Manipulatives: a window into pupils’ mathematical thinking

As teachers, we’ve all experienced that moment when we ask a pupil to explain how they arrived at an answer, only to hear, “I just know!” While this response can be frustrating, it often reflects a lack of language to articulate their thinking or a failure to deeply reflect on the mathematical process.

I like to respond with, “Okay, show me how you know.”

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). are especially useful in this situation — they help pupils break down their thinking, represent the calculation, and serve as scaffolds that prompt them to explain their approach.

In doing so, they offer a valuable window into the pupil’s mathematical understanding.

Making the thinking visible

The EEF’s Improving Mathematics in the Early Years and Key Stage 1 guidance report states how 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). can be used by children to communicate what they know. When pupils use 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)., they externalise their thought process. 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). can support children to communicate their thinking as they narrate the process while representing it.

For example, a child may struggle to articulate how they know that 12 subtract five equals seven. By representing the calculation on a rekenrek, they can visually demonstrate their thinking — first subtracting two to make ten, then subtracting three more to reach seven. The rekenrek offers a clear window into the child’s thought process, allowing the teacher to see whether they have a strong understanding of subtracting through ten.

Observing the ‘how’

We often think about how 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). can be used to teach conceptual understanding, but we can also use them to support our assessment of their understanding. Pupils can use representations to record their thinking and communicate their understanding.

For example, a child using base ten blocks to solve 45 subtract 27 might struggle with regrouping. By physically exchanging a ten for ten ones, the teacher can observe whether the pupil truly understands place value or is simply following a rote procedure. By taking a step back and watching and listening to how the children interact with 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). they can make their thought process explicit. Carefully planned questions can then allow practitioners to probe for deeper thinking where required.

This process reveals more than just whether the child arrives at the correct answer — it provides insight into how they arrived there, deepening practitioner’s understanding of pupils’ mathematical thinking.

Seeing is believing

When misconceptions are identified through this process, 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). can help practitioners address and deepen conceptual understanding. A common misconception in teaching angles is the belief that longer sides make an angle larger. Angle strips can be used to clarify that an angle is defined by the size of the turn, not the length of its sides. By layering angle strips of varying lengths or widths on top of one another, students can clearly see that the turn remains the same, reinforcing the idea that the angles are equal.

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). are more than just teaching tools — they are powerful windows into pupils’ mathematical thinking. By making thought processes visible, they allow practitioners to:

assess understanding,

identify misconceptions,

and support students in articulating their reasoning

whilst supporting pupils to visualise and see mathematical structure

Education Endowment Foundation:Manipulatives: a window into pupils’ mathematical thinking