I walked around and noticed that all the kids had drawn and labeled a diagram, then proceeded to write down x(x + 5) = 42. I asked the class how we would find the area of a triangle, and they said, "Base times height!" And I said, "No, that's for a rectangle, not for a triangle." Within seconds, all the kids who were in my class the previous year shouted out excitedly, "--Divide by 2!!!" Meanwhile, the other kids who didn't learn the triangle area formula via its connection to the rectangle had puzzled looks on their faces; my hint that "base times height" is the procedure for a rectangular area held no apparent meaning to them with regards to generating a triangle's area formula. Rote memorization of "the simple things" is really a problem, because if you start memorizing intuitive things like this, you'll never be able to memorize everything.

The issue here for teachers (not a new discovery, obviously) is that once a teacher just

*gives*the students the formula, the students are no longer motivated to really understand why it works. Once a kid has a teacher who has taught them a certain skill by rote, their motivation to learn the intuition behind that particular skill completely disappears as the kids simply file it lazily into the "already learned, already can do" drawer in their mind. Please don't do this, as it's really hard to un-do down the road. For every concept that you rush through in order get to the "procedural" practice, those topics will never be properly understood, explored, and developed by the child. This not only robs them of the richness of mathematics, but also creates retention issues.

I know, it's not a new discovery by any means. But, I was reminded by this small incident, of the significance of exploratory learning.

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