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 Δ*x = *(*b-a*)* / n, *which determines each specific *x _{i}* value used to find any

*f*(

*x*) where

_{i}*x*varies from

_{i}*a*to

*a+*(

*n-1*)Δ

*x*for left rectangular approximation method (LRAM) estimates and

*a+*Δ

*x*up to

*a+n*Δ

*x*, which equals

*b,*for right rectangular approximation method (RRAM) estimates, as shown in the following figure.

Intuitively, one can see that as *n* increases, Δ*x *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, *x _{i}* varies from

*a+*Δ

*x/2*up to

*a+n*Δ

*x/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 *x _{i}* to

*x*

_{i}_{+1}, or

*x*to

_{i-1}*x*, as the notation may vary. Note that in the following diagram, the specific

_{i}*x*-value used to find its corresponding

_{i}*f*(

*x*)-value is denoted by

_{i}*c*, which is also shown as

_{i}*x**in many calculus texts.

_{i}The following figure more explicitly shows which value of *x* _{i }*corresponds to its

*f(x**value.

_{i})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 Δ*x.* 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.

I’ve noticed the same thing with my students. My gut feeling is that they do not really understand what f(x) means. They have definitely practiced “plugging numbers” (or substituting) values in for x to find the value of f(x) (or y) but I don’t think they understand sufficiently what this means visually. It is probably worth spending more time calculating values of f(x) directly from the graph, and then calculating those values from the function itself, and making the comparison.

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I agree, David, sadly. It is my hope that emphasizing a concept like this repeatedly through algebra 1, algebra 2, and precalculus, would allow more students to be able to recall, and apply, the concept. I believe that is the expectation with a vertically aligned team / department. Do you have any experience there?

In a much broader sense, for any subject, I wonder if we are attempting the impossible when we expect all students to develop skills where they may have no interest given their limited outlook / experience whereas they might recognize the necessity for the skill(s) at some point, ideally in time for it to benefit their life, and more fully invest themselves in the learning process. Students that do so earlier are likely guided strongly by their parents, hence, parental involvement in holding their children accountable for their learning seems to be a factor missing in much of the popular education reform debates of today.

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Dave – I think you are touching on a common theme in secondary math, the necessity of the ability to treat abstractions as objects. From the exposition above, there are several layers of abstraction. Given more time (never enough) you could see at what point are particular students are getting lost. One has to be fluent in the notations used here to follow the logic, and be able to think of delta x in several ways, as a difference, as a width of rectangles, etc. At the most basic level, do you think they understand the basic concept of decomposing the area under a curve into blocks? I wonder what students would say if you cut out these representations and gave them students and then did the dreaded “jigsaw” activity with expert and home groups. I can feel your readers wincing, and so am I a bit. But my point is that each of the pictures tells part of a story. How can we help student reconstruct that story, with peer and other resources. Over time, could they get increasingly quicker and more precise. Yes, I know, there is no time. I do get it.

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Great suggestions, Jack. Next year, I plan to include activities like you suggest to help students develop a more intuitive sense for the abstractions used in this and other concepts, especially as a means to connect between algebraic, graphical, numeric, and verbal representations. My gut is to conduct the activity after I initially introduce the concept then follow up afterwards with a more specific explanation. I’m still getting a feel for how best to sequence different types of instructional methods.

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In the blue pict, are your left and right Riemann sums mislabeled at the bottom? Could cause some confusion.

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Ooh, good catch! I have not used this figure in class, so it’s all good. But I need to practice what I preach in terms of attention to detail! Thanks for pointing this out. It is now correct.

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