When you ask adults what multiplication is, you will often get an answer that is some variant of: “multiplication is something you do with numbers, e.g. 10 times 12 equals 120.” With adults, you usually have to ask the question in a different form, “what is multiplication used for?”, to get at various ideas for what kinds of situations show the kinds of structures that we call multiplicative.
Me, I prefer to think of multiplication as something that has an existence separate (or even before) numbers. But we don’t need to agree on that – nor even be clear on what that might mean – before we can start taking a look at situations that display some kind of multiplicative structure.
I anticipate, when we start looking, that we will find several kinds of structures, and that some of them might be candidates for being “the real multiplicative structure.” I don’t know that it will be important to settle on a single one, though I realize that there is a rather large controversy here that I prefer to skirt around. I have little interest in the controversy as such, but think there is yet room for light rather than heat.
As our starting point, let’s look at some rolls of Mentos. Each Mentos roll has 14 mint candies in it, the same every roll. If we have three rolls, we can determine how many Mentos candies we’ve got. In this situation, we have two quantities, 3 and 14. Yet these quantities are of different ilk. The quantity 3 tells us how many rolls there are, and the quantity 14 tells us about the number of candies in a single roll. If we just look at the numbers, we know that we are multiplying 3 and 14. But this hides something that seems relevant. What we are multiplying is more like 3 rolls multiplied by 14 candies per roll. We might notice a parallel here, when we look at gas mileage. We might wonder how far we travel on 3 gallons of gas in a vehicle that does 14 miles per gallon.
The situations are different, but they do have something in common, and what they have in common is more than just the numbers.
So one way of looking at multiplication is as a number of groups with the same number of items in each group. We’ll see in subsequent posts that this is not the only way to think of multiplication. Are there ways of thinking about multiplication that are intrinsically symmetric with respect to the quantities that we multiply?