Pretend play and the cultural foundations of mathematics

Abstract

The aim of this study is to uncover the emergence of cultural mathematical understandings and communications in young children's spontaneous pretend play. It is based on Vygotskian cultural-historical perspectives and social-semiotic theory, informed by research into ‘funds of knowledge' and considers how children's informal knowledge of family practices enriches their play and cultural mathematical understandings. Longitudinal, ethnographic data were gathered in an inner-city mainstream nursery in the south-west of England. Data include written observation and graphics of seven children aged three to four years of age engaged in social pretend play. The findings reveal that many play episodes included aspects of mathematics and that these increased through the year: they show how the children's home cultural knowledge underpinned their pretend play and informed their mathematics. The children's graphicacy to communicate mathematics within their pretend play was also evident. The findings show also that where children are immersed in mathematical- and graphical-rich environments, bridging home and early childhood cultures becomes a natural feature of their pretend play. They will add to our understanding of cultural mathematical knowledge in young children.

Keywords:

• pretend play
• role play
• mathematics
• funds of knowledge
• early childhood
• children's mathematical graphics

Background to the study

From birth children are immersed in organised cultural environments with a strong propensity to communicate, using a rich and multi-faceted cultural background. Munn and Kleinberg (2003, 51/53) emphasise that children need to learn the cultural rules concerning ‘how to use a system, and what its role is in our culture’ arguing, ‘these cultural rules are possibly the most important things that children learn’: without understanding them children ‘risk becoming stranded in a sea of meaningless activity’.

Where in ontogenesis do the cultural foundations of mathematics originate? Research has highlighted young children's home knowledge (e.g. Aubrey 1997; Carruthers 1997), yet in educational settings there remains ‘such a mystique about maths as a cultural activity’ (Munn and Kleinberg 2003, 109).

Van Oers emphasises children ‘are from the beginning of their life a member of a community that extensively employs embodiments of mathematical knowledge’ (2001, 59/60). Children participate naturally in cultural practices that include mathematical talk and representations, and parents introduce infants and young children to counting and numbers, through toys, songs, games and numbers for birthdays.

We should expect to see evidence of this cultural knowledge in play. However, reflecting on her informal observations Gifford comments: ‘children's role play was concerned with the larger themes of life, like love and power, rather than mundane things like the price of potatoes’, concluding, ‘a laissez-faire approach to children learning maths … does not work’ since children fail to take advantage of opportunities provided (2005, 2).

Other researchers have found a similar lack of mathematics in play. Munn and Schaffer's study of 10 Scottish nurseries found children's use of number without adult involvement was very rare, and observed no maths in role play (1993). Ewers-Rogers and Cowan found children playing ‘fast food’ scenarios ‘used no numbers … ’ arguing, ‘for ordinary English preschoolers, money may have little significance’ (1996). Brannon and van de Walle concluded from their research, that ‘It may be that for young children, number is not automatically a salient dimension of the environment’ (2001, 75). A systematic study of the experiences of children in ten early childhood settings in England confirmed these findings (Moyles and Worthington 2011). However, it appears unlikely that young children could omit one aspect of their experiences such as numbers from their play: where mathematical thinking is not readily observed in pretend play the implication points to dominant discourses and practices of play. An exception to these findings comes from a study by Cook (2006), which, like research by Carruthers and Worthington (2006) began from an emergent literacy perspective. At the onset of her study Cook intervened by adding play resources that included number symbols (such as birthday cards with numerals) to the role play area in a nursery. The most significant outcomes were the consequent increase in the children's use of number symbols in their play. In turn this stimulated the use of child-initiated mathematical utterances and ‘encouraged the children to play about what they knew’ by drawing on existing knowledge (2006, 65, emphasis in original).

In many settings genuine child-initiated pretend play, ‘voluntary, goal-less, spontaneous – but which for children is entirely serious’ (Brooker 2011, 162) has been ‘displaced by a more systematic induction into social patterns of meaning’ (Parker-Rees 1999, 61, cited in Rogers and Evans 2008). This creates tensions between children's ‘natural and powerful propensity to play in ways that transform and find new meanings … the pedagogical imperative to reproduce real life – the café, the shop, the doctor's surgery – so that requirements in literacy and numeracy can be met’ (2008, 37). This highlights practice common in most of the world where adults choose, plan and resource themed role play areas, revealing adults’ perceptions of children's interests, rather than children's authentic and immediate interests that have personal cultural meaning. However, since we support the ideology of child-initiated play, is it realistic to assume it could include aspects of mathematics?

The present study builds on research by Carruthers and Worthington, into children's mathematical graphics (their personal marks, symbols and representations), to focus on the emergence of mathematics in pretend play. Carruthers and Worthington developed the educational concept of children's mathematical graphics from ‘emergent writing’ in which children use their own informal marks and graphical signs to represent and communicate meanings. Their research reveals a continuum from young children's earliest marks to which they attach mathematical meanings, to written calculations (2005, 2006).

Two specific questions are addressed in this article:

1. What evidence of mathematics can be found in young children's free pretend play, and what is the breadth of their mathematical interest?

2. To what extent do children draw on their personal cultural knowledge in their pretend play, and how does this influence their mathematical thinking?

Theoretical framework

Cultural knowledge and mathematics

Mathematics is a human product inseparable from its cultural context (Brandt and Tiedmann 2009). As humans we learn through participating in cultural practices (Rogoff 2008) and through activity humans ‘assimilate the experiences of humankind’ (Leont'ev 1981, 55). Basically, Vygotsky identified two ways of appropriating culture: through directly experiencing cultural situations and practices (leading to spontaneous, empirical, everyday concepts), and through instruction (leading to schooled, scientific concepts). Spontaneous concepts lay a foundation for later elaborations into more scholarly concepts. ‘The development of scientific concepts begins in the domain of conscious awareness and volition … the development of spontaneous concepts begins in the domain of the concrete and empirical. It moves towards the higher characteristics, towards conscious awareness and volition’ (Vygotsky 1987, 220).

Pretend play furnishes opportunities for the development of everyday concepts and, in Vygotsky's view, provides a ‘bridge’ between spontaneous and scientific concepts (1987, 238).

Munn and Kleinberg emphasise that whereas children can readily ‘be taught the mechanics of arithmetic, if they lack any wider sense of purpose of these activities then their spontaneous learning will be hindered’ (2003, 52). However, in many situations home and school cultures are viewed as mutually exclusive (Abreu et al. 1977). Yelland and Kilderry argue ‘it is difficult for children to link mathematical skills and concepts taught only in isolation, since ways concepts are taught in school are frequently very different to their use in everyday life’ (2010, 93).

As active participants of family cultural practices children draw on their personal knowledge in their play, enabling their previously acquired body of cultural knowledge (‘funds of knowledge’) to come to the fore (Moll et al. 1992, 133/4). Taking this focus Riojas-Cortéz analysed role play, reflecting it ‘provided a naturalistic picture of the linguistic and cultural repertoires’ children possess (2000, 305). This highlights how children's current cultural knowledge contributes ‘to their growing content knowledge’ as they move ‘from novice to expert … [emphasising] the importance of constructing new knowledge based on existing evidence’ in collaboration with others (Hedges and Cullen 2005, 4).

Mathematical communication in early childhood

Communication is significant in mathematics, but where formal vocabulary and written notations are introduced without meaningful cultural contexts their use will fail to make sense. From a Vygotskian perspective, mathematical thinking in ontogeny begins within participation in communicative cultural practices that have personal meaning for the children (Van Oers 2012).

Pleas for greater emphasis on the cultural significance of mathematics have been voiced (Bishop 1991; Saxe 1991), and for a more discursive approach that recognises children's own understandings (for example, Krummheuer 2013). Two streams of research have explored early mathematical notations from these perspectives. In the Netherlands van Oers (e.g. 2012) and Poland (2007) have demonstrated how children's ‘schematising’ in play can support abstract thinking that is so important for mathematics. In England Carruthers and Worthington have researched early childhood mathematics from the child's perspective (2006), beginning with the premise that, as with emergent writing, young children can use their own mathematical graphics to explore and communicate their mathematical thinking (e.g. 2006).

Children's need to elaborate ideas by symbolic means is rooted to a great extent in their desire to communicate some aspect of reality and can be realised through their pretend play. As they imitate and explore mathematical ideas and culturally specific tools and symbols in which they have been involved at home, children clarify their ideas and elaborate their goals in culturally meaningful ways.

Pretend play and mathematics

Vygotsky acknowledged play as the ‘leading activity’ for young children, proposing ‘As in the focus of magnifying glass, play contains all developmental tendencies in a condensed form and is itself a major source of development’ (1978, 102). Social pretence and imagination offer potentially rich contexts that ‘situate’ learning and allow children to explore their existing cultural knowledge of mathematics. Vygotsky related play to creating an ‘imaginary situation’:

A reproduction of the real situation takes place … only comprehensible in the light of a real situation that has just occurred. Play is really more nearly recollection of something that has actually happened than imagination. It is more memory in action than a novel imaginary situation. (1978, 103)

Whilst the most effective play appears spontaneous, it does have its own internal ‘rules of behaviour’ that ‘stem from the imaginary situation’ (1978, 95), evident in the examples in this study.

In the following section we describe an empirical study exploring the occurrence of mathematics in children's pretend play related to their personal background knowledge (‘funds of knowledge’). For the purposes of this study pretend play episodes are defined as social play involving two or more children, engaged either in role play or making maps involving elements of pretence and imagination.

Characterisation of the study

In order to answer our research questions we conducted an ethnographic study, focusing on ‘detailed accounts of the concrete experience of life within a particular culture and of the beliefs and social rules that are used as resources within it’ (Hammersley and Atkinson 1995, 9). Ethnographic research allows researchers ‘to get alongside children in their environment’, to ‘enter the world of the participants’, and is a method ‘befitting the exploration of the meanings and constructions held by research participants of their social world’ (Emond 2005, 124/25).

In our study the semiotic focus is on children's graphics and reflects this aspect of social semiotics: the study did not investigate other multimodal dimensions such as body language.

Data gathered are qualitative and suggestive of a ‘thick description’, expressed as ‘a multiplicity of complex conceptual structures, many of them superimposed upon or knotted into one another, which are at once strange, irregular, and inexplicit.’ According to Geertz a researcher must somehow contrive such description in order to be able ‘to grasp and then to render’ (1973, 10).

Research setting and participants

The research setting is a nursery school within a Children's Centre in a large city in the southwest of England, in an area designated one of the 30% most deprived in England. It welcomes families from various ethnic cultures, providing many services to support families, babies and young children in the locality. Sixty children attend nursery sessions each morning and afternoon, and 13 different languages were spoken there at the time of data collection.

The children are free to initiate their own ideas in play, time for play constituting the greater part of each session. It was expected that in a setting in which mathematics and graphicacy have high profiles, children would be likely to explore these aspects in their play. However, at the onset of the study it was not clear if all children would choose to communicate through graphicacy, and the teachers were asked to identify several children who did so. They based their judgments on their knowledge of the children and their previous written observations, identifying Isaac and Shereen, both four-years-old, and Elizabeth, three years and six months.

In addition to these focal children, four additional children (Oliver, David, Ayaan and Tiyanni) were randomly chosen in order to determine if they might also choose to communicate through graphics, providing a total of seven case study children, three boys and four girls. All the children were in their final year at nursery, and at the beginning of the academic year their ages ranged from three years and two months, to four-years-old.

Data sources

The main body of data is taken directly from teachers’ written observations¹ or ‘learning stories’, which ‘document the learning culture … this is what we value here’ (Carr 2001, 103). The data also include graphics from the nursery and children's homes, and transcripts of discussions with the teachers: field notes made in the nursery and during the researcher's home visits to each child's family provided additional information. Six of the seven children were visited at home: it was not possible to visit the remaining child due to personal family circumstances. Visits were informal, the main aim to see the child within the social and cultural context of their home and family, and to observe any spontaneous play or graphics in which they engaged at the time. Short discussions with parents during the home visits focused on the child's play and graphicacy. Data focused on a period of one academic year.

Procedure

The teachers’ established practice is to write ongoing observations of children's play behaviours, actions and talk. As participant observer the researcher also made written observations of play during regular visits to the nursery. As a methodology naturalistic observations used in ethnographic research ‘enable us to study children in situations that have real emotional significance to them … [They] provide invaluable evidence on children's real-life experiences and their reaction to those experiences’ (Dunn 2005, 87).

In order to eliminate bias towards specific research outcomes, two factors are significant. Firstly, the teachers’ standard practice in this nursery is to write observations of all aspects of the children's play and learning throughout each day, writing for children, staff and parents rather than for the purposes of this research. Secondly, to ensure that not only observations of pretend play which included evidence of mathematics were selected, written observations of all pretend play episodes are included as data for analysis.

Data analysis

Analysis involves ‘interpretation of the meanings, functions, and consequences’ (Hammersley and Atkinson 1995, 3) of the children's play and graphicacy. ATLAS-ti software was used to support systematic data analysis of the written observations, and for the purposes of this article, data were coded to identify evidence of pretend play episodes. Within these episodes the following aspects were identified and coded:

1. The children's cultural knowledge, showing the mathematics they explored – coding evident aspects of the children's home funds of knowledge;

2. The mathematics explored – coding the specific mathematics that could be identified, such as children's use of numbers, references to time or measurement;

3. The children's use and understanding of mathematics – coding the role of the mathematics in the context of the children's play and how they appeared to use and understand mathematics to communicate to their peers and to further their play narratives.

Within ATLAS-ti, transcripts of each observation were examined for evidence of the child's known cultural knowledge, based on home visits made by the first author, discussions with the child, their parents and teacher. Relevant sections of text were highlighted and the code ‘a’ assigned: this was repeated for ‘b’, for those observations that contained evidence of mathematical talk or representations. Evidence of mathematics pointing to the child's mathematical understanding, and which appeared to serve a relevant role within the play narrative was coded ‘c’.

In order to strengthen reliability and validity, an additional researcher conducted independent coding of the data, applying it to 10% of randomly chosen observations. Agreement was reached for 94.59% of the codes assigned, a significant level of consensus.

Ethics

Data collection was guided by BERA's ethical principles (2011) and includes voluntary informed consent, openness, the right to withdraw and privacy. The parents were consulted at the onset of the research and their permission sought to observe their child and collect data. Using everyday language, the research was explained to the children and their agreement sought. One family withdrew their child² from the study early in the period of data collection, and none of the data pertaining to this child have been used. Questions of power concerning the first author to gather data is acknowledged: staff at the nursery continuously make written observations in the context of their work, activities with which parents and children are familiar. Parents all gave signed consent for their child's first name to be used in publications relating to the research, whilst confidentiality and anonymity are protected since no surnames are used.

Presentation of findings

Cultural influences and interest in the children's pretend play

Factors influencing variation between children

Analysis of the data shows the children engaged in a total of 146 pretend play episodes over three terms,³ with wide variation from 51 episodes for Isaac, to eight for both Tiyanni and Elizabeth. Variations in social pretend play depends partly on children's confidence in interacting with others to initiate and maintain it. Isaac obviously built this confidence in the year he had already spent in nursery. Tiyanni's lack of confidence may have related to some other difficulties. Elizabeth had attended nursery since she was one-year old and was very confident: in the previous year she had often engaged in pretend play but developed other interests during the year of data collection.

Ayaan's mother described her as ‘very shy’, in marked contrast to her more outgoing siblings. Ayaan's first language is Somali and during her first two terms at nursery she spoke very little English, engaging in no role play during this period. By the third term Ayaan's confidence in speaking English and understanding the nursery's culture had grown, and she often chose to initiate role play with her peers. Cultural differences related to adult attitudes and understandings of play may have been a factor for Ayaan (Somalian) and Tiyanni (West Indian).

The children's play also appeared to be influenced by the extent of their direct involvement in cultural experiences at home or work and was especially marked for Isaac who had accompanied his father to work since he was small and had also been directly involved in the conversion of their home, work that included extensive use of mathematical tools and talk.

Children's interest and involvement

All children showed deep interest and a high level of personal involvement, controlling what Hughes (2001) refers to as ‘the intent and content of their play’ (cited in Brown 2012, 68). Moreover, the children often developed themes and ideas over many days, and in some instances over several terms, allowing increasing complexity of ideas.

Children's play narratives

All pretend play episodes were observed in the nursery: individuals spontaneously initiating their play narratives during the extended free play sessions that are an integral aspect of the nursery's practice. The children's unfolding play narratives revealed that in every episode they drew on their funds of knowledge and suggested also a subtle interweaving of new cultural knowledge from the nursery.

Home cultural influences

Shereen's family is from the Philippines, and shopping, preparing and eating meals together have special cultural importance. Elizabeth enjoyed camping with her family and Oliver was interested in trains.

Shopping with her mum Ayaan knows a lot about paying for goods, and sees her dad count the money he takes each day from working as a taxi driver. At home she loves helping to prepare ingredients and helps care for her siblings. Isaac's father worked as a builder: Isaac sees his dad setting out wood for carpentry involving calculating, helps his father measure timber, mark out squares and angle cuts and understands the use of a spirit level. Isaac's father now runs a local brewery involving deliveries, invoices, payments and counting cash, and Isaac is involved in all these activities. Isaac is also very knowledgeable about a wide range of technologies, vehicles, maps and camping and his father also shares his interest in motorbikes and trains.

The following two observations exemplify aspects of children's cultural funds of knowledge, revealing the embedded nature of their mathematics. Ayaan was just beginning to explore mathematics in her play, whereas Isaac's longer episode contrasts in its complexity. The following two transcripts are taken directly from the teacher Emma's written observations⁴.

For two weeks Ayaan had been playing in the gazebo, offering pretend ice cream through the window to children. Today when a child replied ‘Yes’, Ayaan answered ‘No left’, adding ‘I make more’. Collecting stones and pretending to make ice cream, Ayaan asked Tariq if he wanted any. She passed him an imaginary one, then pressed buttons on the till saying, ‘It's 50 minutes.’ Shortly afterwards Ayaan drew dashes in a notebook without comment.

Next time Ayaan played ice cream shops she asked ‘50 minutes please’. When a child offered ‘£1.00’ Ayaan replied ‘That's £50 please.’

Appreciating his interest in security and locks, Isaac's teacher Emma bought a small safe into the nursery for the children to investigate.

A few days later Jayden and Isaac moved a small cupboard to create a safe, placing a keyboard and clipboard on top. They transported wooden blocks on the trolley and when another child removed one, Jayden wrote wavy lines on his clipboard. Taking his paper, he placed it in the safe, tapping several keys on the keyboard and repeating this each time a child removed a block.

Isaac announced, ‘this is the safe. There's a key, only one – you press it here and it opens. It has a number and no one else knows it, “one, one, eight, seven, zero, six”. It's rather difficult to remember.’

Jayden put some real coins and play cheques in their safe and Isaac stuck a calculator on the cupboard door adding: ‘You need to press the buttons to get in the safe. … it's four, nine, seven, nine’. Jayden pressed some numbers making ‘Beep, beep’ noises as he opened it, then closing the doors asked, ‘What's the closing number?’ saying ‘one, nine, five, two,’ as he pressed buttons on the calculator.

Later Jayden said, ‘you need to give me ‘one, nine, five, two’ and when Emma explained that she didn't have enough cash but could write a cheque, Isaac replied, ‘I need hundreds of pounds!’ Emma managed to find a selection of coins in her purse and Jayden responded: ‘Okay! We need to fill the box: you need to give me 15 hundred and 60 pounds.’

After several days playing with their ‘safe’, Isaac decided to write down the number of blocks being taken from the block areas, ‘one, two, three, gone! Gotta write it down and put it in the safe.’

Reflecting on her observation, Emma wrote:

 …  both boys were using their experiences and understanding of numbers in a real situation. Isaac showed huge awareness of numbers and combinations [and] Jayden, in awe of large numbers in relation to money, knew that £1560.00 was a large amount of cash. Isaac understands that number sequences can be unique for codes on safes and that longer sequences would be harder to remember and less likely that unwanted people will be able to get into the safe.

Sustained play episodes appeared to be most effective in allowing ideas to be explored and developed, and sometimes re-visited over time. Isaac was most likely to do this, the children with whom he played clearly benefitting from his initiatives, contributing their own ideas and peer-models of graphics to their joint narratives.

Mathematical use and understanding

Incidence of mathematics

Over 44% of all pretend play episodes included evidence of mathematical exploration: this exploration was most often evidenced through the children's dialogue, the children using this to communicate mathematical ideas to further their play. Transcripts of Isaac's play showed 19 occasions on which he made reference to mathematics, to only three for Elizabeth. However, Elizabeth often engaged in mathematics independently of pretend play, showing mature understanding that suggested more conscious enquiry. For all children, incidence of mathematics in pretend play increased throughout the year.

The range of mathematics

The transcripts of Ayaan and Isaac's play included above provides full details of their play behaviours and talk as documented by their teacher. Table 1 shows the range and quantity of the children's use of mathematics in all their pretend play episodes.

Table 1. Showing the children's use of mathematics within their play (numbers represent the quantity of play episodes that included references within each category).

Number, quantities counting	money	time	length and distance	direction	speed	weight	temperature	shape, space & capacity	Data handling	Totals
31	24	15	4	2	4	3	3	10	4	100

Table 1 shows that the children included references to number, qualities and counting most frequently. The high number of references to time and money also underscores their significance in the children's home experiences. Isaac for example made 32 references to mathematics within his play (32% of the total made by all children). Whilst Ayaan only began to engage in pretend play during the third term of the year in which data were gathered, she also made a surprising 22 mathematical references.

Tables 2, 3 and 4 summarise children's use of mathematical notions in relation to different pretend contexts.

Table 2. showing the contexts of the children's play and their references to number, quantities and counting.

Contexts	References to mathematical notions
Playing ‘car parks with Isaac in the sandpit, Oliver made a sign for the gate, explaining,	‘These are ticks. When there are three ticks you can go, when there are two you can't go that way. I've made two ticks - that means you are not allowed. People allowed in that way.’
Ayaan playing ‘mother and baby’:	Used numbered raffle tickets as buttons on a remote control for her ‘television’.
Playing ‘deliveries’, Isaac filled a wheelbarrow with bottles, paper and a watering can, explaining:	‘I'm delivering wine, I can't remember which house number but I remember which street and what the house looks like.’
Several children wanted to get inside the car in which David and Remi were playing, but there was no room.	David drew several large crosses on a piece of paper, saying ‘No more children getting in our car.’
Shereen, playing shops,	Counted 20 items on her shopping list with one-to-one correspondence.

Money

Transcript of observation exploring money:

David and Isaac were talking down the phone to each other. Isaac decided to use a diary as a ‘booking book’ for a campsite, and explaining that two people were staying, made two marks in the diary. Isaac used the phone to take more bookings, telling David ‘one hundred million people are staying!’ David replied, ‘I want to stay for two nights’.

But Isaac said, ‘No. I'll put you down for two million nights, but don't worry – it's only £1.00 a night’. He then wrote it down in his ‘booking book’, this time making many marks and David also took a diary and made his own symbols (circles and vertical lines).

Time

Exemplars of the remaining areas of mathematics the children explored in their pretend play are given below:

Table 3. Showing the contexts of the children's play and their references to money.

Isaac and David drew a collaborative, imaginary road map. David explained,	‘You have to pay money. It's £2.00 to park here’.
Isaac with several other children, playing builders in the sandpit.	Isaac wrote a cheque for £500.00 ‘for all the jobs done’.
Isaac, playing builders with several boys in the small house outside: Isaac said:	‘£1000.00 to hire generators’ so we're getting electricity to the building.’
Oliver and Remi have a large cardboard box with interesting shaped holes on the top, into which they post plastic ‘buttons’. Remi announces it's a ‘spaceship’.	Later they changed the function of their ‘spaceship’ to a ‘cash point’ Remi says, ‘We're making money’.
Shereen, playing shops,	Wrote a receipt for a customer.

Table 4. Showing the contexts of the children's play and their references to time.

Playing ‘builders’ with several boys, Isaac said,	‘This is so we can time how much building we're doing.’ ‘We're building a builders’’ yard – we've got 50 minutes left.’
Isaac and David drew a collaborative, imaginary road map with a beach. Isaac explained,	‘Here's where you park your lorries for two hours while you sit on the beach.’
Tiyanni spooned uncooked rice from a baking tray into a cup and put it in the toy oven.	‘This is going to cook for an hour. Hours and hours and hours.’
During imaginary play of builders in the small house outside, Isaac fitted a clock on a hook on the wall, explaining,	‘This is so we can time how much building we're doing. We're building a builders’ yard – we've got 50 minutes left.’
David held a small clock in his hand and pretended to sleep, telling Remi,	‘Five minutes to go … two minutes to go … morning time! We can watch TV now’.

Shape, space and measures

Length and distance:

• Playing builders, Isaac and Jayden estimated the length of the pretend house they were building and Isaac used a tape measure to see ‘How far away to put the next block’ for the house he was building.

• On another occasion (also playing builders), Isaac measured a large box with tape measure, saying ‘60 metres.’

• While drawing a building plan with Isaac, Jayden remarked ‘my house is getting bigger and bigger.’

Speed and direction:

• David referred to the need for a road sign designating the speed limit on the imaginary map he drew with Isaac, showing lorries ‘rushing to the beach’.

• Isaac explained the arrows he'd drawn on their map: ‘these are arrows to say ‘go this way’.

• Isaac used a compass, referring to the compass points ‘north and south’ and related them to locations in the city.

Size/area/capacity:

• Isaac and Jayden built a doorway: Isaac commented. ‘Lefty [a character from ‘Sesame Street’ on television] was too big outside and couldn't get through the door. You'd have to measure it to make sure.’

• An additional child was desperate to sit inside the pretend ‘camper van’ that several children had built, but Isaac explained, ‘There's only room for two – not three!’

• Making a joint plan of a house, Isaac announced, ‘this is my big eating room’. Jayden added, ‘my house is getting bigger and bigger  …  ’.

Weight:

• David, Oliver, and several other boys are outside mixing soil, sand and bark chippings in the wheelbarrow. Oliver said ‘We're making chocolate cake!’ as they stirred their mixture, David commented, ‘It's too heavy – we can't lift it.’

Temperature:

• Isaac explained a feature on the map he'd drawn with David, ‘there's the sandy beach – it's as hot as chicken!’

• Tiyanni removed her pretend cake from oven, ‘It's not warm like yours – it's cool now’

Data handling:

• David and Jayden and Isaac are by the door into the nursery with clipboards, paper, pens and a calendar, checking people in and out. Isaac uses vertical marks for entries, then a letter ‘X’ for a member of staff who he says, will soon be leaving and David writes a mark in her hand saying, ‘That means you work here.’

Children's mathematical graphics

In over 46% of all play episodes that included mathematics, the children also spontaneously used their mathematical graphics to communicate: these included arrows to signify direction; marks and abstract symbols to represent quantities, to show food eaten and to signify cinema open and closed; crosses to signify ‘no more children’ and a personal abstract symbol for ‘£’.

In young children's informal mathematical representations, drawings of arrows and hands have been shown to signify the operations of adding and subtracting (Hughes 1986; Carruthers and Worthington 2005, 2006; Poland, van Oers, and Terwel 2009).

Graphicacy

Shereen's graphics were often mature with clear representational drawings and some use of standard letters of the alphabet, although like all the children she often used scribble-marks as ‘shorthand’ for writing within her play. Isaac had a highly developed interest in signage and writing in the community, and in the nursery and at home frequently drew on this knowledge in play.

The amount of graphicacy the children used is insufficient to allow clear conclusions to be drawn: for example, whilst Elizabeth only once used graphics within her few episodes of pretend play, she showed some mature use in a range of other contexts. All of the seven children used personal abstract symbols to convey meaning, and the remaining four children showed their developing understanding of various symbolic systems.

Adult roles

Parents and other family members influenced children's narratives and concepts.

In some homes pretend play was encouraged and valued, which was evident during home visits. For example, Elizabeth created an elaborate tent with her brother. Isaac set up a complex ‘ice cream van’ in his dad's van, something he often played with his dad. Isaac also made a ‘register' for his family, an aspect of the nursery's culture that had travelled back to home. Graphicacy appeared to be valued in all the children's homes, evident in home scrapbooks and during home visits.

Teachers’ roles

There appear to be contrasting adult responses to children's play: either, without adult intervention, or by adults directly mediating in play.

Whilst teachers’ roles are not a feature of this study, it was clear from the data that adults in this nursery value and support children's free pretend play and interests. Rather than intervening in play they mediate learning in other ways: through creating mathematically- and graphically-rich environments; modelling graphics for authentic purposes throughout each day, and through collaborative dialogue they support and extend children's thinking.

Discussion and conclusion

The aim of this study was to uncover the emergence of cultural mathematical understandings and communications in young children's spontaneous pretend play. As an answer to our first research question, we can say that evidence of mathematics was found within all the children's play and that it extended beyond number and quantity to span the breadth of the mathematics curriculum. No remarkable differences were found between the focal and the randomly selected children.

Drawing on their home cultural mathematical knowledge the children imitated and extended ideas that fused reality with imagination. The children's use of their spontaneous concepts from the mathematical domain contributed to their play in ways that made sense both in the contexts and development of their narratives. These findings are in direct contrast to almost all findings from research into pretend play and mathematics referenced earlier, and may be accounted for as differences between the cultures and philosophies of various early childhood settings.

In the nursery in which the data for this study were gathered, the head teacher and staff have developed an open ethos in which children are encouraged and supported as learners, and their interests and emerging understandings valued. Adults have clear philosophies of young children as learners and of play and mathematics, and have developed deep knowledge of learning and significant pedagogical skills to support children's thinking and learning.

Answering our second research question, analysis of the findings revealed that the children drew extensively on their personal cultural knowledge in their pretend play, exploring and elaborating their mathematical knowledge within the context of their unstructured pretence and imagination. Their cultural knowledge influenced their mathematical thinking by providing coherent contextual and mathematical meanings within their chosen play narratives. Written observations of the three children selected by their teachers for the study showed that they continued to develop their personal interest in graphicacy throughout the year, in both play and a range of child-initiated contexts.

Paradoxically this research reveals that whilst the adults in this nursery did not plan for mathematics within pretend play, the children's self-initiated play triggered their cultural mathematical understandings. Why are these findings in such sharp contrast to those cited earlier? Play scholars have identified the widespread lack of understanding of play, which results in pretend play that lacks clear connections to the children's personal experiences of life (e.g. Rogers 2010; Brooker 2011). Consequently concepts are ‘conceptually disembedded from the practices and the imaginary situation being played out by the children’ (Fleer 2010, 75, italics in the original).

Rogoff highlights learning through participation in cultural activities, involving ‘three inseparable processes’ (2008, 58). Apprenticeship through participation in cultural events is evident in examples such as Ayaan's involvement in shopping with her mother and Shereen's participation in preparing and sharing family meals. Isaac's guided participation in his father's work is unambiguous, as are teacher-models of graphics for authentic purposes. Participatory appropriation allows ‘change through involvement’; visible throughout the children's increasing understandings in pretend play and in other contexts during the year.

As Göncü and Gaskins emphasise, ‘a child's biology, culture and experiences all influence behaviours and play’ (2007, 10): for adults in early childhood settings it is important to understand these influences and maximise opportunities for effective learning. The challenge facing teachers is to determine contexts that will most effectively enable children to employ and explore their cultural knowledge, and to develop ‘spontaneous’ concepts that will gradually connect with scientific mathematical concepts (Vygotsky 1978). Understanding mathematics in cultural practice enables children to ‘bridge’ home and early childhood cultures (Carruthers and Worthington 2006). It begins between people on an interpsychological level, and subsequently within the child on an intrapsychological level: ‘all the higher functions originate as actual relations between human individuals’ (Vygotsky 1978, 57).

On the basis of our findings we can only endorse Brooker's conclusion that ‘we need only to offer children spaces in which they can undertake activities which are important and meaningful to them. … Increasing mediation from the adults and children around them and from the cultural resources’ enables children ‘to increase their participation repertoires, hone their skills and move from being peripheral members of the group to full membership’ (2011, 162). Effective pretend play should be ecologically valid and offer optimum spaces and contexts for the social interactions that will induct children as apprentices and participants into the cultural knowledge of mathematics.

Acknowledgements

With sincere thanks to the children, parents, head teacher and staff of Redcliffe Children's Centre and maintained nursery, Bristol, for sharing their wonderful play and thinking.

Notes

1. Whilst the word ‘teachers’ is used throughout, they work closely with all the qualified early childhood professionals in the nursery, who also contribute to the children's learning diaries.

2. The study began with eight children

3. Of the seven case study children, for one child data were collected for only two terms.

4. The only change made to these transcripts is the substitution of ‘Emma’ for ‘I’ and ‘she’