Happy 3/3 everyone!
I just graded a bunch of proofs that √3 is irrational. The proofs had a lot of holes in them. This didn’t surprise me too much, in large part because the students weren’t math majors; rather, it was for a liberal arts math class taken largely as a gen ed requirement, and the whole proof by contradiction thing is really pretty scary and abstract for most people the first time around under the best of circumstances.
But actually, even when I’ve assigned this to math majors, they struggle. They can have the proof that √2 is irrational right in front of them, be instructed that instead of even numbers they want to look at multiples of 3, and despite my Find and Replace instructions, they still don’t understand what to do. The most common mistake is to replace “even” with “odd”.
In some ways this doesn’t surprise me, but in some ways it does. Why is it such a conceptual leap to go from 2 to 3? It’s a HUGE leap for many people. And so I was pondering this while grading, and Batman suggested it might be because we have a special word for “divisible by 2” but don’t for “divisible by 3”. So you get, what, 10 years of reinforcement that there is just this one special way to divide the integers, and it doesn’t generalize.
What we need is a new word for these numbers.
And fortunately we have one: threeven. So 0, ±3, ±6, ±9, … are all threeven, and the rest are…umm, not. (Maybe we need two new words). This word isn’t mine or even Batman’s; it actually was suggested by one of his students in response to this exact same problem.
As a bonus, it generalizes: there’s fourven, fiven, sixen, seven-en (sev-en? )…as far as you want. Which, admittedly, might not be very far but it still makes for a smoother sounding proof.
Happy threeven day!