Saturday, 21 January 2017

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---------------------


  1. Simon this is such an interesting progression, particularly because it seems "backwards" from how I think as an adult and I think backwards from how we often teach cardinality and ordinality. Teachers often introduce the abstract numbers first. It is so fascinating that you kind of flipped the progression. You focussed on the relationships first. When doing the Hundreds face challenge with older students, I move pretty quickly to the numeric representations of each rod. The way you describe the progression, the students really own the "relationships" first. Then, they bring the numbers. So interesting. I can't wait to hear how this evolves. Do your students "see" that the white "nests inside the read like the number one "nests" inside the number 2? I feel like I want to ask your students a question that gets at this concept, but I don't know what the question is. It sounds like your students stay really grounded in the "color" of the rod. Do you think they intuitively know that the red rod is equal to "2" or have they not made that connection that. I loved reading this. It really made me think. I am still thinking.

    1. Yes, making the hundred faces is atypical with Cuisenaire rods, in that it's a lot about counting. What they really do distinctively is different, and not about counting. End to end and side by side they show arithmetic relationships geometrically. And young children seem to grasp this easily. They don't all always remember what the relationship is called (eg "difference") or what symbol to use (though most do) but they do all see these relationships. And they're all happy to explore them physically, all going off in different directions.

      The nesting thing is interesting. In a sense that's what's removed. The nesting is part of cardinality. Each set of numbers contains all the smaller sets of numbers. Seeing numbers as lengths is very different to seeing them as nested sets. So, with the yellow-pink-white example up above, if we count the white as one, we look and we see the five as being equivalent to the four plus the one. Maybe, in a kind of duck-rabbit way, we also see that five minus four is one, or that five minus one is four. But we're not looking at the ones inside the five particularly, unless we deliberately set out to imagine that. At least that's how I see it, and how it seems to be for older students I've taught. What do you think?

  2. I am laughing because I am so impressed that you actually understood what I was talking about. It felt so clunky when I was trying to explain it, but you definitely got the gist of what I was I wondering. In our district our K-2 teachers hardly use Cuisinaire rods at all. I have been thinking a lot this year about how I can change that because they seem like such an important model of the relationships that you describe above. We use a lot of place value blocks, five frames, ten frames, counters, number tracks, number lines, etc. The K students do a lot of counting activities, but I am starting to wonder if the Cuisinaire rods can deepen understanding. Do you use Cuisinaire rods exclusively?

  3. We use all the manipulatives you'd expect - probably using the C rods less than half the time. Lots of games too.

    I think they would be a brilliant addition to K-2 classes, but it's crucial for teachers to have a good feel and liking for a pedagogy with them. Way back when they were first introduced, they went into classes, but not necessarily with the professional development to use them appropriately. So they went into cupboards...

  4. I love C-rods. They are the one manipulative that I most remember from my own early mathematics learning. They were wooden back then! They offer a powerful visual cue that not only enhances counting concepts (including conservation, which isn't talked about very much) but also addition/subraction, mult/div and fractions and early algebraic thinking. In senior classes I've used them (with a metre ruler) for modelling decimals as well.

    Mr Gregg raises a very good point: without professional development for teachers that focuses on the underlying cognitive and pedagogical concepts the manipulatives lose their power. I've been training teachers for a number of years and found that I can demo the use of C-rods, 10s frames, PV blocks - you name it, we've done it - till the cows come home but it has very little impact on student learning if the teacher doesn't understand WHY they are using them.

    It's all about maintaining what I call the Knowledge Triangle - models + symbols + words. All three are shown in your above example. Models are vital to the development of internal constructs (imaging) and number sense, and the symbols and words are vital to the communication (socialisation) of mathematical concepts for the student.

    I really enjoy your blog, by the way, and regularly share them with my teachers. Keep up the good work!

  5. Thank you Michelle! That Mr Gregg was me too, just another account! Great to hear from another C rod fan - and one who understands that just having the tool isn't enough!
    The ones we use are wood too. I think they work better. ;-)