: Making Meaning with Manipulatives through free play and structured opportunities Mark Williams shares his experiences from a Key Stage One classroom

A five-year-old dressed as Darth Vader watches an episode of Numberblocks through his mask. In the episode, ten aliens are stranded on a planet and must divide themselves between two spaceships. However, one of the ships is too heavy. How will they escape?

Today is a non-uniform day, and after the program, one child heads over to the maths area. She makes a tower of ten cubes, splits it in two parts and starts to write equations, working carefully until she has found them all.

Two more children join her and begin to build a staircase from the cubes — a structure they encountered when the class learned about ​“one more” and ​“one fewer.” A child builds a tower that is far too high, laughs, and cuts it down to size to restore the pattern.

Elsewhere, some children are clicking together squares and triangles to make 3‑D shapes. They have not met these resources before. One child connects a pyramid to a cube to form a house. Another connects triangles to make a bipyramid: I suggest we name it after her.

​“It’s called a diamond. It is the most beautiful diamond in the whole world,” she declares.

Meanwhile, Iron Man is trying to tessellate the squares. He wants to use them all, so I offer him a tray of shape blocks from the shelf instead. He makes a pattern from regular hexagons and starts to add other shapes to it.

The daily maths lesson has not yet begun. These children are choosing to explore mathematical ideas through the manipulation of concrete resources.

As an NCETM Primary Specialist, I support teachers in deepening their understanding of representations in mathematics lessons. Manipulatives — such as place value counters, Dienes blocks, and other concrete resources — are invaluable tools for uncovering mathematical structures in Teaching for Mastery lessons.

When I moved to Year 1, I revisited the Education Endowment Foundation’s guidance report Improving Mathematics in the Early Years and Key Stage 1, and I considered their recommendation to use manipulatives and representations to develop understanding.

The EEF defines a manipulative as an object that children can physically interact with and move to represent mathematical ideas. They can include everyday objects such as pine cones, buttons, small toys as well as resources like pattern blocks, cubes, counters and the rekenrek abacus used in the NCETM’s Mastering Number programme.

The guidance recommends that

The maths area in my classroom is available to children during ​“choosing time”. Further mathematical opportunities are provided in other areas: hoops, dice, giant Numicon and chalk in the outdoor area; coins and prices in the role-play shop; three-dimensional shapes in our small world area. I have noticed that in classrooms where children do not have this opportunity to explore and play with manipulatives, teachers are more reluctant to include them in maths lessons. They fear that the novelty of a concrete resource will distract the children from thinking about mathematics.

The tower of ten cubes represents ten aliens. Splitting the tower in two represents the partitioning of aliens between two ships. The equation records the whole and the parts.

By using a manipulative to represent a problem, the child can strip away the context and focus on the mathematical structure. This process extends their working memory and reduces cognitive load, making it easier to recognise the underlying mathematics and solve the problem.

Not all children enter school with the same experiences to draw on when learning about numbers and shapes. Some may not have had access to resources like blocks, dice, or dominoes at home. Manipulatives in the early years offer a shared reference point, providing a concrete basis for understanding key mathematical principles and relationships.

As my class progresses through Key Stage 1, their learning will become more formal. They will write equations, draw pictures, and create diagrams, but manipulatives will remain a valuable tool for deepening their understanding and demonstrating their growing mathematical knowledge.

Further reading:
Read more about how can manipulatives be used effectively to boost mathematical understanding in this blog written by the EEF’s Grace Coker: Myth-busting mathematical manipulatives

References:
EEF Guidance Report: Improving Mathematics in the Early Years and Key Stage 1
NCETM’s Mastering Number programme