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Exploring Problem-Solving in Early Years Mathematics | EYFS/KS1
This webinar will provide headteachers, mathematics leads, teachers and teaching assistants with practical guidance and methods they can use to nurture and develop pupils’ problem-solving skills in mathematics.
- Learning Outcomes
Webinar Duration: 46 minutes (approx.)
Problem-solving has long been at the heart of the mathematics curriculum. Teaching children how to problem solve in mathematics can support children’s ability to critically evaluate, encourage independence and develop their skills in reasoning and creativity. It is also an essential part of developing mastery of the subject.
In this webinar the Association of Teachers of Mathematics (ATM), who aim to support the teaching and learning of mathematics in the UK, will explore strategies that schools can use to approach problem-solving with the youngest pupils which are creative and engaging and reflect a better understanding of the needs of the learner.
- Understanding how to introduce learning in mathematics which supports children’s ability to problem solve and improve critical thinking skills.
- Recognising successful techniques that can be used in the classroom which improve reasoning in mathematics.
- Appreciating the importance of making problem-solving learning tailored towards the needs of children and ensuring continuous sharing and evaluation of different methods used.
- Understanding what is meant by ‘problem-solving skills’ and how to nurture an environment which encourages curiosity and positive attitudes.
- Building a culture which supports teaching and learning through playing and exploring, active learning and creative and critical thinking.
Since 1994, Helen has been an independent educational consultant specialising in developing the teaching and learning of primary mathematics. In July 2014, she completed her doctorate with the University of Roehampton, London. She is interested in engaging all learners mathematically, and how we might nurture effective and supportive learning communities in classrooms in a current educational climate geared to high-stakes testing. Helen is passionate about all children being given opportunities to become confident mathematical thinkers, through the establishment of a classroom culture that nurtures curious learners.
Helen has taught children across the full primary range and has a particular interest and expertise in early years and KS1 mathematics. Her work involves researching and teaching mathematics alongside colleagues in school and contributing to in-service training courses and conferences. Helen is a long-term, active member of the Association of Teachers of Mathematics (ATM).
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Home > Learning & Development
Learning and Development
Maths problem-solving activities for Early Years settings
- Written By: Judith Dancer
- Subject: Maths
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?
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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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
- 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
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
- 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:
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.
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.
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.
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.
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:
- 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.
- 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.
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.
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.
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.
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.
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.
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
Join Maths — No Problem! CEO Andy Psarianos and experts Adam Gifford and Emily Guille-Marrett as they talk school and home education.
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Paired problem solving in maths
- Classroom Practice Video
- Collaboration |
- Developing effective learners |
- Effective instruction |
Collaborative learning, where pupils learn and attempt tasks in pairs or groups, can be an effective tool to support individual learning. It is important that collaborative learning opportunities are designed carefully and appropriately. While there are major benefits to group work, there are also significant risks if implemented poorly.
Benefits and dangers of group work:
As you watch this video of classroom practice, consider how the teacher:
- Gives clear instructions to the whole class before starting paired work
- Monitors pairs as they attempt tasks
- Provides scaffolds and individual support to pairs
Whether you’re establishing approaches for collaboration for the first time or reviewing your existing methods, take some time to reflect on what the teacher has done, how they’ve done it, what they might have done differently, and how this might influence your own practice.
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- Campbell R and Bokhove C (2019) Building Learning Culture through Effective Uses of Group Work. Impact. Available at: https://impact.chartered.college/article/building-learning-culture-through-effective-group-work/ (accessed February 2020)
- Wiliam D (2018) Collaborative Learning. Available at: https://education.gov.scot/improvement/learning-resources/dylan-wiliam-collaborative-learning (accessed February 2020)
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Infusing mindset through mathematical problem solving and collaboration: studying the impact of a short college intervention.
1.1. problem-based approaches in mathematics, 1.2. infusing mindset and shifting students’ beliefs, 1.3. collaboration, 1.4. a mathematical mindset approach to problem-based learning, 1.5. course design, 2. materials and methods, 2.1. data sources, 2.2. data analysis, 2.2.1. student written reflections, 2.2.2. pre and post mathematical problem-solving assessment, 2.2.3. survey analysis, 3.1. a shift in students’ mindset and beliefs about mathematics.
Upon coming to this class, I began to see math differently, as something creative, something without a right or wrong answer, but rather a means of positive struggle, where I embraced all my missteps as part of a growing experience. I took the growth mindset to heart--not only did I apply it academically, I began going to the gym regularly, taking on a vegan diet, and doing even the little things that used to scare me, such as karaoke night (which, might I add, was extremely fun.) (Michelle, Final reflection)
My relationship with math going in was awful, trash, garbage. I hated it and, to be frank, it didn’t make much sense. The way it had been taught to me hadn’t clicked and it didn’t look like it was going to click anytime soon. I was scared for math in college, and honestly considering going from Chemistry to Linguistics to avoid having to take calculus and physics. I am no longer considering switching. (Ricardo, Final Reflection)
3.2. A Shift in Students’ Problem Solving and Collaborative Skills
I was made sure of this today when we approached a problem nearly identical to the one we attempted to solve on the first day. Instead of jumping straight into the concepts we learned in statistics and calculus, my partner and I made visual representations of our thoughts and identified the patterns within the problem. Even as we were doing it, my partner and I realized the differences in our thought processes and how that led to us finding an equation that could represent what we were trying to solve. (Kim, Final Reflection)
This class taught me how to think. It taught me how to make connections within the realm of calculus and with the people around me. Because of this class, I am looking forward to working collaboratively on work, especially math, because I have found that some of the best learning comes from learning with others. (Tasha, Final Reflections)
3.3. Centering Mindset through Mathematics Problems: An Example
When we began to engage in more of the hands-on activities, I started to shift my mindset and continue to try things out. The moment that I believe changed me the most was the cube task. In that task, we worked so well together, and the work was distributed pretty evenly. I felt accomplished, as well as did my group surrounding me. (Chantelle, Final Reflection)
The first problem that really helped open my eyes was the lemon problem. My group thought really creatively about the three methods that we tried, and actually physically manipulating the lemon helped me see why the different methods worked well. But it was at the end of it, when we discussed the problem as a class, that I saw all of my groups’ solutions were basically just different ways to perform summation/integration. It was the first time that I saw the integration formula/graph, and it actually made sense to me. Since that problem I have been riding a kind of high in the class. I now feel like if I try hard enough, and if I think creatively enough, then I can genuinely figure anything out. (Esther, Final Reflection)
3.4. Students Who Resisted the Messages of the Course
Being forced to learn math at a fundamental level was very frustrating at first because beneath the memorization of formulas and functions, it made me feel like I didn’t actually understand it at all. (Marina, Final Reflection)
4. Discussion and Conclusions
Taking this class has been my best decision at college so far. I feel like I will go into my first-year maths courses knowing a secret that no one else knows: Maths does not have to be intimidating. It is not maths’ fault that it has been portrayed to be an evil subject only conquered by "geniuses" and/or white males. With a little love and understanding, maths can be kind, compassionate, and even fun to be around. (Briaunna, Final Reflection)
Institutional review board statement, informed consent statement, data availability statement, acknowledgments, conflicts of interest.
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Share and Cite
Boaler, J.; Brown, K.; LaMar, T.; Leshin, M.; Selbach-Allen, M. Infusing Mindset through Mathematical Problem Solving and Collaboration: Studying the Impact of a Short College Intervention. Educ. Sci. 2022 , 12 , 694. https://doi.org/10.3390/educsci12100694
Boaler J, Brown K, LaMar T, Leshin M, Selbach-Allen M. Infusing Mindset through Mathematical Problem Solving and Collaboration: Studying the Impact of a Short College Intervention. Education Sciences . 2022; 12(10):694. https://doi.org/10.3390/educsci12100694
Boaler, Jo, Kyalamboka Brown, Tanya LaMar, Miriam Leshin, and Megan Selbach-Allen. 2022. "Infusing Mindset through Mathematical Problem Solving and Collaboration: Studying the Impact of a Short College Intervention" Education Sciences 12, no. 10: 694. https://doi.org/10.3390/educsci12100694
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