Interested readers are urged to check both the discussion on math-teach (see the link above) and to go to the columns Mr. Groves links to in that post, along with Devlin's most recent column on this issue (from January 2010) and two related columns he cites there from December 2008 and January 2009.

With those as background (or at least Devlin's first column on this issue, "It Ain't No Repeated Addition," from June 2008), let me try to extend the discussion.

Devlin's first column closes with the plea, "In the meantime, teachers, please stop telling your pupils that multiplication is repeated addition."

That's not exactly something you'd think would be devastating. But apparently it really shook up a lot of folks, so much so that Devlin realized that many of those responding actually DO believe that multiplication IS repeated addition. Period. And the most troublesome part is that many of those who do are K-5 teachers (that some of them are 6-12 mathematics teachers is beyond troublesome. That there are a few mathematicians who seem to want to defend such folks is nearly beyond belief (though not if you're familiar with the Math Wars).

That's not exactly something you'd think would be devastating. But apparently it really shook up a lot of folks, so much so that Devlin realized that many of those responding actually DO believe that multiplication IS repeated addition. Period. And the most troublesome part is that many of those who do are K-5 teachers (that some of them are 6-12 mathematics teachers is beyond troublesome. That there are a few mathematicians who seem to want to defend such folks is nearly beyond belief (though not if you're familiar with the Math Wars).

Further, even some teachers who know better seem to think there's no problem with telling kids a "white lie." They say that in due time they (or, more likely, some teachers down the line) will say to the students, "Oh, by the way. Not quite" and that this will be completely harmless. (I wonder if some of the thinking that informs this has to do with Tooth Fairies, Santa Claus, and Compassionate Conservatives).

Devlin mentions in several of the columns that he views this sort of thing as far less than harmless, and cites explicitly in the third column, "Multiplication and Those Pesky British Spellings," a body of research that supports his views. I won't attempt to recapitulate that column or the research here, though I have been reading extensively in it, as well as in related work, particularly that of Catherine Sophian of the University of Hawaii in her outstanding book, THE ORIGINS OF MATHEMATICAL KNOWLEDGE IN CHILDHOOD.

I would suggest that this sort of explicit lie or as Devlin calls it "brittle metaphor" - though in the examples below perhaps "brittle rule" would be more on-point - is not exceptional, sad to say. Let me introduce a few more:

1) "You can't take a bigger number from a smaller number," and

2) "You can't divide a smaller number by a bigger number."

Both of these are, of course, false, yet teachers say this every day school is in session and have done so for likely several centuries in this country and elsewhere. I very much doubt there is any American reading this blog who was never told those two things (and probably the third one I will introduce below). When challenged, teachers who know these things are false are likely to offer up the same sorts of arguments that Devlin cites regarding the "multiplication-is-repeated-addition" issue: it really does no harm to tell kids this. They'll "unlearn it" soon enough. My experience with students of a wide variety of ages, including adults, suggests otherwise. I doubt very much that experience is an isolated case.

So what might teachers do instead? Am I about to recommend that they teach students about number systems, modern algebraic notions like complete ordered fields, ordered fields, and integral domains? Or perhaps the Peano axioms (which as Devlin points out are a way to define whole-number arithmetic to first-order logic, not, as are the number systems already mentioned, a descriptive axiom system that tells us how to work within it)? Not likely, though Devlin has been repeatedly accused of wanting to do something along these lines by a number of critics, including at least one mathematician, who know perfectly well that Devlin has explicitly said that is NOT his recommendation).

No, I simply suggest that teachers say to students something like, "We haven't yet learned how to do that, but you will learn [and here the teacher should say something appropriate to the age/grade of the student, though as I will mention, even that may require sensitivity and insight on the part of the teacher, along the lines of "next year" or "in a couple of years"], keeping in mind that there will already be students who know perfectly well that you CAN subtract bigger numbers from smaller ones, or may have at least speculated that this is possible. Some will have already learned outside of school about negative numbers. And so teachers need also be prepared to point students to good resources (on-line, in the library, or perhaps even on the math shelf in class), and to offer to talk with students individually or in a small group if they would like to do so). And obviously, all of the above holds for dividing smaller numbers by bigger ones, given that there will be students who know something about division. (I'm tempted to say that it's not a terribly good idea even to introduce division strictly with problems that have no remainders, but that's one of those things that isn't left open very long at all, and I don't know that teachers actually ever SAY that "You can't divide a number by another number that doesn't "go into" it exactly.)

I think a third commonly-told school mathematics lie brings us to the most general point thus far:

3) "You can't take the square root of a negative number."

But of course, we can. And most high school students have to come to grips with this before they graduate, though generally at a very shallow level.

So what should teachers who aren't teaching about complex solutions to quadratic equations and "imaginary" numbers tell their students instead? By the time students are learning about square roots, it hardly seems like it would be mind-blowing to tell them, "You have not yet learned about the kinds of numbers that we need to solve these kinds of problems [e.g., x^2 + 1 = 0, or just sqr( -16)], but you will in high school" [or whenever is in fact the case].

Thus, what I'm suggesting (nay, calling for) is that we demand that our mathematics teachers in K-12 stop lying to students unnecessarily and instead offer sensible, yet simple responses to questions that require mathematical knowledge that students don't yet have (and indeed, their teachers may not really have very solidly, if at all). Such replies are honest, do not require being "untaught" or "unlearned" at any point down the line. They DO require teachers who have a clue as to how what they teach in mathematics fits into a bigger picture, of course, but it doesn't necessarily require that they have mastered that which they are not actually going to teach. Not that it would be in any way undesirable if they DID have mastery, but I'm not looking to try operating schools with a handful of teachers until we develop enough of them who have such mastery.

In agreement with Professor Devlin, I'm also saying we need teachers to understand the difference between interesting connections between the operations and what those operations actually do. That they know why we really do need multiplication and multiplicative reasoning, not just addition and additive reasoning. And that beyond arithmetic we need exponentiation and exponential reasoning. That some of our models and metaphors don't tell us as much as we may think they do (or have been led to believe they do by others). That every metaphor and model in mathematics breaks down at some point. And that, to quote Korzybski, "the map is not the territory."