Encapsulation
As I go throughout my undergraduate studies, I have gotten really good at digging deeper until I completely understand a subject; however, when I observed around me, I realized that I am in the minority. Often times, students only desired to know how to get from question to answer without any intermediate thinking. When trying to hypothesize an answer, I cannot think of a definitive one. However, I know one possible contribution: encapsulation[1].
The most famous example I came across 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, it's hardly useful information. As a matter of fact you might think of it as piece of trivia — until you see the amount of people that miss it on a quiz or test. And entry-level Calculus is not the only place this is relevant; physics, thermodynamics, mathematics of any kind, computer science, and the like.
A question you might be thinking, 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 conceptually do not 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. ↩︎