Number sense

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What is Number Sense?

Number sense is an emerging construct that refers to a child’s fluidity and flexibility with numbers and what numbers mean as well as an ability to perform mental mathematics and to look at the world and make comparisons.

There may be:

  1. An awareness of the relationship between number and quantity
  2. An understanding of number symbols, vocabulary and meaning
  3. The ability to engage in systematic counting, including notions of cardinality and ordinality
  4. An awareness of magnitude and comparisons between different magnitudes
  5. An understanding of different representations of number
  6. Competence with simple mathematical operations
  7. An awareness of number patterns including recognising missing numbers

Number sense develops gradually over time and at different rates, through exploring numbers, visualising them in a variety of contexts, and relating them in ways that are not limited by formal written methods.


Flexibility with Number Epitomises Number Sense.

Number sense is the ability to be flexible with numbers and to understand both how our number system works and how numbers relate to each other. Children with good number sense have a range of mathematical strategies at their disposal and they know when to use them and how to adapt them to meet different situations.

What Does Good Number Sense Look Like?

Children with good number sense can manipulate numbers to make calculations easier and are flexible in their approach to solving problems. They can assess the reasonableness of an answer, and routinely estimate answers before calculating. They look for connections and readily spot patterns in numbers, which helps them predict future outcomes. These children have several approaches to calculating and problem solving and can use and adapt these for new situations. Children with good number sense enjoy playing with and exploring numbers and number relationships. As a result of these strategies, they can often find the most efficient solution to the problem.

What Does Poor Number Sense Look Like?

Children with poor number sense are procedure focused and will tend to rely on methods that they feel secure with. They apply inefficient and immature strategies to
calculations and fail to spot links and connections that could get them to the answer more quickly. They prefer to use pen and paper rather than work things out in their heads. They are reluctant to estimate an answer before working it out and will generally accept whatever answer they get, without considering whether it is reasonable or not.

This was beautifully illustrated to me by a Year 5 child, who was tackling a question that required her to estimate the sum of two 4-digit numbers before calculating the answer. She approached this task by calculating the answer and then giving an estimate. I asked her why she was doing it that way around and her reply was,

‘It is much easier to find an estimate for the answer after you have worked out what the answer is.’

You have to admire her logic – if nothing else!

Children with poor number sense do not enjoy maths and won’t spend time being creative with and exploring numbers. Ironically, they are doing a harder version of maths, that relies upon remembering and applying procedures, with little understanding of the underlying numerical concepts.


When Does Number Sense Develop?

Psychologists, Klein and Starkey (1988) found that we are born with a sense of number. They measured the focus time of babies looking at pictures of dots and discovered that when the number of dots changed the babies’ focus time changed.

The babies had appreciated a difference in numerical quantity.

This ability to appreciate number quantity is a survival instinct. When our ancestors were out hunting and gathering they needed to be able to perceive danger. So, if one animal approached a couple of hunters, they saw this as an opportunity for a meal. However, if 10 animals approached them, they ran, or they became the meal!

So, we know that very young children can recognise the number of items in a group without having to count them. We call this subitising, and most people can subitise up to six or seven items, when they are randomly arranged.


How Can We Develop Number Sense?

Number sense can be developed over time by providing opportunities to explore and play with numbers; to visualise numbers in different contexts and to spot and predict the patterns and relationships between numbers.

Strategies for Early Years

We can develop number sense from an early age by using visual cluster cards. These are cards showing a number of dots arranged in different ways. This helps the child to develop mental strategies to ascertain the number quantity.

Ten frames are also a useful tool for developing number sense and a simplified version of these can even be introduced at a very early age by looking at a five frame.

For example, let’s place three counters on a five frame. We can encourage the children to talk about what they can see here.

How many counters?
How many spaces are there?
Is three less than five?
How do you know?
Do both pictures show three counters?
How are they the same and how are the different?
If we then look at another arrangement

We can now see that 3 is made up as 

2 + 1

And it is also 1 + 1 + 1 and 3 + 0

Activities like this can help children to understand the ‘threeness’ of three and to understand what the number is made from and how it relates to other numbers, such as five.

Strategies for More Advanced Learners

Explore Different Methods
Give problems with multiple solutions and encourage children to think of different solutions. This helps them move away from the idea that there is only one way to solve a problem and will encourage them to reason mathematically. It also exposes them to ideas and strategies that they may not have previously considered. If children have only been shown one method then their focus will be on remembering and using that method rather than thinking about what the numbers mean and how they work together.

For example, consider this apparently straight-forward question from Mike Askew’s blog

Which number is the odd one out?

15      20      23      25

20?  The only even number.

15?  The only number that is not in the 20’s.

23?  The only prime number.

25?  The only square number.

There are several reasons why each of these numbers could be the odd one out and this type of activity will help to develop number sense, particularly if all the reasons that the children come up with are shared and evaluated. It is also a very inclusive problem as every child will be able to come up with one number that they think is the odd one out and why. The mathematical richness in this activity comes with the discussions and reasons why each number could be the odd one out.

Developing number sense relies on having the opportunity to play with numbers and to explore how they work together. It opens the world of creativity in maths and enables children to enjoy number without the constraints of remembering rules and procedures. Even the most reluctant mathematician can develop their sense of number, as long as they have been given the environment and opportunity to do so.

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