Looking back, looking forwards

There's been loads of developments in my maths lessons over the past year. And a lot of that’s down to the #MTBoS - the Maths Twitter Blogosphere. I'm looking back, and also looking forward to a new Year 4.

I’ve been drawing a lot from the 3-Act lesson, particularly the first act, where there is a stimulus and then space for responding to it. There are the brilliant questions What Do You Notice? and What Do You Wonder? that put the ball in the children’s court. We’re moving in a direction where the children are having a bigger slice of the mathematical authority.

We’ve been estimating like crazy with Andrew Stadel’s Estimation180, going on to develop our own estimation challenges. I owe Joe Schwartz thanks for inspiration with this.

We’ve been using counting circles lots too. Here, it’s the children that share strategies at the end of these brief sessions. They are getting used to explaining their approaches, and looking for other ones.

And we’re relating arithmetic to algebra, looking for general patterns in the way number operations work. Linked with this is valuing the claims that children make, seeking discussion on these claims, and asking if we can justify them. (Thanks: Kristin Gray especially.)

I’m always looking for ‘open middle’ activities, but I’m keen to find more that are student- initiated and open-ended, following up from their own tinkering and questions.

In fact here’s a lot of things I’d like to really start using in the coming year:

• Keeping maths journals
• Always, Sometimes, Never
• Creating polygons and more with Scratch
• True or False questions
• Expressing equations as real life situations/stories
• Modelling real-life situations

Alongside this I’m always keen to try new lessons. For instance I came across a couple by Federico Chialvo that I really liked recently: Number Bracelets and Squarable. And Joel David Hamkin's second booklet, Graph Theory for Kids.  And Iva Sallay's Find the Factors. Then there's Envelopes by Alan Parr, which I came across just today (on nrich too). 

And I’m usually coming up with a few ideas myself. For instance, I’d like to do a lesson discovering Thales theorem (semicircle) together – creating triangles with different angles first.

Caleb Gattegno with Cuisenaire rods

I use Cuisenaire rods lots. But I want to do so more, à la Gattegno/Goutard (thanks; Caroline Ainsworth), to give lots of hands-on awareness of arithmetic, looking for generalisations. And linked with this, I want to go further with the children making and justifying claims.

The problem, the challenge is, where do we make time for all of this? Already, there's not enough time in the year! I’m hoping to recast existing lessons that are too teacher-telling-class, or just not important enough. I've made a quick list of all my lessons last year and I'm going through it, thinking about changes to make, things to drop, add in, or move. That bullet point list will get to work on my list of the lessons. Watch this space.

Can you think of a way to show how that can be true?

Summer holidays. No WiFi. At a café now, so I can blog, but then back to limited quota of 4G.

But there are books! I’ve been going over Connecting Arithmetic to Algebra by Susan Jo Russell, Deborah Schifter and Virginia Bastable a bit more slowly than before and before. It’s such an excellent book. I really like how jargon-free it is. I also like how all the main points are illustrated by, centre around classroom discussions. And of course I like what it’s saying: children can be making generalisations about the arithmetic they do, they can be making claims and justifying them, even proving them, and by doing this they can deepen their understanding of mathematics.

I’ve been reading out this dialogue with 4th graders) to any teacher friend who will listen to me. It’s got so much in it:

Ms Schmidt: What do you know about 327 plus 245? What can you say about the sum?

Angela: It’s more than 500.

Teri: And less than 600.

Mannie: I know it will be an even.

Ms Schmidt: How do you know that?

Audrey: The numbers are both odd and if you add two odd numbers, it will be even.

Ms Schmidt: Does everyone agree? Is that always true?

Fiona: My teacher told us that last year. An odd plus an odd is even.

Samantha: Look, 7 plus 5 equals 12; 5 plus 3 equals 8; 17 plus 7 equals 24. It just is.

Ms Schmidt: But Audrey is saying this works for all pairs of odd numbers, right?

Audrey: Yes, it doesn’t matter what the numbers are.

Joshua: But there are lots and lots of numbers. I don’t think you can ever be sure.

Ms Schmidt: I hear Audrey saying that every time you add two odd numbers, you get an even number. Joshua is saying, if you haven’t tried all the numbers, how can you be sure?

Mannie: It has to be that way. We all know that.

Ms Schmidt: What does Audrey mean that it doesn’t matter what the numbers are? Can you think of a way to show how that can be true? We have been using stories or cubes to make arguments. Take a few minutes, talk to your partner, and see how stories or cubes might help you.

Ms Schmidt [after a few minutes]: I heard Mannie’s group use a story. Would you share?

Mannie: It’s like you had some people in one class and everyone has a work partner except one person. Then you have another class and it’s the same, everyone has a partner except one person. If you put the two classes together, everyone stays with his or her partner and then the ones without a partner pair up. When the two classes are together everyone has a partner.

Audrey: I can show it with cubes. These are both odd numbers. Every cube is paired up except one. I don’t even know what the number is. If every cube is paired except the one at the end, then it’s odd. When I put them together, the two end ones pair up. That makes the total even.

Look at how Ms Schmidt keeps reiterating Audrey’s claim that any two odd numbers will make an even sum. Some of the children want to leave it at “it just is”, but there’s a deeper to go, a why. It’s great that Joshua makes the counter claim – you can’t know for all numbers. Most of the children will accept that Audrey’s claim is true, but now they have to find an analogy, with blocks or a story, that will make the why make sense to themselves and others.

I really like how Ms Schmidt phrases her request for a demonstration (a proof?): “Can you think of a way to show how that can be true?” There’s an open-ness, an ease in the words, and the examples the groups come up with show that she’s put it just right. They also show that she’s given them practice at representing situations with stories and cubes, so that these are familiar tools ready for use.

- - -

This coming year, I’d really love to see more of this kind of thing in the Year 4 classes. We do a lot of using cubes, and Cuisenaire rods of course, to represent equations. But I haven’t really done that thing of getting the children to represent arithmetic in story form. It’s another great tool, that gives meaning to abstract equations, and can be used for the kind of analogy that Mannie’s group produced so brilliantly. I’d like other people in the school to see this too, even if it’s only this short dialogue: it touches on a lot of themes in the book, and has the kind of respectful partnership between the teacher and students that we will recognise in ourselves on our best days.