What is Multiplication? How (Not) to Teach It

Last time we looked at the roles of multiplier and multiplicand from several perspectives. This time, I want to focus on one extended discussion about how children should be taught to think of multiplication.

Times, or multiplied by?

Cindy asked this in 2002:

Defining Multiplication

I just read an article in Phi Delta Kappan (Feb. 2001) by Deirdre Dempsey and John Marshall titled "Dear Verity: Why Are All the Dictionaries Wrong?" They state that Euclid and some dictionaries define multiplication equations like 3x4 to mean "the number 3 four times." I was taught, I teach, and math textbooks say that 3x4 means 3 lots/groups of 4. An array would look like 

   ****
   ****
   ****

Because multiplication is commutative, it may not matter symbolically; however, as the authors mentioned in the article, taking 4 pills a day for 21 days is a lot different from taking 21 pills a day for 4 days. If I want my students to know what multiplication "is," what is the correct explanation?

At the time, I had no access to this magazine; later, after I started teaching at a community college, this question was brought up again and I was able to access it online through the library. I still can’t point you to an online source so anyone can read it. At any rate, at the time I could only answer Cindy’s specific questions, about definitions and about taking pills. I started by focusing on what “times” means grammatically (which, it turns out, the article agreed with, taking the use of the word “times” as problematic):

Your question is really not a math question, but a linguistic question, even though it involves symbols. Since, as you recognize, 3*4 and 4*3 have the same value, there is no mathematical value in the distinction. You are only asking about the underlying image when we say "three times four" in English. (It may have been different in Euclid's Greek.) And I submit that it can be reasonably interpreted either way.

My first thought when analyzing the phrase is that originally '3 times' meant to repeat what follows three times; that is, it would mean the same as '4, taken 3 times'. So 3 times 4 means 4+4+4.

But that seems awkward, considering the way we tend to say the phrase now. We might read through a calculation, saying "3, times 4 is 12, plus 2 is 14, ..." where each operation acts on the first number. We're really saying "3 multiplied by 4." Taken that way, we start with 3 and multiply it by 4, meaning that we repeat it 4 times. So 3*4 = 3+3+3+3.

The difference here is entirely in the grammar: is '3 times' an adjective phrase modifying '4', or is 'times' a preposition equivalent to 'multiplied by', and 'times 4' a modifier of '3'?

I am saying here that we can read \(3 × 4\) either as “3 times 4” (3 groups) or as “3 multiplied by 4” (4 groups), which present different images. The expression as written does not imply anything about which number is the number of groups. (The authors of the article felt that it is better pedagogically to use the “multiplied by” form.)

But what was that about Euclid’s definition? I later found it in Book 7, Definition 15:

A number is said to multiply a number when the latter is added as many times as there are units in the former.

Of course, the Greeks had no symbolic notation for multiplication, so that says nothing about the meaning of \(3 × 4\); they wrote their math out in words, like “A multiplied by B makes C“, leaving no ambiguity about which is the multiplier. The article assumes that “\(×\)” is read as “multiplied by”, so that the second number is the multiplier. But that begs the question! If they had used our notation, might they not have seen \(3 × 4\) as 3 multiplying 4?

Anyway, I continued by considering the issue of pedagogy: Is there, and should there be, only one model of a given multiplication?

When we introduce children to multiplication, it's reasonable to start with one specific meaning, just so they have a concrete image to start with. But we can almost immediately point out that

    ****
    ****
    ****

can be seen equally well as 3 rows of 4, or as 4 columns of 3. Once you've drawn the figure, or once you've written 3*4, no one can really tell which interpretation you meant. The distinction has been abstracted out of the problem.

And that is a good thing, not bad: in mathematics the ability to work abstractly and forget what the concrete problem was is a major advantage, because we can ignore details that don't affect the result, and rearrange the work to make it easier. If I took 4 pills a day for 21 days, and now I want ONLY to know how many pills I took, I can call it 4*21 (I just put the numbers in the order I saw them, without having to think about which is the multiplicand) and re-model the problem as 4 sets of 21, allowing me to add 21+21+21+21, which is easier than adding 21 4's. I can switch freely among different models, because I know that the outcome is equivalent. So I never bother to define whether 4*21 means 4 groups of 21 or 21 groups of 4, because that distinction would only be a hindrance. And if I were doing a calculation for which it made a difference whether I took 4 or 21 pills a day, such as finding the concentration of medication in my blood, I wouldn't be multiplying, because that would be the wrong abstraction!

So "what is multiplication?" It is a commutative operation that can be modeled in two symmetrical ways as repeated addition (when applied to whole numbers).

So, as adults, we don’t want to identify either the first or the second number as necessarily the multiplier, because that is a distinction that is meaningful only in the application, not in the calculation. But for children, it makes sense to start with a concrete model taking it one specific way, as long as we soon help them to see that two different models have the same result, so that they have control over how they do the calculation. (Even though children are described as concrete thinkers, they seem able to handle this “abstract” idea with no trouble, as we have often seen when kids ask about it.)

Doctor Tom also wrote an answer to the question of what multiplication “is”, by stating a mathematical definition of multiplication as an abstract operation, akin to what I have recently mentioned about axioms of arithmetic. It is probably not useful for Cindy’s purposes, but is interesting.

Interacting with the authors

Now, three years later one of the authors of the article cited, John Marshall, wrote to us, chiding me for commenting without having read the article (which, of course, I couldn’t do, so that my comments weren’t about the article at all), and defending his claims. Another three years later, the other author, Deirdre Dempsey, also discovered what I had written and wrote me; by that time I had been able to read the article, and had a better idea what it said.

One comment from co-author Marshall was this:

What I think you need to do is to explain to Cindy, and myself, just what part of the Webster’s 3rd Edition College Dictionary definition, for example, indicates there is a choice. Where in

“Multiplication: the process of finding the number or quantity (product) obtained by repeated additions of a specified number or quantity (multiplicand) a specified number of times (multiplier); symbolized in various ways (ex. 3×4=12 or 3·4=12, which means 3+3+3+3=12, to add the number three together four times).”,

does it make your point? It seems pretty clear to me. In 3×4=12 the 3 is the multiplicand and the 4 the multiplier. Just because the OPERATION of multiplication is commutative it doesn’t follow that the MEANING of multiplication is. AND IT IS MEANINGS WE ARE TALKING ABOUT.

Unfortunately for his argument, it is easy to find other dictionaries that say the opposite; for example, here is Webster’s Third New International:

Multiplication: A mathematical operation commonly indicated by ab, a·b, or a×b and  having various significances according to the type of numbers involved, the simplest being in the case of positive integers where the process is that of repeating b as many times as there are units in a, or vice versa. [my bold]

Clearly they agree with me! (And they correctly note that for numbers other than positive integers, a different definition is needed than “repeated addition”; they are more mathematically astute.) Of course, dictionaries are not the final arbiter of mathematical meaning, as Marshall ought to know. The article quoted other dictionaries that left it properly ambiguous, and just rejected them because of an assumption that no definition can be ambiguous. (Mathematicians, or math educators, may not be the best lexicographers; the latter know all about ambiguity, while the former avoid it.)

He also explained the pill example:

You don’t seem to understand either that, in the pills issue that Cindy quotes, the problem is NOT asking for the total number of pills, but rather how would the pharmacist, or patient, interpret a prescription that used 4×21. It is about the meaning of 4×21 not the product.

When Deirdre Dempsey wrote after I had read the article, I said the following about that point. It turns out that the whole article was centered around students being asked to think of a situation represented by \(6 × 3\); the authors claimed this should be seen only as 3 groups of 6, and not as 6 groups of 3, though most people think it is the latter. The pill example claims that \(4 × 21\) specifically means “21 groups of 4”, and so could be used in a prescription to tell a patient what to take, and not just to calculate the total number of pills.

It seems to me that you have confused the idea of being able to represent a concrete problem as an arithmetic operation (which is an important indication of understanding) with the inverse process, being able to turn an operation into a single kind of concrete problem. The latter is not a valid problem to assign, unless you accept any situation that will yield the same operation. The fact is that an operation like 4×21 does not mean either 4 pills a day for 21 days OR 4 days taking 21 pills a day.

The specific point I dealt with in my original response was the implication that a prescription could be written saying to take 4×21 pills. I certainly hope no one would ever do that, because that phrase simply does not MEAN to take 4 pills a day for 21 days. The phrase “4 times 21” is not a description of a regimen, but an arithmetic operation whose meaning is “84”. All the dictionaries agree on that!

But the main point of the article, as I see it, was that children think concretely, and need to be taught concretely; they are not ready for abstraction. I concluded my answer with this:

Now, there is one point where I agree with you. Replacing the word “times” with “multiplied by” when multiplication is first introduced would be a good idea; it allows you to talk concretely about multiplication with a clear physical model, and it also avoids the unfortunate tendency of so many students to turn “times” into a verb (“I timesed 2 by 3 and got 6”). As long as you then move on quickly to talk about other models of multiplication and to emphasize the commutative property, I’m happy with that change.

The best reason for preferring “multiplied by”, I think, is that it fits into the model of other operations, where we start with a number and operate on it: \(a + b\) means “a increased by b”, \(a – b\) means “a decreased by b”, \(a \div b\) means “a divided by b”, so \(a \times b\) should mean “a multiplied by b”. That consistency is perfectly reasonable, especially in initial teaching. It just happens that it isn’t followed consistently in real life, and is not important in practice. To teach children that only one model is valid is incompatible with everyday usage, and leads to confused questions from students and their parents.

The problem with teaching only one way

Long after those interactions, in 2014, we received one of many examples of this happening in a classroom, which I quoted within my answer to yet another such question:

Defining Multiplication, Redux

My 2nd grader failed her math quiz today because the teacher insists that there is an order as to how a multiplication sentence should be written. 

Here's an example:

   ##   ##   ## 

      Addition Sentence: 2 + 2 + 2 = 6
Multiplication Sentence: 3 x 2 = 6

The teacher said that 2 x 3 = 6 would be wrong.

Another example from the quiz shows a numbered scale with asterisks that represent jumps:

<-0*--1---2---3*--4---5---6*--7---8---9*--10---11---12*--13---14---15*->

Multiplication Sentence: 5 x 3 = 15

The math teacher said that this is the only way that can be written, and that writing it as 3 x 5 = 15 is wrong.

I have already spoken to the teacher and explained that these can be interpreted in two ways, but she insists that this is how 2nd graders should be taught!

Unless I can prove her wrong with a written source of some kind, she will not consider my daughter's answers.

Please help.

One of the things in education that trouble me most is when something is taught in a way that confuses the smart kids who really understand (and maybe have knowledge beyond what is being taught), and even costs them points. Intelligence should never be penalized. Nor should teachers take a position that parents can see makes no sense. But we get questions like this repeatedly.

In my answer, I tried to explain what the teacher is (I think) doing, and make dialogue possible:

I agree fully with you. But there is some reason behind the teacher's position; and a compromise is possible.

Multiplication is commutative, so your point is correct; 2*3 and 3*2 are equal, and either could be used to represent the same calculation or situation. Mature thinkers don't need to pay any attention to the order, but use whichever order is most useful.

On the other hand, it is reasonable for students to INITIALLY be introduced to multiplication using a single model, which in this teacher's curriculum is "this many of those" rather than "this repeated that many times." This simply allows everyone in the class to be on the same page when they give examples.

But they should soon learn the commutative property so that, for example, if they want to find the cost of 50 items that each cost $2, they could just add two 50's rather than add fifty 2's!

Interestingly, many years ago I answered a question about an article in a teacher's magazine that made the exact opposite claim: that 2*3 "really means" 2 multiplied BY 3 (2 taken 3 times):

  Defining Multiplication
  http://mathforum.org/library/drmath/view/61066.html 

Some time later, each of the authors of the article cited (which I didn't have access to at the time) wrote to me complaining that my comments were wrong. They didn't change my mind, but I did then get to read their full article, and it did clarify the point they were trying to make. My conclusion is that it does make sense to teach kids initially to read "2*3" as "2 multiplied by 3," and perhaps to ask them to use that interpretation in class to make examples of where a particular multiplication would be used. They should not, however, be taught that this is the only interpretation.

If nothing else, you can refer your daughter's teacher to that 2001 article from the Phi Delta Kappan, if she has access to it (I had to get it online through my school library site), because it makes the opposite statement about the meaning of multiplication, showing that what your teacher does is not the ONLY way to teach the subject; and moreover that other educators have the opposite opinion. It is not the way students SHOULD be taught multiplication, just the way they ARE taught it in this curriculum.

So here's the compromise I suggest: the teacher can teach one interpretation of the application of multiplication, but make it clear that it is not the ONLY answer, just the one they are using IN THIS CLASS, for now. Your daughter should be commended for seeing that it really doesn't make a difference; and rather than being marked wrong, can be encouraged to learn the way the teacher says to do it merely as a class convention, to help students who need more time to catch on.

It turns out that the Common Core standards (introduced long after the original article) appear to support the interpretation that \(2 × 3 = 3 + 3\), so the teachers in these cases may just be doing what they are told. Here is what the standard says:

CCSS.Math.Content.3.OA.A.1 Interpret products of whole numbers, e.g., interpret \(5 × 7\) as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as \(5 × 7\).

Now, all this actually says is that students should be able to interpret a product in terms of some physical model; the specific instance shown is just an example of one such model, and does not say that this is the only model. So with my understanding of the issue, I would say that they leave it open whether to introduce only one model, or which one to use. But I can see why teachers and curriculum writers might interpret it as requiring this one model. I wish that teachers, teacher educators, and curriculum writers could all have a sufficiently mature understanding of math beyond the level being taught to know what matters, and what doesn’t. But none of us can be expected to do everything right (I myself can’t claim to know all about elementary education), so we have to be gracious.

Unfortunately, the example the standards give is not the “multiplied by” model that the authors of the article convinced me is better as an initial model, but the “times” model that is probably most commonly taught already. Oh, well.

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