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


  1. One thing that strikes me with filling, when observing the children or doing the activity alongside them, is the how satisfying it is and especially the moment when the last space is filled. Does everyone feel this satisfaction? The kids in my class mostly seem to. I wonder if perhaps it was nurtured in me. The result of lots of Sunday afternoon jigsaws with my grandparents - a different kind of filling.
    Also watching the children I notice that there is a kind of mediative quality to this kind of play.

    In a group setting such as our classroom the filling activity usually takes place in parallel with another child/other children. I like the opportunity this offers for sharing (social skills and mathematical) and negotiation. In the Numicon example, one child was filling up big areas using the orange one piece. Alongside, another child had filled their board with bigger Numicon shapes leaving ’one-sized’ gaps. The orange one pieces in the shared tray had run out. I watched them negotiate a way to share some orange ones. The child with many orange pieces agreed to replace some of two orange ones with a blue two piece. They accepted to do this several times. An opportunity for using their good social skills and for equivalence in practise.

    From the overarching concepts referred to in the Diploma Program that you list, Simon, I also wonder if approximation is present in filling up play. I need to think about that some more…perhaps I’m thinking more of estimation.

    So many questions. The water play photo really provokes the most questions for me.
    Water play is so popular and the engagement levels are so high that I can see there are great learning opportunities. But I’d like to dissect it a bit more. How are they exploring capacity? And what else is going on? What can we add to our water play provocations? Perhaps exploring how different tools/objects support the children’s thinking and give us an insight into the learning that’s happening… One idea so far is to have a real purpose for moving water from A to B. Sorry, I think I’m onto another schema now!

  2. Those are really thought-provoking comments, Estelle.

    You know I'm really interested in that first one - the state of consciousness involved. I recognise that kind of meditativeness. A kind of calming. I wonder if it's because there's a goal the children have taken, and they have to get there slowly and keep concentrating. Also, the aesthetic element. Attention isn't flitting around so much, it's calm.

    I like the gentle climb in this - it's mathematical, it's creative, it's achievable, it takes up the attention... but it's not necessarily intense, like hunting or wrestling. It feels like a welcoming kind of slope to be on!

    1. And now I think about being asked questions - how, as Julie Fisher says in Interacting or Interfering, for young children being put on the spot like that can stop them in their tracks, stop them thinking! It's as if the gentle slope is replaced by a rock face.

    2. Good analogy. I like the tree analogy in the blog post too and the idea of looking for mathematical doorways when working with young learners.

  3. The social element - being alongside each other, seeing each other's ideas, sharing in each other's inquiry - that's important and should have been on my list for a lot of maths. Like you, I sometimes fill my own Numicon board or pegboard. Sometimes, if I've say limited myself to particular Numicon pieces someone sitting next to me will start passing me those pieces.

    I'm hoping that as children are more comfortable in class and in English we'll have more naturally arising conversations as we work on this kind of thing together!

  4. I think there must be approximation in there - which as you say is closely linked to estimation. With 'continuous' things like water and sand especially, there's not going to be numbers involved; children will scoop up approximately how much they think they still need to fill the top of the jar. Even the 'discrete' things, like squares and pegs, there's so many of them, I don't think we think in terms of numbers so much as amounts. I've got a long way to go. I'm nearly there. I wonder if anyone ever thinks, oh, I'm about half way now?

    Kelly Darke, replying on Twitter, was thinking about visualisation in this: before we put something in a container we have some idea about whether it will fit or not. That too seems to link to approximation.

  5. And yes - water play. I really like what you say. I need to think about it more. And watch children playing. And play too, I guess. Let's work on this together!