One of the hardest parts of learning math is the vocabulary. I mean, “isda”? “Ibon”? “Punung-kahoy”? What is this, Tagalog?
Wait, sorry, that’s Tagalog.
Still, mathematical words can feel like a foreign tongue. And they’re much harder to acquire than terms in Tagalog or English. To see why, consider how you might learn a new word. A word like, say, “cat.”
First, you might just pick up its meaning from context.
Or, second, someone can just tell you the meaning.
But there’s one situation where learning a definition demands extra effort: when you’ve got no prior experience with the object being defined. In this case, learning the definition is not simply a matter of affixing a name to something you already know.
You need a more elaborate introduction.
It’s this third scenario where mathematical words tend to belong. They name ideas that are invisible, abstract, and yet highly precise. The “derivative,” say, is something exotic and ethereal—even more so than cats.
Which brings me to my point: our mathematical culture gets this exactly backwards.
We tend to define a new term in the abstract, draping it in high-minded language like purple garments—all while nobody has any idea who’s under the robes. A much better method: explore motivating examples, and then give definitions.
How do you define “cat” without a cat? The fact is, you can’t.