A few years ago, I was teaching halves in a 2nd grade class. I’d given each of the children a heap of different-coloured blocks and
one simple challenge:
Can you make halves using any of the blocks?
A visiting principal was observing the lesson. He watched for a little while before asking,
“Don’t you think these students are too old to be using manipulatives?”
He wasn’t being unkind. He was working from a familiar view that concrete materials are there to help children who need extra support, and that, as children become more mathematically capable, they no longer “need” manipulatives.
He expected to see something more like the worksheet to right. The children would identify the whole, recognise that it had been divided into equal parts, shade the required number of parts and move on to the next question.
But that wasn’t what was happening in this classroom.
The students had their blocks and were ready to start exploring.
At first, the children created fairly simple and predictable halves. For example, two blue blocks joined with two yellow blocks, divided neatly down the middle.
After a while, I introduced another question:
How many different ways can you make halves?
Suddenly, the maths became much more interesting.
Some students were stumped at first. Then the discussion started to build. Children began experimenting and rearranging their blocks in different ways. They compared their constructions with their classmates’. They tried different colours, different amounts of blocks and different arrangements, rather than simply making two equal groups cut down the middle.
Some began making a whole and partitioning it in different ways. Others counted out their blocks first to create two halves and then experimented with different designs. Some made patterns; others put their blocks together randomly.
And then something else started to happen.
Children began checking whether someone else’s construction really showed halves.
They discussed, debated, collaborated and reasoned. They explained their thinking and challenged one another’s ideas. Some even began naturally exploring quarters.
They were no longer simply answering the question and looking for a quick right-or-wrong answer. They were investigating a mathematical idea.
- What makes something a half?
- Does this still show halves?
- Can I show halves in a completely different way?
The children had taken a simple mathematical idea and pushed it further than I had explicitly asked them to.
As the lesson continued, I showed the principal what the children were building.
He began walking around the room, asking questions and listening to their explanations. He asked one child how they knew their model represented halves. He compared two students’ models and asked whether they were both correct.
And gradually, I could see the assumption behind his original question beginning to shift.
The students weren’t getting an easier lesson, or using blocks because they were struggling.
By providing the blocks, every child had something concrete to think with. They could build, manipulate, compare, question and explain. The blocks gave them a way to see and talk about their mathematical thinking.
Yes, the materials helped make the concept easier to understand. But they hadn’t lowered the level of mathematics the children were engaging with. If anything, they had pushed the mathematics further than the worksheet alone could have.
The children weren’t simply colouring in a predetermined answer.
They were constructing and testing their own mathematical ideas.
Not only that, the task had become a low-floor, high-ceiling activity. Every student could get started, but there was no obvious point at which they had to stop.
That lesson stayed with me because the principal’s question captures an assumption that is still surprisingly common:
At what point are children supposed to outgrow manipulatives?
We think that’s the wrong question.
Manipulatives aren’t simply a support for children who haven’t yet mastered “real” mathematics. Used purposefully, they can give all students a way to explore mathematical ideas, make connections, communicate their thinking and ask questions that go beyond the original task.
That’s why, at Awesomenicity, we prefer to think,
‘What mathematical thinking could this manipulative make possible?’
