The snippets below capture some of the main points made during the recent MIRA (Multiplication Is Repeated Addition) debate. Believe it or not, this debate is far from over... 

What is conceptual understanding?

(Snippet from September 2007)

A request

"As regular readers probably know, I am a mathematician, not a professional in the field of mathematics education. I know many mathematicians, but far fewer math ed specialists. But I am interested in issues of mathematics education, and I have long felt that mathematicians have something to contribute to the field of mathematics education. (Getting rid of those floating helium balloons would be a valuable first step! Stopping teachers saying that multiplication is repeated addition would be a good second.)"

http://www.maa.org/external_archive/devlin/devlin_09_07.html

It Ain't No Repeated Addition

(Snippet from June 2008)

"In my column for September 2007, which was titled What is conceptual understanding? I remarked that I wished schoolteachers would stop telling pupils that multiplication is repeated addition. It was little more than a throwaway line, albeit one that I feel strongly about. I put it in to provide a further illustration for the overall theme of the column, to indicate that there are examples beyond the ones I had focused on. In the intervening months, however, I've received a number of emails from teachers asking for elaboration. Their puzzlement, they make clear, stems from their understanding that multiplication actually is repeated addition.

If ever there were needed a strong argument that professional mathematicians need to interest themselves in K-12 mathematics education and get involved, this example alone should provide it. The teachers who contact me do so because they genuinely want to know what I mean, having been themselves taught, presumably either in schools of education or else from school textbooks, that multiplication is repeated addition.

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.

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."

http://www.maa.org/external_archive/devlin/devlin_06_08.html

It's Still Not Repeated Addition

(Snippet from July-August 2008)

"The thrust of my earlier column was a plea to mathematics teachers to stop telling students that multiplication is repeated addition. Doing that not only sets up the hapless student for later confusion when they encounter situations where that definition plainly makes no sense (negative numbers, fractions, irrational numbers), it also leaves them with a fundamental lack of understanding of basic arithmetic that those of us who teach at university level encounter with every year's new intake.

Arithmetic for today's world

Let me be plain about it. Addition and multiplication are different operations on numbers. There are, to be sure, connections. One such is that multiplication does provide a quick way of finding the answer to a repeated addition sum. Indeed, if the only thing anyone ever needed to do with numbers is add them, either once or repeatedly, then there would be no need to have something called multiplication; there would simply be a clever shortcut to find the answer to a repeated addition.

But the world has a habit of presenting us with situations where addition simply is not enough. This happens in business, commerce, finance, science, engineering, all over the place. For instance, there is no way to understand a (continuous) volume control on a radio in terms of addition, either singly or repeated. A volume control is not an additive device, it's multiplicative. Indeed, the entire domain of scaling (of which a volume control is just one simple example) is inherently multiplicative, just as combining collections is fundamentally additive.

Addition and multiplication aren't enough for our world either, as it turns out. Biological growth and population growth are inherently exponential and cannot be understood as "repeated multiplication" (which would cash out as "repeated repeated addition" for those who advocate reducing all of arithmetic to addition)."

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

Multiplication and Those Pesky British Spellings

(Snippet from September 2008)

"In my previous two columns, It Ain't No Repeated Addition and It's Still Not Repeated Addition, I explained why I (and many others who are far more knowledgeable about K-8 mathematics education than I) think it is bad to teach multiplication as repeated addition.

To a casual observer, I imagine that the minor firestorm in various math blogs that my columns generated might suggest that my remarks injected something new to the mathematics education field. But, in fact, everything I said has been written about and discussed in the mathematics education community for some forty years or so, and essentially (though not in every detail) agreed upon. Discussed and agreed upon by people who have spent their professional careers studying those issues and have formed carefully thought out conclusions that have been subjected to, and passed, professional peer review.

...

Multiplication is a tricky concept

Here is what Nunes and Bryant have to say about multiplication at the start of Chapter 7 of the book I cited above. (There are more British spellings in their work, so the bloggers won't be able to read this valuable resource, though doing some research before launching into an Internet tirade is clearly not their forte.) The previous chapter deals with addition and subtraction.

According to [a common view] there need be no major change in children's reasoning [after they have mastered addition and subtraction] in order for them to learn how and when to carry out multiplication and division. This view was challenged by Piaget and his colleagues [...] who suggested that understanding multiplication and division represents a significant qualitative change in children's thinking.There certainly are significant discontinuities between addition and subtraction on the one hand and multiplication and division on the other... but there are some significant continuities too... the continuities and discontinuities are as important as each other, and both need to be thoroughly charted if we are to understand the many steps that every child has to take towards a full understanding of multiplication.

After acknowledging that there remains some controversy surrounding multiplication, particularly how you classify types of multiplication, the authors continue - and this is really important:

We must begin with a word of caution. Multiplicative reasoning is a complicated topic because it takes different forms and it deals with many different situations, and that means that the empirical research on this topic is complicated too. So, in order to make sense of the empirical work, we must first spend some time setting up a conceptual framework for the analysis of children's reasoning and only then go on to review the research. [...] Up to now it was possible to build the concepts and the vocabulary needed slowly through the chapter; with multiplication and division we stray so far from common sense and everyday vocabulary that we have to agree on a set of terms and conceptual distinctions at the outset.

...

Incidentally, here is what Nunes and Bryant have to say about repeated addition (p.153):

The common-sense view that multiplication is nothing but repeated addition, and division is nothing but repeated subtraction, does not seem to be sustainable after a careful reflection about situations that involve multiplicative reasoning. There are certainly links between additive and multiplicative reasoning, and the actual calculation of multiplication and division sums can be done through repeated addition and subtraction. [DEVLIN NOTE: They are focusing on beginning math instruction, concentrating on arithmetic on small, positive whole numbers.] But several new concepts emerge in multiplicative reasoning, which are not needed in the understanding of additive situations.

They go on to enumerate and describe some of the more salient complexities of multiplication. The issue is far too complex for me to summarize effectively here. It takes Nunes and Bryant an entire chapter and then some. But note what they are saying in the above quoted passage: Even in the special case of the positive whole numbers, where repeated addition gives the answer to a multiplication sum, the two are not at all the same.

http://www.maa.org/external_archive/devlin/devlin_09_08.html

What Exactly is Multiplication?

(Snippet from January 2011)

...

(Concluding comments.)

"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."

http://www.maa.org/external_archive/devlin/devlin_01_11.html