One of my calculus periods continues to struggle mightily while learning new concepts, especially if they need to recall prior knowledge, or worse connect it to the new material. As an example, when starting to explore integration concepts, students learn to use finite methods to estimate areas under functions, as early mathematicians discovered.
The basic technique uses multiple rectangles to approximate the area under the curve of a function over the interval x = a to x = b as the figure below shows. Since the area of a rectangle is equal to its base times its height, we simply need to determine a specific value of f(x) for each rectangle used, where the number of rectangles is driven by n. In the figure below, n varies from 2 to 4 to 8. Note that Δx = (b-a) / n, which determines each specific xi value used to find any f(xi) where xi varies from a to a+(n-1)Δx for left rectangular approximation method (LRAM) estimates and a+Δx up to a+nΔx, which equals b, for right rectangular approximation method (RRAM) estimates, as shown in the following figure.
Intuitively, one can see that as n increases, Δx decreases enabling the rectangular approximation method to yield a better estimate for the area under f(x). One can also see in the figure above that the two different methods result in either an underestimate, or overestimate, of the area. For a monotonically increasing function LRAM yields an underestimate, while RRAM yields an overestimate. The converse is true for a monotonically decreasing function.
For rectangular approximations, the midpoint of the base of the rectangle may also be used. In this case, xi varies from a+Δx/2 up to a+nΔx/2. This is known as the midpoint rectangular approximation method, or MRAM. The following figure compares LRAM, RRAM, and MRAM, respectively, for the same function. Note that MRAM produces a better estimate than either LRAM or RRAM.
While figures like these, and example problems with specific values for a, b, and n for a given function, were shown to students repeatedly over a three-day period, they still had difficulty when it came to attempting the simplest of problems. When asked how to determine the height of any rectangle, many students seemed perplexed. They were challenged to see what value of x should be used to evaluate f(x) as needed to find the height of a rectangle spanning xi to xi+1, or xi-1 to xi, as the notation may vary. Note that in the following diagram, the specific xi-value used to find its corresponding f(xi)-value is denoted by ci, which is also shown as x*i in many calculus texts.
While my calculus students clearly recall that the area of a rectangle is the product of its base and its height, as drawn above, they struggled determining just how to find its dimensions. The most challenging aspect was determining the height for any rectangle.
Each of these students learned about functions in algebra 1, algebra 2, and precalculus, although I find the latter a misnomer. They all learned how to compute f(x)-values, or y-values, for any x-value and graph the function in the same subjects. They explored finding area through primary school, algebra 1, algebra 2, geometry, and precalculus. They saw several build ups of the concept, which emphasized all the parameters involved, such as f(x), n, a, b, and Δx. They saw how these come together when finding a Riemann Sum, as shown below.
When shown the Riemann Sum notation, each parameter was defined and discussed in detail, to include the Greek capital letter for sigma. In fact, they really did not need to use this notation, per se, to solve the problems given. They did need to have followed the conversation and reasoning to develop a conceptual understanding of how adding up rectangles beneath a curve provides an estimate for the area under the curve.
Yet, they acted as if they were dealing with fire for the first time, as if they were afraid they would be singed by the flames. The class environment is such that most, if not all, seem to feel very comfortable knowing I support their learning, so I do not believe it is the classroom, or me. I do continue to sense that many are afraid of looking foolish to their classmates. However, even the most outgoing and participative students were reserved on these problems. Could it be its the last marking period of the semester? Perhaps compounded by the fact that two-thirds of the class are seniors?
Whatever the reasons, I am surprised this topic was so challenging for these students. When I covered the same material with my AP students four months ago, they did not seem to struggle anywhere as much as this class. I will ask students what seems so difficult for them, however, when I checked for understanding during the three days of instruction, most indicated understanding. Could it be that the “check for understanding” methods do not faithfully capture understanding? Out of necessity, they only probe narrow aspects of a problem, otherwise, it would take too long to use them. I remain puzzled as to why these students struggled so.
Perhaps the pillow will help me tonight.