Calculus and the Mind's Eye

My wife and #3 daughter are teaching/tutoring kids from another home schooling family this afternoon. One of the subjects is arithmetic, a useful skill but not one of my favorite academic disciplines.

Arithmetic isn't quite the same as mathematics, and that's almost another topic. My wife is the math whiz of the family: no surprise, since she's also the one with a degree in computer science. I'm more like my father: much to the frustration of school teachers and administrators who looked at my aptitude test scores, which showed considerable mathematical ability; and my report cards, which definitely didn't.

I think part of the explanation is how our brains 'do the math.'

My father told me about a test he took in calculus class. One of the test's questions described two cylinders which intersected. Students were supposed to pick the intersection's volume from a list of numbers, showing the calculations they used to arrive at that number.

While everyone else started using calculus to determine how much volume the cylinders shared, my father imagined what the cylinders looked like, looked at the cubage where they went through each other, and checked off that number: which was one of the possible values shown on the test.

Then he set about writing down what he hoped was the set of equations that would give that answer.

When the test came back, he'd gotten that problem wrong.

He was the only one in the class to get the right result for the intersection's volume: but his written math wasn't even close to being correct.

I'm the same way. My mind's eye has excellent vision, but I never passed calculus class.