Have you ever wanted to improve the pupils learning and your
teaching of fractions? I have and I have probably taught fractions the same way
as everyone else, but I still had children who did not fully understand what a
fraction is, just like everyone else. How many metaphorical pizzas have been used each day during Maths lessons up and down the land? Yet our pupils have the same misconceptions year after year, despite using their favourite food. What is the
problem? What was I doing wrong? What is to be done?

Another fraction of a pizza |

### The problem

If you give a diagram like the one below and ask the
question what fraction is shaded? What fraction is unshaded? How many children
would give the answers 2/8 and 6/8, probably most, but can they make the leap
to ¼ and ¾? They find this really difficult and need lot of questioning and
prompting to see the equivalence.

Another issue is when you ask a student to share 4 chocolate
bars between say 5 friends and ask how much do they each get. It takes some
time before a 12 year old for example realises it is 4/5. (Perhaps I am being a
bit optimistic there.)

As the pupil get older they are introduced to ratio. Do they
ever see the link between a ratio of 2:3 and the fractions 2/5 and 3/5?

### Different categories of the whole

Reading an article from the nrich website ‘Teaching
fractions with understanding: part-whole concept’ really made me reflect upon
how I had taught fractions over the years. The suggestion, as I understand it,
is to explore fractions in different situations. Think of the quantities, the
types of ‘whole’ where we ask the children to find fraction of it. Like me
you’ve probably used questions like find 3/5 of 400kg with varying success. It
is easy to teach them the method but do they really understand it?

The article suggests that we should think of the quantities,
the wholes as being categorised as four different types, ‘discrete wholes’,
‘continuous wholes’, ‘definite wholes’ and ‘indefinite wholes’.

### Discrete wholes

This is such things as beads, counters, bricks etc. Objects
which are not normally broken up

### Continuous wholes

This category lends itself to food, examples that you could
use include Pizzas, Pies and Cakes. What did we do before Pizzas?

### Definite wholes

This is where the universe on which we are working is
defined but made up of different objects. Like the diagram below which is made
up of white, yellow and blue rectangles.

### Indefinite wholes

‘Indefinite wholes’ are made up objects which have no
apparent beginning or end, you could present them with a picture of a string of
beads but with both ends continuing into the distance or as in the article quoted a diagram that looks like the one below.

My next blog will discuss what we can do in the classroom to
promote deeper understanding of fractions and the part-whole concept.

An excellent book on children’s misconceptions is by Doreen
Drews

*'This practical guide to children’s common errors and misconceptions in mathematics is ideal for anyone training to teach 4-11 year old children and keen to gain a deeper understanding of the difficulties children encounter during their mathematical development. The book is structured around National Curriculum Attainment Targets, and deals with individual misconceptions, in each case providing a description of the error, and an explanation of why the error happens.'*

Children's Errors in Mathematics: Understanding Common Misconceptions (Teaching Handbooks Series)

*Follow me on Twitter croftsr1*