Thursday, August 13, 2015

Apples and oranges: a tutoring anecdote

So I was sitting across the table from a brave young soul who has recently entered the dark swamp of fractions. We were working our way toward the concept of the common denominator, and why you need it for some operations  (like addition and subtraction) and not for others. And in the midst of it, he suddenly asked, "Is that what my teacher means when she keeps saying I can't compare apples and oranges?"
"It might be," I said, temporizing like any good tutor trying to figure out the situation while not saying anything stupid I'll regret or have to walk back later.
A little more back and forth conversation revealed that this was apparently what this particular teacher said whenever a student attempted to add or subtract fractions without putting them into common denominators. It was also what she said in any other context in which she thought students were improperly lumping things together; it was her general warding-off spell against analogies and any time students said one thing was "just like" another. Fifth-graders are natural lumpers; this teacher was a natural splitter, and whenever a lumpy parade started, she would drench it in splitty rain.
"Well," I finally said, "comparison is an operation in informal logic, just as addition is an operation in arithmetic. You can add four apples to five oranges, but what you get isn't apples or oranges; it's nine pieces of fruit. The operation is still meaningful, it's just the result is not denominated -- which is a Latin word for 'named' -- in the same terms."
That led us into a conversation about how a kid with two dogs and three cats has five pets, and a family with five pets and three chickens-for-eggs has eight animals, and the idea of "common" started to emerge, along with "denomination" meaning naming a kind.
At which point he asked me, "So when I'm trying to figure out how to add three sevenths to two sevenths, is that why you ask me 'Well, what's three sheep plus two sheep? What's three stars plus two stars? What's three cups of spaghetti plus two cups of spaghetti?'"
"That is," I agreed.
"And the idea is that sevenths are just the same as cups of spaghetti?"
"Well, not exactly the same," I said. "But they both have in common that you can count them. So if you denominate them the same way, you can add them, but there's no way to denominate sevenths so you can eat them. You can only do an operation on what they have in common, with respect to the operation."
We had a nice little digression about what "with respect to" has in common with "respecting the teacher," which I must not have messed up too badly, because just then the key finally turned.
Suddenly he was explaining everything to me, reinventing a good portion of the Theory of Logical Types, leaping back to the idea that every rational number has an infinite number of names because it can be reached by an infinite number of paths. (That is, 17, 10+7, 8+9, 51/3, XVII, and "the seventh prime" are all names for the same number, and each name is also directions for assembling that particular number—"17" is just more compact than most of the others).
And we zipped through the rest of fractions that day.
Toward the end, we even got as far as the idea that different operations have different input requirements and different expected outputs, and thus to compare operations, you have to compare comparable inputs and outputs.
That led him to finish up with a passionate comparison to Pokemon, which I confessed I was unable to follow. I said as much, and admitted that this made me feel some sympathy for his teacher.
"Oh, you're nothing like her," he said. "That would be comparing apples and oranges."