# How to slice portion-sized problems

MOST CHILDREN understand that they cannot eat an entire cake if they are sharing it with other people, but many still find the concept of fractions difficult.

Researchers from Oxford university observed that most children are able to work with whole numbers relatively easily. But many wrongly transfer the rules applying to whole numbers to fractions. So they assume that a third of a cake must be smaller than a fifth, because three is smaller than five.

Children are often unable to comprehend that the same fraction may refer to different quantities: that half of 12, for example, is not the same as half of 100.

And they do not realise that different fractions can refer to the same quantity: that two-thirds is the same as four-sixths.

Yet children can relate fractions to objects. They realise that if one child is given half a big cake and another half a small cake, the two do not receive equal amounts. Similarly, they seem to understand that a third and two-sixths are equivalent when discussing slices of pizza. However, they fail to see that mixing one glass of orange concentrate with two glasses of water is the same as mixing two glasses of concentrate with four glasses of water.

Working with almost 450 pupils in eight Oxford and London primaries, the researchers observed that many children struggle with conventional methods of teaching fractions.

The most common method involves asking pupils to share a whole into equal parts and then to attach a fraction to each part. For example, if four children share a bar of chocolate and one girl eats one part, she would be eating a quarter. But pupils are often unable to understand the complexities of this calculation.

A more effective method required pupils to use drawings to help their mathematical working. For example, asked what happens when a pizza is shared among six people, they would draw a pizza divided into six parts. But researchers cautioned against this method because the task of drawing pizzas often distracted pupils from the mathematics.

The method recommended by researchers involves capitalising on children's innate ability to relate fractions to real objects. They naturally understand that an object can be shared among different numbers of people by being divided in different ways. This understanding can be applied to teaching fractions using a division-based method that emphasises carving up a single item.

If one bar of chocolate is divided among four children, one becomes the numerator and four the denominator. The fraction therefore refers both to the division (one by four) and to the portion each child receives.

Similarly, if three items were divided between five children, three would become the numerator and five the denominator. Each child would therefore receive three-fifths of the whole.

The researchers said: "Insights from everyday experiences with division can be systematised in the classroom, transforming them into a solid basis for children's further learning of fractions."

* www.edstud.ox.ac.ukresearch childlearningindex.html

## Comments