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April 08
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| January 08
| December
07 | Autumn 07 | August 07 |
July 07 |
June 07 | May 07 | April
07 | March 07 | Feb 07 | Nov 06 | Oct 06
| July / Aug 06 |
June 06 | May 06
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April 2008
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Joe’s Spider
Three-year old Joe had been playing with
some toy spiders and told his teacher that spiders have 8 legs.
Later he decided to draw a spider.
Joe’s drawing gives his sense of a
spider and he represented his idea of many legs. At this age Joe’s
drawing was lively and dynamic since he was unrestricted by
influences of school. When children represent quantities that they
do not count it is their personal sense of quantity that they
represent.
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March 2008
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Data handling: favourite zoo animals
(4-5 year-olds) The children in
this class had visited the local zoo the previous day and when they
arrived at school were engaged in heated discussions about the
‘best’ animals they had seen.
Their teacher capitalized on their
interest by suggesting they might find out what their friends’
favourites were. Several children chose to represent their own
preferences. Bianca drew a tiger, lion, giraffe and - and
then decided to write her friends’ names at the foot of her paper as
each of them told her which their favourite animal was from those
she had drawn. |
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Tommy drew his favourite animals
(left to right) – a lion, a crocodile, a giraffe and a tiger. Each
time he asked one of the children to name their choice, Tommy marked
a cross beneath the animal they had selected.
Later when we looked at the children’s
findings it became clear that some were easier to interpret than
others and this led to a discussion of possible ways of representing
findings so that others could readily interpret and understand the
data.
For young children data handling is
far more meaningful if they can make personal decisions such as
these and come to shared understanding through dialogue. These
examples show individual’s thinking at the time (rather than one
that is ‘better’ than the other) and are equally important in the
children’s development.
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Building on what the children already
understand has much more meaning to them than (for example) being
told to colour in squares on a block graph. It will help them to
understand the value of certain layouts and to make effective
choices about how to represent their data. |
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February 2008
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Miles and the nectarines
Miles was 7.5 years old. The children in
Miles’s class were about to go on a residential trip and were
planning to stop for a picnic on the way. Their teacher used the
opportunity this provided for the children to solve problem, showing
them a pack of three nectarines she had bought and inviting them to
work out:
| How many packs of
nectarines will we need to buy so that all 26 children
can have one each at the picnic? |
Seven-year old Miles decided
to use pen and paper to help him think about how to calculate the
total number of packs they would need.
He chose to orientate his paper in
‘portrait’ format (thereby limiting the length of the empty number
line he drew) but quickly understood that he would run out of space.
His highly adaptive solution was to double several of the jumps that
he made as he worked from right to left.
Providing a real context for
mathematics helps young children’s understanding and leads to
increased levels of involvement. After the children had discussed
how they had worked out their calculations and their findings, their
teacher bought 9 packs of nectarines to add to the food for their
picnic the next day – and the additional nectarine was enjoyed by
their teacher!
Early operations: development
of children’s own written methods
- Calculating with larger
numbers supported by jottings
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January 2008
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Charlotte’s
‘hundreds and pounds’
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Charlotte
(4 years, 2 months)
was with her friend Jessica in the nursery. They had each selected a
piece of paper and chosen coloured pens, holding as many as they
could in each hand and covering their paper with dots. As they
excitedly made marks, Charlotte told her teacher, ‘Look! I’m doing
hundreds and pounds!
Charlotte’s
reference to ‘hundred’s and pounds' meant that she was making
connections with the quantity of dots which seemed a lot to her.
Both a ‘hundred’ and ‘pounds’ fits into her thinking about a lot.
Charlotte used spoken language to express what the marks she made on
paper suggested to her, attending to the link between her marks and
the mathematical vocabulary of quantity in a general sense.
Written
number and quantities:
§
early explorations with marks
§
representing quantities that are not counted
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December 2007
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Nadia (4 years 7 months) found some
squared paper in the technology area and chose to explore symbols in
her own way, using one square for each. It was the abstract symbols
themselves – their appearance and form; their potential functions
and the meanings that she attached to them that interested her.
When she showed this to me, Nadia ‘read’
the ‘E’; several ‘Js’; ‘a star’; ‘T’ ; ‘round’ (gesturing in the air
with a her fore-finger to make a circular movement); ‘umbrella’ (in
the centre of the paper) and ‘square’ (lower right).
Thus, while some symbols she named as
standard letters, for others she focused on the shape or form (i.e.
the circle and square) and two symbols suggested pictorial
representations (the star and umbrella).
The ‘stuff’ that children use to
explore meanings within their play, offers ‘meaning-full’ contexts
for children’s future explorations of symbolic ‘written’ languages
- such as writing and written maths. |
Nadia’s symbols
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Autumn 2007
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Nikita’s Birthday Card
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Nikita (3 years and 4
months) Birthdays are
something with which young children identify and that excite them:
the changing of their age is important to them.
To enhance the graphics area in the
nursery we asked the children to bring in old birthday cards from
their family. We discussed with each child the card they had brought
in and provided additional materials.
Children chose to go to this area if
they wished and adults carefully observed and noted down findings.
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Nikita made a birthday card for herself.
She looked through a box of cardboard numerals and selected a ‘3’
which was her age at the time, then glued the numeral on the paper.
She made some marks with blue and black pens. Nikita was very quiet
whilst she did this and whispered to her teacher ‘This card is for
me – I am three’. Nikita knew
her own age and recognized the numeral ‘3’. Her marks look very much
like early writing and the crosses may have been kisses, although
she did not say. Nikita had showed a great deal of understanding of
birthday cards. |
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August 2007
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Marina (5 years 0 months) and
the Library Van
Child-initiated role play can be wonderfully rich! On this
occasion we had visited the city library, and then two weeks
later had exchanged a box of books at the mobile library van
that had called in the village - that our class had borrowed.
This visit coincided with
Marina’s enclosing and containing schema, and on our
return to school she spontaneously created a narrow, enclosed
space for a ‘library van’. A number of children were also drawn
to this play and explored their understanding about libraries
over a period of several ways.
The children’s role play was rich
in both literacy and mathematics – including using money and
giving change; using a calculator, completing forms and writing
letters.
Marie-Anne made a road safety
poster and attached it to the front of the counter (as she had
seen in the city library). Marina was especially interested in
what she had heard about library fines at the city library and
now drew on this knowledge for her library van role play. She
wrote endless letters to library visitors demanding huge sums
for their overdue books! During this letter writing she chose to
use a calculator and real coins to work out monies owed and to
give change, and also added stamps to the envelopes.
This role play was entirely the
children’s and points to the value of staff providing open-ended
space, opportunities, resources and time for |
children to initiate their own
role play. Additionally it serves to emphasise the importance of
visits and first-hand experience to support schemas (see
Athey, 2006) and role play.
Commenting on Marina's role play,
Wood and Atfield (2005) emphasize
how ‘this vignette shows how literacy and numeracy are social
and cultural practices that children observe in their everyday
environments, and subsequently initiate in their play’ (p.79). |
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July 2007
Jack: adding
grapes
We were using grapes
to explore addition (they would eat the grapes later). Jack
(4 years 11 months): chose two separate amounts of grapes to
add for each of his calculations and decided to use paper and
pen to help him explore the use of symbols.
He left a gap
between the two sets of grapes in each calculation so that it
can be read as ‘4 + 3’ (and below) ‘6 + 6’. We term this use of
a space between two sets to be added or subtracted implicit
symbols, since whilst the child has not at this stage
represented the addition symbol, his layout – and when he reads
it out, shows he has implied the symbol and recognized that an
operant is needed.
Finally Jack drew a
line in each calculation before writing the total: this line
functioned as an equals sign for Jack at this point in time,
helping him move towards a deep understanding of standard
symbols (see the Taxonomy
and exploring symbols)
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June 2007
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Kamrin’s ‘Tweedle
birds’
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This was a
teacher-led lesson in a Reception class: problem solving –
division by sharing
Kamrin (5 years 7 months) invented his own system to
check if 8 could be shared equally between two. He wrote a
question mark by the numeral 8 and then a cross, as at first he
thought it could not be divided equally between two. Then he
invented ‘Tweedle birds’ and shared eight ‘eggs’ equally between
the two birds, adding a tick to show that eight could be divided
equally. Kamrin then went on to explore several other ways to
find if other numbers could be dividing equally in two, finding
increasingly efficient methods of doing this – or using
‘successive shorthand’.
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May 2007
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Julie - Weighing Babies This was
freely chosen play in the ‘baby clinic’ role play area. Julie
(4 years and 3 months) weighed her baby on the scales and
made a mark on her booking sheet in the appropriate space; she
repeated this after using a tape measure to check the baby’s
length. When she wrote the baby’s age Julie said ‘6’ and made 6
distinct tally marks (iconic).
Julie understands that quantity
(representing mass) and numerals can be represented with
symbols. She used her own early explorations with marks
and represented quantities that are counted to
represent her personal meanings as she weighed her baby.
You can see more of the Cambridge
Nursery children’s graphics in Gallery 3
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April 2007
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Nursery: ‘plane tickets’
These 3 and 4 year old
children in the nursery had been playing aeroplanes and the
Nursery Teacher made some ‘tickets’ to which the children could
add their seat numbers.
In these examples Louis, Sarah
and Joe were exploring early marks and using their own early
written numerals. You can see more of the children’s plane
tickets in the Teachers’ Gallery.
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March 2007
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Chloë's Dinner Register
Chloë (5 years) had
brought her own exercise book from home to play with in the
graphics area. She said she was “making register”. She wrote the
names down (as squiggles) for the children in the class and then
counted them to see if she’d made the right number of marks. She
counted to 4 and then counted random numbers to 20. She used
number strips to check and count how many children were in the
class. She chose to use tallies to represent the number of
children who were away.
Chloë has a statement of special
needs and finds articulating language difficult: this was the
first time that she had shown any interest in the graphics area.
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February 2007
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Louisa’s dial
- Written number and
quantities: numerals as labels
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In the
graphics area, Louisa, 4.9 years, made what at first
glance appears to be a clock. In this class the teacher gave
each group of children certain options for them to choose for
play. When Louisa showed me her dial she assigned an activity to
the numerals 1 – 5: bricks, puzzles, role-play, reading and
painting. She paused with her finger on ‘6’, and unable to think
of other play possibilities in her class she smiled and
explained ‘you have to sleep’. As she moved the hand of the dial
she stopped at the letters ‘fo’ (off) and explained ‘this is
where you turn it off’.
Louisa had related what she knew about analogue clocks to that
of her classroom culture: perhaps she was also making links with
her home culture too, where after playing at the end of the day
she goes to sleep.
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November 06
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Sam’s marks
Spontaneous marks that young children choose to make within
their play provide powerful and meaningful contexts.
Bradley (3.5 years) had been playing
in the writing area with the calculator for about 15 minutes,
making marks as he looked at the buttons on the calculator. He
told his teacher ‘my dad’s got one of these’. Sam (3. 6 years
old) was nearby, watching as Bradley used the calculator and
made his own marks on paper as he did so (above). Marks such as
these are an important feature of young children’s early
mathematical development concerning symbols. |
- Written number and
quantities:
early explorations with marks
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October 06
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Mark’s crosses Some of the boys
were very excited about playing in the garage role play area
(next to the graphics area). Mark (4 years) chose a
corner of the playground explaining that this was a ‘no entry
area’ (for cars). He was exploring a sign that children see in
many contexts, in a way that was personally meaningful to him at
the time. Later in school, he will meet ‘+’ and ‘x’ as signs in
mathematics.
We chose this lovely photo for
the cover of our second edition
and would like to thank Louise Glovers, Nursery Teacher at
the Robert Owen Children’s Centre in Greenwich for this lovely
photo of Mark.
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July/August 2006
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Jessica’s Number Line
- Written number and
quantities: early explorations with marks
In the nursery the children had
created a number line together with giant numerals that they had
chosen. Jessie (4 years and 3 months). Jessie was one of
several children who chose to make her own number line on a
strip of paper.
Jessie centred on the letter ‘J’
that was the most important letter to her and used it to stand
for number and written symbols. Jessie’s dots may be
representations of other numerals and ‘line’ may be a literal
translation of the line she had often heard referred to in the
‘number line’.
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June 2006
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Amelie’s Dice game
- Written number and
quantities: numerals as labels; representing
quantities that are counted
- Children’s own written
methods (calculations): exploring symbols
This was a teacher-planned
activity that provided open opportunities for children to
explore their thinking. Each pair of children had two dice
between them and were invited to ‘put something down of paper’
to show what they got each time they rolled the dice.
Amelie (4 years 6 months)
counted the dots each time she rolled the dice and carefully
made the same number of dots with her pen on the right side of
her paper. She has also used letters from her own name
(particularly the capital ‘A’) and her age number ‘4’. She
noticed the marks and symbols that some of her friends used and
added several ‘+’ signs and also a ‘=’ sign at the top of the
paper, with some numerals written within boxes. She ‘read’ the
‘e’ as ‘eight’ (a similar shaped symbol).
Amelie was very proud of this piece – it is dynamic and full of
energy and spontaneity of a young child.
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May 2006
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| Frances
and the train
- Children’s own written
methods (calculations): representing quantities that
are counted; counting continuously & calculating with larger
numbers
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Our
class of 4 – 6 year olds had travelled by train to visit an old
‘pannier’ market in a town in the north of the county and on the
return journey our carriage was very crowded. The following day
Aaron commented “I bet there were a million people on the
train!”
After some discussion about how we might find out, Aaron phoned
the railway station. Returning to the class he told his friends
that there had been 7 carriages on our train and 75 seats in
each carriage. A number of children were intrigued by this large
quantity and wanted to find out how many there were altogether.
Each chose their own ways to work this out (with resources or
through representing their thinking on paper).
Frances (6 years 1 month)
explored a number of ways before drawing representations of the
seats in one carriage (below). She self-checked and found that
she had drawn one too many and crossed it out (above). In order
to represent the remaining 6 carriages, Frances asked if she
could photocopy her first carriage – an interesting solution
using repeated addition. |
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