Number as relations
Ivor collects pencils. Ivor’s mum gave him 7 pencils. His dad gave him 4 more. Ivor gave 6 of his pencils to his sister. Did Ivor end up with more or fewer pencils than he started with? How many?
Both of these ‘problems’ (yes, problems is in scare quotes because they are the type you only meet in school maths, but bear with me, they have a purpose) can be answered by calculating 7 + 4 − 6 (and yes, there are other ways).
Yet children find the second problem much harder than the first. In fact, since we are not told how many stickers Ivor started off with, some children will say that the second problem cannot be answered. Why do they find it more difficult?
The difficulty is in the nature of the answers. In the first problem, five represents a quantity - the number of stickers Ivor ends up with. But the answer five to the second problem represents a relationship - how many more stickers Ivor had than when he started.
We often switch between using numbers as quantities and numbers as relations without paying attention to the distinction. But awareness of the distinction is important in supporting children’s developing number sense.
Take a simple calculation like:
7 − 3 = 4
If we read the ‘−’ as ‘take away’ then the answer four is a quantity - it is the quantity remaining when three things (stickers, marbles, zingbats) are removed from seven. When we ‘take away’ we can hold up the particular quantity that is left behind.
Reading ‘−’ as ‘difference between’, the four then represents a relationship - how many more there are in a collection of seven objects then there are in a collection of three. Unlike ‘taking away’, when comparing two collections there is no particular quantity from the larger collection that specifically makes up the difference. Four expresses an abstract relationship between seven and three.
I often invite teachers to show what picture they might draw to help a child understand 7 − 3. Almost invariably people draw seven objects and strike out, or somehow bracket off, three of them.
It is very rare for someone to draw a set of seven and another set of three and point up the difference. Even adults seem to prefer the concrete, the quantity, over the abstract, the relationship.
To fully develop children’s number sense we need to make sure that they get a rich diet of operating with numbers as relationships as well as quantities. I’ll be exploring this in future blogs.
Meanwhile, if you would like to read more about the research behind this, there is a great, free, set of reports - Key Understandings in Mathematics Learning - that you can download from the Nuffield Foundation website.