Multiplication and division structures

Contexts in the equal sharing structure - part 1

What are some of the contexts in which pupils will meet division in the equal-sharing structure? At first sight we might think that sharing is a very familiar experience for pupils, sharing sweets, sharing pencils, sharing books, sharing toys, and so on. But this idea of sharing a set into subsets corresponds to division only under certain conditions.

First, the set must be shared into equal subsets, which is certainly not always the case in children’s experience of sharing. Second, it is important to note that the language is sharing between rather than sharing with. Children’s normal experience is to share sets of things with a number of friends. Division requires sharing them between a number of people. The division 12 divided by 3 does not correspond to ‘I have 12 marbles and I share them with my 3 friends’. The situation required is, ‘Share 12 marbles equally between 3 people’. This is a somewhat artificial process and may not be encountered as often in the pupils’ experience as we might imagine at first sight. So, sharing does not always correspond to division: it must be not just ‘sharing’, nor just ‘sharing equally’, but ‘sharing equally between’.

In the context of measurement it is not difficult to come up with imaginary situations where we might share a given quantity into a number of equal portions. Cutting up a 750-cm length of wood into 6 equal lengths, or pouring out 750 ml of wine equally into 6 glasses, or sharing out 750g of chocolate equally between 6 children, for example, all correspond to the division 750 divided by 6. But these problems tend to feel like situations contrived for mathematics lessons rather than genuine problems.