Building mathematics

 One of my students, A, has been with me for over a year now. He was just three years old when he started, and without much English. His favourite thing was to get the Playmobil cars out, fill them with people, and quietly act out stories with them on his own.

He did lots of other things too. Here he is in the playground, back then in September 2020, with two other students, making a house with the giant Polydron:

We've watched lots of Numberblocks (see my post on this), where numbers are represented by blocks.

I haven't asked anyone to do this - most of the things that happen in our pre-K (Early Years) classes are child-initiated and developed - but A has been spending a lot of time exploring numbers by building cuboids out of Polydron. While others were content to use our pre-existing interlocking cubes, A wanted to make the cubes himself from squares. He makes other things too, like this house with an interesting floor plan:

But here's 8 (called 'Octoblock' in Numberblocks), built by A from individual cubes:

There is a Numbrblocks episode called Terrible Twos, where 4 splits into two 2s who tickle the other numbers while they sleep which makes them split into ones. Here is A's version in magnetic Polydron, a 4 made of two 2s.

It even comes into his story writing, which is usually about T. rexes and triceratopses:

The building continues:

Recently he's developed an interest in writing. Here he is practicing writing the numbers:

Children also play with the Numicon and tell me equations. They've heard a lot of these on Numberblocks. Here's one from A:

What I really am pleased about is that all this learning is initiated by him (in a climate of appreciation and encouragement of course). It is very much his own and at his pace. I'm pleased that all the four year old students in my class have also found their own different ways to this, enjoying representing numbers and equations. A has also influenced younger students in the class too. T in particular is a real enthusiast for creating the same kind of Polydron 3D numberblocks.

As the students do, T told me some equations linked with what he was doing, and then told me a few more he knew:

I think it's from these landmarks that young students begin to build their number sense, so what he's doing seems just right to me.

Where mathematics comes from

 At the moment I'm reading a little about materiality in learning - how physical materials and the physical environment and what they can do are a kind of teacher as we interact with them.

It made me think about how that is true for learning mathematics. In my last post, I pondered what young children are learning as they construct a play house out of bricks together and move into it.

One of my tentative conclusions is that in arranging matter, they are learning to arrange matters in the broadest sense of the word. They are learning what it is to try things out boldly and get results, what it is to work as a team.

In an older post, I thought about the natural powers that students bring into play when they do mathematics. 

This is from John Mason and Sue Johnston-Wildery:

The kinds of powers that are relevant for learning mathematics are ones that learners will have demonstrated by the time they arrive at school. Learners have innate ability to emphasise or stress some features and to ignore others, enabling them to discern similarity and difference in many subtle ways. They can also specialise by recognising particular instances of generalisations, and they can generalise from a few specific cases. In addition, they can imagine things and express what they imagine in words, actions or pictures, together with labels or symbols.

(my emphasis) 

A couple of things about both agency and these natural powers. First, we learn them by exercising them in the world, interacting with people and things, primarily through play. Secondly, they don't serve us only in mathematics, but in all disciplines and areas of life. 

That second point is important I think. The roots of mathematics are much much broader than the 'trunk' - stretching out into all areas of play and interaction. Squashing playdough, spinning round and round, having a conversation, singing a song, jumping in puddles, listening to a story. And not just because these things have elements of geometry, measure, arrangement and number in them, as they often do - but because all our growing powers are needed for the learning of any discipline.

I often return to this: 

Maryam Mirzakhani, the first - and only! - woman to win the Fields Medal, only got into mathematics near the end of her schooling:

"What are some of your earliest memories of mathematics?"

"As a kid, I dreamt of becoming a writer. My most exciting pastime was reading novels; in fact, I would read anything I could find. I never thought I would pursue mathematics until my last year in high school."

(from here) 

Could Maryam Mirzakhani's love of reading have supplied important ingredients that later went into her mathematical work? I think so.

The foundations of our mathematical learning are in all our learning. 

Maybe - and I'm blundering into unfamiliar territory here - this connects with that old chestnut: 'Is mathematics discovered or invented?'

The question often seems to be answered without thinking of individual development. As if mathematics were a free-standing thing that wasn't created and recreated each time an individual explores and learns about it.

I want to say that it is very much discovered - in our first steps and first reaching we encounter magnitude. It's in the world that we learn the agency, the natural powers and all the other building blocks of mathematical thinking. It's in how we arrange the bricks in that play house, how we create an inside and outside, how we space the toy cars on top. And in the logic of how one step leads to another. The material of the world is full of the different logics of mathematics. And when we abstract it away from its inception in the physical world, again and again we return to the world at further points - we have already felt the inflection points of a curve when cycling in figures of eight on the tricycle in the playground. We know what a plane is from the many floors we have moved across well before we meet the Cartesian plane. In algebra, we often get a whiff of the idea of magnitude that lies behind the cat's cradle of manipulations we do with symbols.

And of course mathematics is invented too. The directions we take are ours. They reflect our economies and social structures. Counting for buying and selling and taxes. Measuring for sowing seeds - and taxes. Another planet would have a very different body of mathematics.

As usual, my ideas around this are half-formed, rough-draft, and probably too briefly expressed. 

Practically, working with 3-4 year olds, I'm confident that all the story reading and writing, all the conversation, all the art work, messy play, construction, games, roleplay, sensory play, small world play - and of course play with mathematically structured materials and moments - is the best thing, not just for learning in general, but for the learning of mathematics too.

Arranging things

Three of my new students (aged 3 and 4) have been settling in this half term. They have been doing lots of playing alongside each other, but this was one of the first times they cooperated on a common project, building a house.

The conversation was in Spanish, so I couldn’t benefit from it. Once it was built, they had encircled themselves and they enjoyed being in the inside they had created and furnished it with sundry items to make it more of a home:

Our early years team have been working with Anne Van Dam, looking at what ‘working theories’ the children are developing in their play. This follows the work of Helen Hedges. 

“Working theories are present from childhood to adulthood. They represent the tentative, evolving ideas and understandings formulated by children (and adults) as they participate in the life of their families, communities and cultures and engage with others to think, ponder, wonder and make sense of the world in order to participate more effectively within it. Working theories are the result of cognitive inquiry, developed as children theorise about the world and their experiences. They are also the ongoing means of further cognitive development, because children are able to use their existing (albeit limited) understandings to create a framework for making sense of new experiences and ideas.” (Hedges & Jones, quoted here)

This concept of working theories builds on the work of Bruner, Claxton, Gopnik, and Vygotsky. It sees children as actively building their understanding through the thinking and play they do. I like it a lot for its respect of children's thinking, seeing the active way in which children make sense of their worlds.

And yet... there's something in the fit of this to the kind of loose parts play that doesn't feel quite right. I can't put my finger on it exactly...

If I were to list some of the ideas and understandings that children might be building in this activity, I might say:

• 'I can make friends playing alongside other children'
• 'I can get better at communicating by doing it'
• ‘I can create a new space by enclosing it’

• ‘I can align bricks and build them up to a wall’

• ‘Here is a place that I belong’

• ‘I can make this place into one I belong in even more’

• ‘I can build’

• ‘I can work together with others on a common goal’

• ‘We can achieve things together, and it feels good’

• ‘We can bring things into our space’

These are not once-for-all understandings that are achieved and then we move on, but ones that grow and grow through students’ time at school. That enclosing is a schema, ie a repeated behaviour, seems to indicate that the understanding that is built up doing it is something that needs to grow incrementally and be connected with lots of other thoughts and experiences.

The more experienced children in the class seem to have this understanding much more firmly, and to build and create together with so much more ease. 

But, like so much of children’s play, it doesn’t seem to be directly pursuing a question (like ‘Where do birds sleep?). Outwardly, it’s more directed towards making, production than it is to finding out. They are, ultimately, finding out, but the immediate impulse is one of creation. In this way the play seems to have more in common with the arts than with the sciences.

This is a thought that I’m still mulling over.

And now this sketchy diagram that I’ve drawn is making me think. The impulse to create, to make, to perform, to challenge oneself allows a feedback loop. It puts the results of our action (beige in the diagram) into the world as something that we can observe. A kind of design cycle. We are building up theories about the kinds of achievements we can have in the world.

Helen Hedges writes ‘Learning may appear somewhat disorganized, perhaps appear to move around and then return later to topics, questions and ideas, may call on invention and imagination to connect ideas…’

To me, most learning is disorganised, full of imagination and invention. So much of learning, of building theories, is devoted not to how the physical world works or even to social and cultural learning considered as learning what’s already there. A huge amount of learning seems devoted to managing possibility. I have these materials - what could I do with them? How might I deploy them, arrange them? I have this time with my friend? How might I fill it? I’ll learn to wildly invent, to put some unheard-of combination boldly into place - and I’ll see what happens then.

New Zealand’s Te Whāriki Early Childhood Curriculum has a section called Learning Dispositions and Working Theories. “Dispositions that can be useful for learning include playfulness, whakatoi (daring), persistence, resilience and imagination. Children also develop dispositions towards domain knowledge…”

What I’m trying to get a grip on doesn’t seem to simply reduce to dispositions though. It is a more disorganised-seeming, less direct way of obtaining knowledge about what daring and playfulness can achieve, what can be done with freedom and within necessity, how the social and physical environment can be remixed. It centres around agency, and uses whatever is at hand to achieve its undefined aims. It achieves its goal of developing capable and skillful being and making in the physical and social world, but its means are more indirect than what comes to mind when we think of theory-building: curiosity -> question -> search -> answers.

This simple house the children make seems to bring so many aspects of learning together. Mathematics is in there too. The straight line of the bricks, the way they are placed in the same orientation, and on top of each other to create layers. The way the lines of bricks are parallel to create the rectangular floor plan, and how the cars are arranged, spaced out evenly.

That word arrange. It means both to place things by some kind of design, and also, metaphorically I suppose, though we're no longer conscious of any metaphor, to organise anything in life. 

Maybe the play follows a similar path to the way the word has travelled? We play at organising loose parts, we learn, ultimately, to organise our lives, whatever the variables it gives us.

And now I'm curious what the roots of that word arrange are:

Ah - putting something in a ring. And there's the word rank, like ranks of soldiers.

It's always fascinating to see the proposed Proto-Indo-European root of a word, and all the offshoots of that:

I find this nexus of related words more than interesting. It's almost as if play recapitulates etymology. We encircle ourselves with loose parts, we line them up in ranks, and doing this we are researching, building working theories about our own agency, our abilities to work with others to build a home, our abilities to arrange matter and matters.

(Adapted from notes we each made for our work with Anne Van Dam)

Numberblocks

In PK, we've been watching the first two series of Numberblocks. These present the numbers zero to ten as different characters who, by the normal rules of arithmetic portrayed dramatically, have the power to transform into one another.

If you're not familiar with Numberblocks, you could watch the first two minutes of this video. (You don't get full episodes on Youtube, you get composites of a number of episodes. We've watched Series 1 and 2 on DVD and on Netflix.)

What do I like so much about it?

Our young three to five-year old children like it so much, want to watch episodes again and again, go home and watch further episodes.

Numbers are represented as arrangement of cubes. All sorts of things become clear like this. Number as area. Addition as combining, subtraction as splitting. Odd and even-ness à la Numicon. Square numbers. Arrays of numbers. All sorts of interesting avenues for investigation are opened up. For instance in the clip above, the numbers are represented as different polyominoes, different arrangements of squares. What tetrominoes are possible? I'm a fan of figurate numbers, of area-as-number (have I talked to you about Cuisenaire?) because I find it gives children a powerful way into number and arithmetic.

Numbers as character and narrative. We all love a story, and the Numberblocks stories are well thought out and entertaining. They also work because of the mathematics. Again, in the Strampolines episode, the story is about what different arrangements are possible. 'One' is sad because she can only make one arrangement, but is cheered up when she learns that copies of herself make up every arrangement (polyomino factorisation anyone?). 

The animation and the music. Yes, we sing along to the songs!

The equations that emerge as numbers transform are represented flexibly. So we don't just see 4+1=5, we see 5=4+1, and also 5-1=4. Children get a sense of what equality means, and they also begin to see how equations relate to arithmetic before having to write them themselves. Often they want to write equations, and it's something we've built on to give the power that writing ones own ideas brings.

Students enjoy recreating and exploring what they watch. Here, for instance, a four-year old student observed that 25 is a square.

I pointed out that there seems to be one missing, and he had a good think about that:

One piece of evidence about how well it's worked for the students, is doing 'Number of the Day' with them on the first day of their next year in Kindergarten. I've been able to pop in and work with them on this. In the past, students have been a bit blank when we've asked them what they know about 1, or have commented on what its shape makes them think of, but this year there were equations.

By day eight, with a bit of help from Cuisenaire rods, the equations were really flowing.

And here are the day 9, here are lots of the ideas the Kindergarten students shared:

So, a big thank you to the Numberblocks team, and also to Debbie Morgan who has been their chief mathematical advisor.

Sleeper effects

There are two studies I keep thinking about. Both involve large numbers of students. Both concern 'sleeper effects' - effects that don't appear straight away, but emerge later.

😴

The first is a study in Boston Massachusetts, and I heard about it via Alison Gopnik. Public preschools had been made available to everyone, but there was such demand that places had to be decided by lottery. Effectively this produced a very large randomised trial, involving more than 4000 students.

The study, The Long-Term Effects of Universal Preschool in Boston, produced some very interesting results. Preschool attendance did not improve test scores in elementary school. It did however have a sleeper effect that emerged later in high school. More students who'd been to preschool finished high school, and more went on to college.

That's fascinating, isn't it? Four-year olds going to preschool didn't effect their test scores in the immediate years following, but it did have a significant positive effect much later. 

What would be your guess about how this works?

😴

The second study was conducted with an even larger number of students, more than 12,000 this time. They are at the other end of their formal education, at a four-year college. Keith Devlin writes about it in this piece, 

Students are randomly assigned to different teachers, and their test results are collected over a long period of time. There are two really interesting results, and I'll just touch on one of them here.

As Devlin puts it:

But here is the first surprising result. Students of professors who as a group perform well in the initial mathematics course perform significantly worse in the (mandatory) follow-on related math, science, and engineering courses. For math and science courses, academic rank, teaching experience, and terminal degree status of professors are negatively correlated with contemporaneous student achievement, but positively related to follow-on course achievement. That is, students of less experienced instructors who do not possess terminal degrees perform better in the contemporaneous course being taught, but perform worse in the follow-on related courses.

Or, in the words of the authors of the study:

We find that less experienced and less qualified professors produce students who perform significantly better in the contemporaneous course being taught, whereas more experienced and highly qualified professors produce students who perform better in the follow-on related curriculum.

Isn't that interesting! Students of more experienced and better qualified teachers get poorer test results that year - - - but better results in subsequent years!

What a fascinating sleeper effect! What would be your guess at an explanation for that?

😴

Copycats

As an EY team we look back at 'Moments in the Day' together - times when something in the children's play and learning strikes us. We share documentation and discuss. In our last time doing this in the school year, Estelle shared this photo with us:

She wrote:

I’m still puzzled about this activity that S., G. and M. engage in regularly.

It is often initiated by M. but not always (I think). I’m not sure I’ve watched closely enough at the right moment. I wonder what skills they are using here and that makes me think that I almost need to try it myself to find out. Perhaps they will allow me to quietly join in…. Otherwise I could have a conversation with them.

There is definitely a quality to this play that is ‘safe’, mindful and we can assume that it is good for their well-being based on the repetition. Maybe for the artist no. 1 there is a feeling of being the leader, being ‘seen’ and valued. For artist no. 2 perhaps the feeling of making a connection in this way has meaning.

Perhaps I can join in and see what is happening; it all happens so fast.

I've noticed children doing things in unison a lot too, and I'm interested. What do we derive from this? 

As teachers, we think of our jobs as being about building individual creativity, individual agency, so where does this leading and following, this doing (almost) the same thing fit in?

Questions like this are quite hard to get a handle on. We have hunches, but they don't feel like the complete story.

Perhaps we should take up Estelle's suggestion and just draw the same thing together, and see what it feels like 'from the inside'. There's no guarantee that we'll feel the same as the students do of course, but it might help.

What else does copying look like in our classes?

Here's some more examples.

At a certain count, friends are jumping off chairs in unison:

Pairs of students making the same as - or here, the reflection of - each other's designs with the square tiles in trays:

We're at the pattern block table and a student says. 'Simon, let's play I make something and you copy it.

Using large foam pattern blocks to make rockets together:

Painting together:

Tap-a-shape:

And of course, the playing music in unison in the forest in the previous blog post.

So what's going on?

I can't say, but there are certain things I sense might be going on. This passage from Sloman and Fernback's The Knowledge Illusion might help orient us:

Sharing attention is a crucial step on the road to being a full collaborator in a group sharing cognitive labor, in a community of knowledge. Once we can share attention, we can do something even more impressive—we can share common ground. We know some things that we know others know, and we know that they know that we know (and of course we know that they know that we know that they know, etc.). The knowledge is not just distributed; it is shared. Once knowledge is shared in this way, we can share intentionality; we can jointly pursue a common goal. A basic human talent is to share intentions with others so that we accomplish things collaboratively.

Let's make a list of some of the things happening: 

• Feeling comfortable with each other,
• Feeling comfortable with an activity,
• Being in the same space with each other,
• Somehow having an idea of doing things in unison,
• Understanding the proposal,
• Accepting the idea together, sharing the intention, having a joint project,
• One  leading, other(s) following (how flexible is this?),
• Monitoring each other's actions,
• Recreating each other's creation,
• Comparing the results,
• Completing the project.

That's part of what happens, and it's a lot. But there's also the significance. What does it mean to do the same thing together? 

That time the children synchronised themselves jumping off the chairs, was such a moment of joy. It seemed like a celebration of friendship and of feeling great in their bodies, in the classroom and together! Not all the examples are so exuberant, but there's a pleasure and significance in not just being in the same place and time, but in the same self-chosen project.

As an adult, I can appreciate this too. In fact, teaching together with the PK team, we plan our activities together. We then, mostly, work in our separate places. But there's a tremendous affirmation in having the same understandings and objectives, in approving of the same resources, environment, activities. And, of course, bringing our stories of what happened back to each other. Our work together is so intertwined that what we do with our students isn't usually the idea of any one of us, it's a kind of team thing.

Then there's singing Beatles songs with friends. We're not exactly doing the same thing: one of us plays piano, another guitar, another ukulele, but mostly we sing the same melody and words. What is it that's so satisfying about it? There's something in there about the whole being more than the sum of the parts.

In our 'Moments' meeting, Nick mentioned that humans succeeded where Neanderthals didn’t because they shared ideas; they didn’t have bigger brains, it was just that they shared their ideas.This is part of what it is to be human, and what has given us our success.

 

I first came across this idea in Rutger Bregman's great book Humankind. Bregman has this chart:

Following anthropologist Joseph Henrich's modeling, Bregman invites us to think of a planet with two tribes. One tribe, the Geniuses, are great at inventing things, but not so good at sharing their ideas; the other, the Copycats are not such great inventors, but do share. The Geniuses are a hundred times better at inventing. The Copycats on the other hand are ten times better at sharing. Which tribe do inventions spread through most?

The Copycats. 

So, I'm coming round to valuing these times when children get into total synch with each other.

And next year, I'm going to copy Estelle's idea, and try to catch more of what is happening as children copy each other. 

Rulers

Dan Meyer tweeted 

My toddlers are convinced that as they get older they'll get bigger (true) and that I'll get smaller and turn into a baby (mostly false). They talk about taking care of me when I'm a baby. I'm in no hurry to correct any of this.

— Dan Meyer (@ddmeyer) April 22, 2021

and Kassia tweeted:

At 4, my son told me he had figured out “why oceans have big waves.” “Whales jumping!” Its was such a lovely theory I couldn’t correct it. My first mom (on on better days, teacher) move for wrong theories is to say nothing. Which is not the same as doing nothing.

— Kassia Wedekind (@kassiaowedekind) April 22, 2021

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

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.

Listening

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.

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.