## Monday, 8 May 2017

### Generalisations

Mike Flynn's Beyond Answers has a nice long list of general claims primary students might make in math class (p158-9):
Begininning with:

### Numbers

• Numbers have a set order.
• Our number system is organised by powers of ten (base ten).
• If 10 is added to a two-digit number, then tens digit increases by 1.
• Numbers can be added and subtracted by their place value.

As Mike says, these structures in numbers might come up incidentally, or the teacher might take a dive into one in particular, according to the interests and needs of the class.

I feel like his list should be more widely available, and expanded upon, so that together we're really looking out for when students come to these structures.

In class I'm on the look out for, if not articulated general claims, at least general explorations. When I see one, I take a picture and bring it to the class's attention. They're getting used to this. On Friday I was getting the class to hold up successively (if w=1) the Cuisenaire rod that's 10, 8, 6, 4... "The even numbers!" someone says. BT comments that you add 2 to each of the even numbers to get to the next. We look at the odds; it's the same.

So how do these general explorations come about? The students know there's a certain latitude in tasks. You know by now that I use Cuisneaire rods a lot. Because they preserve numbers intact and visually show their relationships, they're particularly suited to uncovering and showing structures in our numbers.

For example, back in January (when we were still calling the rods by colour letters; now we're revisiting some of the same situations with numbers) EW wrote how different trains were equivalent to trains of just white rods.
 O + b = 17wO + B = 19wO + O = 20w
So we look at this, and I ask the class to try their own similar equal trains.
It's natural to try the reds after that.
What generalities are there here, now that we're onto using nuimbers?

Some numbers are and some are not equivalent to a certain number of twos.

They alternate, one number that is, then a number that isn't.

Another time, AN made this series of equal trains:
There's an implicit generalisation here:

When you add 1 to an amount, the result is the next counting number.
 What's the genearalisation here?
So, on Friday we laid the staircase of even numbers on the carpet and checked the common difference is two with red rods. We did the same with the odd numbers. We moved on, looking at pairs of rods that equal 11. But a few children were still playing with the odds and evens...

"Look, I've made a Which One Doesn't Belong!"
"Me too!"
Mike is emphasising MP7 - Look for and make use of structure. Here's his summary (p154):
• Recognizing when students are working on an idea connected to structure and bringing the idea to the surface through questions and comments
• Giving students opportunities to see and explore a structure in their own work or in tasks designed to highlight particular structure
• Providing time for students to build an understanding of it
• Supporting students as they apply their knowledge of the structure in novel tasks

How do you look out for generalisations, implicit or spoken? How do you nurture them, promote them?

1. Am going back to my notes from a teacher work session I organized in mid-April and see what generalizations we might have talked about--so I get better at fining them in the moment.

1. Maybe we need to pool all our lists, Judy!

2. I love this idea of pooling lists. When I created the list for Beyond Answers, I knew it wasn't a complete list, but I hadn't thought about any formal way to collect contributions to expand the list. Thanks for suggesting this. Any ideas where we might want to house a list to which others can contribute?

3. Perhaps we could create some kind of shared document, with yours as a starting point, Mike? I don't know how we'd combine ideas, but it would be a resource to go to.

2. Are you familiar with Cathy Fosnot's work? When I think Fosnot, I think "generalize."
I'll have to pull out her Mathematicians At Work series and see if I can get some more details.

1. I keep coming across her, and like what I read lots, but haven't read any of her books. I'll have to add her to my list! Thanks Mark!

3. Thank you for writing about this work. I'm happy to see these ideas take hold and to learn of other approaches teachers take when exploring MP7 with their students.

Sheepishly, I'll admit that I haven't really used Cuisenaire rods, either for myself as a learner or with students. My school didn't have them when I was teaching and I was never given opportunities to use them. However, since reading your blog, I have gained a great appreciation for them as tools to support sensemaking.

I was particularly impressed to see how well they lend themselves to representation-based argument in this post. The challenge for many students when trying to generalize using representations is that they representations they create actually show a specific amount. Therefore, it shows their claim works for one instance, but does not apply to a whole class of instances (e.g. all whole numbers). The Cuisenaire rods seem to lend themselves better for this work because while they do still represent a set amount, they also could represent any amount because the value of w does not necessarily need to be one.

I'm am now seeing Cuisenaire rods as an ideal tool for unpacking the structure of numbers and the operations. This post is a great example of this. Thanks for trying this work with your students and for sharing your thinking about it.

1. I saw you included them as a tool in the book. And yes, representing numbers as lengths rather than with discrete units they have their own special place as a tool.

4. "How do you nurture them, promote them?"
^I'm reminded of Tracy's book in particular the chapter on proof.
It's the trajectory of: notice/wonder ---> make a claim/conjecture ---> test it (always/sometimes/never?) ---> generalize.

Tracy also touches on overgeneralizing and what to do instead of simply saying "no, you can't do that" or giving a rule that itself doesn't generalize.

So perhaps if a student overgeneralizes you could try to see if students could find a time when that's not always true. If serendipity doesn't strike then I'd think about how I could engineer a situation that could illicit a counter-example from the over-generalization and get a discussion going.

1. Yes - I haven't got to that chapter yet, but I've enjoyed the conversations Tracy's had on proving over the past few years.

As for overgeneralizing - I'm happy with first approximations that you accept without quibble - maybe to refine later.

(Did you spot one in Mike's list on Numbers?)

2. Who is Tracy and what book is this? Thanks.

3. The book is Becoming the Math Teacher You Wish You'd Had by Tracy Johnston Zager

I'm reading it at the moment. Wonderful! You can see it here: