Teaching Multiplication: Is it repeated addition?

I agree -- multiplication is not just repeated addition. But the distributive property does connect multiplication to addition in a very powerful intuitive way. As a teacher, I know you've got to start somewhere, and if the kids know addition, using that to teach them the beginnings of multiplication is a good thing.

No teacher would try to explain the US civil war without using background knowledge of geography and slavery, so why should we try to teach addition without using background knowledge of addition. No understanding of any topic is ever "perfect", and no knowledge is ever built in isolation.

The obvious thing to do (and I believe this sort of thing has been suggested by Cambridge mathematician Tim Gowers, in respect to several other issues of primary mathematics education) is to list several properties which define multiplication uniquely, and then determine the specifics of the operation itself--i.e. how to perform multiplicative computations.

So, here goes:

Multiplication (on the integers) is the operation that satisfies the following properties:

1. a*b=b*a
2. 1*a=a*1=a
3. a*(b+c)=a*b+a*c
4. (a*b)*c=a*(b*c)

This uniquely defines multiplication on the integers; to see this, note that (-1)*a=-a (as a+(-1)*a=1*a+(-1)*a=(1-1)*a=0) and that any integer can be written as a sum of 1's or -1's, after which we may apply the distributive law. This is, in fact, repeated addition precisely.

In general, we may define multiplication as anything satisfying these properties; in general this may not uniquely define a single version of multiplication, but in the cases generally taught in grade school, it does. For example, we define the rationals as symbols a/b with a, b integers, where a/b=c/d if ad=bc. Then multiplication is, again, the operation that satisfies the above laws (and is compatible with multiplication on the integers).

It's important to note, of course, that these are just *definitions*. There's no "right" definition--but when we extend multiplication to,say, the reals, all we're saying is: we have a definition on some subset (e.g. the rationals) and we want our extension to have some "nice" property--in this case, continuity. At this point, we might resort to more fuzzy explanations in the primary education case, but you get the idea.

Ultimately, I think the right choice is to move towards this more abstract route--i.e. teach math axiomatically. I'm convinced that if this is done slowly and carefully, giving students the chance to explore the implications of the axioms themselves, it will be (a) understandable, and (b) much more enlightening than the current (rote learning) method.

Daniel

I think you're missing the main gripe, Mark, about why multiplication isn't repeated addition. You didn't give any example above that hits it, and there are an uncountable number of them! Rational numbers work fine with this intuition, since given any rational number times any rational number, we can think of subdividing some circles enough to get what we need. The problem comes in by justifying e times pi, or something of that nature; mathematically, repeated addition would correspond to multiplying infinite series, but that's not technically considered repeated addition, since repeated implies a finite number of applications and that would be a limited process.

@ #2: Axiomatic would be lovely, but one can only grasp the axioms well because of years of intuitive learning! Do you want to start with Peano's axioms to teach them the integers as well? And using a continuous extension to justify moving from the rationals to the reals?! I can't imagine trying to hand wave that to an elementary school child if my life depended on it.

I have to disagree with you Daniel, after a fashion. The axiomatic approach is powerful, but it should come after a more "intuitive" understanding gets built. Even working mathematicians rely on intuitive models and concrete exemplars of the systems they're working with to help guide them through the discovery process. Axiomatizing is essential for rigor, and building a strong foundation for further work, but it still needs an intuitive scaffolding.

For beginning students, the list of axioms won't make much sense---even the ideas of an axiom, a symbol representing variables, an operation, a set, or equality are concepts that are going to have to be learned. Starting with the axioms really is starting in the middle.

Admittedly, I don't know what the best way to teach students is, but I do know that there is a very extensive body of research on math education at the primary level, covering fine levels of detail, such as how students acquire basic concepts like area or function, and how students integrate techniques into their body of knowledge.

Re: Howard

Well, I certainly think that the natural numbers should be motivated by something like the Peano axioms. In fact, I'd say that they already are to a large degree--the Peano axioms just formalize our intuitive notion of counting. That said, I don't think that all of this should necessarily be done in a completely rigorous, formal way--perhaps the best comparison would be the way that integration is generally motivated by Riemann sums in calculus classes, though the fundamental theorem of calculus is rarely proven in high school.

That is, one might say: Here are some properties we might want an operation to satisfy, and here are some reasons why (e.g. take a bunch of boxes, and put the same amount of stuff in each one, and them dump them all out; that motivates the distributive law pretty well).

I think your claim that axioms have to be motivated by intuition, and not vice versa, needs evidence. Since neither of us have learned that way, don't you think it's possible that reversing the order is possible? At the very least, it seems possible to motivate why we'd be interested in looking at objects that satisfy the given axioms, and then go from there.

And, with respect to the reals, I don't think it's really an issue--the "real numbers" are currently not really mentioned in elementary school. That said, I think continuity is quite easy to handwave; say "A small change in the domain causes a small change in the range." Or talk about approximation with rationals (e.g., if one is multiplying reals in decimal form, truncate the decimals at some point.) It does not seem like a stretch to ask young children to have an intuitive understanding for this, really.

And what's the alternative? There's really no other way to multiply real numbers--I'd like to see you add something sqrt{2} times.

What the???

Integer multiplication is identical to repeated addition, functionally, intuitively, logically, in whichever way you want to think about it.

Even fractional multiplication can be trivially converted to a multiplication and a division:

I.e.

5 * 2 * 1/2

Can be broken down to

(5 * 2) / 2.

What's the problem? Of course, integer multiplication fails when dealing with irrational and complex numbers, but are elementary schoolkids learning that these days?

Howard:

I do get it. The thing is, I think that that objection is completely irrelevant in the context where multiplication is being taught. I think that a realistic discussion of how to teach a 3rd grade math topic has to focus on third grade.

We teach young kids stuff that's incomplete all the time. We teach them to read using phonics, even though phonics doesn't work for a lot of english. We teach them
arithmetic, and tell them that they can't subtract a larger number from a smaller one. We teach them about atoms, but don't mention electron orbitals or heisenberg uncertainty.

We teach kids to multiply in second or third grade. We introduce the idea of irrational numbers somewhere between 7th and 9th grade. The idea that we shouldn't use the intuition of repeated addition, because there are going to be examples where it's problematic 5 years later is frankly silly, when there are places where it's going to problematic two weeks later. Multiplication as repeated addition is a really good introductory intuition, but it fails very quickly.

Daniel:

Have you ever tried to teach second or third graders?

I'm not asking that to be obnoxious, but as a real question.

My experience with kids that age is that formalized reasoning is very hard for them. Kids that age seem to be amazing abstract thinkers, but terrible formal thinkers.

The axiomatic structure of numbers is, I think, based on too complex a formal structure to be comprehensible to the average second or third grader. I don't think that kids that age are ready to think that way.

I'd love to be proved wrong about that, because I'd love to try to teach my daughter some axiomatic stuff. But I can't imagine how to do it. If you know of any books or websites that talk about that kind of teaching for young kids, I'd love to see them.

I think that both of these arguments are slightly missing the point. Multiplication isn't repeated addition, but it "comes from" repeated addition. In many mathematical ideas we start with some object that we understand, which is limited to a particular context, and try to generalize it outside of that context. So, for instance, we know a way to find the area of something that has whole number sides: you use repeated addition, and to make the notation shorter we call it multiplication. Then we ask "but what if the sides aren't whole numbers?" and we figure out how we have to extend the idea to rational numbers. And then, because real numbers can be approximated by rational numbers we can define multiplication for real numbers, and complex numbers, and such.

A similar story occurs with exponentiation. Exponentiation likely came from repeated multiplication. But then there is always the question of extending it to rational numbers (leading to roots), and then (again) to real numbers.

So I think that saying that multiplication "isn't" repeated addition is both right and wrong. It is the correct initial intuition, and it is the correct way to think about multiplication to be able to figure out how it should work. For instance, if we want to figure out what 4/5 * 2/3 is, we can say that "2/3 is a number such that, when it is multiplied by 3 we get 2. So the answer should be a number such that, when it is multiplied by 3 we get 4/5 * 2 = 4/5 + 4/5" and figure out what that number has to be. This is a way of thinking about multiplication as repeated addition that actually gives you correct intuition and a correct basis for generalizing it to other contexts.

(And as a random comment: calling matrix multiplication "multiplication" in this context is slightly off, since matrices don't form a field, they form a ring. And matrix multiplication comes from a very different context to number multiplication.)

Last summer I read a book that I think is relevant to this debate. In Where Mathematics Comes From, George Lakoff and Rafael Nuñez lay out some ideas about how "real", abstract, formal math to the embodied human mind. They approach this problem from the perspective of grounding metaphors that extend the natural human ability to subitize small quantities to counting natural numbers, etc.

The metaphors they suggest are all visuo-spatial, like "building" larger numbers out of unit-sized chunks, or counting as iterative motion along a path (which admits the extension of natural numbers to whole numbers including zero and negative numbers). They spend a lot of time discussing how these various metaphors describe addition and multiplication. For instance, under the "object construction" metaphor, addition becomes the combination of multiple number-objects into a single, larger, object.

The work is extremely speculative, but I think it's a step in the right direction even if some of the details don't hold up. Mathematical ideas are not innate, and they have to be taught or discovered somehow. Successful learning of mathematical concepts, from the perspective of Lakoff and Nuñez's work, is the successful development of metaphors linking mathematical concepts to "everyday" concepts, at least until the mathematical concepts can stand on their own, so I think you are correct to differentiate between what multiplication is and how it should be taught.

I agree with Daniel that it is valuable to teach mathematical abstraction, but not at the elementary school level, which is what is under discussion here. Understanding axiomatics requires a certain level of mathematical maturity that small children do not have, and we are talking about small children here, seven or eight years old. If you were to present the list of xxioms for multiplication to a seven year old, you would get a blank stare in return. No, I can't think of a good way to teach multiplication to a child without building some intuition from their prior knowledge of addition.

In high school, on the other hand, once the student has had some time to gain an intuitive understanding of multiplication, I think it would be useful and interesting to present axiomatics and abstraction. Abstraction is covered to some degree in geometry classes, but it could be more heavily emphasized in algebra classes.

(Not sure why I was "Anonymous" above...)

Actually, Devlin is wrong! If you refer to the fundamental theorems of natural numbers (based on the Peano axioms), multiplication is indeed defined as repeated addition! See theorem 28 at:
http://www.ms.uky.edu/~lee/ma502/notes2/node8.html

Derek.

"In the meantime, teachers, please stop telling your pupils that multiplication is repeated addition." But he won't tell those teachers what they should teach their pupils.

Heh - sorry to bring it up, but it's reminding me greatly of the framing debate on SB: "Ok, so you say we're framing it wrong. Now how would YOU frame it, then, in a way that 1) works and 2) doesn't distort the science so much its no longer accurate?" (sounds of crickets follow...)

As for the topic at hand here, I think it's not seeing how the steps of learning multiplication do eventually add up to a practical understanding that is close enough to a true mathematical definition as to suit 90% of the population (given that I'd say only 10% or more ever get to calculus, much less true abstract algebras, for example).

We first learn "multiplication is repeated addition", then we learn the tables 'cause adding up additions is darned tedious, then we learn to apply the tables to other circumstances, and in doing so, our understanding of multiplication changes and yet doesn't. Multiplication of fractions or decimals, geometry, and all of that can be learned because the student *trusts* the abstraction because it remains based in the tables and the tables are based on something proven: multiplication (of integers) is repeated addition (of integers).

The point of education at the lower levels is not to develop an understanding of multiplication (or name a topic) such that the definition in the students head can last forever. The point is to develop the students *trust* in the teacher and the education they are getting by allowing them to relate it to things easily seen.

One doesn't teach gravity to young kids using the Special Relativity definition, which is certainly more correct than Newton's inverse square law, and you don't use that either - you use 32ft/sec/sec. The rest will come since by the time they need to know Newton, you can show where 32 comes from based on the mass of the earth and Newton's constant (and how we're so low-mass compared to Earth that our mass makes no difference in the equation), and then later the physics-bound can learn Einstein in the same way - that ?/c is so close to 0 for most observation levels as to be insignificant and it looks just like Newton.

I do realize that he's trying to avoid the "I hate Algebra" kids that have gotten so much attention in the press recently, kids that do just fine in arithemetic and flunk 4 years of Algebra 1 to the point of never graduating, all because algebra's definitions of operations are unuintuitive compared to the arithmetic stuff they're used to (even though to us, they're the same). He thinks if we teach something closer to algebra's vocabulary earlier, we'll have fewer math dropouts and haters.

But I think that's one of those "address problem child A and you turn perfectly normal child B into problem child B" situations. What we have, while imperfect, *mostly* works, and to throw out 300 years of educational development because it doesn't quite reach a few kids, and replacing with something that's unproven to teach anybody, is a bit much to me.

There's a phrase that's frequently used by physicists: "Lies we tell children." It's basically a realization that physics education is layered - and the things you teach as "laws" early on end up being revealed as mere approximations later on.

For example, you teach 2nd graders about conservation of mass. You don't throw in rest mass vs. relativistic mass and discuss E = m c^2 and mass/energy equivalence and all that. Conservation of mass is a "lie we tell children" (LWTC) that gets them through an awful lot basic science - all the way through most of high school at least (with maybe the occasional non-technical diversion to explain why CERN is interesting or how the sun works).

I don't see why the LWTC model is usable in mathematics as well. "Multiplication is repeated addition" is a useful LWTC that gets them though their first introduction to arithmetic.

ugh - that last post (1011818) somehow lost my contact info on it. sorry.

My recollection of primary school is that we got both the "repeated addition" paradigm AND a quasi-geometric representation in terms of rectangular arrays of dots. Seeing things like eg. "3x4" being expressed as "3+3+3+3" and also depicted as four rows of three dots each, I think, effectively cemented the intuition that the two views are equivalent. Moreover:
- Commutativity is intuitively obvious from the dot-array picture
- Generalizing to multiple-factor multiplication (eg. 2x3x4x5x...) wasn't hard, with the bonus that it pulls in the concept of dimensional spaces of three or higher order.
- It's only a small step to generalize from the integral world (as represented by arrays of discrete dots) to the continuous one (represented by solid rectangles).

Why are we still arguing about this 40 years later? Or was I just too math-smart to appreciate the trouble that some kids have learning this stuff, and for whom we really neeed to agonize over our methods and metaphors?

Why are we still arguing about this 40 years later? Or was I just too math-smart to appreciate the trouble that some kids have learning this stuff, and for whom we really neeed to agonize over our methods and metaphors?

As I wrote above, it all kinda has to do trying to find new methods to reach the kids we lose, the ones the press makes so much of a fuss about when some school system added Algebra 1 to its minimal High School diploma requirements and now 1/3rd of the school system flunks out.

I think Devlin suggested the following in his text:
"Multiplication is useful if you want to know the result of scaling some quantity."
Addition is numeric, multiplication is geometric.

Think of multiplication as scaling. If you scale something up by a factor of 10, it will be magnified to 10 times the original, and yes, that is like starting with zero and adding the thing 10 times, but you can also scale it up by 7/3, which is not like repeated addition. Scaling it by a fraction less than unity shrinks it, as in the case of 1/10 scale or 1:24 scale or 1:10,000 scale.

I didn't come to understand this until I finished my formal education, where I'd been taught all math by rote. Scaling also makes a nice introduction to the scaling laws of physics, engineering, and physiology.

Re: No.16

"Research shows that only 19% of second-graders, 31% of third-graders, 54% of fourth-graders, and 78% of fifth-graders make correct predictions about how many unit squares will cover a rectangle." ( http://www.pdkintl.org/kappan/kbat9902.htm ).

So, not all students will immediately grasp an example of a rectangular array of rows and columns, because understanding how many units are in it, that it stays the same when rows and columns are interchanged, etc, is something that children have to learn, not necessarily innate.

Nobody is trying to argue that the traditional ways of teaching math don't work at all, just that given what we now know about how people learn, it could be much better. Many students do get math now, but many also fail to quite miserably, and always have. Really, it's not that much different than saying that although doctors 40 years ago often cured people, modern medicine can do it better.

In the preface of "The road to reality" R. Penrose talks about a girl who could not "get the hang of cancelling". That and all this bring bad memories to me. I learned multiplication as repeated addition and then I figured out how to extend them to rational numbers. I was good at math. I was in fact the best student at my class. A few years later I learned about square roots and suddenly everybody was adding, subtracting and multiplying strange things like square root of 2. I remember that my mind was clearly messed up about math in high school. I yet had the best GPA of my class but I didn't feel I understand math at all. While I knew the problems involved logarithms or roots at that level are supposed to be very easy, I could not understand what I'm really doing. I guess somehow in my mind, I made axioms in arithmetic that worked well for integers and rationals but I could not extend and generalize them to real numbers. As a results, I never learned trigonometry in high school. I could not understand why everybody in the class could so easily just add or multiply trigonometric functions of x and mix them with all other functions while I could not understand what even adding or multiplying a number by a irrational number means. This lack of understanding of math hunted me all the years in the university. I passed my math courses with B- and Cs. I could not understand anything as I would always get stuck in the basic definitions that did not work out. I did not get to clear my mind about it till years later during my PhD in engineering when I had the opportunity to study, on my own, and finally get to learn axiomatic set theory, peano numbers and such. I feel all my years at the high school and university was ruined because I never learned the basics of mathematics from the beginning. Maybe it was my teachers' fault, but probably they didn't know much about math themselves. Maybe it was my father's fault that kept repeating not to move to next step until understanding everything correctly and truly. I've started to learn math on my own, from scratch. I am 36 now. I hope this year I can finally finish basic calculus and understand it.

Nobody abandons a paradigm merely because it has contradictions and anomalies. A scientist who leaves her paradigm is no longer a scientist. A teacher who abandons a paradigm of pedagogy has to jump onto some new bandwagon. The new paradigm may or may not solve all the problems, but it has to offer hope for people in crisis.

The problem in meta-theory comes before ecponentiation, or multiplication, or even addition. It comes at counting. If the students have not been properly taught what an integer is, they'll never get anything else right.

John Baez elucidates:

I gave an example in "week73". There is a category FinSet whose objects are finite sets and whose morphisms are functions. If we decategorify this, we get the set of natural numbers! Why? Well, two finite sets are isomorphic if they have the same number of elements.

"Counting" is thus the primordial example of decategorification.

I like to think of it in terms of the following fairy tale. Long ago,
if you were a shepherd and wanted to see if two finite sets of sheep were isomorphic, the most obvious way would be to look for an isomorphism. In other words, you would try to match each sheep in herd A with a sheep in herd B. But one day, along came a shepherd who invented decategorification. This person realized you could take each
set and "count" it, setting up an isomorphism between it and some set
of "numbers", which were nonsense words like "one, two, three, four,..." specially designed for this purpose. By comparing the
resulting numbers, you could see if two herds were isomorphic without explicitly establishing an isomorphism!

According to this fairy tale, decategorification started out as the ultimate stroke of mathematical genius. Only later did it become a matter of dumb habit, which we are now struggling to overcome through the process of "categorification".

Okay, so what does this have to do with quantum mechanics?

Well, a Hilbert space is a set with extra bells and whistles, so maybe there is some gadget called a "2-Hilbert space" which is a category with analogous extra bells and whistles. And maybe if we figure out
this notion we will learn something about quantum mechanics.

[John Baez, 15 March 1997, This Week's Finds in Mathematical Physics (Week 99) ]

I've had so many students in urban middle schools and high schools who can't tell a number from a numeral.

I always quote "God created the integers, all the rest is the work of man" and cite the grman author (Google it) and that I am not violating separation of Church and State.

Of course I am not teaching Category Theory to 12-year-olds. But, as you have shown, Mark, it is good for us to know, to get clear on the basics of Math.

Now, I have had an on again, off again, tentative relationship with Category Theory since I was a teenager. There is a gradual and accelerating takeover of parts of Math, including Foundational, by means of Category Theory. This has a bright side and a dark side.

Bright: Category Theory is absolutely NOT based on Set Theory, and thus many of the paradoxes of Godel and Russell and so forth are thrown out.

Dark: Category Theorists have beliefs in things even sillier than the infinities of Set Theory, about which I need not believe in completed infinity, can use the Cantor stuff, have seen the morass deeper in
theory, and don't matter for Science anyway (except as to whether or not space or time are continuous or discrete, which don't use either sets or categories anyway).

Ambiguous: Category Theory is more "gestalt" and less "analytical" in terms of which half of your brain is engaged. This is a paradigm shift, socially, I can say without accepting or denying the claims on
either side of the battle. In any case, Engineering and Science can pretty much watch with detached amusement, or ignore the fight, until the winners start new invasions. The Categorists tend to see Biology and Sociology and Economics as ripe for conquest, due to "networks" and their uses.

Decategorification and recategorification are not merely clumsy big words. They code for an agenda. First, that the real world has structures which set theory throws away. So how do we put the real world structure back into Math?

I have tried making lesson plans based about "Stuff, Structure, and Properties", i.e. the work of Morton.

There is "stuff" in the world. All models are wrong, but some models are useful. Stars are real, but they do things we don't understand. People are real, ditto.

"Structure" and "Properties" are what Math
and Science and Engineering have to grapple with, and that may take new tools and new words.

There is one more issue here: that there are teachers (especially in elementary schools) that do not understand beyond what they are teaching the kids. Part of the problem is the education of the teachers themselves, and partly a systematic problem that teachers are required to teacher almost all subjects especially in elementary levels.

I think you have to introduce the idea of multiplication as repeated addition as (I think) that's how the concept was invented and used by our ancestors. Yes the concept gets more abstract as more math was developed through the ages but I think the idea of introduce ideas as they were developed through history make sense. As long as the concepts get corrected and refined as the kids develop. I think this is called the "genetic approach".

That's where the problem with the teachers gets in the way: if you have a bright kid that are asking tough questions and the teacher has no way of answering or inspire the kid to think more about these concepts, you stifle growth.

Mark W in Vancouver BC

Hmmm.... 3/5th of $25 as repeated adition:

3/5ths is the fifth part of 3 times $25.

So (25+25+25)/5

As long as you understand repeated adition and division, fractions are not a problem.

Of course i*i=-1 _is_ a problem ;-)

Re #19:
Thinking of multiplication as scaling rather than repeated addition is indeed a great way to look at things and it also emphasizes geometrically Devlin's point that numbers come equipped with two operations. Viewing addition as shifting and multiplication as scaling is an elegant, concise, and geometrically intuitive way to understand how the two operations are distinct. The question is, however, can this concept be effectively taught to little kids, kids who don't yet know how to multiply two whole numbers together? Do third graders have intuition regarding the real line developed enough to understand the shift vs. scale idea? I don't remember how I thought about numbers when I was in the third grade. If kids this age do, in fact, understand the number line model then I think its a great idea to emphasize the differences between addition and multiplication in this way. If they don't, then perhaps early math education (first/second grade) should focus more on developing kids' geometric intuition about numbers.

I really don't see the problem with saying that multiplication is repeated addition, because that is what it is. (That is that is how it was originally defined.)

It's true, of course, that the concept has been generalized to more advanced mathematical structures. But does it make sense to teach children these deeper abstractions, without first explaining their original meaning?

Also, if we are worried about advanced definitions, it should be noted that scaling is not the same as multiplication.

5^2 [counting 5 units of 1 in a square which has 2 dimensions(x,y)]:

1 1 1 1 1 = +5
1 1 1 1 1 = +10
1 1 1 1 1 = +15
1 1 1 1 1 = +20
1 1 1 1 1 = +25

5^3 [counting 5 units of 1 in a cube which has 3 dimensions(x,y,z)]. Row count would be: 25, 50, 75, 100, 125.

3/5 of 25: Row 3 (of 5) in above diagram gives correct answer.

I think some of the earlier comments get a little to tied to the abstract mathematics of multiplication. We are dealing with 2nd and 3rd graders here. A big part of it is memorizing the multiplication tables, so they can do basic math in their head. Memorizing the tables works best using the adding approach and doing the 2x2 = 4, 2x3= 6 ... These kids don't even know fractions, so the repeated addition works perfectly. After they learn fractions you can knock out some of addititive intuition because they won't need it any more. This seems like a pointless discussion because I don't think anyone is struggling because they are too attached to the repeated addition they learn in elementary school.

For what little it is worth, my intuitive model of arithmetical multiplication never had to get beyond repeated addition of integers. For example, to multiply the square root of two by pi, I first aproximate them as rational numbers, say 1.41421 and 3.14159. Then I consider that to be the same as adding 141,421 copies of 314,159, and then shifting the decimal place back to just after the first digit of the result. For any given x I could do the same for, say, sin(x)*cos(x), so multiplying functions is just a bit more abstract. For multiplications involving negative numbers, I first get the result for all positive numbers, and then apply the laws of signs to the result. So for me, any arithmetical result I could get from multiplication of two complex numbers can be considered as mainly due to repeated additions. I realize there are more abstract forms of math where multiplication is a defined operation than has nothing to do with arithmetic, but when we are teaching arithmetic I think defining multiplication as repeated addition is quite practical.

Its weird: multiplication feels like repeated addition to me, even when you bring up fractions and such it still feels like the same concept, maybe I've just been told "multiplication is repeated addition" so many times that I cant think "repeated addition" without seeing through it to the broader concept of "multiplication".
I wonder if it's because I was way more interested in algebra then counting as a child. To the point were while I was able to solve simple algebraic equations by second grade and still can't count a group of 6-10 objects both quickly and reliably. (This kills my go game dead.) So its quite possible that multiplication is the more intuitive concept for me, or not

It seems to me that multiplication is the natural generalization of repeated addition, and exponentiation is the natural generalization of repeated multiplication and the argument about whether one is or isn't repeated the other is a very silly argument indeed.

I agree that the question is how we go about teaching it. I think repeated addition is not a bad thing to form part of the motivation for multiplication, but it should not be presented as the only way of understanding it. It will help some people, it won't help others.

I have taught adults with great trepidation about mathematics exponentiation using repeated multiplication and then division as a stepping off point, and then showing how fractional exponents arise quite naturally, before moving to powers with real exponents. Their responses were strongly positive, but that's not to say that one shouldn't be ready with other ways of understanding the topic should it not give them what they need to make each step.

I think it's useful to say "multiplication isn't "just" repeated addition", but that's not to say it isn't useful to understand when it is.

One big problem with repeated addition which might have escaped the purer mathematicians: in the real world we often aren't using pure numbers but quantities.

So, if you are calculating the area of a carpet, it's not 6x8, it's 6ftx8ft or 6mx8m. And the product is 48ft2 or 48m2.

And it doesn't work to shoehorn it into putting the units somewhere else, for example, that it's like a strip 6m long by 1m wide (so 6m2) being added 8 times. Even if that were a valid way of looking at things (I think it's horrrible!), carpet comes in 4m widths, not 1m! (no, you do not cut up a long stair runner to make a room carpet).

Some commenters are showing their mathematical mindset here: why should there be one best way? When I learned to do a man overboard drill for a sailing boat, my instructor described the manoeuvre, drew it out on a board, danced it on the slipway with a colleague, then demonstrated it, then we did it, so there were opportunities for everybody to learn in different ways; what matters is getting to the right understanding, not how you get there.

There's an educational adage: "you can only learn what you almost understand already". I'd say anything that works is fine, and tidy it up later!

Huh
How is 3/5 * 25 not multiple addition?
3/5 + 3/5 + 3/5 + 3/5 ....... (25 times, of course)
or 0,6 + 0,6 + 0,6 + 0,6 .... if you wish.

Same goes for the triangle.

But it's true. In this case, it makes little sense to teach it that way, especially when you get to stuff like 1/2 * 3/4 * 5/6 or so. Just as little as it makes sense to NOT teach it that way when you start off teaching kids about multiplication. You can't go tell a kid "For now, it works absolutely like repeated addition, but in future it won't, so I forbid you to regard this as repeated addition. I say it's not. No matter what it looks like to you."

Because sooner or later, even if you don't explicitly say it, some kid would notice that 2*3 works JUST like 3+3. And then you're going to tell them "No."? Please.

Repeated addition works!

You want to know how to find what 2/5 of $25 is? You can do it as

(2/5)+(2/5)+(2/5)....

You can THEN show that there is a cheeky shortcut that we can do, namely that we do ($25x2)/5=$10 or we can do ($25/5)x5=$10

Devlin has written a followup to the topic:

http://www.maa.org/devlin/devlin_0708_08.html

OK.. that is wierd!

"But how are you going to explain that two a third grader?"

"to", maybe? Seems your brain was stuck on numbers for some reason. :)

In his latest one, he talks about exponentiation as a natural thing in the Reals, but I don't see how it really follows from the Reals' being complete or an ordered field. (Apart from its being in the reals and their being the unique such field.) I learnt exponentiation via the exp function defined as an infinite sum, which I guess being complete allows.

I'm a little late to the party but I'm surprised no one has brought in an ed-psych-nerd perspective. I teach elementary math teachers (my background is in teaching 9-12 math) to keep two things in mind when designing their math instruction:

- hook the concept onto something the children already know
- leave room in their understanding (fancy word: "schema") for the rest of the real number system

(ok ideally the complex number system, but since I teach 11th grade, I'd seriously be happy with the reals. rationals, even, would be awesome.)

So what to do with multiplication? Repeated addition, it can be argued, is destructive if that's the only way the kid understands it. Intuition (for a 3rd grader, we are talking about here), breaks down outside of natural numbers.

So how do they introduce it? Area of a rectangle will get you a long way, and is a model that can accommodate rationals, integers, polynomials, and factoring. Getting students to see one representation of quantity as length of a segment has other uses, too, is aligned with historical development of mathematics, and can be used to drive them into a nice contradiction when introducing irrationals.

Also useful is a sort of intermediate-grouping method. (Remember, the teacher is just introducing the idea of multiplication.) How many little paper cups full of water will it take to fill up a big garbage can? Kids get tired of counting little cups of water pretty quick, but you cleverly leave some intermediate sized containers laying around...understanding of multiplication that does not rely on naturals follows. I've also used this method to remediate 9th graders, but once that rigid schema is set, it's hard to break it down.

Multiplication is not JUST repeated addition.

"Stuff, Structure, and Properties"

The structure of multiplication of integers is vague to students until we include the commutative law.

3 x 20 = 20 x 3.

I give that example because I still remember with shock a student a couple of desks away from me in 6th grade, on being asked by the teacher what 20 x 3 was, saying, slowly and painfully: "3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60."

"That's the right answer," I said, trying to be helpful. "But wouldn't it be faster to say "20, 40, 60?"

He was shocked now. "Is that the same?"

In the Baez story I told, we may understand that integers and counting may to taken as evolved from a different process.

Integers are the "stuff" that we work with in arithmetic, until we build new stuff from them, usually in the order: zero, negative numbers, fractions.

Multiplication is part of the "structure" on that stuff, along with addition.

The "structure" of multiplication has "properties." The commutative law is the property that I've been describing.

If I were teaching multiplication to a child I would simply use a diagrammatic representation of a square to illustrate what I'm talking about. For higher orders of multiplication, I would tell them to use "BIDMAS" and just repeat the square drawing repeatedly.

For example if I wished to have 3x4 I would simply say that this is drawing three dots horizontally and 4 dots vertically... filling all the spaces for dots in to complete the square. This is what multiplication is and I believe why it was invented in the first place- to calculate areas!!! Though if the last statement is true or not I think that this the most intuitive way of teaching children.

Further to the whole question of whether it is repeated addition or not... this is a ridiculous question... just because there are two ways to get to a solution does not necessitate one being absolute and the other incorrect. For example multiplication could also be considered inverse division or even a whole slew of mathematical operation combinations that would still not be wrong, just not as elegant.

To small children, I teach multiplication as "counting sets of things", instead of counting things. I define "x" to mean "of": 3 x 4 = three sets of four.
Then, when they have the idea worked out, they memorize the times tables so they can be efficient (not emphasized so much in elementary school any more).
Then we start talking about multiplying negatives, which is abstract, and fractions, which isn't abstract, but hard to understand for children (most adults don't get it either).
Education is a "process". One of you is talking about a mathematical definition, an the other is talking about a pedagogical one. Sometimes they are not the same thing.

So I'm glad Marc started by making clear that Devlin wrote about how to teach, and then didn't deliver. That part's important.

But what are we going to tell the kids is a problem. I don't want to do it like in chemistry, where we simplify the structure of the atom so that it is unrecognizable, and then every two years or so modify it somewhere closer to the truth. Math shouldn't need to do that.

I teach high school. It is easy for me to say that x^2 = -3 has "no solution in the real numbers" generating the question that all your readers are thinking of right now, and answering, that "yes, there is a set of numbers that includes solutions to this sort of equation"

I can "define" exponentiation in Algebra I as x^n is repeated multiplication of x n times for n gte 2, and n belongs to the natural numbers, and have the obvious discussion.

In geometry I can append the phrase "in the plane" to half of what I say, so that the questions and the discussions come up.

In other words, I can explain further than first definitions, or include caveats that cover the gap. Is it necessary in all cases? Maybe not. But the idea I started with, that math should not need to correct itself, I take that seriously.

So, even though I think Devlin was way off base, I think there is a worthwhile question. Can we teach multiplication through repeated addition, maybe combined with arrays of dots or arrays of unit squares in such a way that we let the nice, physical ways of multiplying inform the kids, build the strong connections in their minds, but so that we don't say:
multiplication = repeated addition or;
multiplication = counting rows or;
multiplication = finding rectangular areas or even;
multiplication = scaling ?
Can we find a way? Is it worth it? Is trying to avoid making the equality pedantic?

I don't know of any case were A + B = -B + A
The world is rich with A x B = -B x A

How does one add rather than multiply to obtain vector cross products like Lorentz force in electrodynamics? Furnish an example of noncommutative addition.

A little late to this blog, but I wanted to point out a few things.

1) Filling a square with dots is not the "area" approach to multiplication. It is just repeated addition all over again. You're counting the dots! If you do 3*5 and make a 3 by 5 square filled with dots, you are graphically presenting 3*5 as 3 rows of 5 dots! If you want to do the real multiplication as area alternative, you must use synthetic geometry and never resort to such formulations of are by decomposing a region into unit squares. THAT certainly would be difficult to teach small children since it would be unprecedentedly abstract and ultra axiomatic.

2) Also, using units falls prey to the very same issue. That is still repeated addition because 3m*3m is 3 columns of 3m units stacked side-by-side. So, that is just repeated addition again unless you understand it through pure synthetic geometry.

3) Multiplying real numbers isn't JUST repeated addition only because it includes the ADDITIONAL notion of a limit. It is true that you cannot just write down pi*pi as pi added it to itself pi times. However, to write down pi at all, you have to write it as a sequence of *rational* numbers. And to write rational numbers down at all you have to write them in terms of integers. And, to write integers down at all you have to write them in terms of 1 added to itself some number of times. Just adding in one concept while 99% of carrying out the actual operation involved is repeated addition doesn't make it fundamentally not repeated addition after all.

4) The same is true about complex numbers. I would tend to concede that i*i=-1 really has nothing to do with repeated addition (which even that is not quite true, actually). But, so what? When you do that one thing it isn't repeated addition, but all the rest of the time...? In other words, to carry out (3+5i)(5+3i), when you get to the i*i part, you have to (let's say) do something completely different from repeated addition, but most of the rest of that calculation is right back to the same old repeated addition. How can one consider that not really pretty much just repeated addition with one little fact thrown in?

5) These so called alternatives to repeated addition just beg the question. Scaling just begs the question. How are you going to teach that again? Do you really expect the student to pull out a ruler every time they want to multiply 3*5? And if they did, what would they do? Count up the ruler to mark a point 3 times the distance from counting up to 5. Or do you just imagine that they will be able to know just exactly what three times that distance would be -- they can just see distances with mathematical precision? And then know that that distance translates to a number?

6) (And, this is what I really wanted to comment on.) No, our understanding isn't ultimately based on the vague heuristic understanding. It is true that you almost surely cannot teach kindergartners abstract axiomatics. And so, you will have to teach them heuristic arithmetic as opposed to Peano arithmetic. But, this is a philosophical point of what is the real truth or the real basis of something, here. It is the most precise understanding of it not the vague heuristics we start with. If you have to clarify, you go from the vague to the precise not the other way around. And, beyond that to the teaching of things, while it is true that children are generally incapable of abstraction when they do become capable of it as adults the model of instruction does a complete 180 and you tend to teach the formality directly. For one thing, there simply is not even remotely enough time to cover the ground that needs to be covered by doing heuristics for a long time and then graduating finally to the formalism. But for another thing, it is because the subject primarily IS that formalism which is not an "empty formalism" but more like the real truth of the matter.

That last point is a philosophical question, not really a mathematical one. Trying to cast formal a priori mathematical subject matter as "empty formalism" and claim that the heuristics is the "real" concept is really just a holdover from the radical empiricism that has swept through society over the last several centuries. (Indeed, it is very much a direct consequence of centuries of British Empiricism.) But, in case you didn't get the memo, that has failed. Math is not reducible to an empty formalism like that. And, science and math cannot together form an alternative to philosophy -- a closed system the tenets of which we can simple choose to accept and that need not rest on a purely philosophical foundation. Math, itself, is not just a convention like that -- that we adopt so that we can use it as a language to discuss meatier subjects like physics. It is a subject unto itself. Mathematical problems are genuine problems not just semantic knots to untie. Mathematical knowledge is genuine, meaty *knowledge* about something (though, perhaps not about "the world"). Mathematics is not just so many purely analytic derivations but is also synthetic.

(Sorry for the long post)

Actually, now that I reread the meters squared comment, it seems like the real point is that meters squared, itself, cannot be understood as repeated addition of a meter to itself. (For instance, even in what I say above, I already have a square meter when I start adding things up.) That may be true, but the meter, itself, is not a mathematical concept or at least not a numerical one. It is ultimately a purely scientific one. So, 3m*3m does have an addional notion built into it that is not repeated addition, but I don't think it does so in a way meaningful to this issue. (And, I think that maybe this also just serves to make my point about how the real issue is probably this centuries old one over empiricism.)

I should also say something about the original blog entry. It is not just true that multiplication can be heuristically understood as repeated addition. In the case of repeated addition, it can also be understood rigorously that way. And, that notion is rigorously extended to create a rigorous understanding of multiplication on other domains. So, teaching multiplication as repeated addition and then extending the concept as you go (what Devlin is complaining about) EXACTLY coincides with the ACTUAL mainstream intellectual development of it. Devlin doesn't just say that multiplication is not JUST repeated addition. He says it is "just false" to characterize it that way. If anything is "just false", here, it is THAT comment of Devlin's.

Devlin seems to think that the rigorous development of the natural numbers is some sort of a game. We do that just to show it can be done. (I guess I'm not surprised a logician might look at it that way.) It is more than a game -- it is the best basis for understanding what it all *really* is. Granted, we don't normally have to descend to that foundational level of understanding very often in much the same way one almost never has to really understand just how their CPU carries out arithmetical operations. But, that doesn't change the fact that it all really comes down to that and only that in a way that it really does not all come down to one's high level heuristic rationales for things.

ALWAYS START WITH THE WHOLE.

Addition is peated multiplication.

IT'S NOT THAT HARD FOLKS!

cheeze luweez.

"In the case of repeated addition, it can also be understood rigorously that way. And, that notion is rigorously extended to create a rigorous understanding of multiplication on other domains."

Prove it.

I did Joshua. See, for instance, p14, Theorem 28 and Definition 6 of Foundations of Analysis by Edmund Landau.

"In the case of repeated addition, it can also be understood rigorously that way. And, that notion is rigorously extended to create a rigorous understanding of multiplication on other domains."

Prove it.
------------------
Ok. Start with the ring of integers. Then extend to the field of rationals by introducing the field of fractions. Then do field extensions once again to eventually obtain the reals.

Presto! :)

Page numbers from other people's books aren't proof.
Sentences vaguely describing what it is you are talking about are not proof.

PROVE. IT.

I think there is a danger of people talking past each other because of different semantics here. I define "teaching arithmetic" as teaching people how to do calculations with rational numbers - because that is all we use in practical calculations, whether by hand, calculator, or computer program. As mentioned above, whenever we calculate with an irrational number, such as pi, we approximate it as a rational number. For rational numbers, as stated by many commenters above, multiplication is the same as repeated additions.

Multiplication as applied to vectors, matrices, or as an abstract operation defined on a group, is not the same animal, even though it may have the same name. When some of us are saying it is fine to teach children that multiplication is repeated addition, we mean specifically arithmetical multiplication, i.e. multiplication of rational numbers. Okay?

(I think I can prove the commutative property of multiplication for integers based on this definition, then extend it to rationals. Take the multiple of 3 by 20: suppose you have 20 boxes, each containing 3 apples, which represents by definition 3*20. Take one apple out of each box and place it in barrel one. There should be two apples left in each box. Take another apple out of each box and put it in barrel two ... yada, yada, yada, you have three barrels each containing 20 apples, so 3*20 = 20*3 ...)

I appreciate Jim's comment. I know that some of what is happening is that we're "talking past each other." I don't know how to solve it from my end.

What's frustrating for me is that as soon as the discussion comes up, people generally start computing with numbers to advance the argument that multiplication is basically kinda sorta repeated addition.

My take is, forget about the numbers. Multiplicative relationships are just fundamentally different from additive relationships. In and of itself, that justifies making a CLEAR distinction between the two operations when teaching kids.

Yes, you can "think about" multiplicative relationships in terms of repeated addition (sometimes; hell, maybe always, I don't care). You can lay out every axiom and theorem in order to show that it is justifiable to "define" multiplication as repeated addition. You can go through the entire history of mathematical thought and show that repeated addition was the chicken and everything after it was the egg. What will still remain is that multiplicative relationships are just fundamentally different from additive relationships--if not "operationally," then certainly conceptually.

If we use Peano, we can "think about" addition as a kind of repeated "moving up 1." But, seriously, does anyone here think that that's what we should be telling students that addition IS? A repeated "moving up 1" on the natural number line?

Isn't it more accurate to say that addition is basically the combination of collections of objects (or something similar)? Sure, you can start with the first object and "move up 1" for each object until you get to the last object in order to find the sum. But that's not what addition IS.

Same argument applies to multiplication.

The fact that it's an idea that might be difficult to implement DOES NOT make it wrong.

Joshua says:

"Yes, you can "think about" multiplicative relationships in terms of repeated addition (sometimes; hell, maybe always, I don't care). You can lay out every axiom and theorem in order to show that it is justifiable to "define" multiplication as repeated addition. You can go through the entire history of mathematical thought and show that repeated addition was the chicken and everything after it was the egg. What will still remain is that multiplicative relationships are just fundamentally different from additive relationships--if not "operationally," then certainly conceptually."

Why Joshua - you may be beginning to see the light!

Please continue to elucidate what you mean by "conceptually" different - or operationally different, or whatever.

The thing is - you have to articulate it precisely to get others to go along. (And that is just another thing that is so wrong with Devlin's two articles - he can't articulate exactly why he feels that the view of multiplication as repeated-addition is totally and utterly false.)

Let's suppose you can make a convincing argument, along with related controlled studies, that they are "conceptually" quite distinct - not just in *you* mind you, but in most if not all people with normal brain functioning. Well, then you would have made an excellent contribution to cognitive psychology, (but not yet to mathematics.) There actually are such studies being done today.

You may (or may not) like the recent book by Lakoff and Nunez - "Where Mathematics Comes From". Whatever.

My point is this: until those sorts of cognitive considerations become embodied in a mathematical theory, they are extraneous to the mathematical validity of defining multiplication via repeated-addition.

The standard today, is to consider mathematical functions as equal when they are extensionally equal (as opposed to intensionally.) So all those other things you posit in your enlightening paragraph that I quote above rule when it comes to deciding mathematical validity, not the hypothesized conceptual difference.

Dear God, with the functions again. I'm tired of arguing about it. I'll go ahead and take my cues about what is mathematically "standard" from a professional mathematician.

Well, I, for one, am perfectly willing to make such a deal. You take your cues from a *mathematician*, and I'll take mine from the mathematics, itself.

Joshua,

I have heard personally and directly from other professional mathematicians that Devlin is just full of it on this matter.

Following just one person on all matters -- that's discipleship, not rationality -- is not a good way to understand what is standard. If the traditional definitions and approach to extending multiplication to rationals, reals, complex etc. were known to be mathematically totally and utterly false, as Devlin says - My God - that would be earth shattering news. There would be standard text's that illustrate the proof of the falsity of this view. Show me the beef- where are these treatises on the subject?

All sorts of otherwise competent people have funny quirks, blind spots, and animuses when it comes to certain subjects. Here Devlin embarrassingly exposes one of his.

And you might try to understand the difference between extensional and intensional -- from what you have said I believe that to be the underlying bugaboo as far as the point you are trying to make.

If multiplication shouldn't be taught as repeated addition, then we need to stop lying about factorials and just teach the gamma function up front. Let's at least try for some consistency if we're going to be absurd.

Oh and derive e starting from a differential equation instead of just waving our hands and pulling some random infinite series that happens to be interesting out of the air.

The question's not whether they're distinct concepts, the question's whether you teach running before or after walking. Running isn't just fast walking, after all--but that provides a pretty good starting point for figuring it out, no?

You get some non-mathematical-but-math-related agenda going (mathed in this case), find yourself in a quasi-mathematical context, and the next thing you know, you start uttering statements that turn out to be mathematical ones without realizing it, let alone making sure they are really true. You just don't have the same standard in a context like that that you would have writing a textbook or something. I think this is why a lot of mathematicians just STFU and stick to the purely professional aspects of their specialty area (especially when you add in the way people misconstrue what they say and/or the possibility of bona fide misstatements on their part).

It's really embarassing for Devlin. I've heard someone even saying that apparently he is just a dumb@ss (despite his apparent reputation) and that this could knock Stanford down a notch. That's certainly a stretch (I think, at any rate), but still that's nothing to take lightly when other professionals are privately reacting that harshly. I hope for everyone's sake, Devlin practices a little more restraint next time.

(Frankly, I originally expected everyone to just side with Devlin because of his notoriety.)

"Let's at least try for some consistency if we're going to be absurd."

LMAO!

Certainly very appropriate in this context....

The other thing is that, quite simply, most people don't need higher math. Many, in fact, will *never* understand the idea that multiplication over the rational numbers forms a "group". Sorry.

But everyone in our society doed need to be able to do arithmetic on numbers, to understand fractions, percentages, and compounding interest. I'd include that they all badly need to get a feel for probability (something that humans are awful at), and for magnitudes: a trillion dollars of government debt is not like a million with a "t" - it's a whole different ball game.

I am one of those high school math teachers who uses the distributive law all the time with my ninth graders to try to get them to be able to multiply without calculators. Addition works well here. Even my low-level kids, however, will not balk when we multiply matrices, or do simple group theory.

What brings these topics up for discussion is the commutative property of multiplication. When I mention that it doesn't always have to be true, someone will want to know when and why. I promise to tell them at chapter N, section Q and tell them to be sure to mark their agendas.

The group theory comes up when we talk about symmetry and what happens to shapes that are flipped or rotated. We make the "multiplication table", but don't pursue it much past that.

OK, let's see someone multiply two 2x2 matrices using repeated addition. Because they can indeed be multiplied.

Introducing multiplication as repeated addition is a classic example of the "lie-to-children" (see http://en.wikipedia.org/wiki/Lie_to_children). It is used in science all the time - you can't start by teaching children the electron cloud model of the structure of the atom, so you start with the idea of protons and neutrons being clusters billiard balls and electrons being other billiard balls that spin around it. You then progress to more "correct" models the further they study physics.

From the original blog entry by MarkCC:
"Is multiplication repeated addition? Sometimes, it is. But multiplication isn't just repeated addition. It includes cases where it makes sense to talk about it as repeated addition, and also cases where it doesn't."

OK, fine - but what has caused the toxic reaction is that Devlin says multiplication is *never* repeated addition, and can *never* be thought of that way, should *never* be taught that way, and even restricted to integers that view is "totally and utterly false".

Lies? How about redefining zero - now *there* is a lie!

From Wikipedia:
"In January 2007, faced with the prospect of an outright ban on the sale of their product, Crisco was reformulated to meet the US FDA definition of "zero grams trans fats per serving" (that is less than one gram per tablespoon) by boosting the saturation and then cutting the resulting solid with oils."

So, zero means < 1!

Re: #19

Thinking of multiplication as scaling is no better than thinking of multiplication as repeated addition. It's got just as many failure mode.

For a couple of quick examples (and these are very much like the kinds of problems that Prof. Devlin is complaining about w/repeated addition):

- I've got a box, 10cm wide, 20 cm long, and 5cm high. What's it's volume? 10cm×20cm×5cm = 1000cm3. How can you explain that with scaling? The failure isn't just the units, either how do you explain that volume is the area of the base scaled by the height?

- How can you describe complex multiplication as scaling?

As I said in the original post, there are a lot of different intuitions that you can use to describe different contexts in which we use multiplication. But they all have cases where they utterly fail. The only definitions multiplication that never collapse are the formal axiomatic ones.

But I'm not convinced that you can teach kids math, right from the start, using axiomatic definitions. I watch my own kids and their classmates - and I can't imagine how you could make axiomatic mathematics make any sense to them. I'd be delighted to be proved wrong - but until someone can show me a textbook, a curriculum, or a lesson plan that shows how to do it, I tend to think that it's probably not doable.

Re #13:

I always worry about arguments that have the general form "You say that X is wrong, so why don't you tell me what's right?". It's a frequent gambit by crackpot types. "You say that vaccines don't cause autism. So what does?" In fact, I've wanted to write a post about that sort of argument, and when it's applicable, but I haven't had time.

But you're definitely right that it's very similar to the framing thing. Ultimately, both come down to discussions of communication and education: how do you present complex scientific or mathematical knowledge to a naive audience?

In both cases, you've got the critics basically saying "You're doing it wrong"; but when you come back with "Ok, I'll accept that I'm doing it wrong. What can I do to fix it?", they basically run away saying "Oh, I'm not an expert in that area, I don't know enough, I can't make any suggestions."

I think that they key differentiating principle between cases where the negative argument without replacement makes sense, and the cases (like this) where it doesn't is related to the failure mode. That is, in a scientific debate like the vaccines cause autism; if you've got a solid argument that autism doesn't result from vaccines, then the failure mode is "do more research". That is, if we know that vaccines don't cause autism, but we don't know what does, we keep doing the same thing that we're currently doing; we keep giving vaccines, and we keep researching the causes of autism. The failure mode is such that continuing to do what we're already doing makes sense and doesn't conflict with the implications of the negative argument.

On the other hand, in cases like framing and education, what's the failure mode? If the way we teach multiplication
is wrong, then what should we do? We can't stop teaching multiplication while someone goes off and does research into how to teach it right. Scientists trying to make people understand global warming can't just stop trying to communicate with non-scientists while Mooney and friends go off doing focus groups. In both cases, accepting the negative argument means that what you're currently doing is wrong. Professor Devlin wants math teachers to stop teaching multiplication as repeated division, *now*. But there's no alternative. We can't keep doing what we're already doing - the entire argument is that we shouldn't do that. Without some suggestion about what the path forward should be, it's a dead-end argument.

Re: #63

As I keep saying, there are always failure modes for any intuitive description of multiplication. But I'd argue that matrix multiplication isn't a reasonable case to bring up here. Matrix multiplication is really very different from standard numeric multiplication. Even the standard axiomatic definition of multiplication doesn't work, because matrix multiplication isn't commutative.

The important point isn't "Is teaching multiplication using intuition X perfect?", but "What's the best way to introduce multiplication to children?"

As I said in the original post, I think that the best approach is to use intuition, but to present more than one intuition, so that it's clear that the idea is bigger than the intuition. So you're giving solid intuitions for some of the standard cases that kids are going to encounter, and you're also making it clear that the intuitions aren't the whole truth.

In my view we should start teaching about "information" (for a lack of a better term) long before teaching numbers as abstractions. Teaching "information" requires teaching both numbers and what they count (e.g., 3 hours and 5 people) and that all numbers are a count of something. This should include teaching ranges of uncertainty (100 to 120 people in a crowd). Addition can be taught as an information operation that is valid only when what is counted is in the same dimension or is the same type of thing, i.e., addition is one-dimensional. Repeated addition is a special case that introduces one form of multiplication that is, in essence, an expression simplification. The extension to multiple dimensions (where the units of measure are different in each dimension) requires teaching a different form of multiplication. Fractions and division are further extensions that also benefit from addressing of units of measure. Once those highly practical forms of information and operations thereon are understood, we can introduce a variable as a generalization that identifies both any number and any unit of measure (or any type of thing). Only then should there be teaching the further abstractions used in mathematics, operations on numbers themselves where we completely ignore any unit of measure.

Re: #68

So you accept the "bait" but don't want to swallow?

I'd say the whole premise of Devlin's article is wrong -

1. He only has some vague, personal, anecdotal evidence that he has identified a problem area (i.e., specifically the introductory teaching of multiplication as repeated addition - which usually is also supported with other visual diagrams such as grids of dots, or pairings of elements from two sets).

2. Saying that the "repeated-addition" understanding of multiplication, even for integers, is mathematically "false" is just an abuse of language.

Re: #69

I'd say that's the consensus opinion, and also pretty much is what's done in schools today.

Perhaps not enough is done to "caveat" the original introduction, as in saying - BUT! there's a lot more to know about about multiplication that will come later...

Done right, it could help, done wrong, it could hurt.

FYI, readers here might enjoy my two most recent posts on the Devlin piece and the conversations it has engendered: "Devlin On Multiplication (or What is the meaning of "is"?)" and "More on Devlin and the Nature of Multiplication." These can be found at and , respectively.

A third post, "Is Mathematics Teaching A Closed Book?" is in the works, dealing with the claim by an anonymous troll calling himself "Haim Pipik," who claims repeatedly that there are no real open questions about mathematics pedagogy. He prowls under a bridge at math-teach@mathforum.org.

On my web page the essay On Math ed appears. I don't know if I agree with everything I said there.

My own children learned to count by 1s (1,2,3,4,5,6,7,8,9,10), then to count by 2s (2,4,6,8,10,12,14,16,18,20), then by 3s, 4s, through 10s. They learned the sequence (7,14,21,28,35,42,49,56,63,70) long before they learned the concept of multiplication. I taught the youngest how to compute squares in his head using x^2 + 2x +1, x^2+4x+4, and x^2+6x +9 upto about 50x50. Then I taught him to compute products of proximate numbers by computing differences of squares. These exercises were conversations in the car as he was driven to school. He not only is good at multiplication he also is good at subtraction. As he says, he subtracts from left to right and then adjusts. So his internal algorithm takes into consideration the most significant figures first. Clearly, the standard subtraction algorithm induces mistakes at the most crucial decimal places.

Multiplication is repeated addition, it is counting by a different base, it is area, and it is one of the structural aspects of a field. Repeated addition defines the Z-module structure of abelian groups. Axioms are great in conjunction with examples. Both are necessary.

The real argument that Devlin, I and others are making is that we need to talk to school teachers, and school teachers need to talk to mathematicians. School teachers often are involved in professional development, but how often do they talk to a professor of literature, a professor of mathematics, or a professor of history?

Even something as simple as this:

"Amy's Mum is making 2 pots of tomato soup. She wants to put 3 tomatoes in each pot of soup. How many tomatoes does she need?"

is a multiplication problem, whereas this:

"Tom has three toy cars. Ann has three dolls. How many toys do they have together?"

is a repeated addition problem.

They are different ideas, fundamentally. The "processes" of finding a product and finding a repeated addition sum are the same for both problems, but the ideas involved--INCLUDING THE MATHEMATICAL IDEAS--are very, very different.

Mathematics is not the notation. Mathematics is the meaning behind the notation. And much of that meaning cannot be captured by the notation itself. No doubt, 3 + 3 = 2 Ã 3. But multiplication and repeated addition are STILL fundamentally different.

What we are doing now in elementary mathematics education is blurring (or even erasing) this distinction when we first introduce multiplication. And, at least in my opinion, that's bad math, not good math.

#28:

Memorizing the tables works best using the adding approach and doing the 2x2 = 4, 2x3= 6...

Actually, my experience with my own kids suggests that memorizing the tables works best by watching the Math Rocks videos from Schoolhouse Rock. My son can tell you 7 x 9 far soon than he can answer 7 + 9, which seems counterintuitive until you know there's music involved.

#74:
While you are correct that addition and multiplication are fundamentally different ideas, I think you both overestimate and underestimate young learners.

I think you overestimate their ability to think abstractly when presented with new concepts. Plenty of people, adult and child, have trouble with absorbing the concrete facts of new ideas and at the same time abstracting them out to different

I also think you underestimate their ability, once they've absorbed the initial 'lie-to-children' (see comment #14), to then understand, with guidance, why, even though multiplication and addition are related, they are fundamentally different.

Teaching from the known to the unknown is a fundamental precept of all education. I'd be willing to bet that you yourself first learned multiplication as repeated addition and came later - maybe even very soon - to an understanding (though probably a subconscious one) of its fundamental difference(s) from its arithmetic cousin.

(Damn - missed finishing a sentence.)

The unfinished sentence should have read (I believe):

Plenty of people, adult and child, have trouble with absorbing the concrete facts of new ideas and at the same time abstracting them out to different, new conceptions of how the world works.

On a very-much related topic, has anyone heard of the JUMP (Junior Undiscovered Math Prodigies) program in Toronto, or its author, John Mighton (Author of The Myth of Ability)?

Just curious.

Wilson,

I don't think making the distinction between the two operations has to be "abstract."

And, yes, it may very well be that what we are doing now is not gravely injuring our kids' brains, but if we can tell them the truth (or at least something closer to the truth) from the beginning and get the same results, we should tell them the truth.

I've got a box, 10cm wide, 20 cm long, and 5cm high. What's it's volume? 10cmÃ20cmÃ5cm = 1000cm3. How can you explain that with scaling? The failure isn't just the units, either how do you explain that volume is the area of the base scaled by the height?

As a classroom demonstration, presumably one could illustrate this by presenting a box and stating it has a volume of 1000cm^3. Then you bring out a significantly larger amount of 1cm^3 cubes (all having length-width-height dimensions of 1x1x1). Then you have kids fill up the 1000cm^3 box with the 1cm^3 cubes, counting how many it takes to fill the larger one as they go.

Presumably, you can do this with 1000cm^3 boxes that vary dimensionally (e.g., 10x10x10 [aka: 10^3]; 10x20x5; 1x1x1000), which seems like a fairly good visuospatial approximation of commutativity. Furthermore, commutativity looks conceptually similar to a cognitive process known as reversibility, which is a cognitive skill that develops around entrance to formal schooling and mathematics learning.

Also, the research literature shows a strong relationship between visuospatial skills and mathematical ability. So presumably, anything showing a fairly direct link between mathematics and visuospatial tasks should be (pedagogically) useful.

I would thus have to agree with Joshua that Mathematics is not the notation. Mathematics is the meaning behind the notation. And much of that meaning cannot be captured by the notation itself. In which case, presumably there's a historical reason why x^2 is called 'square' and x^3 is called 'cube'.

Tony:

Using a box with lots of little pieces doesn't explain how you get from a rectangle to a volume using scaling. The thing is, in that case, you're moving from a plane surface to a volume - and you make that transition by using multiplication.

I agree that there's a lot more to math than notation. But what we're talking about here, in things like comparing multiplication as scaling to multiplication as repeated addition has very little to do with that.

The basic concept of multiplication has a consistent definition in mathematics, which I think comes across best by looking at abstract algebra. Multiplication as a dimension builder, multiplication as scaling, multiplication as repeated addition, multiplication as transformation - those are all valid interpretations of multiplication in the appropriate contexts. But none of them is the whole meaning of multiplication in mathematics.

The question that I'm trying to keep focused on in this discussion isn't so much "what is multiplication?" as "How do we teach children what multiplication means?"

I don't think that the formal, abstract, axiomatic definition of multiplication - which is the one that truly captures the essence of multiplications in all its varied forms - is an appropriate thing to teach to second and third graders. Instead, I think that the right thing to do is to do what one of the earlier commenters called lying to students.

All over subjects in early education, we don't teach children the whole truth about things. In fact, we often teach them things which aren't even incomplete truths, but are utter falsehoods. We teach reading by phonics - when phonics doesn't work a lot of the time. We teach them in the US that our government is a democracy, when in fact it's a republic, which isn't the same thing. In physics, we teach them that mass is a constant, and that energy and matter are different things. We teach them Newton's laws. In chemistry, we teach them that electrons orbit the nucleus of an atom. All of these things, we teach at best incomplete truths at the start, because the whole truth is too complicated to be understood when you're just starting to learn.

I don't see why math should be any different than all of those other topics. We start teaching math using the natural or whole numbers. We don't start off with the full real number system - we start with non-negative integers. We start by saying "you can't subtract a larger number from a smaller one". Then later, we teach them fractions, negative numbers, irrational numbers, complex numbers.

As I keep saying, I like the idea of teaching with intuition. I don't really want to lie to kids - I think that the best way to teach is to be honest about being incomplete. Don't say "You can't subtract a larger number from a smaller one"; say "We haven't learned to do that yet.". Teach the intuition, but show a couple of different ones, to make it clear that the intuition isn't the whole story; but don't spend time explaining the whole story to second-graders - they're not ready to understand it. Tell them what they can understand, and let them know that there's more to learn about it.

Shir Josh - mis-interpret what repeated addition means in this context to (not)make yer point!

It would be nice to sub-threads because there are so many different arguments going on.

Here's just one - and I think its the first one brought up by Devlin's rant:

Is there really a problem (specifically with teaching introductory multiplication as repeated addition - perhaps in combination with other discrete examples (grids, pairings)?

Is there? Devlin has no solid evidence - just personal and anecdotal.

It is in no way the normal usage, or proper usage of the word "lie" to refer to telling part of the truth of a complex subject.

"But I'm not convinced that you can teach kids math, right from the start, using axiomatic definitions. I watch my own kids and their classmates - and I can't imagine how you could make axiomatic mathematics make any sense to them. I'd be delighted to be proved wrong - but until someone can show me a textbook, a curriculum, or a lesson plan that shows how to do it, I tend to think that it's probably not doable."

I tend to agree that you cannot teach K5 children axiomatics. This is largely due to the fact that they are incapable of very much abstraction. But, once you are doing "letter arithmetic" (i.e. high school algebra), you certainly could start teaching axiomatics if you chose to. There are a lot of books that do that, actually, just none of them are in print anymore. It is widely thought that such a thing is "the wrong thing to do", primarily based on the mathematical empiricism of Morris Kline and W W Sawyer. Nevertheless, there are some top notch textbooks out there produced by top notch mathematicians and math ed professionals that do just that. I, myself, use these out of print texts by Frank Allen. And, I have had discussions with my oldest that have resulted in him showing that under Bechenbach's axioms of order in his book Introduction to Inequalities, -1 is not an element of P (which is set up to become the positive numbers). Of course, that's anecdotal and who knows what I mean by that anyway (e.g. if it is really true or if I am just proud of my son). But, you might be surprised at what is possible with older children if you only choose to do it rather than proceed down a more engineering math track.

And, while you cannot teach at the necessary level of abstraction to really do Peano arithmetic or something like that, you can do something else that is quite common in K12 education. You can mimic that concretely. A lot of programs teach students, for instance, to multiply 3*12 by breaking the 2 out, multiplying it by 3 to get 6 and then counting/collecting the tens and adding it all together to get 36. Now that is certainly round about, and it gets more complicated as you start having to carry a one somewhere. And, there are just as many programs that abuse pedagogical tactics like this as there are that seem to successfully employ such methods. However, this tactic is essentially introducing the distributive law concretely while teaching arithmetic heuristically and concretely without ever sayign anything like "Use the Distributive Law to...." Singapore, for instance, employs tactics like this all the time. Long before they actually start calling anything by name and using it in its general case, the student has already, perhaps even for years, been doing problems that employ specific manifestations of the concept.

If you were to teach math to grade school students with an eye toward teaching axiomatics to them when they were capable of letter arithmetic, then you would teach multiplication as primarily being "just" repeated addition. You would mimic the rigorous development of it by starting with repeated addition on the integers and then extending the concept. Of course, you would teach other heuristic ideas about multiplication as they came up and you wouldn't stop anyone from thinking about it however they want to think about it (so long as it works in their context). But, you would be doing something that is more like the opposite of Devlin's entire idea in his two articles (even excepting this separate issue of the correctness of his mathematics).

In fact, if you really want to get historical about all this and "do it right", I would argue that you would not even teach arithmetic. You would just start doing speculative philosophy from the time they can start talking. That is the fastest way to build a person's capacity for abstraction. And, it naturally leads to the epistemological questions that lead to rigor and math. You would just do it Plato-style like it shows up in his Dialogues, and bring up questions like "What is Justice," "What kinds of things 'exist'," "Where did the world come from?" You wouldn't even do math at all. You would just do philosophy until it leads to mathematical logic which would come in time, but probably not much before they are nine or so. Of course, I don't *really* advocate such an impractical method which leaves a kid unable to do even basic arithmetic up to such a late age, but I might if society and circumstances were different. And, perhaps it is important and relevant as to whether or not such a progression is, indeed, the natural intellectual manner of growth while all this other stuff is happening on the side.

Let's be clear though - "axiomatic" is not the same as "the whole truth" either. Gödel showed that.

I'm saying this not in response to Adrian's immediately previous post but rather the remark of MarkCC that only the axiomatic definitions contain "the whole truth". They don't. We'd have to teach our kids Godel incompleteness first - and then of course they would understand that *everything* we say afterwards is in fact a "lie" (if by "lie" we mean not the whole truth.)

The question that I'm trying to keep focused on in this discussion isn't so much "what is multiplication?" as "How do we teach children what multiplication means?"

My assumption is that these questions probably shouldn't be treated as mutually exclusive entities. That is, one's particular understanding of multiplication probably implicitly and explicitly dictates one's underlying belief concerning how multiplication (and other mathematical concepts) should be taught. As an example, Devlin's assertion that one should not teach multiplication as repeated addition is probably based on his understanding of multiplication, and not how a 2nd or 3rd grader might conceptualize multiplication (which I think is the point you're making). So the appropriate question really should be, given one's understanding of children's cognitive development at various ages, what is/are developmentally appropriate ways to teach multiplication (and other math concepts)?

Research indicates that by preschool (3.5-4 years), children grasp cardinality (i.e., the last number in a counting sequence indicates the quantity of items in a set). In mathematical terms, this seems to be a rudimentary form of set theory based on repeated addition of 1 (i.e., counting). So a developmentally appropriate way to introduce multiplication might be to teach it as a repeated addition of sets containing the same number of items (e.g., 5x5 = 5+5+5+5+5; which results in a 5,10,15,20,25 counting sequence).

The other multiplication interpretations (e.g., dimension builder, scaling, transformation) seem like applications of multiplication involving more advanced, probably geometric (i.e., multidimensional) conceptualizations. So these would presumably be taught after the repeated set version. Actually, there's a mathematics program called Number Worlds that recognizes children require certain types of experiences before entering formal schooling, that involve relating concrete experiences to abstract mathematical conceputalizations. Much of these concrete experiences seem to concern visuospatial information processing (which seems like what one would need to understand the various multiplication conceptualizations).

(Important qualifier: My knowledge areas are in cognition, education, and developmental psychology, so I can't make any detailed comments concerning mathematical axioms that distinguish addition from multiplication. My comments here are influenced primarily by this knowledge background).

And, as a cautionary note to the more adventurous minds out there (and not really in any kind of specific response to Joe), Godel did *not* show that we should abandon the axiomatic method and just resort to heuristics. "What is multiplication (of numbers)," is a *mathematical* question not a pedagogical one and not a scientific one or any other kind of question. Landau wrote the book on it. In it, multiplication is originally defined as repeated addition and extended from there. We should be done unless someone can really challenge that. (I can think of about one plausible thing, but no one seems to be championing it, so I wont comment any further on it.)

I don't see anything wrong with the "lies we tell our children" system. The repeated addition scheme works for a lot of things - I remember being introduced to integration by the old "cut out the area on the piece of paper and weigh it" method, and a lot of practical math, from using an abacus to numerical methods, are nothing but advanced arithmetic.
I barely scraped through undergraduate physical chemistry since it was taught "mathematically", without an intuitive grasp of the subject matter. Formula on the black board, mathematical operation, viola, now you can figure out how much gas escapes from a hole in a cylinder. Only later, in graduate QM classes, where the approach was phenomenological - define the problem, show the mathematical solution, and only then teach the math needed to come to the solution, came the insight into, and an appreciation of, the math as abstract but useful method.
While this classifies me as nothing but an advanced 3rd grader, it works as a teaching method. As mentioned previously in the thread, we don't teach advanced electron distribution probabilities and quark theory in basic chemistry. And why should we, 98% of chemistry is organic chemistry, not involving anything more complicated than can be expressed with Bohr's model,refined with s,p,and if we're ambitious, d orbitals.
To me, Devlin seems to try to take the mathematician's way of "disproving a theorem is as much an achievement as proving one" as an approach for real life. Unfortunately, in life you have to provide for an alternative to make true progress.

Does anyone want to take on the question as to whether there is an actual problem that Devlin has identified?

Is the absurdity of a mathematician (of some note) pronouncing there to be a societal problem -- without offering some supporting statistics -- lost on this crowd?

"Is the absurdity of a mathematician (of some note) pronouncing there to be a societal problem -- without offering some supporting statistics -- lost on this crowd?"

Oh puhleeeaaase. Mathematicians do that all the time. You know, the unabomber was a complex analyst... at Berkeley, no less. Apparently, the more notable they are....

"Does anyone want to take on the question as to whether there is an actual problem that Devlin has identified?"

I've wondered that too.

In the absence of an example of a mistake as well as an absence of a rigorous defintion of multiplication in two articles, I've speculated on what possible motivations he has for these articles other than the math.

(I am, of course, joking. I do not think that Devlin=Unabomber. Their similarities end at their talent as mathematicians and their willingness to boldly make assertions that many mathematicians would not.)

Marc,

I have been thinking this for some time, and what stops me is how can one solve a multiplication problem (say 3x2) without resorting to using addition (assuming no multiplication table is available) or at the very least counting.

I understand your points about multiplication branching out to including concepts like volume versus area, but at the level of calculation how does one do multiplication without addition? (Serious question here!)

"Knowing, for example, that multiplying two positive integers was really the same as adding their logarithms revealed an exact correspondence between two algebraic systems that was useful, but not very deep. Seing how a more sophisticated version of the same could be established for a vast array of more complex systems -- from rotations in space to the symmetries of subatomic particles -- unified great tracts of physics and mathematics, without collapsing them all into mere copies of a single example."

["Incandescence", by Greg Egan, Hardcover: 256 pages, Night Shade Books: 16 July 2008,
ISBN-10: 1597801283
ISBN-13: 978-1597801287
# Product Dimensions: 9.1 x 6.1 x 1.1 inches]

The first part of paragraph is:

"'Interesting Truths' referred to a kind of theorem which captured subtle unifying insights between broad classes of athematical structures. In between strict isomorphism -- where the same structure recurred exactly in different guises -- and the loosest of poetic analogies, Interesting Truths gathered together a panoply of apparently disparate systems by showing them all to be reflections of each other, albeit in a suitably warped mirror."

typo(e)s -- "and noe.", "two a third grader"

Yes, agree completely with Mark. (And also with #1 on distributive property.)

Re #2, #3 -- As one of the few for whom the axiomatics of the New Math was good, I will say "repeated addition" is a good start. Since Subtraction was explained as the INVERSE OPERATION to Addition in New Math, In 2nd or third grade I deduced the necessity of an inverse operation to multiplication, which must surely be like repeated subtraction, and so it was. However, the level of axiomatics one can handle in third grade is somewhat limited compared to grad school; it's still motivated by and modeled as repeated addition.

Re #3 and Mark's triangle - even the triangle 1/2 b h can be reduced to addition or counting, while restricted to natural numbers (as we are in Third grade): 1/2 b h = sum of columns of squares in the rectangle (b x h), halved. this is repeated addition, or even mere counting. (Addition reduces to counting by Peano axioms, and New Math approaches addition via counting.)

Addition as grouped counting, muliplication as repeated addition (and likewise disvision as counted subtraction), exponentiation as repeated multiplication all need extension when we extend the underlying number system for completeness -- 2*0=0 but 2/0 undefined, -2*-3=6, 5 * 1/2, 2^(-1/3) defy the simple original intuition that held over N.

Re #3 and "the main gripe" Multiplication of e * pi does of course require extension of *any* definition on NxN to RxR. So? Even if the definition *appears* identical as presented by #2, it must be proved consistent for N subset R. Replicated Addition is a fine first definition for third graders on NxN ... provided we give enough "coming attractions" of the definition -- distributive, identity, inverse, etc -- so that maybe they see Division coming and maybe they see application to fractions coming, or at least aren't shocked by it.

A version of the Peano axioms may work in 1st or 2nd grade (our new math bridge from sets and counting to addition was I suspect inspired by Peano axioms), but Reiman sums won't work in 3rd grade (8 year olds).

#13 "He thinks if we teach something closer to algebra's vocabulary earlier, we'll have fewer math dropouts and haters." There's something to that, but it's been tried. The formalism in the 1960's New Math in elementary school may have lead to a slightly smaller crop of haters in the tail of the baby boom (born post-sputnik) and actually benefited me, but turned off quite a few earlier than alegebra, my recollection. Giving as much of the language earlier as will take is to the good, but no more.

This topic has been pounded to death, but I decided to comment anyway.

Devlin is nuts!

I can't imagine trying to teach 3rd graders the concept of multiplication without showing them examples of repeated addition and geometric examples of blocks forming rectangles. I suppose neither of these examples provide the correct mathematician's definition of multiplication, but they are not wrong either, and kids do learn how to multiply correctly (i.e. they learn the tables and don't perform repeated addition).

I suggest that Devlin might want to read "Why Johnny can't Add" at
http://www.marco-learningsystems.com/pages/kline/johnny.html

RE: #3

"The main gripe" - the main gripe is Devlin's contention that students that are first introduced to multiplication as repeated-addition "inevitably" feel let down, confused, lied to, turned-off-to-math when they learn that to multiply (for example) two fractions you have multiply numerators and denominators.

(Of course, if you somehow teach them "scaling" [because 6 year olds who know nothing about multiplication know all about the concept of "scaling" of course, and can integrate that knowledge seamlessly into their understanding of numbers so far] --then they will be enlightened, happy, curious and above average.)

Its a highly suspect hypothesis on the face of it, and he has no real evidence that it is so. After Devlin is done with them, I'm sure they must be confused.

The confusion lies, I think, in that the objects being added and multiplied are different in the two instances. But problem is that they not different enough. Multiplication is indeed repeated addition when considering just the natural numbers. But multiplication on Q is different than multiplication on N, because the objects being operated on are different. It's just that the 'copies' of N look so much like the old guys that there is some confusion.

If we defined addition and multiplication on the set of all the subsets of a set, they would look quite different, different enough that questions about their similarity would not tend to be the first questions asked. But that's a little advanced. Teaching second and third graders? I'd just say that five apples times three apples is five apples plus five apples plus five apples is fifteen apples . . . and never mind the units.

Personally, I've never seen any confusion caused by 'differing' notions of multiplication.

Re: #98

"Personally, I've never seen any confusion caused by 'differing' notions of multiplication."

Exactly.

It occurs to me that when we try to teach addition of fractions to students, we run into exactly the same problem that Devlin says exists in multiplication - we have to teach them a new way of adding fractions that is a bit more complicated than what they learned before for whole numbers.

Obviously, this makes the notion of addition as the adding together of numbers totally and utterly false! And smart students will no doubt accuse us of changing the rules on them.

We must therefore never use the term "addition" when we refer to - to - well, that operation.

Henceforth, teachers, please stop telling your students that
addition is addition. Please refer to it as the-operation-formerly-known-as-addition.

I find this argument strange, yet oddly interesting, as my daughter in 1st Grade just taught herself multiplication unaided, based on the spontaneous realization that "3 times 4" was 4 added 3 times, and that was actually the same as 3 added 4 times.

Now, I'm not a primary school teacher, so I'm not going to presume to tell people how to teach kids multiplication-- but do you really expect me to tell her she's got it all wrong, and that multiplication is not really repeated addition after all?

Michael:

Yes and no.

I wouldn't say that she's got it wrong, but she's only got part of the picture. But for a first or second grader, it's wonderful.

My daughter figured out negative numbers based on one sentence a friend mentioned to her about numbers smaller than 0. She understands how they work with addition, both with positive and with other negative numbers - all from her own process of thinking it through. She doesn't know how to do multiplication with negatives - her understanding is incomplete. But I wouldn't say that she's understanding negative numbers wrong - it's just not complete.

Part of my problem with Prof. Devlin isn't that he's necessarily wrong - but that he's being an absolutist, and essentially saying that an incomplete understanding is an incorrect understanding. I think that *that* part of his argument is wrong: an incomplete understanding is correct as far as it goes. The important thing is to be clear that it's incorrect. So in the case of your daughter, I'd mention that there are ways of using numbers that you can multiply where repeated addition doesn't work, and that someday she'll learn about them.

Mark, in #102:

I think that *that* part of his argument is wrong: an incomplete understanding is correct as far as it goes. The important thing is to be clear that it's incorrect.

You meant 'incomplete' there at the end, right?

Wilson:

Yes, thank you, I did mean "incomplete" there. "Incorrect" completely undermines my entire argument. :-)

What I meant to say in #3 was that Devlin's gripe, from the point of view of a mathematician, is that multiplication cannot be, in general (even on R), be accurately accepted as repeated addition. I believe he is making the same mistake that I often do, in assuming that non-mathematicians ever really DO work in R; the real world operates with nothing more than Q, and sometimes, in maybe electrical engineering, Q[i] (the complex rationals).

By missing this, Devlin fails to realize that pre-college students can be adequately taught all multiplication as repeated addition. In the context of the standard rational field, the operations of repeated addition and multiplication are ENTIRELY, mathematically even, EQUIVALENT and can be proved so easily (my earlier post waves through it sleepily).

@ #73:
Your child is intelligent. I learned many of those things on my own as a child, too, and without such discussion with adults (in fact just disapproval from teachers: there are no negatives, there are no irrationals, there are no imaginaries, etc) though I'll admit I didn't know as much as he(?). However, don't assume that all kids would be quick to understand those things, or even able after a few weeks of dedicated explanation.

To Mark, WIlson, Michael -

I think if you read the Devlin articles carefully its reasonably clear that he considers understanding multiplication as "repeated-addition" to be incorrect in all and any contexts.

Further communication with him via email made this point abundantly clear.

Joe (#106):

In that case, I've got to conclude that he's an idiot. I hate to be that blunt - but totally denying that understanding multiplication as repeated addition makes sense in any context - well, it's just silly.

One of the classic definitions of multiplication (and this is definition in the sense of "formal mathematical definition of semantics") is the Peano definition, which defines multiplication over the natural numbers as repeated addition. If the Peano definition of multiplication is always wrong, then I don't know what "right" means.

I'm not making sense of those last two posts - is he saying that multiplication on N is _not_ the same as repeated addition on N? That is simply false. Two functions are the same if their domains are the same, their ranges are the same, and identical inputs produce identical outputs always. By definition. So f:NxN->N and g:NxN->N, f(a,b)=axb, and g(a,b)=a+a+. . .+a b times have to be the same. By definition

That multiplication on N is _not_ the same as repeated addition on N is at least implied by saying that universally it is "totally and utterly false" to ever say that multiplication is repeated addition and saying that the fact that when teachers teach it that way when they are teaching arithmetic on the natural numbers is an outrageous miscarriage of math justice.

I beta-tested the New Math textbooks for New York City's unified school district when I was in roughly 3rd grade, absorbing all the texts for 1st through 9th grade in a day. For me, they worked. I wonder how badly I skewed the statistics.

Someone once told me that there were operations beyond addition and multiplication. He explained (however incompletely) exponentiation. I got it right away. Then he explained why a square root was raising a number to the power one half. No problem. Then he said that there were other operations just as important. I racked my brain and drew a blank. "Differentiation and integration," he said, and would not explain, other than to say that I was too young for Calculus. That summer, I bought a used calculus book, by someone called Love, and studied it at the beach.

"Oh," a passerby said, "you're reading a book about love by someone named Calculus?"

But I could apply it to analyzing the trajectory of the model rockets that I launched. Differential equations confused me at first.

I struggled through all 3 volumes of the paperback of Principia Mathematica by Russell and Whitehead. Somewhere in volume 2 they prove that 1 + 1 = 2.

Then someone gave me a copy of Coxeter's "Regular Polytopes" and I taught myself to visualize 4-dimensional shapes. Badly, incompletely, but with some joy.

I saw the word "hypercylinder" and not long after had this message conveyed to a college Math department:

"I must be doing something wrong when I try to visualize a hypercylinder. One can make a cylinder two ways. You can move a circle perpendicular to its plane, and see what it traces out. You can rotate a square about one of its edges. Same shape. By analogy, I visualize moving a sphere perpendicular to the hyperplane where it started, or by rotating a hypercube around one of its faces. These don't look remotely the same to me. What am I missing?"

We learn what we learn. Incomplete explanations didn't hurt me. But the sense of being lied to by teachers always infuriated me. But that's another story.

My experience with school was so non-informative that I only attended on days to take the test. It is a sad system when the teachers can not answer questions honestly. My education actually began after I left the state sanctified system. (Illinois)

Mark,

#106 misleads a bit with this comment. This is the snippet (or maybe it was the whole thing, I don't know) of a reply from Devlin (allegedly) that Joe quoted on the Let's Play Math! thread:

"Sorry, mathematics is not a matter of opinons,
it's either right or wrong. And multiplication
is not repeated addition on any domain."

Devlin did not say in that E-mail (nor does he say in his articles) that UNDERSTANDING multiplication (in some cases) as repeated addition is incorrect:

"(I do think that you need to present simple everyday examples of applications. Teaching a class of elementary school students about axiomatic integral domains is probably not a good idea! . . .

Once you have established that there are two distinct (I don't say unconnected) useful operations on numbers, then it is surely self-evident that repeated addition is not multiplication, it is just addition - repeated!

But now, you have set the stage for that wonderful moment when you can tell kids, or even better maybe they can discover for themselves, this wonderful trick that multiplication gives you a super quick way to calculate a repeated addition sum."

Devlin says that repeated addition does not have the same MEANING as multiplication. And it doesn't. Not on ANY domain. Neither he nor I--nor anyone else, for that matter--would argue that the two are "sometimes" the same, as you mentioned in this post. But I would argue that in every instance that multiplication is "the same" as repeated addition, it has to do with RESULTS, not MEANING. When you consider their respective "meanings," then, no, the two are NEVER the same.

And, no, "meaning" is not the same as "makes sense as." Certainly multiplication "makes sense as" repeated addition on the natural numbers (so long as you consider results, or "process"). Again, not what Devlin is arguing.

But, going back to this "understanding" thing, Mark wrote that Devlin is

"essentially saying that an incomplete understanding is an incorrect understanding"

I would disagree--not necessarily with the statement, but with the focus.

Let me give an analogy. You're a dad or mom, and you're with your three-year-old daughter on vacation to San Diego, visiting the zoo there. You both walk up to the enclosure that contains zebras, and you hear a child right next to you--not your child--say, "Look, Daddy, horsies!"

You might chuckle to yourself a bit, or, if you're a dad, you won't even process that statement as out of the ordinary. You'll forget it as soon as you heard it, because you've heard the same kinds of things from each of your kids.

But imagine instead that the other daddy in that little scenario walked up to the zebra enclosure with HIS daughter and said, "Look, sweetie, horsies!"

Wouldn't it go through your mind--at least for a second--that, "Hey, that guy might be an absolute idiot"?

Re: #107

Mark, I sought clarification via email directly and after some back and forth during which he tried to gently get me to see the light he said:

> sorry, mathematics is not a matter of opinons,
> it's either right or wrong. and multiplication
> is not repeated addition on any domain.

Domain as in integers, rationals, (more precisely ZxZ, QxQ etc.) This was after I tried to explain that people commonly regard multiplication as "repeated-addition" on integers at least, in the extensional sense - and that its perfectly normal and also mathematical use of language to say one function *is* another if they are extensionally equal.

and also he said:

> this was all worked out centuries ago, it just has not yet filtered
> through to the k-12 system. we see the result with entering university
> students who don't understand elementary arithmetic.

Centuries ago? That contradicts his second article in which he says Hilbert's program was the big AHA!

Anyways - *that* is the big reason some of us have been harping on this matter.

Re: #106

Joshua, show us your definition that defines two function as mathematically different, f != g, iff. their MEANINGS are different, even though they are extensionally equal, i.e., f(x) = g(x) for all x in the common domain of f&g.

Define MEANING mathematically - because Devlin says they are mathematically different.

Prove mathematically then, that f (repeated addition) is not
g (multiplication) on the domain ZxZ.

re: #108

Exactly - that is *one* of the key arguments I have with the articles.

Another is his whole premise - that he has truly identified a real definite problem in the way math is taught and that he has the better approach. Whoever mentioned "Why Johnny can't add" -- boy oh boy that hit the nail on the head.

Re: #110
"But the sense of being lied to by teachers always infuriated me. But that's another story."

I never had the feeling of being lied to - at least in any mendacious sense. However, I did very often feel that my teachers had very limited & rigid viewpoints and were not able to shift perspectives easily.

I've since come to understand that as two almost fundamental differences in people, like introvert vs extrovert, thinking vs. feeling, intuitive vs. sensing.

There are also "both/and" type people who can shift perspectives easily and accept multiple descriptions of reality even though they may seem at odds (like the 5 blind men feeling an elephant) and there are "either/or" people who given two descriptions that may seem at odds must decide for themselves (and unfortunately sometimes for others) which is "true" and which is "false".

The people who can accept "repeated-addition" as a partially accurate description of multiplication are "both/and" types.
Devlin and followers are "either/or" -- either its repeated-addition or NOT! (And they say not).
Well, it ain't scaling either, as simple consideration of complex numbers will show, and it ain't just scaling + rotation (as matrix multiplication will show.)

"Both/And" is a better way to go! But I guess you have to be born to it.

Re: #107

Mark, I pressed him on Peano definition via email but rather
than answer me directly on that one he put this in his second article:

"Peano did formulate what are often called the Peano axioms, but their purpose is to show how the positive whole numbers can be defined from first-order logic; they are not a descriptive axiom system that tells you how to work in the system, as are the other axiom systems I just listed."

So he dances around the issue - implying very vaguely and insidiously somehow that Peano axioms and definitions don't count (pun intended) the way his favorites do.

Devlin says:

"Peano did formulate what are often called the Peano axioms, but their purpose is to show how the positive whole numbers can be defined from first-order logic; they are not a descriptive axiom system that tells you how to work in the system, as are the other axiom systems I just listed."

What *is* he trying to say here? - that Peano wasn't really doing math? That Peano is false? It was all a parlor trick? Well, WHAT?

A careful reading shows up all sort of vagueness and dancing around, particularly his analogies which are just substitutes for careful argumentation.

Re: #112

Josh, horses and zebras are the same genus and some learned people do in fact refer to any equid as a horse.

For example, read the first couple lines here:

http://montereybay.com/creagrus/equids.html

Was I just napping when somebody suggested looking at peer-reviewed research on math education to see if teaching multiplication as repeated addition really causes problems?

True to form, Joe comes out with questioning as a defense, rather than an explanation.

As I have said here and elsewhere, Joe Niederberger, notation is not the same thing as mathematics.

I can't give you a mathematical definition of "meaning" in mathematics. To understand that, you're just going to have to tear yourself away from your old college lecture notes and think for yourself.

Read Denise's post for some inspiration in that regard.

"Centuries ago? That contradicts his second article in which he says Hilbert's program was the big AHA!" Anyways - *that* is the big reason some of us have been harping on this matter.

Really? Your big argument is that it was really just ONE century? Joe, you're full of it.

I actually mentioned that on another blog and also to Devlin.

As best I can tell (and I'm no expert) studies being done are typically comparing repeated-addtion versus cartesian-product (another discrete approach that doesn't extend without extra work to reals), or to "correspondence" approaches (ditto), or some other equally elementary and typically discrete approach.

And also best I can tell the results are not universally conclusive in one direction or another.

But I'm not expert - my impressions are just that.

#118: Joe,

Do you think that I didn't know that before I wrote the post? I didn't need a Google search to figure it out.

Good lord! It isn't actually even Peano. It is the standard method of DEFINING multiplication (as repeated addition, and I really don't know what else you could possibly use at that point) FROM Peano's axioms. All Peano gets you is the ability to use mathematical induction. You don't even get addition. So, we should all be referring really to Landau (who wrote the book on this, and pretty much everybody knows him and his book). And, it is not only the only way to do it, but entirely descirptive of what to do and how to do it in terms of actually being able to carry out a calculation. And, more specifically, regardless of how descriptive one feels it is or isn't, there really aren't any alternatives, so that's pretty much what we seem to be stuck with.

Also, Morris Kline was an ass that almost single handedly ruined math education in the US! We were on the verge of a renaissance of mathematics that he destroyed because of a petty rivalry with Edwin Moise. We all deserve to rot in math hell for following him. And, what Devlin suggests is the opposite of New Math and would probably have been totally advocated by the likes of Kline.

I don't have Landau's book but I believe you. I wonder what Devlin would say if asked point-blank about that? I think it would just be "weasel-words" to the rescue again...

Adrian Durham keeps TALKING about Landau (for weeks now), but seems reticent to SHOW anything he's talking about.

Really. I don't think Landau would have some kind of copyright issue.

So show.

joe N (#120):

And also best I can tell the results are not universally conclusive in one direction or another.

But I'm not expert - my impressions are just that.

For what little it's worth, my impressions are in accord with yours. Non-experts unite! :-)

If I suspected that generalizing from nonnegative integer multiplication to fractions, decimals, etc., etc. was a source of particular difficulty, I'd first look to the literature to see (a) if people had investigated that issue and (b) what sort of investigation techniques are commonly used. Failing to find a conclusive answer to my question, I'd reach out to people who do that sort of experimentation and observation. Eventually, I might agitate for new studies to be done, help design such studies, evaluate their results and so forth. It's plodding and a long way from glamorous, but it's a metric Hell of a lot more respectable than pseudo-intellectual punditry.

Blake,

You would actually do a literature search before you made the decision to tell kids that up is down and down is up?

That analogy fails in so many ways I don't know where to begin.

Mark, thank you for your kind words about my daughter in #102. I agree with you when you write: Part of my problem with Prof. Devlin isn't that he's necessarily wrong - but that he's being an absolutist, and essentially saying that an incomplete understanding is an incorrect understanding. I think that *that* part of his argument is wrong: an incomplete understanding is correct as far as it goes.

It seems strange to me that he fails to grasp that one of the most common ways to learn mathematics (or any subject, for that matter) is "successive approximation"-- our understanding is always incomplete, but if we play our cards right (and get lucky), our incomplete understanding gets gradually more complete over time.

In other words, for my daughter, "repeated addition" is a good enough understanding of multiplication, for now. When she's in need of a more sophisticated one, it will come along.

One more data point, by the way-- I taught my oldest daughter multiplication, at around the same age or so, by making rectangles on graph paper. She got the intuition pretty quickly, and that was that. Different strokes, yada yada yada.

Another thing: But neither of those is particularly good for explaining how multiplication can tell you what 3/5ths of $25 is.

What's wrong with rewriting the 25 as 25/1, and then viewing it as (25 + 25 + 25) / (1 + 1 + 1 + 1 + 1)? Sure looks like repeated addition to me (both above and below the bar).

I wouldn't say that she's got it wrong, but she's only got part of the picture. But for a first or second grader, it's wonderful.

I have a PhD, work as a mostly computational biologist, and am posting this while the cluster finishes a batch of analysis. My understanding of multiplication is pretty much the same as this young lady's (with the extension of it to the addition of more abstract quantities an abstract number of times), and I seem to have made it through life pretty well nonetheless.

Frankly, the total absence of common freaking sense that most of you people are displaying is a lot more of a handicap than whatever subtlety of multiplication it is that she and I are missing out on.

#102:

I wouldn't say that she's got it wrong, but she's only got part of the picture. But for a first or second grader, it's wonderful.

I think that statement is correct, and that it has much greater applicability than just the questions addressed in this post.

Issaac Asimov, in an essay title "The Relativity of Wrong" makes a big fuss about this, and about how this essential bit of knowledge is systematically denied in much of modern education, where there is one correct answer that you can tick on this test, and anything else earns you no points.

In fact, I think Devlin makes the same sort of mistake in his criticism, and it also seems to me that he expects a problem to exist with the method of teaching multiplication as repeated addition because he expects students to have been educated exactly this way. He expects students to have learned that whatever method of multiñlication they learned is the right one. Why else would a student feel that they were lied to when they are taught an extension to a learned theory?

I think your suggestion makes a lot more sense.

I have a new way of understanding the loggerheads now - thanks to Joshua and his zebras.

Devlin and his followers only want the word "multiplication" to refer to the particular abstraction they have in mind. They are thinking of something like the genus Equidae (horses, donkeys, asses, and zebras.) In Devlin's case he pretty much says he's really talking about multiplication on real numbers - and therefore only views multipliation on integers as a restriction of the more broadly conceived function.

However, there are species that are distinct (domestic horses - multiplication viewed as a function on ZxZ).

So - Devlin looks at an animal and says - "There is an equid". I look and say - well, OK, but it is also a horsie!

And Devlin (and his disciples) says No! No! No! You are WRONG! Its an equiid!

Its typical of "either/or" type thinkers that they want to control even the ordinarily accepted, correct use of language by other people.

Just because you have a nice abstraction, it doesn't invalidate the reality of the more specific manifestations.

Re: #135

More thoughts..

Of course the situation is compounded by the fact that we have only one word "multiplication" to refer to these different abstractions that exist on different levels.

Imagine if the biological genus were called "Horse" containing
7 species named "Horse", "Horse", ... and "Horse".

I never had the slightest problem with arithmetic when growing up. This put me at quite a disadvantage attempting to get my child through it. But at least he can now be asked, "What's two thirds of three quarters", and he'll get the right answer. And that's better than most adults do.

But formalism had to wait for college. And as it was an engineering school, interested in practical answers, there wasn't even very much there. Really, all we needed to know was how to do it. Formalism can help you know if your answer is right.

It's one half.

Quick - what's a quarter divided by a half?

I wonder if someone could explain Mr. Fisher's story about the tomatoes and the toys. If there are two pots, and each pot has three tomatoes, then we are allowed to call this a multiplication problem. But if there are two children, and each child has three toys, then it is not a multiplication problem. Why is this? Is it because the toys are of different kinds? Or is it because it is only accidental that each child has the same number of toys?

For what it's worth, and although I can't find the reference, recent research suggests that on a neural level (integer) multiplication is handled by a fundamentally different part of the brain than (integer) addition, the latter having its own specialized module and the former being essentially verbal - that is, children learn to multiply by verbally internalizing their times tables! (The evolutionary justification being, more or less, that addition corresponds to unions of sets, which is certainly a concept that merits its own place in the brain, whereas multiplication is a much more "unnatural" concept - how often are exact copies of a set present in a "natural" situation?)

Certainly understanding a computation does not imply understanding a concept, but it's something worth thinking about. As conceptual as you want to get, when it comes down to the multiplication problems that children will inevitably be asked to do, those problems are not done in the repeated-addition paradigm. (I have a bone to pick with the entire state of modern mathematical education in this regard - formal procedures for solving classes of problems replacing actual understanding - but that's another story.)

As far as the comments about meters and meters-squared go, how does one go about explaining the multiplication of two quantities with different units in this paradigm? (Say, mass and acceleration?) In this context, it seems to me that the only thing we can do is define new units in terms of multiplication of old units ("this is a Newton!") and then talk about "scaling," although perhaps this is a little further along in a child's education.

#140:

I have experienced several examples of learning procedures as a prerequisite to acheiving understanding. The earliest one I remember is understanding why multiplication of fractions works after seeing it done with algebraic fractions, but in that case I could not posibly have understood the simple procedure I was watching without boring instruction in simple arithmetic.

A more recent example was introductory mathematics courses in college before getting courses in Physics that used that math. Again, without having learned the boring math first, I'd have been hopelessly lost in physics class.

I don't know if I am representative of normal children learning, but I would not be surprised to find that many children would find themselves in the same situation.

A nice way to explain addition and multiplication is visually, with a ruler and a compass, using Euclides axioms of parallels.

I am a secondary math teacher, I have an MA in pure math, and I have read/skimmed most of these comments. I have a couple of observations.

1) Most elementary teachers do not know how to multiply pi*e. They may have seen a matrix, but most would not remember that multiplication with matrices is not commutative. This is a major stumbling block in changing how this is taught.

2) I have worked in two different high schools with probably 30 math teachers. Maybe 5 of them could have understood the subtlety of this question.

3) If I were going to teach an elementary student how to multiply I would probably show them multiple representations of multiplication and be sure to emphasize that this is how we multiply with natural numbers. I would use repeated addition, a geometric approach and a sequential approach. All have their place and use later in mathematics. Plus I am confident that if a student can understand all three of these then they will have no problem extending to other representations later in their math careers.

Teaching is a lot harder than making mathematically correct statements. Someone hit it on the head when they said math and teaching math are not the same thing. I know Mr. Fisher ripped this guy apart on his blog, but what he fails to understand is that teaching math is a psychological and socialogical endeavor where as math is a scientific.

However, this has been a great discussion to read as a math teacher, but most of my colleagues would have dismissed it a soon as fields, rings and groups were introduced. Twenty years from now maybe someone will be able to turn this into something useful, but for now it seems to be merely academic. Which is where most useful things begin.

By the way, Fisher seems to think I am worried that someone might know my identity or something. It is all over the internet and I haven't really tried to conceal it, but in case anyone is interested, my name is Adrian Durham and I have an MS in Math from Ohio State.

Mr. Fisher didn't rip anyone apart on his blog. In fact, he is mostly wrong about the math. And, Math isn't really scientific. Science is emprical and math is a priori. It is "scientific" only in the metaphorical sense that it is kind of like science because there is generally a formal way of establishing results.

Devlin's point that whatever it is that math teachers teach it has to still be math is a valid one. Before there is the psychological aspect of it, there is the subject itself. What Devlin is saying in his articles is that the teachers are getting the subject, itself, wrong. But, Devlin (and Fisher) has gotten the subject, itself, wrong in what he is saying. Of course, he is a famous logician, so you cannot just take my word for it. You have to look at the math. And, specifically with regard to Fisher's blog, whether you use Peano's axioms or Mikusiniski's equivalent devlopment, they both use the standard recursive definition of multiplication as repeated addition. Long winded discussions about "what it's like" or what your heuristic interpretation of it is do not address the mathematics and are no substitute for the mathematics.

And, for that matter (now that I have flipped through it), Mikusinski's book is really not that good as sort of an official refrence on any of this. He takes the natural numbers and ordering on the natural numbers as primitives! I've tried really hard to figure out how to interpret that in an acceptable way, but my dad, for instance, who is a retired mathematician thinks it is ridiculous, and it is hard to defend it. Furthermore, when you take the well ordering property as an axiom, it is hard to see how it is really any different than Peano's version. (Proving that well-ordering and induction are equivalent is standard fare for undergraduate math majors.) This book is definitely better than the typical freshman calculus, and for an analysis text, I don't think that this development is that critical at all. Baby Rudin (the "standard" for this material) just takes all the properties, including the least upper bound property, as axiomatic only mentioning Dedikind Cuts in an appendix. The analysis really starts with the completeness of the real numbers in the sense of having that least upper bound property. How you get to that point is certainly an important mathematical question, but kind of outside the subject matter. (In other words, I would tend to not judge an analysis text on the way it does this development.)

The bottom line is that no matter how you want to judge the math, it is all coming consistently back to Landau's definitive text starting with Peano's axioms. Everything else is, mathematically speaking, either basically the same (e.g. Mikusinski's book) or is like returning to Aristotle's theory of the spheres for our understanding of astronomy (e.g. using synthetic geometry with straight edge and compass constructions to do geometrical algebra).

This is a mathematical question and as a mathematical question, the fact is that, if anything, multiplication as repeated addition is the original true mathematical basis for multiplying at all. It isn't even just another heuristic. It is how multiplication is originally defined, in the first place. Now, we can either go around talking about pedagogy and heuristics and cherry pick our famous mathematicians to cite or we can do the math. How ironic that these sorts of discussions inevitably revolove around anything but the actual math.

Re: #121
Joshua - you haven't given a matehmatical defintion of anything. A-n-y-t-h-i-n-g.
LOL!

Re: #121 second part
Joshua - yes the fact that Devlin flatly contradicts himself (more than once) is a good indication of the extent of his logic on this matter -- there isn't any. But then how could you be expected to notice?

Re: #145
Joshua - I'm not being fair to you. You haven't given any definition of anything, mathematical or otherwise.

"There's a triangle with a base five inches long, and it's two inches high, with the highest point being three inches in. What's the area of that triangle? 1/2 baseÃheight, in square inches. How can you describe that by repeated addition?"

Oh, but you can. That's sort of what you are doing when you integrate in calculus. On an infintesimal scale an infinite number of times. (Or finite/finite in the case of numerical integration)

Anon, #148: "Oh, but you can. That's sort of what you are doing when you integrate in calculus."

This was pointed out earlier:

As Adriann said, #45: "Multiplying real numbers isn't JUST repeated addition only because it includes the ADDITIONAL notion of a limit."

I resist the temptation to compare Reimann integration with Lebesgue integration.

I've found that even poorly performing middle school students can grasp the idea of a completed infinity as a number, and of (at least in a qualitative way) limits of series.

At which point they have no problem in finding that for any nonzero integer N, that N times Infinity = Infinity. This they can get at by either:

N x Infinity = N + N + N + ... + N [an infinite number of times] = Infinity

and

Infinity times N = Infinity + Infinity + ... + Infinity [N times] = Infinity.

So Multiplication as Repeated Addition, but not JUST repeated addition, still works for them in the extended integers, likewise in the extended reals. They are divided on Cantor's main results, however. Either because the hierarchy of cardinals is too abstract, or because I haven't tried well enough to teach it to the younger students.

My comment turned out a bit long, so I stuck it on my own blog. Note: I am neither a mathematician nor a scientist; I am merely intellectually curious.

One of the best books I have read on teaching arithmetic was Liping Ma's Knowing and Teaching Elementary Mathematics. It compares the teaching in US and Chinese schools. Basically, there is no one right way to teach arithmetic. It is a lot like the elephant as perceived by the infamous blind men. Different children will learn in different ways. The problem is that the TEACHER needs to have some understanding of the elephant, and, in China, the teacher was much more likely to understand the elephant than in the US.

Some children will learn about the elephant starting with its floppy ears, others with its trunk and others still with its great tree like legs. Similarly, they might best learn multiplication as repeated addition, measuring area or as a means of aggregation. It depends on the child, the phase of the moon and what they had for breakfast that morning.

Before we start trying to teach elementary school children arithmetic starting from set theory, we should consider that very few US math teachers actually understood what it means to "borrow" when doing subtraction. Liping Ma actually tested US and Chinese math teachers. The US result was pathetic. I'm not kidding. The Chinese teachers did much better, so they could better explain subtraction to their pupils.

It is interesting following this debate among the mathematically oriented. It would be fascinating to follow this debate among anthropologists. Is multiplication a means of calculating social status or a means of accruing social status? Quite a different question, and each answer leads to a different pedagogy.

Sure, most of what we learn is an approximation. Every one with advanced knowledge in a field looks down at the baser rungs with bemusement or scorn. On the other hand, even the experts sometimes forget the lower rungs. Didn't Linus Pauling forget that deoxyribonucleic acid was an acid in his attempt at ascertaining its structure? Look at how many people confuse the economic concepts of price and value. Of course, the wizards of Wall Street are relearning this distinction even now, and they should have known better.

Late to the party, but it is still on!

I don't think the post is fair to Devlin, he do proposes exactly what he wants to teach first. As superposition and scaling are important and easily demonstrated physical facts, I think he has a point. But in real life we would quickly revert to the simplest and quickest methods for a given purpose, as Mark alludes to.

And why shouldn't we? As opposed to some commenters I do think math is deepened by constantly broaden the use of its definitions and methods. It also points to connections to other sciences, and as shown above useful (for the sciences) such. Math is after all axiomatized to work according to the useful properties of numbers and algorithms that we find out.

I also think axiomatics should be introduced earlier than today; I had a hard time grasp the methods (if not the concept) when I first encountered it, then I had an equally frustrating time trying to junk it when not necessary (most any other theoretical endeavor, such as theoretical physics). But not as a means of learning the basics!

Science is emprical and math is a priori.

Chaitin makes a point of math being quasi-empirical. (AFAIU, along the lines that you can extend your knowledge of Chaitin's number by throwing dice.)

Me, I think as any science it is modeled on observable properties, on numbers and algorithms. So I dunno about how to go about testing any difference to empiricism.

@ Howard, #106:

I believe he is making the same mistake that I often do, in assuming that non-mathematicians ever really DO work in R; the real world operates with nothing more than Q, and sometimes, in maybe electrical engineering, Q[i] (the complex rationals).

I think you mean that our measurements are based on Q as we have measurement standards. But nature itself is continous, Lorentz invariance tells us this must apply on all scales.

So we are trying to approximate R (or C, if you like, whenever phases comes into play), and that and the fact that we must model measurements by probabilistic models rapidly moves us from Q into R by the used algorithms.

Coming so late into this, I figure no one will read it...
Don't mean to be glib or silly but 2 quotes come to mind, one actual, one fractured---

1) Real quote: "I have hardly ever known a mathematician who was capable of reasoning." Plato
Since I see myself as a mathematician, my wife would fully endorse this quote, whereby 'reasoning' translates to commonsense. If she were to read the first few comments above, she would probably close the computer exasperatedly and comment: "Yup, Plato was right. I'd love to hear how a mathematician or scientist would initially explain sex to their child!"

2) Fractured quote: "Now be fruitful and repeatedly add!"

On a serious note, Devlin like anyone who makes extreme comments, has generated a wonderful dialogue on this and many other blogs. In the end, does anyone really believe this discussion will lead to any change in how multiplication will be taught to children in Sep 2008? I am tempted to summarize the 100+ comments on this thread and categorize them but suffice it to say that I'm in the camp of multiplication of whole numbers as repeated addition is just fine. Giving children other models of multiplication to deepen their understanding is also just fine. As they develop, they will come to recognize the distinctions. I could say much more more here but I feel like I would be "repeatedly adding" to what has already been said.
Dave Marain

Alas, I too am a late entrant to this debate, and I don't have my own blog, so here goes nothing - maybe someone will find this on Google one day.

Perhaps I can sum up the discussion a bit as well as inject my own thoughts. It seems to me the point Devlin really wants to make is that multiplication must be viewed and taught as a concept 'sui generis' - a fundamental operation, not merely an extended version of addition or anything else. It simply won't do for school kids or anyone else to look at an expression like 3 x 2 and view it merely as a convenient shorthand for the sum 2 + 2 + 2. But apparently a lot of kids are taught this way and they carry that limited understanding right through to college level. If that's the case, I'd agree with Devlin that there needs to be a revolution in the way mathematics is taught at elementary level.

At the same time, Devlin seems to be overreaching somewhat when he insists that those who understand multiplication of integers as repeated addition have been taught the wrong thing. In order to grasp a concept you need to be able to model that concept. Unless one is to teach the real number system before expecting students to learn multiplication, one has to model multiplication using what the students know up to this point: namely positive integers and - perhaps - the relationship between lengths and areas of rectangles.

Any model for multiplication on positive integers, it seems to me, is going to look a lot like repeated addition, whether you write out the equivalent addition problem, put objects in identically sized groups, or draw dots in a grid. (The latter being the preferred approach in Courant and Robbins' classic "What Is Mathematics?", both in the original and the Ian Stewart revised edition.) All such models seem to me equally easy or difficult to extend conceptually to fractions (though the 'grid' model will call for use of squared paper). And without a doubt, all such models are going to suggest the inappropriate generalisation that multiplication makes things larger, because that's categorically true on the positive integers. In other words, all such models are "brittle metaphors" in Devlin's sense of the term.

So while Devlin is right that teachers need somehow to emphasise the special nature of multiplication and that it's not just shorthand for doing addition over and over, it may not really be possible to avoid suggesting at this level that multiplication is repeated addition. Unless, of course, you simply teach it by rote using the multiplication tables and pretend it doesn't bear any relation to addition at all; not an approach I'd recommend. And of course it's not actually wrong to suggest multiplication on the integers is repeated addition or that this concept permits an extension (of sorts!) to rationals and reals: it's quite true that Landau used this approach in his 1929 "Foundations of Analysis" which it seems was written precisely to explain the new rigorous 'modern approach' to calculus students. (The text of the Prefaces at least is online and a search readily turns this up.)

Perhaps, then, our only hope is to teach as many models for multiplication as possible and hope that the students take the hint. This, more or less, seems to be the nearest thing to consensus I've picked up from all the blog discussions to this point.

So much for multiplication. What I do find more problematic is Devlin insisting with equal force (including in his latest posting) that exponentiation is a 'sui generis' operation too and that it's wrong to teach it as repeated multiplication. In all college-level developments of real analysis I've seen, exponentiation is not fundamental in the same sense as the ring operations of addition and multiplication: rather there is an exponential function, and its inverse the logarithmic function, which can be used to define an operation f(a,b) = exp(b * log a), also denoted a^b. And a^b itself has often been previously defined using repeated multiplication!

For example, when I was an undergrad math major the prescribed text was Bartle and Sherbert's "Introduction to Real Analysis" (2nd ed, 1992). On this particular topic, their approach is as follows:

1. Exponentiation is defined inductively for positive integers n >= 2 and arbitrary real A as A^2 = A*A and A^(n+1) = A*A^n. A convention is adopted that A^1 = A and A^0 = 0.

2. The existence of real nth roots is established: for any positive A this is positive B such that B^n = A. The nth root is written as B = A^(1/n) and this is used to extend exponentiation consistently to all rational indices.

3. The function exp(x) is introduced as the unique function equal to its own derivative and satisfying exp(0) = 1. It is then shown that exp(Q) = e^Q for all rational Q, where e = exp(1).

4. Finally the function log(x) is introduced, and this motivates the new definition X^A = exp(A*log(X)) for all positive real X and all real A. This is called the 'power function with exponent A'.

So while I'd concede Devlin's point on multiplication (with reservations), I think he's genuinely got it wrong on exponentiation - or at least he's sailing against a major established current of thought in pure mathematics.

Ohh, c'mon that's like objecting to the explanation that a gay man is one who is primarily sexually to other men on the grounds that it could also mean a happy man.

The question isn't what the word multiplication is but what the concept that we refer to by multiplication when talking about the integers is and that is repeated multiplication.

#145 included: "what the concept that we refer to by multiplication when talking about the integers is and that is repeated multiplication.."

I'll give the benefit of the doubt and assume that Tru meant:

"what the concept that we refer to by multiplication when talking about the integers is and that is repeated ADDITION."

But that statement misses several of the points that others have made.

And it DOES matter what the word means in common use. It is NOT unconnected with The biblical command "Go forth and multiply" (Genesis 24:2). Which, poetically phrase, by ecclesiastical consensus, contains a hidden clause after the last word. It is commonly interpreted to read "Go forth and multiply" (like rabbits, else be condemned to eternal damnation.)

Which leads to Fibonacci's rabbits, and Malthus. Thomas Robert Malthus FRS [13 February 1766 - 23 December 1834] wrote that societies through history had experienced at one time or another epidemics, famines, or wars: events that masked the fundamental problem of populations overstretching their resource limitations:

"The power of population is so superior to the power of the earth to produce subsistence for man, that premature death must in some shape or other visit the human race. The vices of mankind are active and able ministers of depopulation. They are the precursors in the great army of destruction, and often finish the dreadful work themselves. But should they fail in this war of extermination, sickly seasons, epidemics, pestilence, and plague advance in terrific array, and sweep off their thousands and tens of thousands. Should success be still incomplete, gigantic inevitable famine stalks in the rear, and with one mighty blow levels the population with the food of the world."

To give a mathematical perspective to his observations, Malthus proposed the idea that population, if unchecked, increases at a geometric rate (i.e. 1, 2, 4, 8, 16, etc.), whereas the food-supply grows at an arithmetic rate (i.e. 1, 2, 3, 4, 5 etc.).

Which leads to discussion of exponential growth. And the Cult of The Singularity (tied to the Cult of Transhumanism and the Cult of Nanotechnology) and the Bad Math of "The Singularity is Near" by Ray Kurzweil, who literally misunderstands the mathematical term "singularity" and claims it in any growth curve, without ever suspecting Sigmoidal curves (i.e. logistic function).

Devlin's main statement doesn't even make sense. "Multiplication is not repeated addition". The phrase "multiplication" or "addition" is meaningless unless followed by "of integers" or "of rationals" or "of reals", or something like that. Multiplication on N={0,1,2,3,...} is defined as repeated addition. Certainly any definition of multiplication on N must used addition (I now suppress N), and I cannot conceive of a definition not reliant on repeated addition in some way. Even the geometric, nxm rectangle definition seems to rely on repeated addition when examined closely.

Multiplication RxR->R, say, cannot be talked about as repeated addition. But to restrict this function to define or explain multiply:NxN->N is hopeless, because any rigorous treatment of R, that defines (in one of the many possible ways) R and defines multiply:RxR->R, must *use* multiply:NxN->N.

Certainly, the binary function "multiply:NxN->N" got by repeated addition must be equal as a function NxN->N, to the function NxN->N defined in terms of repeated (this uses the induction axiom when made completely thorough) addition. This simply means that they get the same answers (same inputs, same outputs). I would go further: (1) any definition of multiply:NxN->N must use addition, and (2) any definition of multiply:NxN->N must use repeated addition.

I actually think that anyone who believes they understand multiply:NxN->N as primitive, or in terms not reliant on repeated (inductive) addition, is either lying to themselves, or is simply unaware of the subconcious thought processes going on when thinking about multiplication.

Re: #67

The multiplication by complex numbers can just as well be seen as scaling one complex number by another on the complex number plane and the area being scaled into another direction. I see no problem with multiplication as scaling

I personally think that calculus should be taught as philosophy in the third grade. There seems to be this gap between the science types who believe math is important, and the rest of the world; who thinks math is for geeks. Well, math is important, but it is for geeks; and the problem is philosophy (or rather, the lack of it). Algebra? Come on, everybody knows that's the bad guy in Aladdin, that ain't real; so when somebody like me says, "ya know, 2+2 doesn't necessarily equal 4 in the real world because we actually use algebra..." people look at me like I'm a... mathematician. ;)

Meaning, math is important; but it is not important to everybody, and to find those who find math to be important at a young age (and make the world a better place ;) ), putting the philosophy of math before the boring stuff (rote memorization) is a winner. Doesn't it seem that showing kids stuff like how Cassini found its way to Saturn instead of pimping the quadratic equation might be more like cool and less like geek? We don't have to tell 'em the numbers, we just have to tell 'em the numbers are in there; and they'll go find 'em. Kids ain't dumb, we teach 'em to be dumb; that memorization is better than creative thinking, and then we wonder why the world is full of lawyers. :)

He is wrong. Of course multiplication is repeated addition. You can repeat something half a time, or any fraction of a time. We to it all the time. For instance, repeat after me "I do solemnly swear...". When you say "I do" and do not finish, you have repeated something a fraction of one time. This is actually a semantic issue. Be precise about the type of repetition you mean and you can repeat a fraction of a time, and multiplication most certainly is repeated addition.

So, one mathematician says multiplication is not repeated addition in any situation whatsoever, while another pair says it is exactly that! Whatâs going on? Consider the second quote in contextâ¦
Hard drive recovery

"Actually, Devlin is wrong! If you refer to the fundamental theorems of natural numbers (based on the Peano axioms), multiplication is indeed defined as repeated addition! See theorem 28 at:
http://www.ms.uky.edu/~lee/ma502/notes2/node8.html

Derek.

Posted by: Anonymous | July 25, 2008 4:47 PM"

Nice post :) Thinking multiplication as repeated addition is enough for integers. So it also works for things derived from integers, such as rationals and reals. But no one really think about multiplication as repeated additions any more after encountering fractions as it's just too cumbersome. So repeated additions natually generallizes to rectangular areas.

Some pure mathematician may "natually" think multiplication as the second operation in rings. But it's actually better for children to learn enough concrete examples of rings before learning rings. If you're a pure mathematician and rings are too simple for you, consider that we don't teach children phonology before they can talk well, and we don't teach them music theory before they sing.