Here's my report on one of many lovely sessions at this year's ATM Conference, which I went to for the first time this year.

I've picked it because of the way people with all sorts of depths of mathematical knowledge, working with all sorts of ages, worked with each other. There was no hint of haste or one-upmanship, no competition or even comparison; just a lot of good humour and concentration, and a lot of maths.

As Helen tweeted: One very important guiding principle of @ATMMathematics: "Any possibility of intimidating with mathematical expertise is to be avoided"

Helen and Mike got us working in pairs. Minisha and I worked together. The first thing we had to do was, from a set of blank dominoes, make a set up to 3-3.

Then we moved on to triominoes. We were to construct a 3-3-3 set.

Mike was on hand to answer our question, almost before we'd asked it: was the 1-2-3 domino the same as the 1-3-2 domino? He said we could do it either way, but his bought set had them just one of them. Minisha and I noticed the 0-0-0 set had only one triomino, the 1-1-1 set had four and the 2-2-2 set had ten. It was looking like we were adding triangle numbers, making a kind of pyramid of triangles. I wanted to make shelves so that we could show the pyramid. Looking around... the water glasses...

I've picked it because of the way people with all sorts of depths of mathematical knowledge, working with all sorts of ages, worked with each other. There was no hint of haste or one-upmanship, no competition or even comparison; just a lot of good humour and concentration, and a lot of maths.

As Helen tweeted: One very important guiding principle of @ATMMathematics: "Any possibility of intimidating with mathematical expertise is to be avoided"

Helen and Mike got us working in pairs. Minisha and I worked together. The first thing we had to do was, from a set of blank dominoes, make a set up to 3-3.

We were asked how we did it. Minisha was thinking about whether they would make one big loop. Mike and Helen meanwhile were going round chatting to people about what they were up to.

Other people were trying other things. |

Mike was on hand to answer our question, almost before we'd asked it: was the 1-2-3 domino the same as the 1-3-2 domino? He said we could do it either way, but his bought set had them just one of them. Minisha and I noticed the 0-0-0 set had only one triomino, the 1-1-1 set had four and the 2-2-2 set had ten. It was looking like we were adding triangle numbers, making a kind of pyramid of triangles. I wanted to make shelves so that we could show the pyramid. Looking around... the water glasses...

We organised our pyramid differently to the way we'd created the triominoes. As you moved towards the bottom left there were more ones in the triomino, towards the bottom right more threes, the bottom back had more blanks. And as you moved up you moved towards two-ness. Helen was coming round with her notebook, writing down things that people said.

Mike wondered why we hadn't put the 0-0-0 on the top, so that the second layer would have the additional 1-1-1 set triominoes, and the third layer the additional 2-2-2 set triominoes, etc. I think Minisha and I were both pleased with the way we'd done it, and other people liked the 3D-ness of it.

Helen and Mike stopped us after a while and Helen read out ("re-proposed") some of the things she'd heard said, including my thing about two-ness. She just left a pause after reading them... making me feel like I needed to say more to everyone, which I did. Someone asked how we'd show the next-sized set with our organisation. I said we couldn't. John on our table wasn't happy with that. 'Well, we'd need to go into four dimmensions,' I said. 'We couldn't do it with the glasses.'

I can't remember what Helen and Mike asked us to do next. I think we all had things we wanted to check out. Maybe they told us all to get on with it.

I had a quick look round.

Could the 20 3-3-3 triominoes be fitted, with symmetry onto an icosohedron? Could the icosohedron's edges have the same totals? |

What is the formula for finding the nth tetrahedral number? |

Could you make a game with L-shaped dominoes? |

Minisha and I decided to reorder our layers in the way Mike had mentioned, seeing how that would look.

I remembered a way that tetrahedral numbers were put together to make a cuboid, tried to make it with Cuisenaire rods and failed, googled it and found what I needed.

Six of them fit in the cuboid.

Barbara came over and started looking for a good way of proving that the sets would always follow this pattern, looking at each triangular layer. I peeled off and for a while contemplated the diagonal in Pascal's triangle that has these numbers in.

But I hit the buffers at this point.

What Mike and Helen managed to do was not only, or primarily, show us a fertile area for maths exploration that could be approached by any age, but also to show us a kind of lesson, where they, as teachers, are present more to gently orchestrate a group of people exploring related interesting questions. They watched and listened, enjoyed what people were doing, asked questions, called us all together to share a few things, then sent us off again. Some of you will recognise this pattern. If you're fortunate, you'll recognise it from your own classrooms.

Thank you for your lovely write-up of your experiences of our, collective, workshops. That the 2 by 90 minute sessions sped past was more about the ways different groups of people worked on different developments of the starting tasks of creating the sets of 3-3 dominoes and 3-3-3 set of triominoes. I mainly remember becoming aware of the different outcomes which occurred/appeared and asking occasional questions. Your writing convinces me of the massive importance of planning to start any teaching I do, with a seemingly simple, task which opens up different ways of working on the task and of developing it; some of which I could not have construed. Yes, I did know the number of dominoes in an n-n set form a quadratic sequence and the number of spots form a cubic sequence and the number of n-n-n triominoes form a cubic sequence and the number of spots form a quartic. However, with such a lovely, self-selecting group of people I felt it was a privilege to be part of these workshops. So thanks again Simon for reminding me of what proved to be a most intriguing experience in which I felt more of a 'bit-part player'. Mike

ReplyDeleteSimon, I was following the ATM_2017 hashtag from here in Canada...attending vicariously through folks tweets. I saw many tweets with the triominoes and the pyramid...now thanks to your post I understand more than I could reckon on my own (the glasses covered some of the dots). Sounds like a great session, with the true "craft" of teaching on display by your presenters. Lucky you to have been there in person...lucky us who read your blog, and get a "taste" of the math that emerged that day! As always, Thx for sharing :D @mutchmormath

ReplyDeleteI'm really glad you've got a taste of it, Meagan! It was an example of what I've heard Helen call 'a lot from a little' - a small task, making a small set of dominoes - that can lead off to a lot of investigations.

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