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Children thinkingmathematically: PSRNessential knowledge forEarly Years practitioners

Children thinking mathematically:PSRN essential knowledge forEarly Years practitionersFirst published in 2009Ref: 00861-2009BKT-EN

DisclaimerThe Department for Children, Schools and Familieswishes to make it clear that the Department andits agents accept no responsibility for the actualcontent of any materials suggested as informationsources in this publication, whether these are inthe form of printed publications or on a website.In these materials, icons, logos, software productsand websites are used for contextual and practicalreasons. Their use should not be interpretedas an endorsement of particular companies ortheir products.The websites referred to in these materials existedat the time of going to print.DSI CMM 12-2009Please check all website references carefully tosee if they have changed and substitute otherreferences where appropriate.

The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitioners1Contents1. Introduction32. Enabling environments63. Children’s mathematical graphics124. Numbers as labels and for counting185. Calculating266. Shape, space and measures327. Transitions408. Examples of problem solving in action from PSRN Essential Knowledge479. References51 Crown copyright 200900861-2009BKT-EN

The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitioners31. IntroductionProblem Solving, Reasoning and Numeracy in theEarly Years Foundation StageThis booklet focuses on children’s mathematical development, which is explored in the area of learningand development entitled Problem Solving, Reasoning and Numeracy (PSRN) in the Early YearsFoundation Stage (EYFS) (DCSF, 2008).The EYFS recognises that creativity and critical thinking are important in all areas of learning and are asintegral to mathematics as they are to painting or dance. Reflecting on the narratives and examplesof children’s mathematics throughout this booklet, it is clear that creativity plays a significant role inmathematical thinking and understanding.The four themes of the EYFS, A Unique Child, Positive Relationships, Enabling Environments and Learningand Development, underpin everything that practitioners do with children from birth to five. They alsounderpin every chapter in this booklet.The Practice Guidance for the Early Years Foundation Stage (page 63) and the PSRN card remind practitionersthat A Unique Child will ‘seek patterns, make connections and recognise relationships through findingout about and working with numbers and counting, with sorting and matching and with shape, spaceand measures’ and that ‘children use their knowledge and skill in these areas to solve problems, generatenew questions and make connections across other areas of Learning and Development.’The EYFS guidance then goes on to set out some of the important considerations for children’s learningin PSRN relating to Positive Relationships, Enabling Environments and Learning and Development.Reference is made to these throughout this booklet.What mathematics do you think children are exploring here? How could you enhance this? Crown copyright 200900861-2009BKT-EN

4The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitionersWhy do we need more guidance on PSRN?The independent review of mathematics teaching in Early Years settings and primary schools, known asthe Williams review (DCSF, 2008a), recommended that the DCSF should commission a set of materials onmathematical mark making and children’s mathematical development to be used to support Early Yearspractitioners’ professional development.The review emphasised that although ‘the EYFS provides guidance on developing mathematicalunderstanding through imaginative play opportunities in this area seem to be missed.’This finding is confirmed by the Early Years Foundation Stage Profile (EYFSP) results, which demonstratethat relatively few children (36 per cent) attain point 8: uses developing mathematical ideas and methods tosolve practical problems, in any of the three mathematical assessment scales.The EYFSP scores also reveal that boys do less well than girls especially in the area of calculation: ‘Inpractical activities and discussion, begins to use the vocabulary involved in adding and subtraction’(EYFSP). It is essential therefore that practical opportunities are available on a regular basis withinsmall group activities and throughout the play environment, and that these activities are built on (seeexample below).Developing calculationString beans and containers were set on the carpet for ‘talk time’ and Brandon (4 years 3 months)decided to shell beans and put them in a container which he had chosen. He self-initiated his owncounting. Quietly talking to himself he divided the beans into each section of the container, ‘11 morenow mmm how many more? 5 now’ as he counted the empty spaces. Brandon was using counting tocalculate how many more he needed. Brandon’s teacher regularly puts out quantities of objects to talk00861-2009BKT-EN Crown copyright 2009

The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitioners5about and involves the group in talking about increasingly larger quantities. Mathematical vocabulary isdeliberately used but in meaningful contexts to give the children tools to articulate their knowledge ofmathematics.Building on Mark Making MattersMark Making Matters (DCSF, 2008b) was the first part of the set of materials recommended byWilliams. It aimed to raise awareness of the importance of young children’s mark making as a tool forcommunication and thinking across the six areas of learning and development, while strengthening thequality of provision for mark making in Communication, Language and Literacy and in PSRN.A local education authority advisor, involved in raising the profile of calculation through supportingreception class teachers in their understanding of children thinking mathematically through theirown graphics, stated, ‘Maths is much higher profile now. Teachers and practitioners are talking aboutcalculating significantly more; boys are representing mathematics and are extending themselves muchmore. This has had an impact on the way in which teachers and practitioners work: their expectations ofthe children are much higher.’The Williams review (DCSF, 2008a) also stressed that effective Early Years pedagogy should valueand support children’s own mathematical graphics. Children’s own mathematical graphics help themunderstand the written language of mathematics and how it can be used.What’s in this booklet?This booklet takes Mark Making Matters further and extends and develops the concepts explored therewith particular reference to the three strands of PSRN: Numbers as labels and for countingCalculatingShape space and measures.There are chapters on each of these, as well as the opening chapter on enabling environments for PSRN.Chapter 3: Children’s mathematical graphics lays the foundation for the remainder of the booklet, whichconcludes with consideration of transition between EYFS and Year 1.High-quality experiences of mathematics are theentitlement of every childThe examples in this booklet include children from a range of backgrounds and communities. They havedifferent learning abilities (some with identified special needs and some who are gifted) and includechildren learning English as an additional language. The examples show how all young children have anamazing ability to understand and will often surpass practitioners’ expectations to reach their potential –provided they can explore mathematics through play and in personally meaningful ways.Practitioners need to provide opportunities for all children, ensuring that those less likely to achievewell are fully engaged to support them in developing positive dispositions to learning and realisingbetter outcomes.This booklet aims to help practitioners ‘see’ the mathematics in children’s play. It is underpinned bythe rich legacy of research into young children’s mathematical development, play and learning, andilluminates the sorts of context and adult ‘scaffolding’ that make these experiences so successful. Crown copyright 200900861-2009BKT-EN

6The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitioners2. Enabling environments‘Children’s play reflects their wide-ranging and varied interests and preoccupations.In their play children learn at their highest level’DfES: 2007bThe EYFS requires that:Children must be supported in developing their understanding of Problem Solving, Reasoning andNumeracy in a broad range of contexts in which they can explore, enjoy, learn, practise and talk abouttheir developing understanding. They must be provided with opportunities to practise these skills and togain confidence and competence in their use.The environment (including the indoor, outdoor and emotional environments) that practitionersprovide plays a significant role in supporting young children’s mathematical learning. The EYFS makesit clear that young children learn best through play and that for their learning to be effective, they needsensitive and informed support from adults.All children can be successful with mathematics, provided that they have opportunities to exploremathematical ideas in ways that make personal sense to them and opportunities to develop mathematicalconcepts and understanding. Children need to know that practitioners are interested in their thinking,respect their ideas, are sensitive to their feelings and value their contributions.From birth, babies are keen explorers of the world and need a close bond with their mother, father and/or other key adults to give them the confidence to continue as confident learners. This ‘secure attachment’is promoted in a rich learning environment which provides a safe space for babies to explore and includesfamiliar elements of home, resources that support babies’ physical movement and collections of objectsthat encourage multi-sensory exploration, such as ‘treasure baskets’ and heuristic play.00861-2009BKT-EN Crown copyright 2009

The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitioners7As they get older, babies enjoy having containers to empty and fill, malleable materials to mould, waterand dough, paint play, things to build with and things to knock down. Through these experiences theybegin to develop an understanding of shape, space, and number. They also gain confidence in theirability to control their own learning.Harry (aged 4 years 5 months) came into the nursery one morning shouting, ‘I have a thousand ideas’.His nursery encourages children’s creative thinking and supports children in developing their ideas.It is providing an enabling environment where children can confidently embrace new challenges.The relationship that Harry has with his key person is an important aspect of this trust.For this reason, it is vital that practitioners: share positive beliefs about young children learning mathematicsare aware of the mathematics that arises through children’s self-initiated playhave high expectations of young children’s mathematical understandingunderstand babies’ and young children’s mathematical development, learning from reflecting onobservations and through discussions with their team, and use this knowledge to ‘tune into’ themathematics that children explore within their play. (See also Development Matters, EYFS, 2007a.)ParentsIt is important for practitioners to discuss and share insights with parents about their children’s interestsand play, both at home and in their early childhood setting. This collaborative dialogue can only enrichunderstanding and will provide a rounded view of children’s developing mathematical knowledge andsuccesses. (See EYFS Principles into Practice, Card 2.2.)Extending children’s mathematical learningObserving children’s play will help practitioners to value their growing mathematical understanding andreveal ways to support this development. Children’s interests are powerful catalysts for mathematicalenquiry and will provide a strong starting point to support and extend their mathematical thinking.Opportunities for problem solving, reasoning, critical thinking and reflection are vital if children are tomake the most of their emergent understanding of mathematics. Crown copyright 200900861-2009BKT-EN

8The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitionersSensitive adults who value children’s ideas and support children’s play and mathematical explorationsthrough collaborative dialogue help to ‘scaffold’ children’s thinking. Practitioners can help children gobeyond what they already understand and can do. Thinking, making meanings and understanding aresignificant aspects of mathematics.Engaging in discussion with children means that the adult is genuinely interested in learning fromthe children about their ideas. Using open questions encourages children to talk about their thinking.This will allow ideas to be co-constructed and shared, and meanings to be negotiated and understood.For further information on young children’s talk, see: Mathematical Vocabulary, 2000 and ‘talkinghotspots’.The ‘talking hotspots’ activity helps practitioners to evaluate how the environment supports thespeech, language and communication development of the children and identifies places whereauthentic conversations between children and adults can take place. This will include discussions aboutmathematical issues, for example: the cost of items in the shop, the time a train will arrive and howlate it could be, how long the ribbon needs to be to wrap the present, how many cakes are needed atthe party, how much taller one sunflower is than another, the patterns on pairs of socks. (See ‘talkinghotspots’ from Every Child a Talker: Guidance for Early Language Lead Practitioners, 2008, page 15.)Food experiences help children explore quantitiesand space, shape and measures. These children arefascinated by the movement of the whisk as it goesround and round.Making spaces that children can go inside, up andthrough helps them to explore spaces with their wholebodies. This also gives them opportunities to talkabout positional language in an authentic way.Placing pebbles in a basket outside with different sizedcontainers and calibrated balance scales can supportchildren’s expectations of quantity and size.These children are really interested in building asthey have watched the real builders outside add anextension to their nursery. The nursery provides spiritlevels and other building equipment inside and outsidethe nursery to further this enquiry.00861-2009BKT-EN Crown copyright 2009

The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitionersThe children have made a shelter and have hadto search for sticks and branches of the same anddifferent sizes. Now they have an exciting space toplay in.In this under-threes environment a space has beenset up for the children to crawl in and go up – vitallyimportant for awareness of space, direction anddistance.Ross has built three blocks on top of one another andhas decided to measure them, using a paper tapemeasure.Aimie is using the raffle tickets as birthday invitationsfor her friends. She says the numbers she knows as shegives them one each.In this Reception class the teacher has used a display todraw the children’s attention to numbers around them.Number lines everywhere stimulate discussion aboutnumbers and the children use these within their play. Crown copyright 2009900861-2009BKT-EN

10The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitionersLauren has found a clipboard on which an adult hadbeen marking children’s choices for snack time. Shehas decided to help, making green marks over whatthe adult had written as she asks her friends what theywould like.Large calculators provide use of technology andinteraction with numbers.Providing challenges in the outdoor area stimulatestalk about height: ‘I am going to jump off the highestone.’Numbers on the stairs encourage counting up anddown in a sequence when climbing.Young children love to have their own noticeboardswhere they can add their own graphics.00861-2009BKT-EN Crown copyright 2009

The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitioners11Creating a rich play environment to support PSRNMathematics does not depend on specific mathematical resources but on children having opportunitiesto develop mathematical concepts and understanding. When you have read the other chapters inthe booklet you might find it useful to carry out the activities and audit provided in the ProfessionalDevelopment Meeting on pages 42–50 of Early Years Quality Improvement Support Programme (DCSF,2008) entitled ‘Developing a high quality learning environment, indoors and outdoors, which supportsPSRN’. This can be found at www.standards.dcsf.gov.uk/nationalstrategies. Crown copyright 200900861-2009BKT-EN

12The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitioners3. Children’s mathematicalgraphicsThis chapter builds on the DCSF publication Mark Making Matters (DCSF, 2008b), which emphasises thatopportunities for children’s early graphics should be threaded through every area of learning.What are children’s mathematical graphics?The term children’s mathematical graphics was originated by Carruthers and Worthington (2003).It is used to describe children’s own marks and representations that they use to explore andcommunicate their mathematical thinking. Research into children’s mathematical graphics, (Carruthersand Worthington, 2006) has revealed young children’s development of their early mathematicalthinking as they explore the symbolic ‘written’ language of mathematics. These graphics include:scribble-marks, drawings, writing, tally-type marks, and invented and standard symbols includingnumerals. Young children’s graphical exploration ‘builds on what they already know about marksand symbols and lays the foundations for understanding mathematical symbols and later use ofstandard forms of written mathematics’, Carruthers and Worthington (2006).The EYFS PSRN emphasises that practitioners should:‘Value children’s own graphic and practical explorations of problem solving’ and observe ‘thecontext in which young children use their own graphics.’What do children’s mathematical graphics look like?Children’s mathematical graphics support their developing mathematical thinking, and the followingthree examples illustrate part of this continuous process, making visible the sophistication of children’screative approaches to problem solving, reasoning and numeracy. The first two examples show childrenrepresenting their mathematical thinking as they count. The third example shows how Kamrin, an olderchild, still uses his own more-complex graphics to help him reflect and work on a mathematical problemconcerning division.00861-2009BKT-EN Crown copyright 2009

The National Strategies Early YearsChildren thinking mathematically: PSRN essential knowledge for Early Years practitioners13 E. Carruthers & M. WorthingtonWilliam’s ‘light sabres’William (4 years 6 months) is fascinated by Star Wars and often plays out various scenes with his friend.He has drawn a light sabre for each of his favourite characters from the Star Wars film. The colours areimportant;

children learning English as an additional language. The examples show how all young children have an amazing ability to understand and will often surpass practitioners’ expectations to reach their potential – provided they can explore mathematics through play and in personally meaningful ways. Practitioners need to provide opportunities for all children, ensuring that those less likely to .

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