As with many basic terms in mathematics, there is a commonly held belief that we all know what we mean when we refer to multiplication and division. If pressed for a definition the idea commonly offered is that multiplication is repeated addition, which would mean that division is repeated subtraction.
The idea of multiplication as repeated addition is not far from the mark and can be a useful way of thinking about multiplying whole numbers. However, division problems rarely encourage the use of repeated subtraction. The different styles of division questions encountered emphasise the identification and use of units. Sometimes we are interested in how many are in each group and other times we want to know how many groups there are.
These two types of division questions are described as partitive division and measurement (sometimes called quotitive) division. Partitive division refers to dividing a whole into several equal parts, such as sharing 12 cupcakes equally among 4 people. Partitive division arises from the notion of sharing, or distributing equally into a specified number of parts.
The measurement (or quotitive) interpretation of division determines, for example,how many units of size 4 are contained in the composite unit of 12, a measurement notion.Thus measurement division refers to dividing a whole into groups of a certain number of elements, such as “There are 12 cupcakes and each child gets 4 cupcakes, how many children are there?”
Multiplication can equally be described in terms of groups. Learning multiplication and division is described in this framework in terms of managing groups as units. The progression of increasing sophistication in the identification and coordination of units in multiplication and division is summarised in the following table.