Going off piste

We'd been dividing, looking at remainders with Cuisenaire rods. (See previous post.) There are 17 students in the class, so we looked first at that.

The next day we went onto numbers that the children chose:

I felt that maybe one or two were being "pulled" along with this, not really quite getting it; lots of people were asking me what numbers the different colours were. Did I need to go back and give them a little bit of the play and familiarisation stage?

So I decided that the next day, after doing some quick image work,

I'd give them 20 minutes free play with the rods. I knew that lots of what they did wouldn't be mathematical, but thought maybe just handling and getting used to the rods would help everyone to own them more.

There was lots happening. (click to see gallery of images)

To start with there were lots of patterns and pictures.

Then something interesting happened. One of the students decided to make a game. Soon, games were sweeping across the classroom.

I took some videos:

A lot of these games are a little hard for me to follow. They're not really very mathematical. But they have given me lots of ideas for ones that could be!

I don't know whether I did the right thing or not, but I was so interested by everything that was going on that I let the play carry on for 40 minutes to the end of the lesson.

At the end of the week everyone in the school has "golden time" when they can choose what to do for half an hour. So, I was really pleased when one student said, "Can we play with the Cuisenaire rods during golden time?"

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.

Letting things develop

We started off the term establishing a few ground rules, like, "It's OK to make mistaekes mistakes."

Felix made us laugh. We got talking about how there was a lot of falling off involved when we learnt to ride a bike.

"The first time I rode a bike I didn’t fall off,"

There was a pause, a doubtful but slightly impressed pause, before he added, with perfect timing, "But I did crash into a hedge," and, "At least it wasn’t a brick wall."

Then we began our maths work with a look at our number line.

Our number line, up high

We've put one in both Y4 classes (I think Y5 are going to use it too). It's based on math4love's wonderful Prime Climb game - which we're going to play next week.

We looked at the number line, and then at the hundred square poster:

We looked individually first of all. "What do you notice?" was the question. Then pairs. Then shared a few observations as a class. It was interesting that at first the class weren't seeing the multiples in there. They were looking at how many divisions of the circle there were, and then later beginning to look at how the colours were patterned. Then we wrote down a few of our observations. I accepted anything correct or half correct, even, "there are a lot of colours" and passed the adventurous and sometimes knowledgeable but over-hasty with a light touch of doubt. We left it after fifteen minutes and then returned to it the next day, today. We shared some of what we'd written, and then things started to come together, people were noticing the patterns more and more. Jinmin noticed that 51 had a little 17 in it and it was in the 17 times table! Here's Maryam's observations in her journal:

It's exciting to see how thoughts develop. I haven't really been giving hints. Just asking the class to move between individual looking, pair looking, sharing out loud with the whole class, writing, and then coming back the next day for a little more of the same. Some of the children are going to be familiar with this in a different form; it's great seeing them gradually recognise it again by the sheer power of noticing!

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.

Equality, equivalence, sameness

I've already blogged about equivalence. But there's a philosophical discussion children could have here.

What does it mean when we say things are the same?

Heraclitus famously said, "You can't step into the same river twice." The water, of course, has changed.

This statement, I think, could be a great starting point, stimulus, for discussion. Peter Worley suggests the line could be taken away from Heraclitus and given to Tina, talking to her brother Timmy, while they're visiting a river.

There does seem to be a lot to think about here, and it relates to the idea of equality in maths.
Peter James Jackson in one of his wonderful videos wants us to say = as "is equivalent to".


Why? Maybe to emphasise the idea of the balanced equations, rather than, as with the  = button on the calculator, "gives the answer..." This is good, but I'd like to get equals to be associated with balanced equations too.

Maybe too, to stress that the two are different but the same. But then I'd say that equality has the idea of difference in it.

Parmenides famously "answered" Heraclitus by saying, "You can't step into the same river once."

For a long time I thought he was just being ridiculous. Well, he was in a way, but within that craziness, there is, for me now anyway, a serious kernel.

Whenever we call two things the same, they are also different in some way. They're maybe in a different place, or a different time. They may be in a different form.

For instance, if we say that two celebrities arrived at an event in the same dress, we never mean that they've both squeezed into the one dress. They are wearing two separate dresses in all-too-slightly different places, with all kinds of subtle differences, definitely taking different shapes, possibly different sizes.

So same never means "completely the same". That's what Parmenides is saying: you have to have two (at least) different things to have sameness. Those things will be the same in some important way, but will be different in others.

Now I wouldn't say all this with my class, of course. Or even lead towards any of it particularly. But I know it's there, a linguistic and thought territory to be explored, every bit as "out there" as the little forest behind the houses across the road from school where children come back with all sorts of insects I can't immediately identify.

I think it would be good to have the philosophical discussion and the maths discussion about equality at roughly the same time, preferably near the beginning of year. I'm looking forward to hearing what my new class have to say!

Which One Doesn't Belong?

I really like the Which One Doesn't Belong? idea, that Christopher Danielson has made such a good book of! It's becoming a phenomenon, with a twitter account and a great website created by Mary Bourassa devoted to assembling the growing body of WODBs.

Here's one of mine:

Something's niggling me about them, and I must just get it clear in my head. I wondered if I'd got it wrong, especially when John Golden asked

@Simon_Gregg @WODBMath do you have reasons for all four? BL makes it hard to come up with traits for TR and BR for me.

— John Golden (@mathhombre) June 21, 2015

Perhaps I should be making my differences more different?

 What do I mean? Well take this one, by Barb Seaton:

It works like a good WODB - you can find a reason for any of the four. Take the top right one: you could say, "It's the only brown one. The rest are white." How would that be, if they were all different colours? Like this:

Could you still say, "The top right one - because it's dark brown"? I've been - in my slow way- pondering the difference between these two cases.

 In the first one there's a binary difference: there's the brown dog and considering the category of colour the others are all the same - white. In the second case, as far as colour is concerned, you could pick on any of them and say it's different because it's such and such a colour and the others aren't. It's not binary in quite the same way: there are four values for the category of colour.

Christopher Danielson goes for both kinds of difference in his original example:

He says (my notes in brackets):

• The bottom left shape doesn’t belong because it’s not shaded in. (Binary)
• The top left shape doesn’t belong because it only has three sides, while the others have four. (Binary)
• The top right doesn’t belong because it is the only square. (Not binary - there are three shapes. He could have said, "It has right angles." Somehow that feels a little more binary as right angle- not right angle is such a major distinction with angles.)
• The bottom right doesn’t belong because it’s the only one resting on a side. (Binary)

I'm kind of pleased that even this one has a not binary example in it. It seems to open up the possibilities a bit, to relax the whole thing. Of course, binary is satisfying.

I only did one lesson on Christopher Danielson's book. It was worthwhile, but I didn't pay attention to the distinction I'm making, and how the students were relating to it.

Just recently, Dani  Ruiz Aguilera has posted a good pattern-block example:

Which One Doesn't Belong? #WODB @WODBMath cc @Simon_Gregg @mathhombre pic.twitter.com/0nffUrw6zj

— Daniel Ruiz Aguilera (@druizaguilera) July 13, 2015

I wonder if he had in mind the not-binary category of order of rotational symmetry? Certainly there are lots of other features you could pick as well, some of them binary.

 I'm intrigued by these shapes and see lots of interesting things in them. Like, that you can transform the top right one into the others with a bit of internal rotation:

I wonder, what is the proper formal language for this binary-not binary distinction in differences?
Does it matter? Do you have a preference?

#tiling again

Well, John Golden has posted about his pattern block explorations, so I will too. Along with Dani Ruiz Aguilera he should accept a little of the "blame" for me spending so much time shuffling small shapes around. Dani has been doing some industrial-scale pattern block workshops himself:

Tothom fent mosaics! #patternblocks #17jaem #jaem2015 pic.twitter.com/AYDZv6pn4Z

— Fàtima Romero (@romero_fatima) July 7, 2015

I don't know whether I should put this post here or over on seekecho.blogspot.fr. I put my own things over there usually, including some of the things I've done with pattern blocks. But, although I've kind of "got the bug" with this, hopefully there are spin-offs for the classroom, like Teflon in space travel and the kitchen.

I've posted about some of the work I did in the class with a particular set of pattern blocks, the square, the hexagon and the thin white rhombus, already.

We've used Math Toybox's great Pattern Block tool - and a great thing about this is that you can save your creations onto the gallery, and even edit them again later on. John used it to respond to what we'd done in class:

@Simon_Gregg I see two different patterns merged in your students' work. pic.twitter.com/G7dERbkpw9

— John Golden (@mathhombre) June 18, 2015

I liked it that there were some interesting rules for these line patterns:

We looked at that and one or two other things, and then I asked them to work in bigger groups to extend one of them:

Meanwhile, something was intriguing me: a non-periodic, almost free-form, tiling using these shapes:

It has a really interesting balance between being forced to put the tiles in certain places, and having some freedom. To start with, if you begin with a hexagon, there are a lot of different ways of surrounding it. Here are some of them:

Similarly, there are a lot of ways of surrounding a square (finding them all might be an interesting task). You can see I've restricted myself. I never put two rhombuses against each other. Or two squares. I've got a lot of questions about this, and here are some of them:

1. It feels like it would carry on forever in all sorts of ways. Can we prove it does?
2. Once there's a "line" of squares and rhombuses, the line won't go away. What are the rules for it's behaviour?
3. I've got an intuitive feel for an algorithm for making it carry on. Can we write down an algorithm for making sure it continues to grow without getting into any "impossible" situations?
4. It feels like this could make a good game. What would the rules be?
5. You can get areas of different regular tessellation - how many kinds of these are possible? 
6. Can we - it feels like we can - create writing or pictures with this?
7. Can I have a large public space, St Mark's Square in Venice for instance, to tile in this way? Or failing that, a playground?

Meanwhile, I was exploring other patterns:

Another thing that caught my attention was dodecagons.

Those holes can hold a dodecagon:

These can "point" in one of twelve directions. The idea of embedding these pointers and maybe other dodecagons, in the tessellatin grabbed me, and for a while I thought about how numbers can be patterned in this kind of space.

Here are some rotating dodecagons embedded in a simpler matrix:

John meanwhile had created his own beauty:

(At the same time, I've been reading Roger Penrose on, among other things, his work on non-periodic tiling. I guess I'm thinking about those 17 wallpaper groups, and trying to push other symmetries embedded within a simple translation.)

On a more practical note, I saw that you can get pattern block stickers!

S's learning about symmetry: create & design w/pattern blocks, then transferring w/pb stickers. #RIOSSA #rioschools pic.twitter.com/07IbJ06jF6

— Larry Kelman (@larry_kelman) July 7, 2015

These would really help the students recording a creation directly into their books. And something they could do this for - why hadn't I thought of it before? - popped into my head: pattern block equations:

💡 Just had an idea. Pattern block equations. 💡 Make a shape 2 ways. (Extra: take away what's the same on both sides) pic.twitter.com/pHNdvLgbNk

— Simon Gregg (@Simon_Gregg) July 11, 2015

 As Mary Pardoe tweeted:

@Simon_Gregg Look forward to seeing what they produce (and deduce)!

— Mary Pardoe (@PardoeMary) July 11, 2015