(If you want to understand the title, you can just skip straight to the end.)
When I think back on the inflexible way in which I remember being taught maths, I am often surprised that I became a mathematician. I don’t mean this to be unkind to my teachers — I enjoyed maths in school and I liked my teachers — but I don’t recall being taught to approach problems in multiple ways or to be flexible about which strategy I used.
Mental maths is a perfect example. Until relatively recently I have relied on one strategy — the standard algorithm. That is, I would find the answer by picturing writing the steps on a blank space1 in my head:
The only situation in which I would jump straight to a decomposition strategy was for percentage problems:
It wasn’t until I began teaching the distributive law by starting with numerical examples that I realised the power of the decomposition strategy:
Seeing this as an area model for the first time almost made my head explode. How I had never explicitly made the connection between multiplication and area (or volume or …), I have no idea.
Discovering Fawn’s Math Talks and seeing how other people break these problems apart is captivating, as is looking at visual representations of strategies.
— Fawn Nguyen (@fawnpnguyen) May 19, 2016
This really started to change the way I looked at maths; I now look—always—for visual representations of a mathematical concept. The two images at the bottom of the right figure are student responses to a prompt like: Explore a visual representation of (a-b)2.
So, by now you are probably wondering why it took me so long to catch on to all this, and what it has to do with the tea towel of multiplication.
I’m now very interested in different methods of multiplication and collect them up as I find them. I like the ones that appear more obscure; you look at them and say ‘what?!’. After all, the ‘standard’ algorithm is only conventional to those who know it.
My collection includes various methods of finger multiplication, the line method, Russian peasant multiplication, grid methods, cross methods, paper strip multiplication, and copper plate multiplication. I offer these ‘puzzles’ to my pre-service teachers as a possible project investigation. This is another way to add to my collection!
And check out this method with Genaille-Lucas Rulers to determine 52749×4.
Last year, Twitter offered up this tea towel of multiplication. I have no idea (yet) how the circle method works. How cool is that?
— Tess Maths (@tessmaths) June 14, 2015
 Yes. Ha ha.
 Visual Math Improves Math Performance