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PRE-SCHOOL CHILDREN AS MATHEMATICAL MEANING MAKERS

Agnes MacmillanCharles Sturt University

ABSTRACT

Informal environments usually do not specifically aim to engender mathematical knowledge as a distinctive realm of expertise, and the children, then, would not be expected to be functioning as students of mathematics. However, analyses and examination of the children’s discourses and activities at two pre-school settings in the Hunter Region of New South Wales revealed use of a range of mathematical concepts. Having many opportunities to use language in ways which suited their own imaginative and real-world purposes, the children were able to talk openly and freely, and were able to listen and respond to each others' talk (Nunes, 1996) within the activities of the culture in which they found themselves (Lave & Wenger, 1991)the nature and extent of their linguistic and social knowledge gave them access to the “mathematical” meanings embedded in their play contexts. Abstract thinking processes necessary for literate and numerate knowledge were also evident in the children’s spontaneous discourses. In this paper, one transcript taken during a playdough session at one of the pre-school settings is being used to demonstrate the kinds of mathematical meanings which occurred during play, and to examine the relationships between socially generated meanings and their abstract realisation in the cognitive domain.

Keywords: mathematics, play, young children

INTRODUCTION

One morning recently I went into my local newsagency to buy a birthday card and as I approached the cards display a young pre-school-aged child spoke to me as he picked out a card. “Look at this. It’s a five. I’m not five yet. I’m this one. I comes before five.” He lifted out the card with a shiny “four” on it. I told him, “That’s very clever.” I began my search for a card and was again drawn into the child’s world as he reached for another card higher on the stand: “This is a seven. I won’t be seven for um ... six years.” “Oh, I see” said I. He then pulled out a card showing a small dog surrounded by flowers and said, “Look at this pretty one. I've got a dog like that. He’s nearly the same.” The child’s father then called to him to go with him.

In this informal context, the child was using the social opportunity to communicate what he knew about some abstract ideas. He conveyed through language, numerateor literate and mathematicalknowledge which included number recognition, ordering, and comparing. Not only could he decode the mathematical symbols, but he could apply that

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knowledge to a concept (his age) and an object (his dog) that were interesting and meaningful to him. To express what he knew to another social being was his prime interest and concern. Realisation through linguistic expression of his abstract knowledge occurred through the availability of a socially generated opportunity.

In the curriculum of the informal early childhood sector, mathematics is not an area of specific focus. In fact, no knowledge domain is given a specific focus. Rather, all are conceived within the broad area of cognitive development which holds a place of importance alongside physical, social and emotional development. There are many good reasons for this being the case, and although critiques (Dunn & Kontos, 1997; Fleer, 1995; Katz, 1992; Kostelnik, 1992) and reconceptualisations of developmentally appropriate practice are evolving (Bredekamp & Kopple, 1997; Dockett, 1998; Dockett & Fleer, 1998; Fleer, 1996; Goncu & Fitzgerald, 1994; Halliwell, 1992; Kessler & Swadener, 1992; Siraj-Blatchford, 1997; Wein, 1996), my main purpose here is to facilitate understanding and awareness of the rich potential for mathematical and other abstraction processes within informal contexts, and to make explicit their implications for practice. An outline of relevant aspects of the study’s theoretical framework is followed by a brief explanation of the methodology. The discussions of the playdough episode aim to demonstrate:

the nature of spontaneously occurring mathematical meanings; the links between the linguistic, mathematical, social and cognitive dimensions of

experience, by viewing abstraction as a cognitively realised but socially generated phenomenon allowing children to create what is real from what might be imagined or abstractly expressed;

that learning about the social dimension—how to be a responsibly self-regulative participant in a culture—arises from imaginative or real-world, socially-generated contexts.

ABSTRACT REPRESENTATION AS A COGNITIVELY-REALISED, SOCIALLY-GENERATED PHENOMENON

Play contexts as facilitators of literate meaning making were examined by Rowe (1994) in her seminal work in which spontaneous “literacy events” were identified as significant for pre-schoolers’ realisations of their authoring potential. The unconscious processes of hypothesising generated within playing contexts—realised logically in the conscious system—were explained and demonstrated by Rowe. Her thesis was that the self-monitoring process of hypothesis generation, or testing of anomalies, instigates metacognitive processing. Elsewhere, social representation theory reflects this understanding that semiotic relations are inherent in social representation through processes of signification, or social markings such as role identification, and representation (Lloyd & Duveen, 1993; Moscovici, 1981). Role play, as a feature of imaginative play, is being viewed in the following analyses of children’s playing discourses as a process of decontextualisation derived from the need to represent social knowledge not only through action (perceptually) but also abstractly through language. A semiotic perspective on the role of language in the teaching and learning of mathematics


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extends this conception of mathematics as a system of abstract representation reliant on linguistic expertise (Alrø & Skovsmose, 1996, 1998; Chapman, 1993; Halliday, 1978; Lerman, 1996; Nunes, 1996; Otte, 1997).

Reflexive PositioningPositioning is a metaphorical term used in social psychology and sociocultural theory to explain relations and dynamics associated with power-knowledge relations and identity development. It provides a framework for analysing the potential for members in a community to obtain equitable relations of power and access to the knowledge base and resources of a particular culture. The processes involved in social representation the way we perceive our congruence and acceptance in a particular culture—contribute to the development of two forms of positioning: the intrapersonal aspects of identity development which rely on processes of reflexivity, and the interpersonal meanings and relations which contribute to the development of a collective or group identity (Davies & Harré, 1990). Tan and Moghaddam (1995) defined the concept of reflexive positioning in the following way:

In the same way that autobiographical aspects of conversations are the basic matter of interpersonal positioning (eg. Davies & Harré, 1990, Harré & Van Langenhove, 1992), reflexive positioning is a process by which one intentionally or unintentionally positions oneself in unfolding personal stories to oneself. (p. 389)

In other words, the critical viewpoint from which one reviews and evaluates one’s own social position provides the impetus for metacognitive thinking. People form hypotheses about: intentionally, or how well they have communicated or managed to disguise their intentions; adaptability, or the capacity to change to suit new situations or information; integration, or the capacity to integrate with other’s ideas; transferability, or the capacity to transfer their own understandings and curiosities to a specific social context. Reflexivity is linked to the metacognitive process of self-reflection and refers to the flexibility of the critical functions which precede, accompany or follow social interaction (Tan & Moghaddam, 1995). Hereafter, I refer to these processes as socio-cognitive processes.

Situated CognitionCorsaro (1985) extended Vygotsky’s (1978) theory that cognition moves from the social to the cognitive and referred to children’s cognitive processes in social interaction as “interpretive abilities.” Corsaro’s study of pre-school children during play led to the suggestion that children’s interpretive abilities were used to link cues about how to participate competently, but Corsaro also argued that the children were not yet able to summarise reflexively the significance of the procedures they use. Rather, at a cognitive level, children appear to shape and share in their own developmental experiences, and they take up unresolved problems, tensions and confusions (Corsaro, 1985). According to interactionist (Alrø & Skovmose, 1996, 1998; Lerman, 1996; Nunes, 1996) and social psychology exponents (Davies & Harré, 1990; Harré & Gillett, 1994), children reproduce meanings or redescriptions of experience, and they do not follow some predetermined


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linear path from the cognitive to the abstract or social as Piagetian theory suggests. Rather, intellectual engagement in social interaction leads to new ideas and inquiries (Corsaro, 1985). For reflexivity to occur for young children, then, they would usually need to be encouraged to critically assess a situation from their own vantage point in a given social situation. Some argue (Bruner, 1986; Wertsch, 1985) that this is fundamental to the notion of situated cognition (Lave & Wenger, 1991). Similarly, the zone of proximal development is another interactionist interpretation of the socially-generated nature of abstraction (Vygotsky, 1962).

According to Corsaro (1985), there may, then, be doubts about the extent to which most pre-school children engage in the kinds of detached, objective evaluation processes necessary for critical conscious self-reflection of their own positions in a community, but their interpretations of social experience are irrevocably, inevitably, and unconsciously shaping their images of themselves (Harré & Van Langenhove, 1992; Tan & Moghaddam, 1995). In the normal course of events their self-concepts will be challenged, threatening personal development (or ontogenesis) which in turn threatens interpersonal development (or sociogenesis). Their interpretations facilitate or inhibit their capacities to understand better the meanings of their culture, and ultimately to establish a position of co-participative, responsible self-regulation within that culture (Lave & Wenger, 1991). (See Macmillan, 1998a, for explanations of the relationships between moral and social reasoning, and identity).

Socio-regulative StrategiesThe analyses of the playing episode presented in this paper draw attention to the significance of the intrinsically motivating, and therefore intellectually involved (Malone & Lepper, 1987) nature of play, and its provision for opportunities for social representation and interpretation and management of the rules of the culture. The open-endedness of play offers to the participants the control of communication, socio-regulative behaviours, and resources. Children learn how to position themselves within access to the discourses of the community by using responsive control, or socio-regulative, strategies (modelling, observing, imitating, assisting, improvising, demonstrating, positioning) or restrictive control strategies (exclusion, resistance, denial, closure, transgression, rule-domination) (Bourdieu, 1971; 1977; Foucault, 1971; 1977).

Mathematical Meanings

Socio-regulative Strategies

Socio-cognitiveProcesses

Affective Outcomes

Figure 1. Relationships between the mathematical meanings, socio-regulative processes,


Intentionality Systematicity:

AdaptabilityIntegrationTransferabilityReflection

Modelling:ObservingAssisting Imitating

DemonstratingImprovisingPositioning

CountingMeasuringLocating

ExplainingPlaying

Designing

Engagement

Co-participation

ResponsibleSelf-Regulation

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socio-cognitive, and affective outcomes.

Children respond to the interactive modelling of others by unconsciously adapting and integrating the information, or they resist and reject it (Stone, 1993). Most young children do not consciously evaluate the worth of the meanings and relations being modelled in the communities in which they find themselves, but rely on others to provide them with models which will prepare them for the learning contexts they will be likely to experience in formal schooling.

Relationships between the mathematical meanings, socio-regulative strategies, socio-cognitive processes, and affective outcomes such as co-participation, responsible self-regulation are presented diagrammatically in Figure 1.

METHODOLOGY

Data-collection ProceduresA naturalistic, ethnographic methodology was applied to the study since its main objective was to obtain a view of what was happening in the normal course of events in both informal and formal settings, looking for mathematical meanings which might be present as well the most relevant aspects of the social environment determining the nature of affective outcomes. Two pre-school sites in the Hunter Region of New South Wales, were offered to me through networking with early childhood practitioners and academic colleagues. One centre was a community centre located quite close to a large primary school in a semi-rural community on the outskirts of the region. The other was in a suburb of the region's main city and was one of only a few systemic pre-schools. The episode being used in this paper occurred at the community centre. Participants' identities have been protected and pseudonyms are used.

Non-participant observations were carried out daily for the entire morning's activities during two separate six-week periods at the two pre-schools in the later months of 1994. The observations involved making field notes, tape and video recordings of the verbal and non-verbal interactions of the children and adults as they carried out their usual routine and play activities. Since it was not known at the time of data collection which children would be followed into their school settings, the main strategies undertaken for the observations included obtaining a sample of conversations from each of the activities set out for the day, and to remain with a group of children for as long as the play or conversation was sustained.

Seven children from each of the pre-schools were followed into their first school classes in the two schools located close to the pre-schools. Similar procedures were used during the first two terms of the school year to record class and small-group interactions in the mathematics lessons, except that, due to last-minute requests by the class teachers, these were participant observations with the researcher acting as the teacher in small-group activity sessions. (See Macmillan, 1998b for a report on the instructional episodes of the teacher at the second school site.)


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Data-Analysis Procedures The Mathematical Meanings. Mathematical activity and discourses provide the contexts in which the knowledge base of mathematics and its sociocultural practices can become clear and accessible (Bishop, 1988). “Everyday” mathematical knowledge is accessible to people through artifacts and activity generated by social, linguistic and perceptual interaction. Bishop (1988) identified six “universal” mathematical activities from his study of a number of non-literate cultures around the world. These activities of counting, measuring, locating, designing, playing, and explaining were used to classify and examine the mathematical dimensions of the linguistic and non-linguistic interactions of the children.

Grounded Theory. Strauss and Corbin’s (1990) model of grounded theory was used as the principal justification for the interpretive analytical process, and the non-statistical indexing computer software package, NUD.IST (Richards, 1995), was used to manage more than 100 transcripts of textual data. Categories for analysis of the sociocultural dimension were based on Lave and Wenger’s (1991) model of situated learning (otherwise known as situated cognition, legitimate peripheral participation, or the apprenticeship model). Halliday’s (1978) systemic functional grammar was used to analyse the generic structures (or text types) and to understand what was involved in the acquisition of a register of mathematics.

THE PLAYDOUGH EPISODE

One morning a table was set up outside under a tree with playdough, small trays, containers, and a collection of plastic cutters. Since playdough was usually an indoor activity, it was a novelty for the children to be able to use it outside, and on this particular morning it was so popular that children who managed to locate themselves at the table tended to want to stay there. From the start, children reported to each other about the “cookies,” “gingerbread men,” “bird nests,” and “toffees” they were making. There were interactions demonstrating intentionality (“Don’t you make what I’m making”), integration (“Is it a bird nest?”), transferability (“You don’t know what I’m making.” “You’re not inviting me to your birthday party, are you?”), and reflection (“It’s very small for a bird nest.” “That’s what I told you I was making.”). Unconscious socio-regulative strategies were also evident. Some examples included: observing (“You’re making a bird nest”), and improvising (“[Put] ginger bear in there.”). When one child, Leah, arrived she was accompanied by her mother but found all positions filled at the table, and several other children waiting for a turn. The following conversation between the children was recorded and my italics indicate the use of language which can broadly termed, mathematical:

1. LEAH: Mum, could I do one?2. MUM: I think you have to wait till some one else’s finished.3. TINA: Do you know what I’m making? You gotta guess.4. RUTH: A bird nest.5. TINA: Do you want me to show you how to make a bird nest?6. SASHA: This is going to be my nest [the foil tray].


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7. TINA: Do you know how to make it, do ya [to Sasha who leans on the playdough and rolls and pummels it]?

8. I’ve got six eggs [carefully arranged]. Now look how many eggs I got.9. JAN: Heaps.10. TINA: Yes, I’m making a lot.11. SASHA: Cookies, I’m making.12. JAN: What’s happened to your arm? [One arm of Sasha’s has no forearm or hand.]13. SASHA: I was born like that, wasn’t I [to Tina]? [Jan asks again.] I told her. I was born

like that. 14. Now stop telling me.15. TINA: Look how many I got now [to Jan].16. JAN: Heaps and heaps and heaps.17. TINA: I’ll show you my little eggs [shows Ruth]. I had heaps of playdough.18. SASHA: How do you get these things out?19. JAN: I can get it out.20. SASHA: Can you get it out for me please? 21. JAN [Tries, but can’t do it.] It might need a knife.22. TINA: Do you know what?23. JAN: Do you know how to get them in? You roll them out.

Among the mathematical meanings being used here are numerical quantifiers (“one”, “six,” “how many,” “heaps, “a lot”), a non-numerical qualifier (“little”), and spatial qualifiers (“out,” “in”). Sasha’s explanation about her arm is located in temporal space (13). Central to the use of quantifiers is the social purpose of accessing the resources of the play. “Heaps” and “a lot” of playdough indicates opportunity to use the materials in ways which promote interchange of ideas and imaginative contexts of wide-ranging possibilities. Two complex abstractions were framed firstly as an inquiry, then as an instruction asking for another abstraction were posed and presented by Tina (3). She then posed another inquiry asking for procedural knowledge (5, 7). These abstractions were socially generated, purposeful and provided congealing forces for the play (Lave & Wenger, 1991).

From the perspective of the socio-cognitive processes, Leah engaged in a series of discursive strategies to position herself advantageously in the play. Firstly, she sought assistance from an adult (1). This request for assistance and the mother’s reply modelled a culturally appropriate practice—to wait until someone had finished. Leah’s initial interaction could be interpreted as expressing both intentionality (to be a co-participant) and integration as she sought the opportunity to contribute fully—not peripherally—to the meaning making: intentionality, expressed through language, provided the illocutionary force for Leah’s attempts to gain access legitimately to a position in the play, and underpinned all her negotiations to this end (Lave & Wenger, 1991).

Intentionality was evident later on when Sasha asked for help to remove the playdough from the cutter (18, 20), and after Jan offered her assistance but has some difficulties (19), she deliberated on the appropriate tool to use (20). Use of the responsive control strategy—seeking assistance—is linked again with intentionality in this interchange.


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Tina’s (3, 5, 7) and Jan’s (19, 23) contributions took the form of integrating interactions as they contributed their own procedural knowledge to the context. When Tina inquired, “Do you know what?” (22) she was imitating the spontaneity and curiosity modelled by the adults in the practice. Valuing of open-ended experimentation and improvisations with tools and other resources was evident in Jan’s comment, “It might need a knife” (21).

One example of transferability was evident when Jan’s curiosity was aroused and she sought understanding from Sasha (a child from another unit at the pre-school) about her arm (12). Sasha provides an explanation: one which has probably been given to her. This is an uncomfortable question for Sasha. She overcomes her difficulty by seeking support from Tina, initially, and when the question is repeated, Sasha reiterates her explanation. Her assertion, “now stop telling me,” could be a demonstration of her need for respect regardless of the shape of her arm (14). Her request is received positively and the focus returns to the play.

In relation to other non-verbally realised responsive control strategies, this episode illustrates—albeit without the benefit of a much more illuminating visual medium—how the participating children observed, and demonstrated to each other a range of imaginative, manipulative and mathematical skills. For Leah, being on the periphery of events allowed a vantage point for more vigilant observation, but for little engagement or satisfaction.

As has been noted above, control and management of the technological resources—the playdough, accompanying tools and artifacts—was linked to counting and locating meanings and strategies. This mathematical dimension, spontaneously and unconsciously expressed, continued to be a source of interest and engagement for the participants in the play. A newcomer, Michael, arrived just as another child, Rosie vacated her seat momentarily to pick up tray from the ground. Michael quickly sat in her chair and started playing with her dough. Rosie did not say anything to Michael, but stood beside him, watching him closely when Leah attempts once more to join the play:

24. LEAH: [To Tina] Do you think someone else should have a turn?25. MUM: I think you need to wait until Tina has finished.26. LEAH: I’ll show you how to do it properly. [She rolls some playdough in the palm of

her hand.] 27. I’m waiting, I want a turn, now, so will you give me a turn? [Tina ignores her.]28. TINA: Look how many I got now. They’re the eggs [in the foil tray].29. ALLY: Are you counting the eggs?30. MICHAEL: I need a gingerbread man [as he snatches the cutter from Sasha. She watches and waits, then moves towards Michael to retrieve it. ] 31. I haven’t got it. I threw it over the fence. [He hides it behind his back and then in

his lap.]32. I’ve got it. You’ve had lots of goes. 33. [Sasha returns to her seat and soon afterwards takes a piece of Michael’s

playdough. He notices, but does not respond verbally.]


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34. ALLY: I’ve got two [eggs] here.35. TINA: I’ve got hundreds of eggs. Look how many I got.36. SASHA: We don’t have sleeps.37. TINA: We don’t either. We just have a lie down.38. SASHA: Don’t you?39. CHERYL: [Gazing at Michael as he leaves the table.] He’s mean. He’s very mean and he

took some playdough. 40.TINA: [To Cheryl] Quick. Sit there.41.CHERYL: He’s too mean and he took some playdough. 42. I’m making the same as you [to Tina].

Control of how much and what kind of material is being used (“how many” “counting” “lots” “two” “hundreds,” “the same as”), how much time Leah has to wait for her turn (“turn,” “wait,” “finished”), and the manner of Cheryl’s safe relocation of herself (“quick,” “there”), continue to see counting, measuring, and locating meanings as significant aspects of the experience for the participants. The generalisation, “We don’t have sleeps,” indicates understanding of the negative numerical concept of the “empty set” (36). Cheryl being left without a seat—her experience of “an empty set”—and Michael’s irresponsible self-regulation, excluded her temporarily from engagement and participation in the play.

That Leah has made an attempt to assess the situation objectively—a reflective abstraction—is apparent in her second request to join the play (24). At a broader sociological level it could be interpreted as an appeal to her peers’ sense of moral obligation (“do you think you should”), and relates to the turn-taking pre-school rule. Tension is apparent in this rule, though, when Leah’s mother reiterates that this may mean waiting until a child has finished (25). Leah has realised, perhaps, that neither Tina or her mother are going to respond to appeals for consideration of her need to participate, and has produced another positively framed idea. She has adopted the authoritative role of evaluator, seeking acceptance, respect and understanding of her desire to become a member of the group. Leah then assumes another authoritative role, this time of instructor and expert, demonstrating “how to do it properly” (26). Tina ignores her.

The focus of the interpersonal dynamics turns back to Michael. He broke cultural codes by snatching the cutter from Sasha during a split second interval between her putting it down and picking it up again. Explicit assertion of Michael’s wants accompany this action, as if they provide adequate justification for the breach of the moral code (30). This is followed immediately by two further breaches of the moral code—he lies about having it, and when he realises that he has not hidden it carefully enough, he demonstrates disrespect for Sasha’s capacities to observe his actions by saying that he has thrown it over the fence (31). When Sasha continues to assert the strength of her own position non-verbally by staring at him and standing directly opposite him, he assumes a culturally acceptable code of conduct and uses reasoned explanation to retrieve his moral position. (32). After some moments Sasha resorts to one of Michael’s non-responsible strategies and takes some playdough from him (33).


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A brief interchange between Sasha and Tina demonstrates in a very small way, integration and identification with the pre-school culture as the children from different “rooms” exchange abstract evaluations of a new practice (not having sleeps) (36-38). When Michael leaves, Cheryl’s formerly completely passive responses to Michael’s actions are actively interpreted with reflective comments (39, 41), and at the same time Tina instructs her to return to her position at the table (40). Tina’s gesture is reciprocated by Cheryl’s desire to emulate her creations (42). Having observed a return to the earlier patterns of power-knowledge relations, Leah asks once more for “a go:”

43. LEAH: [To Tina] Could I have a go now? 44. I tell you what, I won’t be your friend. [Tina ignores her.] 45. Just pretend somebody sneaks along and somebody crushed your

[playdough] eggs.46. TINA: No, because we’re not playing a game. We’re not playing a game.47. LEAH: You’ve been here for a long enough time.48. TINA: How about you help me? [Tina shared the playdough with Leah.] 49. Don’t give some to Leanne [a child who had been playing with them earlier].

She’s not doing it. 50. Get the biggest whole bit to make it.

Leah’s frustration is evident with the appearance of restrictive control strategy, a threat to disrupt the bonds of friendship (44)—although friendship is something desired and imagined rather than real in this situation (Corsaro, 1985). This is followed by another proposition from Leah for Tina to imagine another threatening event, this time located in the future (45). Leah receives an explicit rejection—this improvisation for successful access moves into new territory: Leah has begun to unfold a new dramatic possibility: she has drawn on her own repertoire of strategies for “game playing.” Tina seemed to be implying that whereas in imaginative play, exchanges of roles are permissible and rules are generated by all participants and collaboratively formulated, in this situation she was clinging steadfastly to her role of authority figure, offering or denying access according to her own evaluations, and her own rules.

Tina then loosened the reigns of control and offered Leah a role of “assistant,” highlighting the potentially paradoxical nature of power-knowledge relations in an authority-figure role: having the power to control others’ access to the resources and to the imaginative and social dimensions of the context, but being able to abnegate collective responsibility regarding accountability for breaches of cultural codes. This episode illustrates the complexity of role representation afforded by the peer culture—an authority figure role was adopted by Tina: she cleverly and persistently resisted the legitimate and culturally-valued formats of Leah’s negotiations for access to the play. In a classically whimsical strategy displayed by over-controlling authority figures—usually when they realise that their strategies have become transparent to others—Tina released her negative control and allowed Leah entry. One example of mathematical meaning is apparent here in Tina’s last statement, when the ubiquitous theory that “more is better” is evident (50).


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Use of a range of unconscious responsive and restrictive control strategies produced a range of dynamic and volatile meanings and relations: the children’s talk during this episode supported the notion of the situatedness of learning—that the social context allows children to adopt a range of roles to fulfil their immediate purposes, and in the process they ask and respond to their own and each others’ questions. Regardless of whether the discourses are monologic or dialogic discourses, these children are, in effect, the illocutionary forces of their own meaning making, the totality of which is located in the social context (Lave & Wenger, 1991). Moreover, the hypotheses tendered by Bishop (1988) that characteristics of play form a basis for hypothetical thinking, and that decontextualised thinking is involved in imagining other realities, are congruent with the range of thinking processes apparent in this episode.

CONCLUSION

Since social learning is a predominant goal of informal learning environments, it is understandable that the children are encouraged to be as self-regulative as possible regarding the control and management of resources, as well as the generation of meanings which are interesting and relevant to them, for this can provide them with a range of potentially challenging situations. However, I wonder whether the policy of non-intervention is sometimes taken too far in both the practical and theoretical sense, and whether some children find it extremely difficult to manage peer situations to their own advantage. That is, they may need a lot more help to engage in the kind of reflexivity which would assist them to cultivate a repertoire of group entry strategies. This repertoire could be drawn on in preference to waiting patiently on the sidelines, or breaking the social codes, as in Michael’s case. This is not to say, however, that the teachers in this setting did not model and assist with entry-to-play and other necessary strategies, especially when other children explicitly asked for assistance.

For at least two of the children in this playdough episode, access to the resources and discourses of play was problematic, due partly to what seemed to be inequitable social relations. In spite of the use of sophisticated negotiating strategies, when waiting for a turn allows others to remain as long as they wish, interpersonal and intrapersonal positioning strategies could be perceived as having failed, and self-image is likely to be affected. The concern is that while children do not have access to the materials and resources of the practice, they are not included in the playing discourses, and do not have the opportunity to explore their own meaning making potential. Unfortunately, it is not possible here to explore this issue of non-intervention in depth, but I can offer some practical recommendations for early childhood practitioners in formal and informal settings regarding identifying, supporting and valuing potential for mathematical thinking in non-instructional contexts:

Be aware of, and receptive to, the potential for the full range of mathematical abstractions (realised through language and the manipulation of physical objects) particularly within tasks with creative or designing potential, such as all kinds of construction play. Inclusion of accessories which are likely to encourage new ways of perceiving realities, and challenging known perceptions can facilitate mathematical


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learning (Griffiths & Clyne, 1998, 1990, 1994; Macmillan, 1998c; Neuman & Roskos, 1990, 1993, 1997).

Since conceptualisation can be viewed as developing from the social to the cognitive, it is essential that all children are as actively involved in the meaning making as possible by having access to the resources. In the playdough episode, for example, another table and more materials could have been provided to respond to the demand. In other words, more flexible frameworks for providing all children with resources would address the issue of access and equity which underlies any cultural code allowing full participation to be restricted in any consistent way. This is not to say that opportunities for the children to negotiate and ask for more or different materials should be taken away from them, but rather a matter of awareness of the negative positioning of children by other children, and sensitive responses which protect and encourage both parties' self-regulative potential (Masselos, 1994).

Be conscious of the fact that the metacognitive processes necessary for abstract or symbolic representation are present in social meanings and relations, and continue the valuing of dramatic play as a facilitator of literate and numerate knowledge. In order to take the developing ideas to a metacognitive level, document aspects of the play which are significant to the children (through captions, photographs, charts, big class story books, tables, diagrams, graphs, etc). Allow them to contribute as much as possible to these documentations, and encourage valuing of the recordings within the home environment as well as in the centre shared home-centre use of the recordings (New, 1994; Nimmo, 1994; Raban & Ure, 1997).

Respond to children’s expressions of mathematical knowledge by posing very simple problems within the same context, such as, “I wonder what would happen if ...”, “If you changed this ... what would it be?” Model and encourage mathematical processes of hypothesising, verifying, questioning. The concern is not so much related to numerical operations per se, or giving the “right answer,” but on mental engagement and exploration of mathematical ideas and processes.

Children need assistance to think reflectively about what they are thinking and doing, and it needs to happen without their losing a sense of ownership of the context and the outcome. In the story of the child looking at the birthday cards, for example, had this conversation occurred in an educational setting, and had there been a sixth birthday card between the five and seven, it may have been appropriate to ask if he would like to try counting on from the four (and touching each card as he counted) until he reached seven. The adult could offer to assist by putting up a finger for each card/year he counted.

Children need the opportunity for imaginative games and play right through their early years. There is high potential for improvisation, and engagement in a myriad of abstract representation processes and understanding of codes and rule systems. Children model and improvise their own games on those they play with others (Bishop, 1988; Lave & Wenger, 1991).


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Remember that for children—and indeed any person seeking membership in a community—reading, understanding and applying the rules of the culture is linked with the need for acceptance and belonging, whether it is the peer culture of informal contexts, the culture of the school classroom, or other family cultures (Lave & Wenger, 1991). In other words, literate and numerate meanings—highly valued in our society—grow only when and if children’s sense of identity is congruent with that culture. In the case of Michael, for example, his valuing of the pre-school culture did not extend to accepting the rule of waiting for a turn or to ask in a socially acceptable way for the cutter. Ultimately, his lack of interest in the rules of the dominant culture will deny him access to its most valued meanings and relations.

Concluding CommentsOnly one transcript of many possible choices was used here to attempt to highlight the most significant outcome of the first phase of the study—that mathematical meanings were used extensively and pervasively, and were being used by the children to enrich their understanding of physical and perceptual experience. Counting, measuring and locating meanings evolved in conjunction with the mathematical activities of explaining, playing and designing. These mathematical meanings—abstractions of reality—were most prolific during construction play: because the design process requires many spatial, quantifying and qualifying meanings; and during dramatic play: because of the challenges involved in interpreting, expressing and applying roles, relationships and social rules within an interesting cohesive framework for exchange of ideas and purposes.

Finally, a comment on the problem of subjectivity in interpretations of discourse. Though these interpretations were validated through sound research practices and theoretical frameworks in a range of fields (Macmillan, 1997), the children's talk was their talk interpreted by another. Moreover, this other person has biases, conceptual and ideological "blind spots" of her own which undoubtedly coloured her evaluation of the events which unfolded during play. This being the case, others will have other ways of perceiving and interpreting the situations presented, and should, indeed, take a critical view of the playdough episode.

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AUTHORAgnes Macmillan, lecturer, Charles Sturt University, Albury, NSW, 2640. Specialisations: Semiotic and sociocultural perspectives on mathematics education; socio-political issues in early education and care. Email: [email protected]

ACKNOWLEDGMENT The author gratefully acknowledges the funding of this project from an Australian Postgraduate Award Scholarship. She is also appreciative of the early childhood professionals' willingness to allow her to observe their worlds so closely.


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