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I used to think teaching division through the concept of sharing was the best way to instill division concepts. Then I ran into a problem with fractions. For whole numbers, the idea is pretty straightforward: there are eight juice boxes that will be shared equally among four friends. How many does each friend get?
Clearly they each get two juice boxes.
8 ÷ 4 = 2
or
eight items shared among four people results in two apiece.
All well and good. The concept of sharing is something children can relate to and thus have the motivation and wherewithal to solve.
But the idea of division as sharing begins to break down at some point. Take, for example, 8 ÷ 1/2. With our juice boxes, it would look something like this:
And the real-life scenario would read:
There are eight juice boxes which will be shared among 1/2 of a person.
How many juice boxes does the half person get?
Most chldren would answer 4, or maybe even 8. Most children, too, would be utterly surprised to find out that the answer is 16! "How," they ask, "Can the half guy get more juice boxes than there were to begin with?"
They are surprised. They are confused. And rightly so. Half people make no sense. "Sharing" with anything less than two people makes no sense. Objects suddenly doubling in quantity when being shared makes no sense.
Perhaps, then, a better approach is the "how many of this are in that" idea. For the initial problem above, it could be asked, "How many fours are in eight?" Of course, at that point we could no longer use the concept of sharing, but conceptually it seems more mathematically sound.
When applied to 8 ÷ 1/2 it works wonderfully: How many half juice boxes are in eight juice boxes?
As the image shows, eight juice boxes are made up of 16 half juice boxes. In this visual example the result makes a great deal more sense intuitively than the sharing example.
Of course, a mathematically rigorous definition of division would go quite a bit further than this. But for children who are just learning division, what is the best way to go about it?
The short answer is that there is no best way. Like any new concept in math, division must be introduced as simply and inuitively as possible, and then move on from there.
It is, therefore, entirely appropriate to teach division as sharing at the formative stages. In fact, the way you teach it advances as the student advances:
Division as sharing
Division as grouping
Division as repeated subtraction
Division as component parts
Division as the inverse of multiplication
No matter what, the key is to use the strategy that is developmentally appropriate to the student and most fitting to the problem at hand. We wouldn't want to invoke the concept of sharing when trying to divide pi by two in a geometric formula, and neither would we want to bring in multiplicative inverse when figuring out how many cupcakes each person gets at a birthday party.
The important thing is to start with what is most simple and natural. Building a strong foundation of multiplication is also part of the process, as the inverse property of multiplication will ultimately (we're talking years) be the bedrock of division.
