Making Numberblocks

We have two students in our pre-K classes, Y in Star Class and G in Moon Class, both five years old now, who are fanatical about fairly large Numberblocks numbers. "I love rectangles!" G exclaimed the other day!

I've blogged about this series before and I - along with all our students - think it's brilliant! There's a page I go to to access all the episodes. As my students are between three and five years old, we're leaning towards the small numbers.

At the start of the year G's parents told me he really loved the series, and moreover, knew the square numbers and 'Step Squad' numbers (Numberblocks-speak for triangle numbers). I waited, but he didn't seem to be showing it much in class. It was really in December that he discovered a fellow enthusiast in Y, that he started to talk about it.

Most of what happens in Star and Moon classes is voluntary - the students choose which play they engage in. We adults put out things that we think will interest and engage them, and we also have a couple of 'meetings' a day with our classes, but for most of the day the students choose from what's provided.

We've noticed that influence is a really important factor in what the students try out, and what they persist in. I wrote a blog post about this - 'Copycats'. So it's to be expected that it would be social factors that would really bring the enthusiasm out into the open.

At first they were asking to see pictures of Numberblocks numbers on the Internet. Then they were making them with Multilink cubes.

On the 19th December, G made these:

That's 24, seen as two twelves, 18, seen as two nines, and 27, seen as three nines.

On the same day, Y was telling Estelle a story to write down, all about Numberblocks.

And also putting together 30, 40 and 48:

These are regular Numberblocks colours. 30 is yellow because 3 is yellow, 40 is green because four is green. And eight is pink.

There followed a flurry of number-building, which continued in the new year.

We allowed them to display their creations.

There's always a mathematical structure in these. For instance, here's 64 as a cube:

There's the sixty, which should be purple really, but we don't have many purple cubes, and the four, which is green.

A new kind of story began:

We act out lots of our stories - and this one was a challenge - but we did it! A whole series of similar 'times table' stories followed from both students.

A concern though. These students are outliers. Would there be any way for the 'copycat' thing to happen? These two were so deep into their number inquiry - would there be a way for the others to access what they were doing? I was giving a lot of time to them- a pleasure for me - but I wanted to be giving that time to more students.

Luckily - the answer seems to be... yes.

W, who is only telling brief stories, told me this:

Ar., one of our three-year olds, told me this one:

G and Y were pressing on... with 125, as a cube. You can see it's structure here:

and 49, as a square:

One way to spread the goodness was to put out the Unifix cube stairs. They seem to always get filled with the Numberblocks colours.

Our furry versions help too!

Then we hit gold. I put out squared paper and black pens. Somehow it was a lot easier to draw them than make them.

Al. drew this one:

Students were enjoying just drawing the grids - five or six new ones joined in, some of them just enjoying reproducing the grid:

An. wrote this story:

We scanned the drawings and let the students colour in digitally. And now there seems to be momentum building, with lots of them engaging in some way. 

W. drew this one - I helped him start off as he's done hardly any drawing this year:

An. did these:

K was very pleased with this. Though he needed me to tell him the size of the rectangles, it was G and Y who advised on the colouring:

I'm excited about the contagion and I'm hopeful that we'll find more ways to build bridges to allow the student-to-student influence that happens in our classes to do its thing.

I'll keep you posted.

things counter, original, spare, strange

As an IB school, "inquiring" is pretty much the first word that comes up as a statement of principles. But somehow, in mathematics especially, it doesn't usually end up in first place.

It's that old thing of teachers feeling they need to cover the material (in our case it's called the 'scope and sequence') and not knowing that inquiry will go much deeper and in fact cover more.

Teachers do of course ask students to discuss things before moving on, activating prior knowledge, sharing vocabulary, and bringing learnt-but-a-little-forgotten concepts back to the front of the mind.

But this time of discussion can also be one of the best places to find the starting points for student-initiated inquiry.

I was visiting a Grade 3 class (Year 4 in terms of the English system) a few weeks ago. The teacher was getting the students to talk about what they remembered from their investigations into 2D and 3D shapes so far. They had made 2D nets for 3D shapes.

One student, L, asked, 'What about 1D shapes... and 4D shapes?'

The teacher is very attentive and responsive and saw an opportunity here. 'We should write down that question.'

It's a question I love, and I suggested I come back the next morning and address it a little. The teacher welcomed that and so I did.

I started by writing L's question, and complementing her for taking something the class were learning about and going on another step with it. Then I asked what the students had to say about this. S said that a line was 1D. He also wondered what 1.5D might be (funnily enough, this is not a crazy question, as I learnt watching this 3Blue1Brown video a while back).

A said that a circle was 1D, and I agreed that the line part of the circle was. L said she thought that a point was either 0D or 1D, and the consensus was that 0D was correct. N said that he'd heard that 4D was a 3D thing that interacts with you.

I then did a bit of talking and showing. I said we could look at the cube and at how the number of points goes up as we go up dimensions, the ones we know about, going from a point to a line to a square to a cube as in the diagram below. I'd brought the straws and connectors along, and I used those to show this. Some of the students could see it was doubling, so we might expect a four dimensional version of the cube to have 16 points or vertices on it.

I said we only have 3 dimensions in our space, and went through what they were in that room.

I asked if they wanted to see a 2D or 3D picture of one. There was a definite desire to do that, and we looked at some representations of the tesseract (the 4D cube).

images from the Wikipedia Tesseract page

It is a kind of wow thing, I think you'll agree.

We talked about a few other things: touched very lightly on Einstein and spacetime. And then I had to go back to Moon Class. I left the straws and connectors and they tried to make their own versions of the tesseract. The teacher sent me some pictures later:

It wasn't an ideal lesson - there was too much talking from me.

But there were some very good aspects to it:

1. The teacher was creating space for conversation where students thinking and questions could emerge. A lot of us are doing this. Some also get students using whiteboards so that the thinking isn't only verbal but diagrammatic and written too.
2. The teacher documented some individual thinking that wasn't in the direction of the planned lesson, but 90° to it. Fewer of us are doing this. We tend to have a plan in mind that we're getting on with and moreover that time of sharing takes quite a lot of attention to orchestrate. There isn't a lot of headspace for things counter, original, spare, strange.
3. The teacher thought some follow-up on the question was worth giving time to. Admittedly, things 4D isn't everyone's expertise, but that is one of the powers of documentation, of writing questions down in this case - it buys time - to talk to colleagues, to think, to google.
4. My best moves in the event were asking the students for their answers to L's question, and bringing the straws along. Those were two things that put the ball in the court of the students themselves.

If there is one thing I think we need to move forward on as a team, it's probably number 2 in the list. We need to be documenting more, to be preserving more of what the students say and do for future discussion and exploration. 

Taking some things with me from Early Years

I'm leaving the three, four and five year olds, and heading up to Grade 2 (which in UK is called Year 3).

There's so much I love about the way we work with the youngest children, and I'm hoping I'll take some of that with me.

Just two of the things for now, illustrated with two photos from the last week of the summer term.

Estelle on Sports Day with students

We join in

We don't want that 'I am the knowing, observing, and assessing adult' hanging over the students. We break that down by playing alongside the children. Plus, we have fun, and see what it's like to do the things they're doing, and we get ideas.

Rachel at the making table with two students

We are quiet

(Not always of course)

We can sometimes be larger than life, charismatic and inspiring, but lots of the time we are quiet. The students have the agency. We are with them, but not dominating the situation. The students know what they want to achieve. It helps to have us there, and there can be good conversations, but mostly it's the students concentrating on what they're doing.

These are two of the things I want to take with me.

Play as the foundation

I get to see the richness of play on a daily basis with the 3, 4 and 5 year olds I work with. They learn through this play, even though it often looks very different to many an adult's idea of learning. It's through this play that their understanding of themselves and the physical and social world around them develops. They also learn what their powers are, how they can arrange things, make things happen, create important moments with others.

I've been thinking about how this is the perfect course to set off on to develop the kinds of abilities, or "competencies" that they will need as adults.

Play is notoriously difficult to define, but I think Peter Gray's definition makes sense. Play is:

1. Freely chosen & directed by the players
2. Intrinsically motivated 
3. Structured by rules within the player’s mind
4. Always creative & usually imaginative
5. Conducted in an active, alert, relatively non-stressed frame of mind

He elaborates on this, to show the power each of these characteristics for children's development and learning (my emphasis, in bold):

1. Because it is freely chosen and directed by the players, play is a major force for children’s learning how to take initiative, direct their own behavior, negotiate with and get along with playmates, and solve their own problems.
2. Because it is intrinsically motivated, play is how children discover, pursue, and become skilled at what they love to do.
3. Because it is guided by mental rules, play is how children learn to plan, structure, and create the boundaries (rules) for activities that engage them.
4. Because it is always creative and often highly imaginative, play is how children exercise and build their capacities for creativity and imagination.
5. Finally, the mental state of play—active and alert but relatively non-stressed—has been shown in many studies to be the ideal state of mind for learning anything new or doing anything that requires creativity or the generation of new insights.

I also came across this 'Learning Compass' from the OECD:

"Developed as part of our Future of Education and Skills 2030 project, the Learning Compass puts forth a shared vision of what students should learn to be ready for tomorrow."

There's some other important elements around the compass, which I've taken off; I wanted to focus on the compass circle itself.

Looking at the dark cyan ring - the 'transformative competencies' -

• Taking responsibility
• Reconciling tensions and dilemmas
• Creating new value

- I'm struck by how similar these are to the Gray's elaborations of aspects of play.

Just to make it clear, let's put them side by side:

Play	Transformative Competencies
play is how children learn to plan, structure, and create the boundaries (rules) for activities that engage them	Taking responsibility
take initiative, direct their own behaviour, negotiate with and get along with playmates, and solve their own problems	Reconciling tensions and dilemmas
play is how children exercise and build their capacities for creativity and imagination.	Creating new value

Play is developing the kinds of competencies with which we might hope students finish their schooling, those with which they not only know about the world, but can make positive changes to it.

It makes me think that play could be right in the middle of that compass! Play develops skills, knowledge, values and attitudes.

I'm thinking about play for young children here, but I'd like to see playful learning continuing beyond that age to maintain the self-directed approaches children have learnt so much from.

I think it's always useful to think about what that learning is like in the specific case. Luckily, I've blogged about some of these concrete examples. Here's a couple of links to earlier posts:

Arranging things - "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."

Folding, cutting, sticking, drawing - "...it's really not necessary for me to be adding anything in to this process: there's so much happening already: theories being refined, interests pursued, skills honed, and much more."

Mathematics Lessons to Look Forward To!

Jim's book is out!

Jim Noble is my friend and colleague in secondary. His classroom just a few metres from our Early Years playground, no doubt he's had to shut his door many times because of the racket we're making! I see him out there too. 

'If the world were a hundred people...'

Human Loci: Creating a parabola

These are part of two of the lessons described in Jim's Mathematics Lessons to Look Forward To! The book details twenty lessons that Jim returns to, experiments with, hones and polishes. 

"Every time I revisit a topic for myself or in preparation for teaching or mostly during teaching, I always notice something I haven’t seen before and this is often pointed out by a student."

Jim is a great storyteller. He's often called on to be the one who puts important rites of passage in the life of the school into words: leaving speeches, introducing speakers and celebrations. He's always assured, natural, entertaining and considered in what he says. 

This book is the same. And Jim is letting us into the heart of his teaching here, there's a vulnerability, at times, a touch of self-doubt or self-mockery. In the process, he takes us back to what makes the lessons tick for him, why they became exciting and vital.

"Deep down I have convinced myself that the roots of ideas are an important part of them. I think the journey from first idea to activity is a really enjoyable, reflective part of the job."

His lessons are not always outdoors of course, but they are all out of the routine, they all stand out as being alive, practical where possible... experiences as well as lessons. And fun.

"It is fun. I have made no apology about this. I have found that I need this as much as students do. Something that adds variety to the global experience, something practical that gets students out of their seats and sometimes out of the classroom and something that makes us laugh a little is always welcome."

Reading a draft of the book, I was struck again by how we have so much in common in our outlook towards lessons. Is it that we've spent a lot of time together since 2004 when Jim came to the International School of Toulouse?

It's not just that we both photograph drain covers for their mathematical patterns, both enjoy seeing and making Islamic geometrical patterns, both value Seymore Pappert's seminal book Mindstorms...

Many - most! - of his twenty lessons do actually have their counterpart in the primary school.

We've both enjoyed 'Numbersearch' and these feature in the book:

If the white triangle is 1, what other numbers can you see?

Jim got pi involved:

If the white square is one...

Like Jim in secondary, I love to get my primary students coding the path of a robot using Scratch:

“If I need to turn 5 equal turns that make a total of 360 then I need to turn 72 degrees”, which generalises to “for an n sided regular polygon, the turning angle is 360/n”. The thrill of making this conclusion is the same thrill as solving a puzzle. The word ‘discovery’ is much maligned and probably inappropriate as it implies a kind of wandering around until you find something then pick it up. This is much more mathematical in nature. You have a problem that needs solving, you have knowledge of the scenario at the ready and you put bits of this knowledge together to deduce new knowledge. Now that is doing mathematics.

Or using dynamic geometry:

How would you construct this rectangle? The others? I really do recommend having a go here. It is really interesting to focus on the different ways it can be done and there are some surprising challenges hidden away in there.

In this activity, the constructed dynamic rectangle is, in a sense, every rectangle. Students get to explore the notion that each construction has a degree of freedom that is entirely defined by the elements that were used to construct it and the order in which they were used. It is a profound mathematical idea that goes beyond geometry into set theory and the anatomy of a variable. In many ways it is a much more natural way to see mathematics that a set of static images might be and really helps get our heads around the idea of generalisation. “Many things here can vary, but the following will always be true”

Jim asks the reader to have a go at parts the activities. Some of them I tried. I had a go for instance at constructing a rectangle four different ways using Geogebra. Each one has a different 'skeleton', the way it's constructed.

(I've come to the conclusion there's an infinite way of constructing a rectangle that can be stretched into any rectangle and these skeletons don't have to involve parallel sides, or perpendicular sides either.)

There is really no other book like this! Jim takes each lesson from his treasure chest of pedagogical subject knowledge and turns it every way in the light for us. Which different ways could the lesson go? How does it relate to key ideas in mathematics? How does it relate to our understanding of what it means to know? How does it engage students? What connections are there with other subjects?

Jim showing piles of rice to Helen and Mike

You can tell I'm recommending it!

Great teacher, great book!

The lesson chapters:

 1 - What’s in the box

 2 - Cones

 3 - If the world was a village of 100 people

 4 - Goodness Gracious Great Piles of Rice

 5 - How do I love thee, let me count the ways

 6 - Number Searches

 7 - Human Loci

 8 - Statistics telling stories

 9 - Match Point

 10 - Prime Pictures

 11 - Population Growth

 12 - Starting from scratch

 13 - Indestructible

 14 - Dancing Quadratics

 15 - Hot Wheels

 16 - Maxbox

 17 - Dancing Vectors

 18 - Pleasure at the Fairground

 19 - Impossible Diagrams

 20 - Cubism

Folding, cutting, sticking, drawing

I want to write a little about one of the hubs of the classroom.

It's what we call the writing table or drawing table. Which is maybe not the right name for it. A lot more happens than writing and drawing. It could perhaps be called the paper table. It's got a lot of stationery on it. Bits of paper of various sizes, glues, scissors. A lot of cutting, gluing, sticking, folding, stamping and printing happens. A lot of colouring in too. But, these names and simple descriptions aren't really adequate.

It’s 'continuous provision', as we call it: it’s always there, and used every day. I imagine that it extends beyond school too: children often have stationery at home.

Most early years classes have got something like this table. Certainly all four of our pre-K and Kindergarten classes have. This is what continuous provision is all about: a place where children can return again and again and make something, trying out new ideas, combining things they’ve done before, learning from each other.

Since they came to the school when they were three, R and K have been doing this. They're not the only ones, but let's focus on them for now. They're four years old; they've been in Moon class for 15 months. R at first stood out as leader of the duo, always inventive, always relishing what she does. But K seems to be inspired by her to be similarly creative, making things that are distinctive to her, having her own strengths and emphases.

An example, back in September: R's envelope-picture:

What kind of mathematics are present in creating this? An awareness of bringing the corners into the middle of the paper to reorient the square and create triangular flaps. A lot of spatial thinking. An example might be the awareness that when you fold the paper over once, the back of the folded paper faces the same way as the front.  She is probably aware that the orientation of the square changes too: first it was in a 'diamond' orientation, now it's in the conventional orientation. She'll be aware that the small square is made up of four triangles. And that there are diagonal lines across the square that meet in the centre. She's aware that some things can be undone, or almost undone. Pencil can be rubbed out. Cuts can be taped together again. And some things can't be undone. The felt pen drawing can't be rubbed out very easily.

At the same time, K was doing some folding too:

These paper explorations contrast with art activities that use specifically 'art' materials, painting in particular. There seems to be more of a tinkering feel, more mixing. Take R here, where she’s decided to draw round the scissors, drawn and colored in a pill shape, written a little, filled a rectangle…

There's a really strong social element in this. There was a group of girls in Star Class two years ago who all tuned into each other with their drawing and colouring, got more and more confident in that, and continued it into Kindergarten.

There’s also the sense of self-efficacy, of choosing a project, seeing it through to completion, working alongside others and learning from each other. There’s a kind of joy in the workshop ambience, in having control and making together and separately.

Here's some more, this time involving cut-outs:

With this must come some sense of how when you fold and cut, the hole you achieve is not like the cut you made. And a developing understanding of the relationship between the two.

There can be folded-and-cut shapes inside other folded-and-cut shapes:

The smaller shape suggested a watermelon to the girls. It's rare for these creations to be completely abstract; they usually represent something. This is a general feature of a lot of play - mathematics is mixed with creation is mixed with representation is mixed with narrative is mixed is mixed with language is mixed with sociability...

Another day, a butterfly:

^

Another day, a bird:

Another day, flowers composed of four punched hearts rotated:

What is the role of the adult here? Obviously, we keep the table stocked, and help the students to keep it tidy and organised. In the moment, we chat if it doesn't interrupt the flow of the play and conversation. We appreciate what the students are doing, how they're thinking and experimenting, again in a way that doesn't distract from the flow. We document and share with parents on Seesaw, and often with the class in our meeting times. Sometimes we play alongside too; this usually doesn't lead to much in itself, but allows us to be in the workshop too.

This time I started playing with R's leftovers (I'd asked if that was OK). I started making little 'windows' with the heart holes. R quite liked what I was doing this time, and together we made a picture, incorporating a bear on a trampoline, and also some of the folded and cut squares  that were being made at the table at the same time.

But, it's really not necessary for me to be adding anything in to this process: there's so much happening already: theories being refined, interests pursued, skills honed, and much more. 

We leave approximately the same materials on the table most of the time, and that's its power really. The little squares, the A4 sheets, the scissors, glue, tape and pens are enough for an endless range of operations, and combinations of operations that, the way children use them playfully, become more and more sophisticated.

Other things we provide in the class are more one-off. Putting some flowers in a vase to be sketched, along with the sketching materials. This is valid too, but is not a familiar arena that encourages the independence and agency of the students to develop.

In November, R gave a folded-and cut-out character to P, a boy she hasn't had much direct play or conversation with. One of them stuck the character to a sheet of paper, and P added lots of line drawing background. He carried it around with him for half the day.

I was surprised and delighted that this paper play had become a way of reaching out in friendship.

But maybe I shouldn't have been so surprised. These spaces that the students own, which become for them both a laboratory and a language are the natural places for the real events of the class to happen in.