Many teachers find that when they are trying to teach the concept of area, children frequently become confused between the terms "area" and "perimeter." To try to clear the confusion, a research project was recently set up at the University of London's Institute of Education. We decided to look at the tools that children use to measure area — and to try to find out whether these affect, in any way, their understanding of the concept. We started by choosing two different ways of measuring an area — with a ruler or with 1 cm cubed bricks — and then designed some specific problems for a group of children to solve. In all, 240 children, from grades 4, 5, and 6, took part.
We covered a range of different problems — the first involved two children who had been asked to paint two walls to earn some pocket money. The walls measured 10 x 4 cm and 8 x 5 cm — they had the same area but the perimeter was different. When the children were paid, they couldn't decide how to share the money because the wall painted by one was wider than the other — but the second wall was higher. We asked the pupils taking part in the project to work out whether the children had painted the same area and whether they should each receive the same amount of money.
They drew diagrams of the walls to scale on sheets of paper. As you can imagine, it is hard to tell who has done the most work, because the first wall is wider than the second and the second is higher than the first. The children worked in two separate groups and within these groups were asked to work in pairs. Those in the first group were given rulers to solve the problem; the others were given 1 cm cubed wooded bricks. We deliberately restricted the number of bricks to 20 because we wanted the children to consider a range of options — not just use the bricks to fill the space. They quickly realized that there were not enough bricks to cover the shape — which introduced a problem-solving element. Initially, this caused some dismay but eventually many of the children put a line of bricks along the width and another line along the height, working out their own width x height formula.
We asked the children to decide whether the area was the same or different and for them to agree on one answer. This encouraged a great deal of discussion about the problem — and how it could be solved. Once they had agreed on an answer, we set about finding out whether or not they were correct. We took a piece of ruled card — 4 x 10 cm — and cut it in half to make two pieces — 4 x 5 cm. By fitting the shapes over the outlines of drawings of the walls we could see that if they fitted them, the area was the same. We then gave the children a second problem. What if the walls measured 9 x 3 cm and 8 x 4 cm? Here the perimeter was the same but the area was different. We cut a 9 x 3 cm piece of card into three — one 8 x 3 cm piece and the remainder cut in half — to check their answers, and the children who had given the perimeter as the answer could see their mistake easily. A month later we asked each child to work individually on four more measuring tasks. They now had the choice of using bricks or rulers. We wanted to know if there were any differences in the performance of the children who had used rulers in the first tasks — and those who had used bricks. If there were any differences, we would be able to say quite clearly that this was due to their experience with those first tasks.
We found that most of those who worked with the rulers initially, continued to do so. Those who used the bricks chose them again — and performed the tasks much more successfully and with a better understanding of what was required of them. The project highlighted a number of interesting implications for teachers and caused us to ask ourselves the question: "Why is it that the children using bricks should develop a better understanding of the concept of area?" Our findings seemed to show that measuring area with a ruler involves the child in applying a formula (height x width) without understanding the basis of that formula.
Once the formula is understood, however, a ruler is an efficient tool. But starting with it does not allow the child to understanding fully that the area can be found by multiplying the number of bricks in a row by the number of rows. The limited number of bricks available to the children when they were doing the task forced them to think more divergently about the problem. They could not simply cover the shape and count the bricks. This led some to develop the formula by themselves.
When the children are using a ruler, this kind of thinking is not encouraged — in fact, the mistakes made by the children who only had a ruler were very interesting. And although some already knew the formula "height times width," only very few could transfer this to more complex shapes. Some simply gave perimeter as an answer and stuck to it; others used the ruler as a type of unit of area. They placed the ruler across the shape and edged it slowly across. It looked as if they were adapting the ruler to measure the area — but did not have a tool which was suitable for the job.
The project showed us that by limiting the resources at the children's disposal, we really challenged them to make the leap from counting to computing area measurements.
Article adapted from Junior Education, February 1994