A hundred pattern

Back in Grade 3 (English Year 4) now. We've been doing quick dot images once a week, looking at the different ways the students see the images without having to count every dot.

We've presented it to our parents and the school, from Grade 1 to 5, getting them to talk to each other about how they saw it.

It was time to try and count something bigger, to give the classes a bigger challenge. Could they use their ability to 'conceptually subitise' to help solve a more challenging problem? For a while I thought this would be a good one. The holes in the drain cover are conveniently - almost - in groups of ten

But then I remembered this pattern with Cuisenaire rods. It was one that had started off a wonderful discussion on Twitter (See Dan Finkel's vlog of it). But it had those triangles, a bit like the triangles of dots we'd just looked at. Would the students be able to use some of the pattern in this to make their counting easier? I was not sure whether this would be a bit too much of a leap...

 So we could write on it more easily, I photographed it again on white.

The question: if the white rod = 1, what is the whole thing?

Students worked in pairs to try and work it out.

Some pairs broke the image down to help to see it.

I didn't expect that some pairs would rebuild it...

... and some would rearrange it.

It was really interesting to get an insight into thinking. Tracy Zager has just tweeted a flow chart about mathematical thinking that includes the need for the teacher to see the students' thinking in some way.

This is one way I think about potential tasks, questions, resources, curriculum. It’s a two-step process. Thoughts? pic.twitter.com/R7GLV6P0NK

— Tracy Johnston Zager (@TracyZager) October 18, 2018

This student had worried me a few days before when she'd asked me what half of 20 cm was. But now she was working out what three times 15 and three times 18 were. She didn't quite get to the correct answer, 100, because she didn't include the one in the middle. But she's done all but a tiny bit of the thinking! In fact, most of the students who got it wrong, were just getting one small thing not quite right. Her partner thought it was 102 because he counted the middle one three times!

Another approach was to add up all rods of each colour separately and then total them:

Interesting how they'd written the threes down as twos first of all, and then appear to have compensated for this mistake at the end by adding another nine on.

I pulled the class together and shared these two main strategies. I think in hindsight I would like to have dwelt on this more. I would have liked the class to have looked at what people were doing with the rods too, but it was all really busy! Perhaps we can return for a recap after the two week break.

Someone muttered something about failure. I'm not sure how convincingly I answered this. I said that I thought that even when people didn't get 100, they'd done a lot that was right on the way to 100. He seemed to accept this, but I'm sure there's more work to be done establishing a culture where we focus on showing our thinking rather than thinking about right/wrong.

What we did do was really interesting. I asked the class to make their own hundred pattern, so that it wouldn't need to be counted. (Not just ten orange rods, I said!)

There were all sorts of great approaches:

And some more:

We had a look at that double staircase afterwards; it bothered someone that there was only one orange rod!

The student with the yellow circle was making more hundreds, this time making his fives differently. I wish we'd looked at this too. I would have liked to give him credit for not just doing what I'd asked, but for taking it further and trying a variation. We talked about the person, and some students asked me if we could have a go at doing some like this. There wasn't time this time, but I'm so pleased how many different kinds of creations there were, and also that there was talk of coming back to it in a different form! (Time for some #hudnredface images too perhaps?!) I was pleased that there'd been a lot of choice in the two tasks - first in how they solved the question of what the picture represented, then of how to represent 100 differently. There had been so many different approaches to both!

One thing. There were mistakes. But a lot of them were what you might call silly mistakes, ones that the students realised were wrong as soon as it was mentioned. Counting the central white rod three times. Putting just eight tens in their design rather than ten. We all make mistakes like this, and maybe we can as a class get more forgiving about them. Also, having more practice with seeing numbers and counting flexibly should make the mistakes happen less often.

Cuisenaire around the world

There is a story about Nasrudin:

Terribly afraid one dark night, Mulla Nasrudin travelled with a sword in one hand and a dagger in the other. He had been told that these were a sure means of protection. On the way he was met by a robber, who took his donkey and saddlebags full of valuable books. The next day, as he was bemoaning his fate in the teahouse, someone asked: 'But why did you let him get away with your possessions, Mulla? Did you not have the means to deter him?' 'IF my hands had not been full' said the Mulla, 'it would have been a different story.'

I thought about this when I read  Kim Van Duzer's candid blog post about trying to use tape diagrams (bar modelss) with the class and the lesson going wrong. A big part of the problem, as I see it, was that the method had been dropped on teachers from above and Kim didn't really have a feel for the way teaching with the tape diagrams evolves from early beginnings and a liking for what they can do.

Some people suggested Cuisenaire rods and Kim took up there suggestion:

@AnnEliseRecord Tried new problems today and having kids model with cuisenaire rods. Much, much better! May write a follow up post about it.

— Kim Van Duzer (@kvanduzer) November 22, 2016

@themathgirl Will post some of the kids' work when I get a minute. pic.twitter.com/8WGHbxJMig

— Kim Van Duzer (@kvanduzer) November 22, 2016

I'm not trying to say that tape models are bad and Cuisenaire rods good. More that, like Nasrudin, when someone else gets us using an unfamiliar tool, and we don't know it and don't particularly like it, we're not going to use it  with the required subtlety and skill, and the students are not going to benefit. It helps if the process has been more voluntary, and we've built up our liking and skill with the tool ourselves.

Which is why thousands of Cuisenaire sets languish unloved at the back of cupboards. They were promoted in a similar way to Kim's diagrams (and Nasrudin's sword) and without the understanding and enjoyment of them they're pretty useless, may indeed if you're told to use them stop you doing something better that does make sense to you.

Having said that, yesterday was an amazing day for friends voluntarily using the rods, with understanding and pleasure, some of them just beginning their Cuisenaire journey others trying new things.

In our staff meeting at the International School of Toulouse we all stood in a circle and shared things that have gone well recently. Estelle, who every day is trying all sorts of new and wonderful things with her K2 children (4yos), had the rods and square frames out:

#istlive K2 play with the Cuisenaire rods generating lots of interesting talk on pattern, quantities... @Simon_Gregg pic.twitter.com/c091Epv2gf

— Estelle Ash (@estelleash) November 23, 2016

Amanda told us how her Grade 1s had been making and verifying Hundred Faces. Isobel shared how they had adapted the Hundred Face idea to be about signs:

G3 having fun making signs of value 100 using cuisinaire rods #istlive pic.twitter.com/Gk1PMwlj3G

— Isobel Patrick (@isobel_patrick) November 9, 2016

And that very same yesterday Kristin was trying out the rods with Ks and 3rd Grade!

@Simon_Gregg sorting and equivalence fun in K! pic.twitter.com/iZOJhDzW3Z

— Kristin Gray (@MathMinds) November 22, 2016

@Simon_Gregg initial notice/wonder looks like this pic.twitter.com/dgGTrltRHi

— Kristin Gray (@MathMinds) November 22, 2016

Kristin's careful planning, brilliant collaboration and enthusiasm for the students taking centre-stage make me all the more excited to see how she uses this tool!

 Over in Adelaide meanwhile, David Butler had a set arrive:

Look what arrived! I might die of excitement!! @Simon_Gregg @mathinyourfeet @nomad_penguin @MathMinds pic.twitter.com/ERhBDUy7m9

— David Butler (@DavidKButlerUoA) November 22, 2016

And yesterday he got the people at One Hundred Factorial puzzling with them:

Fibonacci sequence mod 3, 4, 5, 6, 7 with #Cuisenaire rods at #100factorial pic.twitter.com/fplG5eSCyo

— David Butler (@DavidKButlerUoA) November 23, 2016

Meanwhile in Maine, Sarah Caban's Hundred Face posters were up:

Hallway #hundredface ceiling display @Simon_Gregg @mathinyourfeet pic.twitter.com/fKwmsLE7f9

— sarah caban (@csarahj) November 22, 2016

(Read her blog posts about this here and here.)

And if that wasn't already an plenitude of surfeits, over in Winnipeg Geneviève Sprenger published a storify about how she's adapted the Hundred Face idea to be about evolving monsters.

My friends, it gives me a lot of pleasure seeing educators and teachers having a go at these things, not because they have to , but because they can imagine good things happening and know how to guide others to the same kind of curious, open and reflective approach!

How we've begun using Cuisenaire rods

Kassia Wedekind tweeted:

@Simon_Gregg Group of Ts and Ss new to Cuisenaire rods want to explore, investigate. Besides open exploration, what else would u start w?

— Kassia Wedekind (@kassiaowedekind) October 14, 2016

Knowing Kassia as a writer and blogger with a lot of understanding about young children's explorations of mathematics, I'm a little daunted trying to answer. But I'll have a go anyway.

First of all I'm impressed with just open exploration. My K3 class show no sign of losing enthusiasm for it. At first, when I asked them, they had no idea that the rods were linked with numbers or measurements, but there is so much implicit maths in what they do, and just doing it extends their understanding of it. It's an aesthetic enterprise as much or more than a mathematical one, but that seems right. After all, mature mathematicians often describe their motivation as a kind of aesthetic pleasure they find in it.

I document a lot by taking photos. This makes it easier to break up our creations at the end of a session, because they are not "lost", but more importantly we look again at what we've made, and I refer back to particular ones as starting points for new departures.

Students need a lot of rods. This is true for any material you use where students like either a large scale or want to continue a pattern. The little sets that people usually buy are not enough.

One of the things the teacher can then do is to say, did you see the way Ana made that wall? Do you think you could all make a wall of some kind?

And they have been so creative, there's enough for many sessions of this kind of return. You can see it right from our first meeting.

For instance, there's been lots of rectangles and surrounded rectangles:

I've used Cuisenaire trays in sessions after this:

Other things that have appeared are plans, maps really, of roads and carparks, models of houses with walls and chairs of different sizes. All of which could be followed up. Helen Williams suggested using different-sized rods to represent the three bears as the story is told, with students choosing their three sizes of rods, and holding the right one up at the right time. There could be three different chairs made, beds...

Using narrative seems is a great way in. We've been reading and writing stories about rockets, aliens and going to the moon, and T created several beautiful rocket images, which also showed one of the first examples of a staircase. So it got returned to twice. Once for a, "let's all try to make that kind of pattern in some way" session, and once for a "show a way to get to the moon" session!

We've just been making some faces within squares with sides as long as the orange rod - which was challenging but also delightful.

Alongside this, we play games. Rods behind backs, can you take out the red rod? Together and with partners. Or, here's the sandwich with something missing, what  needs to go in to finish it off.

And we're just beginning writing.

What trains have you found?#Cuisenairerods@nrichmaths @mathhombre @mburnsmath pic.twitter.com/TIf2OkZtKL

— Simon Gregg (@Simon_Gregg) October 6, 2016

As you can imagine, there's been some great opportunities for sitting with students and listening to them talk about what they're doing. But here I think I've go a lot more to learn form Kassia and her colleagues and their students... 

Writing!

All our individual work in K3 has been with manipulatives and orally so far, with lots of play and games. Today I tried them with a bit of recording in their books.

"What trains can you make that are the same length as the yellow rod?"

Following Madeleine Goutard's lead, I'm leaning towards - and today I asked for - writing rather than drawing as a way of recording, which I would have gone for more in the past, but I was happy to see the drawing too. The rationale here is that symbols are a quicker and simpler way of recording and so they can allow you to think further.

We talked through using the first letter for the rods:

Impressively, just about everyone got it. (One child who doesn't have much English didn't quite understand what I was asking for. I did sit with him for a while and do one example, but he followed up by drawing around rods in various patterns.)

It was interesting to see the different ways the children demarcated their different trains. We should look together at how they managed that.

Although I'd shown them the + sign before, when on the carpet at first with their whiteboards I asked them to write down a train I showed, that wasn't how children recorded, so we went with a list of letters. One boy did add the +s. He wrote

y+p+w

I suggested that to show p+w was the same as y, he wrote

y=p+w

but this was evidently confusing as he then switched to

y=p=w

Maybe I should have just left it!

Where would you go next with this?

I think they need to do something similar a few more times, look at each other's pages, see how to show what they want to say really clearly.

Any thoughts, suggestions? I'm feeling my way through this...

Nasrudin's Sermon

So, the five year olds in my K3 class made an amazing start creating whatever they wanted with Cuisenaire rods. Here's just some of the creations. In all the creativity there's a lot of implicit maths - ideas of equality and inequality, of arrays and rectangles, of sequences, of enclosing and aligning... And they're bouncing ideas off each other like crazy.

And now I've reached the point where I'm starting to ask for some very particular things. Make a "train" of pink and green rods:

This seems like a much narrower place. I'm directing the activity, I introduced a bit of arbitrary terminology - "train". What I fear is losing all the creativity. 

After free play in Gategno's book, comes trains. 

Trains take us to a lot of good things. But they're a little... one dimensional, after all the splendour of creation.

Perhaps if I was better at this, more sure-footed, I'd be confident to draw all this out from what has already been created. Certainly that's a kind of ideal. That the children are showing each other and learning from each other, and have the sense that they're doing so. 

There's a famous story about the wise fool Nasrudin that came to mind when I was wondering if I was rushing ahead too fast:

Nasrudin's Sermon 

One day the villagers thought they would play a joke on Nasrudin. As he was supposed to be a holy man of some kind, they went to him and asked him to preach a sermon in their mosque. He agreed.
When the day came, Nasrudin mounted the pulpit and spoke:
‘O people! Do you know what I am going to tell you?’
‘No, we do not know,’ they cried.
‘If you don't know, then you're not ready for what I have to tell you,’ said the Mulla. He got down and went home.
Not daunted, a deputation went to his house after a few days later and asked him to preach the following Friday, the day of prayer. Nasrudin agreed.
When the day came Nasrudin climbed the pulpit and started his sermon with the same question as before.
This time the congregation answered, in unison:
‘Yes, we know.’
‘In that case,’ said the Mulla, ‘there is no need for me to keep you longer. You may go.’ And he returned home.
Having been prevailed upon to preach for the third Friday in succession, he started his address as before:
‘Do you know or do you not?’
The congregation was ready: ‘Some of us do, and others do not.’
‘Excellent,’ said Nasrudin, ‘then let the ones who know tell the ones who don't.’

Adapted from Idries Shah's The Exploit's of the Incomparable Nasrudin 

naturally ingenious combining

I'm rereading some of the chapters in Madeleine Goutard's Experiences With Numbers in Colour. Something jumped out at me. It was written in 1964, but in a way it's better suited to now, to 2016. The destiny of many if not most sets of Cuisenaire rods around in the 60s was to end up in the bottom of cupboards, along with the Dienes apparatus and Geoboards. What was not there, the ingredient that was needed to make the pedagogy work, was the idea and practice of number talks. Even now this is maybe not that widespread, but at least it's out there, with wonderful books like Intentional Talk leading the way. Now that we are using quick dot images, asking children to look at groups of dots and tell us how they see the total, we understand better the importance of what Goutard below calls "naturally ingenious combining".

I won't be using this particular part for a while as my five year olds will be doing a lot of playing and other things before we get to this stage. But you'll see what's happening. The class is motivated to explore something together, they are creative and playful in the way they find solutions. The emphasis is on doing and trying out rather than having remembered (although they evidently have a lot of experience with rectangles as products of two numbers).

The passage starts with Goutard introducing some rods:

She then introduced a black rod (7 white rods long)...

And not just flexibility of mental calculation. Flexibility in thinking. And also comfort with manipulating numbers. These days too it's easy enough to take photos of what the kids have made , get them up on the whiteboard, and come together to talk about the different representations together.