I've been getting peppered with requests to comment on a recent argument that's been going on about math education, particularly with respect to multiplication. We've got a fairly prominent guy named Keith Devlin ranting that "multiplication is not repeated addition". I've been getting mail from both sides of this - from people who basically say "This guy's an idiot - of course it's repeated addition", and from people who say "Look how stupid these people are that they don't understand that multiplication isn't repeated addition".
In general, I'm mostly inclined to agree with him, with some major caveats. But since he sidesteps the real fundamental issue here, I'm rather annoyed with him.
You see, the argument isn't really about multiplication, but about math education. The argument isn't really about whether multiplication is repeated addition - it's about whether or not we should teach kids to understand multiplication as repeated addition. And that's a tricky question, because the answer is both yes and noe.
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.
What's exponentiation? Is it repeated multiplication? Sometimes. And sometimes it isn't. Try to give me a simple definition of exponentiation, which is understandable by a fifth or sixth grader, which doesn't at least start by talking about repeated multiplication. Find me a beginners textbook or teachers class plans that explains exponentiation to kids without at least starting with something like "52=5×5, 53=5×5×5."
With respect to multiplication, it's the same question, only with even younger kids: how do you explain multiplication to a third grader? How can you start to tell a kid about 2×2=4 and 2×3=6 without showing them that 2×2 = 2+2, and 2×3 = 2+2+2?
Multiplication isn't really a simple thing. What mathematicians mean by multiplication is, roughly, one of the two fundamental operations over the field of real numbers. Outside of the realm of abstract math, multiplication actually has multiple meanings, which each work in different contexts. But they're all concrete applications derived from the fact that multiplication is the second field operation in the field of real numbers.
But how are you going to explain that two a third grader?
Just think of one of the classic word problems that every kid sees in second or third grade when they start doing multiplication. Every kid in class has three apples; so how many apples does the class have?
When you're using that problem, repeated addition makes excellent sense. It also matches the mechanics of what the kids are doing. So it's a good intuitive way to get them started on understanding multiplication. It's not the whole picture - but it's an initial intuition that provides some concrete handle to grab on to.
Of course, pretty soon, you have to break that intuition at least a little bit - because there are plenty of places where repeated addition just doesn't really make sense. Look at the figure over to the side. 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 base×height, in square inches. How can you describe that by repeated addition?
For the triangle, you can do a geometric explanation of multiplication. The two numbers being multiplied are the sides of a rectangle, and multiplying is creating the area inside the rectangle. You can use that intuition to explain the area of a triangle, by showing how to create a rectangle by cutting the triangle into pieces, and re-arranging them. That gives you a geometric intuition about multiplication.
But neither of those is particularly good for explaining how multiplication can tell you what 3/5ths of $25 is.
So the real question isn't "Is multiplication repeated addition?". The answer to that is "sometimes". The real question is "How do we introduce multiplication to children?"
Professor Devlin doesn't have a good answer for that - and in fact, he weasels out of answering it entirely, which really bugs me. After a long argument about how it's all wrong to teach kids to understand multiplication as repeated addition, and lecturing teachers on how the way that they're teaching is all wrong, he wimps out and says, in essense, "But I don't know anything about teaching, so I can't tell you the right way to do it. All I can do is tell you that you're doing it wrong."
So what are teachers supposed to do? Professor Devlin is very forceful in telling teachers what to do: the last line of his article is: "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.
You can't tell a teacher to change the way that they're teaching math without giving them any clue of what the right way to teach it is. What happens in a classroom if the teacher stops using repeated addition to explain multiplication? One of two things will happen. Either the teacher will switch to a different, and equally incorrect intuition about what multiplication means; or they'll do away with trying to provide any intuition at all.
The right answer is to say that simple multiplication can be understood intuitively in terms of repeated addition. Teachers should do their best to be clear that it's just an intuition, not the full meaning. Ideally, they should show multiple ways of understanding it, so that students understand that no one intuition about multiplication is the whole truth. But given a choice between teaching children no intuition, and teaching them a pretty good beginners intuition, I'll take the latter.
Comments
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.
Posted by: spudbeach | July 25, 2008 3:40 PM
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
Posted by: Daniel Litt | July 25, 2008 4:05 PM
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.
Posted by: Howard | July 25, 2008 4:13 PM
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.
Posted by: Matthew L. | July 25, 2008 4:22 PM
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.
Posted by: Daniel Litt | July 25, 2008 4:32 PM
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?
Posted by: Anonymous | July 25, 2008 4:33 PM
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.
Posted by: Mark C. Chu-Carroll | July 25, 2008 4:37 PM
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.
Posted by: Mark C. Chu-Carroll | July 25, 2008 4:42 PM
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.)
Posted by: Inna | July 25, 2008 4:42 PM
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.
Posted by: Dave | July 25, 2008 4:43 PM
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.
Posted by: jc | July 25, 2008 4:43 PM
(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.
Posted by: Anonymous | July 25, 2008 4:47 PM
"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.
Posted by: Anonymous | July 25, 2008 4:48 PM
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.
Posted by: Alex | July 25, 2008 4:49 PM
ugh - that last post (1011818) somehow lost my contact info on it. sorry.
Posted by: Joe Shelby | July 25, 2008 4:50 PM
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?
Posted by: Eamon Knight | July 25, 2008 4:50 PM
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.
Posted by: Joe Shelby | July 25, 2008 5:02 PM
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.
Posted by: --bill | July 25, 2008 5:03 PM
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.
Posted by: 6EQUJ5 | July 25, 2008 5:33 PM
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.
Posted by: Matthew L. | July 25, 2008 5:34 PM
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.
Posted by: Anonymous | July 25, 2008 5:46 PM
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.
Posted by: Jonathan Vos Post | July 25, 2008 5:48 PM
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
Posted by: Mark W | July 25, 2008 6:01 PM
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 ;-)
Posted by: Roberto Alsina | July 25, 2008 6:22 PM
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.
Posted by: jc | July 25, 2008 7:07 PM
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.
Posted by: Steve W | July 25, 2008 7:47 PM
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.
Posted by: Tony Jeremiah | July 25, 2008 10:01 PM
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.
Posted by: Jim RL | July 25, 2008 10:15 PM
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.
Posted by: JimV | July 25, 2008 11:30 PM
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
Posted by: nolrai | July 26, 2008 12:59 AM
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.
Posted by: efrique | July 26, 2008 3:03 AM
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!
Posted by: Sam C | July 26, 2008 5:15 AM
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.
Posted by: raiko | July 26, 2008 5:17 AM
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
Posted by: Donalbain | July 26, 2008 5:46 AM
Devlin has written a followup to the topic:
http://www.maa.org/devlin/devlin_0708_08.html
Posted by: aa | July 26, 2008 5:55 AM
OK.. that is wierd!
Posted by: Donalbain | July 26, 2008 5:58 AM
"But how are you going to explain that two a third grader?"
"to", maybe? Seems your brain was stuck on numbers for some reason. :)
Posted by: Colin M | July 26, 2008 7:36 AM
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.
Posted by: Volly | July 26, 2008 9:43 AM
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.
Posted by: Kate | July 26, 2008 10:28 AM
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.
Posted by: Jonathan Vos Post | July 26, 2008 11:08 AM
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.
Posted by: Michael | July 26, 2008 11:34 AM
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.
Posted by: Math Chique | July 26, 2008 12:33 PM
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?
Posted by: Jonathan | July 26, 2008 12:43 PM
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.
Posted by: Uncle Al | July 26, 2008 1:00 PM
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)
Posted by: Adrian | July 26, 2008 1:49 PM
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.
Posted by: Adrian | July 26, 2008 2:41 PM
ALWAYS START WITH THE WHOLE.
Addition is peated multiplication.
IT'S NOT THAT HARD FOLKS!
cheeze luweez.
Posted by: DD | July 26, 2008 7:07 PM
"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.
Posted by: Joshua Fisher | July 26, 2008 7:35 PM
I did Joshua. See, for instance, p14, Theorem 28 and Definition 6 of Foundations of Analysis by Edmund Landau.
Posted by: Adrian | July 26, 2008 7:42 PM
"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! :)
Posted by: ollie | July 26, 2008 8:19 PM
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.
Posted by: Joshua Fisher | July 26, 2008 10:49 PM
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 ...)
Posted by: JimV | July 27, 2008 12:09 AM
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.
Posted by: Joshua Fisher | July 27, 2008 2:57 AM
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.
Posted by: Joe N | July 27, 2008 10:57 AM
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.
Posted by: Joshua Fisher | July 27, 2008 1:24 PM
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.
Posted by: Adrian | July 27, 2008 2:07 PM
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.
Posted by: Joe N | July 27, 2008 3:08 PM