Under Threes Thinking Mathematically?

Elizabeth Carruthers

Prague 2007

Introduction

In England since 2003 there has been a national government endorsed framework for supporting children’s learning from birth to three. (DfES, 2003). Recently (2006) the Birth to Three Framework has merged with the Curriculum Guidance for three, four and five year olds to become The Early Years Foundation Stage. The curriculum is now in subject areas which includes mathematics. This has lead to much controversy because perhaps it is perceived as a formal school curriculum. (David and Powell, 2007). What about under threes and mathematics?

Children under Three and Mathematics

Studies of babies (Starkey and Cooper,1980;Ulyn,1992 ) claim that babies as young as three months can distinguish quantities.Fuson (1988) studied children’s counting errors.Gelman and Gallistel’s(1978) research uncovered certain counting principles

Most of the studies have been about children and number and the research methods have been clinically set up tasks. Carruthers (1997) the ability to tune into mathematics. Arnold (1999) focussed on a children’s schemas and interests and found that the child in the study interacted with areas of mathematics through these interests.Butler, S. looked studied children of four years of age and found that children interacted with mathematics on a deep level through their schematic interests.

Vygotsky stated that, ‘children under three learn while following their own programme’(1932)

Vygotsky(1962) described two types of learning experiences –

1. ‘Bottom-up learning’ – the child’s spontaneous attempt to understand aspects of social and physical reality without the direct aid of adult or peer tutoring. Spontaneous concepts

2. ‘Top down learning’ problems are posed for the child by the adult, concepts are presented that are valued by the culture. (encountered in schools) Scientific concepts

Athey – The Froebel Early Education Project(1990) studied Children’s behaviour from 2-5 years.Schemas ‘ The most easily understood meaning of ‘schema’ is that schemas of action are co-ordinated systems of movements and perceptions, which constitute any elementary behaviour capable of being repeated and applied to new situations e.g. grasping, moving, shaking an object (Piaget, 1962, p274). Schemas are patterns of repeatable actions that lead to early categories and then to logical classifications.’ (a constructivist idea)Some studies suggest that many of children’s explorations of schemas also have mathematical content (Athey 1990, Nutbrown,1999, Carruthers and Worthington, 2006, Butler, 2007)

The study

Data came from 7 teachers/ practitioner’s diaries at a Children’s Centre in Bristol, England.The children were age from 18 months to 34 months.The analysis was made on the children’s self-initiated play focussing on their interests/schemas.Conversations were held with the practitioner’s about their observations

The MethodDiaries of the children’s play observations were the main source of data. The diaries were of observations of seven children over a three month period. This was a naturalistic study, based on grounded theory( Glasier and Strauss, 1967) observing the children in everyday play at their nursery.This nursery has a play based curriculum where children choose from the materials and resources on offer, both inside and in the outside play area in the nursery.

Play

Free flow play (Bruce)

Play for children under threes is a serious game (Vygotsky, 1933)

Play is the source of development and creates the zone of proximal development. (Vygotsky, 1933)

Water play, Sand play, manipulative materials, blocks, role play, writing and drawing and painting equipment, climbing apparatus, buggies, dolls, musical instruments, wheeled toys, cars, train tracks.

Outside experiences include, trees, trestle tables, bikes, carts, transporters, painting, blocks, climbing equipment, sand, water, digging.

Method continued

•The practitioner’s make written observations of the children, noting significant moments, especially where the children are very focussed and intent on their play.•This nursery is beginning to look at children’s interests to structure the curriculum.

12 diaries were chosen randomly from 41 diaries and from these only seven revealed a pattern of what the children were interested in/schema.The practitioners said they identified children’s interests but did not always write down the observations they had made. They had what may be termed implicit knowledge of the children. Perhaps also sometimes a possible observation would not fit into the compartments of the diary. i.e. a strong child, a healthy child, a skilful communicator.

The data was coded under three headings.

interest/schema

Examples of what the child did.

Possible Mathematical content.

2.Elizabeth Containing Posting corks into a tissue box.

Size and shape.

Putting a spoon in a doll’s mouth and then her own mouth.

matching

Putting lids in a paper bag and taking them out, repeating this again and again

Size and shape

Putting a book in a bag and carrying it around the nursery

Stacking

Putting rings over a wooden stand.

On top off and quantity

Cause and effect

Placing a ball on the box and watch it roll off again

Properties of a sphere.

Rolling an orange backward and forwards (6 times)

Properties of a sphere,

Putting the ball in the ball run and watching it run down the tube (repeating this)

Sphere,

Dropping instruments on the floor to see what sound they make

investigating

At forest school bending a thin branch and watching it carefully as she did this.

investigating

Findings

Some children had two or three easily identifiable interests/schemasChildren interested in the same schema played in different ways using different equipment and therefore the mathematical content was different e.g. Elizabeth and Rosa both in a containing schema chose different equipment and ways to explore their ‘containing interest’. For example Elizabeth’s explorations uncovered her use of size, space and shape whereas Rosa was finding out more about the mathematics of full and empty and capacity.The seven children identified covered a range of mathematical concepts in their play e.g. space, shape, measurement, area, capacity, length, distance, angles and lines.( Butler, 2007) The items available to them dictated the richness of their experience and the way they used them (string, masking tape) The way in which children use materials often dictate their learning potential, (Nutbrown,1994 )Most of the mathematical content that the children were involved in through their interests were in nearly all the areas of mathematics except number.

Cause and effect – possible link to problem solving

The materials the children used to explore their schemas were non- conventional (not typical educational mathematical equipment, e.g. toilet roll and glitter)

Implications

If one does support children’s interests then it is moving towards a more individualised curriculum (currently the dfsp(2007) in England are very focussed on personalised learning)Practitioners knowledge of children’s interests are crucial to support under threes mathematical development. Tuning into ways of knowing children’s individual concerns and responding to them will support their learning ( David and Powell, 2007)From this study it seems that supporting the child’s interests (‘own programme’) could provide the crucial scaffold that enhances their mathematical thinking.The role of resources influenced the child’s play. Teachers should consider including non-conventional resources to support mathematical development, for example, string, cardboard boxes, masking tape, post –its.

When one follows the child’s own programme it ‘reveals the hidden maths’ (Worthington 2007)

References

Vygotsky,L.S. Thought and language. Cambridge, Mass.:MIT Press, 1962.

Wertsch, J.V. From social interaction to higher psychological processes. A clarification and application of Vygotsky’s theory. Human Development, 1979,22(1),1-22.

Vygotsky, L.S. Mind in Society:The development of higher psychological processes. (M.Cole, V. John Steiner, S.Scribner, & E. Souberman, Eds.). Cambridge, Mass.: Harvard University Press,1978.