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.

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.

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.

1. October 13, 2020 pm31 3:04 pm 3:04 pm

Three things: (1) I enjoyed this post a lot, looking forward to the promised next installment. (2) 1960 is certainly important, but I think the 1880s–1930s dates you mention are also important: a lot of these properties were isolated only as abstract algebra congealed as an area of mathematics, and we started to have abstract/axiomatic definitions of objects like a field (rather than “just” having the examples that people cared about). I tend to associate this with Emmy Noether (1882–1935), who maybe was the first person to write down the axiomatic definition of a field (9 axioms, of which you wrote down most, including 2 commutivities). (3) You said you might need help, but I don’t see where :).

• October 14, 2020 am31 12:04 am 12:04 am

I am nervous about misusing words.

Sputnik is a real boundary – because no one would have thought of introducing “properties” to school aged children before then.

Listen to the uncomfortable laughter in the live recording of “The New Math” by Tom Lehrer…

2. October 14, 2020 am31 9:33 am 9:33 am

JD.
GreAt to see you writing again, my view about the emergence of associative and commutative was that we needed them when we found out about things that were not. Vectors and matrices and quaternions, so Im with the 1860 – 1880 period. I think the Bourbaki movement emerged in the 20’s. But it was sputnik that scared us into change, Keep going

October 17, 2020 am31 10:16 am 10:16 am

what the other guys said. non-commutativity became an issue for math-heads in william rowan hamilton times. of course non-commuting behaviors had been noticed hitherto and some version of the shoes-and-socks lemma (“invert everything in the opposite order”) will’ve been widely known if never explicitly given a name. matrix notations in modern parlance are probably best introduced as *changes of co-ordinates* (with invertible matrices given priority); lots of geometric examples in 2- and 3- D; lots of pencil-to-paper drawings & calculation (“young man, in maths we don’t *understand* things… we just *get used to* them). the new math was good to me. and my cohort at university elementary bloomington. donald adair got it in 5th grade and mary ann di-bagio *totally* got it in 6th; my best teacher ever. it’d’ve been great if teachers had bought in nationwide. but this was unlikely for reasons.

4. September 15, 2021 am30 6:25 am 6:25 am

And here I am a year later, rereading Owen and Pat and Joel in preparation for matrix multiplication on Friday…

I wish I’d used this blog more… But glad I got so much out of it.

• October 4, 2021 am31 9:57 am 9:57 am

Fun!

• October 13, 2021 am31 7:46 am 7:46 am

Oh it occurs to me to mention that I have a new colleague, who specializes in the study of non-commutative algebra — but I’m not sure I can effectively gloss what he does in a few sentences. (He also dabbles on the side in the mathematics of the game Set — but that’s a commutative situation.)