maths problem solving for reception class

Mathematical Problem Solving in the Early Years: Developing Opportunities, Strategies and Confidence

Published 2016 Revised 2019

• familiar contexts
• meaningful purposes
• mathematical complexity.

• which they understand - in familiar contexts,
• where the outcomes matter to them - even if imaginary,
• where they have control of the process,
• involving mathematics with which they are confident.
• taking some from one doll and giving to another, in several moves,
• starting again and dealing, either in ones or twos,
• taking two from each original doll and giving to the new doll,
• collecting the biscuits and crumbling them into a heap, then sharing out handfuls of crumbs.

• brute force: trying to hammer bits so that they fit,
• local correction: adjusting one part, often creating a different problem,
• dismantling: starting all over again,
• holistic review: considering multiple relations or simultaneous adjustments e.g. repairing by insertion and reversal.
• getting a feel for the problem, looking at it holistically, checking they have understood e.g. talking it through or asking questions;
• planning, preparing and predicting outcomes e.g. gathering blocks together before building;
• monitoring progress towards the goal e.g. checking that the bears will fit the houses;
• being systematic, trying possibilities methodically without repetition, rather than at random, e.g. separating shapes tried from those not tried in a puzzle;
• trying alternative approaches and evaluating strategies e.g. trying different positions for shapes;
• refining and improving solutions e.g. solving a puzzle again in fewer moves (Gifford, 2005: 153).
• Getting to grips:     What are we trying to do?    
• Connecting to previous experience:  Have we done anything like this before?
• Planning:      What do we need?
• Considering alternative methods:   Is there another way?
• Monitoring progress:    How does it look so far?
• Evaluating solutions:    Does it work?  How can we check?  Could we make it even better?

• Construction - finding shapes which fit together or balance
• Pattern-making - creating a rule to create a repeating pattern
• Shape pictures - selecting shapes with properties to represent something
• Puzzles - finding ways of fitting shapes to fit a puzzle
• Role-play areas - working out how much to pay in a shop
• Measuring tools - finding out how different kinds of scales work
• Nesting, posting, ordering - especially if they are not obvious
• Robots - e.g. beebots: directing and making routes
• preparing, getting the right number e.g. scissors, paper for creative activities
• sharing equal amounts e.g. at snack time
• tidying up, checking nothing is lost
• gardening and cooking  e.g. working out how many bulbs to plant where, measuring amounts in a recipe using scales or jugs
• games, developing rules, variations and scoring
• PE: organising in groups, timing and recording

• Decision making - what shall we call the new guinea pig?
• Parties, picnics and trips e.g. how much lemonade shall we make?
• Design Projects - the role play area, new outdoor gardens or circuits
• Hiding games - feely bags with shapes, the 'Box' game
• Story problems - e.g. unfair sharing, with remainders and fractions, making things to fit giants or fairies
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Home > Learning & Development

Learning and Development

Maths problem-solving activities for Early Years settings

• Written By: Judith Dancer
• Subject: Maths

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Critical thinking doesn’t have to be a daunting prospect. There are simple, effective and exciting ways to encourage children’s mathematical investigation and exploration, says Judith Dancer…

Maths is a subject many adults lack confidence in. Having struggled with it at school they often avoid it, wherever possible, when grown up.

But if maths seems scary for some people, then problem solving in mathematics can cause even more anxiety. There is no ‘safety net’ of knowing the ‘correct answer’ beforehand as problem solving lends itself to investigation and exploration with lots of possible tangents.

Understandably this is often the area of maths where many practitioners feel least confident, and where young children, who are not restrained by right answers, feel the most enthused and animated.

The non-statutory Development Matters Guidance , as part of ‘creating and thinking critically’ in the Characteristics of Effective Learning, identifies that practitioners need to observe how a child is learning, noting how a child is:

● thinking of ideas;

● finding ways to solve problems;

● finding new ways to do things;

● making links and noticing patterns in their experience;

● making predictions;

● testing their ideas;

● developing ideas of grouping, sequences, cause and effect;

● planning, making decisions about how to approach a task, solve a problem and reach a goal;

● checking how well their activities are going;

● changing strategy as needed;

● reviewing how well the approach worked.

All of these elements are, at one time or another, part of the problem identifying and solving process – although not at the same time and in the same problem.

Role of the adult

Problem solving in mathematics for young children involves them understanding and using two kinds of maths:

● Maths knowledge – learning and applying an aspect of maths such as counting, calculating or measuring.

● Maths thinking skills – reasoning, predicting, talking the problem through, making connections, generalising, identifying patterns and finding solutions.

The best maths problems for children are the ones that they identify themselves – they will be enthused, fascinated and more engaged in these ‘real’, meaningful problems.

Children need opportunities to problem solve together. As they play, they will often find their own mathematical problems.

One of the key roles of practitioners is to provide time, space and support for children. We need to develop situations and provide opportunities in which children can refine their problem-solving skills and apply their mathematical knowledge.

You can effectively support children’s developing problem-solving strategies through:

● Modelling maths talk and discussion – language is part of maths learning because talking problems through is vital. Children need to hear specific mathematical vocabulary in context. You can promote discussion through the use of comments, enabling statements and open- ended questions.

● Providing hands-on problem solving activities across all areas of the setting – children learn maths through all their experiences and need frequent opportunities to take part in creative and engaging experiences. Maths doesn’t just happen in the maths learning zone!

● Identifying potential maths learning indoors and outdoors – providing rich and diverse open-ended resources that children can use in a number of different ways to support their own learning. It is important to include natural and everyday objects and items that have captured children’s imaginations, including popular culture.

Problem solving possibilities

Spell it out.

This experience gives children lots of opportunities to explore calculating, mark making, categorising and decisions about how to approach a task.

What you need to provide:

● Assorted containers filled with natural materials such as leaves, pebbles, gravel, conkers, twigs, shells, fir cones, mud, sand and some ‘treasure’ – sequins, gold nuggets, jewels and glitter.

● Bottles and jugs of water, large mixing bowls, cups, a ‘cauldron’, small bottles, spoons and ladles.

● Cloaks and wizard hats.

● Laminated ‘spells’ – e.g. “To make a disappearing spell, mix 2 smooth pebbles, 2 gold nuggets, 4 fir cones, a pinch of sparkle dust, 3 cups of water”.

● Writing frameworks for children’s own spell recipes, with sparkly marker pens and a shiny ‘Spell Book’ to stick these in and temporary mark-making opportunities such as chalk on slate.

The important thing with open-ended problem-solving experiences like this is to observe, wait and listen and then, if appropriate, join in as a co-player with children, following their play themes.

So if children are mixing potions, note how children sort or categorise the objects, and the strategies they use to solve problems – what happens if they want eight pebbles and they run out? What do they do next?

When supporting children’s problem solving, you need to develop a wide range of strategies and ‘dip into’ these appropriately. Rather than asking questions, it is often more effective to make comments about what you can see – e.g. “Wow, it looks as though there is too much potion for that bottle”.

Acting as a co-player offers lots of opportunities to model mathematical behaviours – e.g. reading recipes for potions and spells out loud, focusing on the numbers – one feather, three shells…

Going, going, gone

We all know that children will engage more fully when involved in experiences that fascinate them. If a particular group has a real passion for cars and trucks, consider introducing problem-solving opportunities that extend this interest.

This activity offers opportunities for classifying, sorting, counting, adding, subtracting, among many other things.

● Some unfamiliar trucks and cars and some old favourites – ensure these include metal, plastic and wooden vehicles that can be sorted in different ways.

● Masking tape and scissors.

● Sticky labels and markers.

Mark out some parking lots on a smooth floor, or huge piece of paper (lining paper is great for this), using masking tape. Line the vehicles up around the edge of the floor area.

Encourage one child to select two vehicles that have something the same about them. Ask the child, “What is the same about them?”. When the children have agreed what is the same – e.g. size, materials, colour, lorries or racing cars – the child selects a ‘parking lot’ to put the vehicles in. So this first parking lot could be for ‘red vehicles’.

Another child chooses two more vehicles that have something the same – do they belong in the same ‘parking lot’, or a different parking lot? E.g. these vehicles could both be racing cars.

What happens when a specific vehicle could belong in both lots? E.g. it could belong in the set of red vehicles and also belongs in the set of racing cars. Support the children as they discuss the vehicles, make new ‘parking lots’ with masking tape, and create labels for the groups, if they choose.

It’s really important to observe the strategies the children use – where appropriate, ask the children to explain what they are doing and why.

If necessary, introduce and model the use of the vocabulary ‘the same as’ and ‘different from’. Follow children’s discussions and interests – if they start talking about registration plates, consider making car number plates for all the wheeled toys outdoors, with the children.

Do the children know the format of registration plates? Can you take photos of cars you can see in the local environment?

Camping out

Constructing camps and dens outdoors is a good way to give children the opportunity to be involved in lots of problem-solving experiences and construction skills learning. This experience offers opportunities for using the language of position, shape and space, and finding solutions to practical problems.

● Materials to construct a tent or den such as sheets, curtains, poles, clips, string.

● Rucksacks, water bottles, compass and maps.

● Oven shelf and bricks to build a campfire or barbecue.

● Buckets and bowls and water for washing up.

Encourage the children to explore the resources and decide which materials they need to build the camp, and suggest they source extra resources as they are needed.

Talk with the children about the best place to make a den or erect a tent and barbecue. During the discussion, model the use of positional words and phrases.

Follow children’s play themes – this could include going on a scavenger hunt collecting stones, twigs and leaves and going back to the campsite to sort them out.

Encourage children to try different solutions to the practical problems they identify, and use a running commentary on what is happening without providing the solution to the problem.

Look for opportunities to develop children’s mathematical reasoning skills by making comments such as, “I wonder why Rafit chose that box to go on the top of his den.”

If the children are familiar with traditional tales, you could extend this activity by laying a crumb trail round the outdoor area for children to follow. Make sure that there is something exciting at the end of the trail – it could be a large dinosaur sitting in a puddle, or a bear in a ‘cave’.

Children rarely have opportunities to investigate objects that are really heavy. Sometimes they have two objects and are asked the question, “Which one is heavy?” when both objects are actually light.

This experience gives children the chance to explore really heavy things and explore measures (weight) as well as cooperating and finding new ways to do things.

● A ‘building site’ in the outdoor area – include hard hats, builders’ buckets, small buckets, shovels, spades, water, sand, pebbles, gravel, guttering, building blocks, huge cardboard boxes and fabric (this could be on a tarpaulin).

● Some distance away, builders’ buckets filled with damp sand and large gravel.

● Bucket balances and bathroom scales.

With an open-ended activity such as this, it is even more important to observe, wait and listen as the children explore the building site and the buckets full of sand and gravel.

Listen to the discussions the children have about moving the sand and the gravel to the building site. What language do they use?

Note the strategies they use when they can’t lift the large buckets – who empties some of the sand into smaller buckets? Who works together collaboratively to move the full bucket? Does anyone introduce another strategy, for example, finding a wheelbarrow or pull-along truck?

Where and when appropriate, join in the children’s play as a co-player. You could act in role as a customer or new builder: “How can I get all this sand into my car?”; “How much sand and gravel do we need to make the cement for the foundations?”.

Extend children’s learning by modelling the language of weight: heavy, heavier than, heaviest, light, lighter than, lightest; about the same weight as; as heavy as; balance; weigh.

Judith Dancer is an author, consultant and trainer specialising in communication and language and mathematics. She is co-author, with Carole Skinner, of Foundations of Mathematics – An active approach to number, shape and measures in the Early Years .

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problem solving maths EYFS

• Maths: Open-ended Investigations
• Mathematical Problem-solving

Reception Maths: Open-ended Investigations Mathematical Problem-solving

Problem-solving tasks develop mathematical skills and problem-solving tactics. These open-ended investigations for Reception or Early Years settings are designed to take advantage of outdoor learning environments, but many of them can be adapted to run inside.

Session 1 Shape

Open-ended investigative tasks provide fun, stimulating contexts in which children can connect previous knowledge with new situations, develop mental flexibility, practise mathematical vocabulary and reason mathematically.

Print the sheets and stick them up in suitable play areas. They provide stimulating questions that will enable adults in your classroom to facilitate good mathematical language and learning. Each illustrated activity comes with a list of skills practised that you can use for assessment.

Open-Ended Task

Shape Hunt By looking for and finding shapes, children gain an awareness of similarities of shapes in the environment. They match shapes by recognising similarities and orientation, show curiosity and observation by talking about shapes and begin to use mathematical names for shapes.

More Shapes By looking for and finding shapes formed by windows, children gain an awareness of shapes, practise matching them, and begin to use mathematical names for them. Use language such as ‘square’ to describe the shape of solids and flat shapes.

• More Shapes

Sorting While playing with and arranging twigs, stones, leaves, etc., children can be encouraged to take an interest in shape and space. They can talk about similarity and difference, while sorting objects. Developing mathematical ideas and methods can be used to solve practical problems.

Session 2 Position and Direction

Trails Remember… just about anything you do indoors in maths can be done outside. Some children ‘come alive’ once out of the classroom and may just surprise you with the observations they make or the learning behaviours they show.

Scooters, Bikes, Trikes Riding a scooter, bike or trike prompts counting, consideration of same and different, and position and spatial properties.

• Scooters, Bikes, Trikes
• The Mathematical Journey

Obstacle course Children use everyday language to talk about position, distance and time when running, or walking, an obstacle course. They compare quantities and objects and solve problems.

• Obstacle Course

Milk the Maths: Wellies Encourage children to use everyday language to talk about position whatever they are doing! Putting wellies away is a colourful opportunity.

• Milk the Maths

Session 3 Number and Shape

Holes When digging holes children can use number names in order in familiar contexts. They can use everyday language to talk about size, capacity, position, distance and time. Holes offer fun opportunities to compare quantities and objects and to solve problems.

The Mud Kitchen Ask children questions about shape, space and measure while exploring mud. Consider similarities and differences.

• Mud Kitchen

Planting and Gardening While working in a school garden, children can practise using numbers to identify how many objects there are in a set. They say and use number names in order in familiar contexts, and count everyday objects.

• Planting and Gardening

Hoist Playing with a bucket on a hoist, children can use numbers to identify how many objects there are in a set. They can use everyday language to talk about size, capacity, position, distance and time and compare quantities and objects and to solve problems.

Session 4 Number and the Language of Addition/Subtraction

Leaves When playing with leaves, children have opportunities to see that numbers identify how many objects there are in a set and to say and use number names in order in familiar contexts. They can begin to use the vocabulary involved in adding and subtracting. They can relate addition to combining two and subtraction to ‘taking away’.

Tin Can Alley Play with cans to explore number names in familiar contexts and to.count up to 10 everyday objects. Children can begin to use the vocabulary involved in adding and subtracting and to relate addition to combining two groups of objects and subtraction to ‘taking away’.

• Tin Can Alley

Sand to Sandpit Children can fill a sandpit (or move sand from one place to another) and count up to 10 everyday objects and begin to use the vocabulary involved in adding and subtracting.

• Sand to Sandpit

Logs Put logs onto a trolley and say and use number names in order in familiar contexts. Count and use vocabulary involved in adding and subtracting. Show curiosity and observation by talking about shapes. Begin to use mathematical names for shapes.

More Logs Playing with logs offers countless opportunities to practise counting! Children can also begin to use the vocabulary involved in adding and subtracting and to relate addition to combining two groups of objects and subtraction to ‘taking away’.

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What your child will learn in Reception

In Reception, your child will learn to:

• Count reliably with numbers from 1 to 20, place them in order and say which number is one more or one less than a given number.
• Use quantities and objects to add and subtract 2 single-digit numbers and count on or back to find the answer.
• Use everyday language to talk about size, weight, capacity, position, distance, time, and money to compare quantities and objects and to solve problems.
• Recognise, create, and describe patterns.
• Explore characteristics of everyday objects and shapes and use mathematical language to describe them.
• Age 3–4 (Early Years)
• Age 4–5 (Reception)
• Age 5–6 (Year 1)
• Age 6–7 (Year 2)
• Age 7–8 (Year 3)
• Age 8–9 (Year 4)
• Age 9–10 (Year 5)
• Age 10–11 (Year 6)
• Year 1 (age 5–6)
• Year 2 (age 6–7)
• Year 3 (age 7–8)
• Year 4 (age 8–9)
• Year 5 (age 9–10)
• Year 6 (age 10–11)
• Help with times tables
• Ratio & proportion
• Learning to tell the time
• Numicon parent guide
• MyMaths parent guide
• Maths activity books

Ten of our favourite early years problem-solving activities

A lot of the time when we hear the term ‘problem-solving’, our brain jumps back to the tricky maths teasers from our school days, and we immediately recoil a little. However, problem-solving is much more than number conundrums.   

Problem-solving is a key part of early years development and can support learning across many of the My First Five Years streams. The skill of problem-solving starts developing very early in a child's life and stems from the knowledge of the world that they are constantly building.[1]. For instance, your baby may cry when hungry as they know that crying gets the attention of an adult who can feed them.   

Problem-solving is a part of everyday life for children, from being a baby through to their future adulthood. When children learn how to solve problems, it can support them in building resilience, self-confidence and self-esteem. Taking part in problem-solving activities with others can also help children develop social skills, communication and relationships.[2]   

Psychologist Jean Piaget’s theory of cognitive development also focuses on the importance of problem-solving for early childhood development. In each developmental stage of his theory, the psychologist emphasised the importance of play-based learning for young children when it comes to problem-solving, and in turn building skills across the spectrum.[3]    

Supporting problem-solving  

When thinking about problem-solving activities for your child, it can be difficult to know where to begin.   

To keep children engaged, enabling them to take the lead and follow their interests, is key. Play-based, hands-on learning makes acquiring new skills more interesting and memorable for young children.[4]    

Many activities can support children when developing their problem-solving abilities – the possibilities are wide open. When considering which problem-solving activities are the most effective, it is also important to consider how they can be adapted to multiple interests, abilities and how accessible they are when it comes to using resources and materials.   

To help you out, here are ten of My First Five Years’ favourite problem-solving activities that you can try with your child.   

1) Den-building

Den-building is brilliant for problem-solving as it requires creative and critical-thinking, foresight, and planning. It is also a wonderful way to promote sustained shared thinking with your child. Sustained shared thinking is a way of working together that encourages individuals to evaluate the problem that they are working on and is focused on collaboration, using experiences and prior knowledge.[5]  

When building a den with your child, encourage your child to take the lead. You could provide materials such as boxes and blankets, or you could even ask your child to decide what materials you need before starting, encouraging them to plan out their work. Den-building can also be done both indoors and outdoors and with children from a young age. You may find that people have already started creating these in your local woodland that you can add to, adapt, or just enjoy!  

2) Cooking and baking

Cooking and baking are not only fun activities, but they also focus on mathematical problem-solving. To bring problem-solving into a cooking and baking activity, you can ask your child to count out simple measurements, for instance, cups of flour or sugar. Activities like cooking or baking are great for children to be able to take ownership of what is happening; encourage them to choose what you will make and allow them to do all the elements themselves.   

What’s great about cooking is it really doesn't matter how it turns out! Problems can arise often in cooking or baking, for example, the mixture may turn out too dry, you may be an ingredient short, or your cakes might not rise how you expected them to. If this is the case, talk to your child about what might have gone wrong and how you can rectify it next time! Then when they come to do it again, they can use their prior knowledge to help them.   

3) Playing with patterns

Patterns are a great activity for mathematical problem-solving. You can create patterns of any objects that you can find! For example, with pieces of fruit, pebbles from the garden, building blocks or even snacks! You could encourage your child to continue patterns, fill in the missing pieces or even create their own for you to solve problems with as they grow more confident. 

4) Sorting and categorising

Sorting and categorising objects is an activity that supports children in mathematical problem - solving and can be easily adapted to individual children’s abilities . You could encourage your child to sort by shape, size, colour, or better yet , their interests . For example, if they are a dinosaur enthusiast, they could classify them by wh ich is their favourite or least favourite , or order them by the size of their feet. They may even find enjoyment in helping you with daily sorting such as recycling or washing!  

Puzzles are a fun resource that can be used with children from a very young age. There are a wide variety of puzzles for children to access , such as chunky wooden puzzles or traditional shape sorters. When playing with puzzles, children will have to use their prior knowledge and experience of shape, space and measure whil e also experimenting with different angles and placements. They will use trial and error to find the best way to complete the puzzle and then will use this knowledge in future attempts.  

6) Ice rescue

As well as being a great problem-solving activity, ice rescue enables children to explore seasonal changes, temperatures and develop their fine and gross motor skills using tools. To play ice rescue, freeze toys inside ice overnight. This could be in cake moulds or small bowls. Use toys that will motivate your child, for instance, their favourite small figurines.   

Once frozen, place your blocks of ice in a big bowl or tray, and encourage your child to think about how they can get the items out. You could provide tools, or even get your child to find tools themselves.  

7) Obstacle courses

Obstacle courses are versatile and can be made with a wide variety of resources. When setting up an obstacle course for your child, try to include sections where your child will have to stop and think about how they will have to adapt their body to move through it , for example, something that they must climb over or under, or a section where they have to move differently. You could even include them in trying to create the obstacle course and allow them to make it the most challenging they can.  

8) Filling, emptying and investigation

Many children enjoy filling and emptying during play. Investigating this way helps children to get a sense of size, capacity and explore predicting and estimation. For instance, if your child likes playing with sand, you could ask them to guess how many scoops they will need to fill a container, or if they like water play you could challenge them to find a way to move the water between two containers as quickly as possible , or from one tray to another.  

9) Story problems

Stories are an effective way of introducing problem-solving and they can be a highly engaging way to promote creative and critical-thinking. You could use familiar or traditional stories to help scaffold play opportunities for your child. For example, you could try building a house for the three little pigs that cannot be knocked over. You could test out different methods using materials that you can find around your home.   

If you are feeling creative, you could also make up a little story using your child’s favourite toys. An example of this could be figuring out how to share food between their favourite teddies during a picnic and making sure that everyone gets enough.   

10) Playing with loose parts or open-ended resources

Natural materials such as leaves, conkers, sticks, acorns, and pinecones are all brilliant open-ended play opportunities (if supervised). You can also use household objects like bottle caps, curtain rings, tubes, tins, boxes, buttons etcetera in this sort of play. All it requires is a tray of different objects that you've collected and time to explore them. Your child will have to think creatively about how to utilise the objects and in doing so will be challenging their cognitive capacity by problem-solving to achieve the desired outcomes.   

References 

[1]  Rachel Keen. (2011). The Development of Problem Solving in Young Children: A Critical Cognitive Skill. Available: https://www.annualreviews.org/doi/full/10.1146/annurev.psych.031809.130730#_i22 .  

[2] Sheila Ebbutt. (2009). EYFS best practice - All about ... problem-solving . Available: https://www.nurseryworld.co.uk/features/article/eyfs-best-practice-all-about-problem-solving .  

[3] Piaget, J. (1983). Piaget's Theory. In P. Mussen (ed). Handbook of Child Psychology. 4th edition. Vol. 1. New York: Wiley.  

[4] Unicef. (2018). Learning Through Play. Available: https://www.unicef.org/sites/default/files/2018-12/UNICEF-Lego-Foundation-Learning-through-Play.pd .  

[5] Kathy Sylva, Edward Melhuish, Pam Sammons, Iram Siraj-Blatchford and Brenda Taggar. (2004). The Effective Provision of Pre-School Education (EPPE) Project: Findings from Pre-school to end of Key Stage1. Available: https://dera.ioe.ac.uk/8543/7/SSU-SF-2004-01.pdf .  

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EYFS best practice - All about ... problem-solving

Sheila Ebbutt, a freelance consultant and was formerly managing director of BEAM (Be A Mathematician) Tuesday, July 7, 2009

Responding to challenges and finding solutions is not confined to mathematics but arises in all areas of learning, says Sheila Ebbutt.

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Introduction

The first few years of a child’s life are especially important for mathematics development . For many education experts, no other group represents a greater opportunity to improve mathematical standards than children in the early years.

The more grounded in mathematical concepts young children become, the better their later outcomes. Conversely, research shows that children who start behind in mathematics tend to stay behind throughout their educational journey.

On this page, we’ll examine:

• What do we mean by Early Years?
• What does learning look like in the Early Years
• Why is Cognitive Load Theory so important?
• What mastery strategies are available for Early Years?

What do we mean when we talk about Early Years?

The UK government published the Statutory Framework for the early years foundation stage in March 2017. It sets standards for the learning, development and care of children from birth to five years old.

Areas of learning

The EYFS framework outlines seven areas of learning :

• Communication and language
• Physical development
• Personal, social and emotional development
• Mathematics
• Understanding the world
• Expressive art and design

Mathematics in EYFS

In the context of mathematics, the framework says children must be given opportunities to develop their skills in the following areas:

• Understanding and using numbers
• Calculating simple addition and subtraction problems
• Describing shapes, spaces, and measure

Revised guidance

The DfE published revised guidance in March 2021 to take effect in September 2021.

The mathematics component now incorporates many elements of the mastery approach.

Specifically, the revised framework says:

By providing frequent and varied opportunities to build and apply this understanding — such as using manipulatives, including small pebbles and tens frames for organising counting — children will develop a secure base of knowledge and vocabulary from which mastery of mathematics is built.

In addition, it is important that the curriculum includes rich opportunities for children to develop their spatial reasoning skills across all areas of mathematics including shape, space and measures.

Early Learning Goals

The latest framework has the following early learning goals for mathematics:

Children at the expected level of development will:

• Have a deep understanding of number to 10, including the composition of each number
• Subitise (recognise quantities without counting) up to five
• Automatically recall (without reference to rhymes, counting or other aids) number bonds up to five (including subtraction facts) and some number bonds to 10, including double facts

Numerical patterns

• Verbally count beyond 20, recognising the pattern of the counting system
• Compare quantities up to 10 in different contexts, recognising when one quantity is greater than, less than or the same as the other quantity
• Explore and represent patterns within numbers up to 10, including evens and odds, double facts and how quantities can be distributed equally

Learning in the early years

The first few years of a child’s life are especially important for mathematics development , says the National Center for Excellence in the Teaching of Mathematics.

Research shows that early mathematical knowledge predicts later reading ability and general education and social progress.

As young as eight months old, children are developing an awareness of number names , and include these in their speech, as soon as they begin to talk. As children listen to the talk around them, they are introduced to numbers through opportunities that occur in everyday life, and experience a variety of number rhymes. This supports their growing knowledge of number names.

According to the NCETM, there are:

Six key areas of mathematical learning

Cardinality and counting, composition.

• Shape and Space

Looking briefly at each in turn:

When children understand the cardinality of numbers , they know what the numbers mean in terms of knowing how many things they refer to.

Comparing numbers involves knowing which numbers are worth more or less than each other.

Learning to ‘see’ a whole number and its parts at the same time is a key development in children’s number understanding.

Developing an awareness of pattern helps young children to notice and understand mathematical relationships.

Shape and space

Mathematically, the areas of shape and space are about developing visualising skills and understanding relationships, such as the effects of movement and combining shapes

Measuring in mathematics is based on the idea of using numbers of units in order to compare attributes , such as length or capacity.

Learning to count in the early years is a fundamental skill and key to mastering mathematical concepts in the future, but there’s more to it than you might think, says Sabrina Pinnock, a primary school teacher in Yorkshire.

According to researchers Rochel Gelman and C.R. Gallistel, these are the steps needed to successfully count :

• The one-to-one principle: children must name each object they count and understand there are two groups: the one that has been counted and the one that hasn’t yet been counted
• The stable order principle: children must know how to count in the right order
• The cardinal principle: children need to understand the last number in the set is the total amount
• Counting anything: children need to realise that anything can be counted, not just objects that can be touched, but also things like claps and jumps
• Order of counting doesn’t matter: children need to understand that the order of counting in the set is irrelevant and will still lead to the same amount

Assessing children to find out which step they are struggling with is key to helping them overcome difficulties and become confident counters.

How do children develop counting skills?

Very young children start to count spontaneously and later begin to refine their skills by pointing their finger at the objects they are counting.

They will often try to get all the names of the numbers they know into their count as they pass their finger along the objects. They also reuse numbers. If they have not finished and they have used up all their known numbers, they will begin using the same numbers again. For example, a child might decide to count eight shells she collects at the beach. She might line them up carefully, tag numbers to them by pointing as she slides her finger along the shells, quickly counting out loud, “one, two, three, four, five, one, two, three, four, five, one, two, three.”

In their drive to make meaning, children are eager to experiment as they acquire new small bits of mathematical knowledge. It is extremely important to respect their developing understanding and not expect “perfect” counting sequences.

By valuing children’s partial understanding, children will develop enthusiasm for numbers and become confident mathematicians.

Activities to boost number sense in Reception Year

Children need lots of opportunities to develop number sense and deepen their conceptual understanding. Here are some simple activities to get your Reception Year learners counting:

Crowd control

Display the number of children allowed in each area using pictorial representations of cubes on a 10 frame. Once the children begin to realise how many are allowed in the area, they start to discuss the meaning of more and less. For example, “no more children are allowed in,” or “you can come in because one more than three is four.”

Encourage children to show numbers using their fingers above their head. “Bunny ears six” means they place their fingers above their head to show six. They may decide to use three fingers on each hand. As they become more confident, you could introduce swapping, where they show the same number but with a different configuration of fingers, in this case two and four, or five and one.

Grouping straws

Each morning, drop different amounts of art straws all over the carpet. Say something like, “oh no class, I can’t believe it. I’ve dropped all my straws again. They were all in 10s. Can you help me?” This activity helps children consolidate counting objects and gets them to think about stopping after they have made 10. Providing elastic bands helps them to keep track of their groups of 10.

Fastest 10 frames

This game can help distinguish between those who have developed a good understanding of number sense and those who need further support. Give each child their own frame and cubes. Tell them a number and observe how they place the cubes on the frame. If the children are working with the number eight, do they say each number name as they place the cube on the frame, or do they realise eight is two less than 10? If so, they should be able to place the cubes down faster than other children.

What do they do when you say the next number? For example, for the number five, do they automatically remove three cubes, or do they remove all of the cubes and start over counting from one to five?

Everyday questions to develop number sense

These questions for children aged five to six help develop their number sense and let them practice using mathematical terms.

When prepping lunch or a snack, count out the different types of food with your child, and as you lay the table, count out the different items. Ask your child questions like:

• How many grapes are there?
• How many tomatoes are there?
• How many plates are there?

Practice using the terms more than, fewer than and as many as by asking:

• Are there more grapes than tomatoes?
• Are there fewer tomatoes than grapes?
• Are there as many plates as people eating?

Remember to practice each sentence:

• There are more grapes than tomatoes
• There are fewer tomatoes than grapes
• There are as many plates as family members eating

When counting, make sure that you count one number for one item to strengthen your child’s sense of one-to-one correspondence.

Number Rhymes

Carefully select number rhymes to include those that children are familiar with from home. Make sure the rhymes include:

• Counting back and counting forward
• “No” or “none” (Five little ducks went swimming one day)
• Counting in pairs (two, four, six, eight, Mary at the cottage gate)
• Counting to five, 10 and beyond

Problem solving, reasoning and numeracy

The EYFS requires children to be supported in developing their understanding of problem solving, reasoning and numeracy in a broad range of contexts in which they can explore, enjoy, learn, practise and talk about their developing understanding. They must be provided with opportunities to practise these skills and gain confidence.

Young children learn best through play. For their learning to be effective, they need sensitive and informed support from adults.

All children can be successful with mathematics, provided they have opportunities to explore ideas in ways that make personal sense to them and opportunities to develop concepts and understanding. Children need to know that practitioners are interested in their thinking and respect their ideas.

Foundations

Maths — No Problem! Foundations is designed with all the theory and rigor that underpins a true mastery approach. It meets all the requirements of the national curriculum’s Early Years Foundation Stage. But Maths — No Problem! Foundations doesn’t shy away from embedding learning through play in Reception.

Genuine learning through play in the early years is something the team at Maths — No Problem! gets very excited about. What may appear to be simple games are actually carefully designed activities that have a deep maths mastery focus.

Maths — No Problem! Foundations is a complete Reception programme that includes Workbook Journals, Picture Books, and online Teacher Guides with printable resource sheets, all in one package.

The Maths — No Problem! suite of products — including textbooks, workbooks, a revolutionary online assessment tool, world-class teacher training, and much more — is based on the Singapore method, which combines 30 years of international research with painstaking craftsmanship and constant refinement.

Mark making

Research from Carruthers and Worthington into children’s mathematical graphics reveals young children use their own marks and representations to explore and communicate their mathematical thinking. These graphics include:

• Scribble-marks
• Tally-type marks
• Invented and standard symbols including numerals

Young children’s graphical exploration “builds on what they already know about marks and symbols and lays the foundations for understanding mathematical symbols and later use of standard forms of written mathematics,” the researchers said.

In a 2009 publication, the UK Department for Children, Schools and Families, says practitioners should: “Value children’s own graphic and practical explorations of problem solving” and observe “the context in which young children use their own graphics.”

Developing understanding with careful questioning

When children play and interact with other children, there are always opportunities for maths talk to help them develop a deep understanding, says Sabinra Pinnock.

For instance:

• I have made a pattern. What’s your pattern?
• How many blocks taller is my model compared to yours?
• How do we know this area is full?
• I have three cars, how many do you have?
• Do you have more?
• How do you know?

Give learners long enough to think about their answer and give their response, but not so long that it disrupts the flow of play.

Adding maths talk activities to your daily routine

Developing maths talk in your daily routine gives learners a chance to understand concepts while using real-life concepts. It also means that children can consolidate what they have learned.

The following activities can get you started:

How many children are at school?

Get your class to work out how many children are at school by placing a picture of themselves or a counter representation on large 10 frames. Ask them questions like:

• How do we know this 10 frame is full?
• How many children are absent?
• What can you tell me about number seven?

Sorting and grouping objects as a class

Sorting and grouping objects as a class helps children learn to reason and look for patterns. Give them a variety of buttons each day and ask open-ended questions like, “how can we sort the buttons?” They can use critical-thinking skills to come up with a range of ideas like sorting by size, colour, pattern, and shape.

Vote for a story

First, ask a child to pick two books. Everyone in the class gets to vote (using a piece of lego, for instance) on which of the books should be read. Tally the votes at the end of the day to determine the winner. This can lead to questions such as:

• How many more votes did one book have than the other?

The key to introducing mastery in the early years is to keep activities fun and part of your daily routine. The more learners explore maths through play, the more engaged they become.

Pattern Awareness

Dr. Sue Gifford, emeritus fellow at University of Roehampton, says recent research shows a child’s ability to spot mathematical patterns can predict later mathematical achievement, more so than other abilities such as counting. It also shows pattern awareness can vary a great deal between individuals.

Australian researchers, Papic, Mulligan and Mitchelmore have found pattern awareness can be taught effectively to preschoolers, with positive effects on their later number understanding.

Explicitly teaching pattern awareness links to encouraging “pattern sniffing” with older children in order to develop mathematical understanding and thinking.

What is mathematical pattern awareness?

Patterns are basically relationships with some kind of regularity between the elements. In the early years, Papic et al suggest there are three main kinds:

• Shapes with regular features, such as a square or triangles with equal sides and angles, and shapes made with some equally spaced dots
• A repeated sequence: the most common examples are AB sequences, like a red, blue, red blue pattern with cubes. More challenging are ABC or ABB patterns with repeating units like red, green, blue or red, blue, blue
• a growing pattern, such as a staircase with equal steps

Children who are highly pattern aware can spot this kind of regularity: they can reproduce patterns and predict how they will continue.

Why is pattern awareness important?

Spotting underlying patterns is important for identifying many different kinds of mathematical relationships. It underpins memorization of the counting sequence and understanding number operations, for instance recognizing that if you add numbers in a different order their total stays the same.

Pattern awareness has been described as early algebraic thinking, which involves:

• Noticing mathematical features
• Identifying the relationship between elements
• Observing regularities

The activity Pattern Making focuses on repeating patterns and suggests some engaging ways of developing pattern awareness, with prompts for considering children’s responses. Children can make trains with assorted toys, make patterns with twigs and leaves outside or create printing and sticking patterns in design activities.

Repeating Patterns

It is important to introduce children to a variety of repeating patterns, progressing from ABC and ABB to ABBC.

Focusing on alternating AB patterns can result in some young children thinking that ‘blue, red, red’ can’t make a pattern. They say things like, “That’s not a pattern, because you can’t have two of the same colour next to each other.”

Foundations — Your Reception Solution

This Early Years mastery programme encourages learning through play and sets children on a path to a deep understanding of maths.

Cognitive Load Theory

Cognitive Load Theory has gained a lot of traction in recent years as educators embrace evidence-based research to inform their evolving practice, says Ross Deans, a KS2 teacher and maths lead in Bournemouth, England.

What is Cognitive Load Theory and why is it important?

Why are new teachers so overwhelmed by tasks that more experienced teachers can juggle alongside multiple other responsibilities?

The answer is simple — new skills demand more attention.

This logic can be applied to any situation. When learning to drive, for example, you focus carefully on every small detail. That mental exertion can be very demanding. Compare that to the feeling of driving after you’ve been doing it for years; you may barely remember the drive, the process is so familiar.

Now put yourself in the shoes of your pupils. Each lesson provides fresh learning and new skills to master. Consider what happens inside your learners’ heads when they encounter new information, new skills and new vocabulary.

Working memory

Cognitive Load Theory , originated by John Sweller, acknowledges that working memory is very limited.

Working memory is the information we hold in our minds while we’re learning. The number of things that we can keep in working memory at one time is approximately four, plus or minus one, and perhaps even less for children.

It’s important to keep this in mind when planning and delivering lessons. If our learners cannot balance more than four things in their working memory, then we need to be very careful about the information we choose to present to them.

Intrinsic versus extraneous load

Intrinsic load includes anything that is necessary to learn a desired skill. In other words, the essential stuff.

Extraneous load is anything that will detract from desired learning. In other words, the stuff that should be reduced as much as possible.

It can be tempting while teaching to embellish lessons with child-friendly imagery and gimmicks. While It’s important to foster enjoyment, we should avoid distracting learners from the essential components of a lesson.

Supporting the transition to long-term memory

While acknowledging the impact of Cognitive Load Theory, we can consider the following to support our learners:

Focused learning objective

First and foremost, we must have a very clear idea of what we want our learners to achieve. Keep the limitations of the working memory in mind and let this guide the content you choose to include in a lesson.

Activate prior learning

At the start of the lesson, you may choose to design a task that encourages learners to retrieve essential skills. This means their working memory can hold on to new learning during the lesson.

Present information clearly

Take time when designing lessons to make sure information is presented clearly. Avoid unnecessary extras which may detract from the learning goal. Keep slides clean and similar in style.

Avoid cognitive overload

In maths, problems are often detailed and complex. Consider breaking questions up into chunks so that learners can digest each part separately. By taking away the final question, you can make a maths problem goal-free.

Maths mastery for Early Years

Given the importance of developing sound mathematical understanding in the early years, the maths mastery approach can be especially useful, considering its focus on problem solving and whole-class learning.

Early Years and CPA

If you’re teaching the Concrete, Pictorial, Abstract (CPA) approach in the early years, it’s best to focus on C and P. Here’s how to use concrete and pictorial representations effectively.

The CPA model works brilliantly in the primary years but for the youngest learners, moving onto abstract concepts too soon causes difficulties. Spending as much time as possible with concrete objects and pictorial representations helps children master number skills.

By the time they reach Key Stage 2, children need to develop their understanding of numbers by being able to visualise what the concrete looks like in their heads. Therefore, it’s positive that the revised EYFS framework focuses on numbers just to 10, from 20 previously.

If learners develop a deep understanding of numbers to 10, their chances of understanding larger numbers increases significantly.

C is for concrete

Concrete is the “doing” stage. During this stage, students use concrete objects to model problems. Unlike traditional maths teaching methods where teachers demonstrate how to solve a problem, the CPA approach brings concepts to life by allowing children to experience and handle physical (concrete) objects.

Spending time with real-life objects

The theorist Jerome Bruner stresses the importance of children spending time learning maths through tangible items. Spending lots of time using real-life objects, solving real-life problems, and manipulating abstract concrete objects (when ready) such as cubes and counters is essential in the early years.

Ideas include counting out fruit for snack time, comparing, sorting and counting a range of different buttons, pasta, and even ‘magic beans’ linked to specific topics.

Early years and number bonds

By mastering number bonds early on, pupils build the foundations needed for subsequent learning and are better equipped to develop mental strategies and mathematical fluency. By building a strong number sense, pupils can decide what action to take when trying to solve problems in their head.

How to teach number bonds

Children are usually introduced to number bonds through the Concrete, Pictorial, Abstract approach . Here’s just one way to introduce and teach number bonds.

Concrete step

Children start out by counting familiar real-world objects that they can interact with. They then use counters to represent the real-world objects. From here, they progress to grouping counters into two groups.

By putting five counters into two groups, children learn the different ways that five can be made. For example, 3 and 2 as illustrated below. With further exploration, children work out other ways to break numbers into two groups.

Pictorial step

Now that they understand the concept with hands-on objects and experience, children progress to writing number bonds in workbooks or on whiteboards. Early number bond explorations might simply reflect the two groups of counters that they created during the concrete step, along with other combinations.

Abstract step

With the concrete and pictorial steps done and dusted, children progress to representing abstract problems using mathematical notation (for example, 3 + 2 = 5).

Early Years and place value

Number and place value are foundational concepts for all mathematics learning. This means we need to address how to teach place value as early as possible so that pupils can secure their knowledge of the concept.

How do you develop an early understanding of place value in the primary school classroom? Let’s start by defining place value. It is a system for writing numerals where the position of each digit determines its value. Each value is a multiple of a common base of 10 in our decimal system.

Here are some teaching strategies I’ve found useful when helping learners develop an early understanding of place value.

Progress through concepts systematically

Developing an understanding of place value requires systematic progression. Each new concept should build on previous learning experiences so that pupils can gain deeper, relational understanding as they go.

This approach ensures knowledge is developed, refined and applied correctly as numbers become meaningful tools for solving problems rather than just a series of symbols on a page. Most importantly, this starts our learners on the path to becoming confident problem solvers and pattern spotters.

Use the CPA approach to establish meaning

The CPA ( Concrete, Pictorial, Abstract ) approach helps pupils connect a physical representation of a number (concrete manipulatives) to that same quantity as shown in drawings or graphics (pictorial), and finally to the actual written name and symbol for that number (abstract).

Concrete resources are meaning makers. They add meaning to abstract representations of numbers so that when learners progress to the abstract phase, they know what those numbers stand for, what they mean, and how they relate to each other.

If a pupil can identify the meaning of each component in a problem, they are far more confident in how they work to solve it.

Teach the ‘10-ness of 10’

At an early level, spend as much time as possible studying the numbers from 0 to 10, as understanding the 10-ness of 10 is crucial for maths attainment, and it cannot be rushed.

Once this understanding is locked-in, follow this with an introduction to number bonds. Start with the additive relationships between numbers less than 10, then progress to adding and subtracting up to 10. This ensures that learners see 10 as an important ‘base’ number in all of their future maths applications.

Progress to 20, then to 40

I make sure to take my time teaching 10 and teen numbers so that a solid understanding of place value with numbers up to 20 is properly established.

I then extend the place value concept by working with numbers up to 40 — followed by addition and subtraction to 40.

Because pupils have learned to make 10 and use number bonds, they are ready to begin working with multi-digit numbers and regrouping. Focusing on numbers to 40 while developing the concept of place value also allows learners to associate numbers with easily-managed, physical quantities (meaning makers).

Use base 10 blocks for 100 and 1000

The work we’ve done building a gradual understanding of place value will have prepared pupils to progress to three-digit numbers. So we can now move on to studying up to 100.

We start here by developing an understanding of numbers in multiple place value representations. For example, one thousand five hundred is 15 hundreds or 150 tens.

Once they get the hang of that, learners then sharpen their counting, reading, and writing skills for numbers up to 1,000. Moving into addition and subtraction with numbers up to 1,000 — with and without regrouping — is the next step.

Here is where our work establishing an early understanding of place value is key, because pupils will intrinsically know why these algorithms work for three and four-digit numbers. Base 10 blocks are a great tool to help solidify those earlier place value ideas when working with numbers up to the thousands.

Approach larger numbers the same way

The CPA approach is once again our answer to learning place value in larger numbers. Apply those skills and always be on the lookout for chances to extend number and place value concepts.

For example, you can identify and complete number patterns or find missing digits on a number line.

From there you can explore strategies for mental mathematics as well as addition and subtraction for numbers up to 10,000. Take learners even deeper by having them explore place value with an emphasis on multiplication, division, and decimals.

Mastering maths concepts like place value in the early years is not just key to success in the classroom. It prepares learners for a lifetime of deep mathematical understanding by giving them invaluable real-world tools like resilience and problem-solving ability.

And a confident problem solver in maths is a confident problem solver in life.

Well done on making it to the end of our Ultimate Guide to Early Years.

We’ve looked at the definition of Early Years and what the government recommends in its revised guidance, and we’ve taken a deep dive into some of the most-effective strategies for teaching mathematics mastery in the Early Years.

We’ve also discussed Cognitive Load Theory and what it means for teachers in the Early Years classroom.

If you’d like to learn more about Early Years, we recommend checking out the following links:

• NCETM: How Early Years children develop mathematical thinking (Podcast)
• NRICH: Early Years Foundation Stage Homepage
• The School of School: Episode 17 Play and early years (Podcast)
• Maths — No Problem! CPA approach

Also, don’t miss our other Ultimate Guides:

• The Maths — No Problem! Ultimate Guide to Maths Mastery
• The Maths — No Problem! Ultimate Guide to Assessment

School of School Podcast

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