Earlier this week, I was reminded of a passage from How Not to Be Wrong by Jordan Ellenberg:
Here’s my earliest mathematical memory. I’m lying on the floor in my parents’ house, my cheek pressed against the shag rug, looking at the stereo. Very probably I am listening to side two of the Beatles’ Blue Album. Maybe I’m six. This is the seventies, and therefore the stereo is encased in a pressed wood panel, which has a rectangular array of airholes punched into the side. Eight holes across, six holes up and down. So I’m lying there, looking at the airholes. The six rows of holes. The eight columns of holes. By focusing my gaze in and out I could make my mind flip back and forth between seeing the rows and seeing the solumns. Six rows with eight holes each. Eight columns with six holes each.
And then I had it—eight groups of six were the same as six groups of eight. Not because it was a rule I’m been told, but because it could not be any other way. The number of holes in the panel was the number of holes in the panel, no matter which way you counted them.
Whether 5×3 is five groups of three or three groups of five was at the center of one of those "Common Core math" outrage posts earlier this week. (I put Common Core in quotes because although these problems are usually shared as examples of “Common Core” math, the Common Core is a set of standards, not a curriculum. For more on that difference, see the Common Core website.)
I usually try to avoid commenting too much on pro- or anti- “Common Core” math posts I see shared on Facebook or Twitter. I don’t have children or much experience teaching children, so I feel unqualified to make sweeping statements about the merits of different methods of teaching or learning in elementary school.
That said, I do watch the Common Core posts with some interest. Math education matters to me, and I am interested in seeing it done well. In general, I tend sympathize with the views expressed in this Medium post by Brett Berry. It seems to me that sometimes parents who are frustrated with "Common Core math" did not have a good experience in their own math classes and don’t have some of the facility with mathematics that Common Core aims to develop in children. It seems short-sighted to object to a new teaching method if the old one didn’t work very well.
In the most recent flare-up, a student was asked to solve 5×3 by repeated addition and wrote 5+5+5=15. The “correct answer” was supposed to be 3+3+3+3+3=15. The next question asked for the student to solve 4x6 by using an array. The student drew an array with 6 rows and 4 columns instead of one with 4 rows and 6 columns and lost a point.
As young Jordan Ellenberg discovered while staring at a stereo, it is a basic fact of math that it doesn’t matter what order you multiply numbers in. Multiplication commutes: three groups of five has the same number of objects as five groups of three. It’s easy to feel outraged that the student was punished when their answer was correct. Hemant Mehta, author of the Friendly Atheist blog, wrote a lengthy post laying out some reasons the teacher may have had for marking 5+5+5 incorrect. Mehta’s basic argument is that teachers are not only teaching students how to do problems now but preparing them for future math classes. (In case you’re curious, he does link to the Common Core standard that mentions grouping in multiplication.)
I agree with some of what Mehta wrote, but I have two main objections. One is about semantics, and one is about broader pedagogical goals.
First, semantics. What does 5x3 mean? I think there’s an equally (un)convincing case to be made for either five groups of three or three groups of five. If we say “five times three,” the literal meaning of the words is more like five groups of three than three groups of five, but it’s not the way we would usually say it in English. I think the five groups of three interpretation has an edge, but it’s only a slight one.
If we look at the mathematical notation, on the other hand, there is a case to be made that three groups of five scales up better. Specifically, if we want to think of multiplication as repeated addition, exponentiation as repeated multiplication, and ↑↑ as repeated exponentiation, three groups of five is the way to go. 53 means 5×5×5, not 3×3×3×3×3. Treating 5×3 as 5+5+5 is more consistent with this notation. (You might argue that our notation for exponentiation should change, but you’ll be fighting an uphill battle.)
All in all, it seems to me that 5x3 is a draw between five groups of three and three groups of five from a semantics point of view. It's worth noting that the Common Core standard mentioned above does say, "Interpret products of whole numbers, e.g.,interpret 5×7 as the total number of objects in 5 groups of 7 objects each." It's not clear to me whether that is the only way we are supposed to interpret 5×7.
This brings me to my next objection, which may be more substantial but may also be more context-dependent. I was not in this teacher’s class, so I do not know why treating 5x3 as five groups of three rather than three groups of five was important to him or her. I don’t know whether the students just learned multiplication or were reviewing it or what emphasis the teacher put on the row x column view of multiplication.
With those caveats out of the way, my objection is to Mehta’s suggestion that treating 5x3 as five groups of three will help students in the future with either division or matrix multiplication.
First, I just don’t buy the argument when it comes to division. Mehta writes,
When they see a problem that says 5 x ___ = 15, they’ll be thinking 'I need five groups of SOME NUMBER to get to 15.' In other words, they’ll be able to pick up division a little more quickly because they’re learning the proper way to think now.
I’m not so sure. If instead they see a problem that says 5×___=15 and think “I need SOME NUMBER of groups of five to get 15,” I feel like they are just as prepared for division. It's not clear to me that either way of thinking builds better intuition for division.
What about matrix multiplication? It’s true that when it comes to matrices, A×B is generally different from B×A. (It may even be a different size matrix.) Students sometimes struggle with this harsh reality, and I agree that it’s worth trying to help prevent them from making that mistake. In the case of the rectangular array problem, I'm not convinced that insisting that the first number must be the number of rows and the second the number of columns would help students with matrix multiplication in the future. As Andy Kiersz writes, "describing matrices as rows by columns is essentially arbitrary. We could have just as easily chosen to write them as columns by rows." Even if it would help students remember which way matrix multiplication works, I worry that a rigid focus on which number is a column and which is a row sacrifices number sense now for the sake of matrix multiplication later, and it’s hard for me to believe the tradeoff is worth it.
I remember being in elementary school and discovering that if I multiplied a number by 10 and then divided it by 2, the end result was multiplying by 5. I could multiply by 10 and divide by 2 in my head, so it saved me the trouble of writing down the algorithm for multiplication by hand. I thought I was cheating. Multiplication was something you did by writing two numbers down in a column and then applying a bunch of memorized facts to them in a prescribed order. It made sense to me that my way gave me the right answer, but I thought you just weren’t supposed to do it that way. It took a while before I understood that having a robust set of methods for tackling arithmetic problems was helpful, not a cheat.
In general, it seems that the Common Core encourage the development of number sense and the use of multiple methods to solve problems, which would have eased my elementary school conscience. An important part of number sense is the ability to exploit the associative and commutative properties of addition and multiplication to make problems easier. 45+38 is kind of hard, but I can transfer 2 from 45 over to the 38 to get 43+40, which I can do in my head. I worry that if we are rigid about what 5×3 means, we won’t encourage students to build the fluency with numbers that will help them find easy ways through arithmetic problems.
Do you think emphasizing the row and column view of multiplication would keep students from having so much trouble with matrix multiplication in the future? Do you think there is a good pedagogical reason to keep five threes and three fives distinct for students? Is it important that students can explain why five threes and three fives are the same before they can use them interchangeably? Do you think it's more natural to interpret 5×3 as five threes or three fives? I'm curious what you think.
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