So much fantastic work has been done on using manipulatives within maths lessons. For example, counters in ten frames can be used to model addition calculations that border ten. This highlights efficient mental strategies, helping children to move beyond counting strategies. Similarly, place value counters can be used to conceptually understand a column subtraction, modelling the process of regrouping. In these instances, the manipulatives can bring to life the key mathematical ideas, leading children to a deeper understanding.
Often, though, manipulatives can be used less frequently as children move through KS2. Also, their use can be primarily in developing number fluency, rather than being used as a tool for problem-solving.
How, then, can manipulatives be used in KS2 for all children, to develop a deep understanding of more complex mathematical ideas?
Seeing the structure
The EEF’s Improving mathematics in key stages 2 and 3 guidance report says that ‘Manipulatives should be used to provide insights into increasingly sophisticated mathematics’.
To introduce a new problem-solving concept, I will often start using a question with relatively small numbers. Then, the question can often be represented with double-sided counters.
Also, I like to hide one number when a question is first presented. This means that children have to think about the question, without giving an answer, before the whole question is revealed.
Consider this question:
There are eight children at the party. There are __ more boys than girls. How many boys are there at the party? How many girls?
At this stage, children generate different possible answers to the question. For example, there could be six boys and two girls.
Then, the full question is revealed:
There are eight children at the party. There are two more boys than girls. How many boys are there at the party? How many girls?
The use of manipulatives can encourage discussion as we view and unpick the different ways that the children have used the counters to represent the question.
We can then use the counters to draw out the most efficient method. This will help children to answer similar questions when the challenge in the calculation increases.
For our question, point out that there are two more boys than girls:
How many ‘children’ are hidden? How many ‘boys’ and how many ‘girls’? Then, we see the solution:
This uncovers an efficient method for answering the question: subtract the number of ‘extra boys’ from the total number of children and divide by 2!
Same structure, different context
The next stage is removing the counters and seeing how this question can be represented with a bar model:
Here, the representations have helped the children to develop a deep understanding of the question structure, enabling them to transition into more challenging questions. Then, similar questions can be explored, using larger numbers and moving towards more efficient strategies. Here is a follow-up problem:
Altogether, Rose and Beth have £30. Rose has £6 more than Beth. How much money does Rose have?
This bar model gives an example of how this question can be broken down:
Manipulatives can be so powerful in helping to bring to life the key thought processes in problem-solving. By making children’s understanding concrete in this way, children are armed with the strategies to tackle more complex tasks and to grow as mathematical problem-solvers.