One of the biggest sins one can commit inside instruction of any kind—curriculum or live teaching—is to simply instantiate a concept or skill seemingly out of thin air, like magic. “Here’s how to solve a proportion using the means-and-extremes method,” for example—no functional connection to prior learning, no motivating the need for the knowledge, just poof. (Unfortunately, it is this degenerate caricature of telling that is wrongly held up as the only form of telling possible in teaching.)
The force of this moral commitment against magic in instruction is not absolute in anyone’s mind. But it is also strong enough, I think, to make us desperate to include, for a given concept, any prior-knowledge connection we can lay our hands on, some kind of motivating idea. And not only is it possible for us to choose poorly in this regard, it is certainly possible that such desperation makes bad choices more likely.
This is what I think has happened with the concept of multiplication. Desperate to find some way of making multiplication make sense to students, we made the mistake of connecting it to students’ prior knowledge of addition, and, by doing so, have created a situation where generations of students confuse the two operations at multiple points in their educational journeys. Even with less “additive” connections like area models or equal groups, most students (and then adults) still maintain a notion of multiplication as a kind of fast addition. Here’s how I think the operations are schematized in most students’ minds:
Addition and subtraction are conceptualized correctly, not only in themselves as appending and taking away quantities, but also in relation to each other: taking away is, more or less intuitively, the inverse of appending. But because multiplication is just turbo-charged addition, the only good candidate for its inverse would be turbo-charged subtraction. Yet, we have decided, collectively, that a different connection underlies division: equal shares. This leaves us in a state where we sort of know, intellectually, that multiplication and division are inverses; we just don’t really think of them or treat them that way. This schema also leaves us with the notion that division doesn’t increase or decrease a quantity—it just spreads it out. And that multiplication only increases a quantity. It’s a mess.
Multiplication as Cloning
Students’ schemas around multiplication certainly don’t have to match the “official” organization of concepts in mathematics, but it is somewhat concerning that we have drifted so far from the official version. Ideally, the schema should have two rows, one for addition and its inverse, subtraction, and one for multiplication and its inverse, division. Connections can be drawn among all the operations, of course, but these two rows should be distinct. Multiplication and division do different things to numbers than addition and subtraction.
The best example of this distinction I have come across, for adults, is to imagine a number line and then what each operation does to all the numbers on that line. Adding 2 to every number on the number line translates all the numbers 2 units to the right. Subtraction by 2 moves them all back. Multiplication, on the other hand, does something very different. With 0 fixed, it stretches (dilates) all the rest of the numbers into place. Division compresses them back.
All of that is pretty abstract for students. So, what if we started like this instead, with the notion of multiplication as a kind of cloning device:
We’re not simply appending a quantity to a starting quantity, we’re cloning what we started with (which is why we can’t go anywhere when we start with 0). And we also have a code that we can consistently decipher: for integers, \(\mathtt{a\times b}\) means that every 1 unit in \(\mathtt{a}\) expands to \(\mathtt{b}\). We can do this in different ways, because we can change what we mean by “1 unit.” The quantity 3 on the left expands to 6, using 1 square consistently as 1 unit. On the right, we essentially have \(\mathtt{\left(2+1\right)\times 2}\), using 2 as the unit in 1 case and 1 as the unit in the other. The code simply tells us that every unit expands to \(\mathtt{b}\) units. With the right perspective on multiplication, we pick up the Distributive Property for free (and axiomatically free it is in mathematics).
The best part of this code is that it doesn’t have to change when we reach fractions (yet, the additive model has to be thrown away at this point; good luck, students!).
Even for integers written as fractions, what \(\mathtt{4\times 3}\) means is that every 1 unit (the denominator of the multiplier) in 4 expands to 3 units. Given this interpretation, I’m hoping you can feel, in this next image, how satisfying it is to see not only how fraction multiplication and division use the same code, but also see division in its rightful place as the inverse of multiplication:
Despite its potential benefits, this approach will no doubt face resistance from some educators due to its unconventional nature. But in my mind, this is how curriculum improves: here’s an idea, let’s talk with the experts. Experts (of classroom teaching): the realities of the classroom mean that I can only take the idea this far. Let’s work out the solution! (More examples welcome!)