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Thompson, P. W., & Silverman, J. (2007). The Concept of accumulation in calculus. In M. Carlson & C. Rasmussen (Eds.),
Making the connection: Research and teaching in undergraduate mathematics (pp. 117-131). Washington, DC:
Mathematical Association of America.
The Concept of Accumulation in Calculus
Patrick W. Thompson, Arizona State University
Jason Silverman, Drexel University
The concept of accumulation is central to the idea of integration, and therefore is at the
core of understanding many ideas and applications in calculus. On one hand, the idea of
accumulation is trivial. You accumulate a quantity by getting more of it. We accumulate injuries
as we exercise. We accumulate junk as we grow older. We accumulate wealth by gaining more
of it. There are some details to consider, such as whether it makes sense to think of accumulating
a negative amount of a quantity, but the main idea is straightforward.
On the other hand, the idea of accumulation is anything but straightforward. First,
students find it is hard to think of something accumulating when they cannot conceptualize the
“bits” that accumulate. To understand the idea of accomplished work, for example, as accruing
incrementally means that one must think of each momentary total amount of work as the sum of
past increments, and of every additional incremental bit of work as being composed of a force
applied through a distance. Second, the mathematical idea of an accumulation function,
represented as F x
= f t
dt
a
x
!
, involves so many moving parts that it is understandable that
students have difficulty understanding and employing it.
Readers already sophisticated in reasoning about accumulations may find it surprising
that many students are challenged to think mathematically about them. The ways in which it is
difficult, though, are instructive for a larger set of issues in calculus. As such, our intention in
this chapter is to:
(1) Explicate the complex composition of a well-structured understanding of
accumulation functions,
(2) Illustrate students’ difficulties in understanding accumulation mathematically,
(3) Point out promising approaches in helping students conceptualize accumulation
functions, and
(4) Place students’ understandings of accumulation functions within the calculus as a
whole.
COMPOSITION OF A WELL-STRUCTURED UNDERSTANDING
OF ACCUMULATION FUNCTIONS
Accumulation functions can be represented generally by f t
dt
a
x
!
1
It is worthwhile to
unpack the meanings behind this formula in order to see all that it entails. We will do this in two
passes, first without addressing ideas of Riemann sums, then addressing them.
1
For purposes of this chapter, we will speak only of Riemann integrals over an interval.
ACCUMULATION FUNCTIONS
For sake of illustration, let f ( x ) = 2 e
! cos ( x )
! 2.5. To understand an accumulation function
involving f , students must have a process conception of the formula 2 e
! cos ( x )
2
This means
that they must hold the perspective that though it might require actual effort to calculate any
particular value of this formula, in the end it represents a number, and the number it represents
depends only on the value of x (Breidenbach, Dubinsky, Hawks, & Nichols, 1992; Dubinsky &
Harel, 1992; see also Oehrtman, Carlson, & Thompson, this volume). To have a process
conception of a function’s defining formula implies that one has what Gray and Tall (1994) call a
proceptual understanding of what the formula represents. One has in mind a well-structured set
of procedures for evaluating the formula together with ability and inclination to see the formula
as “self-evaluating” (P. W. Thompson, 1994b), meaning that one sees it as evaluating itself
instantaneously for any number.
3
To understand an accumulation function, students also need a covariational understanding
of the relationship between x and f (Carlson, Jacobs, Coe, Larsen, & Hsu, 2002; Saldanha &
Thompson, 1998; P. W. Thompson, 1994b, 1994c). In the case of the current example, this
means understanding that as the value of x varies, the value of 2 e
! cos ( x )
! 2.5varies accordingly.
It also entails creating an image of how the value of 2 e
! cos ( x )
! 2.5varies as the value of x varies,
thus generating the relationship expressed by the graph in Figure 1.
x
y
Figure 1 : Graph that depicts values of x and 2 e
! cos ( x )
! 2.5 varying simultaneously.
Students who have mastered the process conceptions of formulae and covariational
conceptions of function must then coordinate a third aspect with them—imagining accumulation
and its quantification. Students must coordinate the value of x as it varies from some starting
point, the value of 2 e
! cos x ( )
! 2.5as it varies accordingly, and, in addition, imagine the bounded
2
Briedenbach et al_._ , Dubinsky and Harel, and Carlson speak of a process conception of function.
We also speak of a process conception of formulae. To us, for students to have a process
conception of " f ( x ) = …" requires that they have a process conception of the right hand side. A
process conception of a function entails more than does a process conception of a formula. Our
intent is to avoid adopting an "all or none" stance toward what it means to understand a function.
3
We will give specific examples later in this article of students having and not having a process
conception of an integral.
serves a conceptual role in making sense of the expression f ( t ) dt
a
x
!
. In Figure 2 , f cannot be
thought of as having the same argument as does f ( t ) dt
a
x
!
. In a sense, the graph of f must “pre-
exist” when imagining the accumulation of area between it and an axis. Thinking of t as already
having varied through f ’s domain prior to x varying through a subset of f ’s domain then allows
one to think of
f ( t ) dt
a
x
!
as representing the accumulation of area within an already bounded
region.
RIEMANN SUMS
Calculus texts typically offer Riemann sums as a way to approximate areas bounded by a
curve. The question of how a bounded area itself can represent a quantity other than area requires
us to examine ways to understand Riemann sums and how they arise.
If f is a function whose values provide measures of a quantity, and x also is a measure of
a quantity, then f ( c )∆ x , where c ε [ x , x +∆ x ], is a measure of a derived quantity. The simplest case
is when f ( x ) is a measure of length and x is a measure of length. Then f ( c )∆ x is a measure of area.
If f ( x ) is a measure of speed and x is a measure of time, then f ( c )∆ x is a measure of distance. If
f ( x ) is a measure of force and x is a measure of distance, then f ( c )∆ x is a measure of work. If f ( x )
is a measure of cross-sectional area and x is a measure of height, then f ( c )∆x is a measure of
volume. If f ( x )is a measure of electric current and x is a measure of time, then f ( c )∆ x is a measure
of electric charge. A Riemann sum, then, made by a sum of incremental bits each of which is
made multiplicatively of two quantities, represents a total amount of the derived quantity whose
bits are defined by f ( c )∆ x. Therefore, for students to see “area under a curve” as representing a
quantity other than area, it is imperative that they conceive of the quantities being accumulated
as being created by accruing incremental bits that are formed multiplicatively.
Our account of how “area under a curve” comes to represent quantities other than area
clearly holds an undertone of thinking with infinitesimals. Though a large portion of 19
th
activity in the foundations of mathematics was motivated by the desire to eliminate
infinitesimals, we see no way around explicitly supporting students’ reasoning about them as part
of their path to understanding accumulation functions (and functions in general). Much of this
support should be given in middle school and high school, but given that this does not happen in
the United States, it must be addressed in introductory calculus courses.
6
STUDENTS’ DIFFICULTIES IN UNDERSTANDING ACCUMULATION MATHEMATICALLY
While our analysis of the ideas entailed in understanding accumulation functions also
points to ways that students have difficulty with them, we must also note that the major source of
students’ problems with accumulation functions is that the idea is rarely taught, and when taught
6
The legacy of infinitesimal reasoning in calculus is reflected in the continued use of the integral
notation that Leibniz introduced in 1675, when he used “ dt ” in f ( t ) dt
!
to represent the
difference between successive values of t (O'Connor & Robertson, 2005).
it is rarely taught with the intention that students actually understand it. We anticipate the
objection that definite integrals receive clear and explicit attention in every calculus textbook.
Our reply is that a definite integral is to an accumulation function as 4 is to x
2
. No one would
claim to teach the idea of function by having students calculate specific values of one. Similarly,
we should not think that we are teaching the idea of accumulation function by having students
calculate specific definite integrals.
We say this without hubris. In a teaching experiment (P. W. Thompson, 1994a) conducted
with the intention of investigating advanced undergraduates’ difficulties forming the ideas of
accumulation and rate of change of accumulation, the first author failed to anticipate the
difficulty they would have conceptualizing accumulation functions and failed to anticipate the
importance that they actually do so for understanding the Fundamental Theorem of Calculus
(FTC).
7
Lastly, the idea of limit and the use of notation are two of the most subtle and complex
aspects of understanding accumulation functions. Research on students’ understanding of limit
(Cornu, 1991; Davis & Vinner, 1986; Ferrini-Mundy & Graham, 1994; Tall, 1992; Tall &
Vinner, 1981; Williams, 1991) shows consistently that high school and undergraduate students
understand limits poorly, even after explicit instruction on them. We located only two empirical
studies that addressed students’ reasoning about limit in the context of integration (Oehrtman,
2002; P. W. Thompson, 1994a). Thompson studied advanced undergraduate and graduate
students’ understanding of the FTC, and in that context found students concluding, for example,
that the rate of change of volume with respect to height in a cone was equal to the cross-sectional
area at that height because as you make an increment in height smaller, the incremental cylinder
of volume gets closer and closer to an area (P. W. Thompson, 1994a, p. 34). Oehrtman (2002)
named this way of thinking “the collapsing metaphor,” meaning that students reasoned that the
object being considered (e.g., a cone, a secant, etc.) approached another object having one less
dimension. He found one-third of his subjects (first-year calculus students after instruction)
employing this metaphor in one setting or another.
Oehrtman points out that though the collapsing metaphor is mathematically incorrect, it
sometimes enables students to educe mathematically correct results from incorrect reasoning.
Students sometimes justify the FTC by the incorrect reasoning that as the interval width
decreases, the rectangle collapses to its height (Figure 4 ). Put another way, students reason that
∆ x →0 implies that f ( c )∆ x → f ( c ). They were thinking of an image (e.g., a rectangle) instead of the
quantity (e.g., electrical charge) and the value of its measure. We do note that although the
collapsing metaphor enables students’ intuitive, albeit incorrect, “justification” of the FTC, it
also divorces their understanding of the fundamental theorem from any idea of rate of change.
7
This was a classic case of an outcome being harder than someone expected even though he
anticipated it would be harder than he expected.
Figure 5 : A student’s image of a region filling up with paint as she moves one edge.
f ( x ) dx
a
b
The area between
a and b
of the region bounded by
the graph of this function
Figure 6 : Referential understanding of integral notation.
We mentioned earlier a teaching experiment that investigated difficulties inherent in
coming to understand the FTC (P. W. Thompson, 1994a). It involved 19 advanced undergraduate
mathematics and masters mathematics education students. One aspect of the teaching experiment
emphasized students’ development of a process conception of integrals that entailed ideas of
accumulation, variation, and Riemann sums as the root ideas of integration. We joined these
ideas by defining Riemann sums as one would for fixed intervals, but modifying the definition so
that ∆ x was a parameter and x was a variable. That is, we held ∆ x constant and let x vary instead
of holding x constant and letting ∆ x vary.
8
This can be expressed generally as
F
! x , a
( x ) = f ( i! x + a )! x
i = 0
x " a
! x
$
%
&
%
, a ( x ( b. However, we did not provide this general
representation of the Riemann accumulation function. Instead, we worked with students to model
the accumulation of a quantity that accrued in bits created by joining two of its constituent
attributes (like work from power and time) and develop a representation of the total
accumulation.
At the end of the teaching experiment we used Item 6, among others, to assess the extent
to which students had developed a process conception of Riemann accumulation functions.
8
Our justification for this approach, fixing ∆ x and letting x vary, was our intention to have
students understand integral accumulation functions as being rooted in Riemann accumulation
functions. In earlier attempts to probe students’ understandings of accumulation functions, we
got only the “paint filling” metaphor alluded to in the discussion of Figure 5. We will say more
about the benefits of this approach in the last section.
Item 6
Let q ( x ) be defined by
q ( t ) = f ( i! t )! t
i = 1
t
! t
"
, 0 ≤ t ≤ b. Explain the process by which the
expression
f ( i! t )! t
i = 1
t
! t
"
assigns a value to q ( t ) for each value of t in the domain of f.
Responses to this item showed that understanding Riemann sums as functions was a
complex act for students. Student 1 wrote:
First the value of a certain chunk is measured by i ∆ t. This is then
multiplied by the change which is ∆ t. This is repeated for every
value of t and then added up. Each value of t is cut up into t /∆ t
intervals, and added. t /∆ t is the number of intervals the piece is to
be divided up into.
Student 1 had a number of problems, one being that he was imagining a “chunk” of a
quantity independently of an analytic expression that established its measure— i ∆ t does not
“measure” the chunk, it just puts you at the right place to make it. The student also failed to note
the role played by the function f in creating a “chunk”— it is f ( i ∆ t )∆ t that gives the chunk’s
measure. Also, the student was unclear about what was being summed: “Each value of t is cut up
into t / ∆t intervals, and then added.” However, the subintervals are not summed. A more serious
problem, though, is that this student appeared to be imagining t and i varying simultaneously
instead of, first, varying i from 1 to t! t "
for a fixed value of t , and then varying t.
Student 2 wrote:
- Here ∆ t represents the size of each interval that f is being broken up
into.
- So t /∆ t equals the number of intervals the graph of f is broken up into.
- So our i starts out at 1 and then goes to t /∆ t.
- The expression first finds f and then it finds the i th interval of f that we
are dealing with. Then it finds the value of the function f at that
interval and then multiplies by ∆ t. This finds the area of that particular
rectangle. Then we add it to the previous areas found and plot that
point. You then connect all the points to get your curve.
The process that Student 2 understood is much more well-structured than Student 1’s.
While some of her phrasing is imprecise (“… the size of each interval that f is being broken up
into ”) and suggests that she is reasoning about a graph, she does seem to be imagining the
process being played out for each value of t. One missing element in this student’s explanation is
that the value of t does not vary. Rather, she seems to imagine that she “samples” values of t and
then connects the points that get plotted for each value. This suggests that she was imagining a
Riemann sum over a fixed interval, which would normally correspond to an approximation of a
definite integral f ( t ) dt
a
b
!
instead of the indefinite integral f ( x ) dx
a
t
!
. When students do not see t
as varying, it is difficult, if not impossible, for them to conceive that the accumulation function
has a rate of change for every value of t at which it is defined (Smith, in preparation).
At the same time that Carlson et al.’s (2003) findings point to the promise of building
students’ understanding and skill with calculus on a strong conceptual foundation of covariation,
function, rate of change, and accumulation, their post-instruction interviews suggest that students
had not clarified some important issues. For example, one student, Chad, was shown a graph of a
piecewise-linear function f whose values gave the rate in gallons per hour at which water filled a
container. He was asked to explain the meaning of g ( x ) = f ( t ) dt
0
x
!
and how to evaluate g (9).
Carlson et al. reported that Chad gave an acceptable explanation of the meaning of g ( x ) and also
provided Chad’s explanation of how to evaluate g (9).
- Interviewer: So, how do you think about evaluating g (9)?
- Chad: I see that as finding the time that passes from 0 to 9 and thinking about how much
area gets added under the curve as I move along. I see that water is coming into the
tank, first at an increasing rate, then at a decreasing rate. Then after 4½ hours, water
starts to go out of the tank. As you add up the area under the curve you see that the
same amount of water comes in between 0 and 4 ½ that goes out between time 4½ and
9 …. so, the result is that there is no water in the tank after 9 hours have passed.
- Interviewer: How are g and f related?
- Chad: The derivative of g gives the graph of f. What I don’t get is why t is the variable that is
used in f. I never really understood this on some of the other problems we did either.
(Carlson et al., 2003, p. 270)
Paragraph 2 suggests that Chad could think about accumulation functions and rate of
change to support his evaluation of g (9). He also was thinking about the quantities that x , f ( x ),
and g ( x ) represented ( viz. , number of hours, the rate at which water filled the container, and the
amount of water in the container). He also appeared to attend to how g changes while imagining
changes in x and f. However, we observe that Chad’s statements in paragraph 2 also are
reminiscent of the “paint filling” notion of accumulation discussed in conjunction with Figure 5.
As a result, without querying Chad further, we have no way of knowing if he is using the “paint
metaphor” in a pseudo conceptual way—i.e., does he understand that infinitesimal amounts of
multiplicative bits are being accrued as x varies. In addition, paragraph 4 suggests that Chad had
not worked through the conceptual issues behind the use of t in the accumulation function’s
definition.
In the spirit of our earlier discussion of the need for widely accepted criterion tasks and
widely accepted standards of evidence for the level of students’ understandings, the extent to
which Chad’s understanding was rooted in Riemann sums or the extent to which his
understanding went beyond the paint-filling metaphor is unclear. We see this as pointing once
again to the need for further analysis of what it means to understand accumulation functions and
how to assess levels of understanding. It also points to the need for further investigation into the
implications that various ways of understanding accumulation have for learning related ideas in
the calculus, and the kinds of instruction that will support students in developing those
understandings.
STUDENTS’ UNDERSTANDINGS OF ACCUMULATION FUNCTIONS WITHIN THE CALCULUS AS A
WHOLE
Accumulation functions would not be important if understanding them well did not pay
off elsewhere. In this section we argue that the kind of understanding we have depicted as well-
structured not only pays off in other areas, they are part of understanding many related ideas and
they are essential for understanding many advanced ideas in the calculus. But even beyond the
connections with other ideas that we will outline here, we feel that the precise thinking and
thoughtful use of notation required to understand accumulation functions well is in itself
valuable mathematical activity.
CONNECTIONS WITH OTHER IDEAS
Rate of change. The idea of accumulation both grows out of and contributes to a coherent
understanding of rate of change (Carlson et al., 2003; P. W. Thompson, 1994a). When something
changes, something accumulates. When something accumulates, it accumulates at some rate. To
understand rate of change well, then, means that one sees accumulation and its rate of change as
two sides of a coin. Thus, students’ success in the integral calculus can begin in middle school if
rate of change is taught substantively (A. G. Thompson & Thompson, 1996; P. W. Thompson,
1994a, 1994c; P. W. Thompson & Thompson, 1994).
Function. The obvious connection between the ideas of accumulation functions and
function is that an accumulation function is precisely that, a function. It is nontrivial for students
to understand this. There are three additional important connections that can be exploited in a
calculus curriculum.
- Accumulation functions are, in all likelihood, the first functions students meet that are
defined in terms of a complex process instead of in terms of an algebraic, trigonometric,
or exponential expression. The challenge is to avoid leading them to pseudo-conceptual
understandings of accumulation (see discussion of Figure 5 ) and pseudo-analytic
interpretations of the notation (see Figure 6 ).
- Accumulation is the root of accumulation functions, and hence covariational conceptions
of functions must be a key connection.
- Riemann accumulation functions that are specified by the formula
g ( x ) = f ( i! x + a )! x
i = 0
x " a
! x
$
%
&
%
'
, a ( x ( b are step functions. Computer programs that allow
Riemann sums as defined here will also support students’ explorations of convergence.
10
The issue of convergence, however, expresses itself differently in this context than it does
in typical treatments of Riemann sums. In the typical case, the issue is whether there is a
number that is the limit of a Riemann sum as ∆ x →0. The accumulation function
f t
dt
a
x
!
is then defined so that each value of the function is a pointwise limit. In the
case of a Riemann accumulation function, the issue is whether there is a function that is
10
We used Graphing Calculator from PacificTech to generate the graphs of accumulation
functions contained in Figures 2-5. The functions graphed in those figures were specified as
Riemann accumulation functions.
CONCLUSION
As we mentioned in the beginning of this chapter, the concept of accumulation is, at once,
almost trivial and quite complex. One aspect of the complexity described in this chapter was the
focus on accumulation functions as opposed to the traditional focus on the calculation of a
number representing the area bound by the curve over a specific interval. The emphasis of this
chapter, though, was not on the differences between these two related notions of integral
calculus; it was on the underlying images students bring to bear on such problems and the
implications of those images. The first image involved covering a region, where the result was a
number equivalent to “the amount of paint needed” to cover the area between the x - axis and the
function on the interval [a,b]. The second image involved measuring the accumulation of a
quantity that is created from bits that themselves are made from measures of two quantities, one
whose measure is a function of the other on the interval [a,b], by summing values of f ( c )∆ x , c ∈
[ i ∆ x ,( i +1)∆ x ). The connection between the second image and area is simply that if f ( c ) and ∆ x are
represented by lengths, then f ( c )∆ x gives the area of a rectangle made from those lengths. The
former image is difficult to apply to quantities other than area, while the second necessitates
understandings that both f ( c ) and ∆ x can be measures of quantities (for example force and
distance) and f ( c )∆ x is a measure of a derived quantity (work).
It could appear that these images (painted area and accumulated quantities) are the same.
We note in reply that they are only the same when one has constructed a scheme of
understandings within which area can represent a quantity other than area. Further, the intricacies
of understanding accumulation are often reduced to calculating products and limits without
understanding the significance of either. Without additional focus on constructing, representing,
and understanding Riemann Sums, there is little reason to believe that students will understand
accumulation functions as playing a central role in the FTC.
This paper presents a call for increased emphasis on the FTC as explicating an inherent
relationship between accumulation of quantities in bits and the rates at which an incremental bit
accumulates. Understanding this relationship entails a clear emphasis on covariation as a
foundational idea in calculus instruction. We make this call with awareness of the difficulties
involved in developing a well-structured understanding of accumulation functions, and that this
difficulty stands in contrast with the efficiency of teaching students to calculate definite integrals
as area under a curve. We believe that the benefits make the effort worthwhile. Understanding
f ( x ) dx
a
b
!
as an expression that yields the area bound by the x - axis and f ( x ) is efficient but not
generative. It supports a superficial understanding of f ( t ) dt
a
x
!
. We believe that understanding
accumulation so that f ( x ) dx
a
b
!
is simply f ( t ) dt
a
x
!
evaluated at x = a , where f ( t ) dt
a
x
!
itself has
a well-developed meaning, can be part of a coherent calculus that focuses on having students see
connections among rates of change of quantities, accumulation of quantities, functions as
models, limits, antiderivatives, pointwise and uniform convergence, and functions of two (or
more) variables. Though more work is needed to flesh out instruction that achieves this, we
believe that a focus on accumulation functions as discussed in this chapter will be central to it.
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