One of the hardest parts in learning math is vocabulary. I mean, “isda”? “Ibon”? “Punung-kahoy”? What is it, Tagalog?
Wait, sorry, this is Tagalog.
Still, math words can feel like a foreign language. And they are much more difficult to acquire than Tagalog or English terms. Think about ways to learn a new word to see why. A word like “cat”.
First of all, you can just take the meaning from the context.
Or second, someone can just tell you the meaning.
However, there is a situation where learning a definition takes extra effort: if you have no experience with the object to be defined. In that case, as you learn the definition, you don’t have to just put a name on something you already know.
You need a more thorough introduction.
It is in this third scenario that math words belong. You name ideas that are invisible, abstract and yet highly precise. The “derivative”, for example, is something exotic and ethereal – even more so than cats.
That brings me to my point: our math culture brings this exactly backwards.
We tend to define a new term abstractly and wrap it in high-minded language like purple garments – while no one has any idea who is under the robe. A much better method: examine motivational examples and then Give definitions.
How do you define “cat” without a cat? The fact is, you can’t.