Supporting Your Primary Age Child with Maths Using Manipulatives and Concrete-Pictorial Methods
A child can look as if they have “forgotten” maths when the real problem is that the idea has never become visible enough.
That is why many primary schools now teach maths with objects, drawings and symbols together. Children do not jump straight to written sums because written sums can hide what is happening. Before a child can confidently solve `34 + 28`, they need to understand what 34 is, what 28 is, why regrouping works, and why the answer is reasonable.
At home, parents often feel stuck. The methods may look different from the ones remembered from school. A child may say, “That’s not how my teacher does it.” Homework can become tense, especially when everyone is tired.
The good news is that supporting maths at primary age does not require specialist equipment or hours of extra practice. It usually means slowing things down, making ideas concrete, drawing what is happening, and talking about the thinking behind an answer.
Why primary maths feels different now
Many adults learnt maths as a set of procedures. Line up the numbers, carry the one, borrow from the next column, learn the times tables, practise until it sticks.
Those skills still matter. Children do need efficient written methods. They do need fluency. They do need to remember number facts. The difference is that schools now place more emphasis on understanding the structure underneath those methods.
For example, an adult might solve:
```text
47 + 25
```
by writing it in columns and adding digits. A child might first build 47 with tens and ones, then add 25, then see that 12 ones can become 1 ten and 2 ones. That may look slower, but it helps the child understand why the column method works.
This matters because primary maths builds in layers. Weak understanding in Year 2 can make Year 4 fractions harder. Poor place value can affect column addition, subtraction, decimals and measures. A shaky sense of multiplication can make division, fractions, area and ratio feel confusing later on.
Primary maths is not just about getting the answer. It is about noticing patterns, explaining reasoning and choosing sensible strategies.
A child who understands maths can say things like:
“I know 8 + 7 is 15 because 8 + 2 makes 10, then 5 more makes 15.”
“I know 6 × 4 is 24, so 60 × 4 is 240.”
“Three quarters is more than one half because if the whole is split into four, three parts is more than two parts.”
“The answer can’t be 103 because I’m adding two numbers that are both about 30.”
These comments show understanding. They also show confidence.
At home, the aim is not to reteach the whole maths curriculum. It is to give children chances to handle, see, say and practise ideas in a calm way.
The concrete-pictorial-abstract journey helps ideas stick
A helpful way to think about primary maths is the concrete-pictorial-abstract approach, often shortened to CPA.
It gives children three connected ways to understand a mathematical idea.
Stage | What it means | What it might look like at home |
Concrete | Children use real objects they can move and touch | Counters, pasta, coins, building bricks, buttons, beads, snacks |
Pictorial | Children draw or look at pictures to represent the maths | Dots, bar models, number lines, arrays, part-whole diagrams |
Abstract | Children use numbers and symbols | `8 + 5 = 13`, `4 × 6 = 24`, `3/4` |
The stages are not a race. Children often move back and forwards between them. Even confident mathematicians use diagrams and objects when ideas become more complex.
Concrete methods use real objects
Concrete learning means using physical items to show the maths. These are often called manipulatives because children can move and arrange them.
A manipulative does not have to be a specialist classroom tool. At home, useful objects include:
Lego or building bricks
Pasta shapes
Buttons
Coins
Playing cards
Dice
Counters from board games
Beads
Small toys
Cereal pieces
Straws
Egg boxes
Measuring jugs
Kitchen scales
The key is that the object should match the idea. If a child is learning to count, any small object may work. If they are learning place value, objects that can be grouped into tens are more helpful. If they are learning fractions, paper, food or shapes that can be split into equal parts often work well.
Concrete resources help children see that maths is not magic. It is something they can build and test.
For example, to show `6 + 4`, place 6 counters on the table, then add 4 more. Count the total. Then ask:
“What did we start with?”
“What did we add?”
“How many are there now?”
“Can you show it another way?”
That final question matters. If your child can arrange the counters as 5 and 5, or 7 and 3, they are beginning to see number relationships.
Pictorial methods help children draw the maths
A pictorial method is not just a pretty drawing. It is a representation of the maths.
A child might draw:
Dots to stand for objects
A number line to show jumps
A bar model to compare amounts
An array to show multiplication
Circles split into equal parts for fractions
Tens and ones drawings for place value
Take `23 + 14`. A child could draw two tens sticks and three ones, then one ten stick and four ones. They can see 3 tens and 7 ones, so the total is 37.
For multiplication, `4 × 3` can be shown as 4 groups of 3 dots, or as an array with 4 rows and 3 columns. This makes it easier to see why `4 × 3` and `3 × 4` give the same total.
For division, 12 counters can be shared into 3 groups. Then the child can draw 3 circles and put 4 dots in each. After that, the abstract calculation `12 ÷ 3 = 4` has meaning.
Pictorial methods are especially useful when objects become too slow or cluttered. A child may not want to count out 84 pasta shapes, but they can draw 8 tens and 4 ones.
Abstract methods are symbols with meaning
Abstract maths is the written number sentence, calculation or formal method. It is where children use digits and symbols without needing every object in front of them.
The abstract stage includes:
Number sentences
Column methods
Fraction notation
Times table facts
Algebra-like missing number problems
Written division methods
Children need this stage. The goal is not to keep them using counters forever. The goal is to make sure that the symbols mean something.
A useful home routine is:
Build it
Use objects to show the problem.
Draw it
Make a quick picture or diagram.
Write it
Record the calculation with numbers and symbols.
For example, if the question is `15 - 7`, build 15 using objects, physically take away 7, draw what happened, then write `15 - 7 = 8`.
This may feel slow at first. It becomes quicker with practice, and it reduces guessing.
Everyday manipulatives that work well at home
The best manipulatives are the ones you can find easily and use often. A short, relaxed activity with familiar objects is usually more useful than a complicated setup.
Use counters for counting, addition and subtraction
Counters can be anything small and safe. Buttons, beads, dried pasta, small bricks or coins all work.
Try these simple activities:
Make a number
Say a number up to 20, 50 or 100 and ask your child to build it.
One more and one less
Build 16. Add one. Take one away. Say what changed.
Addition stories
“There are 9 grapes on a plate. I add 5 more. How many now?”
Subtraction stories
“There are 14 toy cars. 6 drive away. How many are left?”
Missing number problems
Build 12. Hide some under a cup. If 7 are still showing, how many are hidden?
Missing number work is powerful because it builds flexible thinking. Children often find `12 - ? = 7` harder than `12 - 5 = ?`, even though the same numbers are involved.
Use ten frames to build number sense
A ten frame is a rectangle with 10 spaces, usually arranged as 2 rows of 5. You can draw one on paper.
Ten frames help children see numbers without counting every item one by one. For example, 8 is seen as 5 and 3. 9 is one less than 10. 6 is 5 and 1.
Try placing counters on a ten frame and asking:
“How many do you see?”
“How do you know without counting?”
“How many spaces are empty?”
“How many more to make 10?”
This supports addition, subtraction and number bonds. Number bonds to 10 and 20 are a major foundation for primary maths.
If a child knows 8 needs 2 more to make 10, then `8 + 5` becomes easier. They can think, “8 plus 2 is 10, then 3 more is 13.”
Use coins for place value and real-life maths
Coins make maths feel purposeful. They help with counting, addition, subtraction and place value.
For younger children, sort coins by size, colour and value. Practise making small amounts, such as 10p, 20p or 50p, in different ways.
For older primary children, use coins to compare strategies:
“Can you make 37p?”
“Can you make it using the fewest coins?”
“Can you make it three different ways?”
“If something costs 65p and I pay with £1, what change should I get?”
Money is useful because children quickly see that ten 1p coins can be exchanged for one 10p coin. This links to place value and regrouping.
Be aware that some children find money surprisingly tricky because coin size does not match value. A 10p coin is smaller than a 2p coin, so children need time to learn that the printed value matters.
Use building bricks for place value and fractions
Building bricks can show groups, arrays, fractions and patterns.
For place value, choose single bricks as ones and joined groups of 10 as tens. Build numbers such as 34, 52 and 109.
For fractions, equal-sized bricks are useful. If an 8-stud brick is the whole, then a 4-stud brick can show one half, a 2-stud brick can show one quarter, and two 2-stud bricks can show two quarters.
Ask questions such as:
“How do you know these are equal parts?”
“Can you make the same whole in a different way?”
“Which is larger, one half or one quarter?”
“Can you prove it?”
The word equal is central to fractions. Children need to understand that fractions are equal parts of a whole, not just any pieces.
Use number lines for movement and comparison
A number line helps children see numbers in order and understand distance between numbers.
You can draw a number line on paper, use chalk outside, or create one with sticky notes on the floor.
Number lines support:
Counting forwards and backwards
Addition as jumping on
Subtraction as jumping back
Finding the difference
Rounding
Negative numbers for older primary children
Simple fractions and decimals
For `36 + 27`, your child might start at 36, jump 20 to 56, then jump 7 to 63.
For `52 - 28`, they might count up from 28 to 52:
28 to 30 is 2
30 to 50 is 20
50 to 52 is 2
The difference is 24.
This method can be easier than “taking away” when the numbers are close together or when subtraction involves regrouping.
How to support key maths areas without creating pressure
The most useful home support often happens in short moments. Five calm minutes can do more than 30 minutes of frustration.
Place value means understanding what digits are worth
Place value is the idea that a digit’s value depends on its position. In 47, the 4 means 4 tens, not 4 ones.
To support place value, build numbers using tens and ones. If you do not have base ten equipment, use bundles of straws, sticks, bricks or drawings.
Try this sequence:
Say the number 36.
Build 3 tens and 6 ones.
Draw 3 tens and 6 ones.
Write `36 = 30 + 6`.
Ask, “What would happen if we added 10?”
Ask, “What would happen if we took away 1?”
Older children can extend this to hundreds, thousands, tenths and hundredths.
Place value also helps children understand zero. In 304, the zero shows there are no tens. It is not an empty decoration.
Addition and subtraction need strategy, not just speed
Many children think good maths means fast maths. Speed has a place, but strategy matters more.
Teach children to ask, “What method makes sense here?”
For `99 + 36`, a child might think:
99 is close to 100.
Add 1 to make 100.
36 becomes 35.
100 + 35 = 135.
For `43 - 19`, they might think:
19 is close to 20.
43 - 20 = 23.
Add 1 back.
The answer is 24.
These mental strategies depend on number sense. Manipulatives and drawings help children see why they work.
If using column methods, connect them to place value. When exchanging in subtraction, do not say, “Borrow from next door” without meaning. Say, “We exchange 1 ten for 10 ones.”
For example, in `42 - 17`, 42 has 4 tens and 2 ones. To subtract 7 ones, exchange 1 ten for 10 ones. Now there are 3 tens and 12 ones. Subtract 7 ones and 1 ten to get 25.
This language makes the method less mysterious.
Multiplication is equal groups, arrays and repeated addition
Multiplication is not only chanting tables. It means equal groups.
Show `5 × 3` as:
5 groups of 3 objects
3 + 3 + 3 + 3 + 3
An array with 5 rows of 3
A number line with five jumps of 3
Arrays are especially helpful because they show the structure clearly. They also help children see that multiplication can be turned around:
`5 × 3 = 15`
`3 × 5 = 15`
This is called commutativity, though children do not need the formal word straight away.
Times table practice works best when children understand what the facts mean. Use rhythm, songs and quick recall, but connect facts to visuals too.
For example:
2 times table
Pairs, socks, eyes, wheels on bicycles
5 times table
Hands, tally marks, minutes on a clock
10 times table
Tens, place value, money
4 times table
Double the 2 times table
8 times table
Double the 4 times table
These links reduce the memory load.
Division is sharing and grouping
Children often meet division in two forms.
Sharing means splitting a total into equal parts. For example, 12 sweets shared between 3 children gives 4 each.
Grouping means finding how many groups can be made. For example, 12 sweets put into groups of 3 makes 4 groups.
Both are division, but they feel different.
Use objects to act out both meanings:
Sharing
“Share 20 counters equally between 4 plates. How many on each plate?”
Grouping
“Put 20 counters into groups of 4. How many groups can you make?”
This helps later when children meet remainders. If 22 counters are shared between 4 plates, each plate gets 5 and 2 are left over. Your child can see the remainder instead of treating it as a random extra number.
Fractions need equal parts and a clear whole
Fractions cause confusion when the whole is unclear. One half of a small biscuit is not the same size as one half of a large pizza, but both are one half of their own whole.
Always ask, “What is the whole?”
Use paper folding, food, bricks, drawings and measuring jugs. Make halves, quarters and thirds. Compare them.
Children may assume that larger denominator means larger fraction because 8 is bigger than 4. Concrete and pictorial methods help correct this. If the same-sized whole is split into 8 equal parts, each part is smaller than when it is split into 4 equal parts.
A good question is:
“Would you rather have one half of this cake or one eighth of the same cake? Why?”
Draw it. Cut paper. Build it. Let the child see the difference.
What to say when your child is stuck
The way adults respond to struggle can shape how children feel about maths.
A child who says “I can’t do it” may be feeling embarrassed, overloaded or afraid of getting it wrong. A calm response helps them stay open to thinking.
Useful phrases include:
“Let’s build it first.”
“Can you draw what is happening?”
“What do you already know?”
“Can we try smaller numbers?”
“Is there a fact that could help?”
“How could we check?”
“Tell me where it started to feel tricky.”
These phrases move attention away from panic and back towards problem-solving.
Try not to rush in with the full method. If an adult explains too much, the child may become passive. Instead, offer a small prompt and let the child do the next step.
For example, if the question is `16 + 7`, you might ask:
“What could you add to 16 to make 20?”
If the child says 4, you can ask:
“So if 7 is split into 4 and something, what is left?”
This guides without taking over.
Keep practice short and regular
Maths confidence grows through repeated success. It does not need to feel like a lesson every time.
Short activities can include:
Counting steps as you go upstairs
Doubling numbers while setting the table
Reading prices in a shop
Weighing ingredients when baking
Sharing snacks equally
Asking the time on an analogue clock
Spotting shapes on a walk
Estimating how many items are in a jar
Playing dice games
Keeping score in a board game
Real-life maths helps children understand why numbers matter. It also shows that maths is part of everyday life, not just something on a worksheet.
Use games to build fluency
Games are ideal for practice because they include repetition without feeling like a test.
Try:
Dice addition
Roll two dice and add the numbers. Older children can multiply them.
Make 10
Use playing cards. Turn over cards and find pairs that make 10.
Number target
Choose a target number, such as 50. Roll dice and add, subtract or multiply to get as close as possible.
Shop role play
Price toys or snacks with sticky notes. Use coins to buy and give change.
Array hunt
Look for arrays at home, such as egg boxes, chocolate squares, tiles or window panes.
Fraction plates
Cut paper plates into equal parts and build wholes from halves, quarters and eighths.
Games also reveal thinking. Listen to the strategy your child uses. Do they count all? Count on? Use known facts? Group items? Estimate?
That information helps you choose the next prompt.
Praise thinking, not just answers
Children need to know that effort, strategy and explanation matter.
Instead of only saying “Well done, that’s right,” try:
“I like how you used the ten frame.”
“You changed strategy when the first one did not work.”
“Your drawing made the problem clearer.”
“You checked your answer in a sensible way.”
“You used a fact you already knew.”
This helps children see themselves as learners who can improve.
Be careful with phrases like “I was never good at maths.” Many adults say this to reassure children, but it can send the message that maths ability is fixed. A better phrase is, “Some maths takes time. We can work it out step by step.”
Working with school and knowing when to step back
Home support works best when it sits alongside school, rather than pulling in a different direction.
If your child is confused by homework, ask them to show you how they did something in class. If they cannot remember, it is fine to write a short note to the teacher explaining where they got stuck. Teachers would usually rather know that a child struggled than receive homework completed by an adult.
If methods look unfamiliar, ask the school for examples. Many schools share calculation policies or parent guides that show the progression from objects and drawings to written methods. These can be very helpful because they show the language your child hears in class.
When to pause homework
Sometimes the best support is stopping for the evening.
Pause if your child is:
Becoming tearful or angry
Guessing wildly
Saying they are stupid
Unable to explain any part of the task
Too tired to focus
Repeating the same error without noticing
Write down what happened in simple terms. For example, “We tried questions 1 to 4. The exchanging in subtraction was difficult, even with counters.”
That gives the teacher useful information.
A calm ending protects confidence. You might say, “This one needs a bit more practice. We’ll let your teacher know where it got tricky.”
When a child needs more support
Some children need more time, more repetition or a different explanation. That is normal. Maths develops at different rates.
It may be worth speaking to the teacher if your child often:
Finds counting sequences difficult
Struggles to recognise small amounts without counting
Mixes up number order
Finds place value confusing after repeated practice
Has high anxiety around maths
Avoids homework completely
Cannot remember number facts despite regular practice
Seems much less confident than in other subjects
This does not mean anything is “wrong”. It simply means the adults need a clearer picture of what support will help.
What a good home maths routine can look like
A simple weekly rhythm is enough for many families. It might look like this:
Time | Activity | Purpose |
5 minutes, a few times a week | Number facts, counting or times tables | Builds recall |
10 minutes once or twice a week | Homework with objects or drawings nearby | Supports school learning |
During everyday life | Money, measures, time, sharing or estimating | Makes maths meaningful |
Once a week | Dice, cards or board game | Builds fluency without pressure |
The routine should feel manageable. If it becomes a battle, reduce the amount and focus on confidence first.
The most powerful question you can ask is often, “Can you show me?”
Ask your child to show the problem with objects. Then ask them to draw it. Then ask them to write the number sentence. This one routine supports counting, place value, operations, fractions and problem-solving.
Supporting Your Primary Age Child with Maths Using Manipulatives and Concrete-Pictorial Methods is really about helping maths make sense before expecting it to become quick. When children can touch, see and explain an idea, written methods become less frightening.
The takeaway is simple. Keep maths visible. Use objects. Draw pictures. Talk about thinking. Celebrate sensible strategies. Short, calm practice at home can make a lasting difference to how a child sees themselves as a mathematician.
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