# Who knows how multiplication is taught? Not me, not Keith.

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.

I posted some of the state standards that mention repeated addition and multiplication together here. And here I repeated Park and Nunes’ citation of the 1999 English National Numeracy Strategy, which, according to the authors, “suggests that pupils should be taught to understand multiplication as repeated addition.”

The relevant section of the 2005 English National Numeracy Strategy is here (PDF). This updated version mentions both arrays and repeated addition as “metaphors” for multiplication.

The best I could do for a textbook was to scan a page of a third-grade book I worked on way back in 2000-2001. It’s the very first lesson on multiplication at that grade level.

Keep in mind that when a state or national standard

doesmention a certain methodology for teaching something (e.g., using repeated addition to teach multiplication), that methodology is almost certainly present in the curriculum, but when a standarddoes notmention a specific methodology, it could be there or not. The page I scanned above is from a third-grade book for California (one that is probably still being used in some places there). Although the lesson definitely ties multiplication and repeated addition together at the beginning (the next page has students making equal groups with counters), no California standard (at the time of that book’s production, 2001) mentions repeated addition. It has been my experience that books present (and teachers teach) multiplication as repeated addition whether or not a set of state standards prescribes that methodology.You’re likely going to disagree that any of the above are relevant, because they mostly point to teaching multiplication using

multiplerepresentations, not just repeated addition. And this idea–that multiplication is not “just” repeated addition–seems to be the basis for your “strawman” post.Devlin’s rebuttal, of course, doesn’t address the “multiple representations” character of your argument–I suspect he disagrees with it, but it may be that he just re-framed your argument. What he wants to say, I think, is that most adults think of multiplication as repeated addition, so they must have been taught that it is. You say that most adults think of multiplication as repeated addition, because that representation is the

strongestof ALL the ways they learned to represent multiplication, not because they were taught that repeated addition IS multiplication.I don’t know that I see much of a difference between “strongest” representation and “only” representation, as far as learning goes. Of course, I’m pretty “hard-nosed” on this subject, as MPG would say (has said). But obviously I can’t (and no one can) force people to do something that they think doesn’t matter.

Devlin’s ‘rebuttal’ doesn’t address multiple representations. It’s a shame, because concerned teachers would have been interested in what he was proposing. But he wasn’t proposing, he was taking a swipe.

Repeated addition connects more strongly to what most people do with multiplication as they grow up than any of the other representations – of course they remember it best. Sorry, no teacher, no textbook is going to prevent people recalling most clearly the meaning that makes the most everyday sense to them.

Let me ask you this. Do you think it is possible for me to write up a lesson (or series of lessons) to introduce multiplication without the use of repeated addition? And, if so, what criteria would I have to meet with that lesson (or series of lessons) in order to convince you–or anyone else–that repeated addition doesn’t have to EVER be used as a representation when

introducing/explainingmultiplication to students?But no one (except maybe you now) was proposing to do away with using repeated addition. Devlin was explicit about that. He just wanted teachers not to say that “multiplication is repeated addition.”

“Stopping teachers saying that multiplication is repeated addition”

“I wished schoolteachers would stop telling pupils that multiplication is repeated addition”

He actually suggests something that none of the commenters, not you, not me, not anyone else, mentioned: “The “learn the technique first and understand later” approach is very definitely the only way to learn chess… Why not accept that math has to be learned the same way?”

And yup, we could teach multiplication by drilling tables and then introducing rules for multi-digit computation… Is this what you are suggesting?

Devlin really just got it wrong.

He claimed that textbooks get it wrong, and he made the claim in a general way, universally. “I assume the reason for the present state of affairs is that teachers (which really means their instructors or the writers of the textbooks those teachers have to use) feel that children will be unable to cope with the fact that there are two basic operations you can perform on numbers.”

By the way, there’s a reason we won’t discuss California’s previous set of math standards, right?

And when you write “It has been my experience that books present … multiplication as repeated addition…” I have to wonder. Was I just looking at the wrong books?

“And yup, we could teach multiplication by drilling tables and then introducing rules for multi-digit computation… Is this what you are suggesting?”

No, but it would be closer to the “technique first – understand later” approach.

“By the way, there’s a reason we won’t discuss California’s previous set of math standards, right?”

I’m not sure what you mean here.

“Was I just looking at the wrong books?”

No, but I thought that you had said that all the books you looked at DID present repeated addition as one “meaning” of multiplication. The fact that this meaning (a wrong one) is surrounded by others doesn’t mean that it’s not taught–or learned, especially since it survives into adulthood as one of the “strongest” meanings of multiplication.

I don’t believe you are still arguing what Devlin wrote. Maybe you should go write about how multiplication

shouldbe taught. But I’m done here.I recall still my first encounter with a mechanical rotary calculator, a hand-cranked one. I had just finished high school, and I was surprised to see how this machine did multiplication and division: repeated place shifted addition for multiplication, repeated place shifted subtraction for division. Twenty-spme years later (about 1972) I was learning computer programming with a small machine, and I learned it worked the same way. I don’t know much about teaching elementary arithmetic, all I know is I learned it very well by rote. Understanding came much later for me. There are all sorts of multiplication around: vector scalar products, vector vector products, matrix pro9ducts, and so on. But it seems to me that in fact ordinary finite accuracy decimal multiplication is repeated addition.

REH