**Teaching fractions is critical**. Mastery of

*fractions*is one of the strongest predictors of success in high-school mathematics and helps in areas such as algebra.

Yet, **students often struggle to master the concept of fractions**. They hold

*misconceptions*about

*fractions*and therefore find it difficult to add, subtract, multiply, divide, compare and order them.

** Errors occur at the surface level**. Common errors include:

*Adding*(or*subtracting*) both the*numerator*and*denominator*(wrong)- Not finding a
*common denominator*before*adding*(or*subtracting*)*fractions*(wrong) - Finding a
*common denominator*when*multiplying fractions*(unnecessary) *Dividing*the*denominator*by the*numerator*when converting to*decimals*(wrong)

When teaching *fractions*, many teachers have seen students make these sorts of *common errors*. And, many teachers *correct *their students by telling their students that they are wrong (or that a step is unnecessary) and repeating the correct way to do things.

There is nothing wrong with what these teachers have done. However, **many errors are merely surface-level symptoms of an underlying misconception**

*.*Research shows that students struggle because they hold deeper

*misconceptions*about

*fractions*.

So, what should you do?

When students hold *misconceptions*, especially *deep misconceptions*, you should take advantage of the teaching for conceptual change process. Teaching *fractions* is an ideal time to use this process.

## Understanding the Underlying Misconception

**One frequent and deep misconception is rooted in students’ prior knowledge about whole numbers. It is known as whole number bias**.

*Prior knowledge* can often make learning easier, but conceptually, the way *whole numbers* work is entirely different to the way *fractions* work. Here are some crucial differences.

While many students represent *fractions* correctly, they mistakenly (and often subconsciously) use their understanding of *whole numbers* to try and understand *fractions*. It just doesn’t work!

For example, the whole number **8 means 8**. It is bigger than 7 and smaller than 9. However, the value of 8 in a fraction depends on its relationship with the other number in the fraction. Consider these statements:

In these examples, **you cannot figure out the value of 8 by its place in the whole number counting system**. It is not that students’ understanding of

*whole numbers*is wrong. Rather, the problem is they cannot make use of this

*prior understanding*when working with

*fractions*.

Numbers in *fractions* work in a totally different way. 8 can be bigger than 9, and smaller than 7. **In a fraction, you cannot know what any number is worth (e.g. 8, 26, 3) without knowing the other number in the fraction**.

**With whole numbers, the number has a value in its own right**. 8 is always 8. It is always less than 9 and more than 4. 28 is always 28, etc. In other words, you can understand the number in isolation.

**With fractions, you can only understand each number in relation to the other number**. More specifically, you can only understand

*fractions*based on the

*ratio*between the

*numerator*and the

*denominator*.

### Many Students Don’t Get This

For most teachers and some students, this new understanding comes quite easily. However, for many students, it doesn’t.

*Conceptual change* is not likely to happen unless you clearly show the inadequacy of their *existing beliefs*. Therefore, when teaching *fractions*, you need to explicitly highlight the fact that:

- Their
*existing way of thinking*about the value of numbers doesn’t work (i.e. finding value by looking at numbers in isolation) - A new way of thinking about numbers is needed

**Only after this has been done, should you start to teach your students about the fact that the value of a fraction (i.e. how big it is) comes from the relationship between 2 numbers – the numerator and the denominator**

*.*

## When To Teach About Whole Number Bias

When should this be taught?

After looking closely at the Australian Curriculum, I believe it should be taught in Year 4.

In Year 4, students start:

- Working with
*fraction walls*, which you can use to show the inadequacy of*whole number thinking* - Make connections between
*fractions*and their*decimal equivalents*, which you can use to show*ratio thinking*

However, if you teach older year levels, and your students struggle with *fractions*, it is worth going through this *conceptual change *process with them.

This is only one example of *conceptual change *with *fractions*. You can go through similar processes to dismiss other *misconceptions*, such as the notion that:

*Multiplying*numbers always makes them bigger*Fractions*are always less than 1 whole

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