Rima was sitting with her dad Varun, trying to put a huge jig­saw puzzle together. Rima was looking for a piece and getting quite impatient. She was about to get up when Varun asked, “Rima what do you think this shape looks like?” Rima replied, “it does not look like a shape. It’s a jigsaw piece, it’s a piece, not a shape.” Won­dering why Rima thought it was not a shape, Varun asked her, “So what’s a shape?” Rima said, “A square or a rectan­gle is a shape — something which has a length and a breadth. This is not a shape, its sides are uneven”

Varun was surprised that Rima, who had been working with different shapes right from a young age be­lieved so strongly that the jigsaw piece was not a shape. He realised he needed to understand what Rima thought of different shapes and what made her decide that something could be called a shape. He asked, “So you mean to say uneven shapes do not have a length?”

Rima thought for a moment and replied, “Yes, how can it have a length? Length is a straight line, something we use to calculate the area of a shape. Miss Mira, in our school, draws a big rectangle on the board and then marks “l” and “b” on it. That’s length and breadth and then when you multi­ply those two you get the area.”

Varun could not make sense of the fact that his daughter who re­membered the formula did not un­derstand what those attributes meant. For her, it was more like sub­stituting a letter for a number on a shape, and the shape had to be simi­lar to a square or a rectangle. He wondered if she knew what area was. On probing further, she said, `Area is ‘l x b’.

Nine thousand students across the country were asked the question below. The ques­tion here aims at checking the students’ skill in calculating and comparing attributes such as the perimeter of simple figures. Here is how the students performed on this question. As indicated in the graph above, only around 28% of the students were able to answer this question correctly while close to 48% of students thought that all the figures would have the same perimeter.

Last week we discussed some misconceptions Rima had about shapes, their attributes, and area. Students like Rima are likely to select option D.

Why did a large number of students select the wrong option D? Is it because they didn’t know what perime­ter was? Is it because they made a mistake in calculat­ing the perimeter?

Students learn different terminologies and often tend to confuse them. In the absence of a core under­standing of these terms, they start using one for the other and tend to make such mistakes.

In this question, they seem to be referring to the `area’ of the shape instead of the ‘perimeter’. There are 3 unit squares in each of the figures. This implies that the total area of all the figures is the same. The perimeter of figures in op­tions A and B is the same.

But if we calculate the perimeter of the figure in option C, we will find that it is 1 unit more than that of the other two. This is be­cause though all the outer sides remain as they are in the other two figures, in the third figure one of the in­ner sides gets exposed be­cause of the shifting which in turn increases its perimeter.

Students simply count the number of squares in the figures and since they are the same, conclude that the perimeter will be the same. The squares shown in the question might be arranged in different ways but as long as they do not overlap, the area will re­main the same. This might not be the same for perime­ter. Perimeter is an attrib­ute related to the boundary of the figure and it depends on how they are arranged. As seen in figure C, one of the squares has been moved a little above the other two. This has added to the bor­der. Students, however, don’t seem to have inter­nalised this.

Are students getting lost in definitions without un­derstanding what they mean? Extensive research conducted by the team at EI shows that students tend to think of all these terms as labels and often inter­change one for the other due to lack of understand­ing.

A common practice in classrooms is to introduce students to the definitions of a term and then teach the concept.

A suggestion would be to avoid this. Al­low the students to grasp the concept first.

They should understand that area is a two-dimen­sional attribute of a figure, which indicates what por­tion of a region it covers. All figures — rectangles, squares, triangles, parallel­ograms, or irregular ones— can cover a region and so all of them have an area.

Similarly, a perimeter is a one-dimensional attribute of the figure, the length of the boundary of the shape. It can be calculated by measuring how long the boundary is and like area, all figures can have a perimeter. Avoid using the terms ‘area’ and ‘perimeter’ while teaching these con­cepts. Once students are comfortable with the con­cept, define the terms. This will ensure that the defini­tion is an explanation of the concept they have just learned.

### Nishchal Shukla and Anar Shukla

Nishchal Shukla leads the Pedagogical Research division at Educational Initiatives (EI). Pedagogical Research team goes deeper into understanding how students think and how they learn different concepts, and different subjects.

Anar Shukla has worked as the Project Manager in the Marketing team of Educational Initiatives.