Education Endowment Foundation:Mathematical manipulatives: from familiarity to fluency

At Alexandra Park Primary, teachers noticed something curious in Year 2: many children were still relying heavily on their fingers to solve addition problems like 7 + 5.

While using fingers is a useful early strategy, which we encourage, some children seemed stuck at this stage, despite exposure to a range of 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).. This prompted us to explore why they hadn’t progressed to use other strategies.

Exploring efficiency

We found that although pupils had been taught various strategies, many lacked understanding of their purpose or efficiency. For example, when calculating 8 + 6, most would ​‘count on’ rather than use known facts or number bonds — highlighting a need to strengthen efficiency and flexible thinking.

We know from the EEF’s Improving mathematics in EY and KS1 guidance report (recommendation three), the powerful ways in which 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). supports children’s developing mathematical understanding. We are also aware that as children move through primary school, many 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). that have been used should be temporary ​‘scaffolds’ which should be removed once independence has been gained (EEF Improving Mathematics in KS2‑3, page 10).

With this guidance in mind, how could we move children away from the familiarity of their fingers to more complex supports?

We began by presenting the children with two addition problems: 8 + 7 and 10 + 5. We asked them to reflect on which one felt ​‘easier’ or more efficient to solve. Almost all agreed that 10 + 5 was easier — because they could confidently add ones to a ten. This opened up a discussion about efficiency in mental calculation.

Understanding the ​‘why’

We modelled solving 8 + 7 in several ways:

• using a number line to jump on 7 from 8,
• holding 8 in our heads and counting on, and
• bridging to 10 and adding the remainder.

We then asked the children to apply these same strategies to the calculation 9 + 7. They used a number line, then tried counting on from 9, and finally used the strategy of making 10 and adding the rest.

The response was unanimous — making 10 was quicker and ​“less fiddly.”

This comparison helped the children understand why
the strategy was being taught. By being explicit and modelling not just the how but the why behind our teaching, we found children were far less likely to revert to less efficient methods like finger counting.

We spent the rest of the session focusing on adding to 8 and 9. Through structured practice, children began to spot patterns and reach generalisations: when adding to 8, subtract 2 from the second number to make 10; when adding to 9, subtract 1.

‘Seeing’ the solution

To support this, children used double-sided counters and tens frames. They built their second addend on a second frame and ​‘moved’ the amount needed to complete 10. These visual representations led to several lightbulb moments — children could suddenly see
the solution, deepening fluency.

By encouraging children to reflect on efficiency and compare approaches, we empowered them to move beyond their initial use of 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 choose strategies with greater purpose. The use of visual representations, structured exploration, and discussion supported children moving from familiarity to fluency.