Conceptual Calculus

When I was in high school, roughly a senior, it came to my attention that there was a set of ideas known as calculus that was somehow related to change over time (until then, I had always thought of calculus as an intimidating and far-off college class that made sure only intelligent, hardworking people became engineers and scientists). While reading Feynman’s Six Not-So-Easy Pieces I encountered dy/dx notation for the first time and was thoroughly stumped in my attempt to follow his derivations. I made it my goal to learn calculus by whatever means possible, and set off to the library. I read through a variety of textbooks, David Berlinski’s A Tour of the Calculus, and part of an OCW calculus course, but in the end my attempts to teach myself calculus were stymied. I learned the power rule, encountered the quotient rule, learned more than I ever wanted to know about the inner personalities of functions (thank you, Mr. Berlinski), and learned that derivatives could be used to describe how things change, but I was still no closer to understanding what a derivative was.

This breakthrough wouldn’t come until I was a freshman in college, taking Calculus I: Differential Calculus with a math teacher who had retired several years before. He spent the first lecture explaining how far the Greeks had gone without ever figuring out how to a) find the slope of a curve at any point and b)find the area inside any shape. He then explained how the Cartesian coordinate system enabled mathematical representations of the shapes with which the Greeks had been fascinated, and then digressed to the topic of limits. Within three or four lectures, we had a secant line of a curve described mathematically, then found the limit as the section of the curve marked off by the secant line approached zero. Before I knew what was happening, we had found what the Greeks in all of the glory only dreamed about. We had taken a derivative! I became so obsessed with the limit definition of a derivative that I wrote it constantly on chalkboards in random classrooms, on every page of my notebook, and possibly on a bathroom stall. . .

The rules that applied to derivatives and made them easy to compute didn’t interest me much (except for the chain rule – I was fascinated by the chain rule), but the limit definition opened up the whole world to me. I could now understand Feynman, and the rest of the world. Anything that changed, really. Of course, it takes the fundamental theorem to really put it all together, but that came soon enough.

Now I’m coming to my point. You’ve already impressed me with your patience, so hold on a little longer. It is my opinion, based on my experience with Feynman, that a truly well-educated person needs an understanding of calculus, at least of differentiation and integration and how they relate to each other. Conventional wisdom says, “most college students can’t even pass algebra; if you add calculus to the general education requirements no one will ever graduate.” The real difficulty in learning calculus, though, is in remembering the rules and knowing how and when to apply them. Differentiating isn’t too bad, but when you get to integration and all of the glorious guesswork involved there, it becomes taxing on even the most intrepid math student. My rebuttal is this: there is no reason for the average well-educated person to be proficient at integrating and differentiating (what one of my math teachers referred to as “computational ability”). In fact, due to the prevalence of computer algebra systems, I would argue it’s hardly necessary for engineers to be proficient at any of this, including solving DEs. So, what’s left? Merely the concepts.

Conventional wisdom would then say, “you can’t learn math without solving problems.” Can’t you? How many engineering students truly come to an understanding of the Wronskian after finding a dozen pairs of linearly independent solutions to a DE? They merely learn to apply algorithms efficiently, something a computer will always do far better than them. I’m not arguing that engineers shouldn’t solve problems, but I’m arguing that business majors, social scientists and the like should be introduced to the concepts of calculus without being expected to solve many problems.

My proposal is this: develop a “Conceptual Calculus” course that covers integration, differentiation and the Fundamental Theorem in a non-computationally-intensive way, to be taught in one semester to non-science majors who have a basic understanding of algebra. This course could (should) then be introduced as part of the core curriculum at universities (or liberal arts colleges, more likely) dedicated to producing well-rounded individuals capable of understanding topics across disciplines.

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2 Responses to Conceptual Calculus

  1. Josh L says:

    I’m going to err on the side that says that math of any higher sort is painful. I’m perfectly okay with being a one-trick-pony and nothing more. Leave me to my literature and psychology and the occassional venture into science, but advanced math makes me die a little inside each time I encouner it :(. Still, you’ve got a good idea; it just terrifies me a little bit.

  2. AJ says:

    Agreed, for the most part, to all of it. The underlying concept is that this sort of higher mathematics teaches you to think critically, and, really, that’s the most important thing. Honestly, you could get along in life just fine (discounting the economic security of a college degree) without a lot of the classes you take in college, but schooling isn’t all about academics. There’s the idea of socialization and learning how to think, because, if you thought it was a conspiracy when you were little, you were right. School is society’s way of producing efficient members.
    Josh also has a point: those out there who consider themselves “one-trick ponies” would definitely have a problem with calculus. I know too many English majors who have trouble with Business Algebra and almost pee themselves around Trig. I can’t imagine them getting the hang of differential equations. Perhaps it’s the expectation that since they’re English majors, they can’t do math, so they defeat themselves before they even try? Or perhaps, the conventional teaching methods of math don’t make sense to those of a more literary leaning, so if the methods were more tailored. . .? However, your point about the underlying knowledge of calculus– thinking critically and applying it– is a very incisive point.

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