Exploding Dots and Math Bumps

Last week saw the end of the first ever Global Math Week and what a successful week it was: over 2 million teachers and students took part! I am proud to have been an ambassador for this and to have brought the joy of exploding dots to many teachers and students. If you haven't heard about James Tanton's Exploding Dots then you need to check out this site. Basically, exploding dots are a way of visualising a journey of mathematical ideas from primary to senior grades.
In preparation for Global Math Week, my colleague, Dan Allen, and I held a number of sessions for interested teachers to introduce them to the idea of exploding dots and how they could incorporate this into one of their lessons that week:

For Global Math Week itself, I went into Grade 2, 3, 4, and 7 classes. The students really liked the idea that they were part of a worldwide event and that they were solving the same problems as students in Australia, China, India, Germany, Tanzania or wherever they had friends and relatives.
I started with simple 2 to 1 and 3 to 1 machines. Here, a grade 4 student is writing 7 using a 2 to 1 machine. Listen to all the kabooms happening in this clip:

We quickly learned that we can't use our normal number words to describe our results so instead of saying "One hundred eleven" we said "one, one, one". Then we used a 10 to 1 machine and found out that twenty-three could be written as...23! Here, I could explain to students that nearly all of the math they have learned so far has been in a 10 to 1 machine so in this case we could use our number words 'twenty three'. 
For the Grade 4 and 7 students, I then tried a 3 to 2 machine. 

Kabooms galore! It was neat to see the students taking care to make sure that they did the explosions correctly and checking with each other to see if they got the same result. Where they were discrepancies, they sought to convince each other of the correct answer. 
Finally for the Grade 7s, I tried an operation with them using a 3 to 1 machine:

This they did with no further instructions from me:

From a personal point of view, there is something about Exploding Dots that brings out a beauty I'd never considered in polynomial division. When Sunil Singh first introduced us to exploding dots last year, he challenged us to do 1÷(1–x) and 1÷(1-x^2) using this method. Even though I knew how to do these using more conventional methods, I was gobsmacked by the visuals produced:

I tweeted my excitement to James Tanton who then sent me another challenge. When I got stuck into this, something so surprising and wonderful happened, that I experienced what can only be described as 'Math Bumps':

Joyous maths indeed. 
My sincere thanks go to James Tanton and all at the Global Math Project for helping to spread joyous math everywhere.