Multiplication and division structures

The operation of division

What are the different kinds of situation to which the operation of division applies?

Remember division is not just ‘sharing’. Equal sharing is only one division structure. Teachers should not overemphasize the language and imagery of sharing at the expense of the other important language and imagery that is associated with division, particularly division as the inverse of multiplication.

There is a wide range of situations in which we have to learn to recognise that the appropriate operation is division. These can be categorized into at least the following three structures:

• The equal-sharing structure
• The inverse-of-multiplication structure; and
• The ratio structure

As with the other operations of addition, subtraction and multiplication, it helps us to connect these mathematical structures with the operation of division if we ask ourselves the question: What is the calculation I would enter on a calculator in order to solve this problem? In each case the answer will involve using the division key on the calculator. As we saw earlier, one of the difficulties in understanding the meanings of the symbols we use in mathematics is that one symbol can have a number of strikingly different meanings. This is certainly the case with the division symbol.