## two ways of thinking about division

20 07 2010

I may have covered this, but while tutoring it became crystal clear. Consider:

8 / 2 = 4

Actually, when I look at it like this, it immediately smacks of fraction. However, if we were using the more normal divide sign, there are two ways of interpreting it. At least, perhaps.

1. “how many two’s are in eight?”

2. “what is eight divided by two?”

The difference becomes particularly graphic when we consider fractions:

2 and 1/2 divided by 1/2… is 5

This is a bit of a jump, mentally, for most people. Which of the above interpretations fits? ie

1. “how many halves are there in two and a half?”

2. “what is two and a half divided by half?”

In my mind, the first makes sense. It’s to do with shapes, or words even.

“how many half-slices of pizza are in two full pizzas and one half-slice?”

“how many pairs in eight?”     :  :  :  :

This is very different from thinking about division as cutting, ie

“what is eight divided by two?” or ” what is eight divided into two groups?”   : :      : :

Now consider a bunch of kids in a room, and at one time they may be thinking of division one way, and then there’s another way being explained to do something. The way of thinking must be contextualised, rather than there being a rule that is true for all time and for all things.

Of course, this is simply the difference between 4×2 versus 2×4. Wow. I thought these were the same. But they aren’t. At least, not in terms of division. In the first way, we have four groups of pairs, and in the second way we have two groups of four. In terms of XQ, and the two different ways of thinking about addition (the action being on the plus, the action being on the equal sign), the action here is in terms of the mind. The divisor determines the number within each group, or, the number of groups. Again, counting in terms of things, and counting more in terms of a higher order, cuts, divisions.

OMG. Can we teach primary teachers to have this sensitivity? Why not? It’s pretty simple really. It’s a matter of listening to what a kid is doing. Noticing, that’s all. And the best way to show that you understand is by giving them more that they can do well, which gives them confidence, and then giving them things that they can’t seem to get with the same thinking methodology. Or perhaps getting two kids who happen to be doing the two different ways confidently to engage and see what they make of it.

ZOMG.