As seen in the previous article, “Equal or not equal”, dated March 13, 2008, a very basic concept like fractions is not fully understood by children.

There are very few students like Diana who understand the concept of fractions. But there are a majority who fall under the category to which Zubin or Danny belongs to. For students like Danny, it is just the number of parts selected upon the total number of parts that matters. The size of the parts, whether they are equal or not is not important. And so for him the answer to the question (shown below) is either 1 or 2.

However, there are students like Zubin, who understand that the size of the parts matter. But they fail when it comes to writing it as a fraction. They think that when it comes to calling a smaller part out of a whole part, a fraction, it will be ¼, irrespective of the fact that the part is smaller. So where does the problem lie?

Probably we need to refer to the way the concept is taught. Most of us will agree that when fractions are taught in school, one of the first things that is taught is the symbolic representation of a fraction. ½ or ¾ are symbolic representations that are taught with stress laid on terminologies like numerator, denominator, proper fraction, improper fraction etc.

So the question is, how important is this symbolic representation or these terminologies without the understanding of what a fraction means?

The problem that we just saw, from the responses of Danny and Zubin, might be common. But it is just one of the many challenges faced by students while learning fractions. Here are a few of them.

Students are not able to relate a fraction to the physical quantity it represents. They might recognise a fraction written as 2/3 as ‘two upon three’ or ‘two-thirds’ or even write a fraction called out as ‘one-fourth’ as ¼. But when asked to relate the same fraction to a physical quantity, they may not be able to do so. A quantity representing three-fourth of something might not be related easily to ¾.

Students seem to misunderstand that a fraction represents two different quantities. They think that a fraction like ¾ represents two different quantities, 3 and 4. They fail to see that it is a single quantity, three quarters.

And the above two challenges somewhere combine together and make it difficult for a child to compare fractions.

While comparing two fractions like 3/3 and 2/2, they tend to apply the whole number reasoning to say that 3/3 is greater than 2/2. And when asked why, they say that because 3 is greater than 2, 3/3 will be greater than 2/2.

They fail to relate the fractions to the quantity it represents; in this case, 3/3 represents 3 equal parts selected out of 3, which represents 1 whole and 2/2 which represents 2 equal parts selected out of 2, which also represents 1 whole and so the fractions are equal.

So how do we handle this? It is important for us to realise that what is more important in this case is not how a fraction has to be represented or what the number on top and that at the bottom in a fraction is called. What is more important is to understand that a fraction represents a single quantity, the parts into which a whole is divided have to be equal or else each part will vary in size and so that part when selected out of the whole might not represent the same fraction as some other part of a different size.

We need to expose children to physical objects and take them through this understanding of fractions. They need to understand the relation between the symbol of fraction and the physical quantity represented by that symbol. They should be given various objects that are later divided into parts, equal and unequal, and they should be allowed to find out if the quantity each part represents is the same or not and then relate it to the fraction it might represent.

However, the role of assessment should not be missed out. It is only through tools like such questions, talking to students and understanding how they are thinking, that the effect of teaching can be analysed.

Educational Initiatives Pvt. Ltd. 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.