A bafflingly graded third-grade math quiz caused a firestorm on Reddit last week and has spread across the internet, causing several people to question Common Core math standards and the teacher’s implementation of them.
The quiz is on some of the most fundamental aspects of basic whole-number multiplication and how students encountering multiplication for the first time can think about problems.
The first question asks the student to calculate 5 x 3 using repeated addition. The student wrote 5 + 5 + 5 = 15, and was marked wrong, with the teacher writing in the “correct” solution of 3 + 3 + 3 + 3 + 3 = 15.
The second question prompts the student to calculate 4 x 6 using an array. The student drew an array with six rows and four columns, getting the answer that 4 x 6 = 24. The teacher again marked the question wrong and drew in a nearly identical array of four rows and six columns:
This, naturally, has a lot of people riled up.
Here’s what’s going on.
This is bad math
It is obvious to anyone who looks at this problem that the student did precisely what was asked. For the first question, they interpreted multiplication as repeated addition. For the second question, they went with a graphic interpretation of multiplication, looking at a stylized version of finding the area of a rectangle.
One of the most basic properties of whole-number – and integer, rational, real, and complex number, along with many more – multiplication is commutativity: for any two numbers A and B, A x B = B x A. Order does not matter in multiplication; adding 5 together 3 times is exactly the same as adding 3 together 5 times.
To a third-grader encountering arithmetic for the first time, that might not be immediately obvious. This actually gives one of the strengths of looking at more visual illustrations of multiplication like the arrays in the second problem: Many students will quickly see that an array with four rows of six columns has the same size as an array with six rows of four columns.
A teacher penalizing a student for recognizing and applying commutativity is extremely harsh and unwarranted. One possible rationale for the grading scheme could be a formalistic issue: The curriculum or teacher might have formally defined multiplying together two whole numbers A and B as the total number of objects in a collection of A groups of B objects each. In that case, 5 x 3 would be defined officially to be 5 groups of 3, or 3 + 3 + 3 + 3 + 3.
It still makes no sense to penalize that student, even in this case. Commutativity is one of the first properties that emerge from that definition, and the student is still, in both problems, capturing the essence of what multiplication is.
This is NOT a Common Core standard
This example, along with so many other viral math problems that baffle students and parents (like this subtraction problem or this check mocking a first-grade counting exercise), is being used as an example of Common Core math being unduly confusing or frustrating.
While this worksheet does present a frustrating situation, it has nothing to do with Common Core. Common Core lays out a set of objectives for what students should be learning in each grade level. It’s still up to individual states, districts, and teachers to come up with the curricula and lesson plans to achieve those objectives.
As New York City high-school math and physics teacher Frank Noschese told Tech Insider’s Madison Malone Kircher, “The standards just lay out what kids should know and be able to do, not actual lessons. Nothing in Common Core forces the specific interpretation these teachers used.”
The two Common Core standards listed at the top of the quiz ask students to “interpret products of whole numbers” and to “use multiplication and division within 100 to solve word problems.”Absent from those standards is an insistence on slavish devotion to a pedantic hyper-formal definition with no particular mathematical meaning.
Indeed, penalizing the student for their recognition that 5 x 3 = 5 + 5 + 5 just as much as it does 3 + 3 + 3 + 3 + 3, or that a grid with six rows and four columns is the same size as one with four rows and six columns, goes against the deeper spirit of much of the Common Core math standards: to reinforce a fundamental understanding of what numbers and operations are and how they interact with each other to provide a solid foundation for further mathematical study.
However this grading mishap happened, it wasn’t Common Core’s fault.
This has nothing to do with higher-order math
One explanation that was floated by several Redditors (ninjakiti here is one example) and blogger Hemant Mehta revolves around the fact that for arrays in computer science and matrices in linear algebra, order does matter.
The idea is that, in the second problem, an array or matrix with four rows and six columns is in fact a very different thing than an array or matrix with six rows and four columns. It’s conventional to describe the dimensions of a matrix by putting the number of rows first and the number of columns second. In that case, the 4 x 6 array asked for on the quiz would be different than the 6 x 4 array the student drew.
I do not think this is the solution here, for two reasons.
First, this is a problem involving basic single-digit whole-number multiplication. It is likely one of the first times students are encountering multiplication. Students are unlikely to encounter matrices until their later high-school years, and it would seem a rather odd pedagogical choice to enforce a convention in linear algebra and computer programming that students aren’t going to see for nearly a decade, without any explanation of why that convention matters, at the same time students are expected to grasp the most basic properties of integer multiplication, like commutativity.
Second, commutativity is a bedrock mathematical fact, while matrix dimension notation is an arbitrary convention. One of the few things I am willing to accept as an absolute, immutable, universal truth that holds in all times and places is that the order in which two whole numbers are multiplied together does not matter. 5 x 3 = 3 x 5 is written into the fundamental fabric of reality in a way very few other things are.
Meanwhile, describing matrices as rows by columns is essentially arbitrary. We could have just as easily chosen to write them as columns by rows. As it happens, modern mathematics settled on the former rather than the latter. This is similar to how English is read from left to right, while Arabic is read from right to left. Neither of those are fundamental, inevitable aspects of these languages; they are just how things worked out.
The idea that a student should be punished for recognizing and applying the fundamental truth of commutative multiplication in service of drilling in a completely arbitrary convention that they can easily learn when they need it 10 years later strikes me as borderline insane.