# data[].attributes.attachment_link.title
Nursery Maths Block 1 including guidance

# data[].attributes.attachment_link.readable_text
Taking a look at how to help children with SEN engage with mathematics.
Teaching mathematics to children with SEN tends to concentrate on the teaching of basic skills, which are easy to teach and assess. Teachers often believe that these children cannot assimilate and retain information so there is a need for constant repetition and drills.

Research tells us, though, that children can solve problems and play with maths before they have mastered skills used to solve problems (National Council of Teachers of Mathematics, 2000). If they are given opportunities to do so, their conceptual understanding and ability to apply their knowledge is increased (Carpenter, Franke, Jacobs, Fennema and Empson, 1997).

Effective classroom practice and development of conceptual understanding is a combination of:

building on children's prior experience

the creation of practical opportunities for children to practice

allowing children to investigate and choose their best strategy

building a climate of collaboration and talk-based learning.

Children with SEN benefit from help searching for, finding, and using patterns in learning the basic number combinations and arithmetic strategies (Baroody, 1996).

Hands-on learning

Using manipulatives is a useful way for all children to play and discover mathematical relationships and explore concepts. A maths manipulative is a concrete or visual object that allows a child to investigate maths concepts using a hands-on approach. These objects can include blocks, shapes, cubes, money or counters. Increasingly, the use of technology is providing children with a variety of manipulatives, including virtual manipulatives - digital "forms" that resemble physical objects and can be manipulated, usually with a mouse. Examples of virtual manipulatives are computer games and the use of interactive whiteboards.

Using concrete materials can enable all children to think and reason in a meaningful way. Stein and Bovalino (2001) state that by providing manipulatives, teachers can create a positive experience for children by offering a concrete form from which children can then build their conceptual understanding.

Children who use concrete and virtual manipulatives demonstrate a greater mathematical understanding than those who use physical manipulatives alone (Olson, 1988). Manipulatives that are meaningful to the child, provide control and flexibility to the learner, have characteristics that mirror, or are consistent with cognitive and mathematical structures, and assist the learner in making connections between various pieces and types of knowledge, are the most successful. For example, virtual software can connect pictured objects, such as base ten blocks, to symbolic representations. It helps children generalise and connect their game playing or their abstract experiences with concrete materials.

Conceptual understanding

Children who have difficulty in mathematics need additional resources to support and consolidate the underlying concepts and skills being learned. They need multiple experiences with models and repetition of the relevance of models with abstract, numerical manipulations.

We need to give more time to mathematics, as the curriculum does not allow enough time for the many instructional and learning strategies necessary for the maths success of the child with SEN (Lerner, 1997).

We need to make a conscious effort to build on what children know how to do, relying on children's own strengths to address their deficits. Informal strategies provide a starting place for developing both concepts and procedures. Projects, games, collaborative and talk based activities can all help guide children's learning. We should use direct instruction only when children are unable to invent their own strategies.

We need to ensure maths is inclusive, practical and fun for everybody. By using a talk-led, active and collaborative approach, maths is within the reach of all our children.

Further information

Melody Lowe is Innovation Director and Lead Consultant at TT Education:
www.tteducation.co.uk

# data[].attributes.attachment_link.description
Taking a look at how to help children with SEN engage with mathematics.

# data[].attributes.headline
Reception Talk about Measure and Pattern

# data[].attributes.subject
Reception Talk about Measure and Pattern

# data[].attributes.body
Year 1 Place Value within 10

# data[].attributes.headline
Year 1 Addition and Subtraction within 10.

# data[].attributes.subject
Year 1 Addition and Subtraction within 10.

# data[].attributes.attachment_link.readable_text
There are many reasons why maths is a core part of the curriculum worldwide. It provides us with skills and knowledge that can be used in our daily lives. From the moment we wake up, we are constantly estimating, problem-solving and making quick judgements about quantities and amounts. For example, you may need to check you have the exact change for a bus or wonder if you can still fit into your trousers after several days of a festive celebration.
To help you think and plan maths experiences outdoors here are some practical suggestions: Getting ready to go outside provides many mathematical moments:

Time the class to get ready. This can be using a non-standard unit of measurement, such as a song for little children. With older children, this will be using a stopwatch or other timer.

Use lining up to reinforce key data handling skills. For example, request children make two lines, e.g. those who are wearing green, those who are not wearing green. This creates a human line graph and can be used for counting and discussing differences between the length of each line. Change the attributes each time you go out. Your children will have plenty of suggestions here.

Problem-solve with your class about ways of getting ready quickly and without fuss. Link these to the strategies used to solve problems, so children can see how a skill learned has real life applications.

Maths on the move. Make the most of the distance between your class and your outdoor space:

Estimate the number of steps it takes to get outside. Discuss afterwards why everyone has a different answer. Is it possible to standardise this distance and how would we do this?

Count aloud and chant in multiples, e.g. multiples of three on each step: 3, 6, 9, etc.

What happens to your counting when you take five steps forward and one step back? Consider how to create links between numbers and the pattern of walking forwards and backwards.

Creating a gathering circle in mathematical ways
Explore the size of the circle made when children hold hands, stretch out and touch each other's fingertips or huddle together shoulder-to-shoulder. Discuss and explore how the size could be measured. This may include:

Pacing around the outside of the circle as a non-standard approach.

Using a trundle wheel for noting metres or yards.

Using a long piece or rope or string. If you put a mark at every metre or yard on the rope then it becomes a giant measuring tape.

Estimating everything
Children need lots of practice at estimating so they are able to make reasonable guesses based upon experience and knowledge. It is a basic strategy for problem solving and enquiry work as well as a useful life skill. Being outside provides a real context for estimating. It is hard to tell the number of birds in a flock, bricks in a wall or exactly how long it will take to walk to the shops. There is a constant need for everyone to be making estimates of amounts and activities based upon our experiences. Teachers can encourage the children estimate and then to check:

Number: having a guess before counting the flock of birds flying overhead - we count ten birds and then use this to count the rest in chunks of ten.

Money: evaluating whether we have enough money to buy something we need.

Distance: estimating how far away the end of
the playing field is.

Volume: thinking about the volume of water in one bucket or watering can compared to another.

Weight and mass: wondering how much food the birds will eat at a bird table.

Time: considering how long it will take to complete a task.

It can help to make group estimates where there is a consensus. With older children, the skill of rounding up or down is a natural progression within estimation.
Playing maths games
All around the world there are strategy games, which were developed using locally found materials on a board that can be drawn onto an outdoor surface. Games involve looking for patterns and knowing the cause and effect of moves undertaken in particular sequences. This usually involves playing the game lots of times and experimenting with different moves. Some basic points include:

Children need time to learn each game by just enjoying the experience of playing it. Older children can assist younger ones. Hold a games session so that parents and carers can learn different games too.

If a game isn't going well, ask the children for their ideas about making it better. What rules could be adapted or changed? How can they make the game more exciting?

Games can be adapted to help the children acquire specific skills in many areas of maths. When you do this, it can be helpful to seek the children's thoughts and suggestions. This gives them ownership of their learning and facilitates a personal interest.

Children enjoy inventing their own games. Whether you have a pile of stones or a few leaves lying under a tree, challenge them to create a game to help them learn a specific maths concept or skill.

By Juliet Robertson, fo under of Creative STAR Learning, UK

Many of these ideas are expanded upon in her book: Messy Maths: An Outdoor and Playful Approach for Early Years. There are also more than 750 outdoor ideas and suggestions on her blog, I'm a Teacher Get Me OUTSIDE Here! www.creativestarlearning.co.uk/blog

If you would like to participate in Outdoor Classroom Day, register your class or whole school, and share your day's activities on social media using the #OutdoorClassroomDay hashtag.

# data[].attributes.attachment_link.description
There are many reasons why maths is a core part of the curriculum worldwide. It provides us with skills and knowledge that can be used in our daily lives. From the moment we wake up, we are constantly estimating, problem-solving and making quick judgements about quantities and amounts.

# data[].attributes.attachment_link.readable_text
Skip to main content

Cookies on GOV.UK

We use some essential cookies to make this website work.

We'd like to set additional cookies to understand how you use GOV.UK, remember your settings and improve government services.

We also use cookies set by other sites to help us deliver content from their services.

You have accepted additional cookies. You can change your cookie settings at any time.

You have rejected additional cookies. You can change your cookie settings at any time.

Accept additional cookies Reject additional cookies View cookies

Non-statutory curriculum guidance for the early years foundation stage.

Applies to England

Documents

Details

Development Matters offers a top-level view of how children develop and learn for all early years practitioners, including:

childminders

staff in nurseries

nursery schools

school nursery and reception classes in school

The guidance can help meet the requirements of the early years foundation stage (EYFS) statutory framework.

Thank you for your feedback

Report a problem with this page

# data[].attributes.attachment_link.description
Non-statutory curriculum guidance for the early years foundation stage.

# data[].attributes.attachment_link.readable_text
Will the premium resources, workbooks and videos also be updated?

Yes! All supporting material will also be updated in line with our new schemes of learning. These, along with the schemes themselves, will be released on a rolling basis across the next academic year.

Are you producing new assessments?

Yes, we are also writing new assessments, both end of block and end of term, to work alongside the new version of the schemes. Like the above, both versions will be available depending on which route you choose to follow.

When will the new schemes be available?

The autumn term content will begin to become available from June 2022 in preparation for schools starting with them in September. You can download some sample material here.

# data[].attributes.attachment_link.title
White Rose Maths - New schemes for 2022/23

# data[].attributes.attachment_link.description
Yes! All supporting material will also be updated in line with our new schemes of learning. These, along with the schemes themselves, will be released on a rolling basis across the next academic year. Yes, we are also writing new assessments, both end of block and end of term, to work alongside the new version of the schemes.

# data[].attributes.attachment_link.readable_text
We use cookies to help provide a better website experience for you, and help us to understand how people use our website. Our partners will also collect data and use cookies for ad personalisation and measurement.

Clicking "Accept" will allow us and our partners to use cookies, learn more in our cookie policy or to change your cookie preferences, click "Manage".

# data[].attributes.attachment_link.title
Maths resources for teachers | White Rose Maths

# data[].attributes.attachment_link.description
We use cookies to help provide a better website experience for you, and help us to understand how people use our website. Our partners will also collect data and use cookies for ad personalisation and measurement. Clicking "Accept" will allow us and our partners to use cookies, learn more in our cookie policy or to change your cookie preferences, click "Manage".

# data[].attributes.attachment_link.provider_name
Creative STAR Learning | I'm a teacher, get me OUTSIDE here!

# data[].attributes.attachment_link.readable_text
There's about 90 blog posts with ideas and suggestions to support you to develop your maths provision outside, be this in the early years or primary school. Most are open-ended and can be easily adjusted to the level at which you work.

Note: for some visitors to this website, the photos in each blog post take an age to upload. They will appear. Be patient.

Maths Week Scotland webinar

25 Outdoor Maths Games for Everyone. A range of ideas and games for making maths accessible to all. Below is a brief video with some top tips for getting outside and enjoying maths in a more expansive space.

The Freebies

Useful posts to get you started teaching maths outside

Outdoor number explorations

Outdoor measurement

Outdoor patterns and relationships

Properties of 2D shapes and 3D objects - outside

Outdoor angles, symmetry and transformations

Handling data outside

Daisy footprints. Does the size of your footprint or the place where your boot lands that will flatten the most daisies? Useful open-ended investigation for a range of ages.

Information or data handling outdoors. Exploring a range of possibilities with a group of teachers.

Resource-based outdoor maths explorations

Environmental maths explorations

Broader explorations of maths

Sketchnoting Capturing maths and other educational thoughts. Examples that include maths concepts. I also advocate teachers showing children how to sketch note so they can do their maths journals using this technique.

Can I go on the computer, Mum? A parent's experience of playing a game and giving her son the opportunity to create a game and change the rules. The outcome was much more positive.

Making environmental print meaningful at Stramash Fort William How one outdoor nursery created signs that were personalised and relevant to each child at nursery. It's about using non-standard measurements to measure distance.

Amazing mazes. A wonderful exploration and creation of mazes by a group of P6 children with additional support needs. Thank goodness for brain diversity and creativity.

STEM explorations

Maths celebrations and events

If you are looking for more maths resources and ideas

# data[].attributes.attachment_link.description
You will have fan-ta-stick time exploring maths outside! There's about 90 blog posts with ideas and suggestions to support you to develop your maths provision outside, be this in the early ye...

# data[].attributes.attachment_link.title
National curriculum in England: mathematics programmes of study

# data[].attributes.attachment_link.description
The statutory programmes of study and attainment targets for mathematics at key stages 1 to 4.

# data[].attributes.attachment_link.readable_text
search

Enriching mathematics for all learners

A free-to-use collection of mathematics activities, lessons and problems designed to nurture curious, resourceful and confident learners of mathematics. Produced by The University of Cambridge.

# data[].attributes.attachment_link.description
The Nrich Maths Project Cambridge,England. Mathematics resources for children,parents and teachers to enrich learning. Problems,children's solutions,interactivities,games,articles.

# data[].attributes.attachment_link.readable_text
This Year 4 Analogue to Digital - 24 Hour lesson covers the prior learning of telling the time on a 24-hour digital clock, before moving onto the main skill of converting between analogue and digital times.

The lesson starts with a prior learning worksheet to check pupils' understanding. If they need more practice, you can choose other resources for that step. The interactive lesson slides recap the prior learning before moving on to the main skill. Children can then practise further by completing the activities and can extend their learning by completing an engaging extension task.

National Curriculum Objectives
Mathematics Year 4: (4M4b) Read, write and convert time between analogue and digital 12- and 24-hour clocks
Mathematics Year 4: (4M4c) Solve problems involving converting from hours to minutes; minutes to seconds; years to months; weeks to days

Get the most from lessons!

# data[].attributes.attachment_link.title
Year 4 Analogue to Digital - 24 Hour Lesson - Classroom Secrets

# data[].attributes.attachment_link.description
This Year 4 Analogue to Digital - 24 Hour lesson covers the prior learning of telling the time on a 24-hour digital clock, before moving onto the main skill of converting between analogue and digital times. The lesson starts with a prior learning worksheet to check pupils' understanding.

# data[].attributes.attachment_link.readable_text
Resonating with the work of early years pioneers such as Rousseau and Froebel - as well as Montessori - there has been a recent resurgence of interest in the UK in the potential of the outdoor environment for supporting children's learning.

One of the central tenets of the Montessori philosophy is the connection with the natural world of the outdoors (Montessori, 1946). Children develop a curiosity about and a love for nature through having the daily opportunity to experience the outdoors with structure and purpose. They will use their hands in everyday endeavours focused on the earth and the outdoor environment around them, enabling them to be close to nature.

The outdoor favourable environment is as important as the indoor favourable environment, both working in unison as natural extensions of each other. To support the learning and development of the children effortlessly, we only need to look to see the opportunities the outdoors provides, both with resources supplied by the directress and resources supplied by nature. This environment brings with it nature's maths: the beauty of shape, symmetry, colour, classification, order, patterning and quantity.

This directly supports Maria Montessori's description of the mathematical mind: the impulse to produce order out of disorder. Dr Montessori borrowed this phrase from the works of Blaise Pascal, a French philosopher and mathematician, noting that he "...said that man's mind was mathematical by nature, and that knowledge and progress came from accurate observation." (Montessori, 1988, p169). Montessori described it as a mind that has the spontaneous ability to organize, classify and quantify patterns and relationships within the context of daily experiences. She recognized that the characteristics of the mathematical mind, order - observation, precision and imagination - can be sharpened with careful nurturing, and nature provides us with the means to do this.

The Mathematical Mind is active in a young child from the very beginning: "in our tiny children the evidence of a mathematical bent shows itself in many striking and spontaneous ways" (Montessori, 1988, p169). The experiences we provide supporting the Mathematical Mind in the first plane of development helps to build the foundations on which all later learning will be built as the child's development proceeds. We are curious creatures: curiosity drives us to exploration of all types and to understand the world around us in all its aspects. this stimulates essential questions such as why, how, when and what, and this is the curiosity we need to embrace as this is the foundation for creative mathematical learning. Such questioning fuels scientific inquiry, the continuing quest for new, more accurate, or more complete knowledge.

Experiences within wild spaces are also vital to effective environmental education. When we talk about wild spaces, we are not always talking of forests and wood areas - consider letting part of your garden grow wild, replace some of your paved area with raised wildflower beds, give nature the freedom to create the outdoor environment for you. What you end up with may not be tidy but will be a great learning resource. The increasing interest in environmental issues has raised the profile of children's use of the outdoor space in terms of the potential for them to develop positive and caring attitudes to the wider environment (Rivkin, 2000). This rich, sensory, natural environment not only supports children's own investigations (Fjortoft 2004; Waite et al., 2006) but also provides an ideal context for group activities in which the development of knowledge, concepts and skills from across the curriculum are embedded within authentic, purposeful and often real‐life tasks. All of us in our settings will have the basis of Montessori philosophy embedded in our practice, but it is important to also remember that knowledge of children's development has been increasing over the years. We must use this knowledge to inform our outdoor practice as we support the children on a daily basis in their mathematical development.

Pratt (2011) argues that in order for children to become effective mathematicians they should develop a 'mathematical disposition'. This, he suggests, can be supported by the doing of maths outdoors, enabling children to develop an awareness that maths is all around them, allowing them to make connections in their learning and experiences of natural materials in situ. Knight (2011) suggests that children will become more creative outdoors where they have freedom to be "innovative, flexible and adaptable" and that this environment will stimulate creative mathematical thinking. When using the outdoors environment the children can construct on a larger scale, explore the world first hand and experience natural phenomena such as weather, the changing seasons and shadows (Maynard and Waters, 2007). Children naturally use mathematical thinking and learn mathematical skills as part of their outdoor experiences. They will count, measure, explore shapes and develop mathematical ideas through their imagination and creative play. They will extend their learning from indoors using nature's bounty. We should recognise the mathematical potential of the outdoors environment for children to discover things about shape, distance and measures through their physical activity - just look outside for a wealth of 'mathematical' resources that children can really get inspired by.

A quick guide to doing maths outdoors

The first tenet in developing children's competencies is Dewey's "children learn from doing" theory (Mooney, 2000). In the outdoor learning environment children are active and interactive participants in their learning processes. Dewey thought that rather than saying "the children will enjoy this", teachers need to ask the following questions when they plan activities for children:

How does this expand on what these children already know?

How will this activity help this child grow?

What skills are being developed?

How will this activity help these children know more about their world?

How does this activity prepare these children to live more fully?

Exploring mathematics outdoors is beneficial as it will:

offer a sense of freedom without the constraints of the indoor environment

encourage a mathematical disposition and enable children to make useful links in their learning

Be memorable in that it offers mathematical pursuits in a context not normally associated with mathematical learning

encourage exploration and risk taking

Support emotional wellbeing

Contribute to children's self-image of themselves as mathematicians

Conclusion

Maria Montessori recognised the scope for learning provided by nature - by using your vision you can provide the same opportunities nature does in the setting, even in the inner city.

When you look closely, the everyday living world is intriguing and magical, and full of awe and wonder: think of the excitement when a child finds their first ladybird, how many times as a child did you count the dots on the ladybirds back? Young children feel this need for exploration, discovery and creative learning strongly and we will have done our job if we can help them to retain this throughout their lives.

"The future will belong to the nature-smart - those individuals, families, businesses, and political leaders who develop a deeper understanding of the transformative power of the natural world and who balance the virtual with the real. The more high-tech we become, the more nature we need."

Richard Louv

A few ideas to get you started with a group of children outside

Build a den and then talk together about the shape and size of a shelter that could be built; how wide, tall, or long do we want to make it? How many people do we want to fit inside? Children can make suggestions for the size and use estimation. What size sticks do we need to collect? How many sticks will we need?

Create an obstacle course. Experience climbing up, around, on, under and over, and talk about size, height, and shape of obstacles. How many children can sit on them?

What patterns can be seen in the clouds? Are they changing? How are they changing? How fast or slow are the clouds moving?

Look at the shadows outside, and draw around the shadow made by one of the children. Measure the shadow outline and the child, and discuss the difference; repeat this in the morning, at midday and in the afternoon, and discuss these differences.

Wild area

Games with grasses: how tall, how many different types, counting

Games with flowers and sticks: counting, number of petals, different patterns

Creating shapes and trails by walking through the long grass

Make a collection of natural objects, and then count them

How many are there? (maybe in a spindle box kind of way)

Sort them using their size and shape

Digging in the Sand

How deep is the hole? (Depth)

Measure using a ruler

How many more spades to dig to the bottom of the sandpit? (estimating)

Can you shovel fast with a small spade or a big spade?(time)

Hoist

How high is the bucket?

Lower the bucket.

How much sand have you got in your bucket?

Is it full or empty?

Is the bucket heavy or light?

Obstacle Course

Directional language (over, under or through)

Positional language (forwards or backwards)

Time taken to complete (stop-watch, sand timer)

Sand Timer

Timing

Turn-taking

How does the sand get through the small hole?

Different measures of time 2, 10, and fifteen minutes

time passing fast and slow

Moving Blocks

How many blocks fit on a trolley? (counting)

Is it heavy to push?

Sand Moving

How many more barrows to empty the sack? (estimating)

Is the barrow heavy or light to push?

What makes it easier to move? (problem solving)

Gardening

Are there enough spades for every one? (counting)

Planting seeds (counting seeds, planting in a row)

leaf shapes

Growing (size, shape)

How much water do we need? (estimating)

Have they grown and by how much? (measuring)

Guttering and marbles

How far do the marbles roll?

What happens if we change direction (prediction and experimentation)

What helps them to roll (problem solving)

Directional language

Leaf Sweeping

How many leaves are in your barrow?

How many leaves on your spade? (counting)

Are the leaves heavy or light?

Logs

How many logs can you build up?

Can you make them balance?

Do you need to pull slowly so the logs don't topple off?

What happens if you build too high? (prediction and experimentation)

Bringing the inside out

Children need the freedom to work with the material where they like

Bringing the environments together to support each other

Bring the physical and the academic together

Resources

Fjortoft, I. (2004) Landscape as playscape: the effects of natural environments on children's play and motor development, Children, Youth and Environments, 14(2), 21-44.

Malone, K. & Tranter, P. (2003) School grounds as sites for learning: making the most of environmental opportunities, Environmental Education Research, 9(3), 283-303.

Montessori, M. (1988). The Absorbent Mind, Clio Press

Standing, E. M. Maria Montessori: Her Life and Her Work, Penguin, 344

Tucker, K. (2014) Mathematics through play in the Early Years, Sage, London

Waite, S., Davies, B. & Brown, K. (2006) Five stories of outdoor learning from settings for 2-11 year olds in Devon (Plymouth, University of Plymouth).
This article was first published in Montessori International autumn 2015.

# data[].attributes.attachment_link.description
Resonating with the work of early years pioneers such as Rousseau and Froebel - as well as Montessori - there has been a recent resurgence of interest in the UK in the potential of the outdoor environment for supporting children's learning.

# data[].attributes.attachment_link.readable_text
​Whether you are an experienced teacher, a teacher trainee, a teacher educator, a parent, a researcher or a policy maker, we hope you will find this website interesting and useful. Specifically, we want to share with you the benefits of te﻿aching and learning mathematics through stories and creative writing, as well as to offer practical ideas for teachers and parents on how to embrace this approach.

This website has been designed and continuously developed based on the feedback we receive from the key stakeholders as noted above. There are plenty for you to do on this website, including, but not limited to:
Please take your time to explore the website using either the menu bar at the top of the page or from the menu below.

Finally, consider s ubscribing to our free quarterly newsletters, and please help spread the word about our project and its website :-)

Best wishes,

Dr. Natthapoj Vincent Trakulphadetkrai B.A. (Hons), M.Sc. (Oxford), M.Sc. (London, UCL), Ph.D. (Cambridge), QTS

@NatthapojVinceT Natthapoj.org

Associate Professor of Mathematics Education
​University of Reading (UK)

Founder & Principal Investigator
​MathsThroughStories.org
​

# data[].attributes.attachment_link.readable_text
Deborah Lambert provides flexible and creative outdoor CPD for teachers and schools, working with the DfE Teachers' Professional Development Standards to promote effective teaching and learning for all curricular subjects.

This includes in-school face-to-face staff training, supported and resourced self-delivery CPD training packages with

structured, collaborative activities to refine ideas, providing the skills, knowledge and confidence to deliver the curriculum outdoors in meaningful and memorable ways.

# data[].attributes.attachment_link.description
Deborah Lambert provides flexible and creative outdoor CPD for teachers and schools, working with the DfE Teachers' Professional Development Standards to promote effective teaching and learning for all curricular subjects. This includes in-school face-to-face staff training, supported and resourced self-delivery CPD training packages with structured, collaborative activities to refine ideas, providing the skills, knowledge and confidence to deliver the curriculum outdoors in meaningful and memorable ways.

# data[].attributes.attachment_link.readable_text
Introduction

Mathematics, a universal language that enables understanding of the world, is an integral part of the curriculum. Beyond the study of numbers, shapes and patterns, it also provides important tools for work in fields such as engineering, physics, architecture, medicine and business. It nurtures the development of a logical and methodical mindset, as well helping to inculcate focus and the ability to solve all manner of problems. Attainment in the subject is also the key to opening new doors to further study and employment. However, despite its importance, for many the subject remains mysterious and difficult, the preserve of those who seem to be 'naturals'. The education inspection framework ( EIF) makes it clear that schools are expected to ensure that the mathematics curriculum 'helps pupils to gain enjoyment through a growing self-confidence in their ability'.

This review explores the literature relating to the field of maths education. Its purpose is to identify factors that can contribute to high-quality school maths curriculums, assessment, pedagogy and systems. We will use this understanding of subject quality to examine how maths is taught in England's schools from Reception onwards. We will then publish a subject report to share what we have learned.

The purpose of this research review and the intended audience is outlined more fully in the 'Principles behind Ofsted's research reviews and subject reports'.

Since there are a variety of ways that schools can construct and teach a high-quality maths curriculum, it is important to recognise that there is no singular way of achieving high-quality maths education.

In this review, we have:

outlined the national context in relation to maths

summarised our review of research into factors that can affect quality of education in maths

considered curriculum progression in maths, pedagogy, assessment and the impact of school leaders' decisions on provision

The review draws on a range of sources, including our 'Education inspection framework: overview of research' and our 3 phases of curriculum research.

We hope that through this work, we will contribute to raising the quality of maths education for all young people.

Ambition for all

Summary

This review identifies that, despite English pupils achieving, on average, higher attainment than pupils in many other countries, the attainment gap between low and high achievers in England is wide. Therefore, in addition to shining a light on approaches that could raise the attainment of all pupils still further, a core theme of this review is how we might prevent struggling pupils from falling further behind their peers.

Context

revisions to A-level mathematics specifications

redesigning of GCSE mathematics specifications to increase their rigour and challenge

establishing the National Centre for Excellence in the Teaching of Mathematics

setting higher targets for teacher recruitment and creating professional development programmes for teachers

There is still more that could be done to enhance mathematics education, such as reducing the shortage of specialist mathematics teachers. Additionally, the gap between the lowest and highest achievers in England is wider than the Organisation for Economic Co-operation and Development ( OECD) average. The average attainment gap between disadvantaged and advantaged pupils is also wide. Disadvantaged pupils in England are much less likely than their advantaged peers to achieve a grade 4 at GCSE[footnote 10] or to meet the expected standards at key stages 1 and 2 or at the end of the early years foundation stage ( EYFS).[footnote 11] The factors that explain the variation in quality of mathematics education in England are therefore likely to contribute to a long tail of underachievement and a wide spread of attainment level, as well as to overall success.

Recent influences in mathematics education include:

'Mastery' pedagogical approaches that have influenced English mathematics education tend to require pupils to demonstrate high levels of achievement before they are moved on to new content. Some mastery approaches place a greater emphasis on problem-solving and on deepening pupils' understanding.[footnote 15]

It is also important to consider that high attainment and proficiency of older pupils may be due to historical curricular and pedagogical approaches, rather than the educational approaches of that time. Finland is a good example of an education system where success in the OECD's Programme for International Student Assessment ( PISA) is thought to be the result of historical approaches.[footnote 18] Approaches that teachers and pupils are familiar with can take time to change. They may, therefore, influence pupils' educational experiences, even after an official change of curriculum or pedagogical approach.[footnote 19]

Principles underpinning the review process

This research has been informed by the evidence and principles underpinning the EIF, which include:

scope, content and sequencing of the curriculum

specification and ordering of component parts that make up composite skills

the value of teachers' subject knowledge

promotion of a range of quality interactions with pupils

quality and pacing of instruction

how to avoid overloading working memory

These have become a useful lens through which to scrutinise subject-specific themes.

Mathematics research tends to use a wide variety of overlapping terms. 'Fluency' is a good example. It has multiple meanings in literature:

How the review classifies mathematics curriculum content

For this review, we have classified mathematical curriculum content into declarative, procedural and conditional knowledge.

Declarative knowledge is static in nature and consists of facts, formulae, concepts, principles and rules.

All content in this category can be prefaced with the sentence stem 'I know that'.

Procedural knowledge is recalled as a sequence of steps. The category includes methods, algorithms and procedures: everything from long division, ways of setting out calculations in workbooks to the familiar step-by-step approaches to solving quadratic equations.

All content in this category can be prefaced by the sentence stem 'I know how'.

Conditional knowledge gives pupils the ability to reason and solve problems. Useful combinations of declarative and procedural knowledge are transformed into strategies when pupils learn to match the problem types that they can be used for.

All content in this category can be prefaced by the sentence stem 'I know when'.

When pupils learn and use declarative, procedural and conditional knowledge, their knowledge of relationships between concepts develops over time.[footnote 25] This knowledge is classified within the 'type 2' sub-category of content (see table below). For example, recognition of the deep mathematical structures of problems and their connection to core strategies is the type 2 form of conditional knowledge.

Summary table of content categories considered in the review:

Declarative
'I know that'

Facts and formulae

Relationship between facts (conceptual understanding)

Procedural
'I know how'

Methods

Relationship between facts, procedures and missing facts (principles/mechanisms)

Conditional
'I know when'

Strategies

Relationship between information, strategies and missing information (reasoning)

Curriculum progression: the planned and purposeful journey to expertise

Summary

The evidence presented here supports careful consideration of sequencing and content that makes a mathematics curriculum a guarantee of long-term learning. Useful facts and efficient and accurate methods are ideally paired within a topic sequence. Strategies for solving problem types are then best taught and learned once pupils can recall and deploy facts and methods with speed and accuracy. When planning curriculum content, teachers also need to prioritise 'forward-facing' knowledge. This goes beyond important facts of number. It includes the mathematical methods that pupils will take with them on their journey. The ideal aim is for pupils to attain proficiency, not just collective moments of understanding, familiarity or experience. This will help pupils to develop motivation in the subject.

Selecting and sequencing core declarative, procedural and conditional knowledge

Problem-solving requires pupils to hold a line of thought. It is not easy to learn, rehearse or experience if the facts and methods that form part of a strategy for solving a problem type are unfamiliar and take up too much working memory. For example, pupils are unlikely to be able to solve an area word problem that requires them to multiply 2 lengths with different units of measurement if they do not realise that the question asks them to use a strategy to find an area. They are also unlikely to be successful if they do not know many number bonds, unit measurement facts, conversion formula or an efficient method of multiplication to automaticity. Therefore, the initial focus of any sequence of learning should be that pupils are familiar with the facts and methods that will form the strategies taught and applied later in the topic sequence.[footnote 29]

The relationship between core mathematical facts and powerful methods

Linked declarative and procedural knowledge are ideally sequenced together to reflect the reciprocal learning relationship between them. This is because:

As a simple example, a pupil can better understand connections of number and the concepts of addition and quantity if they have declarative knowledge of number bonds and procedural knowledge of column addition, which both reinforce each other. In terms of curriculum sequencing, pupils are able to retain knowledge and ability to use core methods when teachers take an iterative approach to teaching and rehearsing concepts and core methods.[footnote 30]

The long-term impact of an early and thorough emphasis on core content

Acquiring new foundational knowledge takes time and effort. However, the rewards go beyond the immediate benefits of being able to recall and apply useful facts and methods.

Furthermore, if this core content has been sequenced well and pupils have learned it thoroughly, they are less likely to forget and are therefore unlikely to need to 're-learn' it later.[footnote 34] A focus on core knowledge in younger year groups can be achieved by focusing on depth over breadth, covering fewer core topics but in more detail.

Amplifying the curriculum through instruction, rehearsal and assessment plans

Successful curriculums illustrate the importance of detail, sequencing and alignment of content, instruction, rehearsal, assessment and mechanisms to continually upgrade.[footnote 35]

The approach outlined above is very different to a curricular offer that does not feature systems for documenting quality sequences of instruction and rehearsal and that may result in more variable rates of learning and outcomes. For example, younger pupils may achieve proficiency through more informal opportunities to learn and where teachers respond to their interests, but leaders should note that disadvantaged novice mathematicians benefit from proactive approaches that can be as simple as ensuring that they are given dedicated time to learn and rehearse mathematics every day.[footnote 39]

A positive attitude towards mathematics is the outcome of success in the subject

Curriculum sequencing: declarative knowledge

Maths facts, vocabulary and symbols at the start of the school journey

Studies indicate that this early acquisition of knowledge significantly predicts later success.[footnote 51] They also suggest that much of this success is reliant on pupils knowing the code for number (for example, the Arabic numerals), rather than relying on a general sense of quantity.[footnote 52] Early lack of knowledge also predicts later struggle[footnote 53] and a later diagnosis of disability.[footnote 54] Pupils who are not able to quickly and easily recall maths facts struggle with calculations due to their working memory being overloaded.[footnote 55] For example, a child who does not know number bonds will be stuck using various forms of 'counting on' when performing simple addition. Even at key stage 3, a pupil's lack of knowledge of the basics in number will have a detrimental effect on their learning of algebra.[footnote 56] Taken together, this information points to the prioritising core declarative knowledge in mathematics from an early age to level the playing field, particularly for pupils with special educational needs.[footnote 57]

Maths facts, vocabulary and symbols at the start of a sequence

Based on the above, high-quality maths education may have the following features

Teachers engineer the best possible start for pupils by closing the school-entry gap in knowledge of the early mathematical code: facts, concepts, vocabulary and symbols.

Pupils are taught core facts, formulae and concepts that are useful now and in the next stage of education.

Teachers help pupils develop their automatic recall of core declarative knowledge, rather than rely on derivation, guesswork or casting around for clues.

Curriculum sequencing: procedural knowledge

Planned obsolescence of early methods

Ideally, pupils gradually cease to depend on some methods of counting and calculating, and associated resources, that they were taught earlier on. This is because reliance on some early counting and calculation methods, in the absence of learning valuable number facts, can hinder later progress.[footnote 71]

Pupils can be helped with simple everyday objects and semi-concrete representations, such as Numicon, but the aim should be that pupils move to working with symbols and abstract representations.[footnote 72] The use of manipulatives, for example, does not always guarantee that a pupil will understand[footnote 73] and their use may distract pupils from thinking about content to be learned.[footnote 74] A method of calculation that relies on derivation may be useful in the short term and as a bridge to formal methods of written calculation that require pupils to accurately recall number bonds.[footnote 75] In the absence of learning this core knowledge, pupils may rely too much on estimation and looking around for clues, or they may develop the habits of guessing and copying.[footnote 76]

Methods for more complex measurements and calculations

Additional risks arise from mixing and matching a toolkit of informal and self-generated methods for working with larger numbers and more complex calculations as pupils progress through the curriculum. This increases the likelihood of pupils generating errors and structuring written records poorly, which may lead to confusion.[footnote 84]

Methods for working algebraically

The message of quality over quantity of procedural knowledge also applies throughout key stages 3 and 4.

If pupils learn high-quality, useful and efficient procedural knowledge, they can then apply this to setting out and using formulae, from calculating areas and perimeters of different classes of polygon in key stage 2, to using the trigonometric formulae such as the cosine rule in key stage 4.

Curriculum sequencing: conditional knowledge

The importance of a curricular approach

This has implications for how problem-solving as an activity is implemented in classrooms where teachers expect pupils to learn how to problem-solve by problem-solving.

Teachers could use a curricular approach that better engineers success in problem-solving by teaching:

Pupils need to be fluent with the relevant facts and methods before being expected to learn how to apply them to problem-solving conditions.[footnote 94]

Giving younger pupils the ability to understand word problems

In the primary phases, pupils' experience of problem-solving often involves solving word problems. The first barrier to overcome is language. Pupils therefore need to be proficient readers at the required level.[footnote 95]

However, even when pupils can read and understand the context, they need to look beyond the surface features of the problem towards the deep structure that signals the strategy to be used. Some pupils who are quick to learn new procedural knowledge, such as how to find equivalent fractions, might be given linked problem-solving by way of differentiation. These pupils are then able to intuit the connections between recently learned methods and certain types of (word) problems. When they learn the conditions for a method's wider use, this builds a bank of strategies over time. They are then able to use that bank of strategies to classify and solve varied problems presented in test papers. However, pupils who have not had this experience will struggle to make the right connections when presented with problem-solving and reasoning test papers.

Teachers should therefore ensure that more pupils experience success in solving word problems, by sequencing the teaching of strategies to 'convert' the deep structure of word problems into simple equations.[footnote 96]

Strategies for solving classes of problem

This contrasts with the view of problem-solving as a generic skill that pupils can transfer to multiple topics and sub-domains.[footnote 99] Applying generic strategies to find examples, look for relationships or weigh up features could yield accurate and inaccurate information. However, pupils who are struggling might not know which information to choose to use.

Curriculum planning should seek to identify and sequence the most useful combinations of facts and methods for solving sub-classes of problems, as well as the features of conditions that these strategies would be useful for.

Based on the above, high-quality maths education may have the following features

Teachers teach useful, topic-specific strategies to pupils, as well as how to match them to types of problem.

Pupils are confident using linked facts and methods that are the building blocks of strategies, before strategies are taught.

Teachers encourage pupils to use core, systematic strategies rather than resorting to guesswork or unstructured trial and error.

Curriculum sequencing: meeting pupils' needs

Planning for what pupils will be thinking about

Close examination of lesson planning and teachers' thoughts about lesson planning in education systems where pupils do well reveal an intense focus on underlying knowledge structures and connections rather than the surface coherence of activities and teaching. This means that teachers are planning for what pupils will be thinking about or with, not what they will be 'doing'.

Further, the option of problem-solving as part of task differentiation does not guarantee that all pupils will learn problem-solving strategies. Leaders and teachers should ideally view learning of all core content, including the links between content, as an entitlement and therefore a pre-planned pathway for all pupils.

Balancing new learning and rehearsal of learning

A moment of understanding does not guarantee long-term learning. Pupils benefit from studying worked examples in addition to practising solving similar types of problems.[footnote 106]

Therefore, teachers need to balance introducing new content with pupils' need to spend time revisiting content.[footnote 107] There should be space within the curriculum for planned consolidation. Pupils should not be rushed through content.

This is easier if the mathematics curriculum focuses on core content early and leaders prioritise and value consolidation. Minimising off-task behaviour may also help to maximise the amount of time available for retrieval, rehearsal and consolidation of learning. Pupils who do well tend to have spent more time on the subject.[footnote 108]

Equity

Teachers and leaders should try to strike a balance between curricular approaches that enable pupils to keep up with their peers and reactive approaches that identify, help and support pupils after they have fallen behind. These reactive approaches are more likely to rely on assessment, diagnoses, personalisation and interventions.

In the English mathematics education system, emphases on reactive approaches are associated with a wide attainment spread and a long tail of under-achievement. Almost 180,000 students had to re-sit GCSE mathematics in 2019. Of these, only 22.3% achieved a standard pass (grade 4) or above.[footnote 109]

In East Asian classrooms, there appears to be little differentiation.[footnote 110] It might be assumed that this is the result of a pedagogical decision to keep pupils learning and doing the same thing. Teachers may worry that high attainers are being held back or that pupils with special educational needs and/or disabilities ( SEND) are not being given enough support. However, in countries like Singapore, all groups of pupils do well. Fifty-one per cent of Singaporean pupils met the advanced international benchmark versus just 11% of English pupils. For the intermediate benchmark, described as the ability to 'apply basic mathematical knowledge in a variety of situations', only 8% of Singaporean children did not meet this standard, compared with 31% of English children.[footnote 111] The reason for this success is because a powerful curriculum and plenty of opportunities to engage in purposeful, intelligent practice lead to better outcomes for pupils.[footnote 112]

Leaders could consider this strategy as a way to promote proficiency in the subject, where pupils stay together not because higher attainers are being held back, but because lower attainers can 'keep up'.

Inclusivity

Pupils with SEND benefit hugely from explicit, systematic instruction and systematic rehearsal of declarative and procedural knowledge.[footnote 113] The benefits of these approaches extend beyond enhanced academic attainment and proficiency. The relationship between cognitive ability and academic attainment, including in numeracy, is in fact bidirectional.[footnote 114] Therefore, educational outcomes for pupils with SEND are likely to improve if teachers use systematic instruction and rehearsal to help pupils learn planned content.[footnote 115]

This approach is particularly useful for pupils with moderate learning difficulties who have slower cognitive processing speed.[footnote 116] Systematic approaches increase the amount of content considered per unit of time. These approaches are also highly beneficial in enhancing the progress, attainment and self-esteem of disadvantaged pupils.[footnote 117] Systematic curricular approaches give pupils with SEND and disadvantaged pupils a better chance of success, of keeping up and therefore of feeling included.

Playing to pupils' strengths: the powerful declarative memory systems of pupils with autism

Many pupils with autism have 'normal to above average algorithmic thinking ability' but can struggle with reasoning and problem-solving because of:

language processing deficits

difficulties in classifying problems by type

lack of knowledge of strategies

Teachers can fill these gaps in knowledge with systematic curriculums, teaching approaches and rehearsal. For example, teaching efficient algorithms to pupils with autism speeds up their calculations. They then have more time to learn strategies for solving classes of problem.

However, research also shows that the unique organisation and powerful declarative memory systems of many people with autism help them study, and develop proficiency in, the subject.[footnote 119] Potentially, a powerful declarative memory system can take on a compensatory role; thus many pupils with autism might benefit from a deliberate focus on memorisation of core facts and methods.

Leaders should therefore consider ways to give autistic pupils more time to study core content so that they can close gaps in learning through deliberate memorisation. Leaders should also make sure pupils' lesson time is used efficiently and effectively.

Pedagogy: new learning

Summary

In this section, we discuss the instructional needs of pupils as they progress through the curriculum. When pupils are at the start of school life or starting a new sequence of learning, they need more instruction than pupils who are already competent in that topic area. Throughout sequences of learning, pupils benefit from teaching that is systematic and clear. Pupils can also develop further understanding when sets of exercises are curated to present new and useful number connections, as pupils rehearse recently taught content.

The novice needs more instruction, not less

'Novice learners' of new mathematics content need systematic instructional approaches similar to those used to teach early reading and writing. Teachers need to ensure daily dedicated time for teaching and practising component parts.[footnote 120] Like the 'code' for language, it is useful to think about early mathematical content as also being a 'code'. Not all pupils will 'crack', discover or invent this code for themselves. An approach that comes closest to guaranteeing foundational success in mathematics is one that acknowledges that:

To most effectively develop more comprehensive and abstract thinking about mathematics, children often need more than their natural, spontaneous learning.[footnote 121]

Use of intelligent variation in sets of exercises

There is a difference between content that pupils have recently learned and content that develops further in their minds through practice. Both can be planned for. Variation within sets of exercises can help pupils to learn:

Leaders need to make sure they curate and control this approach. This is evident in the systematic use of variation in collections of tasks given to pupils in China, Hong Kong and Taiwan.[footnote 126]

Systematic instructional approaches also work well for all ages and stages

Without the adult, even where content and sequence of content might be ideal, the learning of at-risk groups of pupils is compromised.[footnote 131] Evidence shows that pupils can learn from worked examples,[footnote 132] particularly if teachers help pupils to make sense of worked examples.[footnote 133] Questioning, as long as teachers take care with language and timing, can also aid instruction.[footnote 134] Teaching pupils how to construct and use visual representations can help pupils to convert information presented in a problem into symbolic equations.[footnote 135]

Systematic instruction might also offer up benefits beyond enhanced learning of facts, methods and strategies because pupils who are more successful develop better learning behaviours.[footnote 136]

Pedagogy: consolidation of learning

Summary

As pupils progress through the curriculum, they need regular opportunities to rehearse and apply the important facts, concepts, methods and strategies that they have learned. When designing sequences of rehearsal, teachers need to consider both the quality and the quantity of practice that pupils need to develop their understanding and to make core content firm and precise in the mind. Practice needs to go beyond immediate accuracy and understanding. Sequences of rehearsal should help to prevent pupils forgetting content over time.
Categories of rehearsal/consolidation of learning

Content category

Type 1 practice

Type 2 practice

Declarative

Fact retrieval (recall)

Explaining relationships between facts (derivation and parsing of number)

Procedural

Method rehearsal (exercises)

Explaining principles, proving conceptual understanding (such as, use of informal methods, creating bar models and interpreting context)

Conditional

Strategies rehearsal (collections of problems with the same deep structure)

Describing relationships between the problem and choices of strategy (proof/reasoning)

Quantity

Some pupils are quick to grasp new content, while others might need more time to think, practise, recall and apply. Given that proficiency in mathematics requires pupils to attain a level of procedural fluency,[footnote 137] teachers should ensure that they give pupils adequate opportunities to practise. This is more likely to increase pupils' levels of procedural fluency.

In the most successful systems of mathematics education, systematic rehearsal is given more time and focus than in England. Powerful teaching and learning in classrooms where pupils do well are supported by regular homework assignments that require pupils to systematically rehearse content at home.[footnote 142] Teachers in these systems can plan future sequences of learning confident that pupils' foundational knowledge is secure. Under these conditions, consolidation of learning is personalised by time taken to complete assignments where:

every child will have had to attend to every word, every problem, and every exercise included in every textbook.[footnote 143]

Quality

Textbook analysis can provide useful information about quality of rehearsal. In contrast to the ideal of systematic rehearsal aligned to sequences of learning:

English primary textbooks tend instead to move around rapidly and to constantly recapitulate.[footnote 151]

Lack of coherence and evidence-informed features in textbooks given to the youngest pupils could potentially influence the likelihood of later SEND.[footnote 152] This may be compounded by the fact that textbooks are generally seen by teachers in England as a supplemental resource rather than a potential and valuable system of rehearsal.[footnote 153] This is different to how textbooks are used in countries where pupils do best, not just in terms of volume of questions, but also in terms of sequencing and alignment with the curriculum sequence.[footnote 154]

Systematic rehearsal does not always require textbooks, pencil and paper. For younger pupils, rehearsal of number bonds and sequences can draw on a canon of games and songs involving dice, dominoes and counting sequences.[footnote 158] Computing technology can also help pupils acquire number facts by providing them with enough repetitions and direct feedback in ways that they enjoy.[footnote 159] Pupils also experience more progress and enjoyment of computer maths games when core content is introduced as separate learning components that are systematically followed by 'mini-games' than if content is entirely subsumed into the gaming conditions.[footnote 160] This approach can also be a useful intervention for pupils with SEND.[footnote 161] However, teachers should take care with this approach because not all children make the same progress when learning with computers.[footnote 162]

Tasks that are content-focused and achievable

In mathematics, studies suggest that long-term retrieval of core content should be a focus of teachers' and leaders' planning.[footnote 163] This means teachers should set pupils tasks that focus on rehearsal of facts, methods and strategies in addition to tasks that develop pupils' understanding.

Pupils are more likely to engage in disruptive behaviours if they are expected to complete tasks that they have not mastered the component parts of yet. They are more likely to stay on task and be motivated if tasks are achievable.[footnote 166] In turn, sustained completion of tasks helps pupils to improve their ability to focus.[footnote 167] It is better for pupils to initially learn and rehearse content as component parts before learning the conditions for its use within a composite skill.[footnote 168] This has implications for 'challenge' because pupils tend to resort to using the methods they have most facility with, rather than those that are most valid and that have been recently taught, when faced with unfamiliar or demanding tasks. For example, pupils tackling tricky arithmetic problems will default to addition,[footnote 169] pupils who are new to working with algebra will default to arithmetic methods or trial and error,[footnote 170] and new learners of calculus will fall back on familiar concepts that are visually similar, but unrelated to the question.[footnote 171]

When pupils are ready to solve problems, they need to be able to hold a line of thought and to concentrate.[footnote 172] Background noise and general chit-chat have been shown to negatively affect pupils' ability to understand what the teacher is saying, to maintain appropriate behaviours and to concentrate. The children who are most affected are those who are under 13 and children with SEND.[footnote 173] Studies have also shown that the ideal environment for periods of independent work is one that is not just quiet but is in fact near silent.[footnote 174] That is not to say that all rehearsal experiences should be silent. Group work can aid pupils' development of explanations, providing it is tightly managed.[footnote 175] However, there are limits to its impact on learning as it does not always improve attainment and is difficult to implement.[footnote 176] Teachers should balance opportunities for discussion with pupils' needs for quiet periods of time to think.

Scaffolds as aids, not crutches

Teachers need to give careful consideration to how they use scaffolds, frames, physical apparatus and alternative information sources for pupils identified as needing extra support. There is a distinction to be made between using physical apparatus to reveal useful information[footnote 177] and its habitual use as an outsourced memory.

Reliance and subsequent dependence on manipulatives and associated aids can hinder progression through the curriculum.[footnote 178] The implications are that teachers need to give pupils enough time to consolidate learning and they need to plan for how pupils will move away from using the manipulative. This will help to avoid pupils relying on manipulatives to work around gaps in core knowledge that might become barriers to learning later.

Balancing rehearsal of proof and explanations with rehearsal of facts, methods and strategies

There are 2 'types' of practice:

'type 1' involves the rehearsal of core facts, methods and strategies that can be used to complete exercises and solve problems now and in the next stage of education

'type 2' includes explaining, justifying and proving concepts using informal and diagrammatic methods, parsing and derivation of number

Based on the above, high-quality maths education may have the following features

Educators plan to give pupils opportunities to consolidate learning that:

go beyond immediately answering questions correctly

involve overlearning

align with the detail and sequence of the curriculum

are free of distraction and disruption

strike a balance between type 1 and type 2 practices

avoid creating a reliance on outsourced memory aids or physical resources

help pupils to avoid relying on guesswork, casting around for clues or the use of unstructured trial and error

Assessment

Summary

Assessment during the learning journey is most useful when it focuses on the component knowledge that pupils have learned. This approach aids pupils' confidence and makes it easier to analyse and respond to gaps in learning. In mathematics, pupils benefit from timed practice of knowledge that should be easily recalled, such as maths facts. The timing element gives assurance that pupils are not reliant on derivation.

Frequent low-stakes testing helps to prepare pupils for the final performance

Leaders should, however, avoid conflating the 2 concepts with frequent use of summative tests, such as past papers. These can cause lower attaining pupils and pupils with SEND to be regularly reminded about what they do not know and cannot do (which may inculcate guesswork, misconception rehearsal or avoidance tactics). Over time, pupils who do not experience success can then become demotivated, which may negatively impact on their chances of attaining a pass when re-taking courses between 16 and 18.[footnote 184]

However, it is not the tests but lack of proficiency that causes this performance anxiety. Lack of proficiency can also be compounded with the use of 'realistic' settings of story problems that present a language barrier for disadvantaged children.[footnote 185] Teachers can ensure that pupils come to see tests and testing as moments to shine by adopting the principles underpinning the causal pathway to motivation and enjoyment in the subject. This can be achieved by seeking to engineer proficiency and initial success in the subject.

Frequent, low-stakes testing of taught content can help prepare pupils for summative tests by providing memory-enhancing opportunities to recall and apply taught content.[footnote 189] Low-stakes testing also works well when tests of component parts, such as mathematics facts, are timed.[footnote 190] If teachers give honest feedback, pupils' interest and sense of self-efficacy also increases.[footnote 191] Teachers can also set benchmarks for mastery of facts and methods so that they can be assured that pupils are recalling rather than guessing or deriving.[footnote 192] Tests should therefore be aligned closely with curriculum sequences because generic tests are not able to give this feedback.[footnote 193]

Based on the above, high-quality maths education may have the following features

Pupils are well prepared for assessments through having learned all the facts, methods and strategies that are likely to be tested.

Teachers plan frequent, low-stakes testing to help pupils to remember content.

Lessons incorporate timed testing to help pupils learn maths facts to automaticity.

Systems at the school level

Summary

Leaders can support pupils' progression through the mathematics curriculum by ensuring that pupils' bookwork is of a high quality. This is important because when pupils' calculations are systematic and orderly, they are better able to see the connections of number and to spot errors that can be corrected. Leaders can also plan to develop teachers' subject and subject-pedagogic knowledge through giving teachers opportunities to work with and learn from each other. This, for example, helps new teachers to see and adopt useful ways of explaining core concepts, methods and strategies to the pupils they teach.

Calculation and presentation

Accurate calculations and careful presentation give pupils the ability to spot important and interesting patterns of number, as well as errors that need to be corrected. Calculation methods and presentation rules are procedural knowledge that need to be taught and rehearsed to automaticity. Some pupils might naturally develop 'neatness' and subsequent accuracy, but teaching and rehearsing this procedural knowledge gives greater assurance that more pupils will be able to see errors and spot patterns of number, as well experience a sense of accomplishment.

That is not to say that more 'messy' experimental workings should never be allowed. However, teachers can help to engineer calculation and presentation success by balancing experimental approaches with opportunities to learn how to be systematic, logical and accurate when applying taught facts, methods and strategies.

Proactive professional development: the planned and purposeful pathway to expertise

The need for firm foundations applies to all novices, including novice mathematics teachers. Regular observations are often viewed as a main driver of professional development, where teachers are given feedback on aspects to improve. Use of the teacher standards for these reactive approaches prioritises on-the-ground development of pedagogical and subject-pedagogical expertise, highlighting features that are absent or in need of correcting along the way.

Leaders could consider incorporating more proactive approaches that close gaps and allow novice teachers to adopt and improve expert teaching methods, rather than develop their own aspects of effective mathematics teaching from scratch.[footnote 196] Such approaches could include:

regular opportunities to observe and be mentored by experienced and successful teachers of mathematics

systematic plans to build these models of instruction and rehearsal over time so that future generations of teachers can benefit

collaborative planning with more experienced and successful teachers of mathematics

Based on the above, high-quality maths education may have the following features

Conclusion

Throughout the review, the theme of engineering success, underpinned by systems thinking, predominates. These approaches seek to transform an offer of content into more of a guarantee that content can and will be learned. The outcomes of this systems thinking are the observed features and approaches of successful mathematics education:

detailed codification and sequencing of the facts, methods and strategies that pupils will acquire

instructional coherence and aligned rehearsal that increase the chances of understanding and remembering while minimising the need for guesswork or trial and error

Within these powerful mathematics education systems, the textbooks, teacher guides and workbooks are seen as a vital part of the infrastructure for efficiently transmitting subject knowledge and subject-pedagogical knowledge to new generations of pupils and teachers. This signals a need for teachers and leaders to avoid installing features and approaches in the absence of the 'infrastructure' underpinning their efficacy. It is also likely that the features that tend not to be observed or selected, such as the less glamorous quality and quantity of practice, are also integral to the overall success of novice mathematicians.

Quality and quantity of practice is a vital key that unlocks the development of dual tracks of conceptual understanding and procedural fluency. Further, in observing pupils' relative expertise and proficiency, such as in a problem-solving lesson, teachers and leaders should be mindful of the journey that pupils took to achieve problem-solving proficiency. This journey will have involved more than the features and activities of the lessons that proficient mathematicians are taking part in at the time. Variation in the quality of mathematics education in England is likely to be the result of the absence of systems and systems thinking, as well as possible gaps in content, instruction, rehearsal, assessment and the plans for their evolution over time.

Print or save to PDF

To print the content you can:

use the 'Print this page' button under the Contents menu

right-click anywhere on the page using a mouse and choose 'Print' in the menu

press Ctrl + P on a Windows keyboard or Command + P on a Mac

You can also use these options and change the printer destination to save the content as a PDF.

Instructions may vary depending on which internet browser you use, such as Internet Explorer or Google Chrome, and the type of device you use, such as a phone or laptop. You can find your print and save options in your browser's menu.

# data[].attributes.attachment_link.description
Mathematics, a universal language that enables understanding of the world, is an integral part of the curriculum. Beyond the study of numbers, shapes and patterns, it also provides important tools for work in fields such as engineering, physics, architecture, medicine and business.

# data[].attributes.attachment_link.title
Blue Gate Fields Infant School Implementation

# data[].attributes.headline
Early Years Evidence Store Early Mathematics

# data[].attributes.subject
Early Years Evidence Store Early Mathematics

# data[].attributes.attachment_link.readable_text
The evidence also revealed broader strategies to improve children's maths skills. These include:

structuring maths learning;

professional development for educators; and

using education technology (EdTech).

Structuring maths learning
Evidence suggests that there are different ways to implement early mathematics approaches successfully. This means that early mathematics can be introduced into early years settings in different ways depending on how your setting operates. It can be effective to introduce a whole maths curriculum, supplemental activities - shorter activities that are in addition to usual activities, and integrating maths into everyday routines. However, the impact of maths activities varies from study to study. It is important for educators to monitor whether their strategy is working.

Evidence indicates that maths approaches are similarly effective in different group sizes, including in large groups, small groups, and one to -one., However, the latter two require more staff time. If delivering small group or individual sessions, educators may want to prioritise children at greater risk of not meeting expected levels of development.

Children's mathematics skills can develop play as well as through more focused, explicit teaching Formal educator-directed approaches in which educators explicitly support children to develop specific ideas and skills. . Explicit teaching Formal educator-directed approaches in which educators explicitly support children to develop specific ideas and skills. is when educators explicitly support children to develop specific ideas and skills. Maths learning that combines explicit teaching Formal educator-directed approaches in which educators explicitly support children to develop specific ideas and skills. and play is likely to be beneficial. The evidence shows that embedding maths activities in daily routines throughout the day can also be effective.

The effectiveness of additional maths instruction (sometimes called ​'tutoring') for young children is uncertain. There is a lack of evidence that focuses just on pre-schoolers. There is some indication that children supporting each other to learn maths (sometimes called ​'peer tutoring') can be effective.
Professional development for educators
There is widespread consensus that both educators' knowledge of mathematics and of how young children's maths skills develop influences children's early mathematical learning. However, there is a lack of research on professional development focuses on maths in the early years. More research would help to find out what is most effective for educators working in the early years sector and the children they work with. Until we know more about maths-specific professional development, the EEF's guide to Effective Professional Development in the Early Years and accompanying systematic review may be helpful.

An example of an effective professional development programme for educators working in the early years is Maths Champions. Children aged three to four whose setting participated in the programme made an extra three months' progress in maths compared to children who did not receive the programme. Children from lower income families made even greater progress.

Many of the studies underpinning the maths Evidence Store included an element of training for educators and found a positive effect on maths outcomes. Training for educators to deliver a specific set of activities or programme can be a part of introducing effective approaches to maths teaching in early years settings.
Education technology (EdTech)
'EdTech' encompasses many different approaches and devices, from robots to interactive apps and games. EdTech can have a positive impact on maths outcomes though more evidence is needed.

Some promising types of EdTech include:

EdTech that teaches maths or provides opportunities to practice (for example, apps or computer-assisted instruction Apps or computer-assisted instruction include computer- or tablet-based software. The software provides a controlled environment, such as a game. It often gives children feedback to help them learn as they go. ); and

EdTech that helps children to be aware of their own learning.

When EdTech is used, it works better if adults support children to use it. Not all kinds of EdTech have been researched with children in the early years. More research is needed to understand the impact of tools like calculators and robots. EdTech is a useful tool, but it is not always the most effective way to teach maths skills. Educators should monitor whether different uses of EdTech are working to support children's maths learning in their setting.

EdTech can support the learning of children who are at greater risk of not meeting expected levels of development and children from lower income families. However, there is a risk that children who have more experience using technology will learn more from opportunities involving EdTech. When using EdTech, educators should make sure that all children are supported to access the learning.

# data[].attributes.attachment_link.description
The EEF is an independent charity dedicated to breaking the link between family income and educational achievement.

# data[].attributes.attachment_link.keywords[]
'improving mathematics in the early years and key stage 1' reviews the best available evidence to offer five recommendations for developing the maths skills of 3-7-year olds.  recommendations include integrating maths into different activities throughout the day – for example

# data[].attributes.attachment_link.keywords[]
at registration and snack time – to familiarise children with maths language and make the most of the school day.  it also highlights that story and picture books can be a powerful tool for engaging children with basic maths concepts

# data[].attributes.attachment_link.keywords[]
while board games (such as snakes and ladders) are particularly beneficial to developing understanding of numbers.  based on the best available international research – and drawing on the expertise of teachers and academics – this report is packed with examples and case studies to support practitioners.

# data[].attributes.attachment_link.title
Improving Mathematics in the Early Years and Key Stage 1

# data[].attributes.attachment_link.description
Five recommendations to support practitioners in developing the maths skills of 3-7 year-olds