Danny and Diana, famous as the ‘Double Dhamaka’ among their friends, were celebrating their 10th birthday. They were twins, famous for everything they did, right from thinking totally differently to playing pranks on others. And for these very reasons they became the centre of attraction in all the places they frequented.
Two cakes, similar in size and shape, had been ordered and each one was set to cut his/her own cake when Diana came up with an idea. She told Danny to cut the cakes in such a way that each part represents one-third of the whole cake. Danny laughed and said it was a silly idea although he was ready to do it.
They were done with their task and looked at each other’s cakes. Soon Diana shouted with excitement, “Hey! I won! I have divided the cake with each part representing 1/3rd of the whole. You haven’t done that!” Danny too responded with equal passion, “How can you say that?” and with him, joined many of his friends and a few elders too. He continued, “Both of us have divided the cakes into 3 parts and so each part in our cakes represent 1/3rd.” Diana said, “Yes, we have divided our cakes into 3 parts but if you notice, in my cake, the 3 parts are equal in size. In your cake, they are not. And so they don’t represent 1/3rd of the whole cake.” Danny got a bit annoyed and said, “That’s your misunderstanding! I don’t trust you. Let us ask our teacher tomorrow.”
The next day, they could not stop themselves from running to their Maths teacher in school to ask her the same question. Their teacher, who had been teaching the same concept for several years, was literally shocked to hear the responses given by Danny and Diana. Probably she had never realised that the students might have perceived what she had taught them in different ways.
The teacher was nevertheless a fighter! She decided to study this understanding by probing and exploring further to check how her students actually thought about this concept. She took the following question from an ASSET paper and showed it to her students.
As expected, most of the students came up with the wrong answer B instead of the correct one A. And among them was Danny too. She started probing more and asked students why they thought B was correct. This is the dialogue that took place in the class:
Danny: I think it is be B because in both 1 and 2, the triangle is divided into 3 parts and one of them is shaded and so the shaded part in both of them will represent 1/3.
Diana: But Danny, the parts are not equal in triangle 2. It’s the same case as that of our cake.”
Zubin: Yeah Diana. You are right. The parts are not equal. And it matters if we have to select a part. But when we have to say what fraction it represents, we will just have to look at the number of parts shaded or selected upon the total number of parts. So in both 1 and 2, the fraction that the shaded part represents will be 1/3.
The teacher was trying to analyse the responses they were giving and immediately came up with another question.
Teacher: So Zubin, what is a fraction?
Zubin: When we divide something, the number of parts selected upon the total number of parts, represents a fraction.
Teacher: Ok. So what does 1/3 mean?
Zubin: It means that we divide the object into 3 parts and select 1 part. Whether we divide it into equal sized parts or not is not important when we call it a fraction.
Diana: No Teacher. I think the parts have to be equal. Otherwise how can we say that the sharing is done equally?
Teacher: Fine. But what does 1/3 mean? Does it mean a single number?
Danny: No. It means two different numbers – 1 and 3. 1 is the numerator and 3 is the denominator and they mean different quantities. So 1/3 represents two different quantities.
The teacher was just not able to absorb the shocking responses of her students. Are you able to absorb them?
The question shown was taken by around 22,000 Class 4 and Class 5 students in a nation-wide diagnostic test, ASSET. The graph (below the question) gives the data on how the students answered the question.
We will return in the next issue on this series. In the meantime, do send in your reasons on why you think these problems might have cropped up. You can email us at 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 by a special team that travelled to 28 schools across the metros and interviewed over 2,500 students on over 30 questions from ASSET.
