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."