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.
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.
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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.