Beyond finger counting: mental addition and subtraction strategies

Watch a six-year-old work out 8 + 5. Many will hold up eight fingers, then count five more, lips moving. It works, and there is nothing wrong with it as a starting point. But the same child at ten, facing 48 + 37, cannot possibly use fingers. The children who do well in the later grades are not the ones who memorised the most facts; they are the ones who learned to think about numbers, using a handful of flexible mental strategies. This article describes those strategies, explains why they work, and offers ways to practise them at home without a worksheet.

Written for children aged 6 to 9.

From counting all, to counting on, to thinking in chunks

Children generally pass through a predictable path. First they count all: for 8 + 5, they show eight, show five, and count the whole lot from one. Then they count on: they start at 8 and say "nine, ten, eleven, twelve, thirteen". This is a real advance, because the child has understood that the 8 does not need to be counted again. Finally, they begin to use derived facts: they work from something they already know to something they do not. This last stage is where mental arithmetic really lives.

You can help your child move along this path by asking questions that invite them to use what they know: "What do you already know that could help?" is more useful than "What's the answer?"

The core strategies

Make ten. For 8 + 5, think "8 needs 2 to make 10; take 2 from the 5; that leaves 3; so 10 + 3 = 13." This is the most important strategy in early addition, and it relies on knowing the pairs that make 10 (see our article on number bonds). It also leads naturally into place value, because the child sees ten as a new unit.

Doubles and near-doubles. Doubles such as 6 + 6 = 12 tend to be learned early because they are rhythmic and visual. Once known, they open up neighbours: 6 + 7 is one more than 6 + 6, so 13; 6 + 5 is one less, so 11. Children enjoy discovering that one fact can unlock several.

Counting on from the larger number. For 3 + 9, it is far easier to start at 9 and count on three than to start at 3 and count on nine. Children can discover for themselves that addition can be done in either order (it is called commutativity) and use it to their advantage.

Using a friendly number. Adding 9 or 8 is awkward, but adding 10 is easy. So 36 + 9 becomes 36 + 10 − 1 = 45. Similarly, 47 + 28 might become 47 + 30 − 2. Rounding to a friendly number and adjusting is a powerful and general technique.

Splitting by place value. To add 34 + 25, split each number into tens and ones: 30 + 20 = 50 and 4 + 5 = 9, so 59. This strategy is a mental form of the written column method, and understanding it makes the written method far less mysterious later. It is also the natural strategy to use when no carrying is needed.

Jumping along a number line. To work out 46 + 28, start at 46, jump by 20 to 66, then by 8: 66 → 70 → 74. Drawing the line helps at first; later it happens in the head. Number-line jumps in friendly steps (to the next ten, then the rest) are one of the best-researched mental methods.

Subtraction: thinking of it as a gap

Subtraction is harder for children than addition, and many get stuck on it. One of the best ideas for subtraction is that it can be seen as the distance between two numbers, not only as "taking away".

Counting up to find the difference. For 52 − 47, taking away is awkward, but the gap between 47 and 52 is clearly five. Counting up from the smaller number (47 to 50 is 3, then 2 more, so 5) is often quicker and less error-prone than taking away, particularly when the two numbers are close.

Taking away through ten. For 15 − 8, think "take 5 to get to 10, then take 3 more, so 7." The make-ten idea works in both directions.

Using the related addition. If a child knows that 7 + 8 = 15, then 15 − 8 = 7. Encouraging children to think of subtraction as "what do I add to 8 to get 15?" links the two operations.

Subtracting a friendly number. Taking away 9 is easier as "take away 10, then add 1 back": 63 − 9 = 54.

What the research says

Studies of children's arithmetic repeatedly find that flexible thinkers do better than children who rely on one procedure. Children taught a variety of strategies, who are asked to explain which strategy they chose and why, develop deeper number sense and are more resilient when faced with unfamiliar numbers. By contrast, teaching one fixed method too early can lead to children who apply it blindly, for example carrying out a long column subtraction for 1000 − 1 when the answer is obvious.

This is why it is so valuable to ask your child how they got an answer, and to accept different routes. "I did 8 + 2 = 10 and then 3 more" and "I counted on from 8" are both honest answers; the first is just further along the path.

Talking maths at home

Mental strategies grow in conversation. A few habits help.

Ask for the method, not just the answer. "How did you get that?" is the single most useful question a parent can ask. Listen without correcting at once.

Share your own thinking aloud. When you work something out, say it: "I'm paying for 19; I'll think of it as 20 take away 1." Children learn a great deal from hearing adults think.

Compare methods. "Your brother did it by making ten; you counted on. Which do you find easier here?" There is rarely one right way, and choosing a method for the numbers at hand is itself a skill.

Use real situations. Shopping, cooking, sports scores and travel times provide endless natural addition and subtraction. "We have 27 minutes until the show and the drive takes 15. How long can we wait?"

Keep it brief and playful. Five minutes of "number of the day" talk ("What's 7 + something that makes 20?") is better than half an hour of grudging drill.

Let fingers go gradually. When a child is ready, they will use fingers less. If you see them hide their hands under the table, that is usually progress, not cheating.

A word on speed and accuracy

Children and parents often worry about speed. In the early years it matters less than reasoning. Flexible methods get quicker with practice, and children who understand why a strategy works remember it far better than those who memorised the steps. When speed is important, as in the later grades, fluent strategies tend to deliver it naturally.

So the next time your child reaches for their fingers, don't take them away. Instead ask what they already know, and watch what happens when you invite them to use it. Over a few months, you will notice the fingers coming out less often, and something far better in their place: a child who sees numbers as things they can move, split and rearrange.

Try it in the app

Practise this at the screen too: Add and subtract ten in your head, Doubles and near doubles, Add and subtract two-digit numbers mentally. You may also like our articles on number bonds and place value.