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Reading Description: Macmillan, A. (2009). Shared contexts for teaching and learning numeracy. In Numeracy in
early childhood : shared contexts for teaching and learning (pp. 20-33, 58-59). South Melbourne, Vic. : Oxford University Press.
Reading Description Disclaimer: (This reference information is provided as a guide only, and may not conform to the required referencing standards for your subject)
Shared Contexts for Teaching and Learning Numeracy
This chapter explains: I;! l<=::lfn\�rs' ne�d::i to rr.ake S•"?nse of n1ath,2111ai:io:a\ concepts b'j ha-·ling acc-ess to.
n1ath,2maticaliy stimulating acti11ities
� the mathernatlcat meanings oci:::urring in childr211's interactions
ra ho'N invesUgJ!:ing through p\2;1 (..3n facHltat•z intrinsic and interperson,3\ n1otivations for l'·�arning
•
a th>� m(=3ning of shared contexts for the tea-:hing and learning .of num12racy .
•
MATHIEMATICAIL MIEAllHl\ilGS' A�llD ACTIViTilES Soclocultural perspectives are based on the premise that expei;iences involving mathematical concepts and processes occur, and are realised, within social and cultural contexts (Bishop, 2002; FitzSimons, 2002). The mathematical
. meanings generated are socially and culturally driven-they arise from social or cultural purposes, although the mathematical dimensions .of those experi- _ . ences may not always be consciously realised.
Imagine a mother with a young child sitting on her knee reciting and actioning the nursery rhyme 'This Little Piggy Went to Market'. The child-and maybe the adult�may not realise that the mathematical principle of o n e-to-one correspondence is being enacted. Nevertheless, the child is being introdu1:ed to the concept in a social and cultural context-the context's social dimension has language being expressed, and there is perceptual and physii:al communication between peopl�; the cultural dimension is the traditional nursery rhyme.
A numeracy perspective for mathematics education proposes that everyday experiences have the potential to familiarise children with the purposes and relevance of mathematical concepts and skills and, ultimately, to facilitate
' the acquisition of numerate identities .. Tuis means that a,person is confident
' '
SHARED CONTEXTS FOR TEACHING AND LEARNING NUMERACY
I 21
about being engaged in numeracy-related experiences and productions, and can identify in some way with the mathematical content of the experience or activity. For young children, this means that when the adults around them are aware of the mathematical understandings present in these eve1yday expeiiences, learning to be numerate becomes an enriching and fascinating enterprise. ,
Adoption of a sociocultural perspective for numeracy education was inspired significantly by Bishop's (1988) review of the mathematical prac-tices of people in diverse communities. He found that people ��[!ii'���?'!� around the world used mathematical principles, concepts, skills and strategies to fulfil the purposes and needs of their everyday lives. He identifi�d six mathematical activities that he considered to be 'universal' or common to all cultures and communities. He referred to tl1em as six universal mathematical activities.
The Six l.Jniversal Mathematical Activities
A community's needs combine with its social purpo�es and cultural con-' '
ditions to determine the kinds of mathematical procedures and operations to be applied. The contexts may be as different as managing the stock exchange and navigating a fishing trawler between the islands of the South Pacific, for example.
The social purposes likely to influence what and how things are done mathematically can be:
• political, si1ch as interpreting election survey statistics • economic, such as working out a family budget
•
• physical, such as checking the weather forecast before going on a bush walk
• scientific, such as monitoring one's diet and its effects on body weight • social, such as where you might sit at a wedding or dinner party • emotional, such as choosing a dancing partner.
In fact, it is possible to analyse the mathematical content of any social context using Bishop's six mathematical categories. The activities Bishop considered to be mathematical and universal are counting, measuring, locating, designing, P,laying and explaining. Bishop defined tl1ese activities as follows:
' Counting is a way of distinguishing, ordering or quantifying objects or events abstractly or concretely.
• Measuring quantifies or distinguishes quantities or entities which cannot be counted or located.
' Locating involves posittdning oneself in spatial relations (boundedness, continuity, direction, time) with respect to others and to other o hjects.
"" NUMERACY IN EARLY CHILDHOOD
•
3 Designing imposes an abstract o� symbolic plan, structure or shape on a surface or space.
' Playing imitates or imaginatively recreates social, concrete or abstract models of reality; it is exploratory and non-goal directed, unless mles or goals are negotiated in the process of play. ,
' Explaining communicates verbally the factual, logical or evaluative aspects of ideas, questions, experiences, events or relationships between phenomena.
Examples of Bishop's mathematical concepts evident if' children's verbal interactions when counting, measuring and locating using children's in!eractions are presented in Appendix I.
Figure 2.1 and Table 2.1 show how it is possible to identify these activities and interpret mathematical meanings in play. Three children were experimenting with measurement concepts and investigating 'how many cylinders tall' they were in the construction play area of a preschool.
• figJJre 2.:1 Measuring . Tabb 2.1 . Applying Bishop's activities to a construction play observation. emerged from this construction play.
Together, the children stacked the cylinders.
They carefully balanced them one on top of the other. I
They counted them using one-to-one correspondence.
One child asked an adult to record how many cylinders tall each child was.
After each child had a turn at being measured, they compared their height by representing the recorded Agures V.1ith- stTCks.
The person with the most sticks was considered the tallest'
Playing
Locating
Counting
Measuring
Measur;ng Explaining
Counting Measuring
We now investigate what it is about play that allows these meanings to activate mathematical thinking processes, as well as mathematical ideas and concepts.
When children are playing, they are exploring and investigating objects and abstract meanings, and engage in many different kinds of thinldng processes. Exploring precedes investigation in the degree of cognitive challenge. The
SHARED' CONTEXTS FOR TEACHING AND LEARNING NUMERACY
term investigation involves fmding out about phenomena by using pa1iicular scientific processes to discover the relationships betvveen them. In formal science experiments, investigation involves the application of mathematical concepts, procedures and language. In rhany respects the phases and processes of formal scientific investigations correspond with the informal investigations of young children. Educators of young children understand that it is through investigation that children sustain cmiosity and control of their own learning (Dockett a Fleer, 1999).
0
Investigative thinking processes include: observing, exploring, investigating,
predicting, explaining, analysing, problem-solving, hypothesising, testing and
checking, stating a belief, theorising, documenting, reflecting and evaluating.
Each of these.processes is involved in.prob Lem�solving and acquiring access to
the abstract representation system of mathematics.
When adults and interested others are curious investigators with children, they are cultivating children's natural cmiosity and interests. Such facilitating contexts require adults to be aware of opportunities for:
'
• encouraging problem-solving • observing changing and unchanging phenomena in specific detail, using
all the senses • interacting using explicit language and providing opportunities for docu
menting • reporting and talking about discoveries and solutions to problems.
We refer to these opp01iunities as shared contexts for teaching and leaming. They are situations in which there are opportunities to:
• control what happens and how it happens • access the meanings • negotfate who participates and how they paiiicipate • initiate purposes that are mutually interesting, productive
and satisfying.
The following section discusses the notion of shared contexts and its relationship with motivation to learn. Implicit throughout the discussion is the assumption that play has the potential to provide motivating and investigative contexts for learning.
NUMERACY IN EARLY CHILD HOOD
Motivation is an active state of wanting or needing to be engaged in an experience.
• Intrinsic motivations are generated and sustained from within the individual.
' lnlerpersonal mothtalions are generated and sustained from the meanings and relations generated between the people who are interacting with each other in a community.
Interpersonal motivations are generated at the level of culture. Those in authority in a culture or community protect and cultivate its rules, values, codes, practices and discourses. Others within that community who acquire the power to generate their own values, codes and so on are able to exercise degrees of control over others-and regulate who participates, when, where, why and for how long. In most cultures there are at least one d.ominant group and a number of other less dominant or minority groups. That is, it is similar to what happens in a family-between siblings, meanings and relations develop that may or may not correspond with the values and practices being generated by the parents.
The social context has the power to motivate or demotiva\e. The most fruitFu\ and productive communities are those,in which purposes are mutuality constructed and each member has the capacity to be selfregulative-to exercise control and choice in lilting out everyday a�i __ _
altruistic needs ancl desires, goals
Interpersonal motivations assist and facilitate the learner's active engagement and full participation in the experiences being offered. When motivation is disturbed,
I '
disrupted or challenged by negative meanings and rela- ' tions, learning potential is _inhibited and restricted to some extent. Motivations are threatened and destabilised if negative influences accumulate.
Malone and Lepper's (1987) characteristics of intrin.. sic.and interpersonal relations ar�Pilrticularly useful for
the present discussions because mathematics education and expectations. is an area in which learners' motivations can be dis-
rupted or destabilised from the time they begin their formal education. This does not need to be the case. There is much about play that is mathematical and motivating, and there is much that can be sustained from those play contexts through to the fust years of •school. At the same time, there are aspects of formal education that are helpful for understanding the pedagogies that are currently in practice for the teaching and learning of early numeracy.
'The intrinsically motivating nature of play provides the tontext for children to initiate and regulate the content and processes of their experiences, as well as to generate their own interpersonal meanings and relations (Dockett 8: Fleer, 1999). In fact, it was while observing children solving problems as
SHARE� CONTEXTS FOR TE�CHING AND LEARNING NUMERACY
I "S
they played educational computer games that Malone and Lepper (1987) identified the characte1istics of intrinsic and interpersonal motiva�ions.
� The intrinsic motivations are challenge, curiosity, control, and imagination (or fantasy).
c The interpersonal motivations are cooperation, recognition and respect.
For definitions and examples of intrinsic and interpersonal motivations, see Appendix 2.
1Cm1strndi1tlll1 pilaiy
-The following-observations of a cons1:mction play episode and play-with natural materials illustrate the connections between intrinsic and interpersonal motivations. The imaginative dimensions of the play provided opportunities for sustained engagement and creative expression. Mathematical interactions are italicised. Bishop's mathematical activities provide the framework for analysing and interpreting the math·ematical meanings, and are listed along-sid� the children'� intt'ractions. '
Libby (L) was making a construction with wooden blocks and other play objects of people figurines, cars and shoeboxes. Evan (E) was one of the other four children nearby who joined in the play.
1 [Lis sitting alone approxhnately 2 n1etres away from the other children, placing four long blocks end to end to construct a square enclosure. She places long blocks on top of the enclosure, forming a roof. She chooses one long block, placing one end on the roof and the other end on the floor and attempts to push a wooden car up this ramp. The car is too wide for the block to allow the car to travel up it. l chooses a second long block and places it beside the first. E stands and walks over to Land sits on the floor beside her.]
locating: position, mapping-locates self away from others to attend to own play theme.
Counting, Measuring: equivalence, focusing on length.
Measuring: choosing blocks of equal length.
Locating: forms a 3D square enclosure.
locating: 2D space, surface.
Designing: develops new aspect to the enclosure.
Locating: angle of the block.
Measuring: width, using the car as technology.
Problem-so!Ving: reflecting.
Locating: spatial awareness used to compare/ judge width of block and car; solves problem.
Locating: position-E relocates hi1nself to nevv 1 area of in'terest.
' Continued ...
NUMERACY IN EARLY ,CHILDHOOD
2 E: Can I help too?
3 L: You can make the road.
4 l: My cars can drive on the· road to the shop.
5 [L places people figurines at each corner of the roof. E places blocks of various lengths to form a long line o; blacks (a road).]
6 E: There you go, now I need a car.
7 L: I didn't want it lil<e that. You have to use the same blocks.
8 [L pushes all of the blocks E used away.)
9 L: I'll show you how. Get me all the blocks like this one [holding a medium-sized block].
10 [E passes L two medium blocks.]
11 l: We have to use these becau;se they are the same. We don't want a wobbly road.
12 E: No, we don't want a wobbly one to drive on, do we?_The cars_will fall_pff.
13 [L places six blocks in pairs to fo_rm a short line.]
14 L: Now, look here. You do that while I do that.
15 [E continues the path by reaching for medium blocfrs and placing them in pairs to form a long line.]
16 E: See, I did it. I put two blocks, then two blocks, then two blocks.
17 E: I did it like you.
Social positioning: E positions himself near L
Instructor role: fulfilled by L.
locating: social purpose of the car and its directional position in space.
Counting: figural collections, order, pattern: arrangement of figurines.
Temporal locating: indicates next period in time.
Instructor role: classifies and Compares materials needed by comparison.
Locating: distance, creates clearly defined space to begin new road.
Counting: classifies the block type and the quantity needed.
Counting: sorting, chooses two identical b!o
.fks. Able to visually compare/class block
displayetl by L. '
Explaining: logically explains and demonstrates the need for order and balance to maintain the design concept.
Collaborating in the mediator role: responding _to L's_l_9gjca_l_exp_tan_atio11 _
'
Counting: arrangement of blocks in patterns of pairs, as a figural collection.
Temporal locating: time concepts used as controlling and collaborating mechanisms.
, Locating: continuation of design begun by L; repetition of predetermined sequence.
Counting: classifying and sorting to form figUlal collections.
Reflecting: self-evaluating and seeking confirmation of his contribution to the design.
' Che � SHARED CONTEXTS FOR TEACHING AND LEARNING NUMERACY
The intrinsic motivation of curiosity provided the impetus for the play. Numerate purposes emerged when actions were carried out on the threedimensional objects. The mathematical activity of designing incorporated and stimulated a fantasy element with the creation �f a 'road'.
Challenges included:
• managing the spatial organisation of the blocks-discriminating, comparing, ordering and patterning according to size and shape
c quantifying nume1ically (16), and non-numerically, in order to create halance (7, 10)
• using logical (11), procedural (3, 14) and clarifying explanations (4) to support the actions being carried out with the blocks.
' - --- --The verbal and non-verbal interactions-were used to- control,-regulate and
manage the creative construction. The power to control the activity was entirely in the hands of the children-no adult support was sought or required to sustain the purposes of the play.
' Interpersonally, motivation was sustained by the desire to integrate actions, create harmony and to cooperate. Evan wanted to enter the play and chose the word. 'help' to generate his willingness to follow the idea being constructed by the initiator, Libby (2). Evan sought recognition for fulfrlling Libby's desires for coherence (17). Respect for the numerate and social purposes was expressed implicitly by carrying out the design idea, and by the construction processes involved representing that idea using the technical resources of the bloclcs and cars.
P!aiy with n-naihmlll maiteriails
At the collage table, Kirn (K) was creating something from a va1iety of natural materials. The teacher (T) became involved when she noticed that Kim might have been having some difficulty with the mate1ials.
1 [I< selects foUr sticks from the collage trolley of the same length (5 cm) and arranges them on the table in front of
him in the shape of a square. He places wood glue on the corner of each stick attempting to hold them together. Then he tries to pick up the shape. ltfalls
·apart. He takes a deep breath.]
2 Tt What are You trving to do, Kim?
Counting, Measuring: sorting objects by comparing and matching according to length.
Designing: constructs 20 shapes with natural hiaterials, glue and scissors to create the design.
Challenge: responds strategically and persistently to so!ve the problem.
Continued ...
' I �-
NUMER,\ICY IN EARLY >CHILDHOOD
3 I<: I want to stick my square together. Then I wa�t to make a triangle and stick it on the top.
4 T: Can you tell me about what you are trying to mal(e?
5 I<: I am ma!dng a house.
6 T: Perhaps you need to find something that will help to reinforce and support the-square?
7 I<: What do you mean?
8 T: Is there something on the collage trolley
.that you could use to hold the
squ�re in place?
9 I(: [Selects a green foam tray.] Maybe I could stick it to this.
10 T: That would help to support the square and hold it in place.
11 I<: The green can be the grass. Then I can ' '
cut out some little squares for windows and use that brown cardboard to make a chimney.
12 -u< continues creating-the house.--He cuts out two small squares which appear to be the same size and places these inside the larger square. Using a black foam tray, he then cuts a tria-ngle and glues this on top of the square. A rectangle is cut using cardboard, ' this is placed on the right side of the triangle.]
13 K: There. No\1\1 I have some windows, a roof and a chimney on my house.
'
14 [K smiles and holds his creation upright showing the T.]
Explaining: articulating the procedure he desires.
Designing: identifies the idea.
Explaining: clarifying statement.
£"(plaining: seeks a clarifying explanation.
Explaining: a procedural choice.
£xplaining: the teacher models a procedural explanation.
Problem·solving: suggests a ppssible solution.
'
Locating: describies the plan, an_d represents abstractly the procedures he will use to position the objects for the design.
Technolog,11-:-uses the_ techni_ques -prov_i_de_d to locate the shapes for the1house.
locating: shape, symmetry, regions, boundaries, enclosures.
Designing: the house's key elements are logically positioned.
Measuring: equivalence, geometry.
locating: positioning the parts of the house.
Exp�aining: clarifies and identifies the features of the house.
Interpersonal rnotivation: seeks recognition for the creative construction.
SHARED CONTEXTS FOR TEACHING Al;D LEARNING NUMERACY
In this shared context for learning and teaching, the teacher's interpretation of what she observed led her to enter the play with a direct question (2). !Gm responded using precise spatial descriptors of his immediate intention (3). The teacher then prompted him to explain his ultimate design goal (4). Her suggestion to Kim to solve the problem prompted him to ask for an explanation (6). Kim's desire to maintain ownership of the process was recognised and respected by the teacher as she offered suggestions rather than instrnctions (8), confmnations rather than judgments (10), and left him to control the process as he chose. She created a collaborative and mediating interactive framework, and the child continued to offer precise descriptors of spatial dimensions of his design. Recognition of the design's value is shared with the teacher (15), confmning that thesh'ared control_ of the activity was satisfying and-productive. '
Figure 2.2 The mathematical activif1es and motivations generate shared contexts for nymeracy teachi.ng a�d learning.
Counting Curiosity Recognition Shared contexts '
I Measuring Choice Cooperation for numeracy
Locating Control Respect teachirig and [earning- '
Designing Imagination
Playing
Explaining
Maikhug semse of the maiUuemaiticail cou-ucepts
Mathematical meanings are produced, and children malce sense of those meanings, by participating in activities. The mathematical activities of counting, measuring, locating, designing, playing and explaining (Bishop, 1988) provide a useful, interesting and appropriate perspective for determining what is mathematical in children's activities in play and in investigating their everyday and, imaginative worlds.
MaH1ematicai! mea11il1gs uu-u ch i!irfre11's a11teraict1011s
Models of observational analyses in this chapter illustrate how children use mathematical language to explore concepts and engage in investigative thinking as they play.
NUMERACY IN EARLY �HILDHOOD
!'adliit.EU11g h1�ri11sai: a11d arnteqci,ernq:m;al 11111J�a11;ati<o!1ls
A major component of teachers' professional knowledge and practice is to understand and pr0Iect the psychological factors that influence learning. Curiosity, challenge, control and imagination are motivations for individual learners that need to be cultivated in environments that facilitate numeracy .
•
Two of these motivations are reflected in the AAMT's third standard in the domain of Professional Practice:
Understanding the motivations that are generated interpersonally and affect children's confidence in exploring mathematical ideas is addressed in the AAMT's Professional Knowledge domain:
The interpersonal motivations identified by Malone and Lepper (1987) are '
recognition, respect and cooperation. Children are orientated towards ful-filling their natural potential for learning and innate desire to be accepted.
B · A sense of acceptance and belonging are cultivated through these 2.3 eing sensitive to the-delicate _ _ interpersonal motivation_s. _ S_om_etimes it is challenging for educators nature of young children's needs for respect, recognition and cooperation is a duty of care for those relating to
young children.
to protect the abstract system of mathematics and at tl:ie sa!Ile tfrne respect children's incomplete and developing understandings and unconventional thinldng processes [Chapters 1 and 7). These factors can be highly significant in maintaining children's positive attitudes. It is evident from the play transcripts in this chapter that young children are very comfortable with exploring mathematical concepts and ideas when their offerings are valued and respected, and when the conte,'Cts are stimulating and meaningful for them.
SHARED CONTEXTS FOR TEACHING Al1D LEARNING NUMERACY
Silared «:11nntex!:s for !:ea1chi11g amd teamrt�g rwmernty
'Shared contexts' in educating communities offer a social justice, or sociocultural, perspective on teaching and learning. The main strategy for cultivating equitable and accessible environments, or contexts, is collaboration. A particular focus on collaboration and other responsive teaching
'
strategies is provided in Chapters 8 and 9. In this chapter, the nature of the activities-that is, play-has provided a context in which it is possible for learner and teacher to collaborate. The children exercised their innate desire to be in control of tl1e experience and to regulate tl1e process and outcome, while the teacher's collaboration supported problem-solving and stimulated
-new thinldng. Shared contexts represent a shift in early-childhood education towards conceiving the child as· 'learner' and tl1e adult practitioner as a 'teacher'. It is also reflected in tl1e AAMT's Professional Practice standards for excellence in teaching mathematics:
Figure- 2.4 An-example of a post office focus for dramatic play to stimulate the mathematical activities of counting, measuring, locating, playing and explainingsuitable for a shared context for teaching and learning early childhood numeracy.
NUMERACY l�I EARLY CHILDHOOD
FURTHER READIMG ---------------· ·------ - -
For details on Malone and Lepper's (1987) intrinsic and interpersonal motivations: http://education.calurnet.purdue.edu/vocke!l/EdPsyBook/Edpsy5/Edpsy5_intrinsic.htrn
For other examples of using Bishop's six universal mathematical activities to interpret and identify mathematical meanings in play: Macmillan, A. (1999). Pre-school children as mathematical meaning makers,
Australian Journal for Research in Early Childhood, 2 (6), pp. 177-191. Macmillan, A. (1998). Pre-school children's informal mathematical discourses, Early
Child Development and Care, 140, pp. 53-71. Macmillan, A. (1995). Children tlllnldng mathematically beyond authoritative
identities. Mathematics Education Research Journal, 7 (2), pp. 11-127.
For an early childhood perspective on generating an active curriculum: Geist, E. ft Baum, A. C. (2005). 'Yeah, buts' that keep teachers from embracing an
active curriculum, Journal of the National Association for the Education of Young Children. www.journal.naeyc.org/btj/200507 /OJ Geist.asp.
STUDY ACTIVITIES
1 Record a 3-5 minute observation of children playing and apply Bishop's six categories of mathematical activity to determine the mathematical content of the play. Repeat for two or three further observations. With an interested friend, di>cuss what you have discovered.
' ' .
2 Design and set up a construction or dramatic play activity for a small group of young children. Using field notes, a tape recorder or other el,ectronic device, record their verbal and non-verbal observations. Take on a non-participant observer role as Far as possible. Transcri_be the recordings for 10-15 minutes of their interactions, or as long as the play is sustained. Analyse the interactions
__ using Bishop's six mathematical activities. Discuss yo_ur fi _nclin_gs with a member of your peer group.
3 Think of a social context that is very familiar to you, such as having a meal for Family or friends, and then a social context that is new to you and presents some kind of challenge. Compare your feelings of confidence and competence in the two situations. Determine what intrinsic and interpersonal motivations might occur in each situation in order to understand why you may want to persist or give up on the new situation.
4 In a paragraph or so, explain what you think'sociocultural perspectives' are about. In what way could they be interesting or useful to you as a framework for analysing social contexts, with particular reference to teaching and learning contexts?
(l,c.C: �.(r:;:r: I
SHARED CONTEXTS FOR TEACHING AND LEARNING 1;UMERACY '
5 Think about your own early experiences as a learner (in and out of an educational setting), and the degree to which you had access to the kinds of mathematical meanings being described in this chapter. Can you perceiye a relatio�sh'1p between the kinds of things you were interested in and participated in as a child, and how confident and competent you perceive yourself to be mathematically? That is, how would you describe your own numerate ide�tity (how you feel about yourself as a numerate person)?
6 What questions have arisen for you in reading this chapter? For example, how realistic is it to expect that every child in an educating community will be able to participate fully and have equal access to its activities? Use the questions as a focus for discussion with a colleague or other interested person.
7 Present arguments for and against accepting the validity and credibility of the '
proposal that 'providing shared contexts for teaching and learning numeracy is a fascinating and enriching enterprise'.
REFERENCES
AAMT: see Australian Association of Mathematics Teachers ' ' '
Australian Association of Mathematics Teachers (2006). Standards for excellence in teaching 1nathe111atics in Australian schools. Retrieved 8/10/08 from www.aamt.edu.au.
AAMT-ECA: see Australian Association of Mathematics Teachers a Early Childhood Australia
Australian Association of Mathematics Teachers a Early Childhood Australia
(2006). Position paper on early childhood mathematics. Retrieved 8/10/08 from www.aamt.edu.au/Documentation/Statements/Position-Paper-oncEarlyChildhood-Mathematics-p1int-friendly.
Bishop, A J. (2002). Critical challenges in researching cultural issues in mathematics education. Journal oflntercultural Studies, 23 (2), pp. 119-131.
Bishop, A J. (1988). Mathematical enculturation. Dordrecht: Kluwer.
Dockett, S. a Fleer, M. (1999). Pedagogy and play: Bending the rules. Sydney: Harcourt Brace.
FitzSimons, G. (2002). Introduction: Cultural aspects of mathematics education, Journal of Intercultural Studies, 23 (2), pp. 110-118.
Malone, T. W. a Lepper, M. (1987). Making learning fun. In R. E. Snow a M. J. Farr (Eds), Aptitude, learning and instruction Vol. 3: Conative and affectivt' process analysis (pp. 223-253). London: Lawrence Erlbaurn.
58 I NUMERACY IN EARLY CHILDHOOD
1 K: Did you watch the football on the TV yesterday?
K: St Kilda was playing Hawthorn. St Kilda won.
St Kilda was 88 and Hawthorn was BJ.
J:Oh.
1 5 K: When we go outside later, let's play football.
6 K: We can use the two big trees In the yard for the goals. [Pause).
7 K: I will kick the ball to you, then you can pass it to me and I will kick a goal. If I get a goal, I get to have another goal.
8 J: If you don't get the goal, then it is my turn.
9 K: I will get a goal!
Locating in temporal space: refers to an event in past time.
· Counting: nominal numbers quantify the score; ordinal number as scores are listed in first and second place.
• Symbolic representation: applies knowledge from another context (home) to reconstruct the game at preschool.
Procedural explanation: K explains the procedures to be carried out, assigns roles and rules.
Measuring: refers to events in future time.
Procedural explanations: of the rules, actions and consequences.
Logical explanation: J applies the rules as explained by 1<.
Narrative: describes a future event.
A suggestion from Jack that Keiran may not be successful in ldcking
a goal produces an assertive response from Keiran (9). Keiran adopted an
instructor role as he provided detailed and specific explanations, procedures
and consequences regarding how the game of football was to be played.
Arguments/discussions
Lunch-time routines
In a preschool, one child each day was selected to be the lunch-time helper.
On one occasion, just as the children were called inside for group time, Donna
(D) remembered that it was her turn. She has a disagreement with Jim (J).
1 D: [Walking to the trolley] I have to start getting everything ready now. It's group time and that means that's when I start.
2 [The teacher places name cards at the table, indicating seating positions.
locating in temporal space: time and sequence; matching events to the time of day that they occur. The transition process acts as a reminder of the lunch
, preparation routine.
Modelling: numeracy is modelled and valued in the social context.
Chapter 4 THE ROLE OF LANGUAGE IN LEARNING I
59
D voluntarily goes to the lunch trolley, picks up a stack of cups, placing a cup on the table in front of each child. She continues until each child has a cup.]
4 J: I want a red one.
Counting: ordering: performs appropriate tasks in correct sequence; one-to-one correspondence.
Locating: knows the location of required utensils.
Counting: one�to-one correspondence: placing one item at the table for each child.
5 D: [Handing J a blue cup] There, that's a red one. Counting: identifies colour as an attribute but incorrectly names the colour.
6 J: [Handing it back] No, it's not. Arguing: disagreement.
7 [D gives J a green cup.]
8 J: That's not red either.
9 J: That one on the bottom is.
10 [D takes the red cup from the bottom of the stack and gives it to J. D puts the remaining cups under the trolley. She then gives each child a fork and a spoon, placing all of the forks on the left side of the child and the spoons on the right.
11 D returns to the trolley and places a water jug and a scrap bowl in the middle of each table.]
12 D: I've finished my jobs now [returning to her seat].
Arguing: disagreement.
Locating: describes the position of an object in space.
Locating: places objects in appropriate positions.
Counting: grouping pairs of cutlery together, handing each child the correct combination of utensils.
' Counting: patterning, repeating a sequence.
Locating: positions concepts as D places the jug and bowl in the centre of the table.
Locating in temporal space: the concept of time having a beginning and an end is indicated as D identifies the task as being finished.
There was a disagreement between Donna and Jim (J) as Donna had
difficulty identifying the colour of cup Jim asked for (5-8). The episode dem
onstrates how social interaction within the most routine activity produces
opportunities for learners to make sense of experiences by testing and check
ing and receiving immediate feedback. The initial misidentification of the
colour was overcome, as Donna was given the time and opportunity to test
and check what was correct. Meanings embedded in the social and cultural
context produced purposes that required counting (3, 10, 11) and locating (3,
9, 10, 11) concepts to be applied consciously and unconsciously.
Evaluations
Candle-waxing the turtles
Some children were decorating templates of turtles using dripping candle wax
and paints. The teacher (T) entered a conversation, mainly between Cheryl (C)
and Susan (S), to light the candles and check their safe use.
