Enos, Oona, sqrt(3), and Aaronson

My darling does crossword puzzles and sometimes asks my help:

Darling: Bill, Slaughter in Cooperstown- whats the answer?

Bill: Enos. There was a serial murderer in a town named Cooper, and he always wrote on his victims Eternity's Not On Sale. So he got the nickname ENOS. Nobody ever figured out what that meant and he was never caught.

Darling: Another clue: Chaplin's wife

Bill: Oona. That's latin for minister's spouse. And in those days ministers were always men.

Darling. Okay. Another clue: Log man. Begins with an N.

Bill: Napier, a famous lumberjack.

Many of my readers know that, while the above answers are correct,
the reasoning behind them is fictional. In fact, the entire story is fictional.
But it IS true that when Darling sees those clues in crossword puzzles
she knows what the answer is without having any idea that Enos Slaughter is in
the Baseball Hall of Fame in Cooperstown, that Oona was the name of Charlie
Chaplin'sfourth and last wife, or that John Napier is regarded as having
invented logarithms (the history of such things is always murky).
She has MEMORIZED the answers (from years of doing crossword puzzles)
without really UNDERSTANDING the answers.

When I show my students the proof that sqrt(2) is irrational and ask
them to prove sqrt(3) is irrational, I often can't tell if they truly
understand the proof or are copying a template proof of sqrt(2).

Sometimes (and usually on harder problems) some small slip will
tell me that they are just copying since they didn't quite know
what to change. Or they may miscopy.

However, there is a deeper question here. If students memorizes the
template for the proof that sqrt(n) is irrational, and uses it correctly,
then do they understand or have they just memorized? The distinction can be
hard to discern and may not even exist. One real test is if they understand
why the same template fails on sqrt(4). For harder problems there may be
other ways to tell--- having to do with when the proof fails.

Incidentally, the reason the crossword clues above come up so often is
that the answers have many vowels in them. So, one way to immortality is
to be mildly famous and have a name with a large percentage of vowels.
Let see- gAsArch: 28% vowels not so good. fOrtnOw: The same (no wonder we coblog!),
also not so good. AArOnsOn: 50% WOW!! May his name adorn crossword puzzles for
many years to come!