The Abiding ‘Mystery’ of Calculus

I first encountered calculus in the eleventh grade. A mysterious symbol had made an appearance in my physics text–in the section on dynamics–as we studied displacement, velocity and acceleration. What was this ds/dt thing anyway? I had, at that point in time, never studied calculus of any variety; to suddenly encounter a derivative was to be confronted with mystery of the highest kind. I asked for explanation and clarification; I received less than satisfactory obfuscation in response. Something about ‘instantaneous rate of change’, whatever that was.

A few months later, having encountered differential calculus in the mathematics syllabus, I was considerably, if not totally, edified. Functions, curves, graphs, tangents; somehow, I was able to partially relate the material we had studied in the physics class to this mathematical paraphernalia. And then, a little later, in the twelfth grade, having encountered integral calculus and then differential equations, other pieces of the puzzle fell into place as the relationship between mathematical apparatus, the models they comprised, and the physical world became a little clearer.

But as the story of my introduction to calculus–an abrupt exposure to its application and formalism in dynamical analysis–shows, calculus had an initial air of mystery that took some shaking. It had been suddenly introduced as a mathematical tool to enable grappling with a problem of physical mechanics, but the formal insights that lay at its core–especially the concept of a limit–were decidedly unfamiliar. More to the point, its use seemed utterly gratuitous; I could not see how my understanding of the physical details of velocity and acceleration had been improved in any way. And even when I did study differential calculus, I felt as if I became an expert manipulator of its many recipes and techniques well before I understood what my activity entailed. Syntactical manipulation, the transformation of one set of mathematical symbols into another according to a well-specified algorithmic procedure, was easy enough; understanding what those meant, and how they underwrote our understanding of the world of becoming and change, was a different matter.

We were science students in high school, ostensibly preparing ourselves for careers in engineering, medicine, and perhaps even basic research in the physical sciences; calculus was one of our most important tools. But we remained befuddled by its place in the conceptual apparatus of our studies for a very long time. This should be, and was then, a matter of some perplexity, especially when I consider how enlightened I felt when I better understood its place in making a changing world comprehensible.

Years on, when I became embroiled in debates over curricula in computer science undergraduate education, it occurred to me little had changed; many students remained perplexed by calculus’ importance in their education, by its most foundational presumptions and applications.Nothing quite exercises pedagogues like mathematics education, and in their catalog of perplexities, the failure to properly contextualize calculus should rank especially high. I’m almost tempted to describe it as a civilizational failure, so convinced am I of the judgment of any extraterrestrial visitors when confronted with this peculiar combination of indispensability and incomprehensibility in our epistemic scheme of things.