The mathematical brain - how children learn to count
Counting is easily taken for granted but there's a lot of fascinating research into how we learn to count - and there's more to it than you may think.
The mathematical brain
It’s first worth considering where our capacity to do mathematics comes from.
Neuropsychologist Brian Butterworth in his book The Mathematical Brain, suggests we’re born with an innate sense of number hard-wired into our brain and he attributes this to a small region of the brain behind the left ear that he calls 'the number module'. He compares this idea to colour – in the same way we perceive the greenness of a leaf, we can also perceive the 'twoness' or 'threeness' of a group of objects.
Take counting: like times tables and algebra, we tend to think it's something kids have to be taught. Wrong, says Butterworth - it's an instinct. Sure, we have to learn the names and symbols of numbers to develop that instinct, but, because the number module is hardwired into the brain, basic counting comes naturally.
Remote tribes can count even when they have no words for numbers. In maths as in language, he believes, "kids start off with little starter kits." And their maths starter kit is the number module.
There are other theories too - such as maths being an extension of our spatial awareness – but there’s something nice about the idea of a little maths starter kit.
A word of warning: All this doesn't mean a child is predestined to be either good at maths or not. Far from it. We’re all born ready to learn maths, and it’s what happens in the first 10 years or so that sets us up for success.
Counting with toddlers
Research suggests that toddlers as young as 12 months have a sense of how many there are in a set up to around three objects. This comes from their innate sense of number.
Counting is learned when the toddler starts making the connection between this innate sense of how many there are and the language we use to count, for example 'one, two, buckle my shoe'. This is the first stage in learning maths and it's the building block for many early concepts.
Should parents count with their toddlers? Absolutely: count stairs and steps, count feet going into socks, count birds on the garden wall, count everything you come across in daily life. This is the sort of learning that doesn't need books (although reading books about numbers and counting is great too!).
Counting - early learning milestones
Here are some stages of learning to count that you may notice your child going through at ages three to five:
Knowing that each number word represents a different quantity.
Children 'count' from a very early age (think of all the kids songs and rhymes that involve counting), but don't realise that the words they are learning are associated with different quantities until they are around 4 years old.
Knowing the 'number words' must be said in a specific order
Realising that numbers come in a specific order which never changes is called the Stable Order Principle.
In the beginning, it's normal for children to skip numbers or get them muddled up as they count. But as their number skills develop, they come to realise that the order is important, and soon they will have it memorised.
Knowing that each object must be counted only once
This is known as the one-to-one correspondence principle. The process of managing how to count a group of objects without missing any and without counting any twice, is something that your child will develop with lots of practice.
Knowing that the last number counted is equal to the quantity of the set.
This is called the cardinality principle. If your child counts six oranges- 1, 2, 3, 4, 5, 6 - and then you ask "How many oranges are there?" and they start counting them again, then they haven't grasped the cardinality principle.
Realising that the number of objects in a set stays the same unless any are added or removed
If your child counts six cans of beans in a straight line, and you rearrange the beans (in front of their eyes) into two stacks of three, they will know there's still six cans without recounting. This is the linked to the Order Irrelevance Principle, which is when kids understand that the order they count things (right to left, or in a random way) doesn't change the number of items there are.
Counting non-visible objects
Your child will realise they can count things they can't touch or see - such as sounds, members of their family, or even ideas. This is called the abstraction principle, and is an understanding that we count everything in the same way, no matter whether it's apples or noises.
They will also understand that any group of objects can be counted as one set, regardless of whether or not all the objects in the set are the same. For example, they can count three piles of assorted items on the floor of the toy room.
Recognising how many objects are in a small set without counting
As mentioned, children are born with an innate ability to tell the difference between small numbers of objects. This ability develops so that by the age of four or five, they should be able to tell you there are five of something without counting them one by one. Subitising is a good indicator of how well children have grasped the other principles of counting.
Counting on - as a step towards adding
Learning to add comes as an extension of counting. Here are some stages a child goes through to make this connection:
- Counting all - For 3 + 5, children will count "one, two, three" and then "one, two, three, four, five" to establish the quantity of the sets to be added – for example, three fingers on one hand and five fingers on the other. The child will then count all the objects "one, two, three, four, five, six, seven, eight"
- Counting on from the first number - Some children come to realise that it is not necessary to count the first number to add. They can start with three, and then count on another five to get the solution. Using finger counting, the child will no longer count out the first set, but start with the word ‘Three’, and then use a hand to count on the second added: ‘Four, five, six, seven, eight’.
- Counting on from the larger number - It's more efficient when the smaller of the two numbers is counted. The child now selects the biggest number to start with which is "five", and then counts on "six, seven, eight".
- The final stage isn't really counting - it's where learners know their number facts and skip the time-consuming counting altogether.
Number lines are great visual tools for making this connection between "counting on" and addition or subtraction - we use them in Komodo a lot. Here's an earlier blog article all about number lines.
Beyond basic counting
Counting is the first mathematical pattern learners encounter. From here they soon begin to count backwards which is a step towards subtraction and they'll also count in twos, fives and tens which are a foundation for multiplication.
The next big step is the idea of place value and counting to base 10. Learners often make this leap simply because it's an obvious and efficient way to count large numbers. In Komodo, we use practice examples like this to help learners make the connection to counting in tens and ones.
It's easy to forget that counting is a key concept in maths with many stages before it's mastered. There's certainly a lot more to it than one, two, three!
I'm Ged, Co-founder of Komodo, ex-maths teacher and dad. If you have any questions please get in touch.
About Komodo - Komodo is a fun and effective way to boost primary maths skills. Designed for 4 to 11 year olds to use in the home, Komodo uses a little and often approach to learning maths (20 minutes, three to five times per week) that fits into the busy routine. Komodo users develop fluency and confidence in maths - without keeping them at the screen for long.
And now we've got Komodo English too - check it out here.
