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How to record fractions using the CPA approach

How to record fractions using the CPA approach

Many adults and children find the concept of fractions difficult. And with good reason. Maths mastery expert Dr Yeap Ban Har explains how to teach this abstract convention by using the CPA approach.

A few years ago, I was fortunate enough to attend a professional development course taught by Dr Yeap Ban Har. He explained that one of the major reasons children struggle with conceptually understanding fractions simply comes down to the way we write them.

And yet, the importance of recording fractions in a way that’s easy for children to understand is fundamental to teaching maths for mastery.

Perhaps sharing his advice can help you and your teaching practice too.

Why difficulties can arise from how we record fractions

One of the major difficulties when it comes to understanding fractions is the convention of how we record the information.
To unpick this, one of the first things we need to understand is that in mathematics there are four kinds of numbers:

  1. Cardinal Numbers
    Cardinal numbers denote a quantity (1, 2, 3, 4, 5).
  2. Ordinal Numbers
    Ordinal numbers define the position of something in a series, such as first, second or third.
  3. Measurement Numbers
    Measurement numbers only make sense if there is a unit. It doesn’t make sense to say “I weigh 50,” there needs to be a unit attached to the number: “I weigh 50 kgs.”
  4. Nominal Numbers
    A nominal number is a number used only as a name, for example, the 23 bus.

Why is knowing this important when teaching fractions?

If I asked you to write 3 quarters on your paper, most of us would likely record the symbolic notation to represent the fraction. Meaning, we would write the numerical symbol 3 to represent the numerator and the numerical symbol 4 to represent the denominator.

What causes the issue for many learners is the way we record the denominator.

When we record the denominator we use a numerical symbol — so a number — but we are using it in a nominal way. By representing the denominator with a numerical symbol, many learners assume it is a cardinal number like the numerator when in fact it is not. The denominator is a noun, it’s the name of the fraction.

Let’s use an example: saying 3/4 is the same as saying 3 apples. What we mean is ‘I have 3 of something, I have 3 of that thing, I have 3 quarters’.

Children often struggle because we don’t explicitly teach them that the denominator is not a cardinal number, it’s a nominal number when we’re helping children learn to record fractions.

What’s the most common mistake children make when recording fractions?

This lack of understanding is illustrated when learners begin adding fractions. Often the common mistake children make is they will add the denominators together the same way they have added the numerators. This reflects a lack of understanding that the denominator is a noun, it’s the name of the fraction not a quantity like the numerator.

We need to give learners an understanding that the denominator and the numerator are different, even though we are representing them both in the same way.

Easy ways to help learners record fractions

The symbolic notation of fractions can create problems for learners, and often as teachers, we introduce it too early which unintentionally creates further confusion. So how do we help them develop an understanding that the denominator and the numerator are different?

Here are some easy ways to record fractions and give your learners a deep understanding of how they work.

1. Use concrete resources

Concrete resources are critical for giving learners a conceptual understanding of fractions. Using resources to support teaching the concept of adding fractions is an easy way to record fractions and help children see that only the numerator is cardinal and represents a quantity.

2. Don’t introduce the concept of fractions with the symbolic notation

Fractions follow the natural language convention of ‘number plus noun’, like place value. For example, 3/4 follows the same language convention as 3 tens, both mean we have 3 of that thing.

When we introduce fractions to younger learners, explicitly following this convention will develop their understanding of the numerator and denominator. Rather than recording fractions with the symbolic notation, record the numerator with the numerical symbol and the denominator as the written word.

For example: 3/4 can be recorded as 3 quarters.

Recording fractions initially as the number plus the noun will help learners develop an understanding that the numerator is cardinal and the denominator is nominal.

3. Use pictorial and visual representations to support the abstract algorithms

Similar to using concrete resources, it’s important to support the abstract maths with visual representations. Drawing diagrams to represent 3/4 added together with 2/4 will help learners develop an understanding that only the numerator represents a quantity.

Visually representing the abstract concept of fractions is important for developing your learners conceptual understanding of fractions throughout their primary school learning. The visual representations will help learners understand conceptually what the abstract notations and procedures mean.

Information in mathematics is highly encoded — really we’re using a bunch of symbols to represent abstract concepts. So to ensure we give our learners a conceptual understanding of fractions, we need to consider how we’re presenting the information to them.