Saturday, 24 October 2020

Seeing the mathematical: Filling

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

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

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


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

What are the tiny mathematical doorways for young children?

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

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

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

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

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

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

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

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

Tuesday, 20 October 2020

Positioning and enclosing

I often find it hard to fit the play and creation of my students into an overall map of development. Is this not something that's possible? Perhaps we have to trust that a rich ecosystem of experiences, play and creation will provide the environment for all the learning that needs to take place? In all this, can we identify thresholds, turning points, key moments? I don't know. Perhaps it's best to try and document and ponder certain things students are giving time to, having success with?

Here's a kind of success story. Though I'm not sure if I can discern a story arc in it, we could call it a months-long high point:

One of my 3 year old students is finding it hard to get used to life in school. One of her 'safe spaces' is to sit and paint or draw a picture. Here are some of her drawings:

She takes her time with them, often watching what's happening in the classroom from the vantage point of her creation. This one above is an early example. To me it's fascinating: how she has divided up the paper into sections with the orange lines and then experimented with dots in lines, dots in circles, circle parts, dots in groups, scribbly lines. 

She has a stencil here and she's adding areas of black using the stencil, on top of the orange and black lines and areas that she's already laid down.

There's a discipline in her investigations. This one has stuck to roughly oval areas of colour each separated from the other.

Here she has created her circular areas with dramatic movements of the pen, and then has carefully filled them with red.

More wild movement and careful colouring. I wish I could ask her about her pictures but we only share a few words of English. I'm left to interpret them myself, and, who knows? I'm likely to be missing things. But I see a concern with positioning elements, with spacing them out on the paper. Also a balance of wild and careful, of pen stroke and filled colour.

This one surprised me - hands! She'd done this after I'd been cutting out outlines of my hands to record ways that we could hold up three fingers.
She and another student caught the offcuts as I cut and made sure they went in the bin! I didn't imagine she would take the idea into her drawings.

How do I "assess" this? What development is there?

I think the answer is that I'm in no hurry for any development. This exploration seems so beautiful, so complete as it is. I wish it was all in one book. I'd love to be able to talk about the images. I'm glad I've got photos of many of them, before she folded them so that they'd fit in her bag to take home.

How do we analyse work like this? I'm not sure. The play schemas give us some small handle:
  • Transporting
  • Enveloping
  • Enclosing
  • Trajectory
  • Rotation
  • Connecting
  • Positioning
  • Transforming
Which do you see in these drawings? I see a lot of concern with positioning, and some with enclosing. Add in a little rotation and connecting. But this hasn't taken me a lot further. It does help me to see how  drawing links with other learning, opens my eyes to possible connections.

I'm pretty sure there will be development. I don't expect it to follow an obvious path - I can't predict how it will go. Will she keep that fine balance between the wild scribble and the careful positioning and filling? Will she continue to take her time on each piece? I hope so.