Keith Devlin wrote “stop telling your pupils that multiplication is repeated addition” and all hell broke loose.

There was a storm at Let’s Play Math, and then Denise wrote a second post. There is still a storm raging at Good Math Bad Math. And a bunch of places I don’t normally go. And then Josh at Text Savvy has written 11 posts (they start here) – but he disabled commenting, which turns the conversation into an echo chamber. (Josh, it looks bad if you complain about your comments not being published if you run a site where comments are not allowed)

So, my two cents.

Devlin’s point was directed to how we teach little kids math, and he blew it. So we stop telling kids that x = + (rep) and we tell them what exactly instead? Don’t ask Devlin. He devoted a second column to the issue, and never got there.

Doubling back, what was his objection? That we say “math is repeated addition” and that somehow this ruins kids’ ability to handle arithmetic: “Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not.” He’s wrong.

What should we do?

Teach multiplication through repeated addition, skip counting, counting arrays, finding unit areas of rectangles, Cartesian product, scaling… That’s too many to start, isn’t it? Pick one, then one more. And tell the kids that multiplication will manifest itself in many other ways as well… Add more… and remind them… No big deal here.

By the way, congratulations to Denise for handling this in an intelligent way. And for, ever so briefly, becoming the central blog in a math ed tempest.

1. August 2, 2008 pm31 8:17 pm 8:17 pm

I know people are miffed that I’m not publishing any comments at my site anymore. Though, if I were, and some of them were not getting through, I would not consider it out of bounds for someone to complain to me–no matter what kind of site they ran.

This issue is by no means an easy issue for education. Raymond Duval gets at the problem this way:

Any learner is faced with two quite opposite requirements for getting into mathematical thinking:

(1) In order to do any mathematical activity, semiotic representations must necessarily be used even if there is the choice of the kind of semiotic representation. (2) But the mathematical objects must never be confused with the semiotic representations that are used.

The crucial problem of mathematics comprehension for learners, at each stage of the curriculum, arises from the cognitive conflict between these two opposite requirements: how can they distinguish the represented object from the semiotic representation used if they cannot get access to the mathematical object apart from the semiotic representations? And that manifests itself in the fact that the ability to change from one representation system to another is very often the critical threshold for progress in learning and for problem solving.

2. August 2, 2008 pm31 8:19 pm 8:19 pm

Oh, also, quick FYI: “Keith” Devlin.

3. August 2, 2008 pm31 8:32 pm 8:32 pm

Fixed it. Thanks!

4. August 3, 2008 am31 5:40 am 5:40 am

It was a fun roller-coaster ride, but I’m glad the tempest has moved on. I’m still not totally sure what to think about all the different arguments, but it has forced me to clarify my own thinking a bit, to notice a distinction I hadn’t noticed before. I’m not sure, however, that the distinction makes as much difference to how we teach multiplication as Devlin seems to think.

5. August 3, 2008 am31 7:00 am 7:00 am

And then, a great question, how should we change or modify how we introduce multiplication to younger children… that got lost. Shame.

6. August 3, 2008 am31 7:56 am 7:56 am

I don’t think it completely got lost. I’m sure several questions will bounce around in my head for awhile, as I continue to mull over what I’ve read. For young children: How can we help them recognize multiplication situations, so they don’t just stare blankly at a story problem? For middle school students: Is there a way to make sure that they understand the extension of multiplication to rational numbers, rather than letting them get by with just memorizing the rules?

And for high school: How do we get them to understand exponents and logarithms? (I think that question did get lost in the tempest.)

7. August 4, 2008 am31 1:34 am 1:34 am

Exponents!

Now, here’s something interesting. Eyes go saucer-wide when I explain that our quick definition works only for counting numbers greater than 1, and that we need to define $x^1 = x$, that it doesn’t just proceed from “repeated multiplication.” But I get good buy in after I let the argument run a bit. (bright 9th graders)

8. August 4, 2008 pm31 5:43 pm 5:43 pm

I have several problems with all of this.

Firstly, nobody is arguing that we shouldn’t tell kids (or let them find) that multiplication produces the same results as repeated addition where applicable. The difference between being told, as a small child, that X *is* the same as Y, as opposed to X *does* the same as Y, is not in my opinion significant. So the whole thing’s a red herring.

Secondly, the historical and imo only comprehensible approach is to say that multiplication *is* repeated addition on the integers, and that as you expand your number systems (to systems all of which are obtained by smoothing and mangling integers anyway) you have to expand your definitions. Multiplication on the fractions is just the only possible smooth continuous extension of that on the integers, and so on to the reals, and onwards.

The only alternative – to say “this is just abstract stuff that happens, and happens to model these phenomena” – is excellent and necessary as a reeducation for undergraduates, but is absolutely not tenable for pre-teens (based on the opinions of just about everybody with any experience at all, anyway). Undergraduate induction is probably the right time for this, as well as the traditional time.

Thirdly, the argument that kids cannot survive this change in definitions at a later point is just nonsense. I did. You did. Devlin did.

Fourthly, Devlin’s argument from authority (I am a mathematician, you are teachers) fails on people like, say, me – I am a multiply-published mathematician, albeit less eminent than him. For that matter, I know Israel Gelfand expounds multiplication as repeated addition, and he’s a great deal more eminent than either of us.

9. August 7, 2008 am31 9:09 am 9:09 am

The only alternative – to say “this is just abstract stuff that happens, and happens to model these phenomena” – is excellent and necessary as a reeducation for undergraduates, but is absolutely not tenable for pre-teens (based on the opinions of just about everybody with any experience at all, anyway).

Let me just clarify for what seems like the billionth time that Devlin is NOT arguing that repeated addition never ever be spoken of again in relation to multiplication. Some people have been all over the Web arguing about Devlin’s articles for nearly a month and continue to pretend that this is what Devlin is saying. It’s not.

10. August 7, 2008 pm31 9:15 pm 9:15 pm

You are arguing, in bold, against nobody. Please relax.

11. August 8, 2008 am31 3:23 am 3:23 am

Oh, all right.

12. August 9, 2008 am31 12:22 am 12:22 am

Colour me puzzled. I’m being shouted at (by a man who apparently is cross because he’s said it before on a blog I don’t read) for saying something I didn’t, and that the quote he’s shouting at pretty clearly doesn’t say (“…happens to model these phenomena”, yeah?). Indeed, “Firstly, nobody is arguing…”, my second paragraph, makes it really *really( clear that I know that Devlin’s not saying that. I know you know this, JD – it’s just fun when people accuse me of not reading things in response to a post they clearly haven’t read :).

The internet. It’s a whole new universe of discourse.

13. August 9, 2008 am31 12:45 am 12:45 am

Sorry.

I didn’t mean to use boldface to indicate shouting–only to show that it was a quote.

My disagreement is primarily with the first part of the third paragraph in your comment–that saying “this is just abstract stuff that happens, and happens to model these phenomena” is the “only alternative.”

14. August 9, 2008 am31 5:32 am 5:32 am

Perspectives from a non-math person:

1) No one in the Let’s Play Math comments noted that the stretch/shrink-and-rotate interpretation of complex number multiplication DOES work with the one-dimensional number line, if you think of flipping (multiplying by -1) as rotation by 180 degrees.

2) Defining multiplication as repeated addition omits the conceptual issue of the multiplicative identity, which is going to make division as the opposite of multiplication harder.

3) I’m not sure what to think about the multidimensional interpretation of multiplication. It’s a good application with positive reals, but it’s much harder for me to grasp that as definitional than multiplication as stretching/shrinking.

15. August 10, 2008 pm31 8:59 pm 8:59 pm

Joshua – I agree that was a poor phrasing (very poor, in fact, since I did not say what I meant to say!), and apologise for it. There are of course other alternatives, notably the “scaling” that Devlin talks about – but I confess to being sceptical as to how well understood they can be by kids at a just-learning-their-tables stage. I would point out, though, that he says in the original essay “Why not say that there are (at least) two basic things you can do to numbers: you can add them and you can multiply them […] Adding and multiplying are just things you do to numbers – they come with the package”. So that was, actually, what he recommended.

The grown-up mathematician’s point of view is of course that multiplication *is* an abstract thing that happens to model a whole bunch of different (sometimes wildly different) concepts. Like a lot of people, I’m leery of the idea of teaching through heavy abstraction to infants (and not just because “new math” crashed and burned so horribly), and I don’t think that the fact that kids can learn chess is any sort of reason to think otherwise. (I do agree that technique first and understanding afterwards is a natural and effective way to teach arithmetic, of course.)

When kids ask me why we define things the way we do – for example, with fractional powers and negative powers- my answer is along the lines of “we want something that is nice and smooth and well-behaved, but that still fits the old, natural definition and rules we had before – and if we want that, this is all there is”. That’s how all these abstractions arise, and it is at least *a* way to be comfortable with them, and it is at least as “true” as anything else.

16. August 11, 2008 am31 5:47 am 5:47 am

You point out, with that quote, a good place where Devlin is a little, shall we say, confusing. And it’s probably what people are thinking of when they say he’s being disingenuous–after all, he appears to be making a suggestion in one part of the article and then declaring himself unfit to make suggestions at the end.

I have had many similar experiences pitching lesson plans or lesson ideas to other textbook editors. Unfortunately, it seems that no matter how much I stressed that “these are just ideas; I’m not suggesting that this is what gets put on the page” or “I’m just trying to provide a different perspective here; I’m not saying that one must follow this word for word,” inevitably someone would declare that all discussion and reflection must cease because it was simply impossible to implement the ideas or perspectives exactly as I laid them out. (Of course, sometimes my ideas were just bad ones.)

However, when I read Denise’s suggestions and Maria’s ideas about making a clear distinction between repeated addition and multiplication, my sense was that they don’t go far enough. (I left my own suggestion in the thread at Denise’s post.) And I imagined that, given his volume control and scaling examples (and the quote you mentioned), that Devlin would think that NONE of these suggestions goes far enough.

Now, I should say that I’ve had some E-mails with him back and forth that would suggest he doesn’t, in fact, think this. But even if he did, tough. He’s not teaching these kids; we are.

For me, (and, until recently, I assumed for everyone else) it’s just a given that we don’t take pedagogical advice from mathematicians. But if we can pull out ideas from anywhere to improve our instruction–in some way, any way–to be more consistent with mathematical truth, that’s a good thing.

17. August 14, 2008 am31 4:07 am 4:07 am

Uh…I can’t let this one go unnoted:

“the historical and imo only comprehensible approach is to say that multiplication *is* repeated addition on the integers, and that as you expand your number systems…”

This isn’t historical, not at all. I’d love to be presented with an example of a culture which has a mathematics, and also has no notion of fractions or parts of wholes or changing units to measure things more conveniently or some such.

Somehow the notion that “we all did four function arithmetic with just natural numbers until the day that some dude thought of extending the number system, and then WHOA! all bets were off!” is a lovely tale that gets told and retold and retold, but sadly is not supported by the historical literature.

The earliest mathematical cultures for which we have significant artifacts (“significant” = “enough material to determine what the symbols actually represent in context”) already have relatively sophisticated mathematical content: place value number systems, means of representing fractional quantities, means of solving indeterminate equations (e.g. quadratics in one variable, linear systems of two variables, special nonlinear systems). [Mesopotamia, c. 2000BCE]

To find mathematical culture prior to the notion of “fractions” might force you to go to Ishanto, or perhaps Karel Absolon’s wolf bone discovery dating to 40,000BCE.

I understand wanting to introduce multiplication in the context of only one number system, one representation. I even understand wanting to avoid worrying about the need to change one’s definitions once one is working with integers, or rationals. [I disagree, but understand.]

The claim that this follows some sort of historical evolution is false, however.

18. August 14, 2008 pm31 5:33 pm 5:33 pm

Fascinating take. But even if before written history, we have a number of concepts (0, fraction, negative number, addition, multiplication). Wouldn’t they have arisen in some order? not all at once? (I am assuming that whole numbers and counting were part of our species from the start)

Or is this stuff just way too speculative?

19. August 14, 2008 pm31 10:28 pm 10:28 pm

Devlin tried to deal with the historical argument with his “horse-drawn carriage” analogy and a reference to Hilbert’s work.

If you’re interested in a larger body of work that may relate to the “historical” side of things, look up “historical phenomenology,” “didactical phenomenology,” and/or “Freudenthal,” though Freudenthal himself cautions against using the historical development of a concept as a basis for teaching.

Below is from Chapter 2 of his book, Didactical Phenomenology of Mathematical Structures. He certainly could have been talking about repeated addition here:

“These concretisations, however, are usually false; they are much too rough to reflect the essentials of the concepts that are to be embodied, even if by a variety of embodiments one wishes to account for more than one facet. Their level is too low, far below that of the target concept. Didactically, it means the cart before the horse: teaching abstractions by concretising them.”

20. August 15, 2008 pm31 5:34 pm 5:34 pm

“The claim that this follows some sort of historical evolution is false, however.”

I’ll certainly bow to your superior knowledge and agree that there’s no evidence for it. You wouldn’t really expect there to be at this remove, after all. Do you really doubt, though, that integers arose as arithmetic concepts before fractions?

21. August 15, 2008 pm31 6:58 pm 6:58 pm

Surely counting is the starting point, going back (if Absolom and Marshack etc… are to be belived) some 40,000 years. I have a harder time identifying meaningful arithmetic operations in that early context.

By the time the historical record is unambiguous [c. 4000 years ago], we already have a thorough treatment of four function arithmetic with counting numbers, positive rationals, and to some extent positive reals.

The ontological status of 0 and negative quantities was still being debated as recently as the start of the 19th century. No similar qualms were expressed regarding the (positive) rationals.

I’ll grant that there are good pedagogical reasons to present natural number arithmetic as a stand-alone system before extending arithmetic to larger rings and fields. But I don’t see any justification for labeling those as “historical” reasons rather than pedagogical reasons.

October 1, 2008 am31 1:00 am 1:00 am

Devlin articles (now 3) are wrong in many ways:

Doesn’t provide evidence beyond anecdotal that there is in fact any real pedagogical crisis or problem.

Doesn’t provide much on his suggested solution. Very sketchy.

Suggested solution (teach as “scaling”) in fact fails in the same way that he critcizes “repeated addition” for failing. Scaling does not accurately describe complex multiplication anymore than repeated addition accurately describes real multiplication.

Says things like “multiplication is not repeated addition on any domain” (meaning not even restricted to integers for example) without defining his terms. By simple function theory a definition of a function naturally called “repeated addition” can be given and shown to be extensionally equal to multiplication. As far as ordinary math parlance goes – extensionally equal means identical — multiplication *is* repeated addition on integers.

5> Disses blogs and people where he had lots of people really hammering on his shortcomings. Argues from appeals to authority rather than logic. Refuses to address meaningful criticisms, or answer direct request for clarification, such as what can he possibly mean by .