Cardinality, ordinality and developments with the Cuisenaire rods in K3

A few posts ago, I talked about asking children to use a particular manipulative, thinking of the Cuisenaire rods in particular. I got some great replies, most of which emphasised asking children to select the appropriate tools is an important part of the problem-solving process. I've been pondering this lots, which is why I haven't replied to the replies. I'm also reading Mike Flynn's brilliant new book Beyond Answers, which outlines how the CCSS Standards for Mathematical Practice can be brought to life in K-2 classrooms (I hope to blog about the book soon).

I'm trying to make problem-solving more and more part of my class, and I want to have more lessons where the children select the right tool, whether it be cubes, number line, hundred square, ten frames or whatever to help the solve problems.

But I've also got the provisional conviction (if such a thing can exist!) that the Cuisenaire rods, used right, can give young children an environment to explore in a more open-ended way, and give them a really robust and flexible number sense. It's a conviction I want to test through my reading and thinking, and also in practice.

There's a few things the rods do really well. One is being solid colourful things that children enjoy making things with. That's a great starting point.

Another is that, in some ways, they bypass counting.

Now counting is important and I've been putting more emphasis on counting collections this year as well as choral counting.

But counting is complicated. As Young Children's Mathematics by Carpenter et al summarise it,

• There's an ordered sequence of counting numbers, and numbers are always assigned to items in a collection in the same order starting with one.
• The one-to-one principle. Exactly one number from the counting sequence is assigned to each item in the collection.
• The cardinal principle. The last number in the counting sequence assigned to the collection represents the number of objects in the collection.

And when it comes to counting for addition or subtraction there are added complications.

Alf Coles contrasts this way of knowing numbers, cardinality, with one that isn't based on a set of objects counted, ordinality, which is represented as a teaching approach in Gattegno's use of Cuisenaire rods:

"One clear hypothesis to emerge is that students’ awareness of ordinality may be distinct from awareness of cardinality and, in terms of developing skills needed for success in mathematics, that ordinality is the more significant."

I see it as, once the different lengths become familiar, children can think about addition and subtraction without having to count. Like here, you can see that the pink plus the white are the same length as the yellow. You get to see pretty quickly that if you switched the white and the pink they'd still equal the yellow in length. And lots more besides. You can hold the whole relationship in your head, without any smaller units distracting.

What I've been able to do is help the class to gradually build up a familiarity with this, and say to the students, 'you go and make something now, and write it down.' And what's really exciting, they're starting to explore patterns in this, starting to systematise what they're exploring.

There were some great developments this week. Here's T looking at how if you repeatedly add a white rod you move up through all the different lengths in turn:

C had a variation on this:

 F was exploring the fact that something equals itself, enjoying the tautology of it:

D knew that he could generate lots of trains equal in length to the orange rod, just by creating the now-familiar staircase that children are making again and again:

 M was looking at what you have to add end-to-end to a pink rod to move up through all the lengths of rod:

I asked if anyone wanted to be videoed reading reading what they'd written - and some did.
The ability to set out all the information systematically is a great skill in maths, aside from any breakthrough that it helps you make. I hadn't asked for this and of course not everyone was doing it. And a few children were getting muddled.

Or needing to go back to the rods and fix what they'd written:

But luckily my TA and I were able to get round to everyone, and I think all of us are getting the basic idea, and that all is one of the my main considerations for when I'm happy to point to new developments and try new things.

One of the things we did move on to this week was based on E's work:

E was measuring these trains with a long line of white rods and counting them up. That's hard to read; it says:

o + b = 17w

o + B = 19w

o + o = 20w

It might look too simple to be a development, but this implicit measuring of the rods by another rod both connects with cardinality and leads onto lots of other things. It can lead to measuring with other rods, and crucially to talking about the rods as a number rather than simply as a colour, things that we'll be doing soon.

So the next day we looked at E's work and I asked the class to do similar things. And off they went:

Interesting with the bigger ones - a few children getting tangled with the troublesome teens - twelve, thirteen, fourteen, fifteen - it's a tongue-twister, and so easy to miscount at this age.

One group wanted to try it with ten oranges!

(We're going to have to come up with a convention to distinguish o from 0.)

We looked at one creation as a class:

I asked the class how we'd write this, and everyone wrote their way on little whiteboards. I chose a few to come to the whiteboard and share their ways:

We also looked at another lovely bit of systematisation, this time from A.

She was measuring all the rods with the whites. We're about ready with this to start talking about the rods "as numbers". A little pinch of cardinality, and our ordinality has new wings...

------------------added a little later---------------------

Gattegno:

Reasoning with Which One Doesn't Belong

I do a 15-minute Which One Doesn't Belong session most weeks. I've blogged about WODB before, and I regularly tweet about it. Last year my Grade 3s were getting really expert. This year, my K3 (5,6 year olds) are all really into them.

Over Christmas my copies of Christopher Danielson's book and teacher's guide arrived. I've been enjoying the teacher's guide; there's a lot of background in there, and a lot of useful advice. I really recommend it. Christopher took the book through its paces with children of  all different ages, and despite the simplicity of the basic idea, there's a lot to think about in its execution.

Christopher looks at development as a geometer using the Van Hiele model:

A lot of my students' thinking is, not surprisingly, at Level 0. But there's also a fair bit of Level 1 emerging. And at Level 1 with WODB you start to get some great reasoning.

Today I used this one that I'd made:

All but one of the 22 students had at least one thing to say:

Christopher suggests using this kind of recording as a reminder, and spur to further thought:

"Simply writing a key word (square) or phrase (all angles the same) or sketching a quick diagram, or circling key features of the shape are all quick ways to maintain visible reminders of the unfolding conversation for everyone to access." (p30)

And it's making me think that I should keep these up, because there's a lot of things that need returning to and developing, and who knows at what pace and at what moment thoughts will come to people?

I made some low-res video of the session. Here's a few moments. In this one you can see one of the "it's like" observations, and also one that describes a property (the red one is the only one with one round bit).

Some of the "it's like" statements need further investigation:

Sometimes an observation of a property leads to a discussion about what exactly that property is, the sort of reasoning I really want to develop in my students:

So maybe we need to pin up that second sketch too, and come back to it?

Tweeting about moving from "it's like" to properties, I got some good advice from Christopher and from David Butler:

@Trianglemancsd @Simon_Gregg Would a question like "what is it that makes it like a clam?" help to make the move?

— David Butler (@DavidKButlerUoA) January 12, 2017

We also talked about when to introduce vocabulary. I'd be interested in ideas about how to develop this. How could this kind of lesson be extended in another lesson, maybe in smaller groups or individually? What, in all this is worth developing? What I think would really help my class is to  encourage everyone somehow to look for properties and reason about them. Ideas?

My maths autobiography

This is a brief response to chapter one of Tracy Johnston Zager's wonderful Becoming the Math Teacher You Wish You'd Had. It's not the typical maths learning experience she describes there, but it's mine.

Paddington Green Primary School was a chaotic kind of place. Once, at least, a supply teachers left crying within the hour. And there were a lot of supply teachers. I don't remember a lot about it (except about Ivor Cutler). But I know the maths teaching was practically non-existent. Sometimes loads of sums on the board.

But my mum loved maths. Unlike her brothers she'd not been allowed to go to university, but she did train in electrical engineering at the BBC's college in Evesham.

She didn't 'teach me maths'. We did play a lot of games though.

I said these words about my mum at her funeral:

And yes, we played chess. And battleships, and boxes, and draughts. And backgammon. And Chinese chess, and Go. And Mah-Jong and Hanafuda. Endless games of Totopoly and Cluedo. With Peter too. Monopoly we weren’t so keen on, but we played it when Tad came home. Then there was Diplomacy (you didn’t like the garish board) and of course Scrabble. There was something mathematical in all this, the best kind of mathematical.

When I was in my teens we used to get a magazine called Games and Puzzles. We tried out lots of the games, and often the puzzles. My mum had a lot more patience for them than I did.

My secondary school maths was probably a bit better than the average. I got my 'O'-level, got my 'A'-level. But given the average wasn't so hot, I was never A. really excited, B. awed, C. exploring for myself.

No, there was some exploring. When I was 14 (in 1974) there was the possibility of computing in the maths classroom, outside of lesson times. There were two teleprinters. One of them was connected to a telephone. You could dial the City of London Polytechnic mainframe, there'd be a wrrr-weee-wbong-wbong noise and you'd put the receiver into a wooden box with a receiver-shaped insides, and the teleprinter was connected. About half a dozen of us became obsessed. We were writing programs in BASIC, learning it all by trial and error, no books, no teacher. We stayed in at breaktimes, and after school until about 8pm when the caretaker would come in shocked to find us still there and shout at us to get out of the building.  I wrote things like a program to play a race game. You printed out the track, rewound the paper and then ran the race. You could accelerate or decelerate either left-right or up-down by one each time.

While one person was on the connected teleprinter the other was on the other, typing their program. You then printed out as punched tape ready to feed in to the online teleprinter when it was your turn.

punched tape - source

 There was always a bug when you ran the program; I'd go home scrutinising the lines of code on the 36 bus. It was really compulsive, the adrenaline of the hunt, if that doesn't sound too overblown. I still have some of this stuff. Here's a page of the program for that game:

The first of two pages of the program to run the race game

The game:

The race (coloured in after)

I wrote a program to play Monopoly (!), one to print out a graph of weather data we'd collected on our balcony at home (I had to ask my maths teacher how to smooth the graph out - the only time I asked any teacher how to do something I wanted to  do! - he knew too: I needed a moving average), there was also a game where spaceships were in 3D space, accelerating and decelerating and shooting at each other (I worked out that I needed what I later learnt was Pythagoras' theorem).

Because it was a bit obsessive and I wanted to give more time to other things, I gave it up when I was 17. But I learnt so much from it.

I did 'Natural Sciences' at Cambridge, mainly biological kinds of things, with some history and philosophy of science thrown in. There were some maths lectures, on the basis of the statistics we were using, but I didn't really keep up. One thing I did do however, I read Martin Gardner's pieces in the Scientific American, then bought his books. That was the kind of maths I actually enjoyed.

So when I began teaching in primary, I had a fairly strong feel about what the real thing is: play, invention, the thrill of the hunt. I'd also been given Seymour Papert's 'Mindstorms' (pdf) to read at teacher training college, which, taking kids' free exploration of the Logo programming language as its theme, strengthened what I felt already. All this got worn down by all sorts of pressures, tests, National Curriculums, Numeracy Strategies, and most of all by having virtually no training or real ongoing professional development as a teacher.

Of course, all that has changed now, with the #MTBoS...

Thanks, Tracy.