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.

@MarkChubb3 Everyone should have a ladder of abstraction handy. https://t.co/9j0XlN5xMD

— Simon Gregg (@Simon_Gregg) October 23, 2015

and then it went like this:

@MarkChubb3 Can you give me a particular instance of that? ;-)

— Simon Gregg (@Simon_Gregg) October 23, 2015

and

@Simon_Gregg Dividing fractions... How many understand what's going on here? Some don't see need for concrete if Ss can get an answer

— Mark Chubb (@MarkChubb3) October 24, 2015

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:

And more Year 2 maths free writing that they are bringing in from home. @stocklandschool #YesUCan pic.twitter.com/HDvP6SLPUR

— stocklandmaths (@stocklandmaths) October 14, 2015

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:

& these are the models Arnaud Chéritat made to understand how to turn a sphere inside out! (https://t.co/havucDig4Z) pic.twitter.com/EpfSdioxjn

— Simon Gregg (@Simon_Gregg) September 26, 2015

____________________________________________________________________________

Illustrations added for my reply to Joe's comment below:

Illustration A: What can you say about this picture?

Illustration B: What can you say about this picture?

Madeleine Goutard and Cuisenaire rods

Impressed by Caroline Ainsworth's researches into using Cuisenaire rods, I got hold of Madeleine Goutard's 1963 booklet, Talks for Primary School Teachers. Gattegno writes of Goutard in the foreword thus:

Not the least of her talents is the ability to express simply, accurately and concisely ideas that in my words remain obscure to many.

And they are sometimes, his words. I'm convinced that Gattegno was proposing an approach to teaching that put the learner at the centre and encouraged their initiative, understanding, communication. But it doesn't necessarily come across in that very directive video, and his wealth of ideas to mine doesn't necessarily come across in his books:

which - despite the treasures in them - somehow put me off with their layout and wording. 

And there are really all sorts of good things. Like this, taking thirteen and seeing which rods "fit" in it:

I usually go straight for representing factors:

but this made me see, there was a step in understanding I was missing out, looking at which numbers line up in the 'wall" of a number and which ones don't.  This realisation came just in time, as we've just begun to look at factors and primes. 

There are seventeen children in the class, so we started off talking about getting into groups, and children being left over. Then I began to show how we could look at this with the rods. 

And off we went with the rods themselves:

I'm interested in how much we've absorbed of this, and how to explore further, how to write about it, how to digest it. (Q: Is it worthwhile getting the kids to take such trouble over the drawing? I'm inclined not use colour in future.)

Anyway, some snippets from Madeleine Goutard that show, much better than Gattegno does, how exploratory all this might be:

We must avoid an over-emphasis on teaching: that is, we must avoid showing the child things he can find out for himself in his own way. (p3) 

... he must be given the greatest possible initiative in building the mathematical edifice. Instead of always beginning the lesson with: "Get out your rods; do this, do that"... the start may be in the form of a question:- "Who remembers what we were looking at yesterday? And what did we find...?" (even if it is thought to be insufficiently grasped, and that there will be incorrect answers). "Let's get our rods and make sure that what we are saying is true." Leave the initiative for conversion to the children themselves. "But did we have to use that particular rod? Which other one could we have used...?" Make them work out what would happen if: "We never  worked out if that number had factors? Does it? Why? Which ones? Let's see if we guessed right." In this way the mind flies ahead of the facts, instead of lagging in their wake and being led to a docile submission to what is obvious. (p5) 

What I want is to see children who express themselves and use their knowledge, such as it may be, creatively.  (p44) 

It is not the passive possession of knowledge which is important, but rather the ability to acquire it: that is, to make use of what one has already to lead to further knowledge in abundance. (p48)

Caleb Gattegno and Cuisenaire rods

A couple of years back I uploaded a brief video about Cuisenaire rods for Mathagogy.

I've done a lot more with the rods since then, and would make a different video. But I still think, suspect most of the learning is ahead. One of the things that makes me feel like that is this 1961 video of Caleb Gattegno:

What do you think when you see this?

Me, I'm very attracted to the possibility that young children can learn so well using the rods. But I must admit to some discomfort. 

I know it was a demonstration, not a lesson but...

Those children! They obviously get the mathematics, but... they seem so, well so quiet, these five year olds! And so clever. Were they selected from among many to be in the film? (The children in the French version of this film seem a little more like the five year olds that I know.)

And Mr Gattegno, he seems like the giant that the tiny students follow, and maybe we teachers follow. The demonstration is very directed. Was that how he taught? I would love to see a proper lesson.

Really, what do you think?

My feeling is that Gattegno's real lessons would have had more input from the children themselves, would have allowed them to explore and to talk. I like this diagram that Gattegno drew to explain how he sees the teaching situation he wants, on the right:

I'm not sure from his books how the lessons went exactly - I know he started off with free exploration and play - and the impression I get is that children were playing games with each other, trying things out, explaining things. This seems to be part of his philosophy. And I think to get to the understanding we see in the film, they must have been doing and discovering things for themselves.