Two kinds of division

If you had to explain what the expression “10 ÷ 5” (that is “10 divided by 5”) meant, what would you say? To be clear, I’m not asking for the answer, I’m asking for a story that will give it meaning.

I’ve been asking people this for the last few days and there are two main stories:
1. I have 10 things to split into 5 groups; 10 ÷ 5 is how much is in each group.
2. I have 10 things to split into groups with 5 in each group; 10 ÷ 5 is how many groups there are.

Most people only say one of these two, which is interesting because only knowing one of them can get you into all sorts of trouble when it comes to solving actual problems.

If you only know it as “how many groups of 5 fit into 10” then you’re going to have to think quite hard to figure out how many each person gets when you share 10 among 5. And it would be even worse if it wasn’t a whole number of objects shared among a whole number of people but, say, a number of moles of chemical shared across a number of litres of water to make a concentration. Indeed, both perspectives on division are often needed in the same drug calculation problem in nursing and medicine!

As a teacher you can get into trouble too: consider the meaning of “10 ÷ 1/2”. The first interpretation would give you “I have 10 things and I split them into half a group; 10 ÷ 1/2 is how much in each group.” While this is correct (and quite interesting actually), it makes much less sense than “I have 10 things and I split them into groups with half a thing in each group; 10 ÷ 1/2 is how many groups there are”.

Mathematicians have the tendency to say that division is simply the inverse of multiplication (so that “10 ÷ 5” means “the solution to 10 = 5x”). But this denies that the understanding and use of maths is deeply connected to how we picture it. When two pictures explain the same maths, we’ve got to be both aware and careful!

(PS: For those interested in a bit of Maths Education terminology, Meaning 1 listed above is called “Partitive Division” and Meaning 2 is called “Quotative Division“. It took me ages to figure out what they were going on about at my first Maths Education conference! Oh, and there are in fact more ways than these to think about division, corresponding to the many ways there are to think about multiplication!)

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