Quantitative Reasoning with Definite Integrals
By: David Bressoud @dbressoud
David Bressoud is DeWitt Wallace Professor Emeritus at Macalester College and former Director of the Conference Board of the Mathematical Sciences. davidbressoud.org
The April 2023 issue of the International Journal of Research in Undergraduate Mathematics Education (IJRUME) is devoted to “The Teaching and Learning of Definite Integrals.” It contains seven informative articles spanning how students understand definite integrals, the progression of student learning about these integrals, how to help students reason with definite integrals, and the potential for confusion at the intersection of mathematics and physics. In this column I will discuss two of the papers that look at the nature of quantitative reasoning—as expressed in my April column “What is Quantitative Reasoning ?”— in the context of modeling with definite integrals.
In “Approaches to Integration Based on Quantitative Reasoning,” Steven R. Jones and Robert Ely survey the research literature on the teaching and learning of definite integrals, separating them according to two broad approaches: Adding Up versus Accumulation from Rate. Adding Up is probably the more familiar approach, frequently introduced by means of geometric questions such as finding areas, volumes, or lengths of curves. There is a target value that one approximates by chopping the geometric object into small pieces, adding them up into a Riemann sum as an approximation to the desired value, and then taking the limit as the size of the pieces approaches zero.
Accumulation from Rate focuses on the integrand as a rate of change with the goal being to find the total accumulation. Again, the situation must be chopped into small pieces, but the difference is that we are thinking about the problem in terms of covarying variables such as time and distance and focusing on accumulations over small intervals. In fact, time is almost always, at least initially, one of the variables. This is the approach chosen by Pat Thompson and his group. It leads naturally to the concept of an accumulation function, a definite integral with a variable upper limit, and from there easily into the Fundamental Theorem of Integral Calculus (See “Restore the Integral to the Fundamental Theorem of Calculus,” Launchings, May 2009).
For mathematicians, there is no distinction. We understand how the two are different manifestations of the same idea. You see this very clearly in Newton’s Principia. Almost all his integrals involve accumulation given a rate of change, but his justifications are entirely geometric.
The problem is that students do not always make this connection. Pat Thompson has illustrated this with a comment from one of his students. Having learned that integrals can be thought of as the area under a curve and then later taught that integrating velocity yields distance, the student complained that he could not see how an area can give you a distance.
Student difficulty connecting the two approaches justifies the considerable depth into which Jones and Ely explore the literature on each approach. They look at three modeling situations and explain the quantitative reasoning that lies behind Adding Up versus Accumulation from Rate. They effectively illustrate how strict adherence to Accumulation from Rate can complicate many modeling situations. I believe that it is important to start with Accumulation from Rate. The first time students see a definite integral it should represent an accumulation function, with a variable as the upper limit. But at some point early on the connection needs to be made to Adding Up. I believe that the transition from Accumulation from Rate to Adding Up is much easier than the other direction.
In “Emergent Quantitative Models for Definite Integrals,” Michael Oehrtman and Courtney Simmons examine student work on five modeling tasks that ultimately lead to definite integrals, teasing out how each individual or team went about identifying the appropriate quantitative variables and their relationships. Their paper is well worth reading just for the detailed record of the variety of ways that students thought about these problems and went about solving them.
Among the tasks they set their students were
Suppose a 10-m chain with a total uniform mass of 15 kg is freely hanging from the roof of a building. Write an integral that represents the total energy required to lift the chain to the top of the building.
The gravitational force between two particles of mass m_1 and m_2 at a distance r apart is F = G m_1 m_2/r^2 where G is a universal constant. Write a definite integral that that gives the gravitational force between a thin uniform rod of mass M and length l and a particle of mass m lying in the same line as the rod at a distance a from one end.
When pollen from red cedar trees is released from their cones it travels through the air. Pollen from a mature tree settles on the ground with an estimated density of delta(r) = 37∕(10 + r) g/m^2 a distance r meters from the tree. Write an integral that gives the mass of pollen distributed within 100 m from a mature tree.
The kinetic energy of an object with mass m and constant speed v is K = (1/2)mv^2, at least in the case where the entire object is moving at the same speed, v. Suppose a 0.1 m rod has a mass of 0.03 kg with uniform density. It is rotating around one of its ends at a rate of one revolution per minute (like the second hand of a clock). What is the total kinetic energy of the rod?
Creating the definite integral involves three levels of modeling (see Figure 1). An important reason for separating these levels is that students often invoke different quantitative meanings for each one. In a personal correspondence, Michael Oehrtman pointed out that instructor recognition of this can help attend to the different ways of reasoning that students employ, making it possible to support their learning more effectively. I will not get into the subcategories of each of the models. To fully appreciate these terms, it is helpful to see how they are used in the context of the student work.
Figure 1. Emergent Quantitative Models for definite integrals. Oehrtman and Simmons describe three types of basic models and five types each of local and global models.
The “basic” model consists of a student’s quantitative interpretation of the fundamental relationship such as “rate times time” or an “inverse square law.” The first step is the recognition that the basic model is insufficient because one or more of the quantities vary within the given situation.
The second step involves the creation of a “local” model by partitioning the situation into disjoint regions in which the variation is sufficiently limited that a local version of the basic model can be employed.
The third step involves accumulating the results of the local model to build a “global” model that could be a Riemann sum or could move directly into an integral representation, either way reflecting what the authors refer to as a “parts-of-a-whole” symbolic form.
These three steps are a reasonable description of how students are taught to approach modeling with definite integrals. What I find thought-provoking in this paper is that Oehrtman and Simmons continue with four additional pieces that get at the quantitative understandings that underlie this process and support student comprehension of how and why the model works. To make certain that I get them exactly right, I quote them.
An image of reconstructing a local model consistent with a differential form f (x) ⋅ dx , including (i) conceptualization of a single variable, x, on which all other variation within the basic model continuously depends, (ii) a single factor within the quantitative structure of the local model corresponding to the differential dx or Archimedean Δx , (iii) a functional representation of all other factors within the local model corresponding to the integrand f (x) , and (iv) the interval [a, b] of values assumed by x.
A conception of the accuracy of the local model generally increasing as the magnitude of the differential quantity decreases.
Coordination of refinement of a global model with a limiting value of the exact desired quantity. Note this limiting process may be conceived as applied to the global model (resulting in the normative interpretation of a limit of Riemann sums) or to the local model (resulting in a “nonstandard” interpretation of a continuous sum of infinitesimals).
A symbolic form for definite integrals that associates the symbolic template, the integral from A to B of C, with a conceptual schema in which C is a differential form representing the local model, aligned with the quantitative structure of the basic model, and whose independent variable varies from A to B.
Modeling with definite integrals is arguably the most useful tool that is made available to students in the first year of calculus. Too often we breeze through it, leaving students with the impression that the most important thing they learn about integration is how to find antiderivatives. Those toiling in undergraduate mathematics education research know better. It is heartening to see the emergence of this work.