A recent post on the continuing conversations, fights, and debates inspired by Keith Devlin's recent trio of columns on problems he and some mathematics educators (in England, if not elsewhere) see with teaching multiplication-as repeated-addition (or, variously ONLY teaching it that way and never leading students to see that multiplication either is not repeated addition or is a lot else besides repeated addition) to elementary students, let me to write a response that gets at some of what I've been thinking about as I read those British researchers. Below is a re-edited version of my reply:

A crucial point I've tried to make previously in various places: there is a serious problem with the absolutist language a lot of folks in this debate insist upon using and which Prof. Devlin falls into at times (and hardly just at math-teach: check the comments at lets play math; good math, bad math, and any number of other blogs where this issue has been debated for several months. Some folks on those blogs make the argument on this thread at math-teach seem tame. See in particular comments by

**Joe Niederberger**). However, as I've come to appreciate Prof. Devlin's style and intentions, I'm not so worried about his habit of overstating his case as I was initially. I think he wanted to draw attention to a number of important issues in the teaching of school mathematics, perhaps most broadly the likelihood (or at least the possibility) that sloppy teaching of elementary mathematics from a perspective that doesn't consider things from only slightly higher mathematics causes difficulties for a lot of kids as they progress into college mathematics courses. That would certainly be worth thinking about if true, and given the many posts we get on this list about how the sky is falling, too many kids aren't ready for college math, and so forth, one might expect that Prof. Devlin's columns would meet with approval from some of the educationally conservative thinkers here. Indeed, I was caught off guard at the anti-Devlin sentiments from Wayne Bishop and, later, Haim Pipik (though perhaps given that Devlin is on speaking terms with mathematics educators and doesn't find all math education theory or research to be garbage, I should have expected their knee-jerk dismissals).

I tried initially in my first blog post on Devlin's columns (before his third one appeared on this issue) to call into question his very strong assertion that there would be inevitable problems for students if they were taught that multiplication was repeated addition. But by now, anyone who is really paying attention to what he's said knows that he modified the claim near the end of his third column in a way I find acceptable:

Meanwhile, now you know why (or at least you know where to start finding out why) it is crucially important that we not teach children that multiplication is repeated addition. (Or, if you prefer, why we should not teach them in a way that leaves them believing this!)

That last parenthetical remark is precisely what I think most reasonable thinkers on this issue would likely conclude if they had taken the trouble to read the references. And the language in those articles and books, if you look at what Devlin himself quotes, is actually very modest. It's only Devlin who goes a bit heavy-handed for much of his three columns (though he gives plenty of explanation for both his perspective and his concerns), but ultimately I'm unable to fathom how anyone could argue vehemently and unrelentingly against the very last sentence in the quotation above.

For example (but many other examples are similar),

Research SUGGESTS that such teaching MAY BE at the root of later misconceptions. Alternative models for teaching have been shown more effective in experimental studies "(Clark & Nunes, 1998) but EVIDENCE IS STILL LIMITED AND MORE RESEARCH IS URGENTLY NEEDED. [emphasis added]

I'd add that this research would likely be most profitable if it were of the kind Ball describes when she and her colleague talk about research "inside education." The interaction between a number of factors needs to be considered in judging whether there really is a meaningful disadvantage in even including, let alone teaching exclusively, the multiplication-as-repeated-addition idea, or a meaningful advantage in the "alternative models" just mentioned. It's not merely a question of materials or merely of teacher moves or MERELY any one thing in isolation.

I think Prof. Devlin has gone on to point to a lot more than what he raised in his first two pieces. I doubt, for example, that he meant that we MERELY say that addition and subtraction are just two things we do with numbers, but rather that among other things we teach students, we make clear that two main things we do with numbers (or "to them") are to add them and multiply them and that the two are not identical. I think he's right that students should see that and continue to be exposed to the differences, just as we can't be satisfied with tell students that exponentiation IS repeated multiplication.

I would argue that students will, as they gain mathematical maturity, be increasingly able to see different ways in which there are deep differences between addition/subtraction and multiplication/addition, even if they ALSO consider some similarities. One issue Devlin raises that I think is worth noting is his concern that we're hobbling kids by teaching them that "multiplication makes things bigger," and I agree that it's a problem for a lot of students that arises repeatedly (one glaringly obvious example is on the typical SAT question where students are asked: for x > 0, which of the following is greatest - A) 1/x B) x C) x^2 D) sqr(x) or E) the relationship cannot be determined. Even when the problem is modified to exclude x = 1, many students are all-too-quick to pick "C" and I think this reflects the "multiplication (and exponentiation, which is "merely repeated multiplication") makes things bigger" idea. But I would point out that we see the same confusion about addition/subtraction: students get stuck at the whole number (or even natural number) assumption that addition always makes things bigger; subtraction always makes things smaller. So I believe we need to address a related though probably less complex problem with how we present addition and subtraction to students as if they are NEVER going to have to deal with integers. If no one else things any of this is important, the folks who write the SAT, ACT, GMAT, and similar test where math is at issue obviously do, because they exploit student misconceptions about all this relentlessly.

Returning to Devlin, he repeatedly states that one can arrive at the correct answer to many (though not all) multiplication problems through the act of doing repeated addition, but that doesn't mean that the former IS the latter or is "merely" the latter. I would have balked at the second notion, without ever having read Devlin, but I think he was trying to break through a lot of inertia amongst K-12 teachers, especially in the lower grades. And clearly he was right, given the reactions his columns engendered, as many teachers did (and do) seem to think that multiplication IS repeated addition and that's all ye know or need to know. Or at least all THEY want to know or teach. And that is really not acceptable.

Frank, the fellow whose post inspired this entry wrote,

The rest of the first article and the second article were mainly devoted to explaining why equating multiplication and repeated addition is mathematically incorrect (without providing any clear examples of why it was incorrect or important for K-12 education.) At the end of the second article, Devlin pleads:Let me stress again that I am not suggesting we teach children arithmetic the way professional mathematicians view it. Rather, my point is that, however you teach it (and I defer to professional teachers in figuring out the how), don't do anything that is counter to the way the mathematicians do it .... some of your pupils may well end up in universities where they will HAVE to do it the right way.

In other words, don't create any misconceptions in K-12 that might make my job harder.

That doesn't seem to be Devlin's point, but rather not to make those students' lives harder. I don't think he is worried about the difficulty of his job, but the intractability of some student conceptions.

Frank continues:

In his third article (Multiplication and Those Pesky British Spellings), Devlin finally gets around to explaining why K-12 teachers and students (as opposed to university mathematicians) should care whether or not multiplication is exactly the same process as repeated addition: It may make it difficult to separate the concepts of addition and multiplication when students do proportional reasoning. Devlin states that"Arguably Nunes and Bryant know more about multiplication and how to teach it than any other researchers in the world", so he quotes them on the subject of repeated addition:

"The common-sense view that multiplication is NOTHING BUT [Frank's emphasis] repeated addition, and division is nothing but repeated subtraction, does not seem to be sustainable after a careful reflection about situations that involve multiplicative reasoning. There are certainly links between additive and multiplicative reasoning, and the actual calculation of multiplication and division sums can be done through repeated addition and subtraction. [DEVLIN NOTE: They are focusing on beginning math instruction, concentrating on arithmetic on small, positive whole numbers.] But several new concepts emerge in multiplicative reasoning, which are not needed in the understanding of additive situations.

After reading the portion of their book that is available on the Internet, Nunes and Bryant make a great deal of sense to me. Notice that Nunes and Bryant say that multiplication is MORE than repeated addition, while Devlin says that "It Ain't Not Repeated Addition". Nunes and Bryant refer to repeated addition a "common-sense view" (that is incomplete), while Devlin equates it with the level of ignorance represented by the phrase "ain't not".

Do you think Professor Devlin has done a good job of educating the mathematics community about this issue?

I think Devlin was citing sources earlier than in the third article, but I won't quibble. He clearly was informed by that research, and it's interesting that it's pretty much all BRITISH research. He pokes some fun at the fact that it seemed to be news to a lot of the blog respondents that this body of work existed. I'm not sure many of them would be any more aware of it had it been American research. We have a few members of this list who don't seem to much care what research is out there if it doesn't confirm their prior beliefs.

And yes, I think Prof. Devlin has done everyone a wonderful service with these columns. Was he absolutely right in every particular? Probably not, though I think all three columns bear rereading with great care: he answers a lot of questions that folks claim he does not, and he certainly provides support for his ideas, though as he's told me, that didn't stop some some folks from excoriating him for NOT providing support for his ideas. Go figure.

I know that there aren't many people in mathematics or mathematics education who can reach as many people as Keith Devlin can, and as the issue he's raised is clearly of great interest and likely importance, he deserves everyone's thanks and respect, though that hardly means that everything or anything he says should be accepted. He meant to provoke thought and has done so, though unfortunately he has also provoked a lot of thoughtlessness. That's the nature of things these days, of course, and I have no doubt he can handle the heat.

Postscript: for those of you puzzled about the title of this post and the accompanying photo: the actor Tom Bower played a character name Cecil Skell in a terrific movie with James Woods and Robert Downey, Jr. called TRUE BELIEVER. Skell is a schizophrenic who believes that the phone company killed JFK (because he wanted to break up the phone company). He happens to be a key eyewitness in a murder who is never called by the defense in the original trial of Woods' client, a Korean who has now killed an "Aryan Brother" in self-defense while in prison for the first crime. Skell insists that the murderer "wasn't Chinese, because Asians have a different aura." When Woods and Downey, Jr. discover proof that in fact Skell is right and the murderer was not Asian but merely looked like he might have been, Downey, Jr. says to Woods, "Does this mean that the phone company DID kill Kennedy?" To me, it's the all-time greatest movie line to use when you realize that something that initially struck you as a bit crazy is in fact probably true.

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