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 d

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_lemma https://en.m.wikipedia.org/wiki/Newman%27s_lemma#Diamond_lem...