# 1 What’s so difficult about fractions?

One of the reasons fractions can seem so difficult is that there is a lot to understand. For example, half of something can be smaller than a quarter of something else. An example of this is ‘half of six is three’ and ‘a quarter of sixteen is four’. So learning about fractions by folding pieces of paper or by dividing circles may mislead students, especially if the paper is always the same size. Students must be taught to ask ‘A fraction of what?’

Developing an understanding of fractions is not so different from learning to understand other mathematical concepts. For example, very young children are offered many different experiences as they learn to generalise the concept of ‘three’.

Despite being older when they learn about fractions, elementary students similarly will need a great many rich and varied experiences if they are to begin to develop a good understanding of fractions.

Many students will have had experiences that help them to develop some understanding of fractions. In her research, Nunes (2006) found that primary school students already have insights into fractions when solving division problems:

• They understand the relative nature of fractions: if one student gets half of a big cake and the other gets half of a small one, they do not receive the same amount. They also realise, for example, that you can share something by cutting it in different ways: this makes it ‘different fractions but not different amounts’. Finally, they understand the inverse relation between the denominator and the quantity: the more people there are sharing something, the less each one will get.

## Talking fractions: using the language

Encouraging the students to talk about fractions and use the vocabulary will help them understand some of the difficult vocabulary associated with fractions. The questions you use should show the students how important the correct vocabulary is, so that everyone knows what is being referred to.

First, model some ways of talking about fractions and drawing attention to how words are used. Then focus on getting your students talking. The more the students use the words themselves, the more they will build their understanding of fractions. Asking the students to make up questions to ask one another is a good way to get them talking. Another way is to ask the students to explain the reasoning they used to arrive at their answers.

The first activity is for you to think about issues of learning fractions in your classroom.

Think about what your students need to know about fractions, and make some notes on the different ideas. Use your textbooks. If you have a multigrade class, you will need to think about what different students need to know about fractions:

• how to find out a fraction of a quantity
• what fraction one quantity is of another
• how to add fractions together.

For each of the ideas associated with fractions, write down how the vocabulary associated with those ideas and the way it is used to express ideas. For example ‘half of ten’ means ‘divide 10 by 2’, but it can also mean ‘multiply 10 by ’. The students might also see , which has the same outcome and is thus equivalent in meaning but which may also be expressed as ‘10 divided by 2’ or ‘10 shared between 2 people’.

Think about some specific students in your class. What activities might help them to understand the different ways that fractions can be expressed and the different meanings given to those interpretations?

What you can learn in this unit

2 Developing an understanding of fractions