“Multiplication is Not Repeated Addition” Revisited

The story so far

Last year I wrote about a kerfuffle of the mathematical portion of the Internet over a series of three articles by Keith Devlin that claimed not only that teaching multiplication as repeated addition was not only wrong, but outright harmful.

I mentioned I was working on an ultimate post that delivered a final verdict. This is still not that post, because the original got caught in a cycle known as “looking for more research”, “rewriting”, “reconsidering”, and then a touch of old fashioned “procrastination”.

However, Mr. Devlin has returned yet again to the topic in a January 2010 article, so to reduce my guilt over what will likely be an endless process, I’m going to put forth my main point.

My main contention

One common argument pattern (in education, politics, or anything really) is when two sides take different sets of premises but don’t argue about their premises; they instead argue about their ramifications. In an effort to get to the heart of the matter, in answering “is multiplication ever repeated addition?” I attest:

It depends on your philosophy of mathematics.

Two quotes from mathematicians

For newcomers, I’ll begin by saying it one more time: multiplication is not, repeat not, repeated addition. Not even for positive whole numbers. This is the one case where you do get the same answer, but getting the same answer to two procedures does not make them the same procedure. —Keith Devlin

Just as addition can be thought of as repeated counting, multiplication can be thought of as repeated addition. — Melving Fitting and Greer Fitting

Wait, what?

So, one mathematician says multiplication is not repeated addition in any situation whatsoever, while another pair says it is exactly that! What’s going on? Consider the second quote in context…

Adding and multiplying defined from scratch: the executive summary

While there are many books on basic number theory, I’m going to discuss the one from the second quote, which happens to be online.

1. Start with the natural number system, by establishing counting numbers starting with 1, which have successors (eg the successor of 4 is 5). This is pre-addition, so the notation for “3 successors after x” is x^{+++}.

2. The addition x + y is defined as taking the yth successor after x, or as they put it, “We might think of 4 + 3 as a set of instructions to us, telling us to start with the number 4 and count off the next 3 numbers.” In their notation that would mean 4^{+++} is the same as 4 + 3. In other words, addition can be thought of as repeated counting.

3. Now, multiplication.

Our informal guide is: to multiply 4 by 3 we write down 3 4’s and add them. But how can we express this without dragging in machinery to count things? We look for an alternative definition which will result in 4 \cdot 3 being the result of adding 4, three times, but won’t make it necessary to count 4’s.

This follows with an official definition of multiplication, notated by \cdot, as that operation such that

x \cdot 1 = x
x \cdot y^{+} = x \cdot y + x

The first part the identity operation (5 times 1 is 5) but the second is considerably more clever. Rather than defining x times y, it defines x times y+1. This is best illustrated in steps:

Step 1. 5 \cdot 1 = 5 by the first definition.
Step 2. 5 \cdot 2 = 5 \cdot 1 + 5, adding 5 to the answer from Step 1.
Step 3. 5 \cdot 3 = 5 \cdot 2 + 5, adding 5 to the answer from Step 2.
Step 4. 5 \cdot 3 = 5 \cdot 3 + 5, adding 5 to the answer from Step 3.
and so on.

Stepping back, while the “recursive programming” nature of the definition is odd, it is clear the definition of multiplication is causing repeated addition.

4. Add 0 to get the whole number system. Addition is unproblematic: just define 0^+ to be 1.

5. Multiplication, though, gets messier:

The idea we followed in the last chapter was that multiplication was repeated addition, and so 2 \cdot 3 told us to add 3 2’s together. This idea easy covers 0 \cdot 3 It tells us to add 3 0’s together . . . There is more a problem with 3 \cdot 0 however. If we apply our basic idea, we find ourselves adding together 0 3’s. We have no 3’s (or anything else) to work with, so we certainly can’t get a number for an answer.

This is neatly handled by special-case defining x \cdot 0 = 0 \cdot x = 0.
Note, however, that we are not strictly following the “repeated addition” algorithm anymore, although the special case is a “natural extension” of the idea.

Now, one can go on to fractions and so forth, but given that Mr. Devlin is arguing that multiplication is not repeated addition even in the case of positive whole numbers I’m going to stop here; just keep in mind each step can be a “natural extension” of the previous idea but the tweaks make them not formally identical.

The argument

If something is a “natural extension” (in this case, having to change things with 0) is it still “the same as” the original?

But more importantly: is the original formal method made to define addition, once changed, still “a part of” the fuller picture of mathematics? Is the formalism a seed still planted within the mathematical framework, or does the addition of an axiom (or two axioms, or five axioms) make any thinking about the “internal” model wrongheaded?

It is no accident many ardent defenders of multiplication as repeated addition come from computer science; in programming, old code has a framework built around it that does not change the nature of the original code. Whereas when mathematicians add to a theory, even a single extra axiom may radically shift the nature of a structure (take for example mathematics with or without the Axiom of Choice).

Further reading

Want to mess with your brain further? Try the Doyle and Conway paper Division by three with the grand first line:

In this paper we show that it is possible to divide by three.

and the ominous:

There is more to division than repeated subtraction.

EDIT: Comments are closed for now. I will be posting more about this topic later at which point I will re-open for comments.


50 Responses

  1. Boooo!

    Oh, wait, sorry. Force of habit.

    Nice, Jason. We’ve been wrestling over the topic again at the mathforum, so I think I’m sharp on all this again. Your main point has bubbled up there as well, and I think it’s an interesting way for the conversation to turn.

  2. If you want to think generally enough, then it’s pretty clear that multiplication isn’t repeated addition. In the category of sets, multiplication corresponds to the categorical product and addition corresponds to the categorical coproduct. The special property of the category of sets is that every set is a coproduct of some number of copies of 1, which is why it’s possible to view multiplication as repeated addition in this sense. This fails miserably for more general categories, even the ones where the product distributes over the coproduct, and even for categories which look a lot like \text{Set} such as topoi.

  3. […] 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 […]

  4. What’s so great about this debate–and I would disagree that it depends on one’s philosophy–is that we don’t NEED repeated addition, either mathematically or pedagogically (as a meaning, anyway; of course we need it for “reckoning”).

    But people hold on to it so desperately, so passionately. Multiplication can be taught to the same children we introduce it to now, and mathematics can chug along quite undisturbed (as it has) without–in the case of mathematics, the nonexistent formal definition of–repeated addition.

    Jonathan seems to have reversed who exactly it was that got called out–and who between himself and Devlin is the mathematician and the “math columnist.”

  5. Thanks for the background. I would expect mathematicians to have different takes on this, depending on their point of view, depending on what subfield of math they’re in.

  6. I’ve been meaning to jump into this argument for a while. I think we do need repeated addition for defining multiplication of real numbers. We can hide it away behind a bunch of symbolism, but the property that 1+1=2 leads inevitably to repeated addition as multiplication for whole numbers. Everything else comes from reasoning that starts there: we get negative integers by extending whole numbers in a consistent way, we get rational numbers by extending integers in a consistent way, and we get real numbers by extending rationals in a consistent way. It’s the way the real numbers are built (and complex numbers, etc). Generalizing from real number multiplication to get the sort of definition you use in abstract algebra is useful mathematically, but it seems like a dead end for elementary pedagogy. I think the winning move pedagogically (for K-10 students, at least) isn’t to avoid repeated addition, but to make the connections between that definition and its generalizations as apparent and clear as possible.

    (Hm–I probably need to work on making my rants sound less rant-like. Sorry.)

    • You don’t need addition to define multiplication. You can consider the categorical product in a category without ever thinking about the categorical coproduct.

      • Qiaochu Yuan:

        Exactly. I think this is where I’m going in the future with this. What the MIRA folks argue is that we NEED repeated addition as a definition (or “aspect”) of multiplication.

        It’s pretty obvious that we don’t need it mathematically. The folks who absolutely desperately need it pedagogically should have the burden of explaining this discrepancy–with something other than anecdotes and lack of imagination.

  7. I’m trying to figure out how this debate actually serves any purpose (except for gratuitous self gratification of the proponents on either side) for anybody who’s not going into pure math.

    I made it through a whole buttload of calculus classes and got an engineering degree without ever having to worry about the difference. I welcome either side to come in and try their hand with my 5th period remedial class, and see whether they can get my students to give one little bit of crap about this argument.

  8. I can’t speak to Mr. K’s question regarding pure math. I very much doubt, however, that Keith Devlin’s concern was about pure math (I know it wasn’t, in fact), and I am 100% sure that’s not my concern. I coach high school math teachers in Detroit, and have coached upper elementary teachers in Pontiac, MI (which is in some ways a smaller version of Detroit). And this is most definitely not an academic argument (or at least not merely so). Dismissing it too lightly as “merely” academic or merely an ego-contest is a mistake, on my view.

    Take Laurel Langford’s post. She appears comfortable stating that the “multiplication is repeated addition” (MIRA) definition is adequate or necessary for “real numbers.”

    But of course, the issue is probably NOT one of definitions, first of all, as Devlin and others have tried to point out. In particular, it’s much more about what sorts of things multiplication DOES that addition DOES NOT. After all, mathematicians are very well-known for being parsimonious about fundamental axioms and undefined terms. Would they REALLY make it so clear that we need two fundamental operations for arithmetic – addition and multiplication – along with their inverses, if one would suffice? Let’s call everything some version of addition in that case: addition = addition; multiplication = repeated addition; exponentiation = hyper-repeated addition, etc. (taking roots would be “inverse hyper-repeated addition”).

    But even so, I’m a little surprised to see Laurel seem to make the distinction on when we might not want that MIRA definition to hold: she draws the line not between the natural or whole numbers and the rest of the real numbers and their subsets, but between the real numbers and abstract algebra. So Laurel: are you happy with the idea that pi * e = pi iterations of e (or e iterations of pi)? I have a really hard time wrapping my head around that, but if you can explain it to me in a way that makes sense, I’ll be excited to hear that.

    Usually, MIRA defenders tend to admit that SOMEWHERE within the real numbers (maybe when we go beyond rational numbers), that repeated addition thing gets a bit dicey.

    But this is just the tip of the iceberg, really. I suggest interested readers look at the comments on my last blog entry (which is on this very topic) from “Burt” (who is Burt Furuta, from Hawaii. They introduce some really important issues I think are being missed in the conversation here. The link to the post is http://rationalmathed.blogspot.com/2010/02/keith-devlin-extended.html.

    Burt sent me a follow-up version of his ideas today that I think explicate the ideas in his comments on my blog that goes even deeper. Should he give me permission to do so (as these are an excerpt from a book chapter in progress), I will post them on my blog and post a link or reminder here.

    One VERY key issue he raises has to do with units of measure, and probably the heart of all the concern he, I, Josh Fisher, and a number of others have really IS about what happens to a large number of kids (though perhaps not ANYONE reading this blog entry and its comments) regarding their understanding of rational numbers, proportional reasoning, the mathematics they need to do when they’re solving problems in science, economics, etc.) and related issues. I don’t think Burt or anyone else I’ve been reading seriously the last two years on how we might better teach arithmetic is primarily worried about definitions or whether we’re hobbling kids so they can’t be professional mathematicians. The damage we are worried about strikes way too many people and cuts them off from a great deal of important mathematics at the level of calculus and BELOW. And it hits well before they reach calculus, if indeed they ever do.

      • Too bad someone with an interesting blog can’t say something a bit more substantive. If your “QED” comment is intended to be profound or clever, I have to say that it appears nothing more than snide to me. That’s rather unfortunate. And I, for one, don’t appreciate it. I think you can do better.

        Did you bother to read Burt Furuta’s comments, for example? Somehow, dismissing those smugly, if you indeed read them at all, speaks much more poorly of you than of those who are trying to make a positive difference in how we teach a fundamental concept to kids.

  9. Special K has now commented twice on an issue that no one cares about.

    QED, indeed.

  10. @Everyone: Diplomacy, please.

    @Mr. K: I tend to agree with Sue’s contention this doesn’t make the top ten of math ed issues, or even top twenty, but that doesn’t mean it isn’t useful / interesting to talk about. I’m not going to resolve “If you’re going to teach math, you need to enjoy it.” in a single blog post.

    @Qiaochu Yuan: I’ve been meaning to write a “Category Theory for Laymen” post sometime (the attempts I’ve seen so far have been, hm, not so clear) so I might try “Category Theory Up To Understanding That Multiplication Thing” sometime.

    @Michael: Someone commented on handling pi * pi at my other thread. To quote his (her?) standpoint “As for irrational numbers, they are limits of sequences of rational numbers, and their products are limits of sequences of products of rational numbers… in others words of repeated addition.”

    • If you do, I highly recommend that you read Lawvere and Schanuel’s Conceptual Mathematics. It’s an attempt to explain the conceptual content of category theory in very elementary terms and I think the first few chapters in particular would suit your purposes well.

    • Hmm. “the limits of sequences of products of rational numbers” are an example of repeated addition?

      Why does that sound like either nonsense or an incredible stretch to me?

      And by “incredible stretch” I mean that it appears at best to be tormenting the notion of repeated addition to desperately avoid the idea that multiplication has something to it that ISN’T additive. Do you believe that’s really the case?

      • Let me make an analogy with logic:

        Normally, we use AND, OR, and NOT, and that’s enough to get all our logical connectives. However, NAND alone (or NOR alone) is sufficient to produce all the needed effects (truth-preservation, falsity-preservation, linearity, monotonicity, self-duality) to have a complete logical system.

        Now, one needs to go through many more contortions if restricting to NAND and NOR, but it can be useful in a computational sense; if I recall correctly one of the Apollo computers used NAND only to simplify the circuits.

        So if in a raw, computational sense, one can get pi * pi with solely addition, then by the philsophical standpoint I cite above it would be considered a form of repeated addition. “Tortured”, to be sure, but not necessarily useless: consider in the new areas of quantum and biological computation that circuits may end up with particular restrictions, so (just like how the Apollo computer simplified to NAND for technical reasons) it may be handy to restrict one’s computations solely to addition.

        I agree that the computational sense does not include everything from the mathematical sense, but to do a really full treatment requires category theory, as per Qiaochu Yuan’s comments.

  11. > doesn’t mean it isn’t useful / interesting

    I’m willing to agree with that. And your original post actually does a very nice job of that. But I will still contend that the ensuing arguments and “discussion” is more about ego and less about decent pedagogy.

    FWIW, I’m in the “multiplication isn’t (always) repeated addition” camp. You figure that out in the first two weeks of linear algebra. More simply, repeated addition of linear inches will never give you an answer in square inches.

    The part that makes math neat, to me as a teacher, is that despite all protestation that multiplication isn’t repeated addition, skip counting will still find you the area of a rectangle with integer sides. The interest isn’t in a narrow definition, it is in the same pattern showing up and behaving consistently across a variety of different situations.

    I am having a difficult time clarifying my feelings about the debate (though my issue is clearly with the debate, rather than the topic thereof). I think in essence it is that I feel one side underestimates the abilities of our students, while the other ignores their need for incremental learning, and with neither side actually seeming to advance the actual pedagogy at all.

    • Reposting a comment from below, it is easy, in passionate debate, to assume motivations of those on the opposing side. Let us stick to the topic itself and not the people involved with it.

      I’d love to hear your thoughts on the ed issue when you have them fully fleshed out. My own thoughts are still tangled up with my eternally-delayed post mentioned above.

      neither side actually seeming to advance the actual pedagogy at all.

      Does Maria Droujkova’s poster count? I think it came directly from all the discussion.

    • I don’t see how those of us who have been trying to look at possible connections between teaching multiplication to kids as if it’s either essentially repeated addition, nothing more than repeated addition, or anything along those lines are doing either of the things you say at the end. One very open question here is whether being able to multiply is the same as understanding multiplicative reasoning, and I think there’s serious concern that this is not necessarily so. But moreover, there is ample reason to ask about those who do NOT succeed at multiplication and whether there are important connections between that difficulty and the issue on the table.

      As for “their need for incremental learning,” I don’t necessarily buy that as an unquestioned assumption. But be that as it may, Davydov, grounding himself in Vygotsky’s work, took a very different tack on all that and the implications for pedagogy surrounding arithmetic. He and Vygotsky may have been utterly wrong, of course, but I think their evidence and thinking is just as worth looking at as Piaget’s (and will remind readers here that Piaget’s work isn’t popular with a lot of folks who are adamant that teaching kids MIRA is perfectly fine (maybe no one here feels that way, but over on math-teach, that’s very much the case).

  12. >one side underestimates the abilities of our students, while the other ignores their need for incremental learning,

    I like this point. Taking sides is not the way. The more we can find to agree with in the words of someone we are disagreeing with, the more we can learn from one another.

    • “one side underestimates the abilities of our students, while the other ignores their need for incremental learning”

      “I like this point. Taking sides is not the way.”

      Nor was that the point, which is something you slyly avoid and then pretend to agree with, which I don’t like (it’s rotten).

      But, to Mr. K’s point (and this has been raised several times):

      When we tell students that multiplication is just repeated addition (as we do) and then move on to inches x inches = square inches, how on earth is that satisfying students’ “need for incremental learning”?

      Or, hey, for that matter, tell me why “need for incremental learning” should trump “true.”

      • It is easy, in passionate debate, to assume motivations of those on the opposing side. Let us stick to the topic itself and not the people involved with it.

        For example, it is interesting to consider if a notion of repeated addition causes issue later with inches x inches = square inches. (I should point out that the original post above is considering solely the mathematical issue.)

      • I just want to offer my apologies to SueVanHattum for any part of my comments that insinuated that SHE was sly or rotten. I retract those sentiments.

        She is a more-than-qualified participant of this debate, which includes teachers, mathematicians, educators, etc., of all stripes. And Sue has many of these stripes.

        This is an important debate. Viewpoints will clash and the ideas therein will inspire and perhaps disgust. But we should not forget that we are debating ideas and not personal qualifications.

        –Joshua Fisher (Text Savvy)

    • A former mentor of mine when I was in the doctoral program in English at U of Florida, William R. Robinson, used to say that the further someone was from getting it right, the more useful it is to find something in what they have to say that is “heuristic” (by which he meant ‘thought-provoking’) and that the closer someone is to ‘getting it right,’ the more significant are the ways in which they ‘get it wrong.’

      I’m not saying I agree with that down the line or know what “getting it right” or “getting it wrong” is much of the time. But somehow I think there really is something very fundamentally off-base in seeing multiplication as repeated addition for one’s starting point or primary model. And those of us who, like Devlin, have been waging battles against complacency about that notion, would very much like to see everyone seriously question that model, no matter how hard s/he wishes to accept it and adhere to it, at least long enough to look at the implications for mathematics education if it turns out to be more than a bit flawed. In particular, whether we’re truly undermining generations of kids when it comes to their understanding of quite a number of key mathematical ideas that arise well before calculus: rational numbers, real numbers, ratios, multiplicative and proportional reasoning, etc.

      Of course, for those not keeping score at home, I was completely aghast at Devlin’s original two columns when I first saw them and thought he was way off base AND way out of line. I am now of a very different mind on these questions. A very key reason is that I’ve read a good deal that makes sense along these lines by Davydov, Catherine Sophian, Terezinha Nunes, Burt Furuta, and others, some of it theoretical, some of it about research results. Seems to me that if this is a worthwhile question, folks need to do homework before digging their feet in once and for all, and I’m not sure that very many, particularly not over on math-teach, have done much, if anything, along those lines.

  13. p.s. My post on this hasn’t been linked to here yet, has it? Mr. K, if you can stand to read more about this, I wrote something I think you might like.

  14. In Definition 15 of Book VII of the Elements, Euclid states that multiplication is, “when that which is multiplied is added to itself as many times as there are units in the other.”


    I haven’t seen anyone mention the Euclid connection.

    I know that many things have changed since the days of Euclid, but he was pretty sharp intellectually and I think that his conception of multiplication as repeated addition indicates that people (ie children) have a natural inclination to view multiplication as repeated addition.

  15. Euclid wasn’t exactly known for his rigorous or clear definitions: his definition of straight line is “that which lies evenly with the points on itself.” Not sure what to make of that. His definition of a point was similarly opaque: “that which has no part.” That’s why modern mathematicians consider both of those terms as better left undefined. I’m not saying multiplication therefore should be undefined, but I’m not too enthusiastic about Euclid being the fellow to look towards in this regard. YMMV.

  16. Oh, and I’m very skeptical of “a natural inclination” on the part of children as presented by someone today, let alone in Euclid’s time. Adults are prone to assume they know what’s natural for kids when quite often that only reflects their own adult perspective founded on very little that has to do with children.

  17. Note, by the way, that there are no “units in the other” for repeated addition. There is no other. It’s just x + x + x . . . You can be sneaky and say that “the other” is “the number of times,” but that’s not a real variable here.

    (Thanks are owed to Devlin for that little nugget.)

    • Just for my own writing sanity, I wish there were an easy way to avoid the likely confusion between “units” in the sense of “What’s the unit here?” and “units of measure.”

  18. And since repeated addition has no “other” variable and multiplication does, it’s almost like the latter is a real, true-blue binary operation, whereas the former is just a calculation procedure.

    Isn’t math fun?

  19. is anything *ever* context-independent?


    (not in *this* context, anyway….)

  20. very informative article Jason. Keep it up. I think, this blog deserves to be in my blogroll. 🙂

  21. Devlin says: “getting the same answer to TWO procedures does not make them the same procedure.”

    TWO procedures?
    OK so where’s the other contender? We’ve got one guy in the ring — Devlin says he’s fake, but he won’t have the real champion show his face? Ah! Fuggetaboutit.

    • Joe’s approach to this argument seems to be to mock, joke, and posture. Once in a while, he actually says something substantive, appears to hear what those with whom he disagrees have to say, and even appears capable of replying thoughtfully. But when it comes to Dr. Devlin, it’s as if we’re witnessing Joe Ahab and Moby Keith. And it’s not really useful, informative, or entertaining.

      Joe, if you have a problem with Keith Devlin (and obviously you do), you need to write to him. And since it’s clear that he has no interest in writing to you (and with cause, I think), why not leave him out of it? He has a VERY large audience, mostly of people who have reason to think he knows a great deal of mathematics and isn’t a rash, thoughtless fellow.

      Many very contemplative folks also think he has a point or two to make on all this. But some people, you amongst them, insist on making it all so very personal.

      The two procedures to which you mockingly refer, though I would say, and suspect Devlin would, too, that we’re talking about two operations, are obviously iterated addition and multiplication, as if you didn’t know.

      And of course the point is that you can use iterated addition to get the same result as you get from doing multiplication, at least for integers, without having to think too hard about what you’re doing or what the underlying ideas might be, as long as you keep the numbers relatively small or are willing to truly USE iterated addition to calculate ridiculously large products. I personally don’t recommend that anyone do so, however, and neither do you.

      But you and a host of like-minded people, for diverse reasons all of which I find unconvincing and many of which I find distressing coming from ostensibly smart, knowledgeable people, play at the notion that there’s simply no difference between these two things, methods, procedures, operations, concepts, modes of mathematical thinking, etc., and you ground that primarily or solely in the fact that you can reach the same ANSWER with each, if you have world enough and time, of course.

      And rather than even CONTEMPLATE that there’s some important reason or reasons not to think of these operations as merely operation A and operation “Do A a lot of times until you’ve gotten to where you would have gotten had you done M instead” rather than A and M, distinct though related, you dig your heels in.

      You may have come over here to post today, seemingly out of the blue, because of Joshua Fisher’s excellent post on math-teach. Perhaps he got to you sufficiently to make you feel you needed friendlier ground where nearly everyone would agree with you. Or think your little jabs above were amusing. Or clever.

      Devlin has made his points, Your dopey boxing analogy isn’t even true. I and some others think you got whipped before you even started. But you’re still claiming you wuz robbed. Kind of sad.

  22. He is confused between several mathematical function(s), and a method that computes that function on whole numbers. It *is* sad!

  23. Joe,

    We all have sympathy for punch-drunk fighters, but we don’t particularly enjoy being drooled upon by them.

    There’s an old saying that you can lead a crank to logic, but you can’t make him think. I will wager that you haven’t bothered to read a single word of the complex research that informs and underlies the questions Devlin and those he cites in some of his columns raise. You continue to believe that you can claim that kids think a certain way, that it’s OBVIOUS that they do (from your naive, uninformed adult, ‘common sense’ perspective), as if we make decisions about how humans grow and develop based on what ‘should’ be the case made in hindsight.

    In hindsight, kids should know about object permanence. After all, it’s just common sense. But they don’t until a certain point in their development. There’s a ton of research to back this. And that is the tip of the iceberg in terms of such research and the theoretical frameworks that emerge from it.

    Sophian, Nunes, Bryant, Davydov, and many, many others, are part of that research and theoretical tradition. You’re not. And you insist upon both staying ignorant of it and in joining people who are perversely proud of so doing (who just happen to be the Usual Suspects when it comes to matters involving mathematics education and education in any arena. That such people are almost to a person social conservatives, at the very least, might bother you if you attend to the intellectual company you keep. But perhaps not.)

    What can we say regarding people who consciously choose to weigh in on a complex subject while remaining willfully ignorant of all the relevant research? Who deny that such research matters? Who lump such research, which isn’t coming for the most part from the much derided world of math education but rather from psychologists of various stripes?

    I can’t speak for anyone else, but for my part, I find it impossible to take the silly, ad hoc arguments being offered up by such people seriously. Every time some post is founded upon the idea that X must be the case because it’s OBVIOUS that X must be the case, and the person pushing X is speaking about how kids MUST think and learn, and that it’s after all just a matter of common sense, I am tempted to stop reading anything that person has to say on the subject.

    Unfortunately, the shrillest, most hateful (in your case, for who knows what bizarre reasons, of Keith Devlin) voices in this debate seem all to operate the same way: ignore the cognitive and perceptual psych research, pretend it doesn’t exist or is irrelevant, and repeat that it’s just obvious that X is the case.

    Now Joe, in all seriousness: whom do you think an informed audience is going to listen to when it comes to an argument about mathematics: Keith Devlin, a renowned mathematician with an internationally renowned body of work and a well-deserved reputation for being able to make deep mathematical ideas accessible to the public, or some random crank from New Jersey with an Ahab complex, particularly when the latter rides in on a hobby horse and won’t read or cite any research whatsoever about kids perceptions and thinking?

    Whom would YOU believe if you weren’t that crank and Devlin weren’t the expert you’ve turned into your personal White Whale?

  24. Left out a phrase in the sentence “Who lump such research, which isn’t coming for the most part from the much derided world of math education but rather from psychologists of various stripes?”

    It should read, “Who lump such research, . . . but rather from psychologists of various stripes, [ in with much weaker, less scientific research (or at least less quantitatively-based research) from that disrespected area of inquiry.” ]

    Sorry for the omission.

  25. Michael, I left this discussion because of ad hominem attacks on me. (And you pulled me back. Good job.) ;^) It is possible to refute Joe without using those sorts of arguments yourself.

    You made a particular comment that I feel a need to respond to:
    That such people are almost to a person social conservatives, at the very least, might bother you if you attend to the intellectual company you keep.

    Yikes! Do you only wish to learn from people who already see the world through the same lens you do? I deeply respect a few people who see themselves on the traditionalist side of the ‘math wars’, in part because they don’t care about ‘the intellectual company they keep’. They are looking at the situation carefully, and who-says-what is not of interest to them.

    [Before the right wing cooped the phrase, I used to have a pin saying ‘Politically Incorrect’. (I got it at the People’s Food Coop. Damn, I miss Ann Arbor sometimes!) I am proud of not fitting in other people’s boxes.]

    I am tickled when I become friends with people who have quite different views from mine – because I know I’ll learn from them. Digression: Long ago, I taught women’s studies. The ‘debate’ over abortion is uglier than this debate can possibly get, as lives are at stake (from both points of view). I have strong opinions on the matter, and have blogged about them. But it wouldn’t be fair to push my opinions, regardless of the social cost the other side would cause, on my students. So I set some ground rules for the discussion. We were not going to debate. We were going to sit in a circle and tell personal stories. I started us off by telling a personal story that one might expect to come from someone on the other side. End digression.

    I know this debate is different, and facts may be more important than anecdotal stories in this case. However, the repeated mention of research, without details to clarify, is frustrating to me. If there is good research out there, please describe it in detail.

    You wrote:
    [Kids] don’t [know about object permanence] until a certain point in their development. There’s a ton of research to back this.

    The research starts with Piaget, right? However, I’ve read about newer research that brings his results into question. Have you seen the study (maybe it was Dehaene? sorry i don’t know how to find it) where babies shown 3 objects put behind a screen and two pulled out register surprise that they don’t register when the number stays the same? I also remember someone describing an experiment more like those Piaget did, but with candies the kids would get after – m&ms, perhaps – pushing them closer did not make the kids think there were less of them. My guess (conflicting research doesn’t give an answer) is that, for young kids, object permanence comes and goes.

    So research on what kids are really thinking is hard to do well. As is research on what’s right and wrong in a classroom. I’m sure you saw the quite bogus ‘research’ that allegedly ‘shows’ that teaching with abstract examples is better than teaching with concrete examples. It was so badly done. But people will say “research shows”. If a teacher uses just MIRA, we know s/he is likely to have other problems with her/his math teaching. Maybe those other problems are what cause the differences in test scores people observe.

    The research Devlin quotes in this article does not back up his claim that multiplication as repeated addition is just plain wrong, it only shows (maybe) that the correspondence (huh, what does he mean here?) model is better. (My stand is that we should teach multiplication many ways, MINJRA.)

    One last objection to your comment: Since Devlin does not accept comments where his articles are posted, it is reasonable to expect people will want to offer their objections at public discussions of the issue, rather than writing to him privately. When I posted on this at my blog, I also wrote him to point to it. He wrote me a lovely reply, but he does decline to discuss publicly.

    • Sue, re: the M&M experiment, I speak partly of what you mean here. (The same chapter mentions the M&Ms. Rather than asking “which has more” they set up the candies and asked the kids to pick one. Invariably they picked the larger set no matter how the candies were arranged. This got around the effect of them trying to meet the adult’s expectations in their answer.)

      The other baby counting experiment you mention is more about number sense than object permanence. I don’t know of any research overturning the general wisdom on object permanence.

    • “Michael, I left this discussion because of ad hominem attacks on me.”

      Wrong. If a person says, “You don’t know what you’re talking about, and you’re a poo-poo head,” then that’s just two arguments, which require substantiation. If a person says, “You don’t know what you’re talking about BECAUSE you’re a poo-poo head,” then that’s an ad hominem argument.

      “Yikes! Do you only wish to learn from people who already see the world through the same lens you do?”

      Well, you’re the one that tuned me out the second things got difficult for you.

      “I know this debate is different, and facts may be more important than anecdotal stories in this case.”

      Good point. Wonder who else might have said that in this debate? (Hint: It’s I.)

      “However, the repeated mention of research, without details to clarify, is frustrating to me. If there is good research out there, please describe it in detail.”

      Right after this paragraph, there’s this:

      “However, I’ve read about newer research that brings his results into question. Have you seen the study (maybe it was Dehaene? sorry i don’t know how to find it).”

      Can you describe THIS in detail, please? Jeez.

      “The research Devlin quotes in this article does not back up his claim that multiplication as repeated addition is just plain wrong, it only shows (maybe) that the correspondence (huh, what does he mean here?) model is better.”

      Tell us why. Have you even read that research? If you have, then how on earth could you question what correspondence is? And that’s just one idea mentioned. I don’t believe you. How could anyone believe you?

      “But people will say ‘research shows’.”

      A phrase right from the KTM archive. Good to see you’re thinking for yourself.

      To be honest, I will take Joe and Pam in the math-teach forum any day over this. At least we get somewhere with them. This is just looking for attention.

  26. Jason, thanks for giving us more detail on those m&ms. I am fascinated once again. (The joys of a bad memory.) ;^)

    Hmm, I see what you’re saying about the baby one. Hmm, I need to understand object permanence better… Ok, here’s Wikipedia: Object permanence is the understanding that objects continue to exist even when they cannot be seen, heard, or touched.

    I understand that the main point of the baby experiment was the number sense issue. (I kind of conflated the two issues.) But I think it does also display some sense of object permanence. If the original objects didn’t ‘continue to exist’, why would it matter that the ‘new’ ones were of a different number?

  27. I slipped. I didn’t mean “object permanence” which I’m well-aware has been called into serious question. I meant the far more on-point issue of conservation of number and conservation of volume. There has been additional research which confirms and elaborates upon Piaget’s work there. I know of no one in the field at this point who calls those ideas into question. Kids of varying ages are not born with a clear understanding of these, but develop understanding with time. Whether instruction can hasten that process is always an important question, of course.

    As to the rest of your post, Sue, I’m not going to go ’round that Mulberry Bush again. You have your methods and I have mine and they are not a matched set. Vive la difference.

  28. […] the multiplication/repeated addition debate breaks out again here, here, here, here, and here. I appreciated Michael’s analysis of Children Doing Mathematics, Terezinha Nunes […]

  29. […] No comment. Grade 3: Understand that a unit fraction corresponds to a point on a number line. For example, 1/3 represents the point obtained by decomposing the interval from 0 to 1 into three equal parts and taking the right-hand endpoint of the first part. In Grade 3, all number lines begin with zero. 2. Understand that fractions are built from unit fractions. For example, 5/4 represents the point on a number line obtained by marking off five lengths of ¼ to the right of 0. […]

  30. […] is Not Repeated Addition” Research) and solidified my opinion on the mathematical end (“Multiplication is Not Repeated Addition” Revisited) but I had yet to finalize what I thought of the educational issue: does teaching multiplication as […]

  31. […] 6×4, konsep tanda ‘x’nya saja itu lho diperdebatkan. Contohnya silakan kunjungi https://numberwarrior.wordpress.com/2010/02/26/multiplication-is-not-repeated-addition-revisited/ Mohon diperhatikan bahwa yang berdiskusi di sana punya latar pendidikan matematika yang tinggi. […]

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