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. 

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.

Mathematics Inquiry

We're reading Kath Murdoch's great The Power of Inquiry, my friends-and-colleagues Rachel and Estelle and I. We've had some really nice Zoom meetings chatting over chapters.

The Primary Years Programme of the International Baccalaureate, which our school follows, gives a central place to inquiry - and yet, and yet, on the ground, old habits die hard. The book challenges us to think how inquiry works in practice, what we can do to make it more of a reality in the classroom.

We've jumped about. We're reading Chapter 3 at the moment - 'Beyond Topics'.

On page 40, Kath Murdoch lists features that characterise most journeys of inquiry:

• They are generally driven by questions - both teacher and student generated.
• They require active research/investigation.
• They most often seek to connect learning with students real-life experiences.
• They are as much about process as they are about content - and content is conceptual.
• Students experience connected learning episodes - one task is clearly linked to the next rather than being simply an 'activity'.
• The learning is responsively planned (rather than fully mapped ahead in detail).
• Aspects of the learning tasks / assessments are co-constructed with students.
• The planning is emergent - the details of the process unfolds rather than being pre-determined.

One of the things to strike me was that these describe well how I see lessons that involve thinking mathematically.

(For the third point, about real-life experiences, I have a qualification to make: mathematics, as well as describing many real-life situations, is an abstract study in its own right - a little like some art and music are. It's not always, or even mostly, going to be about real life.)

Kath Murdoch gives some examples of Mathematics inquiry questions:

• How do we measure time?
• What is long?
• What makes a pattern?

...and many more.

These are big questions, but on a much smaller scale, small moments in the mathematics lesson can also show the same features. I want to write briefly about some examples, in this case with young learners in playful, child-initiated contexts.

Here are some of our PK children (4 and 5 years old)  making a kind of carpet. They've put lots of square magnetic Polydron together, and now they're adding a triangle border. It was initiated, and completed by the children without any input from the teacher.

They haven't stated a question in words, but play like this is purposeful, and you could say it's investigating an implicit question. Something like, 'Is it possible to make a really big rug out of squares and then round it off with a border of triangles?'

If I was to extend this with the children, I might want to show the class the image of what was created and invite comment. I might ask if other sizes could be made. What is the smallest rug like this that's possible? How about if there was some pattern to the colours? What would that look like? I might keep the image up somewhere for reference. And wait and see if the inquiry took off again. Connected learning episodes allow students to tune in, to make variations on a theme, create a base of shared experience for discussion.

In one of our K3 classes (5 and 6 year olds) recently, a boy, I'll call him Y, was making a similar pattern in an odd moment, this time out of pattern blocks. He often chooses to create patterns with manipulatives. I sat with him. He doesn't generally want to chat much or even answer questions; he sometimes has his own monologue about what he's doing. I knew I wouldn't be able to 'steer' his play much, but still I wanted to be there, maybe throw something into the mix. Here's what he made:

I admired his square and how he'd surrounded it with a border of triangles (might this be a good way into investigating perimeter with older students?)

Y then broke that, and started making a bigger one. Then he made a bigger one still.

This seems like the essence of a lot of mathematical inquiry. First of all his attention is on a whole - the square surrounded by triangles. It's quite a satisfying whole. Then he wants to see how that extends. Does it just work for his particular case, or is there a general pattern? I got the iPad  out and recorded what he'd done. I also started making the ones he'd broken so we could see them all together.

Concepts here:

• We don't just look at individual cases, we look for patterns or regularities in the situation.
• We find ways of keeping the pattern as well as the individual case in mind.

As a teacher, in a playful child-initiated situation, I want a light touch. I want to understand his interest, the unspoken 'question' he's answering for himself, and do what I can to enhance that inquiry, including if necessary, staying at am's length!

I did direct Y's attention to the pattern of numbers of triangles in the whole family afterwards: 4,8,12,16,20... but I don't think he was that keen on me 'directing' anything!  I didn't push it, knowing that if we valued his 'question', his beginning, this was something that could be a focus later.

I enthused about his work a little and posted it on Seesaw. A few weeks later, I popped into his classroom and saw that he was making the same patterns. And another time, he made it with Polydron.

It was obviously something that Y was getting satisfaction returning to.

This process - of actively returning, of varying, of building on previous experience - is so much part of the process of mathematics - whether it's child-directed play, or teacher-led investigation. I would love all our teaching to centre on and respond to this process of inquiry, whether it be with a big explicit question, or as in Y's case, a small implicit question.

Becoming the Math Teacher You Wish You'd Had

Tracy Johnston Zager's book, Becoming the Math Teacher You Wish You'd Had: Ideas and Strategies from Vibrant Classrooms is a - perhaps I should say the - book that I'd recommend to all math/maths teachers, and that includes all of us primary/elementary teachers. And for those of us working in inquriy-led IB PYP classrooms, the fit is perfect.

Tracy takes a range of things that real mathematicians do as her starting point. Chapter 7 for instance is titled, Mathematicians ask questions. She starts with a quote from Jo Boaler's book What's Math Got to Do with It?:

Peter Hilton, an algebraic topologist, has said: “Computation involves going from a question to an answer. Mathematics involves going from an answer to a question.” Such work requires creativity, original thinking, and ingenuity. All the mathematical methods and relationships that are now known and taught to schoolchildren started as questions, yet students do not see the questions. Instead, they are taught content that often appears as a long list of answers to questions that nobody has ever asked. 

She gives examples of resources for encouraging questioning, like 101questions and Notice and Wonder.

My favourite part of the chapter is one of the dips into real (and yes, vibrant) classrooms, this time with Deborah Nichols' first and second grades (p152). The question had come up, 'Are shapes math?'
The first step was to find out what the students wondered about shape.

And here are their questions. What an amazing set:

I want to ask some of these students what they meant by some of these! That first one, 'How big can circles go?' - is that about the practical constraints on us creating circles? Or is it about how circles start to look straight when they're really big? Like us walking on our planet. Or is she asking about circles in space? Or something else? Is the question 'How round can a circle be?' related?

Also, 'Are shapes fragile?' Did the questioner mean can shapes be distorted easily? Like the way a triangle or a tetrahedron model made of edges is quite robust, but a square or cube can be deformed easily?

Anyhow, what the teacher did was to arrange a sequence of experiences, with shapes that allowed for there to be a real dialogue between these questions, the shapes themselves and what the teacher needed to be learnt. Without allowing the inquiry to go off in directions that wouldn't really answer the questions, the students' questions - and the answers that came - stayed to the fore. Bit by bit the students built up the knowledge and vocabulary they needed to answer the more mathematical sides of their questions. And they thoroughly covered the learning that's set down for those grades.

As Tracy writes:

Perhaps knowing that students' inquisitiveness leads to the same ideas that mathematicians study and standard-writers emphasize can help us feel less pressure to tell and cover and explain. If we allow students to ask, we will likely end up in the same place but with much more engaged, empowered students.

Arranging things

We've just finished our first week of the new year, me and my class of five-year olds.

I've been thinking about Graeme Anshaw's blog post where he asks, What type of maths inquiry do we employ the most and the least in our classrooms?

He outlines these different forms of inquiry:

	Demonstrated Inquiry	Structured Inquiry	Guided Inquiry	Open Inquiry
Posing the question	Teacher	Teacher	Teacher	Student
Planning the procedure	Teacher	Teacher	Student	Student
Drawing conclusions	Teacher	Student	Student	Student

How do we find a place for open inquiry, where the students are asking the questions? More specifically, how do I do it, with my class, especially as most of the students don't have English as a first language?

For young children like mine, how they ask questions is often through their play. If I pile these up here, what would it look like? How could I arrange them more satisfyingly?

Here's a couple of students arranging wooden blocks in a line, then arranging frisbees and bats on top. 

You need lots of components to get good patterns going - we need more bats and frisbees! We've just got more magnetic Polydron and the building is impressive. Here's T asking how he can transform shapes, and what happens if he uses triangles to add star-points to other shapes?

Others were enjoying our new straws and connectors, starting with squares, building cubes, and then puting them together to produce a tall tower:

Arranging squares:

Cutting holes in folded paper:

 

Making balls and worms of play dough:

Creating train track networks:

and playing with Cuisenaire staircases:

EL made this last series of staircases. Actually she had to make it three times. The first time someone slid some rods into it, and it was too late to repair, the second time it was standing up and got knocked over during lunch time. I made sure there was time when she could get it finished and photographed.

I think it's important that everyone recognises that there is maths implicit in all these things. Many people still think of maths as needing to be about counting or numbers or sums, and feel anxious that these should happen. But these things are there in the physical things that the students are doing already! And also:

• Spatial and geometrical awareness
• Categorising and sorting
• Creating patterns
• Investigating the results of processes

There are also the social skills being exercised in much of the group and paired work. All of the IB PYP social skills are needed at various points:

• Accepting responsibility
• Respecting others
• Cooperating
• Resolving conflict
• Group decision-making
• Adopting a variety of group roles

Of course, it's great to connect all this making to language, to talking about what we've done and reflecting on it, and ultimately to symbols too. And to start to abstract from the particular. But I don't want to be hasty with this. A lot of the student's creations are so pregnant with mathematical possibilities that I want to (as Helen says) re-propose them to the class, perhaps along with what they said to me at the time. I'm also this year, annotating photographs of the students' creations with them.

And there will be time to start making connections to some of the big concepts in maths. Like the way Graeme starts with the big central idea, "Our base 10 number system evolved for a variety of reasons and led to place values that extend infinitely in both directions." What these big ideas are varies from list to list. I see Jo Boaler and her team have just developed some:

Others have come up with big overarching ideas that stretch right over the years. Like Mike Askew's:

In addition to considering this, in the IB PYP we work with transdisciplinary themes, that bring out the connections between traditional subject areas. Including maths of course.

So, I'm hoping, for instance, to connect, for our current unit of inquiry on transport systems, the train track building that's going on with some discrete maths. How places are linked, how are nodes linked by edges. Moving from very concrete things like trains and cars, onto networks, reasons for networks, connections. I tried this last week, asking students to make roads between every pair of two houses, and after a few asking them to guess how many roads there will be:

What happens if the roads are not allowed to cross?

— Justin Lanier (@j_lanier) September 8, 2017

What I really want to keep though in all this, while trying to develop the big ideas, is the wonderful, individual, confident play and creativity.