Two factors are critical in supporting children to be confident and capable mathematics learners - pedagogical leadership and teacher knowledge (subject and pedagogical).1
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Pedagogical leadership Strong pedagogical leaders understand how to integrate mathematics throughout their service's curriculum to best support children's learning. They have extensive subject knowledge and use this to support teachers to improve their knowledge and practice and to teach mathematics more intentionally as part of the curriculum. Leaders promote a culture where teachers reflect on their teaching practice to continually make improvements. In services with strong pedagogical leadership, teachers are more likely to be up to date with current research and good practice about mathematics 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. |
Teacher knowledge Effective teachers successfully integrate the different strands of mathematics of Te Kakano in the curriculum and provide each child with rich learning experiences across the breadth of mathematics. The curriculum reflects a balance of child-initiated learning experiences and deliberately planned activities. Teachers understand how cultural concepts such as kowhaiwhai, rangoli, and tapa designs, relate to mathematics and children's learning. They know how to integrate these into their teaching practice. They share children's progress and learning with parents and whanau in wall displays and in assessment information such as portfolios. |
New Zealand and international research highlights the need for a balance between spontaneous child-initiated play and planned mathematical learning.2 When supporting children's mathematical learning, teachers need to be clear about what they are trying to achieve and then deliberately reflect on the actions they can take to help the child achieve the intended outcome. Epstein points out the need for both approaches.
"Children need opportunities to initiate 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 of activities."3
In this resource we have used the term deliberate teaching but acknowledge that sometimes the term intentional teaching is used. Esptein explains what intentional teaching is.
"Intentional teaching means teachers act with specific outcomes or goals in mind for children's development and learning. Teachers must know when to use a given strategy to accommodate the different ways that individual children learn and the specific content they are learning."4
In the Best Evidence Synthesis, Anthony and Walshaw state:
There is now strong evidence that the most effective settings for young learners provide a balance between opportunities for children to benefit from teacher-initiated group work and freely chosen, yet potentially instructive, play activities. Teachers in early childhood settings need a sound understanding of mathematics to effectively capture the learning opportunities within the child's environment and make available a range of appropriate resources and purposeful and challenging activities. Using this knowledge, effective teachers provide scaffolding that extends the child's mathematical thinking while simultaneously valuing the child's contribution.5
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, how they integrate mathematics throughout the curriculum, and build on individual children's knowledge in both spontaneous and planned ways. Teaching of mathematics is intentional and embedded in a holistic curriculum.
Leaders play a significant role in promoting deliberate teaching. They access professional learning for themselves and teachers, encourage and lead reflective practice (such as peer observations, videoed practice to review, and collaborative planning), professional reading, action research and internal evaluation (self review). This leadership means teachers can keep up to date with current research and good practice and have the knowledge necessary to be innovative in their teaching of mathematics.
In services with a balance of deliberate teaching and spontaneous learning, children have access to high quality mathematics resources. Teachers are skilled at using interesting and challenging resources to scaffold children's learning and extend mathematical concepts. Teachers consider:
In the following example, ERO observed how a leader and teachers used their pedagogical and subject knowledge to deliberately plan for and extend children's mathematical learning. In this service, teachers empowered children to make choices about their learning. They also fostered children's exploration and experimentation by respecting and encouraging the children's thinking and ideas. With this support, children in this service developed elaborate, imaginative and deep interests that extended well beyond their initial ideas.
| Curriculum and teaching practice |
A four-year-old at this service had seen a YouTube video about 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. |
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| 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. |
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| Curriculum and teaching practice | In the middle of his project, the teachers and children attended a light show at a local art gallery. The use of light fascinated him, and he 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. | |
| Teacher knowledge |
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| Assessment |
The following comments show the children's development of and reflection on their mathematical understanding - not only about shape, but also pattern, locating and counting.
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| Curriculum and teaching practice |
The children went on a trip to the local library, where the children's 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.
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| Teacher knowledge | One child was especially excited by the big numbers and, back at the centre, shared that his favourite number was 100. He wrote the number on the whiteboard to show his friends, and also wrote the number 100 on a birthday card for his mum. |
| Teacher knowledge |
He was having a conversation 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. |
In another service, teachers supported new learning about measurement by planning to extend this learning in several different and meaningful contexts.
| Curriculum and teaching practice |
The children experimented with multiple ways of measuring things, 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. |
| Teacher knowledge | One child took the tape measure and began stretching it alongside pieces of wood. He told the teacher that he wanted to make two 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. |
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.
| Curriculum and teaching practice |
Children at the centre had been exploring size. Teachers provided three rectangular mats, labelled big, medium and small; and a 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 that helped her to sort and classify based on more specific attributes, 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. |
| Teacher knowledge |
The teacher then showed her how to measure the length of a line 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 teaching practice |
The teacher planned to continue this child's interest in measuring by exploring other non-standard ways of measuring. |
When spontaneous learning is happening well, children experience a curriculum in which mathematics is included in meaningful contexts that recognise their strengths and interests and build on these. Leaders and teachers understand how to integrate mathematics into the child's interests. Teachers may also integrate mathematics into their bicultural practice, for example, learning mathematics though waiata, creating and exploring kowhaiwhai patterns and Maori carvings, and using resources that reflect the cultures of children in the centre.6
The example below highlights how one child's learning was extended as a result of the collaborative culture evident in the service she attended. Teachers in the service worked collegially to reflect on their practices, children's interests and new learning. This collaboration meant that a child's success seen by one teacher was extended later by another teacher.
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Curriculum and teaching practice |
The teacher noticed a child's interest in arranging blocks on a baseboard. After sitting together for a while making patterns, the teacher suggested she could try to cover the whole board with blocks. This was a challenge, as there were a number of nodules on the board and the blocks were all different sizes. The child carefully started putting the blocks on from one side, making sure the blocks lined up. She then started adding more blocks from the opposite side of the board, and found herself stuck with a gap in the middle that was smaller than any of her blocks. She asked her teacher if there were any smaller blocks to fill the gap, and was prompted to look for some other holes in her block arrangement. She tried to put a block in one of these spaces, and commented that her block was too big. |
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Teaching practice |
The teacher then helped her to count the number of nodules on the board in the space, and the number of nodules on her smallest block. They compared the numbers, and realised that they were different. The teacher suggested making a bigger hole might help, so after considering for a while, the child moved the pieces across to fill one hole, and enlarge another. She then counted the nodules in the new, bigger hole, and compared the number to the nodules on her brick. They were the same, so she was able to fill the board completely. |
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Teacher knowledge |
A few weeks later, this child was observed by another teacher repeating the activity by herself, and doing a great job of it. The teacher noticed this, and thought it was time to extend the girl's thinking, by introducing different shapes, as well as different sizes. The teacher got the number-teaching blocks for the girl, and straight away, she started trying to cover the board with them. She got nearly to the end, before realising that the blocks she had left would not fit the remaining spaces on the board. This time, she was able to use her problem-solving skills from the earlier activity. She moved some pieces, looked at the different sizes and shapes of the blocks and the holes, and tried different ways of putting in her remaining blocks, until she solved the puzzle. |
The following examples show the ways in which teachers were supporting and extending children's interests in mathematics.
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Child initiates activity |
One child was playing with magnetic shapes. "I'm making shapes.” She sorted the shapes into groups and then organised them into patterns. The teacher asked what she knew about shapes, and pointing to each group in turn, the child said, "Circles, squares, triangles and rectangles.” |
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Teacher notices learning happening |
As part of this the teacher recognised the child's learning and identified an opportunity to extend the mathematical learning by engaging the child further in sorting, ordering and creating with shapes. The teacher recognised shape and spatial awareness, patterns, and sorting and organising into categories as some of the mathematical concepts being explored by the child. The teacher then supported the child to create artwork using a complex mosaic technique. |
The Playcentre, in the example below, is led by a group of Playcentre-qualified parents who meaningfully integrate mathematical concepts into children's play.
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Noticing of mathematics occurring naturally |
Parents talked about mathematics with the children as they played in the sandpit, mixed play-dough, played with water using measuring cups, read books, and shared finger plays, rhymes and favourite songs. "Maths is a natural conversation that we have with children all the time... there are lots of opportunities for questioning.” (Supervisor) |
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Questioning prompts transfer of concepts |
A group of three boys went to the children's growth chart. They began to measure themselves and compare their heights: "Where are you up to?” One child reads the ruler - "I am six-nine, see. That's taller then you, ay?” The boys continued to compare each other against the chart. Next to the growth chart is another type of chart, used to measure growth using photos of children over time. The children moved from the height chart to comment on the photo growth chart. "Look at how small I was, now I am six-nine and much bigger. I am bigger sideways too.” A parent asked the boys what other things they could track the growth of. One child remembered that they had a metre ruler in the vegetable garden and they had been tracking the growth of the bean plants. The children raced off to investigate and measure the beans. |
In the third example, the teacher offers the child a pen and paper to record his thinking while at the same time giving him the opportunity to decide how he wants to record his findings.
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Children initiate activity |
The child and his friend were working together to explore weight and counting. They had a set of scales and were watching the number change as they carefully added more objects. He used his finger to point to each of the numbers, counting as he went. |
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Teacher notices and extends the learning |
The teacher noticed his counting and pointed out the pens and paper, suggesting he might like to record his counting. He used his finger to count some rocks in a container he had been using on the scales. He got to 10, and then tried writing 10 lines on a piece of paper. This was not working as he wanted, so he changed his approach. He laid out the rocks in a line, and drew a line for each rock. He then counted the lines. The child had problem solved independently, and had used his number skills in ways that were meaningful to him. |
Some leaders and teachers advocate a 'child-centred' curriculum and consider that because mathematics is everywhere children will engage spontaneously with mathematical ideas at their own pace and that there is little need to deliberately plan, instigate or extend specific experiences into deeper learning.
This can be a result of limited teacher knowledge, a lack of ongoing professional learning, or simply that the prevailing teaching culture discourages deliberate teacher intervention in children's play. Teachers may have good subject knowledge in mathematics, but over-reliance on children to manage their own curriculum means that teachers feel constrained from using their knowledge and teaching practice to extend children's learning.
An absence of deliberate teaching means that opportunities for deeper engagement in or recognition of mathematical learning are missed.
Children have many and varied opportunities to explore the six strands of mathematics. Teachers provide lots of resources focused on mathematics, and use these and open-ended questioning to stimulate children's immediate interest and learning. They may be aware of home mathematical experiences, parents' expectations and aspirations, and incorporate these, particularly those relating to the child's cultural identity, into hands-on mathematical activities.
Nevertheless, despite the curriculum providing many opportunities to learn about mathematical concepts, children's engagement with these remains largely incidental and relies upon them taking an interest in the activities provided. Significant learning opportunities are frequently overlooked because mathematical experiences are often one-off events, rather than enabling children to explore concepts in more depth or take their new learning into other contexts.
Without a balance of deliberate teaching and spontaneous learning, a 'hands off' approach does not benefit children's learning. When teachers do not deliberately or intentionally extend children's interests and build on their learning over time, children are disadvantaged.In reflecting on how well mathematics is implemented as part of their curriculum, leaders should consider not only how effectively teachers notice, recognise, and respond to children's immediate and obvious interests, but also how they deliberately extend and scaffold children's deeper learning and understandings. Teachers have a critical role to inspire and engage children in mathematics, rather than leave the child to initiate engagement independently and at random.
Teachers document children's learning and progress in mathematics as part of their assessment practices. Rich assessment information shows the mathematical learning in the children's play and in conversations with their peers and teachers. In the best instances, assessment information can show the development of children's working theories about mathematical concepts. Assessment information includes next steps to build children's mathematical learning. Such assessment informs planning and highlights the child's achievement and progress to parents, whānau and the child.
The following examples show how two services recorded children's developing understanding of mathematics and used this documentation to extend their learning. The first example is of a child developing confidence with balancing and comparing weights. Carefully scaffolding his learning gave the child a sense of achievement and the confidence to teach his new skills to a friend.
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Curriculum and teaching practice |
The child had devised his own see-saw with wooden blocks, and was balancing different combinations of items at each end, commenting that heavier things went down and lighter things went up. He compared the weight of different types of objects too - one cow was heavier than two blocks; plastic lids were lighter than anything else. |
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Assessment |
The assessment information showed the teacher planned to extend this learning by helping him to make predictions about weight, then seeing if his predictions were accurate. |
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Teacher knowledge |
The next learning story has the boy working with his teacher and a set of balance scales. The teacher prompted him to hold objects in his hands and asked which he thought was lighter. He looked at the objects, and felt their weight, before replying "the blue block is lighter... and it's smaller”. He was then able to check his prediction by placing the objects in the cups on each end of the balance scale, and use his earlier observation of "heavier goes down, lighter goes up” to see which of all the objects was lightest. |
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Curriculum and teaching practice |
Later on, a teacher noticed that the boy was sharing some pule (samoan shell beads), dividing them up in different ways, and showing another boy how the balance scales work. He prompted the other boy to predict which was heavier, then showed him how to check. The boys discussed which they thought would be heavier, then put the shells in the cups on the balance scale. He explained to his friend "This one is heavier. It's down low, so it's the heavy one.” |
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Learning partnerships |
The boy's mum commented in his portfolio that he shared his learning with his family when he got home, explaining about heavy and light, and how heavy things went down and light things went up. He was very proud of his learning! |
The next example shows one child's mathematical progress in sorting and patterning and how she is working in increasingly complex ways. The assessment information documented the child's learning over almost a year.
| Curriculum and teaching practice | The first record shows the child sorting beads into groups by their colour. She sang a counting song in Maori as she did this. The teacher noticed the girl's interest in patterning, and planned to include more opportunities for sorting and patterning in future |
| Teacher knowledge |
A week later, the teacher noticed the child creating more complex groupings, by both colour and shape. The child arranged these groups all in a line, and stuck the pieces together with tape. The teacher extended the girl's understanding by exploring the environment with the child, looking for patterns. |
| Assessment |
The next assessment record shows the same child again grouping objects by their characteristics and explaining why each object belongs in the group: "these ones are all squares” and "these are the ones you can look through”. The objects in each group had both similar and different characteristics, but she had decided what feature she was basing her grouping on, and did this accurately. The teacher thought that the child was ready to try adding to a pre-existing pattern. The child sat with cards that had patterns printed on them, with an empty spot. She looked at the card carefully, and talked about the different features. She decided the pattern was by colour, so pointed to each pattern and named the colour. When she got to the empty spot, she immediately knew what colour she would need to put there, and went looking for a pattern of the right colour. A few months later, the girl was arranging driftwood, shells and stones into a pattern on the floor, when another child noticed what she was doing. She explained what she was doing, and asked the other child questions to guide her in creating her own pattern. She offered advice when needed, and commented on the other child's pattern. The complexity of her patterning has increased over time, and she understands the concept well, as shown by her willingness and ability to support others to create patterns of their own. |
Such informative assessments highlight the child's development of understandings and dispositions as well as their strengths and interests. This type of information can be shared with the child's primary school teacher to support their continuity of learning.
Children's portfolios also provide a useful source of information for parents to see their child's development, enabling them to support and extend their child's learning in authentic ways at home. Leaders and teachers share their knowledge and resources with parents and whanau, including explaining the links between Te Wharikiand The New Zealand Curriculum so that parents can better understand and support their child's learning.
Many services need to move from just having mathematics visible in children's assessment records, where it is not necessarily the focus of the assessment. The challenge for leaders in these services is to deepen teachers' assessments of children's strengths, capabilities and interests, and to highlight children's developing understanding of mathematics concepts in subsequent curriculum planning and assessment records.
This guide provides examples of deliberate teaching, assessment that underpins deeper learning, and teacher response to spontaneous learning. Despite evidence of good practice in many settings, ERO has found that mathematical teaching practice in the early years remains variable.
In considering how to improve mathematics in the curriculum, leaders and teachers in the early years should be aware of the following issues which have the potential to limit or disengage children's propensity for learning mathematics:
In some services, overly teacher-directed activities are undertaken in response to parental pressures to introduce formal programmes perceived to prepare children for school.
Such practices have good intentions but are misguided. In the worst cases, these activities are inappropriate and demotivating for children. They are not aligned to the objectives of Te Whāriki or The New Zealand Curriculum as they are not based on children's interests and inquiries, and therefore have the potential to lead to children developing negative attitudes to mathematics.
Leaders and teachers with the most deliberate and knowledgeable approach to mathematical learning for children avoid such misunderstandings about best practice. They do this by sharing their knowledge and resources with parents and whānau to support their child's mathematical learning in authentic ways. Assessment information, such as that documented in portfolios, empowers children, enhances their identities as learners, is a resource to connect knowledge at home with new learning, and fosters a sense of belonging and engagement.7 Teachers provide parents and whānau with useful information to share, during their early years and as children make their next steps to school. They work with local schools to facilitate collaborative relationships and sharing of responsibility, becoming familiar with both curriculum documents, and making comparisons and links to better support children's learning.
Teachers in the early years have the potential to implement a curriculum that will better support children's life-long mathematical learning.
This guide is intended to support leaders and teachers to undertake internal evaluation (self review) of their leadership and the curriculum, teaching practices and assessment processes that underpin mathematical learning in the early years.
As a leader, you can think about how you can encourage teachers to:
ERO has developed two resources that will help with your internal evaluation:
Leaders and teachers may like to use these prompts and indicators to guide an internal evaluation of mathematics in their curriculum.
Investigative prompts
Indicators
Investigative prompts
Indicators
Curriculum design and implementation
Investigative prompts
Indicators
Investigative prompts
Indicators
