Via Mark Chu-Carroll, I just finished reading this article by mathematician Keith Devlin. He writes:
Let's start with the underlying fact. Multiplication simply is not repeated addition, and telling young pupils it is inevitably leads to problems when they subsequently learn that it is not. Multiplication of natural numbers certainly gives the same result as repeated addition, but that does not make it the same. Riding my bicycle gets me to my office in about the same time as taking my car, but the two processes are very different. Telling students falsehoods on the assumption that they can be corrected later is rarely a good idea. And telling them that multiplication is repeated addition definitely requires undoing later.
Reminds me of a joke.
A man facing a mid-life crisis sells everything he has to finance a trip to Tibet. He seeks the counsel of the High Lama, certain that the Holy One will tell him the meaning of life. His friends and family implore him not to do anything so foolish, but he ignores them, certain that he is doing the right thing. His travels to Tibet are a litany of trials. Bad weather, flight delays, passage through war zones, attacks by wild animals, but he perseveres. He knows the wisdom of the Lama is worth any sacrifice. Finally he enters the chamber of the great Lama and says, “Holy One! What is the meaning of life?” The Lama strikes a dramatic pose, and after a moment's thought says, “Life, is like a tree limb!” Then he falls silent. The man is horrified. Greatly annoyed he says, “After all the trials, the ridicule, the personal sacrifice, you tell me that life is like a tree limb?” The Lama gives him a puzzled look. He says, “You mean it isn't like a tree limb?”
That was my initial reaction to Devlin's statement. It's news to me that multiplication isn't repeated addition. It certainly is that, sometimes. Picking up where the last quote leaves off:
How much later? As soon as the child progresses from whole-number multiplication to multiplication by fractions (or arbitrary real numbers). At that point, you have to tell a different story.
“Oh, so multiplication of fractions is a DIFFERENT kind of multiplication, is it?” a bright kid will say, wondering how many more times you are going to switch the rules. No wonder so many people end up thinking mathematics is just a bunch of arbitrary, illogical rules that cannot be figured out but simply have to be learned - only for them to have the rug pulled from under them when the rule they just learned is replaced by some other (seemingly) arbitrary, illogical rule.
Pretending there is just one basic operation on numbers (be they whole numbers fractions, or whatever) will surely lead to pupils assuming that numbers are simply an additive system and nothing more. Why not do it right from the start?
Why not say that there are (at least) two basic things you can do to numbers: you can add them and you can multiply them. (I am discounting subtraction and division here, since they are simply the inverses to addition and multiplication, and thus not “basic” operations. This does not mean that teaching them is not difficult; it is.) Adding and multiplying are just things you do to numbers - they come with the package. We include them because there are lots of useful things we can do when we can add and multiply numbers. For example, adding numbers tells you how many things (or parts of things) you have when you combine collections. Multiplication is useful if you want to know the result of scaling some quantity.
I still don't see the problem. When teachers tell elementary school students that multiplication is repeated addition, they are not making a philosophical statement about the nature of multiplication generally. They are giving instruction on how to evaluate problems involving the multiplication of natural numbers. The statement “Multiplication is repeated addition,” captures something very important about multiplication, and it is something that young children, not yet ready for abstract reasoning, can understand.
If there are any elementary school mathematics teachers reading this, I'd be curious to know if Devlin's fear has any merit. Do students feel they have been lied to upon learning that multiplication of fractions is different in fundamental ways from multiplication of natural numbers? Do students come away from their early lessons thinking that numbers are just an additive system? I'd be very surprised if that were the case.
If your whole world is positive integers, then you will never go wrong thinking that multiplication is repeated addition. When you enlarge your world, why should it be confusing that the old concepts have to expand as well? Telling students that multiplication of natural numbers can be understood in terms of addition in no way suggests that it is not also a full-fledged operation in its own right, or that it serves different purposes from addition.
So how do we answer the question in the title of this post? Abstractly speaking, multiplication is one of the basic operations that define the algebraic structure known as a “ring.” Rings are required to have two binary operations. One of these operations is called addition, and it is required to satisfy certain axioms. Specifically, we require that addition be commutative, associative, possess an identity (think 0 in the integers), and possess inverses (think negative numbers, again in the integers). Multiplication is a second operation that is required only to be associative, and to interact with addition via the distributive property.
The mechanics for carrying out these operations will depend on the specific ring you are studying. Multiplication of integers is one thing, matrices something different, and continuous functions something else still.
Regardless, if we are to give a definition of what multiplication really is we would have to say that it is an abstract binary operation on a set that satisfies certain axioms.
There was a time when people seriously thought it was appropriate to explain this sort of thing to young students. Devlin makes it clear he has no sympathy for that approach:
Teaching a class of elementary school students about axiomatic integral domains is probably not a good idea! This column is not a rant in favor of the “New Math”, a term that I use here to denote the popular conception of the log-ago aborted education reform that bears that name.
Quite right.
Having read Devlin's column several times, I still have no idea what teachers are supposed to be telling their students. Devlin seems to have had some teaching experiences very different form my own:
Of course, there are not just two basic operations you can do on numbers. I mentioned a third basic operation a moment ago: exponentiation. University professors of mathematics struggle valiantly to rid students of the false belief that exponentiation is “repeated multiplication.” Hey, if you can confuse pupils once with a falsehood, why not pull the same stunt again? I'm teasing here. But with the best intentions of drawing attention to something that I think needs to be fixed.
Really? Exponentiation is repeated multiplication, so long as we are talking about positive integer exponents. Personally I have never found it difficult to explain to students that if you wish to discuss non-positive-integer exponents, you have to alter your notion of exponentiation.
I think what Devlin is getting at here is that there is a distinction between understanding conceptually what mathematics is all about on the one hand, and being able to carry out the mechanics of solving actual problems on the other. When teachers say, “Multiplication is repeated addition,” or “Exponentiation is repeated multiplication,” they are addressing the mechanical aspects of the subject. Devlin's concern, I think, is that these mechanical considerations sometimes transgress their boundaries, and get improperly applied to the conceptual side of things. This makes it difficult to give students a broader understanding of what these operations are really all about.
I do not believe this is correct. I think the reason students have so much trouble with abstract mathematics has nothing to do with conceptual barriers we put in their way by poor arithmetical pedagogy. Rather, they find these abstractions difficult because they are difficult. They require a style of thinking that simply does not come naturally to people.
Ideally we want to balance mechanical and conceptual understandings of mathematics. Both are important, and both have a big roll to play in a proper mathematical education. But after many frustrating years of trying to teach calculus to freshmen, I've come to the conclusion that conceptual understanding is vastly overrated. Give me robots who can carry-out basic arithmetical and algebraic operations with quickness and accuracy and I'll be happy. If they have strong mechanics there is a platform for teaching the concepts. Without strong mechanics there is little hope.
I don't know a whole lot about how mathematics gets taught in elementary and middle school. The bits and pieces I've picked up on the street suggest to me that things have gone too far in the direction of conceptual understanding, without enough emphasis on mechanics. The irony is that in calculus classes things have probably gone too far the other way. We spend a lot of time with the mechanics of cranking out derivatives, learning techniques of integration, and grinding away at Taylor series, which makes it difficult to make students take a big picture approach to the subject.
For elementary and middle schol students I want proficiency in arithmetic and algebra, even if that proficiency is entirely at the level of mindless repetition. (I would prefer deep conceptual understanding too, of course, but I will take what i can get.) For college students not specifically majoring in mathematics or science, I want them to be able to explain what calculus is all about, what sorts of problems it addresses, and what it's big ideas are, but I don't care so much whether they remember the formula for integration by parts. Likewise for other branches of higher mathematics.
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Saying 'repeated addition' seems to work at first, but bogs down when the multiplier is something like 2.5, just as exponentiation seems like repeated multiplication, until the exponent is 'e', for example.
I was schooled in arithmetic-by-rote and had no real understanding of mathematics until after college, when I discovered by accident that math had power. After dealing with some scaling laws from physiology and engineering, it occurred to me that multiplication is simply direct scaling, whether up or down, and that exponentiation is dimensional scaling, again either up or down.
but hang on, multiplication by fractions is simply multiplication followed by division. or rather fractions are a representation of inconvenient (or unfinished) division. it still works out to repeated addition (or subtraction) either way. I'm not sure how the abstraction doesn't still fit...
well if you're going to bring 'e' into it...
This whole crapshoot mathematical thing started with introducting "negative" numbers, y'all! Yeah! I *still* want someone to hand me -1 dollar bills! Ain't no such thing!!! So THERE!
Of course, if anyone else has read Isaac Asimov's little screed on numbers (1/2 piece of chalk, anyone), you'll know where I'm coming from. Even concepts like "number" has to be modified and expanded to include negatives, rationals, reals, ..., and heaven forbid imaginaries or transfinites, blah blah blah.
heh.
The Bible does not say, "Go forth and add repeatedly" - though that is what a lot of people seem to have been doing...
It seems to me Devlin is exponentiating a mountain out of a mole hill, here.
I was following all of the thrashing around in the comments following Mark's post on Good Math, Bad Math.
Personally, I don't really see a problem. I recall being taught early on that multiplication was repeated addition. That was an easy way for me at the age of five to conceptualize and understand it. When we started getting to fractions and other "rational numbery" kinds of things, the teacher only had to say once that, "OK. You were taught early on that multiplication was repeated addition. That's true in some cases and not true in other cases." And off we went into more complexity. It did not confuse me in the least. Quite the contrary. I was able to grasp a difficult concept very early and move forward.
All you need to do is make sure that future teachers know what was said by past teachers so that extending multiplication to scaling and other concepts is really that--an extension and not a complete, abrupt redefinition.
Multiply one bite of an apple by the number of bites needed to eat it and you still get one apple, except it has no future. Such is the meaning of life.
I think Devlin's point is that addition and multiplication are fundamentally different operations when defined on the reals, and that these operations have an interesting relationship when restricted to positive integers. Devlin seems to me to be saying that teachers, when defining multiplication, focus too much on this relationship between addition and multiplication, and not enough on situations in which multiplication is not interpreted as repeated additon. Devlin explicitly mentions geometric scaling as an aspect of multiplication that is not repeated addition--I get students in college who don't know that `making a triangle three times as big' is the same thing as `multiply the lengths of each side by 3'.
The bits and pieces I've picked up on the street suggest to me that things have gone too far in the direction of conceptual understanding, without enough emphasis on mechanics.
I have never had that impression myself. It seems to me that most elementary school teachers think that math is *all* mechanics. They don't understand what real math is...it's just something you use to balance your checkbook. Good students are those who can memorize tables without actually understanding the underlying concepts such as grouping for multiplication. I went through this as a kid (being constantly told I was bad at math because I'm not good at computation). Yet when I got to high school geometry and we were doing proofs, I came out of the class with a perfect score. Over time I practiced and became better at computation, but I actually started getting a good handle on the conceptual aspects of elementary math when I was homeschooling my own son...which we were doing, in part, because his teachers kept saying he was bad at math.
The history of mathematics is generalization of concepts, sometimes with modification. Integration isn't the same as addition, either. But neither unrelated to it.
BTW, if I were asked to make a triangle three times as big, I might multiply the lengths of each side by sqrt(3). ;-)
meh. semantics.
No one has brought up that multiplication is addition abstracted into the logarithmic domain. Am I the only one who uses a slide rule? Exponentiation (one step further) is multiplication in logs, or addition in log-log. Simple.
But then, I recently had trivial cause to figure out how much gold is a billion dollars (just under a 4 foot cube, 61.9 cubic feet). But the (somewhat innumerate) guy that I was discussing it with was amazed that a mere mortal like myself could work that out off the cuff. Except for the cube root at the end, the rest was just multiplication.
The real problem is that children are never taught the value of abstraction when they are still young enough to intuit such relationships. There is too much identity training: The idea that a word is an object, that a number symbol is the quantity, that a word is the deity.
OK, this probably will make me look stupid, but ...
I feel I'm unversed enough in mathematics to comment. I liked math just fine through high school, and I took calculus in college and thought it was way cool, and, had I paid more attention, I might have actually learned it.
The thing is, I never thought of multiplication as anything but repeated addition, and I never thought of exponentiation as anything but repeated multiplication, and throwing in fractions and negative numbers and even irrational numbers never made me think that the underlying concepts had failed to work.
In fact, it made *perfect* sense -- and I guess I'm missing something, because it still does.
Not just adding, but counting ...
If you add 1 pie to 1 pie, you have two pies, so you count 1 pie twice: 1 x 2 = 2.
If you add 1/2 a pie to 1 pie, you have 1.5 pies, so counting 1 pie one-and-a-half times is 1 x 1.5 = 1.5. If you count a half-a-pie twice, you have 1 pie, so 2 x 0.5 is 1. I fail to see how using fractions is NOT repeated addition.
Negative numbers: If I owe you 3 pies a day, I owe you 6 pies after 2 days. -3 x 2 = -6. In what way is this not repeated addition (assuming subtraction is a subset of addition)?
Fractions times fractions: If you stop halfway through counting half-a-pie, you have a quarter-of-a-pie, so 1/2 x 1/2 = 1/4.
I thought that's why the x symbol is called "times": You perform the operation that many times.
MarkCC used the example:
I don't understand how you couldn't. Stop halfway through adding 5 twice. (Or stop halfway through adding 2 five times.) (Or add 5 twice and give me back half.)
So what am I missing?
This has already been debated ad nauseam over at Mark's blog, but...
Multiplication is repeated addition. This isn't just some useful approximation used to teach young children. In the Peano axioms, multiplication is defined recursively as repeated addition. That's where it all starts. Now, all sorts of subtle and deep properties flow out of that simple definition, but that doesn't somehow change the fact of what it is.
When we move from the natural numbers to other domains, the definition gets generalized, but it is still at its core repeated addition. How could it not be? As we move further away from its origins, it may become useful to conceptualize multiplication in other ways, but that doesn't change where it comes from.
Multiplying by fractions is repeated addition. If I want 3/8 of something, I divide it into 8 equal pieces and then repeatedly add 3 of them together. This is how fractions should be taught. Too often students see operations with fractions as just formal manipulations without understanding what those manipulation mean--emphasizing the 'repeated additions' part would eliminate a lot of that lack of comprehension.
Multiplication by real numbers is still based on repeated addition--it is the limit of a sequence of repeated additions. We don't typically think of it in those terms, but it's important to be able to drop down to that level of detail when needed. And so on.
I don't believe that students feel like they've been lied to each time the operation gets generalized; that's just downright silly. If anything, it's a good life lesson that everything is more subtle and complicated than it first appears. This phenomenon is certainly not limited to mathematics.
I do have one little quibble with Jason's description of what multiplication really is, as the second operation for a ring. This is simply a different meaning of the word. Mathematicians overload the meanings of common words all the time, but in this case the meanings are so close that it's easy to confuse them as being the same thing. For this definition, multiplication may or may not be repeated addition, depending on the ring in question. But by the time the student is mature enough to learn about the algebraists' notion of multiplication, the idea that the word "multiplication" sometimes refers to operations that are not repeated additions should not be too troubling.
I teach math at the Middle School level. My approach has always been to tell my students that multiplication is simply a faster way of counting, and that all arithmetic is a shortcut for finding a faster result.
To me, the math I teach is a layering of new skills and concepts on top of old ones. Students are certainly bright enough to figure this out. By the time they reach this level, they have experienced this in all disciplines. I can show them how fraction multiplication is an extension of repeated addition, but quickly becomes more than addition can handle. It's a mini-lesson on life: things become more complex and require new skills.
I can recall 4th grade math. There were actually some who resented the fact that there was more to multiplication than basic facts. It would be funny if I couldn't see where these guys were headed.
What Devlin seems to be missing is that multiplication on Natural numbers is repeated addition -- by definition. At least, according to Guiseppe Peano. When teaching to school children who's arithmetic domain is already restricted to natural numbers (or even positive integers) there is no loss of generality or specificity to teach it as repeated addition.
When you move to rationals, reals, and negative numbers, of course you need to introduce new rules and methods, simply because multiplication is more complex on these more complicated notions of "numbers".
What's the problem again?
Bill: you say: "I get students in college who don't know that `making a triangle three times as big' is the same thing as `multiply the lengths of each side by 3'"
Funny, if we look at multiplication in terms of area, and we know that the area of a triangle with base "b" and height "h" is:
A = (1/2)bh
then we make the AREA of the triangle 3 times as much if we multiply both base and height by the square root of 3.
b' = b(sqrt(3))
h' = h(sqrt(3))
A' = (1/2)b'h' = (1/2)b(sqrt(3))h(sqrt(3))
= (1/2)bh (sqrt(3) (sqrt(3) = 3 (1/2)bh
= 3A.
If you want a slice of pizza three times as big, don't you want three times as much pizza?
That kind of carelessness with terms such as "three times as big" is part of what confuses so many students who suffer through bad math teachers.
I refer you also to the book:
Twice As Less: Black English and the Performance of Black Students in Mathematics and Science,
by Eleanor Wilson Orr, W. W. Norton & Company, October 1997, Paperback: 242 pages
ISBN-10: 0393317412
ISBN-13: 978-0393317411
I think it may be pedagogically useful when introducing multiplication to say that it is equivalent to finding the area of a rectangle. The links to repeated addition and how you handle non-integers should then be fairly apparent visually.
I don't know, Jason. There's an analagous situation in science teaching, where pretty much every year you're told that what you learned the previous year wasn't quite true and this is how it really works. A combination of that approach and extra-curricular reading beyond my then understanding of the scientific method caused me to more or less lose faith in science at the age of 15 or so. I had previously been a total maths/science student, but I went on to study English literature at university, only taking a maths A-level. I didn't really get back into science until half way through my degree, by which time it was too late to do anything about it, career wise.
Patrick said,
I agree. Aren't there bigger problems worth addressing? We have a whole metric bagful, I'd say. Pick your own favorites. What about the parents and teachers who keep saying that girls can't do math — shouldn't we be spending our energy taking an evidence-based look at the propagation of such negative stereotypes? What about iron deficiency anemia, which has been implicated in causing irreversible cognitive impairment and for which children from poor families are a high-risk group?
Part of this problem, it seems to me, is that some teachers define multiplication as repeated addition without any suggestion that there might be more to it than that. Saying, `multiplication is repeated addition; that's all it is,' and then later teaching them `multiplication is also other stuff' seems much less effective than introducing multiplication by saying `multiplication and repeated addition give the same results; there are some other ways to think about multiplication, though.'
I regard the Peano's axioms stuff in the comments as a red herring; except for folks in logic and foundations, I don't know any mathematicians who do proofs from Peano's axioms, nor do I know many mathematicians who think about concepts in terms of the derivation of the concept from Peano's axioms. I think of multiplication as a scaling/rotation and addition as translation, 'cause my field is complex analysis--multiplcation by 1 + i is, to me, a scaling by root 2 and a rotation by pi/4. I never think of multiplication by 1 + i in terms of Peano's axioms, and I would be very surprised if any mathematican focusing on complex analysis did.
And, for Jonathan Vos Post above, I didn't include the context, unfortunately: given three sides of a triangle, one of whose sides is 6, and a similar triangle, one of whose sides is given as 18 (and there's a picture of the two triangles, properly labelled), what are the other two sides? In any case, I think your comment about `bad math teaching' is exactly what Devlin is getting at--telling children that multiplication is ONLY repeated addition, without suggesting that there are alternative models for multiplication, is bad math teaching.
Who learns anything by starting at the most general...ie a general description of multiplication. You start with a small example like repeated adding and work up from there. It's not a problem. You don't have to unlearn to incorporate another domain. I don't remember having to unlearn how to multiply numbers when I learned to multiply vectors.
Next I suppose we will have to teach someone how to play pool by discussing how the wave function of the ball is going to spread out across the table. (nod to G Gamow)
bill: no implication intended as to your own teaching.
"we will have to teach someone how to play pool by discussing how the wave function of the ball is going to spread out across the table."
Quantum Soccer [by Greg Egan]
http://gregegan.customer.netspace.net.au/BORDER/Soccer/Soccer.html
This page contains an interactive applet, which will only be run if your browser supports the Java programming language. The applet is about 32 K, so it will take some time to load.
In the game of Quantum Soccer, the aim is to shape the wave function of a quantum-mechanical 'ball' so that the probability of it being inside one of the goals rises above a set threshold. This is achieved by using the motion of the players to alter the energy spectrum of the wave function: when a player moves across the field, the energy that this action provides (or absorbs) enables transitions between certain modes of the wave function. The pairs of modes involved depend on the player's velocity; the exact rules are spelt out in the mathematical details, but it's easy to experiment using trial and error.
But --bill, but why do you have to multiply the other sides? You've added 12 to one side of the triangle, why don't you add 12 to the other side of the triangle? (Or you've raised one side of the triangle to the 1.61314719th power, why not raise the other side to the 1.61314719th power?)
That is, you have to teach the students that scaling means multiplication, and not addition or exponentiation or something else. You also have to teach that if you have two similar (in the mathematical sense) shapes with different sized sides, that is a scaling. And then per J.V.P., you have to teach them that if you scale lengths, that's a very different thing than scaling areas or volumes, so the area of a triangle with side of length 18 is not simply three times the area of a similar triangle with a side of length 6.
You only get part of the way with teaching people that mathematics = scaling. In addition, you now have the problem that Johnny can't figure out how many apples Mrs. Smith needs to bake 12 pies, if each pie needs 6 apples (like he could when multiplication was repeated addition). And yes, you could teach him that he can use multiplication as a "trick" for that case, with Johnny's problem being inadequate education. But just flip the argument around and you can equally well apply it to teaching the scaling triangle case as a "trick".
All of this misses the point of the original article, though. Reading the follow on article (http://www.maa.org/devlin/devlin_0708_08.html), Devlin makes it abundantly clear that he opposes *any* rationalization of multiplication. 'Multiplication is what it is,' and students should understand it as a deus ex nihilo, without resource to filling in the blank after "multiplication is _____" (e.g. "repeated addition"/"scaling")
Arrrggg!
"You only get part of the way with teaching people that multiplication = scaling."
And I previewed it several times too ... (grumble, grumble)
Surely addition as well as multiplication depend upon the objects, right? Multiplication on objects in N is just repeated addition, just as(per Peano) addition is mth successor of some number n. But what about other objects? If you look at the set of all the possible subsets of a given set, 'addition' is the finite difference of two subsets(the union of the two subsets less their intersection.) And 'multiplication' is simply the intersection of two subsets. The problem seems to be that when one starts to multiply fractions, they sort of 'look like' something already familiar, that is, it is not that multiplication as an operation is too different, but that the operands are too alike.
Yeah, putting in a few weasel words might be a good idea: "multiplication can be thought of as repeated addition to start out with"... or something on those lines.
However, at the initial ZF &c. VonNeumann integer construction: isn't multiplication essentially defined as repeated addition?
a*0 = 0
a*(b) = (a*b)+a
Okay now, teach the strengths and weaknesses of multiplication as repeated addition, but only as theory, not as fact; then let the children decide.
I recall being taught about some "number line"--but it was not pointed out that the numbers need not be spaced on a linear scale. That could confuse a lot of kids too.
My recollection of grade school is that the breakdown of the "repeated addition" notion came when it became time to multiply two negative numbers and get a positive. I remember visualizing it as repeated addition of 1-dimensional vectors, with the negative sign as a direction-reversal operator--except that since we hadn't yet vectors, I couldn't figure out how to put the concept into words to explain it to any of the other kids.
However, I don't think that it is a big problem. In many ways, it follows the historical development of mathematical concepts as generalizations of practical computational operations. I think that often the history of a concept provides a bottom-up approach that tracks rather well the easy road to comprehension, as compared to the top-down approach of starting with an abstract generalization and then developing the specifics.
Bill's got it right.
I really don't get the obsession with Peano. The end-result of running an operation (which is really "defined" beforehand) through the hurdles of induction is that either (a) the operation applies to all natural numbers or (b) it doesn't.
So, x(y + 1) = xy + x applies to, or works for, all natural numbers. Great. No one argues about that deft bit of symbolic reading comprehension. But do you want to have that equation tattooed on your arm to express your belief that repeated addition is the very essence, the very core, the very foundation, the absolute MEANING of multiplication--on ANY domain--for now and evermore? Me, not so much.
The "historical development" is irrelevant. By that logic, we should start out calendar instruction by teaching kids the Julian calendar. (And then we could "extend" it to the Gregorian, right?)
Multiplication and repeated addition are not one and the same--not on the positive integers, not "conceptually," not "extensionally," not "intentionally," not at all.
They get the same results on the positive integers, and we, as trained adults, easily blend them together in our minds. Still not the same.
""Arrrggg!
"You only get part of the way with teaching people that multiplication = scaling."
And I previewed it several times too ... (grumble, grumble)"""
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