I had not thought about introducing division to Katie, figuring that at some point after playing a lot with multiplication there would be a natural place/time for it. That being said, I was very surprised when at Katie’s enrichment program they introduced division before multiplication. However, when I looked at the way they were doing it, it looked no harder, and perhaps even easier, than multiplication.
When we think of say 12 divided by 3 we interpret it as “if we divide 12 objects into 3 equal groups, how many objects will there be in each group?” While this is intuitively what it means, it is not the easiest way for a kid to calculate it! To find an easier way, we need to appeal to the commutativity of multiplication. Now the question becomes “if we divide 12 into groups of 3, how many groups will we have?” Kids can easily draw circles around groups of 3 and then count how many groups there are. Katie had absolutely no trouble with this, and after we showed her how to do the first one she could easily do the rest.
Perhaps this is how many people introduce division and this isn’t very enlightening. However, I haven’t thought of it in this way before, and so I thought I’d share. Some time later I would like to show her why the two questions lead to the same answer, as follows: label the objects in each group 1, 2, and 3. Now group all the 1’s, all the 2’s and all the 3’s and we have 3 groups of 4, precisely because we had 4 groups of 3. This is a standard (but beautiful) way of showing that multiplication is commutative.