# Devlin (repost)

Two years ago Keith Devlin stirred up a storm by writing that teachers should stop saying “multiplication is repeated addition.” At that time I wrote several posts about it. Essentially the math is a little interesting, but pointing fingers at teachers is uncalled for. And I caught him being awfully sloppy. Now he’s written again. Math Mama and Number Warrior both have interesting ripostes. But all I have is a repost:

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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.

The problem that is (provably) widespread is teaching addition years before even introducing multiplication, and then taking years to introduce exponentiation. A few curricula, none of them with a very wide following, introduce additive and multiplicative work simultaneously. I am yet to see a curriculum to introduce additive, multiplicative and exponential work at the same time. My small-scale experiments can’t be called “a curriculum” yet, as I am the only one using the materials with just a few children.

All sides of that war start from the assumption that of course students learn multiplication (years) after addition. Well, this assumption is a variable, and it can be changed. There are at least dozens, probably hundreds depending on broadness of your net, studies done on the subject of “additive misconceptions” – kids and adults inappropriately using addition when they should use multiplication. A case study I did with a kid who worked on multiplicative tasks before being introduced to additive tasks (even counting) ended up with the kid having multiplicative misconceptions, that is, using multiplication or exponents in situations where addition was called for. The effect, much like additive misconceptions, persisted for several years after addition was introduced. I consider this effect problematic.

For my part, I would like to see more experimental studies comparing effects of introducing multiplication after (and through) addition, and exponentiation after (and through) multiplication, to grounding these operations in their own metaphors and introducing them more or less simultaneously, with rich connections. And then studies on comparing grounding operations in metaphors to the non-grounded approach recommended by Kevin.

In 1913 Henry Sheffer proved all math is reducible to addition. This proof gices us NAND logic and is the basis for all digital computers. In short, if there is any math that cannot be reduced to a special case of addition, it could not be represented on a binary computer.

Sheffer, H. M. (1913), “A set of five independent postulates for Boolean algebras, with application to logical constants”, Transactions of the American Mathematical Society, 14: 481–488

From the Common Core Standards (that’s like just about *everyone* isn’t it?):

“Understand that multiplication of whole numbers is repeated addition. For example, 5 x 7 means 7 added to itself 5 times.”

Georgia State Standards for Mathematics Grades 3-5:

“Describe the relationship between addition and multiplication, i.e. multiplication is defined as repeated addition.”

I read “multiplication of whole numbers is repeated addition” as a metaphoric “is.” Something like: “A garden inclosed is my sister, my spouse; a spring shut up, a fountain sealed.” http://kingjbible.com/songs/4.htm

Solomon didn’t really think the lady was a garden, though. I realize the metaphor may not be intentional in the multiplication quotes, I just choose to read it this way.

I don’t think that the problem is widespread in the UK. I haven’t hard the phrase “multiplication is repeated addition” in my circles before.

In 1913 Henry Sheffer proved all math is reducible to addition. This proof gices us NAND logic and is the basis for all digital computers. In short, if there is any math that cannot be reduced to a special case of addition, it could not be represented on a binary computer.

Sheffer, H. M. (1913), “A set of five independent postulates for Boolean algebras, with application to logical constants”, Transactions of the American Mathematical Society, 14: 481–488