Neurolinguistic hackers (would-be) sure wig out over some bakeem* things.
I just was thinking about how to present the "When you have this stuff memorized, you will start to recognize patterns without thinking about it." I was punching numbers into the microwave, and thought about how I'd figured that the most effective way for me to heat the item for one and a half minutes was for me to input 90 seconds. How I automatically knew that thirty was half of sixty. Because I'd done that so much. Because I recognized it without thinking about it. And then I thought: Worm. Because, when we say that Little Fayoumis should always think about what he is doing, and then we tell him that it is a virtue to memorize things that have been proven so that he will not have to think about it -- what does that teach him to do? Does that teach him to not think about anything he is told to memorize? Does that teach him that when he is exposed to something a thousand times, he should turn his brain off? That the more a new idea is presented to him, the less it bears examination?
And how do we define things that should remain unchanging? Simple enough to say, "If it is established that X + Y = Z, every single time, without exception, then it is safe to memorize X + Y = Z as a fact." But, where do we draw the line to a kid who hasn't input a significant fraction of the world yet, what is solid and what is not? He's going to encounter negative numbers, and often they zonk out the minds of kids who don't live in subarctic climates. As teachers, we have the responsibility to say, "Learn this for now, but know that there are refinements that we'll introduce later, but now you can accept it, and we'll teach you where the exceptions are later. But first, figure it as if it doesn't vary."
When one's accepted that something is Always This Way, Forever And Ever Amen, one tends to ignore data that contradicts the hypothesis. Like, 2+2 = 5. Where the 2s are abstractions, that's absolutely fallacious, given something like base 10, not base ... erm ... the base where 2 + 2 = 5 is true. But for real-world situations, like if there are mice involved, 4 or 5 may morph into 25 with no warning. Granted, it takes a very closed mind to argue that two plus two equals four, when confronted with the squeaking, stinking reality of 21 baby mice and four adults in one suddenly-too-small cage. But surely there are more subtle applications, such as social rules or wierd physics moments or economics or any other science that deals with things as they are, while hemmed in by expectations of things as someone once thought that they probably were or should be.
*bakeem: from a created language, meaning odd, unusual, unique, weird.