Early mathematical concepts

early grade mathematics assessment and early mathematics education
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EARLY MATHEMATICS: A GUIDE FOR IMPROVING TEACHING AND LEARNING FEBRUARY 2016 www.ero.govt.nzEarly mathematics: a guide for improving teaching and learning Published 2016 © Crown copyright ISBN 978-0-478-43832-1 Except for the Education Review Office’s logo and the photographs of people used throughout this report, this copyright work is licensed under Creative Commons Attribution 3.0 New Zealand licence. In essence, you are free to copy, distribute and adapt the work, as long as you attribute the work to the Education Review Office and abide by the other licence terms. In your attribution, use the wording ‘Education Review Office’, not the Education Review Office logo or the New Zealand Government logo.Early mathematics: a guide for improving teaching and learning 3 Contents Introduction 4 Improving mathematics teaching and learning 4 Background 5 The importance of early mathematics 5 Creating a disposition for mathematical learning 6 Mathematics in Te Whäriki7 Summary of Effective Pedagogy in Mathematics/Pängarau Best Evidence Synthesis Iteration (BES) 8 A guide to children’s early mathematics learning 10 A balance of deliberate teaching and spontaneous learning across the breadth of mathematics 11 Assessment for learning 24 Improving teaching of mathematics in the early years 28 Internal evaluation to improve practice 29 Resource One: A continuum for improving early mathematical learning for children 30 Resource Two: Investigative prompts and indicators of practice 31 Conclusion 35 What needs to happen to ensure that all children have rich mathematics learning? 35 4 Education Review Office Introduction This guide includes examples where there are Improving mathematics both deliberate and responsive mathematical teaching and learning learning opportunities for children, and where the children have opportunities for learning from This guide describes how children can be across the six mathematics strands. The focus supported to become confident and capable is on: mathematical learners in the early years. how teachers in the early years continually ERO has found two factors that are critical – reflect on and improve what they do pedagogical leadership and teacher knowledge. the importance of leaders and teachers’ Effective pedagogical leaders understand how understanding of how mathematics is to integrate mathematics into the curriculum to implemented as part of their curriculum best support children’s learning. They support their professional knowledge, ongoing teachers to develop an in-depth and broad learning and deliberate teaching that understanding of early mathematics, and supports mathematical learning associated practice, and promote a culture where teachers can reflect on their practice to their use of assessment information so continually make improvements. Where there is that they can confidently notice, recognise, strong pedagogical leadership teachers are more respond to, and extend children’s likely to be up to date with current research and understanding of mathematical concepts. good practice about early mathematics. They are This guide also outlines aspects of practice for able to use this knowledge to be innovative and leaders to watch out for or be concerned about intentional in their planning and teaching. Such when thinking about mathematics within their teachers confidently recognise and extend service’s curriculum. In particular, a ‘hands off’ children’s understanding of mathematical philosophy, where teachers rely solely on concepts in a range of contexts. children to take the lead in their own learning, The curriculum in most early learning settings can result in missed teaching and learning provides children with a wide range of opportunities. This misinterpretation of the opportunities to explore mathematical concepts. notion of a ‘child-centred’ curriculum fails to A balance of child-initiated learning experiences appreciate the critical role of the teacher in and deliberately planned activities provide a deliberately extending and scaffolding children’s platform for teachers to extend children’s learning. Leaders wanting to provide an in-depth developing mathematical understanding. mathematical curriculum should focus on Children can develop as confident and capable developing teachers’ subject and pedagogical mathematical learners through learning knowledge, so that they can confidently engage opportunities that reflect the six strands of with mathematics in ways that support and 1 Te Käkano – patterning, measuring, sorting, extend children’s learning. locating, counting and grouping, and shape. Skilled teachers intentionally plan experiences to extend children’s developing mathematical understandings. 1. Ministry of Education (2012). Te Aho Tukutuku – Early Mathematics. Wellington: Ministry of Education. This resourse has previously been distributed to all early childhood services and will be made available online later in 2016.Early mathematics: a guide for improving teaching and learning 5 Background The importance of early mathematics Children’s early experiences of mathematics development of a group of learners from early form the foundation for their future mathematics childhood education, through school and into learning and success. Mathematics enables adulthood. It shows that at age 10, the quality children to think logically, strategically, creatively of early childhood education still influences and critically. Mathematical knowledge and skills children’s competencies that lead to a successful provide building blocks for success in many areas adulthood, and that mathematical ability is one 2 of life and work. of the most influential factors. The study shows that most children acquire basic mathematical New Zealand and international research on knowledge and skills before the age of eight children’s learning in the early years confirms the years. Duncan and Murnane suggest that early importance of early experiences in mathematics mathematics skills are highly predictive of later 3 for future educational success. New Zealand’s 4 academic achievement. Competent Children study has tracked the 2. Ministry of Education. (2015). Spotlight on Mathematics/Pängarau. Retrieved from www.educationcounts.govt.nz/topics/BES/spotlight-on/ spotlight-on-mathematics-pangarau 3. Duncan, G. et al. (2007). School readiness and later achievement, Developmental Psychology, 43, 6: pp 1428-1446. Clements, D. et al (Eds). (2004). Engaging Young Children in Mathematics, New Jersey: Lawrence Erlbaum. Wylie, C. (2001, April). Competent children: Findings and issues from the first 7 years. Paper presented at the Ministry of Social Policy seminar, The long road to knowledge: Longitudinal research and  social policy, Wellington: Ministry of Education. 4. Duncan, G. and Murnane, R. (2011). Introduction: the American dream, then and now, Duncan, G. and Murnane, R. (Eds) Whither  Opportunity? Rising Inequality, Schools, and Children’s Life Chances. New York. Russell Sage Foundation: Ministry of Education. (1996)6 Education Review Office Creating a disposition for mathematical learning New Zealand’s early childhood curriculum, Peters and Rameka, in an article about Te Whäriki, focuses on supporting children’s Te Käkano, explain that dispositions influence learning dispositions and broad competencies how people tend to invest their capabilities. that can be readily transferred to new situations They state that motivation is crucial to the (such as starting school). The attitudes and development of children’s early mathematical expectations that are formed at an early concepts and it is important that the disposition age continue to influence a child’s learning to explore these concepts is not lost. 5 throughout life. The ECE Taskforce report, “Pedagogical approaches play a key role in  An Agenda for Amazing Children in 2011 ensuring that the inclination to use mathematics  highlighted the importance of learning is fostered and teachers’ own positive  dispositions, such as curiosity and perseverance, dispositions towards mathematics are likely to   6 7 to later educational success. These are critical be central to this process. Peters and Rameka ” dispositions for children’s early learning argue that inappropriate teaching practice, such experiences of mathematics. as rote learning in isolation of any meaningful contexts, may result in children developing The concept of Te Käkano – the seed – has been negative attitudes to mathematics. used to describe teaching and learning of mathematics in early childhood. See Figure 1. Figure 1: Te Käkano Source: Te Aho Tukutuku   Image credit: Ministry of Education © Crown 2016 5. Ministry of Education (1996). Te Whäriki, He Whäriki Mätauranga mö ngä Mokopuna o Aotearoa, Early Childhood Curriculum. Retrieved from http://www.education.govt.nz/early-childhood/teaching-and-learning/ece-curriculum/te-whariki/ 6. ECE Taskforce report, An Agenda for Amazing Children: p107. Retrieved from www.taskforce.ece.govt.nz/wp-content/uploads/2011/06/ Final_Report_ECE_Taskforce.pdf 7. Peters, S. and Rameka, L. (2010). Te Käkano (the seed): growing rich mathematics in ECE settings. Early Childhood Folio, 14, 2. Pp8-14.Early mathematics: a guide for improving teaching and learning 7 Te Aho Tukutuku/Early Mathematics Mathematics in Te Whäriki Te Aho Tukutuku is a Ministry of Education (the Mathematics is woven throughout the strands Ministry) resource designed to support and of Te Whäriki, specifically in Mana Reo – strengthen the teaching and learning of Communication and Mana Aotüroa – Exploration. mathematics in early childhood education. However, there is potential for mathematics in The Ministry intended the material to spark all strands. mathematical discussions and investigations, provide information and examples to support Mana Reo – Communication ‘noticing, recognising and responding’ to Mathematical learning outcomes relating children’s mathematical learning, and be a useful to the goals of communication include: resource for teachers and leaders to explore. familiarity with numbers and their uses The resource is grounded in Te Whäriki and skill in using the counting system and the Te Käkano framework in Kei Tua o te Pae/ mathematical symbols and concepts such Assessment for Learning: Early Childhood  9 as numbers, length, weight, volume, shape Exemplars Book 18: Mathematics Pängarau . and pattern. It includes short papers as well as information about mathematical learning and examples of young children putting mathematical ideas Mana Aotüroa – Exploration into action. Mathematical learning outcomes relating to the goals of exploration include: Te Aho Tukutuku uses the Te Käkano framework setting and solving problems as a metaphor for growing rich mathematics. It recognises the movement and unfolding looking for patterns from “te kore, ki te pö, ki te ao märama” (from classifying things for a purpose nothingness, to the night, to the world of light) guessing and the dynamic, integrated lifelong nature of using trial and error mathematical learning. 8 thinking logically and making comparisons. 8. Ministry of Education. (1996). Te Whäriki, He Whäriki Mätauranga mö ngä Mokopuna o Aotearoa, Early Childhood Curriculum. Retrieved from http://www.education.govt.nz/early-childhood/teaching-and-learning/ece-curriculum/te-whariki/ 9. Ministry of Education (2009). Kei Tua o te Pae. Assessment for Learning. Early Childhood Exemplars Mathematics (18). Retrieved from www.education.govt.nz/assets/Documents/Early-Childhood/Kei-Tua-o-te-Pae/ECEBk18Full.pdf8 Education Review Office Nurturing the seed Summary of Effective Pedagogy Many things nurture the seed: teacher pedagogy, in Mathematics/Pängarau Best teacher content knowledge, family/whänau 10 Evidence Synthesis Iteration (BES) knowledge, and resources. The Te Aho Tukutuku resource provides information on the importance Key findings: The early years of these aspects, and offers a way to strengthen Young children are powerful mathematics and enrich, foster and develop mathematics learners. Research has consistently teaching and learning. demonstrated how a wide range of children’s everyday activities, play and interests can be Strands of early mathematics used to engage, challenge and extend children’s Te Aho Tukutuku outlines six strands of mathematical knowledge and skills. There is now early mathematics: strong evidence that the most effective settings Pattern - the process of exploring, making for young learners provide a balance between and using patterns. opportunities for children to benefit from Measuring - answering the question teacher-initiated group work and freely chosen, “How big is it?” yet potentially instructive play activities. Teachers in early childhood settings need a sound Sorting - separating objects into groups understanding of mathematics to effectively with similar characteristics. capture the learning opportunities within the Locating - exploring space or finding or child’s environment and make available a range ‘locating’ something, such as a place of appropriate resources and purposeful and (location), or an item in space. challenging activities. Using this knowledge, Counting and grouping - the process for effective teachers provide scaffolding that working out the answer to a question about extends the child’s mathematical thinking while “How many?” Grouping involves putting simultaneously valuing the child’s contribution. things together. Shape - naming shapes and identifying the unique specific properties or features of shapes. 10. Anthony, G. and Walshaw, M. (2007). Effective Pedagogy in Mathematics/Pängarau Best Evidence Synthesis Iteration BES. Retrieved from www.educationcounts.govt.nz/publications/series/2515/5951Early mathematics: a guide for improving teaching and learning 9 Key ideas: The early years Issues identified in early mathematics teaching All children can be powerful mathematics learners. Low levels of content knowledge and the resulting lack of confidence limits the Children have their own purpose for activities. ability of teachers to engage children in the Children’s involvement in mathematical mathematics learning opportunities present learning experiences depends on interest. in existing activities. This low-level knowledge Mathematics learning experiences should also limits the ability of teachers to introduce be both planned and informal/spontaneous. more focused, interventional activities designed Everyday activities and play situations provide to cater for diverse learners. Other issues a wealth of mathematical experiences. identified included teachers: Teachers can extend the child-initiated being unaware of the need to cater for activities by scaffolding, thematic instruction, children’s interests and mathematical or instruction. abilities, and to engage children in Teachers need to cater for children’s interests challenging learning experiences and mathematical abilities and to engage missing many opportunities for sustained, children in challenging learning experiences. shared cognitive engagement Content matter is important. seeing mathematics as numeracy only being unaware of home mathematical experiences, expectations and aspirations not following their planning for mathematics teaching and learning.10 Education Review Office A guide to children’s early mathematics learning Two factors are critical in supporting children to be confident and capable mathematics learners – 11 pedagogical leadership and teacher knowledge (subject and pedagogical). Pedagogical leadership Teacher knowledge Strong pedagogical leaders Effective teachers successfully understand how to integrate integrate the different strands of mathematics throughout their mathematics of Te Käkano in the service’s curriculum to best support curriculum and provide each child children’s learning. They have with rich learning experiences extensive subject knowledge and across the breadth of mathematics. use this to support teachers to The curriculum reflects a balance of improve their knowledge and child-initiated learning experiences practice and to teach mathematics and deliberately planned activities. more intentionally as part of Teachers understand how cultural the curriculum. concepts such as kowhaiwhai, Leaders promote a culture where rangoli, and tapa designs, relate to teachers reflect on their teaching mathematics and children’s learning. practice to continually make They know how to integrate these improvements. In services with into their teaching practice. They strong pedagogical leadership, share children’s progress and teachers are more likely to be up learning with parents and whänau to date with current research and in wall displays and in assessment good practice about mathematics information such as portfolios. in early childhood. They are able to use this knowledge and high quality resources to be innovative and intentional in their planning and teaching. As a result, teachers confidently recognise and extend children’s understanding of mathematical concepts in a range of contexts. 11. Pedagogical knowledge – knowledge, skills, attitudes and attributes needed to provide clear, purposeful, stimulating and effective learning experiences. Lee, S. (2010). Toddlers and maths: fact, fiction of just fun? The First Years: Ngä Tau Tuatahi. New Zealand Journal of Infant and  Toddler Education. 12(2), pp38-43:p40.Early mathematics: a guide for improving teaching and learning 11 A balance of deliberate teaching and spontaneous learning across the breadth of mathematics New Zealand and international research In this resource we have used the term highlights the need for a balance between deliberate teaching but acknowledge that spontaneous child-initiated play and planned sometimes the term intentional teaching is used. 12 mathematical learning. When supporting Esptein explains what intentional teaching is. children’s mathematical learning, teachers need “Intentional teaching means teachers act    to be clear about what they are trying to achieve with specific outcomes or goals in mind for   and then deliberately reflect on the actions they children’s development and learning. Teachers   can take to help the child achieve the intended must know when to use a given strategy   to outcome. Epstein points out the need for both accommodate the different ways that   approaches. individual children learn and the specific   14 “Children need opportunities to initiate   content they are learning.” activities and follow their interests, but   teachers are not passive during these child   initiated and directed activities. Similarly,   children should be actively engaged and   responsive during teacher-initiated and   directed activities. Good teachers help    support the child’s learning in both types    13 of activities.” 12. Anthony, G. and Walshaw, M. (2007). Effective Pedagogy in Mathematics/Pängarau Best Evidence Synthesis Iteration BES. Retrieved from www.educationcounts.govt.nz/publications/series/2515/5951 O’Brien, N. (2015). Strategies for teachers developing strategies for mathematics in early childhood education. He Kupu 4(1), pp18-22. Retrieved from www.hekupu.ac.nz/index.php?type=journal&issue=21&journal=341. Bobis, J., Clarke, B., Clarke, D., Thomas, G., Wright, R., Young-Loveridge, J., & Gould, P. (2005). Supporting teachers in the development of young children’s mathematical thinking: Three large scale cases. Mathematics Education Research Journal, 16(3), pp27-57. Edens, K., M., & Potter, E. F. (2013). An exploratory look at the relationships among math skills, motivational factors and activity choice. Early Childhood Education Journal, 41, 235-243. 13. Epstein, A. (2007) The intentional teacher: Choosing the best strategies for young children’s learning. Washington DC. National Association for the Education of Young Children, pp8-9 14. Epstein, A. (2007) The intentional teacher: Choosing the best strategies for young children’s learning. Washington DC. National Association for the Education of Young Children, p1.12 Education Review Office Balancing spontaneous learning with deliberate teaching Spontaneous Deliberate Learning Teaching Assessment for learning In the Best Evidence Synthesis, Anthony and how they integrate mathematics throughout the Walshaw state: curriculum, and build on individual children’s knowledge in both spontaneous and planned There is now strong evidence that the most   ways. Teaching of mathematics is intentional effective settings for young learners provide   a and embedded in a holistic curriculum. balance between opportunities for children  to benefit from teacher-initiated group work and   Leaders play a significant role in promoting freely chosen, yet potentially instructive, play   deliberate teaching. They access professional activities. Teachers in early childhood settings   learning for themselves and teachers, encourage need a sound understanding of mathematics   and lead reflective practice (such as peer to effectively capture the learning opportunities   observations, videoed practice to review, and within the child’s environment and make   collaborative planning), professional reading, available a range of appropriate resources and   action research and internal evaluation (self purposeful and challenging activities. Using   review). This leadership means teachers can this knowledge, effective teachers provide   keep up to date with current research and good scaffolding that extends the child’s   practice and have the knowledge necessary to mathematical thinking while simultaneously   be innovative in their teaching of mathematics. 15 valuing the child’s contribution. Deliberate teaching Teachers’ planning guides their deliberate use of teaching strategies as part of their broader curriculum. Teachers know how to provide a range of experiences for children to support and extend their learning. They can explain and discuss their pedagogy related to mathematics, 15. Anthony, G. and Walshaw, M. (2007). Effective Pedagogy in Mathematics/Pängarau Best Evidence Synthesis Iteration BES, p2. Retrieved from www.educationcounts.govt.nz/publications/series/2515/5951Early mathematics: a guide for improving teaching and learning 13 In services with a balance of deliberate teaching In the following example, ERO observed how a and spontaneous learning, children have access leader and teachers used their pedagogical and to high quality mathematics resources. Teachers subject knowledge to deliberately plan for and are skilled at using interesting and challenging extend children’s mathematical learning. In this resources to scaffold children’s learning and service, teachers empowered children to make extend mathematical concepts. Teachers choices about their learning. They also fostered consider: children’s exploration and experimentation by respecting and encouraging the children’s children’s interests thinking and ideas. With this support, children in parent aspirations this service developed elaborate, imaginative and their service’s priorities for children’s learning deep interests that extended well beyond their current research about mathematics initial ideas. next steps identified in assessment information. Creating 3D structures Curriculum and A four-year-old at this service had seen a YouTube video about teaching practice constructing with plastic cups and he had shared his excitement about making some similar constructions with his teachers. A series of learning stories show the development of his ideas and the deliberate extension of his mathematical exploration by his teachers, which culminates in a long-term project not just for him but for many of the children in his group.14 Education Review Office Assessment The learning stories in his portfolio showed him making complex and accurate 2D drawings of the 3D structures he wanted to create. Teachers placed these drawings on display for him to refer to as he began to construct in 3D. In the middle of his project, the teachers and children attended a light Curriculum and show at a local art gallery. The use of light fascinated him, and he teaching practice then wanted to add light to his structures. Teachers provided torches and helped to make the room dark, and he added another dimension to his structures. By this stage, other children in the group had caught his enthusiasm Teacher for designing and creating 3D structures from the plastic cups and knowledge had started building their own – with and without light. The teachers’ planning shows a clear focus on supporting this mathematical learning. They made sure that resources were available to prompt children’s interests through construction sets, blocks of a variety of shapes, and a variety of games with a mathematical focus; and they intentionally used mathematical language with children. Children were urged to draw their ideas and then to transfer their 2D drawings into a 3D reality.Early mathematics: a guide for improving teaching and learning 15 The following comments show the children’s development of and Assessment reflection on their mathematical understanding – not only about shape, but also pattern, locating and counting. “It’s at the apex.” “It’s about making the cups balance.” “Remember the one where it went along the floor for a long way, and then went up the wall? I had to angle the cups to start going up the wall to make it balance.” “I had to use equal numbers of red and green cups to make the pattern. The yellow ones are the contrast, but they are all the same size. The yellow ones just look bigger because they are nearer when you stand in front.” “Can you see how it alters the dimensions of my face when I stand behind the tower and you can see me in front?” “I had to work hard to get the balance right. Sometimes the cups are tricky, but I just had to concentrate to get it right. First I just made a triangle shape, but now I’m making more linear shapes that can go across the floor.”16 Education Review Office In this next example, teachers noticed that a trip to the library provoked one child’s interest in large numbers. The teacher recognised opportunities for extending this interest when they returned to the centre. Open-ended questioning encouraged the children to explain and extend their thinking. Big numbers Curriculum and The children went on a trip to the local library, where the children’s teaching practice librarian explained how the Dewey Decimal System used numbers to categorise books so people could find books they were interested in. They went for a number hunt around the library, and found each of the big sorting numbers 000, 100, 200, 300, 400, 500, 600, 700, 800 and 900. One child was especially excited by the big numbers and, back at the Teacher centre, shared that his favourite number was 100. He wrote the knowledge number on the whiteboard to show his friends, and also wrote the number 100 on a birthday card for his mum.Early mathematics: a guide for improving teaching and learning 17 Teacher He was having a conversation knowledge with his friend, and said he wanted to be 100 years old. His friend said he wanted to be 102. The teacher noticed the conversation, and asked why the second child chose 102 – he responded that it was because it was bigger than 100. Later, the children were sitting with the number puzzles, where each piece is a digit. The teacher asked them what big numbers they could make, suggesting 90 to start with. This was easily done, so the teacher suggested an even bigger number – 100. The children did this, and then continued changing only the first digit, so they made all the hundreds up to 900. One child was unsure about naming these numbers, saying “twenty hundred, thirty hundred”. The teacher guided him to name them correctly, and he quickly caught on, confidently counting in hundreds up to 900.18 Education Review Office In another service, teachers supported new learning about measurement by planning to extend this learning in several different and meaningful contexts. Measuring, estimating and comparing Curriculum and The children experimented with multiple ways of measuring things, teaching practice such as how much water their gumboots could hold, how long they could hang off a beam, and how far playdough could bounce. Teachers extended this interest by showing them different ways the measurement information could be presented. They used the tape measure to measure each other’s height and recorded these on a height chart. They counted how many segments were in their mandarins each day and recorded this data in a pictograph. One child took the tape measure and began stretching it alongside Teacher pieces of wood. He told the teacher that he wanted to make two knowledge pieces of wood that were the same length. The teacher helped him by explaining that he would need to make sure the number on the tape measure was the same for each piece. There were not any pieces already the same length, so he realised he would have to cut one to match the length of another. The teacher prompted him to find the same number on his measuring tape as before, and mark the wood with a pencil to show where he needed to cut it. The boy was able to do this, and learnt a practical application for his new measuring skills.Early mathematics: a guide for improving teaching and learning 19 In the following service, teachers took a planned approach to support children’s learning in a way that responded to the different interests, strengths and ages of children in the group. Comparing size Children at the centre had been exploring size. Teachers provided Curriculum and three rectangular mats, labelled big, medium and small; and a teaching practice collection of objects, with a big, medium and small one for each object type. The teacher demonstrated the activity, placing the biggest pompom on the mat labelled ‘big’, the next biggest pompom on the mat labelled ‘medium’, and the smallest pompom on the mat labelled ‘small’. A child quickly picked up on the task, and placed the appropriate lock, bell, square etc. on each mat, describing the biggest objects as the ‘daddy’ ones, the medium ones as the ‘mummy’ ones, and the smallest ones as the ‘baby’ ones. This tied in to the language and comparative sizing used earlier in the Goldilocks and the Three Bears story. The teachers planned to extend that child’s learning with activities Curriculum and that helped her to sort and classify based on more specific attributes, teaching practice such as length, weight or height. An older child was comparing her shoe size to that of her teacher. The teacher suggested measuring their feet as a way of describing the length more accurately. They got a ruler and measured the teacher’s foot, followed by the girl’s foot. They compared the numbers, and saw that the teacher had the bigger foot and the bigger number.20 Education Review Office Teacher The teacher then showed her how to measure the length of a line knowledge with her feet, carefully balancing and putting one foot exactly in front of the other. The teacher counted how many foot-lengths a line was, and then the girl counted how many of her foot-lengths the line was. The numbers were different, and the teacher asked why that might be. The child replied that the teacher used fewer foot-lengths because his feet were bigger. Curriculum and The teacher planned to continue this child’s interest in measuring by teaching practice exploring other non-standard ways of measuring.

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