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.

It's:

- 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.

*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.'

*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.

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 |

- 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?

- approximation,
- change,
- equivalence,
- generalisation,
- modelling,
- patterns,
- quantity,
- relationships,
- representation,
- space,
- systems,
- validity

*filling*?