sense to be made of the mathematics at all. Here U2 discussed his understanding of the algebraic process of integration, prompted by his use of the power rule:

U2: So math gives us these sort of weird tools, and they behave differently than any, like, physical tool we know of, and so it doesn't really make sense to ask why they work or how they work, because they work mathematically, not physically. So this mathematical tool called the integral allows us to change functions, to apply this operation that changes functions into other functions. And so if we Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 351

have a function describing a line, where g looks like something like that, where this is x and this is y. We can apply this operation to it, and this operation is the integral. So we've applied this operation, and so this entire thing is the integral, just like a plus sign. It's just a, it's an operation. [He boxes off the integral sign and the dx within one boundary, omitting the g(x) integrand. See Fig. 5.] And the-.

JFW: The integral symbol with the dx?

U2: Yes. Yes. They're both part of the same symbol, and neither one makes sense without the other. So I guess, the dx kinda makes sense, but you have to remember that it doesn't, like, make complete sense, and it's just basically an artifact that's there to remind us that we're multiplying of sorts.

U2 clearly experienced a conflict whenever he tried to interpret the symbol for the differential: Bthe dx kinda makes sense, but you have to remember that it doesn’t, like, make complete sense.^ He could use it Bto remind us that we’re multiplying of sorts,^ because he could associate it with the multiplication of Riemann sum-based reasoning, yet he had to include the qualifier Bof sorts,^ because the algebraic transformation process is not really doing any such thing. In noting that Bweird tools … work mathematically, not physically,^ he acknowledged that the symbolic process cannot be subjected to sense-making in the way that other algebraic representations may be (via Sherin’s 2001 symbolic forms, for example), but his ultimate conclusion was that the transformation process is one of mathematics’ Bweird tools^ for which Bit doesn’t really make sense to ask why they work or how they work.^ U2 most clearly expressed his belief that integration was ultimately not subject to sense-making in a later interview when he was asked to explain why the integral of a mass density function over a volume computed the mass of the volume. During our conversation, I mistakenly thought he had suggested that integrating the density function was a means of adding up densities:

U2: No, you're not adding up densities, because you are, you are taking these two little bits of information and you are combining them. You are finding the mass of an infinitesimally small little object, and you're using, you're finding the-. You're taking this infinitesimally small object-. Not even that. Like, it's impossible to actually accurately explain what this integral is

Fig. 5 U2’s depiction of the integral as an operation 352 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356

conceptually. It's impossible to do it. You can only do it using, like-. It's a, it's a limit. You are taking, you are describing the, the density, you're, you're applying a limit to a function, to get you-. And you're taking this, you're-. [...] OK, I'm glad I-. I'm not trying to explain it conceptually anymore, because it's not possible, like, it's, it, to talk about an infinitesimal volume and an infinitesimal density. That doesn't make sense. So you're starting with some- thing, you're starting with a density function. You're taking that density function, and you're taking this imaginary volume, because it is a volume right now, dV is a volume. And then you're applying this operation that gives you another function, and this function tells you the-. And you tell the function however many volumes you want to do and the location of every single one of those vol-, it, no, no, no, no. The function-, you provide it with bounds, and then for you it automatically calculates exactly every single dV, it finds them and it tells you the mass of every single one. You aren't, I mean, all you're doing is you're finding a new function. And you do or do not know what it does. And it happens to have a ton of real world applications.

In this excerpt (as well as several others), U2 interpreted the integration process as Bcombining^ the integrand and the differential (in this case, the density function ρ and the differential volume dV). U2 appeared to prefer this language over the language of Riemann sum-based multiplication, because antidifferentiation as transformation was not subject to such multiplicative reasoning. At this point, however, U2 got so frustrated with the language of infinitesimals, that he simply concluded that Bit's impossible to actually accurately explain what this integral is conceptually.^ He re- emphasized: BIt’s impossible to do it.^ He was not simply abdicating his responsibility to explain himself; rather, he indicated that he was Bglad^ to reach this point, because it led him to conclude that no satisfactory or sensible explanation could be given for the transformation process. BI’m not trying to explain it conceptually anymore, because it’s not possible … to talk about an infinitesimal volume and an infinitesimal density. That doesn’tmakesense.^ Riemann sum-based reasoning did not reflect what actually took place in the transformation, and algebraic reasoning was not applicable to the rules of antidifferentiation.

Summary

These examples show that even students who demonstrate an ability to interpret integration using Riemann sum-based reasoning may be still be reluctant to do so. An appeal to infinitesimals can seem Bhokey^ or even Bnot possible^ to talk about meaningfully. Just as with the beginning students discussed earlier, however, the difficulty arises when students attempt to explain exactly what they are doing when they are applying the FTC. These more advanced students show clear evidence of advancement over the beginning students, yet they are not yet able to reconcile their knowledge of Riemann sums with their knowledge of integration using antidifferentiation. One student held out hope that he might come to understanding through the study of more advanced mathematics, while the other came to the conclu- sion that, unlike physical tools, some mathematical tools cannot be subjected to sense- making. Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 353

Discussion and Conclusion

What all of these students have in common is a difficulty in finding meaning or sense in the symbolic manipulations of integration through antidifferentiation. Some beginning students appear to expect that such manipulations can be subjected to familiar algebraic and geometric sense-making. For these students, there is no reason to expect that anyone might impose Riemann sum-interpretations on the definite integral, since the symbolic processes of computing such integrals do not correspond to the algebraic and geometric sense that they can use to interpret Riemann sums. Some upper-level students, on the other hand, can learn to use Riemann sum-based reasoning quite successfully to interpret integrals, but find the logic of doing so highly dubious—again, because Riemann sum-based reasoning simply does not meaningfully correspond to solution processes based on the FTC. Riemann sum-based reasoning, it would seem, requires students to apply counterintuitive notions of Binfinitesimals^ onto symbolic processes involving mysterious function transformations. It is not my intent to portray any of these students as Bdeficient^ or as using poor reasoning. To the contrary, I think all of the students described above were doing their best to do what we want all students of mathematics to do: to try to make sense of the mathematics they are using. The problem, I think, is that nothing in their mathematical careers has yet offered them the opportunities to develop the tools needed to transition from sense-making in algebra to sense-making in calculus. Simply put, the conceptual tools of algebraic sense-making are of only limited use in doing calculus, and this fact is rarely, if ever, addressed directly in typical calculus curricula. Mathematics education researchers typically promote algebra curricula that support students in developing meaning and sense behind their algebraic constructions and manipulations. In a review of literature on the teaching and learning of algebra, Kieran (2007) observed that Beven though the nature of the meaning that students can draw from algebraic structure can be elusive, this source of meaning is considered by many mathematics educators and researchers to be fundamental to algebra learning^ (p. 711). Kieran also noted that most algebraic activity prior to calculus can be categorized as generational, in which algebraic expressions are meaningfully generated or constructed, and transformational,in which algebra expressions may undergo a variety of types of manipulation, typically with the expectation of maintaining equivalence. This point is key. A central value of algebraic processes is that, while the form of an algebraic expression might change, it remains fundamentally equivalent to the expression from which it was derived. The processes of differentiation and antidifferentiation stand in stark contrast to such algebraic processes, as one function is transformed into a new, non-equivalent function. Physics education researchers have been equally concerned with the meaning that students develop behind the mathematical expressions that they create and manipulate. Sherin (2001) proposed that successful physics students construct for themselves a Bvocabulary^ of symbolic forms, each of which Bassociates a simple conceptual schema with an arrangement of symbols in an equation^ (p. 482). By means of such a vocabulary, students come to associate their conceptual under- standings of certain physical situations with the syntax of physical-mathematical expressions and equations. Sherin did not address directly the meaningfulness of 354 Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 symbolic manipulations that may occur in problem-solving processes, however his arguments concerning how students look for and perceive meaning in mathemat- ical expressions reflect the activities in which the students of this study were engaged. In sum, researchers in both mathematics education and physics education recognize the need for students to develop means of making conceptual (mathe- matical and physical) sense of the algebraic structures that they create and of the algebraic manipulations that they perform. Nothing, however, in the standard calculus curriculum prepares students for the sudden transition from making sense of the symbolic processes of algebra to making sense of the symbolic processes of calculus. It is the case, of course, that because the method of Riemann sums and the method of antidifferentiation both generate the same mathematical result in computing the value of a definite integral, it is entirely legitimate to use Riemann sum-based reasoning to interpret the definite integral. To do so, however, does not parallel any earlier mathe- matical experiences students are likely to have had. Furthermore, even if students do learn to use such reasoning, it does not address the question of why their familiar sense- making tools fail them in the cases of differentiation and antidifferentiation when what might appear to be algebraic manipulations do not result in functional equivalence. To do so requires a student to adopt an expert’s perspective on the definition of the definite integral as the limit of a Riemann sum, as well as to develop a rather deep and subtle understanding and appreciation for the significance and surprising nature of the FTC. Indeed, this research has led me to a deeper appreciation for the theorem than I previously held. All of this may be interpreted to suggest the need to introduce explicitly at least two conversations into the calculus curriculum. Calculus students could benefit from richer accounts of the relationship between Riemann sums and the definite integral. As it stands, too many students dismiss Riemann sums as an unpleasant stepping-stone to be endured in a curriculum whose goal was really to get to the FTC. Once that goal is reached, many students are led to believe, either implicitly or explicitly, that Riemann sums can be left behind. The fact that the underlying structure of Riemann sums as sums of products can be used to make sense of and interpret the definite integral— precisely because they do both find Bthe same thing^—is a gem of mathematical sense- making that is typically overlooked entirely in calculus curricula in favor of a near- obsession with finding areas under curves. The call to support students better in developing Riemann sum-based reasoning within the calculus classroom is not new, of course, and others have begun addressing this matter as well. (See, for example, Doughty et al. 2014; Engelke and Sealey 2009; Kouropatov and Dreyfus 2014; Meredith and Marrongelle 2008; Thompson et al. 2013; Thompson and Silverman 2008.) The present research, however, suggests that supporting students in this way may be more complex than we might have supposed. Students could equally benefit from explicit discussion and investigation of the differences between sense-making in algebra and sense-making in calculus. While much effort has traditionally been given to trying to teach introductory students sophisticated analytical definitions of limits and their use in defining derivatives and integrals, the fact that these processes result in function transformations and the significance of this are overlooked entirely. Similarly, the notion that some symbol Int. J. Res. Undergrad. Math. Ed. (2018) 4:327–356 355 manipulations are used to maintain an expression’s equivalence, while others result in an expression’s transformation is both subtle and essential to understanding the differ- ence between the symbolic manipulations of algebra and the symbolic manipulations of calculus. What a study like this demonstrates, in part, is the consequence of students’ lack of awareness of such distinctions and the need for corresponding curricular change. To date, even among those who have expressed the need for promoting Riemann sum-based reasoning in calculus instruction, no one has called attention to the problems raised in this paper. What can be generalized from a study of this sort? On the one hand, the students interviewed make up a small sample of students, some of whom demonstrated what may well be quite idiosyncratic behavior. Nonetheless, I have argued that, despite a variety of idiosyncratic responses, each of them can be related to challenges that students face in trying to make algebraic sense of antidifferentiation processes, and/or trying to reconcile those processes with Riemann sum-based reasoning skills that they have been taught. If this is the case, then the ease with which these examples emerged from a group of only fifteen students (many with rather strong mathematics back- grounds) suggests that such challenges may well be quite widespread. What we see here may be individually and independently constructed responses to a very common problem.

Acknowledgements This research was supported in part by the National Science Foundation under grant No. PHY-1405616, and in part by a faculty development leave from Xavier University. Any opinions, findings, conclusions, or recommendations expressed in this paper are those of the author and do not necessarily reflect the views of the National Science Foundation or Xavier University. I would like to thank Corinne Manogue, Tevian Dray, John Thompson, and Mike Loverude for their thoughtful conversations and feedback throughout the course this research.

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