Why Does Exploding Dots Work?

FWIW, the answer to the question in the article (why you always end up with the same set of coins if you don't use quarters) is "Diamond Lemma"[1]. The author does mention it in a footnote, but doesn't go into details.

As a mathematician, I love the concept. I love New Math, too, however -- and yet it was not a success.

Why? I believe that the article is asking the wrong question: will the kids get it?

The question to ask is: will the teachers get it?

Given that the answer is hidden in the footnote (and is usually not taught unless you're getting a graduate degree), I am pessimistic about that.

I've spent a long time thinking about how we can teach math better (and quite a bit teaching it, mostly as a TA, but also as an instructor and tutor). I think the answer is -- we have to raise a new generation of teachers before even trying to change the curriculum or methods in any way. As it stands, a college degree in math education (or even mathematics, sadly) does not prepare someone to be able to do mathematics. And that's the thing that needs to change first.

[1]https://en.m.wikipedia.org/wiki/Newman%27s_lemma#Diamond_lem...

I'm confused, isn't exploding dots just binary?

For binary, I like this better: https://youtu.be/zELAfmp3fXY