On Understanding, Not Just Memorizing
Throughout my undergraduate studies, I've become good at digging deeper until I completely understand a subject. But when I look around me, I realize I'm in the minority. Often, students only want to know how to get from question to answer without any intermediate thinking. I can't pinpoint a definitive reason why, but I have one hypothesis: encapsulation[1].
The most famous example I've encountered is the derivative of any trigonometric function besides $\sin x$ or $\cos x$. As any student who's taken Calculus can tell you,
$$\frac{d}{dx} \left(\tan x\right) = \sec^2 x$$
However, most students couldn't tell you:
$$
\begin{align*}
\frac{d}{dx} \left(\tan x\right) &= \frac{d}{dx} \left(\frac{\sin x}{\cos x}\right) \\
&= \frac{\cos x \cos x + \sin x \sin x}{\cos^2 x} \\
&= \frac{\cos^2 x + \sin^2 x}{\cos^2 x} \\
&= \frac{1}{\cos^2 x} \\
&= \sec^2 x
\end{align*}
$$
At first glance, this seems like useless information. You might even think of it as a piece of trivia—until you see how many people miss it on a quiz or test. And entry-level Calculus isn't the only place this matters. This pattern appears everywhere: physics, thermodynamics, mathematics of any kind, computer science, and the like.
You might be wondering: why does it matter? If it's worth a couple of points on a test, is it that harmful? Maybe not—until you realize you don't conceptually understand the subject, only mechanically.
The next time you're learning something, ask yourself: "why?". The outcome might delight you.
- The act of enclosing in a capsule; the growth of a membrane around (any part) so as to enclose it in a capsule. ↩︎