So I first asked M to just talk me through her reasoning. We then took some time to translate that reasoning into proof steps, complete with all of the nitpicky details. It probably took 9 or 10 steps.

And to the follow-up questions, she simply answered: "No."

👑

[4/4]

The message sent here is, "The way you thought about this isn't good enough." 😡😡😡

Sure, there's a time and place to talk about streamlining. But that time and place is NOT when they are first learning to write proofs! [3/4]

My daughter (M) took that to mean that she needed to come up with a four-step proof. She hadn't even done a longer one, because she'd read ahead. That's where she was stuck. She couldn't figure out how to prove it in four steps.

And honestly, I didn't see how to do it in four steps either. [2/4]

Correction: It's one checker, telling me to use the full word "October," but for three different reasons: "Word choice," "Spelling," (?!?) and "Grammar." 🤦

YoU gEt SuMmErS oFf

The only piece I'm a little iffy on is "so every vertex has degree 3." How does that follow?

Wait... a vertex can't have degree 2... and the sum of the degrees is twice the number of edges...

Never mind, I got it! 😁

(Is it possible to prove it without first finding the individual values of a and b?)

Interestingly, I have students who would agree that the sets of even and odd _natural numbers_ are the same size, but quibble over whether the even and odd _integers_ are the same size (see the second image).

The one-to-one correspondence idea kind of arose naturally when we were comparing ℂ to ℝ².

Right. I feel like there exist concrete examples of the proper-subset model that are either explicitly self-contradictory or just don't "feel right." I'll have to think about it.

I’d rather not introduce a new object (“infinite string”). I’m thinking about temporarily limiting the scope to infinite subsets of N. How do they compare?