In this lesson, the teacher introduced division by

by sharing though telling a story about twins who

liked everything the same (scubas the same number

of sweets each). She went on to ask the children if

they could think of numbers that could be shared

equally, and the children began to explore this in

their own ways.

     Kamrin (5 years, 7 months) invented his own

system to check if the number '8' could be shared

equally between two. He invented 'Tweedle birds',

in turn giving an egg to each bird. Originally he

had written a question mark by the '8' (the inverted

'j'), adding a cross by it, but now he scribbled over

the cross and wrote a tick to confirm his finding

that '8' could be shared equally.

   Miles's class were about to leave on a

residential trip, and their teacher used a

pack of three nectarines to encourage

the children to calculate how many packs

would be needed for the class of 26

children. Placing a piece of A4 paper in

portrait format,

    Miles (7 years, 5 months), began by

drawing an empty number-line. Due to

 the orientation of his paper Miles found

that he had to restrict the number of

 jumps he made, changing jumps of 3

to jumps of 6 several times, before

arriving at his answer.

This class had just returned from a trip by

train to visit  a large country market. the

next day, Aaron observed, "I bet there's a 

million seats on the train!", and their teacher

asked how they might find out. Several

suggestions followed (the library, the head

teacher, the computer), and then one child 

suggested they "phone the train people." 

With help in finding the phone number,

Aaron proudly asked for the information he

wanted, and on returning to the classroom,

told the children that there had been 75

seats in each carriage, and seven carriages

on the train. Several children decided they

would like to work on this problem, and did

so, in a variety of ways.

      At first Frances (5 years, 7 months),

wrote '75' seven times, then drew a

carriage (a square) with 76 seats in it. Self

-checking, she found she had one too many

and crossed it out. 

Unsure how to work out how many seats

 altogether on the train, Alison had a

 sudden insight, "the photocopier!" When

the six additional copies were laid out on

the floor with the additional one, the

children were very excited to see this

powerful representation of the complete

'train' with an equal number of seats in each carriage.

 
The children in this class had already been

thinking of multiplication in terms of an array, 

and knew the two, three and 10 times tables.

They also understood how to count in 100s.

Their teacher challenged them to work out 

the nine times table, the children all laughing

at this suggestion, and saying "no!"

    Alison (7 years, 6 months) began by writing

'5 x 99 =', then '2 x 99 followed by a lot of 

crossing-out, and the word 'no'. Beneath it 

she wrote '99 + 99 = 20098' (a logical way

of writing 298), but abandoned this. She then 

used vertical strokes to represent 99, repeating

this again - but then wrote 'rong' and 'no' beside

them. 

   In discussion with Alison, her teacher asked 

if there was anything else she could try, and this

seemed to be a 'eureka' moment for Alison. She 

changed from her iconic method of writing

vertical lines - to using standard symbols written

as repeated addition. Finally she wrote '100 +'

five times, and, subtracting five, arrived at her

answer.