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How maths is taught in primary schools now

If your child's maths book looks nothing like yours did, you're not alone. Primary schools in England now teach children to understand why a method works before they practise it, so you'll see lots of objects, pictures and diagrams along the way. This guide explains the methods your child is most likely to meet, roughly in the order they meet them.

Concrete, pictorial, abstract

Most schools follow an approach called concrete-pictorial-abstract, or CPA:

• Concrete: children handle real things, such as counters, cubes, straws or blocks, to see what is happening.
• Pictorial: they draw it, using dots, bars, number lines or diagrams.
• Abstract: they use only numbers and symbols, like 34 + 25 = 59.

This isn't only for younger children. A Year 5 pupil meeting a new idea may well start with blocks again. Moving back to objects is not a sign that a child is behind.

Ten frames and part-whole models

In Reception and Year 1, two pictures come up again and again.

A ten frame is a 2 by 5 grid. Children see at a glance that 8 is "5 and 3 more" and "2 less than 10", without counting one by one. It builds number bonds to 10, which almost every later sum relies on.

A part-whole model shows a whole split into parts. From one picture, children can read four facts: 8 + 2 = 10, 2 + 8 = 10, 10 − 8 = 2 and 10 − 2 = 8. It shows that addition and subtraction are linked. There's more on this in our guide to number bonds to 20.

Number lines

From Year 1, children count forwards and backwards along number lines. Later they use an empty number line, with no marks except the ones they need. To work out 63 − 28, a child might count up from 28: 2 to get to 30, then 30 to get to 60, then 3 to get to 63. That's 2 + 30 + 3 = 35, so 63 − 28 = 35.

Partitioning

Partitioning means splitting a number into parts, usually tens and ones. To add 47 + 35, a child might partition both numbers: 40 + 30 = 70, and 7 + 5 = 12, so the answer is 70 + 12 = 82. It's a mental method, and it's also the idea underneath every written method that follows.

Column addition and subtraction

The formal written column method is usually taught from Year 3. The numbers are lined up so that ones sit under ones, tens under tens, and so on.

Addition: for 456 + 278, add the ones first: 6 + 8 = 14. Write 4 in the ones column and move the 1 ten into the tens column. Then 5 + 7 + 1 = 13 tens: write 3 and move 1 into the hundreds. Then 4 + 2 + 1 = 7. The answer is 734.

Subtraction: for 532 − 278, the ones are 2 − 8, which can't be done. So the child exchanges one of the tens for ten ones: the 3 tens become 2, and the 2 ones become 12. Now 12 − 8 = 4. In the tens, 2 − 7 can't be done, so exchange a hundred: 5 hundreds become 4, and 2 tens become 12. 12 − 7 = 5. Then 4 − 2 = 2. The answer is 254.

You may have learnt to call this "borrowing" or "carrying". Schools now say exchanging (sometimes "regrouping"), because nothing is borrowed and paid back: one ten really is swapped for ten ones.

The grid method

Many schools use the grid method in Year 3 or 4 as a step towards formal multiplication. The numbers are partitioned and each part is multiplied separately.

It's slower than the methods that come next, but it shows clearly why multiplication works, and children can see where every part of the answer comes from.

Short multiplication

From Year 4, children multiply 2-digit and 3-digit numbers by a 1-digit number using short multiplication, a column method. For 243 × 6: 3 × 6 = 18, so write 8 and carry 1 ten. 4 × 6 = 24 tens, plus the 1 is 25: write 5 and carry 2 hundreds. 2 × 6 = 12 hundreds, plus 2 is 14. The answer is 1,458. In Year 5, this grows into long multiplication, for multiplying by 2-digit numbers.

Short division: the "bus stop" method

Short division is called the bus stop method because the number being divided sits under a line that looks like a bus shelter. It is in the curriculum for Year 5, though some schools start it earlier. For 745 ÷ 5: 7 ÷ 5 = 1 remainder 2, and the 2 moves next to the 4 to make 24. 24 ÷ 5 = 4 remainder 4, and that 4 moves next to the 5 to make 45. 45 ÷ 5 = 9. The answer is 149.

Chunking

Chunking is dividing by taking away big "chunks" of the number you're dividing by. For 156 ÷ 6: take away 10 lots of 6 (60), leaving 96. Take away another 10 lots (60), leaving 36. Then 6 lots of 6 is 36, leaving 0. That's 10 + 10 + 6 = 26 lots, so 156 ÷ 6 = 26. It's useful because children can use facts they're sure of, and it's a way into long division in Year 6.

Bar models

A bar model is a drawing that turns a word problem into a picture. It's used all the way through primary school, and is especially helpful for problems that seem confusing in words.

Why schools teach this way

Many schools in England follow a mastery approach, influenced by the way maths is taught in places such as Shanghai and Singapore. The main ideas are:

• Everyone can learn maths. Being "a maths person" isn't something you're born with.
• The class moves on together. Rather than racing ahead, children who understand quickly are given deeper problems on the same topic, and children who need more help get it quickly, often the same day.
• Understanding first, speed second. More time is spent on fewer topics so that ideas stick.
• Fluency, reasoning and problem solving. These are the three aims of the National Curriculum. Children practise facts until they're quick, but are also asked to explain and to use what they know in new situations.

The written methods are where children end up, not where they start. A child who understands why exchanging works is far less likely to get muddled than one who has learnt the steps by rote.

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