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.
5 thoughts on “The Abiding ‘Mystery’ of Calculus”
It goes back to Zeno, how can you go half way there an infinite number of times and still get there in finite time.
Limits as we approach infinity are counter to our natural sensibilities but the math suggests they may be real.
Having been taught quantum principles on my dad’s knee, the continuum hypothesis was not anything I believed in. So when I was taught calculus, I developed my own proof based on difference tables, not infinate limits. My high school teacher had no comment on my proof, she was in disbelief. So she brought it to the department head, who verified it, and I got an achievement award in mathematics for it, which my Mom has always cherished.
And I think that understanding is why I excelled in differential equations and the beauty of being able to transcribe nature without the mumbo jumbo.
Good to see you here. That’s a great story and somehow so typical of you! Well done – you had the hacker’s spirit even as a kid!
This sounds remarkably similar to what I experienced in my 11th grade. Almost identical. It was the very first Physics lecture wherein the teacher opened a hole new world to us, and it went 10 feet above my head. So much distress it caused that I had started thinking Science wasn’t my up of tea.
Similar to your experience, when I was able to comprehend this monster in the Maths classes, it caused a bit of embarrassment that I was frightened by a truly benign behemoth because of the way it had been made to pounce on me. Calculus was going to be my point-milking cow for the years to come.
I reckon the conclusion is nothing has changed much between the two periods (I am talking about early 2000s). I doubt if it would be any different now.
Thanks for the comment. I think you are right; this is still how calculus is introduced and taught.