Journal of Mathematical Behavior 32 (2013) 122–141
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The Journal of Mathematical Behavior
j ournal homepage: www.elsevier.com/locate/jmathb
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 a b s t r a c t
Article history:
Researchers are currently investigating how calculus students understand the basic con-
Available online 12 January 2013
cepts 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
Keywords: drew on while working on pure mathematics and applied physics problems. This research
Calculus provides evidence that students hold a variety of productive cognitive resources that can be
Integral employed in problem solving, though some of the resources prove more productive than
Student understanding
others, depending on the context. In particular, conceptualizations of the integral as an
Undergraduate mathematics education
addition over many pieces seem especially useful in multivariate and physics contexts. Symbolic form © 2012 Elsevier Inc. All rights reserved.
Accumulation
1. Introduction and relevance
In recent decades, more and more attention has been given to compiling a body of research regarding student understand-
ing 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,
∗
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
S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141 123
2008; Tipler & Mosca, 2008) and students who continue into further calculus courses will encounter the integral more
often than the derivative (Salas, Hille, & Etgen, 2006; Stewart, 2007; Thomas, Weir, & Hass, 2009). However, an overre-
liance 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). Addition-
ally, 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 por-
tion 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 predomi-
nant 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
1 2
properly calculates the integral x dx using an anti-derivative but then fails to adequately interpret the corresponding 0
n 2
n→∞ x x
Riemann sum lim k=1( k) . 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 “selec-
tion” 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.
124 S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141
Avail able symbolic forms in
stud ent’s cognition
(blend of template
Framing by students: + conce ptual schema)
Interview context math/physics context +
interview context
math or Activation of symbolic form physics what is being asked?
problem
what is mathem atical
knowledg e? Observable outcome:
Stud ent 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) and
E
= P + K. (2)
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[ ], [ ] [ ] d[ ], or [ ] d[ ], interview items were created to provide the students an opportunity to discuss each of
[ ] [ ]
S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141 125
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 exhaus-
tive 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
126 S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141
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
f
x − f x dx.
integral “ a 1( ) 2( ) ”
Devon: 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: 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].
Interviewer: Which part is? Just to make sure.
Devon: 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 rep-
resentative 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.
S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141 127
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., “lim x→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: 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.
.
. .
Adam: 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].
128 S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141
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 SP 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
600
explained how the addition happened inside the integral R dt.
0
Ethan: 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.
Interviewer: What do you mean by “add up all the RPMs?”
Ethan: 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 Df(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
S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141 129
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,
2 3 2
Devon and David were given the integral “ 2/x − x dx” from item Math2 and were asked to calculate it. In their first step 1
2 −3 2 2
they broke the integral into two parts, “ 2x − x dx.” David recognized that they had left out the dx from the first of
1 1
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.
3 2
David:Inan integration the dx is always essential, because it shows that this entire thing [waves hand over the integrand, “2/x –x ”] is a
derivative of x.
.
. .
David: The fact that this entire thing is sitting right next to each other, and dx outside, means that basically this entire function [motions hand
3 2
over “2/x –x ”] is the derivative of an original function.
3 2
David had conceptualized the integrand “2/x − x ” 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
3
function. In fact, David wrote the function “2ln(x )/3” as the anti-derivative for the first term (the fact that it is incorrect is
irrelevant here) and said,
3 −3
David: You try to find the derivative of this [points to 2ln(x )/3], which would just be equal to this [points to 2x ].
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
[] ∫ []d[] [] the integrand represents the “top side” the differential’s main purpose is to determine the “bottom side” of the shape
the "∫ " 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.
130 S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141 the “original function” produces integrand
[] “original ∫ []d[] d[] tells you how to function” [] find the “original function”
Universe of Functions the purpose of the integral is to “find” this “original function”
[] the limits “ ∫ ” 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.”
0 x
Devon: This one [points to e dx], you are finding the difference between these two [points fingers to the 2 and 0]. So, regardless of the c, it
2
would just be a difference. So that’s how I think of it, as a difference.
x 2 0
Since Devon later plugged the 2 and the 0 into e to get e and e , 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
0 x
answer symbolic form. Bill and Becky were working with the integral “ sin(x)dx” and were comparing it to e dx. 2
S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141 131
Becky: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 betweensymbolic 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 3 2
/x
− x dx two different curves in the plane. Bill and Becky were explaining the meaning of the integral “ 2 ” and Bill, who
1
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.
3 2
Becky: 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/x ” and the “x .”]. It’s like f of
x [f(x)] plus g of x [g(x)].
3
Bill: Yeah I guess you could do that, if you took 2 over x cubed [2/x ], that’s probably just . . . let’s just say it’s this [draws a graph]. And then you
2
take x squared [also draws the x 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 Df(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: 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].
.
. .
Chris: 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
132 S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141
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
2 2
Math1 and had assigned the functions “f1(x) = x /4” and “f2(x) = x /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
2 2
x / − x / dx −
able to calculate the area using the integral “ −x 4 2 ” (again, the oversight in using “ x” and “x” for the limits is
not important here).
Chris: Now we could just say from negative x to x, take the two integrals, subtract the difference. Or we could just double from 0 to x because it’s x
2 2
simpler [erases–x, writes 2 x /4 − x /2dx]. Because the right side is a mirror of the left.
0
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 mul-
tiplication. 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 Df(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 multi-
dimensional region in space, even when there was no indication of a multivariate function. Adam and Alice were discussing
S.R. Jones / Journal of Mathematical Behavior 32 (2013) 122–141 133
Fig. 9. Sketch (and reproduction) drawn by Aaron and Alice.
the integral “−2 Df(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
Rdxdy]. 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 “ Rdxdy, ”
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)].
.
. .