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?

It's a visual representation of place value-- it doesn't have to just be binary; it can just as easily be base 10, or base 37, or base 1.5, or base x (polynomials).

The full set of lessons is here: http://gdaymath.com/courses/exploding-dots/

I guess it's a way to teach children (or anyone) binary in a visual way, without talking about exponents.

Not just binary. It seems to be teaching the place value idea with a graphical representation. It would work for any base.

In fact, it seems that the representation is also used to study polynomials and other objects.

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