Helping children understand fractions with Numicon

What makes fractions difficult to understand?

‘Fractions’ is often a topic area that many adults (and children) find particularly challenging, leading to uncertainty and maths anxiety. For many of us, if we reflect on our childhood experience of learning about fractions, it was mostly abstract procedural-based learning of tips and tricks (with very little understanding, in many cases). Models, when used, were often a linear fraction wall, pizzas or other circular images - the relatable context was often linked to food. The relationships between these models were not always explored and therefore the connections and relationships were often missed or confused. 

Fractions, decimals, ratios and percentages are essentially different forms of notation for the same ‘rational numbers’ - we represent them symbolically in several different ways. Often these different forms are taught at different times or using different representations, resulting in children not making the connections. Not having a secure understanding of the relevant mathematical language can be a further hindrance for children in explaining their understanding. 

Added to all of this, we now have children in our primary schools who have had at least two years of disrupted teaching and who may have missed core conceptual understanding. Did you know that the Numicon approach develops key mathematical ideas and vocabulary, supporting children to develop a deep understanding? 

The Numicon learning theories and pedagogy

Many of us are familiar with Numicon Shapes, and often these are seen as an integral part to a developed mathematical understanding for the youngest children in school. The Numicon Shapes are one element of the Numicon approach; a way of teaching mathematics that grew out of a classroom-based research project conducted by Ruth Atkinson, Romey Tacon and Dr. Tony Wing.

There are two main learning theories that underpin the Numicon approach: Jerome Bruner’s enactive-iconic-symbolic representations, often referred to as concrete-pictorial-abstract (or the CPA approach) and Numicon’s own pedagogy.

In Numicon, we interpret Bruner’s representations as follows:

• Concrete: the children engage with a physical resource that can be manipulated
• Pictorial: both the development of visualisation that stems from using the concrete resource that can be recalled subsequently, and/or a jotting or drawing
• Abstract: both the spoken word whilst discussing the concrete or pictorial representation, and/or the recording of mathematical understanding using symbols and numerals.

We do not see this as a progressively linear theory. Rather, we believe that by engaging in a concrete activity the children will also be developing their visualisation and describing their mathematical understanding. Sometimes the children might use a concrete resource to prove their thinking after they have worked using abstract notation. 

This theory links into the Numicon pedagogy wheel:

• Communicating mathematically: being active, illustrating, talking. This clearly links to Bruner’s theory. It is also worth noting that all partners (children and adults) are actively involved in the dialogue - not just passively hearing or waiting to speak. This often leads to a lesson structure that has a short teacher input - sometimes just a question - and (as illustrated below) children often begin to reason from the outset.
• Exploring relationships (in a variety of contexts): children explore a variety of connections and relationships through reasoning. If they know 7 ones and 3 ones make 10 ones, they will eventually know what to add to 0.7 to make 1.
• Generalizing: in doing mathematics, exploring relationships, and looking for patterns, children will make new situations predictable. This also links to Benjamin Bloom’s Taxonomy and higher order thinking.

All Numicon planning incorporates these learning theories, as exemplified below.

Adding fractions with Numicon

Let’s consider a Numicon activity taken from Book 3. In this activity children are writing adding sentences with fractions with the same denominator for the first time. If the Numicon planning is being followed in order, they will already have used resources to represent fractions and have met the language of numerator and denominator.

Start by asking the children to look at a six Shape. Ask, what is the relationship between one of the holes and the Numicon Shape? Agree one hole is one out of six or one-sixth. I would reinforce this understanding by counting the voids, one-sixth, two-sixths, three-sixths, four-sixths, five-sixths, six-sixths makes one whole. Next ask the children to cover their six Shape with two red, one green and three yellow pegs. Now ask them to discuss with their partner:

What fraction of the model is covered with red pegs, with green pegs, with yellow pegs?

There is no expectation that the children will discuss equivalent fractions for this model. However, if they do, you should discuss the relationships they can see.

Next ask the children to look closely at their model. Ask, what part of the fraction is represented by the Numicon pegs (The numerator) and what part is represented by the Numicon Shape? (The denominator). How does this help us to write the fraction two-sixths? (The two pegs sit on top of the Shape, i.e. the numerator sits on top of the denominator). Now ask the children to write an adding sentence to record the fraction of each colour in their model.

Ask, what do you notice about the number sentence? Agree that in this example the denominator has remained consistent, but the numerator has been added together. Finally, ask the children to work with a partner to find different ways to fill the six-shape to see if they can form a generalisation for adding fractions with the same denominator. Agree that when we add fractions with the same denominator, what we actually add are the numerators. This will support the children to consider subtraction (and later multiplication) of a fraction by a whole number.

Final thoughts

How can we support children to secure a deeper understanding of fractions?

• We use the Numicon learning theories and pedagogy.
• We maximise opportunities to fully explore the model with the children through questioning.
• We provide children with the specific vocabulary so they can reason and communicate their mathematical understanding.

Finally, a further consideration is that Numicon has a core set of concrete resources which continue to be used as a child progresses through school. The resources are the same, but the mathematics we explore is different. Using the same resources supports the children’s confidence as they have already established the relationships and connections of the resources through play activities in the Early Years.

For example, here we have a baseboard being used to explore fractions, decimals and percentages:

This is also a great activity to explore the equivalence between one-half and two-quarters.

A magnified one Shape becomes a decimal baseboard, here showing 0.1

Using the baseboard to explore percentages and connections between fractions, decimals and percentages e.g. 1/100 = 0.01 = 1% and 35/100 = 0.35 = 35%.

To learn how to get the most out of your school's Numicon resources with your Key Stage 2 pupils, you can sign up to the NCETM-accredited Professional Development course 'Progression with Numicon for ages 8-11 (KS2//P4-7). Click here to learn more.

Rebecca Holland is an independent Educational Consultant who has gained a wealth of experience as a primary teacher, subject leader, chair of governors, primary mathematics consultant and a school improvement consultant both in the UK and internationally. She is a lead maths consultant at Oxford University Press and is delighted to work on the authoring teams for Numicon and My Maths.