Exploding Dots For Global Math Week

A man learns math by exploding dots. Image courtesy of Francisco Barberis via FlickrCC

If you hang around the #MTBoS long enough you can’t help but notice something called exploding dots. Today in a quite moment I took some time to dig in, and I am not disappointed.

Exploding dots is the focus project of Global Math Week, happening Oct. 10-17, 2017. Kind of like Hour of Code, the aim of the Global Math Project is to get hundreds of thousands of people all over the world doing math together at the same time. The architect behind exploding dots is MAA mathematician-at-large James Tanton who hosts G’day Math a blog full of problems, lessons ideas, and mathematical essays.

Today I watched this video of Tanton presenting exploding dots to a general audience. First off, he is a mesmerizing lecturer and the video is worth watching if just as pedagogy inspo. But on top of that, this thing he presents is just so much fun. The idea behind exploding dots is a simple visual representation of base 2, base 10, and eventually base x. The basic idea is that you fill a row of boxes with dots, and the dots represent 1, x, x² and so on depending which box they lie in. If one box gets full, the dots explode, and move over to the next box. Using this representation, Tanton builds arithmetic from the ground up, starting with addition and multiplication, and then adding in subtraction and division — even polynomial long division!

I won’t say any more about the precise details, since it’s better to explain visually. Just go watch the video.

One thing I appreciate immensely about his presentation is that he’s very clear that the point of this isn’t to get answers (we can easily do multiplication on our iPhones), but rather it’s figuring out how to develop a system intuitively and rationally to make it do what you want.

The only thing that is just niggling my brain a little bit is the problem with convergence of power series. In the video, Tanton shows how easy it is to use his exploding dots method to write

1/(1-x)=1+x+x²+x³+x⁴+…

giving an infinite power series representation to the rational function 1/(1-x). But this really gives the impression that you can put any number in for x and the equality holds, like for example, x=2. But this would give

-1=1+2+4+8+16+…

in other words

0=2+2+4+8+16+…

which of course can’t be true, and I imagine it wouldn’t take long for a clever student to pick up on that. I understand that Tanton is presenting this as an elementary alternative to the usual presentation, which is great. But I’m just curious if there’s some obvious way in his construction to see when a function is actually going to be equal to its power series representation. I’m sure the internet will let me know.

You can sign up to be part of Global Math Project and check out all of the great lessons and resources for you, your students, your children, or any unsuspecting friends who are foolish enough to go to happy hour with you on October 10th.