From CHILDREN DOING MATHEMATICS by Terezinha Nunes and Peter Bryant:

One is tempted to stop there. The above comes from the opening of the authors' chapter on multiplication and division, in a section called "Multiplication, Division, and New Number Meanings. For the debate that has been raging all over the internet since Keith Devlin published his first column about multiplication in 2008 called "It Ain't No Repeated Addition," it seems that for many people, the ideas raised in the above quotation are absurd. After all, one can compute products by doing repeated addition and quotients by doing repeated subtraction. If you can get the right answer to a computation by two different operations (which begs the question as to whether repeated addition is actually a well-defined operation), aren't those operations the same? Isn't multiplication, when all is said and done, repeated addition? Certainly this is true for the whole numbers, right? And the analogy used there can readily be made to fit integers, rational numbers, maybe even the real numbers in their entirety. It's merely common sense, and this is, after all, the way CHILDREN think, isn't it?

Let's look further at what Nunes and Bryant have to say before considering the above question again. Summarizing the number meanings and situations for additive reasoning they write:

"[I]t would be wrong to treat multiplication as just another, rather complicated, form of addition or division as just another form of subtraction.

The reason for this is that there is much more to understanding multiplication and division than computing sums. The child must learn about and understand an entirely new set of number meanings and a new set of invariants, all of which are related to multiplication and division but not to addition and subtraction. [p. 144]

One is tempted to stop there. The above comes from the opening of the authors' chapter on multiplication and division, in a section called "Multiplication, Division, and New Number Meanings. For the debate that has been raging all over the internet since Keith Devlin published his first column about multiplication in 2008 called "It Ain't No Repeated Addition," it seems that for many people, the ideas raised in the above quotation are absurd. After all, one can compute products by doing repeated addition and quotients by doing repeated subtraction. If you can get the right answer to a computation by two different operations (which begs the question as to whether repeated addition is actually a well-defined operation), aren't those operations the same? Isn't multiplication, when all is said and done, repeated addition? Certainly this is true for the whole numbers, right? And the analogy used there can readily be made to fit integers, rational numbers, maybe even the real numbers in their entirety. It's merely common sense, and this is, after all, the way CHILDREN think, isn't it?

Let's look further at what Nunes and Bryant have to say before considering the above question again. Summarizing the number meanings and situations for additive reasoning they write:

Additive reasoning is about situations in which objects (or sets of objects) are put together or separated. All the number meanings in additive situations are directly related to set size and to the actions of joining or separating objects and sets. Number as a measure of sets involves putting objects into a set where the starting-point is zero; number as a measure of transformations relates to the set that is joined to/separated from another set; number as a measure of a static relation (in comparison problems) relates to the set that would have to be joined to/separated from another in order to make two sets equal in number. [p.144]

Anything too radical there for the folks who seems to cling so passionately to the Multiplication Is Repeated Addition (MIRA) point of view? I would think not. For all that Nunes and Barnes are doing is reflecting the essential nature of addition itself. And there is no real debate about that raging underneath the battle over multiplication. The trouble comes when Devlin and others suggest that multiplication not only need not be defined as "repeated addition" (even though it often is in many places, including reputable ones), but also that it simply should not be so defined. It is a separate operation that stands on its own, with no need to be defined as something else.

To be sure, as previously stated and as Devlin and others on the other side of the aisle in this particular debate have stated many times, we can readily arrive at the result of multiplying two whole numbers, say, 3 x 2, by adding 2 + 2 + 2. No one will argue that the resulting product and sum are not the same whole number, 6. But is that coincidence (in the literal sense of two things "falling together") sufficient to determine that we are looking at the same operations (indeed, there is good reason to question whether repeated is a well-defined binary mathematical operation (Devlin says it is not)?

Well, if not, why not? What IS different about multiplication and multiplicative reasoning? Nunes and Barnes state:

Situations which give rise to multiplicative reasoning are different because they do not involve the actions of joining and separating. We will distinguish three main kinds of multiplicative situations: (1) one-to-many correspondence situations; (2) situations which involve relationships between variables; and (3) situations which involve sharing, division, and splitting. [pp.144-145]

Those of us familiar with some of the arguments of those in the MIRA camp know that one typical response from such people is to suggest that their antagonists are presenting "distinctions without difference." Is that in fact the case with the analysis of Nunes and Bryant? Let's examine each of the situations they mention above in this light.

**One-to-many correspondence**

Most of us are familiar from ordinary experience with the notion of one-to-many correspondence. Some obvious examples are: a car has four wheels (1-to-4); a hand has five digits (1-to-5); a triangle has three sides (1-to-3); a piano has eighty-eight keys (1-to-88), and so on. As Nunes and Bryant point out:

There are some continuities between these multiplicative situations and additive situations. The most salient is that some of the number meanings here are also connected to sets.

In the examples above, "four wheels," "five digits," etc., do refer to set size. But Nunes and Barnes claim that there are four greatly significant differences:

First, the multiplicative situations involve a constant relation of one-to-many correspondence between two sets. This constant one-to-many correspondence is the invariant in the situation, a type of invariant which is not present in additive reasoning. The one-to-many correspondence is the basis for a new mathematical concept, the concept of ratio. In order to keep, for example, the correspondence '1 car to 4 wheels' constant, each time we add one car to the set of cars we must add 4 wheels to the set of wheels - that is, we add different numbers of objects to each set. This contrasts with the additive situation where, in order to keep the difference between two sets constant, we add the same number of objects to each set. [p.145]

In raising the issue of ratios (and thereby proportional reasoning) quickly, Nunes and Bryant cut to one of the core issues that mathematics educators must take seriously: if multiplication is fundamentally not about joining/separating sets, but about things like ratios, is it reasonable to suspect that one major cause for the well-known difficulties students have with rational numbers and proportional reasoning comes from the propensity of so many teachers to introduce multiplication as if it were simply repeated addition, a viewpoint that is, of course, naively reinforced by those teachers and other adults who were taught to think about multiplication the same way?

It is hard to believe that many young children who run into difficulty grasping multiplication will NOT be told by most adults, "Look, honey: it's just repeated addition. See it now?" Amazingly, I have read some MIRA supporters claim that many kids don't know this "fact" (which of course these same teachers presume is correct), and go on at length about how they must remediate such students by showing them this "obviously true" model. One of them has gone so far as to argue that Devlin and his supporters wouldn't be making claims against MIRA if only he and they had studied with her as a teacher. I'm not confident that this was intended as humor.

Getting back to Nunes and Bryant: they state that this new kind of reasoning

leads us to the second difference: the actions carried out to maintain a ratio invariant are not joining/separating but replicating (to use Kieren's expression) and its inverse. Replicating is not like joining, where any amount can be added to one set. Replicating involves adding to each set the corresponding unit for the set so that the invariant one-to-many correspondence is maintained. For example, in the relation 'one car has four wheels', the unit to be considered in the set of cars is one, whereas the unit in the set of wheels is a composite unit of four wheels. The inverse of replicating is removing corresponding units from each set. If we remove one car we must remove four wheels, in order to maintain the 1 : 4 ratio between cars and wheels. [p. 145]

Here we see one of the most problematic points in trying to convince MIRA supporters that multiplication is not repeated addition. They do not see that in a question like, "If each car has four wheels, how many wheels are there on six cars?" that there are two sets being operated upon differently from the way sets are in additive situations. I suspect strongly that this has something with the difficulty they have in seeing (or simply the refusal to grant) that there are implies or explicitly stated units associated with each set in the problem as stated and they are not the same for each set, whereas in addition, we would be talking about joining or separating sets of the same kind of object (with the same unit, say, 'tires') in each set. With the one-to-many concept grounding this view of multiplication, we have a set of cars with a unit "one car" and a set of wheels with a composite unit, "four wheels per car." Thus, if there are six cars, this problem can be thought of as six cars * four wheels/car. Multiplying the numbers yields

24 cars * wheels/car which equals 24 wheels. In my view, a final non-composite unit, in this case, wheels, can emerge by itself only from multiplication or its inverse. Were this truly repeated addition, we would start with a set containing zero wheels and add four wheels at a time. Joining sets of wheels can only result in wheels. Cars have nothing to do with the case. We could, in fact, add 1 wheel, then 3 wheels and then 2 wheels and then 5 wheels and then 4 wheels and then 6 wheels and then 0 wheels and then 3 wheels and get the same set of 24 wheels. But it would not have the same meaning as 6 cars times 4 wheels/car = 24 wheels. This is the point the authors make when they state that in joining "any amount can be added to one set." And I am convinced that MIRA supporters simply do not see this crucial difference. (There is something here worth looking into regarding how the units behave in rational number arithmetic that I shall delay until another post.)

Turning to the third crucial difference between addition and the one-to-many correspondence situation, Nunes and Bryant state:

[A] ratio remains constant when replication is carried out even if the number of cars and the number of wheels change. In a set where there are 3 cars and 12 wheels, the ratio is still 1 : 4. This is the case because the ratio does not represent the number of objects in either set but is an expression of the relation between the two sets. [pp. 145-156]

This distinction, too, seems utterly absent when reading what MIRA defenders speak about. They harp repeatedly on the fact that the calculations are the same, and thus the operations must be the same or at least that multiplication is reasonably viewed as definable in terms of addition (or the "functions" must be the same, terminology that on my view does absolutely nothing to clarify the underlying meanings or fundamental mathematical ideas under consideration, though perhaps it does help obfuscate those ideas for adults, including some teachers, who aren't sure what a function is or whether calling something a function will help focus, rather than obscure, the issues.)

The last distinction Nunes and Bryant raise for the one-to-many correspondence situation is that:

a new number meaning can be identified in the number of times that a replication is carried out. For example, if we start with the simple situation where we have 1 car and 4 wheels and replicate this starting situation six times, "'6' refers to the number of replications - called the scalar factor. A scalar factor is neither about cars nor about wheels; it does not refer to the number of objects in the sets but to the number of replications relating the two set sizes of the same type. 'Six' expresses the relation between 1 and 6 cars and between 4 and 24 wheels. For the ratio to remain constant, the same scalar factor must be applied to each set.

It is worth pointing out that ratios do not need to involve a unit: for example a recipe may involve a 2 : 3 ration between the number of eggs and cups of flour. When you increase the number of cups, you also need to increase the number of eggs so that the ratio remains constant.

The number meanings in one-to-many correspondence situations are schematically represented in figure 7.1 [Note: the figure consists of two drawings, each with a truck on the left and four wheels on the right. The first drawing has the words "1 truck, 4 wheels: each replication keeps the same ratio." The second has the words "2nd replication of 1 truck, 4 wheels"] In short, one-to-many correspondence situations involve the development of two new number meanings: ratio, which is expressed by a pair of numbers that remains invariant in a situation even if the set size varies, and the scalar factor, that refers to the number of replications applied to both sets maintaining the ratio constant. It should be clear that neither of these meanings relates to set size: the ratio and the scalar factor remain constant even when the set sizes vary. [p. 146]

I will remind the reader that the above four differences are only raised about the first of THREE situations Nunes and Bryant examine. Still to come are those that involve relationships between variables - co-variation, and those that involve sharing and successive splits. We will examine these in subsequent posts. However, I believe that there is enough in just this first situation to seriously damage the idea that multiplication is just repeated addition, even looking strictly at whole numbers. The supporters of MIRA as a reasonable way to introduce children to multiplication and to help children who are struggling with multiplication and multiplicative reasoning have a lot of 'splainin' to do. Or, more likely, explaining away.

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