Our initial exposure to an idea shapes our intuition. And our intuition impacts how much we enjoy a subject. What do I mean?
Suppose we want to define a “cat”:
- Caveman definition: A furry animal with claws, teeth, a tail, 4 legs, that purrs when happy and hisses when angry. . .
- Evolutionary definition: Mammalian descendants of a certain species (F. catus), sharing certain characteristics. . .
- Modern definition: You call those definitions? Cats are animals sharing the following DNA: ACATACATACATACAT. . .
The modern definition is precise, sure. But is it the best? Is it what you’d teach a child learning the word? Does it give better insight into the “catness” of the animal? Not really. The modern definition is useful, but after getting an understanding of what a cat is. It shouldn’t be our starting point.
Unfortunately, math understanding seems to follow the DNA pattern. We’re taught the modern, rigorous definition and not the insights that led up to it. We’re left with arcane formulas (DNA) but little understanding of what the idea is.
I fully “buy into” this type of thinking. I used to give the students the formal definition and try and break it down. For the past couple of years I’ve taken a much more informal path with definitions. It seems like the students are better with building the formal on-top of the informal instead of vice-versa.