Devlin's Angle

January 2011

What Exactly is Multiplication?

After my three articles on the nature of multiplication in 2008 (June, July-August, September), I promised myself to leave the topic alone for some years. In fact, I never set out to write that series at all. The first one was in response to the unexpectedly widespread reaction I received to what I thought was a fairly innocuous, throw-away comment I had made at the end of an earlier column. In that initial posting, I had noted that it was not a good idea to tell students that "multiplication is repeated addition." It turned out that large numbers of teachers thought it was just that, so I was driven to post what ended up as three columns, each one longer and more in-depth than its predecessor, explaining why the "multiplication is repeated addition" meme (MIRA) is not only wrong but an educationally dangerous one to propagate. The danger was demonstrated in dramatic form by the large volume of correspondence I received, and the plethora of public blog-threads that ensued, which showed that a great many citizens fervently believe that MIRA.

Why do I say danger? "After all," some correspondents said, "even if MIRA is wrong, does it really matter if people believe it? As long as they can use a calculator properly and get the right answer, a false belief does no harm." My answer to that is twofold. First, as an educator I have an ethical position. Those of us who teach mathematics, at any level, simply should not be in the business of spreading falsehoods. Sure, we sometimes tell "less than the whole truth" in order to provide our students with a manageable path toward mastery of what can be difficult concepts. But when we do so we have an obligation (1) not to tell a blatant falsehood, and (2) leave open the doorway to later refinement, extension, or other modification when the student progresses further. The trouble with the MIRA story is that, as research shows, many students are left with the firm, but false belief that multiplication actually is repeated addition. Knowing that, we should stop telling that particular story.

The second part of my answer is that in today's world we are faced with a great many decisions that depend upon an understanding of quantity. Some of them are inherently additive, some multiplicative, and some exponential. The behavior of those three different kinds of arithmetical operation differs dramatically, and thinking, say, additively when the problem is multiplicative (or even worse exponential) is likely to lead to some poor decisions that in many cases really are dangerous. To give just two examples, poor numerical thinking about risk can lead to an unnecessary expenditure on airline security, as I pointed out in last month's column, and a lack of appreciation for exponential growth can blind otherwise smart people to the catastrophic dangers of global warming. (I am making a mathematical point here; there are many other factors involved.) It is a fact of life that many people will go through life without being able to do mathematics. When they need the services of a mathematically able person, they can surely find someone. But in a world built on numbers and arithmetic as much as on words and language, for our students to reach adulthood without understanding what the four basic operations of arithmetic are, for heavens' sake, seems to me to be a situation that those of us in mathematics education should not accept.

Though the initial furor my original postings generated seems to have died down, I still receive emails from teachers and parents along the lines of "Okay, I see what you are saying, but can you then tell me exactly what multiplication is?" and from teachers who ask me to suggest how they should teach multiplication to young children. This column is in response to those requests.

By and large, my replies to those teachers who have written to me has been from the same perspective I adopted when I wrote my original three columns. I am a professional mathematician, but not a trained K-12 teacher. My expertise (and my credentials) are in mathematics, not in teaching. As a professional mathematician, I can (and I believe should) provide advice on the mathematics that is taught in schools, and constructive critiques of the way it is taught, but I do not think it is appropriate for me to suggest how to teach. Others are far more expert at that than I am. (I think it is no accident that the majority of professors of mathematics education are former teachers. It is hard to teach others how to do something you have not done yourself.)

Usually, I would refer my correspondents to two books that I myself found very useful in informing myself of issues of mathematics education. One is Adding It Up: Helping Children Learn Mathematics, authored by the Mathematics Learning Study Committee of the National Research Council, and published by the National Academies Press in 2001. The other is Terezina Nunes and Peter Bryant's excellent 1996 book Children Doing Mathematics. There are other books, and a lot of published research articles on the subject. For instance, in 1994, Guershon Harel and Jere Confrey edited a mammoth volume (414 pages) titled The Development of Multiplicative Reasoning in the Learning of Mathematics that contains a wealth of research findings, insights, and useful advice. (All three books are available on Google books.)

The problem with my response is that teachers rarely have the time to wade through a volume of several hundred pages, written primarily for the professoriate. I knew it, but they were the best sources I was familiar with.

What I can (and will) do here is try to explain my own understanding of multiplication. In some cases, I suspect that is what my correspondents were asking for. But in my replies to them, I never did more than make a few vague remarks about "It's based much more on the real-world concept of scaling than on addition." I was reluctant to go further because none of us really knows how we ourselves learn or understand mathematics, let alone how others do so or might find it profitable to do so. As a professional mathematician of many years standing, my understanding of multiplication may be different from that of others, and moreover aspects of it may be hard or impossible to convey to a child learning math for the first time. Having never taught at the K-12 levels, I would not know. The fact is, multiplication and multiplicative reasoning are complex and multifaceted. (That why the Harel-Confrey book I cited above fills 414 pages.)

Interestingly, the complexities of multiplication almost never arise in the daily activities of the professional mathematician. Indeed, a mathematician who has not reflected on how the subject can - or should - be taught at school level may not be aware of those complexities, though will be able to appreciate them at once the moment they are pointed out. (That was certainly my situation for much of my career.) For the mathematician, multiplication is an abstract, binary function on numbers (and other abstract objects) whose behavior is specified by axioms. We never ask the question "What is it?" nor do we seek to define it in terms of "more basic" functions. (I note in passing that at the level of formal, axiomatic mathematics, multiplication simply cannot be defined as repeated addition, since the latter is not a well-defined function, rather it is a meta-schema outside the axiomatic framework.) We just use the function in the way specified by the axioms.

My concept of multiplication

The concept I have of multiplication when I am working within mathematics is operational rather than ontological, and is built on the axiomatically defined, abstract binary function that constitutes one of the two fundamental operations in a field (or more generally a ring). I know the properties of multiplication, both obligatory (such as associativity) and conditional (such as commutativity), how it relates to other functions (e.g. distributivity), and whether or not a particular instance has an inverse. I am comfortable dealing with it. I don't ask what it is; all that matters are its properties.

I realize that my professional's concept of multiplication is abstracted from the everyday notion of multiplication that I learned as a child and use in my everyday life. But my abstract notion of multiplication misses many of the complexities that are part of my far more complex mental concept of multiplication as a cognitive process. That is the whole point of abstraction. Though many non-mathematicians retreat from the mathematicians' level of abstraction, it actually makes things very simple. Mathematics is the ultimate simplifier.

For instance, the mathematician's concept of integer or real number multiplication is commutative: M x N = N x M. (That is one of the axioms.) The order of the numbers does not matter. Nor are there any units involved: the M and the N are pure numbers. But the non-abstract, real-world operation of multiplication is very definitely not commutative and units are a major issue. Three bags of four apples is not the same as four bags of three apples. And taking an elastic band of length 7.5 inches and stretching it by a factor of 3.8 is not the same as taking a band of length 3.8 inches and stretching it by a factor of 7.5.

In fact, the nature of the units is a major distinction between addition and multiplication, and one of several reasons why it is not a good idea to suggest that multiplication is repeated addition, even in the one case where repeated addition makes sense, namely when you are dealing with cardinalities of collections. With addition, the two collections being added have to have the same units. You can add 3 apples to 5 apples to give 8 apples, but you cannot add 3 apples to 5 oranges. In order to add, you need to change the units to make them the same, say by classifying both as fruits, so that 3 fruits plus 5 fruits is 8 fruits. But for multiplication, the two collections are of a very different nature and necessarily have different units. With multiplication you have a multiplicand (written second) multiplied by a multiplier (written first). The unit for the multiplier has to be sets of the unit for the multiplicand. For example, if you have 3 bags each containing 5 apples, then you can multiply to give

Note how the units cancel: BAGS X APPLES/BAG = APPLES

In this example, there is a possibility of performing a repeated addition: you peer into each bag in turn and add. Alternatively, you empty out the 3 bags and count up the number of apples. Either way you will determine that there are 15 apples. Of course you get the same answer if you multiply. It is a fact about integer multiplication that it gives the same answer as repeated addition. But giving the same answer does not make the operations the same.

The MIRA fallacy becomes very apparent when you consider my second example, where I take an elastic band of length 7.5 inches and stretch it by a factor of 3.8. The final length of the band is 28.5 inches. But what are the units? What goes after the number 3.8 in the calculation

[3.8 - - -] x [7.5 INCHES] = 28.5 INCHES ?
The answer is nothing. It has no units. In this case, the 3.8 is a dimensionless scaling factor.

Incidentally, even when the initial length of the band and the scaling factor are positive integers, it makes no sense to view this example as one of repeated addition. If I take an elastic band of length 7 inches and stretch it by a factor of 3, its final length is

[3] x [7 INCHES] = 21 INCHES
But I did not take 3 copies of a 7 inch band and join them together (addition), rather I scaled (stretched) the 7 inch band by a factor of 3.

What about when you use multiplication to compute the area of a rectangle? If the rectangle is pi inches by e inches (where e is the base for natural logarithms), then its area is

[pi in] x [e in] = pi.e
or, in approximate numerical terms, 3.14in x 2.72in = Again, notice the units. (Note too that repeated addition is not going to get you very far with this example.)

There are other applications of multiplication where the multiplier and multiplicand have different domain interpretations, but since I do not want to write a 414 page column, I'll leave it at that with the hope that you get my point.

So what is my mental conception of multiplication? It's a holistic amalgam of all the above and several variants I have not listed. That's why I say multiplication is complex and multi-faceted. The dominant mental image I have is very definitely the continuous one of scaling, and I see all the others in terms of that. This means that my conception of scaling within this context is a very general one, that encompasses examples like my bags of apples. I can view the computation "3 bags each containing 5 apples gives 15 apples altogether" as "scaling" a bag of 5 apples by a factor of 3. In my experience, acquiring the concept of multiplication amounted to creating this mental amalgam - the amalgam that is my concept of multiplication.

I do not know how or when I acquired this scaling-centric concept of multiplication, and cannot really do it justice in words (unless I simply listed all its many facets), but I've had it as long as I can remember and it definitely is a single, holistic concept. To me, that concept is what multiplication is. As such, it is a numerical operation that corresponds to a very general form of scaling. Was I taught it, or did I just develop it over time? I do not know. Do other mathematicians have the same concept? Probably, though as I noted earlier the ontological nature of multiplication rarely arises in professional mathematical activity, so likely very few have bothered to reflect on the matter.

On the other hand, scaling is a natural physical concept, and abstracting from physical scaling to the numerical operation of multiplication is no more difficult than abstracting from the physical activity of combining two lengths together to give the numerical operation of addition. The advantage of approaching multiplication from scaling is that the resulting numerical operation works in all cases, whereas the MIRA approach works only for positive integers. (Sure, you can tell stories to extend the resulting RA notion to rationals, but it is contrived, and the final jump to real numbers is problematic. The scaling approach gets you to real numbers in one go, where real numbers are identified with lengths of lines.)

So, for all those who have asked me, that is what I understand by multiplication: a somewhat generalized notion of scaling built directly on a physical intuition. And though, as I keep stressing, I have no experience teaching elementary mathematics, I cannot for the life of me see why multiplication is not taught that way.

It may be that one reason some of us are more successful in mathematics than others is that we manage to form good mental concepts early on in the learning process. (I hesitate to use the phrase "the right concepts", since I do not know that there is a unique correct concept of multiplication, and am very aware of my own difficulty of articulating mine in words.)

Devlin's Angle is updated at the beginning of each month. Find more columns here. Follow Keith Devlin on Twitter at @nprmathguy.
Mathematician Keith Devlin (email: [email protected]) is the Executive Director of the Human-Sciences and Technologies Advanced Research Institute (H-STAR) at Stanford University and The Math Guy on NPR's Weekend Edition. His most recent book for a general reader is The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, published by Basic Books.