## Saturday, 24 October 2015

### Manipulatives

The #elemmathchat conversation is always at the slightly difficult hour of 3 am. here in France. I did set my alarm this Thursday, and did manage to wake up to turn it off, but that's as far as it went. So I caught up with it on Storify and favourited and commented a little. Here's one I liked.

and then it went like this:
and
Tracy quoted this from Making Sense in her blog post recently:
“In traditional systems of instruction, teachers are asked to provide feedback on students’ responses, to tell them whether or not they are right…this is almost always unnecessary and usually inappropriate. Mathematics is a unique subject because…correctness is not a matter of opinion; it is build into the logic and structure of the subject…There is no need for the teacher to have the final word on correctness. The final word is provided by the logic of the subject and the students’ explanations and justifications that are built on this logic” (Hiebert et al. 1997, 40).
It's a bit like Gattegno had it in his picture I've shown before:
Gattegno was saying, the knowledge (K) doesn't get poured into the student (S) by the teacher (T); the teacher communicates what they want to communicate by pointing them towards something that will give them the knowledge directly.

This is especially true when the affordances of a manipulative and the way the student has been asked to explore it give the student instant feedback.
 27 allows you to fit three nines exactly along next to it. They fit really neatly!
Although Gattegno cuts a slightly odd figure to us now, and his "lesson" is evidently a kind of performance that is at the end of a series of lessons, because he's at the root of how (and that!) we use Cuisenaire rods, we owe him a lot. This second clip is where things really start to get going:

(I wonder, how did Gattegno make that link between "one third of" and x 1/3? Also, how did he get the children comfortable with thirds? I find that's often puzzling for children.)

I feel that his diagram isn't quite right. I want to put conversation into it. Gattegno is talking with the students a lot. But, for me, it's especially student to student conversation - which is notably absent in this video, but needn't be for Cuisenaire rods to be used to give students access to the logic of maths. My diagram would look more like:
 My elaboration of Gattegno's picture
I love how Gattegno goes off from the rods into writing equations about 27. Again, this can be done with students making their own equations.

Caroline Ainsworth, following Madeleine Goutard's lead, gets students to write lots of equations about a number. You can see how this could follow on from some version of that 27 discussion:
Here's a page of a child's writing from Goutard:

This seems a really fruitful direction, that I'd like to make my own. I've headed off that way before, but there's a lot further to go.

And have I answered Mark's question? I'm not sure. But I was struck recently, how at Toulouse's "Nuit des Chercheurs", how even a University Professor, Arnaud Chéritat, and his students are using 3D printed models to understand something that's too illusive without something to handle and look at:

____________________________________________________________________________

1. Thanks once again for highlighting the incredible versatility of cuisenaire rods. They are under-utilized, at least in my school, and I need to get the sets I have in the hands of kids across all grade levels.
The video provided a good model on how to use the rods to find fractional parts of whole numbers. But I found it hard to watch. I'm not at all convinced those kids really knew what was going on. Even if the lesson was a kind of performance, his interactions with the children left me feeling very uneasy.

1. I feel the same as you, Joe. There are things about these videos that are a mystery to me. Things I don't like. Gattegno obviously has the students tuned into him as a teacher, but it's a lot of initiation-response-evaluation, a lot of weight put on the teacher and the one child he's interacting with.

But at the same time, I find them a challenge. I sense that the children are learning to think about numbers flexibly through handling the rods. I suspect he has led them through lots of sessions like this with other numbers too, and they've approached multiplication facts this way. I like the link to writing.

We're lucky because we've got some great tools to reinterpret material like this.

I'm wondering how I'll do it with my Year 4s. I think pick smaller numbers, so that anyone who's not quite sure of a number fact isn't excluded. And I'll do the kind of "What do you notice?" session that I use quite a bit now (see Illustration A above). For this fraction-type discussion, I might slip in an image like Illustration B, and see if anyone responds to those empty fours I've added on.

I'll get the kids using their small whiteboards, perhaps in pairs, and then sharing some of their equations. Then noting them down in their maths journals. Then proposing a similar picture themselves which we might respond to on the next day. I think we might cover a lot of the same ground, like this, without weirdness, and with kids that are a whole lot more relaxed.

2. I think my discomfort is related to the young age of these kids. It's as if they're being used to prove a point about cuisenaire rods, like "Look what complicated math we can get these young kids to do!" There's a lovely moment in the video where he asks, "What's one-third of 27?" and there's a muted response of, "Blue." What an equation! 1/3 x 27 = blue!