Jenni Back worked closely with Sue Gifford and Rose Griffiths on two projects focused on using practical scenarios to manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods).-in-the-foundations-of-arithmetic‑2″ target=“_blank” rel=“noreferrer noopener”>make sense of whole numbers, fractions and decimals, funded by the Nuffield Foundation. Two books ‘Making numbers’ and ‘Making fractions’ were developed alongside the project reports.
“Now I understand fractions!”
Fractions can often be a notoriously tricky topic to teach. It involves seeing two numbers as one quantity and understanding the relationship between them: complex stuff. Developing solid foundational understanding is key. Manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). and representations can play a crucial role in helping pupils to engage and make sense of mathematical ideas. They provide a bridge to abstract thinking. Research evidence also shows they can increase engagement and enjoyment for pupils.
Through using manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). in a way that makes sense to children, they can interpret the mathematics and see how the fractions work in the way that they do. Story setting can provide context for that sense- making and the manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). give the children objects to act on in that context. This is not only motivating for learners but also supports mathematical thinking.
Fish Fingers for Tea
A favourite fraction task that I play with pupils to support understanding of thirds involves penguins sharing fish fingers! (p.47 Griffiths et al., 2023).
The children work in small groups, and each have blank dice marked with these numbers:
They have:
- two penguins
- a plate with ‘fish fingers’ on it
- six whole ones and others divided into thirds to exchange (made from orange kitchen sponge cloths and each measuring 6cm by 2cm).
Each child takes a turn to throw the dice for each penguin. After each throw the children check to see which penguin has the most and how many fish fingers are left on the plate.
- Who is the greediest penguin?
- How many fish fingers did each penguin get for their tea?
The fractions of fish fingers give the children practice in counting up and down in thirds and exchanging one fish finger for three thirds or vice versa.
Watching the children exchange parts for wholes helps adults to see how well they understand the links between the manipulative and the mathematical idea. Encouraging pupils to record their findings in their own way can also support and reveal understanding of fraction notation.
The practical, hands-on experience that manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). provide can help children make sense of mathematical ideas in a meaningful and motivating context.
Further reading:
- Back, J. (2013) Manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). in the primary classroom. http://nrich.maths.org/10461 (accessed 21st February 2025)
- Education Endowment Foundation. (2020). Improving mathematics in Early Years and Key Stage 1, pp. 16 – 17. Available at: EEF_Maths_EY_KS1_Guidance_Report.pdf
- Griffiths, R., Back, J. & Gifford, S. (2016) Making Numbers: using manipulativesobjects that educators and children can move and interact with to represent mathematical ideas (including fingers, everyday objects, such as buttons or pine cones, and mathematical resources such as Numicon, Cuisenaire rods). to teach arithmetic. Oxford: Oxford University Press.
- Griffiths, R., Back, J. & Gifford, S. (2023) Making Fractions: practical approaches to fractions and decimals. Oxford: Oxford University Press.