Matt Enlow
@unsolvedmre.bsky.social
Mathematician, math teacher, math enthusiast, math/art-ist. Universal humanizer.
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]
And to the follow-up questions, she simply answered: "No."
👑
[4/4]
October 28, 2025 at 4:01 PM
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]
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]
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]
October 28, 2025 at 4:01 PM
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]
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]
And honestly, I didn't see how to do it in four steps either. [2/4]
October 28, 2025 at 4:01 PM
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]
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." 🤦
October 17, 2025 at 12:41 PM
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
October 8, 2025 at 8:18 PM
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! 😁
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! 😁
October 8, 2025 at 2:57 PM
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! 😁
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?)
September 27, 2025 at 6:44 PM
(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).
September 27, 2025 at 6:28 PM
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.
September 27, 2025 at 4:25 PM
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.
September 27, 2025 at 4:24 PM
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?
September 27, 2025 at 4:08 PM
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?
After each class I tried to summarize the conversation on a Page in Canvas. Here's where we currently stand. Now I have to think about how best to proceed—and I want to do so WITHOUT any investment in them landing at the "traditional" answers! Which will be tough.
September 27, 2025 at 2:10 PM
After each class I tried to summarize the conversation on a Page in Canvas. Here's where we currently stand. Now I have to think about how best to proceed—and I want to do so WITHOUT any investment in them landing at the "traditional" answers! Which will be tough.