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Gallery 5

    © Copyright M. Worthington & E. Carruthers 2011

 Beginnings in Play

June 2012

Biscuits for bears

Luke and Zainab (both 4-5 years)

The children were having a ‘teddy bears’ picnic’ and there were some surplus biscuits. Their teacher Sara asked the children how they might share the biscuits equally between the six children.

The mathematics: counting, division by sharing, fractions

At first Zainab (whose example is on the left) decided to share the six biscuits between four children. She wrote numbers 1 to 6 and then drew four circles to represent biscuits. She drew a line from each number to a biscuit and had two left over. Zainab was unsure what to do with the remaining two biscuits and decided to divide them into quarters, adding the numerals ‘5’ and ‘6’ (to represent the two remaining children) and drawing four lines beneath each, to represent the four quarters of each biscuit. Although Zainab did not completely solve the problem, her graphics show how she was thinking about fractions. Zainab had reframed the original question, perhaps thinking that sharing four biscuits between six children was too difficult, and that adjusting the question would reduce its complexity.

Luke (whose example is on the right) drew one person to represent the six children, and then drew four biscuits. He decided he would divide the biscuits in half (so that each child could have an equal piece) but then discovered that there would be one biscuit remaining. Drawing a circle round this remaining biscuit, his solution was to put it in his back pocket (a potentially real life and meaningful solution)!

It is really important to look closely at the processes involved in young children’s thinking rather than focusing too closely on correct answers. By valuing the children’s own methods and really listening, teachers can uncover what the children know.

Written number and quantities: representing quantities that are counted, numerals as labels
Calculations – children’s own methods: counting continuously; separating sets


April 2012

Kiera (5 years 2 months) and Lauren (5 years 0 months)

The mathematics: number, quantities, measurement

Kiera, Tanny and Lauren were playing ‘doctors’. Kiera pretended to give some medicine to Tanny, explaining that the number on the cup said ’80’ and explaining, ‘It tells you how much medicine you are giving’.

The photo above shows Lauren pretending to be ‘mummy’ as she made a list of ‘phone numbers. She wrote and read ‘300’ (not shown) for the doctor’s number, then ‘37’, reading it as ‘three’ and ‘seven’ for her friend Oma’s telephone number.

In learning environments where mathematics and children’s own graphical communications are valued and supported, children’s self-initiated role-play often naturally includes mathematics..

Written number and quantities: numerals as labels


March 2012

Mimi's Crosses (3 years 4 months)

The mathematics: quantities, counting, shape and space

Mimi was playing outside: she made a series of crosses on the ground and drew a line around them, enclosing some single crosses. She explained to some children nearby, ‘Now everyone can no come!’ However, one of the children walked across Mimi’s crosses, causing her to emphasise, ‘No! She no come!’ Mimi then drew a boundary around two of the crosses to further emphasise that she wanted the space to be kept free from other children. After an interval, one of the children also took a chalk and drew a long line bypassing Mimi’s symbols and, in response, Mimi also drew a long line creating a wide boundary that enclosed all her crosses – and then stood on guard over them.

Mimi used graphical symbols to communicate and re-emphasise her message to others, using repeated crosses to deny access to the space in which she was playing. Crosses appear to be one of the first symbols that young children consciously create. Children use them flexibly to signify and communicate many things in their drawings and maps, and in their early writing and mathematical graphics.

Making meanings in pretence and imagination: exploring symbols.


February 2012

The mathematics: counting (to 29), measuring length, comparison

Maisie wanted to see how long her Christmas paper-chain was. Although there were several 30cm rulers on her table she decided to make her own ‘ruler’, and wrote as many numbers as she could fit on the strip of paper she’d chosen. When she reached the end of her first strip (with numbers up to ‘9’) Maisie added a second strip (with numbers from 9 – 17) and then when this was full, attached a third strip, writing numbers from 18 – 29. Finally Maisie held her paper ruler against her paper-chain to check that her ruler was sufficiently long, and then measuring her paper-chain announced it was ’27 long’.

Maisie’s teacher supports children’s graphicacy and open ways of working and, since this was a problem that Maisie posed she had ownership of it, her strategies contributed to her growing understanding of linear measurement.

Making meanings in imagination and symbolic play: models with found materials
Written quantities and numerals: numerals as labels; representing quantities that are counted
Calculations: children’s own methods: counting continuously


January 2012

Jazper - 3 years, 6 months.

Images and ideas often appear to combine in ways that can seem unlikely to adults. For example, pointing to different features in his drawing, Jazper explained ‘Red lorry, blue car and road’ then added ‘lots of eyes.’ Jazper’s drawing seemed to combine elements of both a drawing and a map (perhaps too with an unspoken narrative).
In rich learning cultures, children use graphical representations to communicate for a variety of purposes - including imagination, pretend play and everyday contexts. Such rich experiences help them come to see that they can also use their marks and symbols to explore and communicate their mathematical thinking.

Making meanings in imagination and symbolic play: drawings / maps


December 2011

Stacey’s café

This lovely example demonstrates a highly significant aspect of young children’s development, as they begin to use graphical marks and symbols to communicate a range of ideas and purposes, including their mathematical thinking.

Stacey (3 years, 2 months) was playing in the little house outside. Picking up a pen and notebook she asked her friend, ‘What do you like?’

Her friend replied ‘Chicken Tikka please’ and as Stacey began to make circular marks, dots and other marks she asked ‘Chicken and chips?’ adding ‘Fifty pounds please.’ Her friend pretended to give her some money, which Stacey put in her pocket. Stacey drew on her home knowledge of ordering a take-away meal, clear in her understanding that orders were written down in the restaurant and integrating her understanding of asking for and taking orders, and money.

Written number and quantities: early explorations with marks - attaching mathematical meanings


November 2011

Nursery – Rio’s parking tickets

Rio (2 years, 2 months)

Rio was playing outside with his friends: they were talking about parking, and Rio spontaneously made himself some parking tickets, drawing scribble-marks on gummed labels.

Written number and quantities: early explorations with marks


October 2011

Nursery - a spontaneous ball game

Henry, Joe and Thomas: 4 years, 4 months - 4 years 7 months.

 

These are just two of the many graphical signs that these boys generated during the course of their game. These abstract symbols have personal meaning for the children within the context of their play and support their developing understanding of the power of graphical symbols.

'This means you lose'

'This means you double lose'
Written number and quantities: explorations with symbols

September 2011

Max’s card-girl (reception)

Max cut two tiny rectangles of paper and joined them at the top so that he could open and shut them. Drawing a face on the uppermost piece he announced that his artefact was a ‘card-girl’.

Although we might regard this as either a ‘drawing’ or a ‘card’, this hybrid, multimodal artefact suggest it could be moved and played with as a prop in Max’s imaginative play, as a small puppet or person.

We all make meanings in multiple and complex ways and when considering young children’s meanings, it can be helpful to think beyond single definitions such as ‘drawing’, ‘writing’ or ‘models’.

Multimodal meaning making: combining paper model and drawing (with potential for movement and voice - as a puppet or toy)


August 2011

Caravans at snack time (nursery)

At snack time Harry (4 years, 5 months), initiated a conversation about his experience of staying in a caravan. Since some of the children had not stayed in a caravan, Harry decided to represent the one he’d stayed in to help them understand what they were like. He showed the ‘two wheels’ and wrote the ‘long number’ on his caravan ‘12148’.

Written number and quantities: representing quantities that are counted; early written numerals; numerals as labels.

 

Finnian (4 years, 1 month), wanted to represent the caravan he’d stayed in too. He showed that there were many caravans on the site where he’d stayed, and that they were all very close together. The caravan he drew at the top of the whiteboard show him, standing on the sofa and looking out of the window.
 

Written number and quantities: representing quantities that are not counted

 


July 2011

Kyran’s drawing of his mummy (nursery)

Commenting on this drawing, Kyran said‘ she’s got funny hands!’

Developing from the ‘generational structures’ that John Matthews identified (1999), children are influenced by signs that have strong visual impact and distinct cultural uses in their culture. They make choices and decisions about the symbols they use to encode and communicate specific meanings and some of these become signs that others understand and accept.

It appears that some of the most powerful signs that young children use include zigzags, crosses and arrows.

Other examples include children who used crosses to represent an aeroplane; to emphasize ‘No! Keep out!’; to signify ‘shop closed’; to identify someone who has lost in a ball game; as kisses on a birthday card and to signify items on a shopping list.

In communicating mathematical ideas, individuals used crosses to identify peers’ choices (data handling); to cross out items (denoting subtraction) and later as an addition sign. This flexibly also supports understanding of standard abstract signs such as ‘x’ (as a letter and as a multiplication sign); letter ‘T’ and ‘t’ and the numeral ‘4’.

Kyran - making meanings with marks: explorations with symbols

See also: Gallery 4: CM Graphics of Past Months


May 2011

Early bi-literacy: ‘bi-literacy’ refers to children and adults who combine their first and second languages when writing. Drawing and writing are aspects of semiotics that also use abstract symbols.

Lay Hau Yun’s home language is Cantonese. This example shows how she represented her name twice, first as a zigzag line (top left) as many young children do in English, and then on the right, in Cantonese characters. On another occasion whilst playing, she was writing numbers as she counted, beginning ‘1’ and ‘2’ and following this with the written Cantonese character 三 for ‘3’, integrating her knowledge of both written languages.

Emergent writing: explorations with symbols

 

See also: Gallery 5: Beginnings in Play


April 2011

Written number and quantities: early written numerals; numerals as labels; explorations with symbols; representing quantities that are not counted

Finnian’s age (nursery)

Finnian was aware that he was younger and smaller in stature than the other children in his group.

Taking a pen Finn emphasized his exact age on a nearby whiteboard explaining ‘I’m not three and a half! I’m three and three quarters… Look. This is how you write ‘three and a half’ [the line of symbols at the very top] and this is how you write ‘three and three quarters’!’ [the remaining symbols in the centre and lower down].

Not only had Finnian used symbols to explore fractions, he understood (and used) symbols to persuade.

 


March 2011

Amelia's Shopping List

In the nursery Amelia was playing shops, busy organizing things and chatting happily to her friends. As she made marks and letter-like symbols she said, ‘Chocolate biscuits, rice pops, sausages’.

Amelia appeared to be listing foods she liked to eat and that her mummy bought when they went together to the supermarket.

Written number and quantities: Attaching meanings to marks


February 2011

In the nursery Oliver was exploring rulers and set squares with his friends. One of the children used a ruler to draw a triangle and then pivoted the set square, using it as a template to get the shape he wanted. Oliver used the rulers to measure the length of the table, lining them up carefully in a straight line and then using them to draw across the table. Others enjoyed freely drawings across the length of the large sheet of paper.

The children used a great deal of mathematical language as they chatted about their lines, shapes and drawings, developing their understandings of measurement and shape. Their free play enabled them explore aspects of measurement in open-ended ways.

Written number and quantities: early explorations with marks; representing quantities that are not counted


January 2011

Shakkai’s Dad

Shakkai laughed as he showed his friend that he’d drawn his dad with ‘4 eyes’, then decided that he’d draw another, saying, ‘I’m going to add another eye. Look! He has 5 eyes now!’ and the boys laughed and laughed about Shakkai’s drawing.

Written number and quantities: representing quantities that are counted.

Calculations - children’s own methods: counting continuously.


December 2010

Liana’s ‘picture for my mum’

Young children’s drawings are an important aspect of graphicacy. Liana’s drawing began as a story might: ‘it’s a rainy day’ to set the scene, continuing as Liana listed items ‘Here’s a small flower, here’s a bigger flower. This is a scarecrow; a house; a balloon and an apple’.

We can also see that naming items appeared to be important aspect of this, so it almost represents a list or inventory.


November 2010

Mason’s Spy Gadget

Intrigued by new media, new technologies and popular culture, Mason was now a champion of ‘21st century’ play and this example shows how he also drew on his knowledge of password protection and access, numbers and writing. Mason watched Leola who was cutting a piece of card, and finding a piece of yellow card, he also folded and snipped similar cuts around its perimeter. Next he wrote letters and numerals, reading 'sk’ ‘714bp10’ and, lifting it to his face, explained it was ‘a spy gadget… 'sk' is ‘to keep the password safe. To switch it on you have to say '714bp10'’. I asked if there was a way to switch his 'spy gadget' off and picking it up he replied excitedly 'Yeah! You have to read it backwards!' promptly reading, '10 pb417'.

October 2010

Nathan’s astronaut - nursery

Nathan was in the art area, exploring an idea of his own. Taking a white envelope Nathan tucked coloured paper beneath the flap of the envelope, securing it with masking tape. He explained that the coloured paper was an astronaut, and the envelope was his suit: the tape allowed the astronaut to undo his suit (the flap of the envelope) and climb out of the space suit.

Nathan accompanied his spoken explanation with actions, moving his model rapidly above his head in a trajectory to ‘the moon’; saying ‘blast off!’ and making a whooshing sound as, in his imagination, the rocket left earth.

Although the meaning of Nathan’s astronaut was not immediately accessible to adults, the artefact he’d made and his words and vocal sounds combined with his actions and explanation to make symbolic meanings.


September 2010

One is a snail, ten is a crab

After sharing the delightful picture story book One is a Snail, Ten is a Crab with the children in her combined nursery and reception class (4-5 year olds), the teacher suggested that the children choose their own number and work out which combination of creatures’ legs would total their chosen number.

Tyrees burst out ‘I know why 10 is a crab, because it’s got 10 legs – see, 1, 2, 3, … 10.’ Then added, ‘I know, 9 could be an octopus and a snail.’ He explained he was going to work out which ‘800’ and reaching for some paper wrote ‘800’, after a while explaining that he had 4 crabs and 4 snails ‘that’s 10, 20, 30, 40 and 4 more – 44! That’s not the 800 – I need loads more so I think I’ll do more crabs ‘cos they’ve got most legs.’

He continued to draw ‘There, I done 6 more crabs. That’s 10, 20, 30, 40, 50, 60 and 40 – there’s 100 and 4 more snails – that’s 104.’ Pausing, Tyree decided ‘I don’t think I’m going to do any more because it’s too big. I need 6 more so I can do 6 snails, or a dog and 2 snails, or 3 people – but I’m just going to do a fly because that has 6 legs.’

Calculations: children’s own written methods: counting continuously, separating sets, counting with larger quantities

Their teacher described this as a real ‘eye opener’: this was the first time she had tried to support children’s use of their own graphics to support their mathematical thinking. She regarded the children’s self-challenges and their individual problem solving as remarkable.


May 2010

James is a young four-year old and in his first term at school. Registers have personal meaning to young children as they think about their peers and identify each, enacting the teacher’s role. At nursery and school writing ‘registers’ is often a preoccupation in children’s play.

In this instance James has used a range of letter-like signs, written from left to right. He understands that writing in our alphabetic script uses different letters and has thought about some of their features. He also includes short zigzags, perhaps encoding his sense of the appearance of cursive ‘writing’, or capturing the movement of an adult’s hand as she writes,

James - 'Seeing who's here'

Imagination and symbolic play: making meaning with marks for writing


March 2010

Megan: “A very big fast roller coaster!”

Megan is in the reception class (the first year of school in England). This was the first of three drawings Megan drew at home about fairground rides, including a Ferris wheel and ‘a runaway train’.

Megan’s drawing suggests the route the roller coaster took, its undulating and rapid movement and shows its many seats. Megan told her mum ‘This is a very big fast roller coaster!’ Her mother explained ‘Megan was thinking about how much she’d love to go to a funfair again’.

Megan recalled some of the different rides with excitement and used the drawings as a means of persuading her family that they should take her again.

Communicating meanings through graphicacy
Children use their graphics for a wide range of communicative purposes, to sometimes face and explore anxieties and to feel in control or to develop, negotiate and justify their sense of belonging (in their family and peer group).

They reveal how new media, technologies and popular culture exert influence on both their feelings and their representations and – as in Megan’s example – they show how children also use their graphics to persuade.


February 2010

Max (Nursery)– ‘Yoda’s house

Max drew on his personal interest of the ‘Star Wars’ films in his representation of ‘Yoda’s house’. Although he didn’t give any explain for the details he had drawn, his drawing suggests a plan or a map, with various featured identified by their location on the page and their relationship with each other.

The outer green circle suggests that Max was thinking about the inside of Yoda’s house, and the different arrangements and shapes (of lines and other abstract symbols) suggest that he used them to convey very specific meaning).

Max included an arrow pointing inwards (lower right). Our research has shown that arrows play a distinct role in children’s own early calculations. see:

  • Carruthers, E. and Worthington, M. (2008) 'Children's mathematical graphics: young children calculating for meaning' in I. Thompson, (Ed.) (2008) Teaching and Learning Early Number, Maidenhead: Open University Press, (2nd ed.).

December 2009

Daniel's Sign 'Shop Closed'

In the nursery, Daniel had been playing shops and decided to make a sign to show when the shop was ‘open’ and another to show that it was ‘closed’. His teacher had noticed what he was doing and Daniel explained:

Daniel: It’s closed now, the café is closed
Adult: How do I know it’s closed?
Daniel: Look here, see? Closed, that means it’s closed.

Daniel pointed to his drawing of face crossed out on chalk board and rubbing it out he drew a smiling face without a cross:

Daniel: Look! Open that means its open now... Oh dear...

Drawing a cross over his drawing of a face he explained 'it’s closed'.

Young children use crosses in a variety of contexts to signify different meanings in drawings: they also use them to stand for writing. This flexibility of sign-use is highly significant in supporting children as their understanding of the abstract written language of mathematics evolves.


November 2009

Nathan's 'Writing'

See also: Aman’s boat

Children sometimes also use a particular symbol to mean one thing in one context, and then use it to mean something different in another context. For example, on one side of his paper (not shown) Nathan drew a horizontal line with zigzags as his ‘birthday cake’ (his mum made a ‘caterpillar’-shaped birthday cake for his 4th birthday). Turning his paper over, he repeated the same lines and zigzags (figure 4) now referring to them as ‘writing’.

Other children may use zigzags to signify fierce animals (e.g. crocodiles, monsters); lightening; water or stairs. They are generalising about a graphical sign and also understand that they can be used flexibly. You may like to look out for lines, crosses and other symbols in children’s graphics (e.g. drawing, writing, maps and mathematics).

For more examples, see: Worthington. M. (2009) 'Fish in the water of culture: signs and symbols in young children’s drawing', Psychology of Education Review Volume 33, Number 1, March 2009.


October 2009

Felix's Marks

Felix (4 years 1 month) used pens to make these busy marks. Afterwards he explained it was ‘night-time’.

Young children often attach their own meanings to their graphics after noticing something in their marks and representations.

The pedagogical feature that appears to contribute most to imaginative, symbolic play (and to their understanding of symbolic marks and representations) is adults’ interest in children’s meanings (Worthington, 2009, paper submitted for PhD) unpublished.

September 2009

Aman's Complex Symbol

Aman (4 years 3 months) was playing outside near the sandpit. She found a twig and began scratching in the sand that had spilt on the ground, drawing what she described as ‘boats’. She completed the top of each ‘boat’ with a wavy line explaining this was ‘water’.

By combining the curved line of the boat’s hull with the wavy (or zigzag) line it appeared that she was communicating (in one sign) boat-on-water.

Children explore, make, think about, encode, transform and communicate meanings through their own marks and signs in flexible ways. This helps them understand that signs can be used to carry different meanings (in different contexts and for different purposes) - including mathematics.

 

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