Illustrated · 3 min
An anecdote
This year, I took a Calculus class, with no prior experience with the subject. Since the gory details of the class itself will bore you (they bored me), I will summarize my experience of the class in one statement.
Calculus was a very interesting subject, and calculus class was the most boring thing on the planet.
So you ask, but why? Well, for one thing, I didn't truly learn calculus from class ...
From the beginning of the year, I had tried not to listen directly to the teacher and accepting whatever she said as utterly, completely true. Instead, I tried to parse and relearn the concepts my own way: limit, derivative, integral, etc.
I had always believed that calculus, despite the extreme formalities of the notation, could be understood intuitively, by anyone. And I still stand by that statement.
Not only that, but it helped out a lot. My grades in that class were stellar, and not because I studied more than anyone else (I chose to do competition math, which has almost no calculus).
The best way to explain it is that, for example, I thought of derivatives as change. They thought of derivatives as giant, complicated formulas involving fancy, shiny new "concepts" like limits.
Sure, they talked about "change" and whatnot. But only in name; their true grasp of the concept was still those massive formulas. They were too good for intuitive understanding.
A Case Study
I never let myself get obscured by the super technical definitions of slope and whatnot, let alone the infamous "limit definition." For me, derivative is rate of change, and that is that.
At first, my classmates' (and teacher's) paradigm of formulas won. They could easily do computational stuff, accurate to the decimal! The derivative of ? Easy! Just plug it all in.
But soon, they were clearly bogged down by all the confusing definitions and theorems.
For example, take the Fundamental Theorem of Calculus.
In layman's terms, it goes like this: the change in the accumulation (or integral) of a function is the value of the function.
With that statement, even a proof can be intuitive, and if you understand the proof, the concept becomes much easier to understand. So let's take it apart.
What exactly is the change in the integral? An integral is all about summing the values of a function. It is represented with the blue bars. Of course, the rate of change of the integral is the red bar; you have to keep adding on new red bars.
Of course when the value of the function is big, the red bar is big, and the area changes more. Similarly, when the function is small, the area changes less. In fact, the area changes with exactly the area of the function, since the function determines the height of the red bar.
We can carry this further. The function is the change in the accumulation, or in calculus jargon, the function is the derivative of its integral. This establishes that the integral is the inverse of the derivative. (Well, almost. But the details aren't that relevant here, and you are welcome to search it up here.)
For the other students, this was no symphony of imagination that ended with an explosive new revelation. It was just another slew of definitions; apparently, the integral is something involving limits and stuff. And they had to recite the First Fundamental Theorem of Calculus verbatim to "understand" it better.
Of course, only more misery awaited the people who understood calculus as a giant soup of definitions to memorize. As calculus class marched on to more exotic locales, their heads were spinning, and couldn't remember all the topics. All because they couldn't truly understand the topics their own way.
(Ironically, the teachers actually stressed not simply memorizing. But even they couldn't really do anything but memorize.)
But let's not end on a sad note. You can escape this, and all it takes is a few minutes of looking at and rereading the topics, and no small amount of thinking.
And it's not just calculus. You can approach any subject this way, even the non-math subjects! Really, each subject is the product of a framework of a few powerful, simple, ideas, which you can understand your own way.