Friday, 23 April 2021


Dan Meyer tweeted 

and Kassia tweeted:

And I have just read this in Wally's Stories: Conversations in the Kindergarten by Vivian Gussin Paley . It's a lovely example of how Paley is able to write against herself, to document her growing points as a teacher, alongside the learning of the children:


Rulers were another example of the wide gulf separating my beliefs from those the children demonstrated whenever they were allowed to follow their ideas to logical conclusions. I had not realized that "rulers are not really real." We were about to act out "Jack and the Beanstalk" when Wally and Eddie disagreed about the relative size of our two rugs.

Wally: The big rug is the giant's castle. The small one is Jack's house. 

Eddie: Both rugs are the same. 

Wally: They can't be the same.  Watch me. I'll walk around the rug. Now watch: walk, walk, walk, walk, walk, walk, walk, walk, walk - count all these walks. Okay. Now count the other rug. Walk, walk, walk, walk, walk. See? That one has more walks. 

Eddie: No fair. You cheated. You walked faster. 

Wally: I don't have to walk. I can just look.

Eddie: I can look too. But you have to measure it. You need a ruler. About six hundred inches or feet.

Wally: We have a ruler.

Eddie: Not that one. Not the short kind. You have to use the long kind that gets curled up in a box.

Wally: Use people. People's bodies. Lying down in a row.

Eddie: That's a great idea. I never even thought of that.

Wally announces a try-out for "rug measurers." He adds one child at a time until both rugs are covered-four children end to end on one rug and three on the other. Everyone is satisfied, and the play continues with Wally as the giant on the rug henceforth known as the four-person rug. The next day Eddie measures the rugs again. He uses himself, Wally, and two other childen. But this time they do not cover the rug.

Wally: You're too short. I mean someone is too short. We need Warren. Where's Warren?

Teacher: He's not here today.

Eddie: Then we can't measure the rug.

Teacher: You can only measure the rug when Warren is here?

Jill: Because he's longer.

Deana: Turn everyone around. Then it will fit.

(Eddie rearranges the measurers so that each is now in a different position. Their total length is the same.)

Eddie: No, it won't work. We have to wait for Warren.

Deana: Let me have a turn. I can do it.

Jill: You're too big, Deana. Look at your feet sticking out. Here's a rule. Nobody bigger than Warren can measure the rug.

Fred: Wait. Just change Ellen and Deana because Ellen is shorter.

Jill: She sticks out just the same. Wait for Warren.

Fred: Now she's longer than before, that's why.

Teacher: Is there a way to measure the rug so we don't have to worry about people's sizes?

Kenny: Use short people.

Teacher: And if the short people aren't in school?

Rose: Use big people.

Eddie: Some people are too big.

Teacher: Maybe using people is a problem.

Fred: Use three-year-olds.

Teacher: There aren't any three-year-olds in our class.

Deana: Use rulers. Get all the rulers in the room. I'll get the box of rulers.

Eddie: That was my idea, you know.

Deana: This isn't enough rulers.

Eddie: Put a short, short person after the rulers - Andy.

Andy: I'm not short, short. And I'm not playing this game.

Wally: Use the dolls.

Teacher: So this rug is ten rulers and two dolls long? (Silence.) Here's something we can do. We can use one of the rulers over again, this way.

Eddie: Now you made another empty space.

Teacher: Eddie, you mentioned a tape measure before. I have one here.

(We stretch the tape along the edge of the rug, and I show the children that the rug is 156 inches long. The lesson is done. The next day Warren is back in school.)

Wally: Here's Warren. Now we can really measure the rug.

Teacher: Didn't we really measure the rug with the ruler?

Wally: Well, rulers aren't really real, are they?

I recognise this kind of thing from my own teaching: the children are thinking about things a certain way, and I'm eager to present my ready-packaged solution to all their needs. But it's not time yet. The value of a transcript like this is that it puts our teacher noses in it! Are you really wanting to replace this brilliant conversation and thinking with your pale version of progress?

It's interesting here how the children's thinking around measuring the rugs with each other is so rich - there's debate, there's problems, resolutions, ad hoc rules, modifications and concensus. The teacher's tape measure solution is relatively meagre. It may be 'right' from our adult perspective to use a tape measure, but where the children are now, 'Well, rulers aren't really real, are they?'

Young children are learning incredly fast, learning more than we adults are able to. But they don't necessary learn in the chunks of time we would like them too. And they don't necessarily learn in the 'efficient' way we would like them to. They repeat things again and again, seeming to need to do this to realise something or some things. Here they need people lined up on the carpet. That's a lot more interesting to them, a lot more what they need than any next step.

What could the teacher do here if not be the supplier of the answer, the next piece of information? I'd say, enter into the moment without itching for the next step. I'd say, take a picture of it and put it up on the wall. And, document it, to discuss what the learning and theory building is with other teachers, And of course, share it with the parents. A transcript like this is precious. Even a remembered summary of it is something that can help us to think about real learning.

Sunday, 21 March 2021


Simon writes:

Back in 2015, Estelle and I ran a workshop on talk in the classroom. I was in Grade 3 (Year 4), Estelle was in Grade 1 (Year 2), and we were sharing ways we encourage students to talk more in our classrooms. To prepare for it we visited each other's classes, watched some established ways, and tried some new ideas too. 

Estelle, as well as being a wonderful friend, is a fellow edu-geek. We read, we discuss, we even go to see the odd French education film like Le Maître est l'Enfant and Être plutôt qu’avoir?

Some things were working against student conversation in class back then. We teachers have some tradition behind us, and a lot of curriculum to get through. We end up listening for rather than listening to.

This distinction, listening for rather than listening to is one Helen Williams uses lots, but it seems to have had multiple origins. Max Ray-Riek was one of them, and if you have five minutes to think about this a bit more, this is a now classic talk of his:

I'm not sure how much we were thinking about listening to at that point. We were thinking about 'What do you notice? What do you wonder?' to hear what the students actually have to say, but we were very much orchestrating what lessons were all about and the kinds of things that might be talked about in them. I had discovered people in the mathematical Twitter world who were guiding me towards close listening to students. Estelle was and is a brilliant listener. But we were still listening mainly for the matter in hand, the curriculum content.

In June 2016, Estelle and I knew we were moving down into Early Years, Estelle as the coordiantor. We went off to Prague for a great few days on play. Estelle would be leading the Early Years through a lot of change, but I'm not sure if she knew how much change there would be.
Loose parts play in Prague

That September, and for the next few years, I taught in K (5 and 6 yos). We still had specific mathematics lessons, for 45 minutes each morning. At the time, I blogged about the some maths aspects of this:
'There was a lot of space for the students own creations and explorations. I was keen to keep a sense of agency, and tried to respond to any initiative. This was as important, I feel, as the exact direction we went in. That sense of 'this is an inquiry we're following because B started us off with this; let's see where it goes' is something I really want to nurture again, and even more so, next year.'
Last year, Estelle and Rachel were in K and I was popping in, especially for maths lessons, in my role as STEAM coach. From September, they changed the mathematics so that there was a lot more choice in the range of activities available. Then in November, Rachel went off to Ljubljana for a play-based learning course, and when she came back she said they were recommending moving away from specific subjects at specific times of day and onto a continuous provision where children could choose what they engaged in for long periods of time. Estelle listened and was really responsive to the idea and Rachel's enthusiasm for it. In January, the K classes became more truly play-based, with children having a lot more choice.

As STEAM coach, I wanted to keep some record of the changes, so at the end of the month I asked Estelle about the changes. Here's a few minutes taken from that conversation:
It was not easy for me, this change. I had really enjoyed having a time with the K students each day where we would be exploring maths together. But of course, there had been a cost in terms of student autonomy and agency. Now children would choose more how to spend their time, and it would be our job to make sure the mathematics offer was atractive and just right!

The PYP (Primary Years Program of the International Baccalaureate) was changing too. In 2018, the PYP document The Learner described a major shift towards play for young children. Here's part of a table that describes the changes:

Move away from

Move towards

Predetermined time structures and routines

Flexible timeframes and routines that are responsive to the needs of the students

Pedagogy that centres around instructional processes for students and is teacher-led

Play that is co-constructed between students and teachers

Repeated large-group experiences as the basis for all learning

Whole-group experiences at pertinent learning moments

Actually, we were already well on the way to the right hand side, but there was more travelling to be done.

We've been thinking a lot about our pedagogy. During lockdown, we zoomed about Kath Murdock's Power of Inquiry. And when it was over, we met in peron after reading Anna Ephgrave's Planning in the Moment.

This year, we've been doing Saturday morning sessions with Anna Van Dam, debriefing afterwards to apply all the things we'er learning to our classes:
We've also discovered, and are loving, Vivian Gussin Paley's books.

And Rachel, Estelle and I are together in PK, with 3-5 year olds - in Sun, Star and Moon class! It's a kind of homecoming in some ways - finally the choice and playfulness that I tried hard to allow space for in lessons is the actual stuff of our time!

We're becoming more and more interested in listening to children. And finding out that our questioning is often not helping. Julie Fisher's Interacting or Interfering: Improving Interactions in the Early Years crystalised for us how some of the tools we might use in adult conversations, such as questioning, are often actually a hindrance with young children.

And as John Mason writes in the context of teaching mathematics:

"The secret of effective questioning is to be genuinely interested not only in what learners are thinking, but in how they are thinking, in what connections they are making and not making. Genuine interest in the learners produces a positive effect on learners, for in addition to feeling that they are receiving genuine attention, you can escape the use of questions to control and disturb negatively. Instead of asking for answers, which in most cases you probably already know, you can genuinely enquire into their methods, their images, their ways of thinking. In the process, you demonstrate to learners what genuine enquiry is like, placing them in an atmosphere of enquiry which is, after all, one view of what schooling is really intended to be about."

Especially as most of our young students don't have English as a first language, we're finding that watching can be an important part of listening. Seeing what our students do, trying to guess their lines of thiking.

Slowing the pace down, giving students our attention for longer periods of time.

Here's Estelle in the forest the other week. I managed to video part of a much longer conversation. 

Young students like this don't usually respond well to the direct approach, to questioning. It's more about creating the conditions for relaxed conversation. Here Estelle establishes a slow pace, peeling open acorns, seeing that some of them have turned to powder inside, seeing that there are holes in those ones, talking about worms. It's a comfortable situation, and sure enough, a student begins sharing his knowledge about worms. 

After this, Estelle starting delving for acorns that were beginning to grow. 

(We took some back to the classes, and there are some seedlings now!)

There's a quality to the listening which we sometimes get. There's a giving attention to whatever the student wants to say, in their own time.

Estelle, Rachel and I have been documenting 'Moments in the Day' - times when we watch a student at play, document it, try to see what learning is happening, and bring it to the EY group for discussion.

An example:

Estelle writes:
"G is sitting in the sun and holding up a jelly digit. What catches my eye is his quietness and his gaze on the object. I go over and ask what he’s noticed. I try to go carefully and softly with my interactions wanting to avoid taking over.

G says these things at different points in the play and conversation.

“I just cut it in half and it did that.

What is this? (holding up the digit zero)

Every time I do this it does that (points to two bubbles as they move in opposite directions when he presses his finger down.

I made four now (bubbles).”

Observations, sharing, trial and error, comparing across objects that are similar but not exactly the same. Counting the number of bubbles. Thinking about letters and numbers. It was G on the number hunt who was asking about the zero and saying it was an o.

A joins us:

“It’s broken into more!” Bashing it and making loads of tiny bubbles.

...There is a quality of being in the moment, attention and pause which was noticeable."Documenting together (and thanks to Anne Van Dam for encouraging us in this) is making us more keen to listen. Reflecting together on what we've documented is making us realise how much there is in what we hear.

We're all still learners in this art. We have to tell ourselves to leave space, to not do all the work. 

It's kind of odd that it's hard for us. Estelle, for one, has listening as a superpower. I know how good she is at listening to me and other friends, and to colleagues at school. But, the challenge is harder now. Making space for students who, at 3, 4 and 5 years old, and are still building up the confidence to speak in English. Making our interactions carry little weight, to not swamp their tentative beginnings at expressing themselves.

Now at least, and at last, our antenae are twitching, waiting to hear what our students want to say, trying to read in their play the theories they are building. (Thanks again to Anne Van Dam for that emphasis. See the previous post for more on this.) 

"In listening to others, accepting them in their irreducible difference, we help them listen to themselves, to heed the speech of their own body of experience, and to become, each one, the human being he or she most deeply wants to be." from D M Levin, The Listening Self, quoted in this piece by Brent Davis.

Saturday, 20 February 2021

Dinosaurs and thinking

 Rachel, Estelle and I have been on a great course with Anne van Dam on Saturday mornings in January. 

One of the things she asked us to consider was the theory building that we see in children’s play and conversation, and to use this intentionally as the basis for planning.  

I wondered about some of the times play doesn’t seem to be theory building - for instance when some of the students like to get the dinosaurs and bash them together in dinosaur fights. Maybe I needed to look closer, Anne suggested. So I did. I recorded a little of U and V playing with them.
After bashing for a while they say:

U: Le he apretado el cuello.
V: Todos los dinosaurios son fuertes pero este es el mas fuerte, a que si?
U : Vamos a ver los cuellos. Son los mismos?
V: No, el con la boss, el mio es mas grande.

I don't speak Spanish (yet - I'm puting in some Duolingo time on it!) so I asked our colleague Irene to translate from the video, and she kindly sent back:

U: I held his neck.
V: All dinosaurs are strong but this one is the strongest, am I right?
U: Lets see how tall the necks are. Are they the same?
V: No, the one with the hump, mine, is the strongest.
So... thinking about strength, where the strength is, comparison, anatomy… lots going on!

More specifically, they seem to be considering the features of the dinosaur that might contribute to strength, and beginning to measure those features. They are interrogating each other and seeking evidence: ‘Let’s see...Are they the same?”

Anne encouraged us to think about next steps, and as it seemed both to be fascinating to some of the students, and to be a place where they were thinking critically and theory building, I thought it was maybe worth building on. What could the next steps be, I wondered?

I showed this image, and asked what is strong here:
I was very pleased that T wanted to contribute lots. Though he’s 3 and beginning with English, he knows a lot of dinosaur names and has a lot of interest in them.

Alongside this, a question about the relative strength of two pyramids came up. Which is the stronger?
We tried it out with our dot stones, which have graded weights, to see how much each pyramid could support.
It turns out the square-based pyramid is stronger:
Back to dinosaurs...

At some point I tried playing one of the many simulations of triceratops facing up to a tyrannosaurus. Children started asking to watch more of these, and I found some that could work. I was stopping the video at various points and getting lots of observations and conversation.

I also wrote this up and shared it with Estelle and Rachel and the team in our regular Monday meetings where we look back at specific moments of play and learning. Afterwards, I wrote:

I’m finding putting this down and sharing it with colleagues is helping me think of next steps. It’s bringing it into focus for me - I don’t really see the way ahead, but I feel like there are enough clues in what’s happened already, and in our shared knowledge, to come up with some ways forward. It helps to have detailed evidence to work on, and might give us pointers to more general matters about pedagogy too.

Then I saw on Twitter a story about palaeontologist Dr Elsa Panciroli who had stumbled over a fossil Stegosaurus bone on the Scottish island of Eigg. If we could talk to her, it might help us to see that people - scientists - do the work of finding out about these creatures. It might also show the students that they could ask questions and get answers. And of course, tell us more about dinosaurs. I tweeted to her - and she agreed to Zoom with us! 
She became our 'Dr of Dinosaurs' and answered the questions brilliantly. It was great to see children that were just beginning to feel confident at school put whole sentences together in English asking their questions and getting answers.

If I was in any doubt about the impact this had, one of the parents shared how her son had been so animated about our meeting that she'd written down what he'd said:
And when I asked about favourite dinosaurs, Stegosauruses were now the most popular.
I don't know where this will go next. But it feels good to be following up on not just one  on the things that interests some of the students, but on their thinking about that interest, and to be making connections outwards from there.

Monday, 15 February 2021

Number books

  Simon writes:

Estelle has tried helicopter stories before - hearing children's stories, writing them down for them exactly as they tell them, and then giving the class the chance to act out the stories on a classroom 'stage'. (We've discovered Vivian Gussin Paley, who made great use of this approach. We've read her brilliant Mollie is Three, and are moving onto The Girl with the Brown Crayon.)

Z (4 yo) sat down to begin a book. Like this. She tells me what to write in my story book, and wants it on the page too.
The second page followed:
Two other children had joined her:
At some point, W decided this was going to be a number book. There was one cat on the first pages, and two cats on the next pages. It would carry on like that.

She wrote out the numbers, copying them from the wall.
'I've just learned to write five!'
And into the afternoon:
From seven onwards, there was a lot of counting and checking the numbers:

Z was really pleased with her book, smiling and laughing at how much she was writing and drawing, and loving her creations! The whole group spent about an hour on it in the morning and another in the afternoon. It was also a leap forward with writing numbers. Perhaps she needed the reason to use the numbers to be motivated enough to try and write them.

A week later, Z made a little book with some pretend writing. Estelle had been talking to me about encouraging her class that it was OK to do this, but, without a word from me, Z knew this was a good thing to do.

The book grew to be up to ten. Later she read her writing to me:
  1. All about the number 1, because 1 is a tiny number and zero is nothing.
  2. 2 and 2 equals 4.
  3. Number 3 is big enough to be a monster.
  4. Number 4 is big enough to be a zombie.
  5. I feel so alive. You know that you've arrived when you're with number 5.
  6. Look what you do. You look good.
  7. Let's say number 7. Look how I work with this information book.
  8. I'm 8 always. And you can see 5. This number likes 5 because they're friends.
  9. This number is very tricky, but more easy than paint.
  10. And then, let's look at number 10. 10 is big enough to be a dinosaur.
I wrote it all down in my story book, and read it back to her. Later she asked Steph to copy it all into her book. Again, she was really pleased with her book, and keen to take it home.

It's very interesting to me how different people's paths to loving numbers are. Counting is really working for some children, with our How Many? images going well, Pass it On and Numicon games are good with others, Numberblocks episodes are doing it for others (Numberblocks pop into Z's book at number 5, with a line from the number 5 song). For Z, a way in is being an author, and finding out that she could make substantial books that had an integrity of their own and would be exciting to share with others.

Since then, I've shared her work with the class, making some blank zigzag books available with numbers 1-7 on pages. I'm going to get some blank stapled books out on the table too.

Saturday, 21 November 2020

fascinating water play

A short while back I was talking with Estelle and - I can't remember what the subject was - was it play schemas? - anyway, the subject of water play came up. It was something we both wanted to look into a little more.

We put water out, because it fascinates children. We think they must be learning if they're so active and so fascinated. Children will spend half an hour or more pouring and filling and emptying and much more. But what kinds of things are they investigating? What is interesting them in the water play?

I've been watching students play, asking myself what's going on, sometimes asking students but not getting much reply, and asking my colleagues.

If you haven't thought about this already, you might like to stop and think about what the fascination is with water play, before you've read other people's answers.

I thought I'd try Twitter too. I posted a photo of a student playing, and asked, 'What is it about playing with water that makes it so fascinating?'

Syreeta answered the call: 

We enter the world via the amniotic sac of fluid. Perhaps it reminds us of our beginning.

It's true, we are water creatures. Not only that, but we come from a very long line of water creatures.

Once I'd made clear that this wasn't a rhetorical question, answers came flowing in.

Kassia tweetedFilling and pouring seem to interest kids (and adults!) of all ages.

(Filling and emptying had been my first though too: satisfying to get to the end points - full and empty - and then to reverse the process. Maybe satisfying to so easily change the state of something into its opposite. Also, it doesn't have to be water: it can be rice or sand or wood pellets .)

Christopher repliedTrue. At least 40% of fun of home brewing is playing with water. Which, by the way, involves siphoning. Do these children have access to a siphon? Cuz if you're gonna make a tremendous mess, a siphon is a SUPER fascinating way to do it.

Must siphon!

Aston too was clear, adults have the same pleasure: It’s not just children- I’m 38 and been working on our rink in the back yard. Nothing more satisfying then watching the water spread out and freeze.

Jack wroteI think in part it hits a sweet spot between something that acts on its own and thus gives a sense of mystery and something that is controllable and thus reassuring. Also their is the slight drag of moving through it which is wonderful tactile feedback from the world.

The tactile feedback links in with what Steph had said: it's a sensory experience in a way that most of the day isn't. Estelle's impression too, putting her hands into the water, was about the sense of touch: how we felt the cold of the water entering it, and the warmth coming out.

David also commented on the meeting of opposites in water: 

Maybe it’s because water is so paradoxical:

You can see it’s there, but you can see *through* it. You can feel it, but not grasp it. You can make mess with it, but the mess disappears. You can carry it, but it can carry things too.

Michael too saw an oppositionIt is solid enough we can shape and change it but only for a moment, sending us back over and over again to try again.
Also it makes really satisfying sploosh sounds.

Westley thoughtIt's magical, like fire. We can control it but it also has a life of its own.

Justin also had a word about fire: It burns less than fire.
Face with tears of joy
Watching students play should provide some of the answers. One of the things that seemed to fascinate this student, was how you could tilt the container just a little and the water would swill to the other end and start pouring. Shake it, and it comes out in all sorts of ways! 

He spent about 40 minutes with this water. He liked the bubbles too. Sometimes, the bubbles made a kind of noise. Here's Estelle listening to it.

When I got home, I showed Pam some photos from the day, and asked her too what the fascination is in water play. She had a lot to say:

Water is just the most fantastic material. The way it has so many interesting properties, shapes, colours. The way the light passes through it. The way it twists as you’re pouring it. It doesn’t just go from one place to another. When you pour it, it catches the light, it sometimes has a smooth bent surface, it cascades, it’s in drops, it might fall in zigzags through the air.

You can hold it, but you can’t hold it. You can scoop it, but you can’t control it. If you put your hand in to pick something up, it’s not where you think it will be.

There’s something mysterious about it.

If it’s in a transparent container, it’s different according to what side you put it into. There’s nothing boring about water.

And then there’s bubbles! Even in water without squeezy in, there’s a bubble when you drop something in.

It’s funny as well. You splash it, and it goes on your clothes but there’s no harm – it will dry out. Maybe a bit of water on the floor. But it’s just fun.

You’ve got something floating and then it sinks, you can experiment with it just by playing and having a laugh. It’s fun.

Meanwhile, more tweet answers were washing in.

Dan suggestedI wonder if asking why might not get to the heart of what it's *like* to play with water? What’s it like to ... might get closer to the experience?

Amanda wroteIt's the one substance besides air that we have a lot of regular contact with, but it acts differently than air, in very interesting ways. When we go to the beach, it seems like access to a totally different world. It's incredibly powerful.

We don't think much about air because for the most part it's not visible or tangible to us. But water does cool stuff!

Poly tweetedI am totally with this little one: watching water move is fascinating! Might be interested in this book by a marine biologist, all about our fascination with all things water and why it makes us happy.

I am interested! It seems to me that the exploration of water, the experimenting and contemplating is carried by a comfort with water, the pleasure in being close to it. While we're enjoying water, waves of learning splash over us too!

maggie and milly and molly and may
went down to the beach(to play one day)

and maggie discovered a shell that sang
so sweetly she couldn't remember her troubles,and

milly befriended a stranded star
whose rays five languid fingers were;

and molly was chased by a horrible thing
which raced sideways while blowing bubbles:and

may came home with a smooth round stone
as small as a world and as large as alone.

For whatever we lose(like a you or a me)
it's always ourselves we find in the sea

e.e. cummings

MB talked about our attraction to water: And as well as all it does, it’s really good for us to touch the elements of nature. Water, sand, Earth, wood, pebbles etc. Think we are intrinsically drawn to it.

Sarah also spoke about the emotional power of water: When my son was born,he was always unsettled and barely slept. He didn't sleep through the night for four years. Water was the one thing that calmed him.He would immediately relax and was soothed. He's now 19 and still loves water. Water can be restorative as well as fascinating

MariaIt's splishy, splashy fun!

"From one million miles away our planet resembles a small blue marble; from one hundred million miles it’s a tiny, pale blue dot. “How inappropriate to call this planet Earth when it is quite clearly Ocean” Arthur C. Clarke, quoted by Nicole

All this has of course made me only more keen to have water play as a big part of our provision for our 3, 4 and 5 year olds. This week we've had red strawberry-scented water, (a bit too) blue peppermint-scented water

 and yellow lemon-scented water.

It's also made me want to watch closely, an see what it is that children are attending too, and experimenting with. 

Friday, 13 November 2020

Mathematical joy!

Y is the youngest three year old in my class. Though she's becoming more comfortable at school each day, she is still tentative about joining in activities with other children, still holds back from lots. The other day she and I were in front of the big tin cans that hang so they can be hit with bamboo beaters. She watched as I beat out a rhythm on a can - and when I then held the beater out to her, she ran away.

But, when it comes to mathematics, she is in her element. She has a kind of exuberance, of joy playing with numbers and shapes.

I first noticed it when I'd put some Numicon out back in September. Steph - always upping the opportunities for mathematics! - added some jelly numbers. I was amazed when Y quickly matched up the Numicon pieces and the numbers, and seemed to be really enjoying it too!

Doing some mathematics is one of the places Y seems to feel really comfortable alongside other students. Here she is on the left, arranging some of the dot stones:

And here she is in October with the same student, sorting the magnetic Numicon-type shapes:
I'm really pleased that she is happily sociable in her mathematical play. For her, but also for the others - to spread the mathematical joy!

Later in the day, she was at the board with another student, this time putting the pieces up and calling out the numbers as they did. Steph suggested they add the word 'Splat' which seems to give it a bit more drama, making it into more of a game:

We've played this game quite a lot. Now, in November, she's changed it to be 'Splat, two fives!' as she puts a pair of  pieces up with both hands. When it's 'Splat! Two twos!' it makes her laugh.

It's evolving all the time. I brought a pen out and wrote some of the numbers yesterday, and she took the pen from me and enjoyed doing that too:
The idea of equality came up in the Numberblocks episode we watched, and as she was doing this, I was saying how the six and the four together are the same as, are equal to the ten. This might be something we develop more, depending how much it interests her.

Some thoughts:
  • Y's joy in all this is something precious. I don't want it to be dampened by taking any of this out of her hands. She is leading the way, sharing what she's doing with me, and I want to keep it that way.
  • Like I said, the sociability of her enthusiasm is a real asset for the whole class. In these mathematical moments she forgets any diffidence and is happy to take turns or work together with a partner. 
  • I don't think it's important to go too much into the written symbols for numbers at this stage, but Y clearly has an interest in this, and I'm happy to respond to that too.
  • I'm really interested to see how her pleasure and understanding grow, and how it spreads through the class.

Saturday, 24 October 2020

Seeing the mathematical: Filling

I'm teaching in PK this year. It's my first year teaching the 3 and 4 year olds! Luckily, I've got a wonderful team to work alongside, who I'm learning so much from, and who do a lot to compensate for gaps in my knowledge.

One of the things that intrigues me is what mathematics looks like at this age.

I don't know the full answer to this - does anyone?? - but I do know it doesn't look like it does later on. Here are some of its characteristics, from my point of view.


  • woven into all sorts of other activities - art, building, role play, small world play, block play;
  • not mainly about numbers or counting;
  • mostly expressed through spatial means, often with physical objects;
  • hardly ever symbolic (for instance, using the names and written symbols for numbers);
  • often something that happens for a few minutes and then it's over for now;
  • not about trying to remember anything;
  • difficult for us to see, or recognise as mathematics.
We may not even recognise that we're not recognising it. It's a little like the way the substance of what becomes a tree enters the tree - not only are the roots underground, but the tiny root hairs where the uptake happens are hardly visible to us even when we dig. And, that's not all: most of what becomes trunk, branch, leaf, flower, fruit comes from the air, entering the tree through tiny holes in the leaves. It certainly doesn't arrive as wood in any way! And what does enter the tree - the carbon dioxide, the water, the minerals - doesn't enter in any obvious way - it enters through a million invisible doorways.

What are the tiny mathematical doorways for young children?

I came across an interesting list in an interview with author Grace Ling:

'I think the biggest challenge was to get out of the mind frame that “math is numbers.” I kept thinking it had to be kids counting, but after many talks with Marlene Kliman, a senior scientist and math specialist at TERC, she really opened my eyes to how we use math without even knowing it — sorting, sharing, comparing, finding, waiting.'
I was particularly struck by a couple of those items. Finding for instance - how does that link to mathematics? When I tweeted her that as a question, Grace answered, 'I mentioned finding because in “What Will Fit?” Olivia finds something to fit her basket.'
What Will Fit? - one of Grace Lin's Storytelling Math books
Yes, and finding because:
She has set herself a task;
She has set herself constraints;
She has a way of measuring whether what she finds will fit the constraints.
This is especially mathematical in my view, because she has her own inquiry that she is following through on.

In this case, the finding has to do with the size of the pumpkin - that it fits in the basket. There are no numbers involved. Here it's continuous magnitudes that are important, and these manifest themselves by a kind of comparison - does it fit in the basket? (My post on continuous magnitudes is here.)

Do we recognise this kind of fitting as mathematics?
I like blogging about this, because it helps me to get my thinking clearer - to focus in on the mathematics. Fitting in the pumpkin-in-basket case is about filling. Filling seems to link closely with the play schema of enclosing. There's a boundary and you put things inside it. With filling, we often want to completely fill up to the boundary, to fit in as much as possible. Sometimes, we dispense with the boundary, and just try to cover the space without leaving any gaps.

We do a lot of filling in our classes:
Filling a peg board
Filling containers with water
Filling 20 cm square trays with square tiles
Filling space without gaps with magnetic Polydron triangles
Filling triangular holes with pattern blocks
Filling a square tray with Tangram pieces
Filling a chessboard with glass pebbles
Filling a Numicon board
What are some of the qualities of the mathematics here? 
  • There's a rigidity in the frame, just as there was to the basket that had to fit the pumpkin, or sometimes in the way the pieces fit together - that is: there are constraints;
  • There is also freedom - the space can often be filled in a variety of ways. Take this last image of the Numicon. The student chose to try and fill with the light blue "2" pieces and the orange "1" pieces (we talked about them as 2s and 1s) until these ran out. Another time she tried with other pieces. She's also thinking a little about symmetry;
  • There's often a kind of beauty to the finished product;
  • The activity is often quite abstract - it doesn't have to link with a narrative. Children get used to abstraction;
  • The activity is about equivalence, equality - all the parts add up to the whole; and the different ways of filling are equivalent to each other;
  • There are discrete or continuous magnitudes involved - for instance the number of holes in the Numicon pieces or the space they occupy;
  • As well as the final product, there's a process, and the process could be different for the same end product - for instance in the way pegs are added to a pegboard: some students go round the edge first, some start in the middle, some fill randomly. There's time during the process for conversation, and comment on what's being done;
  • Where there's a boundary, there's usually a clear end point - when it can be seen there's no more space. The product, or a photo of it, is an object that can be celebrated, discussed and reflected on. Is there a pattern, symmetry? How is the student's work developing?
At the moment, it's hard for me to do the conversation part much. Most of my children have English as second or third language. Sometimes I'm talking to Spanish speakers in French. But a lot is communicated about the students' intentions in the choices they make during the fitting and filling.

There's probably a lot more to this than I've listed, but already that's quite a lot. A look at the overarching concepts referred to in the Diploma Program (for the 16-18 year olds) of the International Baccalaureate shows surprising links. Or, perhaps they shouldn't be surprising, since the more conceptual we get, the more generality:
  • approximation, 
  • change,
  • equivalence,
  • generalisation,
  • modelling,
  • patterns,
  • quantity,
  • relationships,
  • representation,
  • space,
  • systems,
  • validity
Which ones might crop up in filling?

The ones that jump out to me are equivalence, patterns, quantity and space.

Children aren't necessarily articulating anything about these yet, but they are nevertheless thinking mathematically as they construct examples, thinking for instance implicitly about equivalence. When we ask 5 year olds to make this more explicit, the background they've had of experiencing equal areas filled makes this a small step:
Equal-area Cuisenaire rods
Equal area pattern blocks