What is Multiplication, Anyway?

Yet again, the internet has seized upon an elementary student’s math homework and has decided to argue about Common Core. This time, it’s about a test question. The student was asked to “Use the repeated addition strategy to solve : 5 x 3” (Reddit, via The Telegraph, via Greg Ashman); the student answered “5 + 5 + 5” and was marked wrong because the answer ought to be “3 + 3 + 3 + 3 + 3”.

Witnesses for the Prosecution (such as Ashman and Evelyn Lamb) point out the obvious: The student has used addition, repeatedly, and gotten an expression that is mathematically equal to the question. Witnesses for the Defense (such as Brett Berry and Hemant Mehta) argue that the commutativity of multiplication is not a logical given, and that while “5 x 3” means “3 + 3 + 3 + 3 + 3”, it only happens to equal “5 + 5 + 5”. Indeed, of the four basic operators that students get exposed to in elementary school, two of them (i.e., half) are not commutative: Subtraction and division. So there’s an argument to be had that forcing students to avoid simply assuming commutativity avoids a future confusion of 3/4 = 4/3.

Ultimately, I side more with the Prosecution than for the Defense: The student should not have been marked incorrect. But there are a bunch of different issues being blurred together into one controversy.

First, the matter of Common Core

This is not an exclusive Common Core problem. This is a problem of implementation, and it’s not unique to Common Core. Only slightly more than a year ago, a nearly identical controversy exploded over an Indonesian teacher grading a second grader wrong for writing that 4 + 4 + 4 + 4 + 4 + 4 = 4 x 6 = 24. I wrote some commentary then (in which I came to a different conclusion than I do now). The controversies are so similar, in fact, that I’m not completely convinced that the US version isn’t a hoax created so that people can storm about the “repeated addition” approach to multiplication in a way that doesn’t feel so foreign. Even if the US version is real, though, the existence of the exact same grading strategy in Indonesia supports the idea that this is not the result of Common Core per se.

Lamb notes, correctly, that the Common Core is not a curriculum, it’s a set of standards. Mehta claims that this approach to multiplication is supported by this third-grade Common Core standard: “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.” This is a valid point, although it’s also worth noting that the phrase “repeated addition” on corestandards.org turns up zero hits: It’s simply not a phrase that’s used (Josh Fisher comments that he and Keith Devlin criticized the presence of that phrase in an early draft, though).

I’ve commented in the past that Common Core provides examples that are often wrongly interpreted as directives. Unfortunately, in this specific case, it’s unclear whether the “e.g.” refers to the numbers given (that is, “interpret products of whole numbers” should be taken as “interpret a × b as the total number of objects in a groups of b objects each”) or to the example given (that is, “interpret products of whole numbers” would also be satisfied as “interpret a × b as the total number in b groups of a objects each”). However, in this case, I am inclined to side with Mehta and Berry, and to say that the teacher was satisfying the most literal interpretation of that standard when marking that problem wrong (although see my further comments below).

Point for the Defense, with the preceding caveats.

However, Common Core has another third grade standard, which the Defense is conveniently ignoring: “CCSS.MATH.CONTENT.3.MD.C.7 Relate area to the operations of multiplication and addition.” And while 3.OA.A.1 can be used to defend seeing 5 × 3 as 3 + 3 + 3 + 3 + 3, this second standard undermines the intellectual calisthenics that Mehta and Berry are engaging in. Mehta supports the most literal interpretation of 3.OA.A.1 because teaching multiplication as commutative at the third grade obstructs some hypothetical future understanding in those cases (such as arrays) where multiplication is not commutative, while also jumping the gun on the joy-and-wonder that some future student will bask in upon learning of commutativity. It’s true that the commutativity of multiplication (and addition) is not a theoretical given, but rather falls out from details of how multiplication works. Berry, meanwhile, insists that the student is wrong because of how multiplication is defined in the first place.

However, commutativity is more obvious when applied to area (turn your map!), and Common Core’s two standards create competing definitions of multiplication. Keith Devlin has a few things to say about that, which I’ll get to below.

So, there is a Common Core problem here, but it’s not because Common Core is withholding the teaching of commutativity, even implicitly, until later grades. Rather, there’s an inconsistency between standards at the same grade. And, in my mind, the actual issue relates not to Common Core directly, but rather to teachers of mathematics who are some combination of inept and insecure and who therefore teach a specific lock-step method instead of allowing for basic variation.

So… what is multiplication, anyway?

The two conflicting Common Core standards are rooted in the fact that multiplication in early elementary mathematics has two basic, distinct uses, but it’s usually taught as one operation. For the sake of discussion, I’ll call the two uses aggregation and area. (Edit: Carlos Castillo-Garsow notes that dilation is also included in Common Core elementary standards [CCSS.MATH.CONTENT.5.NF.B.5]. That’s a third usage.)

Aggregation involves finding the number of items in a collection of objects where we have grouped the objects into equal-sized smaller collections. That’s 3.OA.A.1. If three friends all have five cookies each, then they have fifteen cookies total. If five friends all have three cookies each, then they also have fifteen cookies total. Witnesses for the Defense correctly point out that these are two different scenarios, and that the most natural way to add up the first situation is 5 + 5 + 5 (take the number of cookies held by each friend, and add them up). I say “most natural”, because there’s no particular reason why the friends couldn’t line their cookies up, note they have three “first cookies”, three “second cookies”, and so on. Indeed, if three friends had a complete set of alphabet blocks, it would make perfect sense for them to say, “We have 3 As, 3 Bs, 3 Cs,…” and so on. Certainly if we’re counting the cards in a standard poker deck, it’s just as natural to say “Each suit has thirteen cards, and there are four suits” as it is to say “There are four suits, and each suit contains one of each of thirteen cards”.

Area involves finding the area of a region in squared units, given its dimension in plain units. That’s 3.MD.C.7. If a rectangle has dimensions of five inches by three inches, it has an area of fifteen square inches. There is no addition here. This is another place where the Common Core standards do stumble a little, but that’s because they’re written at the student’s level. If the standards were really written in order to prepare students for a decade in the future, addition would not be included in this standard, but it is: “CCSS.MATH.CONTENT.3.MD.C.7.D Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.” While area multiplication, philosophically, involves finding a squared-unit area given unit-measured lengths, Common Core is actively encouraging teaching multiplication-for-area as an aggregation process: Fill an area with unit squares and count the unit squares. This sets students up for Pick’s Theorem later (for whatever that’s worth), but it further blurs the issue of the importance of keeping units in mind as a crucial part of sense-making.

This distinction is hardly trivial. Students often struggle when the area of an object is “less than” its perimeter. A unit square, for instance, has an area of “1” and a perimeter of “4”. I put the numbers in quotes here because I’m actually only providing the numeric value. Because we so often don’t use units, it’s very easy for a student to forget (or even fail to notice in the first place) that the area of a unit square is \(1u^2\) while its perimeter is \(4u\). The values aren’t comparable in a “lesser than” way because the units are different. And while “groups” and “items” are not clearly commutative, the lengths of a rectangle clearly are (any student can see that by simply rotating a piece of rectangular paper).

Conceptually, the two forms of multiplication can collapse back together later. Indeed, what ought to happen is what Devlin suggests ought to be the starting point: Multiplication involves values and units being combined in a specific way. If he had his way, I believe he’d get rid of the aggregate teaching altogether. If three friends have five cookies each, then (3 × 5) @ (groups × cookies) = 15 cookies. If my garden is three yards by five yards, then (3 × 5) @ (yards × yards) = 15 square yards. With a bit of philosophy about abstract (default) units, which we ought to be teaching anyway, we can accomplish aggregation using area-based methods.

However, I don’t support teaching it this way from the outset, because I do believe that very young students would struggle more with this method. I do think teaching the two views of multiplication (per the Common Core standards) is a good idea at that level. We can then introduce, perhaps around middle school, the idea of default units. I’ve discussed that concept before in this blog, but the relevant bits here would be:

• To add values, you must have the same unit. There is a default unit when none is specified. The default unit must be the same.
• When multiplying values, multiply units as well. There are two abstract units: The default unit U and the group unit G. \(U \times U = U^2\), while \(G \times U = U\).

Once students understand this, we should drop the notion of “multiplication is repeated addition” almost entirely (it may revisit briefly if we start calculus with Riemann sums, but only until we’ve moved past those into full-fledged integrals).

But about this teacher, and this problem…

Point for the Prosecution: At the end of the road, multiplication isn’t repeated addition in the first place and multiplication (of scalar values) is commutative in the standard complex number system, so forcing third graders to distinguish 3 + 3 + 3 + 3 + 3 from 5 + 5 + 5 is not truly preparing students for a solid grounding in future mathematics. It’s stressing a trivial point at the cost of frustrating a student over an irrelevancy.

It is certainly true, as Mehta and Berry argue, that 3 + 3 + 3 + 3 + 3 is not equivalent to 5 + 5 + 5, but neither is 5 * 3 equivalent to either. Just as commutavity is the result of  mathematical consequences of the nature of multiplication, it is equally arbitrary that the first value in aggregate multiplication of 5 * 3 is the number of groups and that the second value is the number of items per group. Why are we forcing students to abide one arbitrariness with minimal or no comment, but penalizing them for functioning under the other? “5 * 3” can just as naturally be read in the context “There are five cookies per pile, and three piles” as in the context “There are five piles of three cookies each”.

This redeems the Common Core standard: “Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each.” can also imply an example of “interpret 5 × 7 as the total number of objects when there are 5 objects in each of 7 groups”… i.e., 5 × 7 = 7 + 7 + 7 + 7 + 7 = 5 + 5 + 5 + 5 + 5 + 5 + 5.

Grammatically, as I noted last year, “five times seven” is confusing. Historically, it probably did indeed mean “seven, five times”, and so the interpretation of 5 × 7 = 7 + 7 + 7 + 7 + 7 is the more linguistically defensible one. But I say “more” defensible because, regardless, “five times seven” is not proper modern grammar outside of mathematics. The way it would be said is “seven, five times”, and that’s confusing enough to defend a student interpreting “five times seven” as “five seven times” (as my six-year-old son does). So either interpretation is linguistically legitimate. Which is mathematically “correct” is a matter of opinion.

Furthermore, the argument that two out of the four operators don’t allow commutavity is countered by the point that three of the four operators clearly involve taking the first number and doing something to it according to the second number: 4 + 2 means “starting at 4, add 2 more”; 4 – 2 means “starting at 4, take 2 away”; 4 / 2 means “starting with 4, divide it into 2 groups”. The same is true for exponents: 4 ^ 2 means, “take 4 and multiply it by itself”. So, any English translation aside, 4 × 2 ought to mean “take 4 and repeat it 2 times”, i.e., 4 × 2 = 4 + 4. This is an even more basic issue than commutativity is. So, on this argument, the student is right and the teacher is wrong.

In the end, I am more cynical about this teacher’s motivations than the Witnesses for the Defense are. I am not inclined to think that this teacher is thinking about how allowing students to implicitly commute multiplication (if indeed that’s what they’re doing; perhaps they’re simply parsing “five times seven” differently) will cause them confusion in array multiplication. There are valid arguments for the teacher’s response and valid arguments for the student’s response, and I think the reason the teacher marked this wrong is because the teacher said to do it a certain way and hasn’t thought much of the issues in this blog post through at all. This is Cookbook Mathematics, and it did not start with Common Core: Follow the instructions. Do it this way. It doesn’t matter if your cookies turned out fine, because you added the sugar to the flour when I told you to add the flour to the sugar, so you did it wrong.

Every child knows that sometimes order matters and sometimes it doesn’t: You can put your left sock on, then your left shoe, then your right sock, then your right shoe. Or you can put your left sock on, then your right sock, then your left shoe, then your right shoe. But you can’t put your shoes on before your socks.

So this is not a problem with Common Core, this is a problem with mathematics education. Taken as a whole, Common Core actively discourages “Do it my way or you’re wrong”.

I could understand if the instruction had said to use repeated addition as a strategy and the student had drawn a picture to show that the area of the rectangle was 15. If the teacher has insisted on using repeated addition and proceeded to mark the student wrong for understanding the concept of multiplication but not using something resembling the “right method”, well, okay, sigh, but whatevs. But the student did use repeated addition, just not the way the teacher had taught, and should not have gotten marked wrong for having done so.