Sunday, 29 November 2015

Division without calculation

Rosy tells me that Year 5 (Gr 4) are about to do some work on division.

I've been thinking about some work that could be done with Cuisenaire rods on division, a lesson that would not necessarily help with learning a division algorithm but would develop number sense and reasoning. I looked through the division chapter in Madeleine Goutard's Experiences With Numbers in Colour and this caught my eye.

How many brown rods are in six black rods?

Black rods are 7 cm long; brown rods are 8 cm long.
Here's Goutard's diagram:

If I had the rods with me now, I'd make a little video (in the style of Peter James Jackson's great videos). But just to show the six blacks and ask the question would be enough. And then what? To show them the movement straight away? I think so, because the point of this is that calculation isn't necessary. It's about manipulation, transformation.

I think what I'd ask for next is for children to make another example of the same kind of thing with different rods. "See what you can do."

I'd go round as they were trying it, and maybe select some to show with the document viewer to the whole class. Then, when everyone had got the idea more clearly, we'd work on it a bit more. Would any children make an example where more than one rod has to be moved to the end? That would be great.

Then I'd ask them to draw and write about what they've done on squared paper. I wouldn't be looking for a general description. It would be enough that the children are exploring a number pattern and thinking about a way that numbers can be manipulated. But if there were some observations about what was going on, or questions, so much the better!


Professor Smudge had a tweet with a question on this:

Monday, 9 November 2015

Same difference again

Last school year I blogged about how children in year 4 were proving a generalisation about subtraction. It was time to try the same thing again with my new class. They took a little longer to get the task, but after a while all got to the stage of creating a set with a difference of three:
When I asked what they noticed, a few people saw that the set would continue forever:
But they didn't see mention how the pairs increment. So we left it at that.

When we'd earlier made equations about the number fifteen,
T had written this:

30 - 15 = 15
29 - 14 = 15
28 - 13 = 15

So I took it as my second chance, and a couple of days later showed the class the pattern. They got the idea quickly. What do you notice?
Everyone agreed with these generalisations and they went up on the wall.

I asked them if they were able to explain why these claims were true, using words and pictures. This year however they found it much harder to explain themselves, and I don't know why. Sure, the class has a different character. It's earlier in the year. But I'm puzzled that they found it harder to explain what they thought. Most managed to get something on paper, but it didn't seem to convey the general nature of the situation like it did last year.

Tuesday, 3 November 2015

Can students ask - and answer - vast abstract questions, without being taught? Madeleine Goutard on Free and Conquering Minds and Cuisenaire rods

The Cuisenaire Company has republished Madeleine Goutard's Mathematics and Children, and I've been reading my copy.

Here's something. Back in the 1960s she was training teachers in the Province of Quebec, promoting the use of Cuisenaire rods. And yet her first chapter begins with a warning not to use them too much:
“It is generally agreed that concrete experience must be the foundation of mathematics learning. When children find it difficult to understand arithmetic it is at once suggested that this is because it is too abstract; for small children the study is then simply reduced to the counting of objects. It seems to me that there has perhaps been too great a tendency to make things concrete and that perhaps the difficulties children experience spring from the fact that they are kept too much at the concrete level and are forced to use too empirical a mode of thought.” (p2, my emphasis)
What kind of abstraction is she looking for then? Exactly the kind that Connecting Arithmetic to Algebra is recommending: looking for general patterns in the way simple arithmetic works.
"I find it of limited value to ask children a large number of definite, restricted questions whose answers they obtain through manipulation of the rods. On the other hand, I find it most profitable to start with vast questions which can be seen in a number of ways and which permit a continuous analysis of the dynamics involved. This is why I shall consider here families of equivalent additions, of equivalent subtractions, and of equivalent products and quotients." (p3)
And as well as having a clear idea of the kinds of areas that are fruitful to investigate, Goutard had a very strong view of the role of the teacher.
"The teacher is not the person who teaches him what he does not know. He is the one who reveals the child to himself by making him more conscious of, and more creative with his own mind. The parents of the little girl of six who was using the Cuisenaire rods at school marveled at  her knowledge and asked her: 'Tell us how the teacher teaches you all this', to which the little girl replied: The teacher teaches us nothing. We find everything out for ourselves.
It is evidently very difficult to give the child so complete an impression of non-presence, and to convince him that he alone is the artisan of his own education, but this is the way in which free and conquering minds are formed." (p184)
You can see that Goutard has an abstract way of writing, quite philosophical and psychological. I find I want to ask her, "How did the teacher teach her?? Yes, I know she stood back and gave her space. But how did she set up her sessions? How do you reconcile the seeming oxymoron of having the student genuinely following their own way, and at the same time having the teacher directing them towards the vast questions? How, not just in general terms, but how does a lesson go? What do you say? What do you do?" I want to see dialogues, with just a little bit of analysis, like you find in Connecting Arithmetic to Algebra. I want to get a feel for actual lessons.

But we haven't got that.

Luckily, all is not lost. If we don't have a clear roadmap, we've got a clear destination and a bearing. And I'm getting a more and more clear idea about the details. It helps to have colleagues like Caroline Ainsworth bringing Goutard's ideas to life. And #MTBoS collagues like  Tracy Johnston Zager,  Kristin Gray,  Mike FlynnElham Kazemi and Kassia Wedekind and many others who while they understand the vital importance of the agency of students, also seek ways for them to connect with the 'vast questions'.

I think the square can be circled.

Our beginnings are small:
After which we go off to write our own ones in our journals.

The next step is to take one of these, and focus on it. (For example, I find the relation between the five threes and Jinmin's triangle number way of expressing it worth following. Worth getting the rods out and investigating.) And then you're listening out for the generalisation, the claim. Up on the claims board. Do you agree with it? Then how would you explain it?  Show it with pictures, words, equations, a story...
I hope we can do our bit to reanimate Goutard's brilliant philosophy. I hope too to capture a few of the moments while we're at it, and perhaps share them with you.

Today we got our "Mathematical Claims" board started. To get the ball rolling I showed an image and asked for remarks. After the obvious features, I asked if anyone could say something more general. "What does general mean?" someone asked. We talked about that a little and then Tibo said:
to which Jinmin adedd:

These ones came a bit too quick and easily; they'd talked about them with teachers before. But they went up on the board to get the idea going. I'm looking forward to claims that come out of a more immersive experience of trying things out, reflection, and talking things through, ideas that are a little more hard-won, (and hard-defended).