Commutative? Who Studies “Commutative”?

It could come in any grade. It could come up in almost any mathematics course in the United States today. But why? What is “the Commutative Property” and why do we study it? Has everyone always studied it?

I may need some help from the mathematicians who read this blog. Which probably means Joel. Maybe Owen. Back in this blog’s  heyday I had literally hordes – maybe 8 or 9 – who peeked in. How far I’ve slipped.

A Little Math (skip ahead)

The real numbers (or, for most of us, “numbers”) are commutative under addition. That means that a + b and b + a have the same value, (assuming a and b are numbers, or, in more technical language, “real numbers”). When people say “The Commutative Property” – and by people I mean People who are not Mathematicians – they mean this fact, which educators label “The Commutative Property of Addition.” They label a similar fact “The Commutative Property of Multiplication,” ie ab = ba. Some teachers also teach students that division and subtraction are not commutative, which is usually fine, but sometimes puzzles children who are still wondering why “five minus seven” is different from “take five from seven.”

There are other properties, and they matter just as much. And they all have longer names, or descriptions, than we remember, or than we usually use. We use shorthand. There’s the Associative Property of Addition for Real Numbers, and the Associative Property of Multiplication for Real Numbers. There’s the Distributive Property of Multiplication over Addition or a(b+c) = ab + ac. There’s a special number called the Additive Identity (that’s just zero) and another called the Multiplicative Identity (that’s just one). And there’s a few more fancy sounding properties for pretty simple ideas like Closure and Zero Product.

Back to Reality

So here I am, last week, teaching kids a bit about matrices. We are multiplying them, and I am stalling. This is a new operation, on a new object. They need more practice multiplying, more fluency, before I introduce what comes next. And instead of assigning all the odd exercises, I decide that I will find interesting things to do that will require some multiplying, and give them some practice. And so I decide that we will decide which properties hold for which operations for matrices.

Those would be two by two matrices with real valued entries, but I’m going to stop right there before I bore the both of you who already know this stuff and make the rest of you’s eyes glaze over. But I also stopped right there for the kids.

The Question

Commutative Property? I ask. Why do we study the Commutative Property?

And the clever answer “so that we know 7 + 3 is the same as 3 + 7” is just so wrong, because little kids who can’t pronounce Commutative (communative?) figure that out on their own. And I ask about the other properties, and the attempts to answer are noble, but universally wrong. They don’t know.

Would they be shocked to know that I have really old math books (1880s – 1930s) on my shelf, and that they do not contain the word “commutative”? No, not shocked, and not properly impressed by my old books. Barbarians. But I flip to the place where the properties should be, and I open the index for the books with them, and, not there.

Getting to the Answer

1960 I tell them. 1960 is roughly the dividing line. I type Спутник on the screen. Even the class without native Russian speakers gets it. Sputnik. Horrified to have been beaten into space by this beeping medicine ball, the United States vowed to close the Space Race by adding the words “Associative” and “Commutative” to every preteen’s vocabulary. And by teaching us basic set theory at a very young age. The New Math. Kids got some fancy vocabulary. I did. By second grade I knew what each one of these symbols {⊂, ⊊, ∩, ∪, ⊄, ∈, ∉} meant. I knew them well. I don’t understand why that didn’t get me a job at NASA. Most of my current students did not recognize any of those symbols. Those who’d seen any of them, it was the curly brackets. Or if they’d seen any of the others, it was one or two, and in 8th grade or later.

We do an etymology detour. Who can correct me?  I told them the “S-” in Sputnik is cognate with English “Co-,” that the “-put-” in Sputnik is related to “path” in English, and that the “-nik” means “doer” or “-er” or “person. Thus Sputnik roughly equals “with+path+person” or more naturally, “traveling companion.” If you are good with etymology, I feel shakiest with put~path. Help a fellow out.

One Last Detour

But before I get a chance to ask if Transposition Distributes over Matrix Multiplication, I have students in one class probing further. How could the US catch up in the space race? Wasn’t this “The New Math” thing worth trying? Nope. I don’t think so. The US does just fine going back to its bread and butter when it comes to science and technology – importing scientists. And the kids talked about Operation Paperclip and Werner Von Braun.

The New Math and Werner Von Braun in one discussion? Sounds like a Tom Lehrer playlist. Next week.