Learning Curves

How Differential Calculus and Reading Employ the Same Skill

We build too many walls and not enough bridges. -Sir Isaac Newton

It’s fairly widely known, and confirmed in an old NFL Films piece, that former Pittsburgh Steeler Lynn Swan studied ballet in the off-season. I’m sure some of his teammates who spent their free months flipping over giant tractor tires and running on hot sand asked him, with respect to the ballet: why? When you see clips of Swanny reaching for balls on go routes interspersed with footage of him in tights pirouetting and striking Arabesques, you get it.

Keep this in mind while I transition from the gridiron to the proving ground for another contact sport: the college composition classroom. From time to time, a student’s excuse for poor performance in a reading and writing class is that they’re a “math person.” Unfortunately, by “math person” they often mean someone who’s good at adding and multiplying. But on occasion I’ve encountered the odd engineer or serious science student who’s a math person in the more thorough sense of the term-someone who can manipulate abstractions effectively, for one thing-and I’ve had some success explaining how the calculus they’re learning might help their reading.

Let me describe the relationship between reading and differential calculus as I see it. I have to ask you to stipulate a sort of equivalence between a curve in 2-space and a text[1], insofar as a text is something to be studied when reading and a curve is something you can study when doing calculus.

One thing to define up front: specific definitions for the ordinary words “says” and “does.” At my school, we use these words in reading classes to help move students from basic comprehension toward more rigorous analysis. An example of basic comprehension is something like a 5th grade book report, wherein you summarize a book and maybe add a sentence at the end concerning whether or not you liked it. These assignments ask you to understand and regurgitate what a text “says.” Identifying what a piece of text (a book, a paragraph, or even just a sentence) “does” requires you not only to understand what it says but also identify its function in terms of, for example, building the argument or furthering the plot. When describing what a piece of text does, students have to know and use abstract categories so they can employ accurate descriptive verbs: e.g. “this paragraph reconstructs the events of that fateful evening” or “that paragraph provides three examples for why the new water line will be a burden on taxpayers.”[2]

Now, we can shift to differential calculus. Taking the derivative of a function allows you to determine the slope of the tangent line for the function’s corresponding curve at any given input value. In more plain English, the derivative of a function allows you to determine how a curve’s trending (its rise over its run, its behavior) at a given point. Trending and behavior essentially = doing. The initial function equates to what the curve is saying and the derivative describes what it’s doing.[3]

Just take a quick look at two blue curves:

For the curve on the left, we can see that when x=0, the tangent line (in red) is parallel to the x-axis, and therefore its slope is 0[4]. What’s the curve “doing” at that point? Well, it’s going neither up nor down. It’s flat. The curve on the right has two points at which the slope of the tangent would be 0: somewhere between -1 and -2 and somewhere between 1 and 2[5] (see these tangent lines in red).

These two blue curves look quite different, but at some points they’re doing the same thing. Now consider a couple of argumentative texts that seem wildly different. One, say, argues that the World Bank’s policies amount to usury and the other claims that Blue Velvet is really a retelling of Great Expectations. However different, these texts will probably do some of the same things: somewhere in both will appear, explicitly or implicitly, the central claim being argued. They also may respond to counterarguments or provide evidence and support. Again, these textual doings correspond, in my conceit here, to a certain slope of a tangent which occurs in two curves that look nothing like one another.

If you’re interested in an optional elaboration, see this note[6]. Otherwise, proceed.

Adding one layer of abstraction–by focusing on “does” in reading or the 1st derivative in calculus–helps you compare one text(curve) to another, which allows you to more effectively and efficiently compare, categorize, assess, respond to, contrast, contextualize, etc. them. All those verbs describe, basically, what we nowadays consider critical thinking, which, it hopefully has been demonstrated, doesn’t happen solely in the territory of humanities and reading and writing courses, because the learning curves for reading and calculus are in fact more similar than most may initially believe.


[1] Of course, in all sorts of ways, a curve and a text are different-and I don’t just mean in terms of looks but in terms of what they actually are since a curve can simply be a curve and/or it can represent something in the physical world such as a projectile’s lateral path over an interval of elapsed time, while a text can’t be, in the same way, representative of anything other than itself, really (though probably some post-structuralist texts try (and perhaps succeed)).

[2] Just another quick example of this distinction using a somewhat famous first line:

“Happy families are all alike; every unhappy family is unhappy in its own way.”

We could easily describe what this says by paraphrasing it: all contented nuclear units are similar while malcontented families tend to be different with respect to the source or manifestation of their malcontentedness.

But what does it do? It:

  • Delivers an aphorism.
  • Introduces the novel’s general subject: families.
  • Brilliantly encapsulates what Tolstoy considered to be the raison d’etre for this novel and, for that matter, for any truly great novel, which is to look at the concrete particulars of some people’s lives that are interesting because of the challenges they face.
  • Delivers a claim, presented as fact, to introduce the authoritative authorial tone Tolstoy will employ throughout the text.

There are probably dozens of claims we could make based on that first sentence and what it’s “doing.” And this is how reading gets awfully messy. But I’m here to argue that all these “doings” do have the similarity of being an attempt at abstract explanation of what the text at that place does and how it thereby fits into what comes before and after it.

By the way: In A Tour of the Calculus, David Berlinski claims that “Art is drawn irresistibly to what is singular, but mathematics is drawn irresistibly toward what is generic” (his emphasis). I don’t shy away from the fact that the sort of reading I’m talking about teaching here is more of the New Critic-mathematics sort than the post-structuralist-artistic variety.

[3] We’re limiting our discussion here to real-value functions-whose definition gets us into a larger and mostly fruitless discussion, but I will say that if we expanded beyond real-value f(x)’s, the incredible value of what I’m saying would be revealed with respect to the manner in which this works for post- or even post-post-structural texts with a large amount of pyrotechnics that we could equate with curves on the imaginary number line.

[4] Optional refresher: y=mx + b where m is the slope and b is where the line intersects the y-axis. In this case, y = -2, or more explicitly, y= 0x -2.

[5] Since this curve represents y = sin(x), I can tell you that these points are actually -π/2 and π/2.

[6] When you take the derivative of a function, which I’m arguing here describes the function’s (curve’s) tendencies, you get another function. Then, as I’ve mentioned, you can determine the instantaneous slope of the tangent (slope of the tangent = behavior) at any given input value. In reading, especially when dealing with a challenging text, it’s more common to figure out what a text is doing at several discrete points (say, in several paragraphs) and use those to start to sketch what the text does overall. Then, you can use this general statement of what the text does to check whether each paragraph actually contributes to the argument. Using an example from above, if you believe that what a certain text does overall is trace the World Bank’s failure in helping developing countries, but you discover a paragraph in the text about the Muppets, you had better figure out how that paragraph contributes to the overall argument (what it’s doing there), or you might have to reconsider what the text’s overall argument is.  Regardless, the idea here is that the derivative’s equation is the equivalent of a macro-does for an entire text.  And where in math you can use input values to get the slope at any point, in reading, you can use your macro-does as a way of checking yourself.