Excerpt from “Division”

Division is often the most confusing of all the basic math functions. The language can be even more baffling than it was for multiplication: “3 into 2 won’t go,” “divided by,” “over,” “shared between.” Just how important is long division and why does it cause everyone such problems? And if division is about making things smaller, how come dividing by 0.5 on a calculator gives me a bigger answer?

Common problems children have with division

  1. Not fully appreciating that division is the inverse of multiplication, so not using the multiplication facts that they know to work out related division facts. For example, if you know that 7 × 4 = 28, you also know that 28 ÷ 7 = 4 and 28 ÷ 4 = 7.
  2. Thinking that division is only about “sharing” (“share 42 apples between 6 people”) when it can also about repeated subtraction (“put 42 apples into bags with 7 in each bag”).
  3. Thinking that division always make things smaller: 35 candies shared between 5 children means each child gets 7 candies, but there are 5 children so there are still 35 candies. No candies have been taken away, they have just been rearranged.

What IS division—sharing or subtracting?

Division is usually introduced as the idea of sharing, formally called “partitive division.” Children particularly engage with the idea of sharing candy (and want to be sure that they get their fair share). So if there is a calculation like this:

48 ÷ 8

then it will typically be dressed up as a “real world” problem like: “I have 48 lollipops that I want to divide equally into 8 bags. How many lollipops do I put into each bag?”

But there is another way of interpreting sharing. Compare this problem to the one above: “I have 48 lollipops. I want to put them into bags of 8. How many bags can I fill?” This can also be solved by calculating 48 ÷ 8.

There is a big distinction between the two problems. In the first one, the sharing problem, we know how many lollipops there are and how many bags we want to put them into. What we don’t know is how many lollipops will end up in each bag. To solve this problem you could literally share out 48 objects—set up something to represent the 8 bags and go “one for you, one for you . . .” until all the lollipops were divided.

In the second problem the situation is subtly different. There are still 48 lollipops, but this time you know how many you want to put into each bag, not the number of bags. To solve this practically you would put out 48 objects and then take away 8 for the first bag, 8 for the next one and so on until all the lollipops were used up. This is division as repeated subtraction (sometimes called “measurement”) rather than sharing.

Understanding both “types” of division

It’s important that your child is familiar with both types of division problem, the “sharing” type and the “subtracting” type. There are two reasons for this.

First, interpreting a question as “sharing” or “repeated subtraction” can make a surprising difference to how easy the child finds it to calculate the answer (just as in subtraction, “take away” and “finding the difference” changes the way you think about the sum 2001 − 1998).

Here are a couple of examples that one education expert explored with children:

6,000 ÷ 6            6,000 ÷ 1,000

Children who thought of division as sharing find the first problem easy—they can picture in their minds 6 people and imagine giving 1,000 things to each of them. But they find the second calculation difficult, as they cannot cope with trying to imagine 1,000 people. In contrast, children who treated division as repeated subtraction found the second calculation easy—all they had to do was subtract 1,000 from 6,000 as many times as they could, which was 6 times. But repeatedly subtracting 6 from 6,000, wow, that was going a long time. Being flexible in thinking about which version of division to use makes both calculations easy. As does confidently knowing that 1,000 × 6 = 6,000 and using the relationship between multiplication and division.

The second reason for understanding both types of division is that when (later in school) children start to divide by fractions, it is really only repeated subtraction that makes any kind of sense.

—from “Division”