Understanding the integral: Students’ symbolic forms

Journal of Mathematical Behavior 32 (2013) 122–141

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Understanding the integral: Students’ symbolic forms Steven R. Jones ∗ Curriculum and Instruction, University of Maryland, 2311 Benjamin Building, College Park, MD 20742, United States

a r t i c l e

i n f o

Keywords: Calculus Integral Student understanding Undergraduate mathematics education Symbolic form Accumulation

a b s t r a c t Researchers are currently investigating how calculus students understand the basic concepts of first-year calculus, including the integral. However, much is still unknown regarding the cognitive resources (i.e., stable cognitive units that can be accessed by an individual) that students hold and draw on when thinking about the integral. This paper presents cognitive resources of the integral that a sample of experienced calculus students drew on while working on pure mathematics and applied physics problems. This research provides evidence that students hold a variety of productive cognitive resources that can be employed in problem solving, though some of the resources prove more productive than others, depending on the context. In particular, conceptualizations of the integral as an addition over many pieces seem especially useful in multivariate and physics contexts. © 2012 Elsevier Inc. All rights reserved.

1. Introduction and relevance In recent decades, more and more attention has been given to compiling a body of research regarding student understanding of mathematics at the undergraduate level. Already this research has provided much information about how students learn and understand a variety of concepts from calculus, differential equations, statistics, and mathematical proof. Among calculus concepts, researchers have focused heavily on student thinking about limits (e.g., Bezuidenhout, 2001; Davis & Vinner, 1986; Oehrtman, Carlson, & Thompson, 2008; Oehrtman, 2004; Tall & Vinner, 1981; Williams, 1991), but have also provided insight about how students understand the derivative (e.g., Marrongelle, 2004; Orton, 1983b; Zandieh, 2000) and Riemann sums and the integral (e.g., Bezuidenhout & Olivier, 2000; Hall, 2010; Orton, 1983a; Rasslan & Tall, 2002; Sealey & Oehrtman, 2005; Sealey & Oehrtman, 2007; Sealey, 2006; Thompson & Silverman, 2008; Thompson, 1994). Overall, the concepts of the derivative and the integral are less explored than the idea of the limit. While the limit is fundamental to calculus, the derivative and the integral have additional layers of meaning above and beyond the limit, as well as meanings that do not necessarily require accessing the concept of a limit (Marrongelle, 2004; Sealey & Oehrtman, 2007; Thompson & Silverman, 2008; Zandieh, 2000). Thus, the derivative and the integral need special attention in order to learn how students understand the main ideas of first-year calculus. In particular, students’ understanding of the integral is an especially valuable topic, since integration serves as the basis for many real world applications and subsequent coursework (Sealey & Oehrtman, 2005; Thompson & Silverman, 2008). The integral shows up in a variety of contexts within physics and engineering (Hibbeler, 2004, 2006; Serway & Jewett, 2008; Tipler & Mosca, 2008) and students who continue into further calculus courses will encounter the integral more

∗ Corresponding author. Present address: Sciences and Mathematics Division, Sierra College, V-313B, 5000 Rocklin Road, Rocklin, CA 95765, United States. Tel.: +1 916 660 7987. E-mail address: [email protected] 0732-3123/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jmathb.2012.12.004

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often than the derivative (Salas, Hille, & Etgen, 2006; Stewart, 2007; Thomas, Weir, & Hass, 2009). However, an overreliance on certain interpretations of the integral, such as an “area under a curve,” can limit the integral’s applicability to these other areas (Sealey, 2006). Evidence of student difficulties with the integral has been documented over the years in several studies (Bezuidenhout & Olivier, 2000; Orton, 1983a; Rasslan & Tall, 2002; Tall, 1992; Thompson, 1994). Additionally, researchers have noted the perception among educators that students transitioning into science courses are routinely struggling to apply their mathematical knowledge to the science domain (Fuller, 2002; Gainsburg, 2006; Redish, 2005). This should be of primary concern for instructors of first-year calculus due to its nature as a service course and the large portion of science students enrolled in these classes (Ellis, Williams, Sadid, Bosworth, & Stout, 2004; Ferrini-Mundy & Graham, 1991). Hall (2010) demonstrated several ways that students may interpret the definite and indefinite integral, including “area,” “Riemann sums,” “evaluation,” and “language.” Conceptions of the integral as an area or as a calculation appeared predominant among his students. Hall’s main focus, however, was on the influence of informal language on students’ thinking about the integral and he did not attempt to analyze the composition of their concept images (see Tall & Vinner, 1981). Sealey and others, on the other hand, primarily emphasized students’ conceptualization of the integral as a Riemann sum (Engelke & Sealey, 2009; Sealey & Oehrtman, 2005, 2007; Sealey, 2006). These studies focus on how students connected the Riemann sum to concepts like the limit and how students used it in solving certain problems, such as approximating the force on a dam. Much of the work was centered on how ideas of accumulation and error were entwined with the conception of the Riemann sum. This body of work provides insight into a few pieces of students’ overall concept images of the integral as well as what is being done to develop students’ understanding of accumulation. However, it still leaves open the need to identify the actual cognitive structures that inhabit students’ minds regarding the integral. If a student perceives the integral as an area or a Riemann sum, or in some other way, what does that knowledge look like per se? What ideas do the symbols of the integral evoke in students’ minds? What aspects of a problem cue students into activating a particular interpretation of the integral? There is still little we know about the meaning students place on the various pieces of the integral symbol structure and how these pieces come together to form the overall concept in a student’s cognition. The purpose of this paper is to document cognitive resources of the integral that students hold and draw on in mathematics and physics contexts. In the next section, the reader is acquainted with the theoretical constructs of cognitive resources and a particular type of cognitive resource called a symbolic form. The emergent symbolic forms presented in this paper are analyzed for their impact on student thinking during mathematics and physics problems.

2. Theoretical perspective and framework 2.1. Cognitive resources and framing In this study, the aspect of knowledge that is considered is that of cognitive resources (Hammer, 2000). Cognitive resources are “fine-grained” elements of knowledge in a person’s cognition (Elby & Hammer, 2010). As an example, a student’s concept of integration may not be a single entity, but may rather be made up of smaller units, including ideas of area, anti-derivatives, summations, or differentials. Each of these may be made up of even smaller units. If this is the case, one cannot claim that a student’s concept of integration is one fixed object in their cognition. To illustrate, suppose a student properly calculates the integral

n

1

2

0

x2 dx using an anti-derivative but then fails to adequately interpret the corresponding

Riemann sum limn→∞ k=1 (xk ) x. Calling this student’s conception of integration either correct or incorrect may be too simplistic a view of his or her knowledge (Clement, Brown, & Zietsman, 1989; diSessa, 1993). Within the framework of cognitive resources, inadequate reasoning “differs from the notion of a ‘misconception,’ according to which a student’s incorrect reasoning results from a single cognitive unit, namely the ‘conception,’ which is either consistent or inconsistent with expert understanding” (Hammer, 2000, p. 53). Instead, it may be the selection of certain cognitive resources over others that results in the satisfactory or unsatisfactory reasoning (Elby & Hammer, 2010; Hammer, Elby, Scherr, & Redish, 2005). It is important to note that this investigation does not directly study students’ beliefs about mathematics (Elby & Hammer, 2001), though it is acknowledged that students’ beliefs impact the ways in which they might draw on their cognitive resources during problem solving through the process of framing (see Lunzer, 1989; MacLachlan & Reid, 1994). Framing means “a set of expectations an individual has about the situation in which she finds herself that affect what she notices and how she thinks to act” (Hammer et al., 2005, p. 97). Framing directly influences students’ tacit “selection” of cognitive resources during problem solving and is consequently a key component of interpreting the data in this study. For the purposes of this paper, a cognitive resource that relies on physical phenomena, such as position or movement, does not count as being purely mathematical and would instead be considered a blend of mathematical and physical knowledge (Bing & Redish, 2007; Fauconnier & Turner, 2002). One type of blend that is important to this study is seen in the symbolic form, which is described subsequently.

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Framing by students: Interview context math or physics problem

Available symbolic forms in student’s cognition (blend of template + conceptual schema)

math/physics context + interview context what is being asked? what is mathematical knowledge?

Activation of symbolic form

Observable outcome: Student responses

Fig. 1. Framework: Student symbolic form activation.

2.2. Symbolic forms According to research regarding physics students’ use of equations (Lee & Sherin, 2006; Sherin, 2001, 2006), there appear to be certain types of resources that inhabit students’ cognitions, which can be expressed in terms of symbolic forms. A symbolic form is a blend of two components: a symbol template and a conceptual schema. To describe the two components, consider the following example of two equations from physics, dealing with velocity (1) and energy (2):

v = vo + at,

(1)

E = P + K.

(2)

and

The symbol template is simply the structure or arrangement of the symbols in the equation. The right-hand side of both equations bears the template “[ ] + [ ]” denoting two terms separated by a plus sign. Also, each equation has another template, “[ ] = [ ],” showing two expressions separated by an equals sign. Together, both of these equations have identical templates: “[ ] = [ ] + [ ].” The conceptual schema, on the other hand, refers to the meaning underlying the arrangement of the symbols. In the right-hand side of the first Eq. (1), the two terms refer to a base and a change (Sherin, 2001). That is, vo is the starting point of the velocity and the at is the amount of additional velocity that the object receives after t amount of time. Thus, the symbolic form associated with the first equation is “[amount] = [base] + [change].” For the second Eq. (2), the P and the K in the right-hand side refer to two components of the total energy. They are each a part of a whole. This represents a different symbolic form, “[whole] = [part] + [part]” (Sherin, 2001). From this example it can be seen that two symbolic forms may share the exact same symbol template, but have different conceptual schemas. In this paper, several symbolic forms associated with the integral symbol template are proposed and analyzed in an attempt to address the following questions: What conceptual schemas have students blended onto the entire integral symbol template or onto parts of the integral symbol template? How do these symbolic forms influence problem solving in items involving the integral? 2.3. Framework: activation of symbolic forms Since the beliefs students hold about mathematics, the interview context, and the problems presented to them during the interview can be expected to mediate the selection of cognitive resources, they must be incorporated into the framework regarding symbolic form activation used in this study (Hammer & Elby, 2002; Hammer et al., 2005). The framework is represented in Fig. 1. A student may hold a particular symbolic form in their cognition, but not draw on it because of the set-up of a problem or because of what they think they are being asked to do. However, if it is possible to detect an influence from framing on resource activation, an opportunity is created to shed light on the context-sensitivity of certain symbolic forms of the integral. 3. Student participants and data collection 3.1. Data collection and analysis In order to document and analyze the symbolic forms of the integral that students might possess, I conducted interviews with nine students selected from a major university in California and a major university on the East Coast of the United States. The interviews were done with pairs of students, with the exception of one student who was interviewed by himself. Each pair was interviewed twice, again with the exception of the lone student who was interviewed only once. There was a one week lapse between the first and second interview for each pair. In the interviews, mathematics and physics problems were given to the students, with the expectation that they would work together to solve the problems to the satisfaction of both participants. Based on the various arrangements that the integral symbol template can have, such as



[]

[ ] d[ ], [ ]

 [] []

[ ] d[ ], or



 [] []

[ ] d[ ],

[ ] d[ ], interview items were created to provide the students an opportunity to discuss each of

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these templates. In each interview, an open-ended item was given first, in order to allow the students to invoke the integral of their own accord. (For a complete listing of all interview items used, see Appendix A.) The interviews were conducted in an empty classroom where only the researcher was present with the students in order to ask follow-up questions. The students worked on the board, while verbally discussing their thinking, and their activities and written work were videotaped. The nine videotaped interview sessions and the researcher’s notes were the primary sources of data for the study. The data were analyzed in iterations, starting with an initial pass to mark the places where the students referenced symbols or concepts related to the integral, or where visual depictions of the integral were made. These locations in the data were then examined in detail in an attempt to sketch a conceptual schema that was being applied to the symbols of the integral (see Strauss & Corbin, 1990). Once these schemas were outlined, the data was re-analyzed in its entirety, looking for confirming or disconfirming evidence for the conceptual schemas (see Yin, 1989). The posited symbolic forms were shown to other mathematics educators who challenged and debated their structure. Despite the utility of interviews there are certain limitations to this study. First, analyzing student responses, spoken or written, only approximates the intended goal of capturing knowledge. Second, an interview occurs over a limited amount of time, and therefore only captures what the participant thinks about during that time frame. It can only document those symbolic forms that the students “choose” to activate. Thus, I cannot claim that the symbolic forms described are an exhaustive list of those potentially possessed by the students, nor that the results would necessarily be reproducible across time. Lastly, this study cannot make any claims as to the frequency that the observed symbolic forms would occur within the overall student population. While Hall’s (2010) study does report the frequency of various interpretations of the integral for his students, larger scale statistical studies would be needed to shed light on this question. Another important note regarding the analysis must be made here. In order to properly document one piece of knowledge, it must be artificially isolated. The descriptions in Section 4 of the students’ work focus only on the relevant symbolic form that is being discussed. Hence it may give the false impression that a student holds only a single conception of the integral, which is most certainly not true. It may furthermore give the false impression that students activate and use only one cognitive resource at a time. These false impressions are merely artifacts of the need to isolate each symbolic form under consideration and do not represent the actual structure of the students’ cognitions. Rather, there is ample evidence that the students hold several symbolic forms regarding the integral and can draw on them simultaneously within the context of the same problem. 3.2. Student participants The students selected for this study were intended to be “typical” in two important ways. First, since a large proportion of first-year calculus courses are made up of science students (Ellis et al., 2004; Ferrini-Mundy & Graham, 1991), it is sensible to include students from this category. Second, the participants should be experienced with calculus, so that difficulties cannot simply be attributed to a lack of mathematical experience (Redish, 2005). In order to select participants who satisfied these conditions, students were only recruited who had completed the first calculus course at their university and had either completed or nearly completed the second calculus course. Furthermore, in order to select students who could be considered “successful” in their calculus courses, I required that the participants received a grade of an A or a B in these courses, or that they had passed the relevant AP exam with a score of a five. Lastly, I only selected students who had nearly completed an introductory calculus-based physics course at their university. Thus, all of the students had experience in working with the integral in both mathematics and physics contexts. The pseudonyms given to the students are to help suggest the pairs that they worked in. The students are known in this paper as Adam, Alice, Bill, Becky, Clay, Christopher, Devon, David, and Ethan. Ethan was interviewed only once, by himself. 4. Symbolic forms of the integral In this section, the symbolic forms that emerged from the interview data are discussed. A symbolic form is said to be cognitively “compiled” if there appears to be a stable assignment of conceptual schemas to the pieces of the symbol template. Here, stable means a consistency with which the student blends the schemas and the symbols, within a given task as well as across multiple tasks. Throughout the interviews, there appeared three main symbolic forms that encompassed the entire integral symbol template, “

 [] []

[ ] d[ ].” Each of these symbolic forms can be supported through the work done by individual

students, as well as laterally across the various students that were interviewed. Following the description of these three main symbolic forms, as well as a description of a miscompilation of one of them, several other symbolic forms pertaining to specific parts of the integral symbol template are discussed. 4.1. The adding up pieces symbolic form This symbolic form for the integral deals with thinking that is similar in ways to a Riemann sum (see Sealey, 2006; Sealey & Oehrtman, 2005, 2007), though it carries additional conceptual meanings, some of which diverge from the Riemann sum. Devon and David were working on item Math1 (see Appendix A) and had sketched a picture, labeling the telephone poles

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Fig. 2. Sketch (and reproduction) drawn by Devon and David.

as a and b and the wires as f1 (x) and f2 (x) (see Fig. 2). As they worked on how to determine this area, they wrote down the

b

integral “ Devon:

a

f1 (x) − f2 (x)dx.”

You can’t just put area, you have to somehow divide it into, let’s say the length. Let’s say you slice it this way [motions several vertical lines from top curve to bottom curve], and then you add up all the individual lengths [puts hand on left side of shaded region and sweeps hand across to the right side]. And then that means we have to find the difference between these two curves, that’s why we label it [points to f1 and f2 ]. And by finding the curve and then integrating over them [again sweeps hand from left to right], that’s how we find the area.

Devon started his explanation by stating that in order to understand their integral, he had to “divide” the region of interest. Then these “individual lengths” had to be systematically added up, which he demonstrated by sweeping his hand from the left of the shaded region to the right. The fact that he used the left-to-right motion twice to describe the integration shows that it played a strong role in his thinking. Devon:

Interviewer: Devon:

I would imagine it as, you slice it [draws a thin rectangle, see Fig. 2], like very small pieces and each of them is a dx [draws an arrow from the bottom and writes dx]. And this part [puts fingers along the height of the thin rectangle] is the, is this part right here, this term right here [points to f1 − f2 inside the integral]. Which part is? Just to make sure. This part right here, the length here [underlines f1 –f2 inside the integral and draws an arrow over the height of the rectangle]. And then every little bit [uses finger and thumb to mark the width], I call it a dx.

Devon made a single rectangle that served as a reference for what was happening within the integration. I call this a representative rectangle. (Note that in multivariate cases, the “representative rectangle” may actually be a “representative cube,” but I will use “rectangle” as a generic word for all cases to mean a “generalized representative rectangle.”) It was sufficient for him to create only one rectangle and he was able to conduct much of his reasoning based off of it. The representative rectangle played a significant role in several of the students’ work. Devon described that this rectangle was constructed from the integrand “f1 − f2 ” and the differential dx. I then asked Devon to say what the a and the b meant. Devon:

If you really think why we put it there, like I said, I slice it into little pieces. And all the pieces we’re looking at is from here to here [motions with hand from left of the shaded region to the right], and it has to do with the values of it [says this as he moves his hand from left to right again]. It’s more like an action thing, I think.

Devon used the left-to-right motion a total of four times to indicate the addition that the integral implied. He saw the addition as “more like an action.” Thus, the addition takes place as an active totaling of the amount. An appropriate metaphor might be a clerk with a calculator, keeping a running total from a list of receipts. The limits of the integration, a and b, told him to add up “from here to here,” meaning they served as the “starting” and “ending” places for the active totaling. Both the active addition and the temporal components of starting and ending indicate that this conceptual schema imbues the symbol template with dynamic properties.

4.1.1. The infinite addition In most instances where the students had activated the adding up pieces symbolic form, there was strong evidence that they viewed the rectangles as “infinitely thin” and the addition as happening over “infinitely many” rectangles. For example, when Chris and Clay were working on the same item, I asked them to explain why an integral would give an area. Chris, who had drawn on the adding up pieces symbolic form, explained why the addition produced area. Chris:

We want to find the area, so theoretically we could add up the value of a bunch of rectangles, and add them up. But we’re going to constantly have little gaps [draws rectangles in between the two curves.]. . . So we’re going to be missing this area [points to small gaps in between rectangles and curves]. So we assume, by integrating we assume that dx is infinitesimally small.

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Fig. 3. A visual representation of the adding up pieces symbolic form.

In Chris’ conception, an integral takes as assumed that dx is “infinitesimally small,” so that there are no longer any “gaps” left by the finite rectangles. This type of language was common and consistent across the students. During instances of drawing on the adding up pieces symbolic form, the students described the rectangles as “infinitesimal,” “infinitesimally small,” “infinite number,” “infinite amount,” “infinitesimal rectangles,” and “infinitely many.” It seems clear that for these students, the adding up pieces symbolic form has embedded in it an inherent notion that the rectangles have already achieved the status of being infinitely thin and that the addition process requires an infinite summation over the infinitely many pieces. The adding up pieces symbolic form compiled in this manner diverges in an important way from the traditional Riemann sum process, which constructs a numeric sequence based off of finite additions over finitely many rectangles. The students in the interviews commonly thought of the limiting process (i.e., “limx→0 ” or “limn→∞ ”) as occurring before the addition takes place, not after. This distinction is more than just linguistic. There is evidence that the students tended to separate the finite Riemann sum process from the final, infinite integral process. While Adam and Alice were working on this same problem, they had also drawn in thin rectangles between the two curves, like Chris and Clay. Adam:

... Adam:

So, this goes back to a Riemann sum, where you take small portions of each graph [outlines a thin rectangle with his finger]. Like what I have here, which would be dx. In Riemann sum, you define what the width is, or how many sections per. . . graph you have. With the dx, when you’re integrating, you’re taking an infinite number of lengths of portions. It’s kind of like you’re adding them all up. Going back to Riemann sums, it represents the infinite amount. . . If you had the infinite amount of portions for a Riemann sum, that would represent this, the integral.

Bill produced a similar explanation during his interview. Bill:

I think of an integral as just a way of expressing an infinite Riemann sum. As dx goes to 0, well, as, as the length of each rectangle goes to 0, then it becomes a dx.

Lastly, Clay states at one point that the integral is better than (not the result of) the finite summations. Clay:

If you want an exact answer, integration would be a better choice than cutting it into smaller and smaller pieces.

The integral was often considered by these students as a special case of a Riemann sum, namely a Riemann sum with an “infinite amount of portions.” Since this thinking was so common to the students drawing on this conceptualization, I have incorporated it into the adding up pieces symbolic form. This form is visually represented in Fig. 3.

4.1.2. A problematic miscompilation of the adding up pieces form As students construct cognitive resources, it is possible that the compilation of knowledge may diverge from traditional expert thinking. In this section, a miscompilation of the adding up pieces symbolic form is presented, though one could argue that it represents an attempt to construct the adding up pieces form, simply misunderstanding a key element of it. Ethan was working on item Physics 1 and decided that he needed to look at the density  over the box’s entire volume in order to find the mass. He wrote down an integral of density with respect to volume, “ D dV .” Ethan:

Density is varying. So this integral means that we’re going to add up all the densities [points to D], infinitely small, so that you get the overall idea. You’ll get the exact idea of its density. Once you do that, then you just multiply by volume [points do dV].

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Fig. 4. Graph (and reproduction) constructed by Ethan.

Hence small amounts of density, not mass, are added up under the integral.  At another point in the interview, Ethan was working on item Physics 3, and was trying to describe why the integral S P dA would calculate the total force. A similar description involving the addition occurred. Ethan:

Since it’s a non-uniform pressure, you’re adding up all the pressures that are at each point, each kind of location on the surface, and overall they would tell you the pressures—you can get the whole pressure.

In a third instance, Ethan yet again displayed thinking along these same lines. While working on item Physics 2, he explained how the addition happened inside the integral Ethan: Interviewer: Ethan:

 600 0

R dt.

You’re just adding up all of the, I guess in this context, you’re just, over time, you just add up all the RPMs that happened from 0 minutes to 600 minutes. What do you mean by “add up all the RPMs?” The integral just adds up things, just keeps adding them up [places hand on table to his left and sweeps it across to the right]. I guess if you had it, if you just had a function of RPMs, something like [draws axes], this is time here [points to horizontal axis], maybe this is RPMs [points to vertical axis] and you have something like [draws squiggly graph, see Fig. 4]. This function will tell you the area here, which is just the way you’re just adding it up [draws vertical lines from his graph to horizontal axis, see Fig. 4].

Ethan consistently thought of the addition as happening over the quantity represented by the integrand. Piecing his descriptions together provides the following conceptual schema for the integral template. The differential (dV, dA, dt) acts as an infinitesimally fine-grained partition and the quantity represented by the integrand (D, P, RPMs) is added up over each piece. At the conclusion of this summation, the resultant “total” quantity of the integrand (density, pressure, RPMs) is multiplied by the quantity represented by the differential (volume, area, time). I call this the adding up the integrand view of the integral. Ethan’s stable view of the integral is problematic in that the summation of the integrand does not produce any physically meaningful result and is inconsistent with a standard Riemann sum. Yet it is based on the notion of accumulation, meaning there are still productive ideas he is drawing on. In considering the difference between Ethan’s explanations and the other students’ explanations, there appears to be one key element missing. Ethan never described a representative rectangle (or cube), suggesting that it might serve a role in helping students relate the integrand to the differential. Other students, such as Chris, Clay, Bill, and Devon, who used a representative rectangle during these applied problems were able to link the rectangle (or cube) to a multiplication between pressure and area, density and volume, and force and distance. An example of this is shown in Section 5.3. 4.2. The perimeter and area symbolic form The next symbolic form derives from the common interpretation of the integral as an area,  though the symbols are imbued with additional meaning. Chris and Clay were discussing the integral in item Math5, “−2 D f(x)dx, ” when Chris decided to represent it graphically. Chris:

So if you want to draw a graph [draws axes], um, we have f of x [draws squiggly graph above x-axis]. And then since we’re saying over the domain D, domain is usually when we’re dealing with x, y axes. We assume it’s with respect to the x axis and also the integral deals with x [points to dx] and we have a function of x, so we can assume D is a domain from some point x1 to some point x2 [labels x1 and x2 , see Fig. 5]. . . So we have. . . [draws dotted vertical lines from x1 and x2 ]. And then we’d take the integral from x1 to x2 [said as he shades the area, see Fig. 5].

Chris marked x1 and x2 on the x-axis and used them to create the left and right sides of a shape in the x–y plane by drawing vertical dashed lines to mark off this region. Hence, the limits of integration are not merely numbers, but become actual physical boundaries of a region in the plane. Next, the graph itself creates the “top” of this fixed region, meaning the integrand f(x) also helps create a physical boundary. Thus, in this conceptualization, the shape has to be constructed via the symbols of the integral before the “integral as area” idea can be used. Lastly, part of Chris’ reasoning for using an x-axis was

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Fig. 5. Graph (and reproduction) constructed by Chris and Clay.

the fact that “the integral deals with x.” He said this as he pointed to the dx in the integral, indicating that the dx helped Chris make the decision about what would serve as the “bottom side” of a fixed region. In summary, a key feature of this form is that the individual symbols of the integral carry the conceptual meaning of becoming the physical perimeter of a fixed region. A second key feature of this symbolic form is that the area of the region is taken to be one static whole. It is not subdivided into parts nor measured using successively more accurate approximations, as in a Riemann sum. The perimeter and area symbolic form is visually depicted in Fig. 6. 4.3. The function matching symbolic form This symbolic form is closely linked to the anti-derivative process. Though this form resembles the rote procedure for calculating an integral, I argue that the students are not, in fact, simply invoking calculations, but are giving meaning to the symbols of the integral. This may be similar to the reification of processes into objects (Sfard, 1991). In their first interview, Devon and David were given the integral “

2

21

2/x3 − x2 dx” from item Math2 and were asked to calculate it. In their first step

2

they broke the integral into two parts, “ 1 2x−3 − 1 x2 dx.” David recognized that they had left out the dx from the first of these two integrals and added it in, so I asked them to explain why it needed a dx. David went back to the initial integral in the task. David: ... David:

In an integration the dx is always essential, because it shows that this entire thing [waves hand over the integrand, “2/x3 –x2 ”] is a derivative of x. The fact that this entire thing is sitting right next to each other, and dx outside, means that basically this entire function [motions hand over “2/x3 –x2 ”] is the derivative of an original function.

David had conceptualized the integrand “2/x3 − x2 ” as being the “derivative of an original function.” Hence, it appears that the function may have come from somewhere else. The dx had a role in signaling to him that this “original function” became the integrand via a derivative. The integral could be thought of as a matching game, trying to get back to this original function. In fact, David wrote the function “2ln(x3 )/3” as the anti-derivative for the first term (the fact that it is incorrect is irrelevant here) and said, David:

You try to find the derivative of this [points to 2ln(x3 )/3], which would just be equal to this [points to 2x−3 ].



In the next interview item, David and Devon were discussing the integral “ sin(x)” from item Math3, which had the dx intentionally left off as a means of generating conversation. Devon came up with the function “−cos(x) + c,” so I asked them what the “+c” meant. David interjected his thoughts. David:

You could always have a function added with a constant. The thing is when you derive the entire function the constant just goes away.

The original function could have had a constant, so it needs to be there. Whatever function David came up with for an answer needed to match the integrand under a derivative. David then solidified the connection between the “original []

∫

[]

the integrand represents the “top side”

the

[]d[] the differential’s main purpose is to determine the “bottom side” of the shape

" ∫ " is an area,

taken as a whole

x1

x2

[]

the limits

∫

[]

become the physical “left and right sides” of the region

Fig. 6. A visual representation of the perimeter and area symbolic form.

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S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141 the “original function” produces integrand

[]

∫

“original function”

[]

[]d[]

d[] tells you how to find the “original function”

Universe of Functions the limits “

∫

the purpose of the integral is to “find” this “original function”

[]

[]

” represent a difference of values of the “original function”

Fig. 7. A visual representation of the function matching symbolic form.

function” and the integrand when summarizing his work at the end of the task (note that he frequently used the words “equation” and “function” interchangeably during the interview). David:

It [the integral] just means you’re trying to find the anti-derivative of the equation. The original equation to the derivative inside the integration.

As David stated, the function inside of the integration is a derivative. The purpose of the integral is to match it back to the original function from whence it came.  During their work with this same task, David had added in a dx at the end of the integral, to make it “ sin(x) dx .” He then explained why it was necessary. David:

Again, I guess it matters because if you don’t have the dx, then it’s just going to be like sine x [sin(x)]. . . but, is it the second derivative, or the first derivative or something like that? . . . So I think it’s just for the sake of organization just to have the dx in there, to signify that this is the derivative of the original function.

Twice David linked the dx with an indication that the integrand is a derivative. Essentially, its role appears to be to help one know how the original function became the integrand. If the dx wasn’t there, one wouldn’t know how to work backward to recover the original function. There needs to be one important note made about the function matching symbolic form. It could be easy to dismiss this form by saying that it does not meet the criteria of a conceptual schema. However, it is possible to view this as the reification of the anti-derivative process. Because derivatives are usually taught before integrals, the integral may be thought of as “undoing” a derivative. This means that the integral does not exist in isolation, but rather as an inverse process. David expressed this by casting the integrand as the result of an action done to some “original function.” A metaphor for the meaning of the integral would be a child matching colored play-doh with the appropriately colored jar that it came out of. The red play-doh originates from the red jar, so the color tells the child where to put it back. The dx indicates how to “put the integrand back” with the original function. If the integral is conceptualized in this way, what is the meaning of the limits of integration? I asked David and Devon why the integral of sin(x) ended up with a “+ c” when the other integral they worked on did not have a “+ c.” Devon:

This one [points to

0 2

ex dx], you are finding the difference between these two [points fingers to the 2 and 0]. So, regardless of the c, it

would just be a difference. So that’s how I think of it, as a difference.

Since Devon later plugged the 2 and the 0 into ex to get e2 and e0 , it is reasonable to assume that by pointing at the limits of integration, he does not mean the actual 2 and 0, but the function values represented by plugging those numbers into the function. Devon explained that he is “finding the difference” when limits of integration exist. This language suggests a modest layer of meaning above mere subtraction, such as a measurement between two values, or of competing terms (Sherin, 2001). These pieces are put together to describe what I call the function matching symbolic form, which is visually depicted in Fig. 7.

4.4. Other symbolic forms of the integral template 4.4.1. The integral symbol with no limits as a generic answer While the function matching symbolic form can be easily extended  to the integral with no limits of integration, there emerged another conceptual schema that could be applied to the “ ” symbol with no limits on it, which I call the generic



answer symbolic form. Bill and Becky were working with the integral “ sin(x)dx” and were comparing it to

0 2

ex dx.

S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141 Becky:

131

I would say the one that has numbers [i.e., limits], you’re asking for a specific area or a specific region of, like, whatever y it is. One that doesn’t, you’re just asking for it in general. I kind of interpret that as later on, if you want to know it, what values it’s between, you have a more broad range to put the values into. Whereas when you solve for a specific 0 and 2 you’re giving the answer and you can’t really, like, work on that.

Becky stated that when no limits are present, “you’re just asking for it in general.” She elaborated by saying that “later” you could attach numbers onto it to find out a specific area or a specific numeric value. Thus, the integral without limits is like a generic answer, one that is not necessarily complete until limits are established “later on.” Once limits are provided, the generic answer collapses down to a specific answer, from which no more “work on that” can be done. Hall (2010) mentions a similar conception he saw among his students, which he calls “potential area.” However, I argue that this notion is extendable to conceptions beyond just area. For example, Becky states that the addition of limits could work both for “areas” as well as “values it’s between” suggesting both the perimeter and area and the function matching ideas. Similarly, one could conceptualize a list of numbers waiting to be added, such as in adding up pieces, except the addition cannot take place until the person is told where to start and stop adding. 4.4.2. The area in between symbolic form The symbol template “

 [] []

([ ] − [ ]) d[ ]” is really just a special case of the regular integral template “

 [] []

[ ] d[ ].” However,

several of the students took integrals written in this form and, without prompting, equated it to finding the area in between

2

two different curves in the plane. Bill and Becky were explaining the meaning of the integral “ 1 2/x3 − x2 dx” and Bill, who was drawing heavily on the perimeter and area symbolic form, was trying to explain what the “area” for this integral would look like. Becky then brought in the idea to separate the integrand into two functions. Becky: Bill:

I don’t know if this is correct, but you could do it kind of like the f of x and g of x [f(x) and g(x). Points to the “2/x3 ” and the “x2 .”]. It’s like f of x [f(x)] plus g of x [g(x)]. Yeah I guess you could do that, if you took 2 over x cubed [2/x3 ], that’s probably just . . . let’s just say it’s this [draws a graph]. And then you take x squared [also draws the x2 graph] and then you take all the area here. . . The area is just the difference between those two curves [draws in vertical lines at x = 1 and x = 2, shades area in between graphs].

Bill readily caught onto Becky’s idea and drew two curves in the plane. His usage of the perimeter and area symbolic form allowed him to interpret the integral as the area in between these two graphs. Note that the subtraction sign in the integrand may have played an important role in seeing the integral as the area in between. It is possible that an integral of the form “

 [] []

([ ] + [ ]) d[ ]” might not generate a similar type of conception.



4.4.3. Four ways of understanding the front multiplier “[] ”



Clay and Chris were discussing the meaning of the integral, “−2 D f(x) dx, ” and had approached it by giving D a specific range of values, “D = (x1 ,x2 ).” Activating the perimeter and area form, Chris visually represented the integral by drawing a squiggly graph to represent f(x), marking off x1 and x2 , and shading in the region. He then launched into explaining the effects of the negative two in front of the integral sign. Chris:

... Chris:

If we multiply by negative 2, essentially that means that we’re flipping this over negatively [points to his graph] and we’re multiplying it by a double magnitude. . . So each infinitesimal rectangle [draws in a thin rectangle, see Fig. 8], we’re doubling its y magnitude [doubles length of the rectangle]. Because. . . or we’d be doubling the area of it, and since dx is staying constant then that would mean we’re doubling the y component. So we’d get. . . [draws in “flipped over” graph, see Fig. 8]. And so this would be our resultant value [says this while shading the region]. We could think of it two different ways. We could think of it as being, um, doubling the area and moving it negatively [makes hand motion like he’s flipping something upside down]. Like I did here. Double that. Or you could just move it in and have it be doubling and multiplying by negative 2 on the function itself. So moving this up [puts chalk on the rectangle and moves chalk upward] and negatively [swings chalk down below x-axis] and then multiply by dx.

Chris showed “two different ways” of understanding the multiplier in front of the integral symbol, depending on which conceptualization of the integral he was drawing on. In the first, Chris’ usage of the perimeter and area form created a single object (a bounded shape in the plane) that could be manipulated by the negative two. The fixed object becomes animated through the front multiplier by “moving it negatively,” i.e., flipping it, or stretching it out or compressing it. When this meaning is given to the symbol in front of the integral sign, I call it the stretch or flip symbolic form. It is an action done to the integral as a whole. The other way that Chris described the front multiplier is that it can be “moved in” to interact with the function itself. He demonstrated this by using a representative rectangle (see the adding up pieces symbolic form), which was then altered by the factor of negative two. While this is similar to the stretch or flip symbolic form, there is an important distinction. The representative rectangle depicts what is happening before the integral symbol plays its role. Chris is describing the separate notions of treating the integral as one object that is manipulated as a whole versus the blending of the front multiplier with

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Fig. 8. Graph (and reproduction) constructed by Chris and Clay.

the integrand to alter what is actually added up by the integral. This thinking is clarified by a statement made by Devon while discussing this same integral. Devon:

If you take it as a whole function, the 2, like you would not see the 2, you don’t even see the 2 here, it’s just, like, a part of it [the function].

Here, Devon was thinking of the front multiplier as becoming completely blended with the integrand. The front multiplier no longer exists as a separate entity, but has become “part of [the integrand].” When the symbol representing the front multiplier is seen in this way, I call it the melds with integrand symbolic form. Clay and Chris also present a third schema that can be applied to the front multiplier. They were working on the item Math1 and had assigned the functions “f1 (x) = x2 /4” and “f2 (x) = x2 /2” to the two curves representing the two wires (they later revise these — for the present purposes,the functions they use do not matter). They then agreed that they would be x able to calculate the area using the integral “ −x x2 /4 − x2 /2dx” (again, the oversight in using “−x” and “x” for the limits is not important here). Chris:

Now we could just say fromnegative x to x, take the two integrals, subtract the difference. Or we could just double from 0 to x because it’s

simpler [erases–x, writes 2

x

0

x2 /4 − x2 /2dx]. Because the right side is a mirror of the left.

Chris decided that they could take advantage of the fact that the curves were symmetric from right to left. By using the front multiplier “2” to represent this, Chris shows evidence that it was conceptually connected with the symmetric graph. Thus, this view of the front multiplier is called the symmetric graph symbolic form. Note, however, this conception has an additional requirement that the symbol template have a zero in the lower limit of integration. (It is possible to have it in the

 []

upper limit instead, though it is less common.) The symbol template associated with this form would be “[ ] 0 .” The final interpretation of the front multiplier is essentially a reduction to more basic conceptualizations about multiplication. Here, the front multiplier is seen as one quantityand the integral as another, where the two simply have a multiplicative relationship. While discussing the integral “−2 D f(x) dx, ” Devon explained this way of understanding the negative two. Devon:

Either take the 2 in and then you can, like, take this as a new function, or you take this as some kind of value [points to -2] and this some kind of value [points to the integral]. It has nothing to do with the integration here. . . This and this [points to -2 and then the integral]. They don’t have to have any kind of relation, it’s just a random thing, times them together. Multiply together.

The negative two does not necessarily have to have anything to do with the integration itself. Rather, the front multiplier and the integral are separate values or quantities that are then simply multiplied together. In this conception, the meaning of the front multiplier is reduced to the meanings an individual has about multiplication (see Sherin, 2001). I call this the multiplies the result symbolic form for the front multiplier.



4.4.4. The subscript symbol “ [] ” as representing a region in space This symbolic form regards the subscript symbol that is attached to the bottom of the integral, which is different from the presence of a lower and an upper limit. Some students gave this symbol the possible meaning of representing a multidimensional region in space, even when there was no indication of a multivariate function. Adam and Alice were discussing

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Fig. 9. Sketch (and reproduction) drawn by Aaron and Alice.



the integral “−2 D f(x)dx” and Adam had just explained to Alice that the D could represent any domain for f(x), such as the intervals “D:(1,2)” or “D:(2,3).” However, he then added a different thought about the meaning of that symbol.



Adam:

I guess just as far as the domain goes, sometimes there’s a double integral [writes “

R ”].

There would be an R right here. Which would

be  the region, like right here [draws a triangle in the plane and shades in the region, see Fig. 9]. . . So it would be like. . . dx. . . dy [writes R dxdy]. So the x would go from, the region for the x would be 1, 0, or something, 0 to 1, right here. But you could say, instead of writing these out, you could just put R [writes R underneath the area] for the region.

Adam took the domain, D, to potentially represent a shape in a two-dimensional plane, as opposed to a one-dimensional  line. He claimed that the D, which he renamed R, “would be the region” and then rewrote the integral as “ R dxdy, ” suggesting that he specifically connected the subscript symbol, D (or R), to a two-dimensional region in the plane. Other students’ work often used the subscript symbol to denote regions in three-dimensional space as well. I call this the region in space symbolic form for a subscript placed on the integral symbol.

4.4.5. The differential as a shape symbolic form This symbolic form is specific to the differential, dealing only with the “[ ]d[ ]” part of the symbol template. It appears that the differential is thought of as representing a shape, which may be dependent on the type of function present in the integrand or on the coordinate system being used. Devon was working on item Physics 1 and had produced the integral  “ V (r) dV” for the box’s mass, where (r) represented a function for the box’s density and V represented the box’s volume. He then attempted to describe how his integral fit with the figure shown in the interview item. Devon:

In this case it really depends. . . It’s just dV as corresponding to the  of r [(r)]. Like I said, you could either integrate the horizontal and vertical way, or if. . .it depends on how the trend of the density is. Let’s say, it depends on the distance from the origin to that point, then dV, maybe I’d use the coordinate system, the polar system, so dV would be a different shape. Like I would have a different way to slice it. Generally, I would say that it’s the dV that’s corresponding to the  of r [(r)].

... Devon:

Let’s say this is case 1, and the function  of r [(r)] is something related just to ax plus by plus z [writes “ax + by + z”]. And then I would, it would be easy for me to slice it. . . It would be like, every little dV would be a small piece of box, a small box. . . But like the principle is, you have to be corresponding to the  of r [(r)]. Let’s say if this is not a box, but like a sphere, I can have multiple ways to slice it.

Devon consistently referred to the differential dV as being a shape. If he used the function “ax + by + z” then “it would be easy” to say that dV was a small box. But if he used the polar coordinate system, dV “would be a different shape”. If he were working with a sphere instead of a box then he could have “multiple ways to slice it” to come up with different shapes for dV. During the item Physics 3, he mentioned that the differential dA could be a “piece of pizza” if the domain was a circle. It appears, then, that the actual shape “d[ ]” takes on depends on either the coordinate system being used, the shape of the domain, or the appearance of the function used as the integrand. I call this the differential as a shape symbolic form.

4.5. Two other cognitive resources regarding the integral Two other important cognitive resources showed up in the students’ work that add to the discussion of the integral, though they are not necessarily symbolic forms. The first of these cognitive resources regards another interpretation for



the “

[] ”

symbol. Several students readily interpreted this symbol as a type of shorthand for the symbol template “



b

 [] []

.”

Given the integral “−2 D f(x) dx, ” many of them immediately turned this integral into “ − 2 a f (x) dx.” This type of thinking persisted, however, even in cases where upper and lower limits did not necessarily reflect the problem modeled by the integral. For example, Clay and Chris were working on item Physics 3 and were attempting to sketch out the meaning of  the integral “ S P dA .” During their work, they decided to assume that the surface, S, was rectangular, drew a rectangle on the board, and imposed an x-y coordinate system on it (see Fig. 10). They discussed how the rectangle could be divided into

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Fig. 10. Sketch (and clarified reproduction) drawn by Chris and Clay.

smaller rectangles, which was represented by the dA. They then changed the integral to “ go back and explain the meaning of the s1 and s2 they had written. Clay: Chris:

 s2 s1

P dA, ” so I later asked them to

s1 is the minimum, where you’re starting from, I guess. And s2 would be the last one you’re integrating, but that would be inclusive of all the ones in between. s1 would be the first square here [draws a square in the left-hand corner], and then s2 would be this square here because it’s where both x and y reach the maximum [draws a square in the right-hand corner, see Fig. 10].

While their interpretation of the symbol correctly reflects a Riemann sum addition over a partition of the rectangular area, it nonetheless does not necessarily reflect the traditional meaning of the limits of integration. Despite this, it is apparent



that several students viewed the subscript symbol “

[] ”

as equivalent to writing the limits of integration “

 [] []

.” I call this

the shorthand for limits cognitive resource. The final cognitive resource to document in this paper deals with the idea of negative area. Traditionally, area “underneath” the x-axis is viewed as negative, though it appears this concept has the possibility of becoming generalized to the idea of a graph that “faces the opposite direction.” David and Devon were working on item Math6 and quickly agreed that integration would be helpful, though David warned that this integration needed to be done carefully. David:

I think you could [use an integral], but you’d have to be careful because, say you graphed it like this [draws axis through middle, see Fig. 11]. Then if you use integration without being attentive you just get 0. Because this is positive and this is negative [writes + and–in two halves, see Fig. 11]. I would suppose that this would be like negative f of x [−f(x)] and this would be positive f of x [f(x)].

An important feature in David’s explanation is that the symmetry of the shape seemed to prompt him to think of negative area. There was nothing about the problem itself that said anything about negative area, nor a choice of axes. As they continued to work on this concept, David revealed more about the connection between symmetry and negative area. David:

This [points to top half of figure] is pretty much the same as this one [points to bottom half of figure], just flipped around. So I would suppose that they’re identical, the only thing is that this is negative, so it’s facing the other way. It’s like a complete opposite reflection.

David’s consistent use of phrases such as “flipped around,” “complete opposite reflection,” and “facing the other way” makes it clear that direction was a highly salient feature in assigning positive and negative area. Directionality appeared to motivate his idea that the half that “faces the other way” must yield negative area. Thus, students may perceive a particular part of a figure or graph as being “negative” even in the absence of a coordinate system. This idea about graphs is labeled the facing the other way cognitive resource.

Fig. 11. Sketch (and reproduction) drawn by David and Devon.

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135

Table 1 Essential characteristics of the symbolic forms. Symbol template

Name of symbolic form

Brief description of conceptual schema

[ ] d[ ]

(1) Adding up pieces

The integrand and differential create rectangles, each with a small piece of the resultant quantity. The integral is an active (infinite) totaling of these pieces. The limits are the starting and stopping point of the running total.

[ ] d[ ]

(2) Adding up the integrand (miscompilation)

The differential determines a partition, where a small quantity of the integrand exists in each piece. These small quantities of the integrand are added up and the result is multiplied by the variable of the differential.

[ ] d[ ]

(3) Perimeter and area

The integrand, differential, and limits of integration construct the physical boundaries (perimeter) of a whole, fixed region. The integral represents the area of this region. The region is taken as a static whole.

[ ] d[ ]

(4) Function matching

The integrand comes from some “original function.” The differential indicates the relationship between them. The integral seeks to find this “original function.” The limits represent a measurement between values.

[ ] d[ ]

(5) Generic answer

The integral yields a “generic” version of a result. The generic result is “waiting” for limits to be attached in order for a more specific result to be determined. The generic result can take in a range of possible inputs.

[ ] d[ ]

(6) Function matching (no limits)

See function matching. No limits are attached to this, though, creating no need for a measurement between values. The integral only seeks to find the “original function.”

(7) Area in between

See perimeter and area. However, the unique structure of the integrand suggests that the fixed area is situated in between two curves in the plane.

(8) Stretch or flip

The integral is taken to be a single entity and the front multiplier has the effect of physically manipulating this entity. For example, a fixed region might be stretched out or flipped over the x-axis.

(9) Melds with integrand

The front multiplier can be combined with the integrand function in such a way that the front multiplier and the integrand are no longer distinguishable. They are blended into creating a new integral.

(10) Symmetric graph

The front multiplier represents a symmetry in the graph of the integrand. Thus, the integration is actually occurring over a larger area than that given by the limits of integration.

(11) Multiplies the result

The front multiplier is seen as a value to be multiplied with the value of the integral. This may reduce to symbolic forms for simple multiplication.

(12) Region in space (13) Differential as a shape

The subscript symbol is seen as representing a region in multi-dimensional space. The differential is thought of as a shape in space. This shape may be dependent on either the function of the integrand, the coordinate system used, or the shape of the domain.

(14) Shorthand for limitsa

The symbol “

(15) Facing the other waya

Certain parts of the figure are deemed to be “facing the other way,” which implies they might connote negative area.

 [] []

 [] []

 [] []

 [] []

 

 [] []

[]

[]

 

[]

[]

([ ] − [ ]) d[ ]

 [] 0



 []

[ ]d[ ]

 []

Graphs/Figures



[] ”

is seen as shorthand for “

 [] []

.”

a These two do not represent symbolic forms in that they do not constitute a conceptual schema blended with a symbol template. However, they appear to be stable cognitive resources.

4.6. Contrasting the symbolic forms In order to compare the various symbolic forms documented in this paper, Table 1 presents the essential components of each form. The table lists the symbol templates and conceptual schemas that together make up each symbolic form. This allows the reader to contrast symbolic forms that make use of the same symbol templates, or that have similar conceptual schemas. The names given to these forms are also listed. 5. Analysis of symbolic form activation 5.1. Symbolic form activation per interview item The students demonstrated a significant array of symbolic forms that they possessed and were able to draw on in order to reason about problems involving the integral. While Section 4 scrutinized each form individually, and artificially isolated segments of students’ work in order to do so, the students often drew on more than one symbolic form during a given problem. They were also able to work flexibly back and forth between different conceptualizations. In the following table, the symbolic forms that each student provided evidence of using are listed according to the interview item they were working on. The numbers represent the symbolic forms as presented in Table 1. Some students drew rather heavily on a particular symbolic form, which is denoted by a bold, underlined number. For a list of the interview items used, see Appendix A.

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Table 2 Symbolic form activation broken down by student and interview item.

M1 M2 M3 M4 M5 M6 P1 P2 P3 P4 P5 P6

Adam

Alice

Bill

Becky

Chris

Clay

David

Devon

Ethan

1,3 3,4,5,6, 15 3,4 3,4 3,11,12, 14 n/a 1 4 1,14 4 n/a 1,3

1,3 3,4,5,6, 15 3,4 3,4 3,11 n/a – 4 1,14 4 n/a 3

1,3,7 3,7 3,6,9 3,4,9 1,11,14 n/a 2 1,3 1,2,14 1,3 1,3 n/a

3,4,7 4,7 4,5 4 11 n/a – 4 1,14 4 4 n/a

1,3,7, 10 1,3,7 1,3,4,5 1,4 1,3,9, 11,14 n/a 2,9 1,3,4 1,3,14 1,3,5 1,11 1

1,3,7, 10 1,3,7 1,3,4,5 1,4 1,3,9, 11,14 n/a 2,9 1,3,4 1,3,14 1,3,5 1,11 1

3,4,7 4,7 1,3,4,5,6,13 6 4,9,11, 14 7,15 n/a n/a n/a n/a n/a n/a

1,3,7 4,7 1,3,4,5,6,13 6 4,9,11, 14 7,15 1,12,13 1,4 1,12 1,4,5 1 1

n/a n/a n/a n/a n/a n/a 2, 12 2,3 2,3 n/a n/a n/a

5.2. Framing and symbolic form activation As discussed in Section 2.1, the framing employed by the students affects the (likely unconscious) choice of resource activation. Traditional mathematics courses have been described as mostly containing equations and expressions that do not necessarily carry any physical meaning (Dray & Manogue, 2006; Meel, 1998). Thus, the pure mathematics contexts in certain interview items may influence how students interpret the integral. In fact, Table 2 illustrates that during the items that consisted of a pure mathematics problem, all of the students drew significantly on the perimeter and area symbolic form and the function matching symbolic form. For most of the students, the adding up pieces form was only occasionally invoked and did not serve as the major driving force behind their explanations, with the exceptions of Chris and Clay who relied heavily on it throughout both interviews. By contrast, science courses contain equations and expressions with additional layers of meaning because of the physical quantities the variables represent (Dray & Manogue, 2006; Redish, 2005). When presented with these types of problems, Table 2 shows there was a marked shift for several students toward the adding up pieces (or the adding up the integrand) symbolic form. In particular, during applied physics problems, Clay, Chris, Devon, Adam, and Bill exhibited this shift toward the conception of the integral as an addition. Bill demonstrated an interesting “mid-problem” shift from the perimeter and area symbolic form to the adding up pieces form, which significantly improved his ability to interpret the meaning of an integral. This switch is discussed in the following section. Also, Ethan stably drew upon the adding up the integrand form, which is based on similar thinking to adding up pieces. However, for Alice and Becky the meaning of the integral provided in the function matching form constituted a more stable interpretation even during the physics problems. The consequences of resource activation during the physics problems were marked. The students who transitioned over to drawing mostly on the adding up pieces symbolic form for the physics problems tended to explain the integrals with less difficulty and were able to provide meaningful reasons for why the integral calculated what it was purported to. In contrast, the students who only relied on the perimeter and area and function matching forms often struggled to understand the integrals in the physics problems or to describe how the integral could calculate what it was supposed to. This difficulty cannot be ascribed to a lack of experience with physics since all of the students came from similar situations within calculus-based physics courses. Nor can it be said that these students did not possess something like the adding up pieces form, since every student provided evidence of having it. Rather, it appears that the choice of symbolic form activation may have either enhanced or inhibited the students’ ability to work with integrals in applied physics problems. Additionally, there appeared to be a relationship between the choice of symbolic form activation and the facility the students exhibited while working with integrals with multivariate functions. In general, the perimeter and area symbolic form and the function matching symbolic form became less productive for the students in explaining the meaning of multivariate integrals. The adding up pieces form, however, became significantly productive in explaining the multivariate integral. This is not to say that multivariate integrals cannot be understood through the other symbolic forms, but it emerged from the students’ work that the adding up pieces proved more constructive. An example of the productive nature of the adding up pieces form in both applied physics and multivariate contexts is now provided.

5.3. The utility of the adding up pieces symbolic form Bill had been drawing heavily on the perimeter and areasymbolic form throughout his interviews. He was working on item Physics 3 and was trying to explain the meaning of “F = S P dA, ” which contains a multivariate pressure function, P. Bill had decided to depict this integral in a way that reflected his reliance on the perimeter and area symbolic form. He drew a one-dimensional graph (i.e., the graph itself was a one-dimensional object), labeled it P, and labeled the horizontal axis A.

S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141

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Fig. 12. Graph (and reproduction) constructed by Bill.

He marked off vertical lines for the left and right sides of a bounded region and shaded in the area (see Fig. 12). However, after Bill had produced this graph, he appeared unsatisfied and did not know how to use his figure to explain the integral. Bill: Interviewer: Bill:

So, if we took the integral of that, it would be, it would be all this. . . [shades in the region underneath the P-graph]. And that would be the total force it would exert. . . I feel like I skipped a step. So conceptually, what does dA mean? The change in area [points to “A-axis”]. . . It’s hard to say. . . But I think if you took the, if you measured the pressure and there’s that much area [puts chalk on a point on the “A-axis”], and you took the pressure of some other lesser area [puts chalk on another point on the “A-axis”], and you subtracted, that would be what dA is, the change in area.

Bill’s creation of a one-dimensional graph for the multivariate function P appeared to cause him problems in understanding this integral. In particular, Bill seemed at a loss as to how to reconcile his use of A as an independent variable with the fact that the area of the surface should not be changing. Furthermore, he seemed uncomfortable in describing why the area of the region he had created should represent the total force, stating that he felt like he “skipped a step.” Both Bill and Becky became somewhat frustrated and silent at this point, so I asked them to think about the table I was sitting at as the surface area, S, and asked them whether the integral in the problem could be applied to some non-uniform pressure the table might experience. Bill:

I believe that, uh, I’m just trying to relate this to rectangles. If we just took the area of this piece of the rectangle here, this part of the table, and found the total force exerted on that, you would get some kind of estimate. Because if you added all those. . . it wouldn’t be exact, it would be an estimate. But as you make that area smaller and smaller and smaller, and then it would get better and better, until it gets close to 0 and it would be an integral.

Here Bill had switched his thinking and now showed evidence of drawing on the adding up pieces form, improving his ability to work with the multivariate function over a two-dimensional domain. He now talked about dividing the domain into “rectangles,” relieving his quandary about the fixed nature of the area A. Then as the area of these individual rectangles became “smaller and smaller” it would “become an integral.” After some discussion, Bill summarized his thinking, which showed strong evidence of drawing on the adding up pieces form. Bill:

... Bill:

[Draws a rectangle on the board to represent the table.] Let’s just say this is dA [references a small strip at one end of the rectangle]. This whole thing is dA, this whole area [again references the small strip]. And you have pressure pushing on that, on all that area. So you can multiply P times dA and you get the total force pushed, exerted on that part of the table. Yeah, if you make that area smaller and smaller and smaller and then add up those infinite, those really small areas on the whole table, you get the total force.

By activating the adding up pieces symbolic form and using a representative rectangle (through referencing one small strip of the table), Bill satisfactorily explained the meaning of dA, the relationship between P and dA, and how the integral calculated the total force exerted on the surface. This demonstrates the productivity of this symbolic form when dealing with a multivariate function. The perimeter and area form did not provide Bill with a context for explaining these relationships in the integral, because it focused him on an apparently less useful one-dimensional graph and an ill-defined variable “A” that did not seem to satisfactorily represent the meaning of the integral. This example corroborates the conclusion made by Thompson and Silverman (2008) and Sealey (2006) that the area conception of the integral alone may not be sufficient in understanding the integral in a variety of contexts. In this paper, this conclusion is extended to include mathematics and physics problems involving multivariate functions. Bill’s example promotes the “adding” or “accumulation” aspects of the integral as being essential for robust understanding (Sealey, 2006; Sealey & Oehrtman, 2005; Thompson & Silverman, 2008).

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6. Discussion Tables 1 and 2 demonstrate one important fact about student understanding of the integral. The students interviewed for this study do appear to have a large pool of productive knowledge regarding the integral and were able work with multiple conceptualizations of it during a given task. While much of the research on the integral has been focused on either the misconceptions students have or their lack of appropriate knowledge (Bezuidenhout & Olivier, 2000; Orton, 1983a; Rasslan & Tall, 2002; Tall, 1992), this paper promotes the idea that student difficulties might not necessarily arise from lack of knowledge, but from the activation of less-productive cognitive resources over others. The results suggest that helping students “choose” which interpretations of the integral to use, such as an addition over small pieces, might assist them in applying their knowledge of the integral to other mathematics and physics problems. This coincides with some educators’ call to recast certain “misconceptions” as the activation of cognitive resources that are not effective for the given situation, but that are not necessarily “wrong” (Elby & Hammer, 2010; Hammer, 2000; Hammer et al., 2005; Smith, diSessa, & Roschelle, 1993). The area conceptualization is not wrong, but was less productive for Bill in understanding an applied, multivariate integral. This is not to say that the area conceptualization is a useless interpretation of the integral, or that it does not serve a purpose in some contexts. However, as Sealey (2006) describes, the notion of an “area under a curve is not sufficient for understanding the definite integral” (p. 7). In certain symbolic forms, such as the adding up the integrand symbolic form, it is possible for students to have compiled a problematic resource that deviates from standard thinking. While the adding up the integrand symbolic form could be called a “misconception,” it is important to note that it is rooted in the useful concepts of accumulation. Thus, Ethan’s conception does not need to be removed or replaced, it simply needs to be realigned. The usefulness of the adding up pieces symbolic form in both physics and multivariate contexts offers some pedagogical lessons. In traditional textbooks (see Salas et al., 2006; Stewart, 2007; Thomas et al., 2009) the integral is often motivated by a discussion of the area of a region under a curve, which may lead to the compilation of the perimeter and area symbolic form. Subsequently, the vast majority of attention is dedicated to the computation of the integral as an anti-derivative, which the students might use as a base for constructing the function matching symbolic form. Unfortunately, these traditional textbooks often dedicate only one section to the integral as a Riemann sum, and these sections generally emphasize the Riemann sum only as a computational method for finding the area of a fixed region. Thus, a standard calculus textbook may not expose students much to the idea of the integral as “adding up pieces” or “accumulation.” This interpretation of the integral may need to be highlighted and emphasized in order to provide students the chance to develop this conceptualization and to learn how to effectively apply it to situations where it would be helpful (Sealey, 2006; Thompson & Silverman, 2008). The results in this paper add to the work done by several researchers (Dray & Manogue, 2006; Engelke & Sealey, 2009; Hughes-Hallett et al., 2005; Sealey & Oehrtman, 2005; Sealey & Oehrtman, 2007; Sealey, 2006) in developing ideas of how to use student understanding of the integral to influence curricular materials. By stressing the concepts of “accumulation” and “total amount,” students may have more opportunities for compiling and drawing on the adding up pieces symbolic form. All of the students in this study demonstrated that they had components of this productive interpretation of the integral in their cognitions, yet many students did not employ this interpretation to its full potential, possibly because of the emphasis of “area” and “anti-derivatives” in current curriculum. There is still much research to be done in learning how each interpretation of the integral influences students’ problem solving and which conceptualizations facilitate student understanding in many different contexts. Further work is also needed to fully understanding how instructional intervention affects the assignment of meaning to the integral symbols. This would assist educators in providing students with the tools to construct a robust, multifaceted understanding of this important mathematical concept. Acknowledgements This work was developed through the support of grants from the National Science Foundation, Grant Nos. ESI 0083429 and DRL 0426253. Any opinions, findings, and conclusions or recommendations expressed are those of the author and do not necessarily reflect the view of the National Science Foundation. Additionally, the author wishes to thank Dr. Patricia F. Campbell, Dr. Ann Edwards, Dr. David Hammer, and Dr. Edward Redish for their helpful feedback and insight. Appendix A. Description of Interview Items The following are all of the interview items used during the sessions with the students. Based on the direction of discussion and time constraints, not every item was shown to each pair of students. Ethan was interviewed only once, meaning he did not receive several of the items. Notes have been provided for which students saw each item. ITEM Math1 (presented to all student pairs, excluding Ethan) Two wires are attached to two telephone poles [image of telephone poles provided]. Suppose we wanted to know the area between the two wires. How could you figure that out? ITEM Math2 (presented to all student pairs, excluding Ethan)

2

2 1 x3

− x2 dx Compute and then discuss this integral.

ITEM Math3 (presented to all student pairs, excluding Ethan)

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I want you to look at each of the following integrals and talk about what they mean.





0

ex dx

sin(x) 2

ITEM Math4 (presented to all student pairs, excluding Ethan) I want you to look at each of the following integrals and talk about what they mean.





dx

√ tdx

ITEM Math5 (presented to all student pairs, excluding Ethan) Suppose we had a function f(x) with a domain D. What does this integral mean?

 −2

f (x) dx D

ITEM Math 6 (presented only to David and Devon to produce more data) This picture shows the outline of a violin body [figure of a violin provided]. If you wanted to know the area of this shape, how could you figure that out? ITEM Physics 1 (presented to all student pairs)

This shows a box with varying density (dark = more dense, light = less dense). Suppose you wanted to know the box’s mass. How could you figure that out? ITEM Physics 2 (presented to all student pairs) The durability of a car motor is being tested. The engineers run the motor at varying levels of “revolutions per minute” over a 10 hour period. Denote “revolutions per minute” by R.

 600

What is the meaning of the integral 0 R dt? ITEM Physics 3 (presented to all student pairs) A two-dimensional surface (S) experiences a non-uniform pressure (P) and we want to know the total force exerted. We can use the surface’s area (A) to compute this through the integral:



F=

P dA. S

Why does this integral calculate the total force exerted? ITEM Physics 4 (presented to all student pairs, excluding Ethan) We know from kinematics that acceleration and velocity are related by a(t) = (d v(t)/dt). We can rearrange this equation and integrate to get the equation





a dt =

dv.

What does this equation mean? Why are these two terms equal to each other? ITEM Physics 5 (presented to all student pairs, excluding Adam/Alice and Ethan) Fy is used to denote the amount of a force in the y-direction. U is used to denote the change in potential energy. These two concepts are related through this equation:

yf U = −

Fy dy. yi

Explain this equation. What does each part of the equation/integral mean? ITEM Physics 6 (presented to all student pairs, excluding Bill/Becky and Ethan)

This represents a metal bar with varying mass along its length (lighter = less dense, less mass/darker = more dense, more mass) How could you figure out the center of mass for the bar along its length?

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