Ankita and her classmates had gathered in the classroom for a session on student interviews conducted by the ASSET team. They were asked the following question:

More than half the class thought it was option A. Manan explained, “Since we are removing a square of perimeter 20 cms from a triangle having perimeter 80 cms, we have to subtract 20 from 80 and so we will get 60 cms as the answer.”

Some students thought the answer should be option B. Priya said, “The square that is cut out has a perimeter of 20 cms. So each side will be 5 cms long. Now, the piece that is remaining has a perimeter of 80 cms. Three sides of the square are left out in that piece. So, if we subtract 5+5+5 from 80, the answer will be 65 cms.”

Manan and Priya, when asked what they meant by perimeter were able to define it correctly. But they were unable to apply the definition to this problem. This is very common amongst students. Definitions are learnt without internalising. It is similar to rules being accepted without understanding the reason they were made in the first place.

Why are the students making these mistakes?

There are probably two different reasons due to which students might be selecting option A or option B.
Those selecting option A might be confused between area and perimeter. They seem to know the definitions of area and perimeter. But they do not seem to be able to explain them clearly for a shape like the one above! This could be as both these concepts are taught in conjunction with each other and students extend the more intuitive concept of area to perimeter. (Though they answer many questions about area incorrectly!)

Students selecting option B might have a strong belief that when a part is removed, values of associated quantities must decrease. It is probably the understanding of area that they are extending to perimeter and therefore tend to believe that whether it is area or perimeter it  ought to decrease. During these interviews, we asked some students to follow a step-by-step process of finding the length of each side of the square first and then calculating the perimeter. But the belief is so strong it just seems illogical to them that the perimeter can increase!

However there are some students who are able to relate mathematical concepts to daily life experiences.

After having heard reasons that students gave for Option A and B, the interviewer turned to Ankita for her explanation. Ankita was new in class and a little hesitant to explain. So the interviewer asked her to explain what she had in mind. She said, “Suppose the first triangle is an island and the second triangle is like an island similar to the first one but from which a part was removed. Now, if I had to walk along the borders of these islands, then I would have to walk more on the second island right?”

She thought a little bit; traced her finger on the figures and said, “Yes, I would have to walk more. And perimeter is nothing but the length of the boundary, and the boundary is longer in the second triangle. So it has to be more than 80 cms!” She calculated it in her mind and said that the answer will be option C. Option C is the correct answer.

Ways to ensure that students understand perimeter:

This shows how relating things to practical life can sometimes help students internalise concepts easily. Here are some methods that can be tried while teaching the perimeter.

*Use cardboard cut-outs or cut shapes while teaching area and perimeter

*Make use of questions (like the above) that clearly test for conceptual understanding

*Allow students to experiment with these shapes - making the cut outs, joining different pieces and estimating the area and perimeter in different cases.

*Ask students to answer this question using the cut outs - "What happens to the perimeter when a piece is cut out from a shape?"

*Finally, ask some questions without using the term perimeter and saying “length of the boundary.” (Use of terms sometimes makes students suspend thinking and fall back on 'formulae'.)

Do share your experiences of teaching perimeter and responses to questions like these by writing to misconceptions.dh@ei-india.com

(Educational Initiatives is an Ahmedabad-based organisation working at the forefront of driving change in education through research, large-scale assessment and the development of new learning technologies. Student responses in this article are taken from a series of interviews done by a special team that travelled to 28 schools across the metros and interviewed over 2,500 students on over 30 questions from ASSET.)