# Education Endowment Foundation:EEF blog: The complexity of counting on

## EEF blog: The complexity of counting on

Author
Grace Coker
Content Specialist for Maths

Grace Coker is our maths content specialist. In this blog, she explores the importance of giving young children regular opportunities to practise learned maths strategies in a range of contexts, such as with storybooks, puzzles and games.

Blog •3 minutes •

Farhan, you have three cubes.”

Sahar, you have four cubes.”

How many cubes are there altogether?”

Mrs Dalkin watches as Farhan counts his own cubes again, touching each cube as he does so… one, two, three. He then continues by counting his friend Sahar’s… four, five, six, seven.

There are seven cubes altogether.”

Sahar, do you agree with Farhan?”

Yes, I think there are seven too as I have four and Farhan has three more so five, six, seven.” Sahar lifts three fingers in turn as she counts five, six and seven.

Both children identified the total, but their strategies were different.

Farhan used the counting all’ technique, and Sahar counted on’ from the larger number.

### Does the method matter?

Counting on’ is a more efficient method than counting all’. But it’s also a more complex one. It requires the child to hold one number in their head’ and count on. This is challenging to do at first, as it relies heavily on the child’s working memory.

So how can Mrs Dalkin encourage children to use this more efficient yet more complex strategy?

Well, first it’s about being knowledgeable about the methods children could use to solve this problem.

It’s about being aware that children will be familiar with the counting all’ method as they will have used it in previous years at home and at school.

It’s also about recognising that counting on’ is a more complex method that doesn’t just come naturally to some children. It’s a method that needs to be modelled, repeatedly practised, and regularly promoted.

Spending time talking about the different maths strategies used, even with younger children, can help pupils to see why one method is more efficient than the other.

Boaler (2009)
suggests that children who are involved in talking about different mental calculation strategies can see how thinking flexibly about number can help them approach problems in different ways.

### Counting on in different contexts

Learning does not happen instantaneously, but over time. Children need repeated experiences and practice in many different contexts to support their understanding.

Playing games can be a motivating context for children to embed and reinforce learning.

The second recommendation of our ​‘Improving Mathematics in the Early Years and Key Stage 1’ guidance report states that board games with spaces that are linearly arranged, consecutively numbered, and equal-sized may be particularly beneficial to numerical understanding, by providing opportunities for developing strategies such as counting on.’

### Caterpillar Game

Let’s take the Caterpillar Game taken from Mathematical Reasoning, a whole class programme designed to develop pupils’ understanding of number and reasoning about quantity, as an example.

In this whole-class activity, the teacher projects the slide on the board and the class plays as two teams race to reach the end of the caterpillar. Each team throws the die and pupils work out where their mark will be by counting on from where the marker is.

For example, if a team’s marker was positioned on three, and they rolled a 4 on the die, they would count four, five, six, seven’ whilst lifting four fingers in turn. The team’s marker would then be moved to position seven.

Fingers are an important manipulative for children. They can be useful for supporting counting.

To win the game, the team must end exactly on the last square, requiring them to anticipate what number they need to finish and compare this with the number thrown.

Children are also given the opportunity to play the game in pairs to continue practising the counting on’ strategy.

The complexity of counting on’ can be made easier for children like Farhan, through embedding and reinforcing the learning through motivating contexts. Purposeful, playful practice – you can count on’ it!

Check out this video kindly given from Mathematical Reasoning (University of Oxford) of the Caterpillar Game being played.

Search
your school to explore which EEF-funded projects to support maths in the early years and Key Stage 1 you could take part in.

References

The book referenced in this blog is:Boaler, J (2009) The Elephant in the Classroom: Helping Children Learn and Love Mathematics. London: Souvenir Press.

## Keep-up-to date with our latest news and resources

Our News Alerts are e‑mailed to 45,000+ subscribers regularly.