A meditation on commutativity of multiplication

May 5, 2020

We’re in the final stretch of the Kickstarter campaign now! The last ten days will determine just how successful the campaign is. Keep spreading the word, and hopefully we’ll get to 4x of our campaign target, hit all the stretch goals, and have a fantastic start producing what I believe will prove to be a game-changing educational product.

But today, I don’t want to talk about that. I want to think about multiplication again. Specifically:


I have a memory about commutativity of multiplication. I was in third grade, and I’d been identified as being “good at math.” So one day, they moved me up into the fifth grade classroom during math time. I was nervous, being in a new room, but also excited. The lesson was on commutativity of multiplication. The teacher wrote on the board 4 x 7 = 7 x 4. She went on to tell us that this was a rule of multiplication. A general fact.

Then I burst into tears. They put me back in third grade for math for the rest of the year.

I’m not sure why it made me so sad. But as an adult with a deep interest in math education, I continue to have an axe to grind with the commutative property of multiplication. First of all, it’s crazy to introduce it as a self-evident truth. Symbolically, it seems obvious, but order matters with most things. If a = “put on socks” and b = “put on shoes,” then ab ≠ ba. And don’t even think about changing the order of putting on a parachute and jumping out of an airplane.

So order usually matters. But with multiplication it doesn’t. Is that obvious though? Most kids learn about multiplication as repeated addition or skip counting first. Is it obvious that skip counting by 13 seventeen times is the same as skip counting by 17 thirteen times? Starting off, the numbers sure look different.

13, 26, 39, …

17, 34, 51, …

And magically, they just happen to both arrive 221 at the appointed moment. What gives?

Another approach: is it obvious that 13 bags with 17 chocolates in each is the same as 17 bags with 13 chocolates in each? What if the numbers are bigger?

And here’s another question worth considering: which is bigger, 13% of 17, or 17% of 13? Not even sure if commutativity applies anymore, are you?

It takes a little bit of work to even notice how big a problem commutativity is. And it’s worth noticing, and being stuck on it! Because if you never get stuck, you don’t see how elegant the visual is that shows us that, indeed, order doesn’t matter when it comes to multiplication.

And there it is. One picture perfectly describes why 4 x 9 = 9 x 4. I don’t need to check that they’re both equal to 36. I just need to see that I’m looking at the same array from two different angles!

The leap to see that 13 x 17 = 17 x 13 or  ab = ba in general is right there. You could probably sketch the general argument for why commutativity works now.

But something deeper has happened too. Henri Poincaré said that mathematics is the art of giving the same name to different things. We’ve just done that. The same thing (the array), seen at different angles is given different names (4 x 7 or 7 x 4). There is bridge between them that wasn’t obvious at first! This is real mathematics. And just to drive the point home, I have a challenge for you. Look at the same array above, and see if you can find the right angle to look at it so you see:

1 + 2 + 3 + 4 + 4 + 4 + 4 + 4 + 4 + 3 + 2 + 1

See it? It’s hiding in plain sight.

I’ll even give you a hint. Don’t look at it at a right angle 😉

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David Ruffner
3 years ago

It’s definitely not obvious! I remember being confused about it when I learned about the commutivity property in algebra. Then it was confusing again when I studied quantum mechanics where certain operators do not commute. In both cases these rules are just stated as facts. I agree that we should encourage the questioning of these rules.