This is about an argument about nothing.

A respected math columnist went after teachers for saying that multiplication is repeated addition, but it turns out that he doesn’t know if many teachers do this. I called him on it. And his response came up short.

Background

Yup. One more Devlin post. Synopsis so far for those of you who weren’t watching the whole multiplication vs repeated addition follies.

Keith Devlin, back last Fall, wishes that he could stop teachers from saying multiplication is repeated addition. He elaborates, big time, in “It Ain’t No Repeated Addition” in July. Denise, who teaches math, thinks about it, and asks, then how should we teach multiplication? That’s when the comments get a bit out of hand. Denise posts again. Some other people post. Even I post.

Mostly the posters and commenters were yelling and screaming about whether or not multiplication is repeated addition. In all of this, the question that matters - how should we teach, was pretty much buried.

Question pops up

Fast forward a few days. I am in New Orleans, setting up classrooms. And I stop to skim a variety of elementary and middle school math texts. And I don’t find the error Devlin is chasing. Instead I find books discussing and introducing multiple meanings of mathematics.

Could there be some texts that say Repeated Addition = Multiplication? Sure. But my unscientific sample didn’t find them. Could some teachers ignore the texts and teach Repeated Addition = Multiplication. I know that some do. But I don’t really know if it is very many. So I wondered out loud if Devlin was jousting with a straw man.

Devlin’s rebuttal

His recent column, he’s making one more go of it, attempts to rebut 6 arguments. It is longer because he will “be quoting from some of the leading mathematics education scholars of the twentieth and twenty-first centuries…”

But when he comes to my arguments, um, no. He provides next to nothing. There is one British ed journal article that says teaching multiplication as repeated addition is a problem (from ten years ago, directed to British national policy, looks like the research was a small study in London.)

And his coup de grace? Studies (one British, one Canadian) that show adults, when asked to define multiplication, respond with repeated addition.

(To look for yourself, find the heading “The Problem Is WIdespread” about three quarters of the way down)

Now, think for a moment. Of the various models we may use in teaching multiplication, isn’t repeated addition the strongest? Isn’t that exactly what you would expect an adult, 15 or 30 years removed from grade school to recall first? They remembered what we should expect them to remember - but that doesn’t tell us what they were taught.

Could he have cited something else? Yup. If he found state or national standards telling teachers to teach RA = M, but I don’t think they exist. If he had found studies that said, “teachers do this a lot”… If he could show us texts that do the same… maybe they are there. Josh at TextSavvy might know?

Two things went wrong here.

Like the engineer who comes to a school knowing math but not knowing how to teach it, Keith Devlin arrived to a topic (math ed) that he remembers. He was a student. And he probably remembers better than most. But we are talking memories, not current knowledge here.

And second. Something I recognize. Stubbornness. Look how well he writes. Pick any other column. Pick his recent interview. There’s intellect, there’s quality of expression. He hasn’t poorly defended his position because he argues poorly; it’s just stubbornness without facts supporting it.

I’d be interested in recommendations about multiplication should be taught, but as for this topic, I think this will be my last post.

Keith Devlin, mathematics commentator, wrote: that “Stopping teachers saying that multiplication is repeated addition” would be a good thing. He sharpened it (”I wished schoolteachers would stop telling pupils that multiplication is repeated addition“), and repeated it (”a plea to mathematics teachers to stop telling students that multiplication is repeated addition.”)

Now Josh has an interview with Devlin, and has opened up the comments section.

Kenneth Devlin wrote “stop telling your pupils that multiplication is repeated addition” and all hell broke loose.

There was a storm at Let’s Play Math, and then Denise wrote a second post. There is still a storm raging at Good Math Bad Math. And a bunch of places I don’t normally go. And then Josh at Text Savvy has written 11 posts (they start here) - but he disabled commenting, which turns the conversation into an echo chamber. (Josh, it looks bad if you complain about your comments not being published if you run a site where comments are not allowed)

So, my two cents.

Multiplication is not repeated addition.

Multiplication can represent repeated addition.

Devlin’s point was directed to how we teach little kids math, and he blew it. So we stop telling kids that x = + (rep) and we tell them what exactly instead? Don’t ask Devlin. He devoted a second column to the issue, and never got there.

Doubling back, what was his objection? That we say “math is repeated addition” and that somehow this ruins kids’ ability to handle arithmetic: “Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not.” He’s wrong.

What should we do?

Teach multiplication through repeated addition, skip counting, counting arrays, finding unit areas of rectangles, Cartesian product, scaling… That’s too many to start, isn’t it? Pick one, then one more. And tell the kids that multiplication will manifest itself in many other ways as well… Add more… and remind them… No big deal here.

By the way, congratulations to Denise for handling this in an intelligent way. And for, ever so briefly, becoming the central blog in a math ed tempest.