This image has been going around online for the past few days:
A student was asked to use the “repeated addition strategy” to evaluate 5×3. The student wrote 5+5+5 and got 15 as a final answer. However, points were taken off since the student was apparently supposed to write 3+3+3+3+3. A point was also taken off of the second question for a similar reason (the student made 6 rows of 4 marks, instead of 4 rows of 6).
Some people use this as a critique of Common Core, saying that any set of standards that leads to this must be flawed.
Other people have defended the teacher for various reasons including that the teacher is being intentional about teaching the techniques/thinking that will be most useful later on when students deal with operations that aren’t commutative (subtraction, division, matrix multiplication, etc.) or that “if the teacher has not covered the commutative property, then it might be unwise to let a student continue with this line of thought.”
Both groups of people are wrong.
First, teachers should never take points off for this. If I were the student whose work lost points in the image above, I would probably never forgive this teacher, and not take him or her seriously for the remainder of the school year, which is no one’s best interest. I assume I would not be alone in having that reaction. Similarly, the above argument that this would only be acceptable if the commutative property had already been explicitly taught is absurd. Punishing students for having (correct) mathematical insights that have not yet been taught is counterproductive.
However, just because a teacher should not have taken points off here doesn’t mean the Common Core is flawed. It is a GOOD thing that the Common Core encourages teachers and students to think about math using various mental models (here, thinking of multiplication as repeated addition or as an array). If you haven’t thought about it before, it is actually not immediately obvious why 5+5+5 should be equal to 3+3+3+3+3.
For people fluent in math, we interchange 3×5 and 5×3 automatically in our minds without thinking about it (this called commutative property). For people already fluent in math, it is still valuable to think deeply about why it is actually the case that 3 groups of 5 is actually equivalent to 5 groups 3.
For people not (yet) fluent in math, it is valuable to think deeply about why this actually the case so that this can be integrated into their understanding of math as they develop math fluency. Looking at 3×5 and 5×3 both as repeated addition and as arrays helps students to more deeply understand what is really going on here. The Common Core, like every good math teacher, WANTS kids to realize that these are all different representations of the same thing.
Here’s what could have happened, consistent with both Common Core and common sense.
Teacher asks four questions:
- Evaluate 5×3 by repeatedly adding 5 the proper number of times.
- Evaluate 3×5 by repeatedly adding 3 the proper number of times.
- Compare your answers in 1 and 2. Notice anything surprising? Explain.
- Evaluate 4×7 by repeated addition in two different ways.
or
- Evaluate 5×3 by making an array with 5 rows.
- Evaluate 3×5 by making an array with 3 rows
- Compare your answers in 1 and 2. Notice anything surprising? Explain.
- Evaluate 4×7 by making an array in two different ways.
When the students later discuss division (or WAY later when they discuss matrix multiplication), there can then be separate deep conversations about why those operations are not commutative.
Having New Eyes 
“I assume I would not be alone in having that reaction.” You assume correctly. I still remember with rage similar instances from up to 20 years ago.
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Posted this earlier this morning on the Math Memoirs blog post on these problems:
I think it’s important to point out that nothing in what you wrote is peculiar to so-called “Common Core” math, a mythical being that exists nowhere but is discussed everywhere. There are, of course, Common Core Standards for Mathematical Practice as well as Common Core Mathematical Content Standards, but neither of these documents comprises specific curricular materials (e.g., textbooks, software, video lectures, problem sets, etc.). So in discussing the above issues relating to interpretations and models of multiplication as repeated addition (another highly problematic issue I won’t go into here other than to mention that the Stanford mathematician and prolific author, Keith Devlin, has written a set of excellent columns urging elementary educators to rethink teaching the notion that somehow multiplication IS repeated addition), it is misleading and probably somewhat dangerous to suggest that somehow the issues you discuss are peculiar to “Common Core” math in any way.
These ideas, while important, existed long before anyone even imagined national standards for math and literacy. And they will exist as mathematical reality long after “Common Core” has faded into distant memory and Americans are upset about some new initiative to try to improve the quality of mathematical teaching and learning. As a nation, we fear change in education, particularly when it comes to mathematics. The 20th century is filled with efforts on the part of various groups and organizations to reform what mathematics is taught in schools and what the goals of mathematics education should be for children. These efforts date back to the early years of the previous century, culminating in various publications in the last decade of that century by the National Council of Teachers of Mathematics (NCTM), whose standards volumes beginning in 1989 met passionate opposition from a small but vocal group of hardline educational conservatives and reactionaries, members for the most part of Mathematically Correct and/or NYC-HOLD. Given that much of the philosophy that informed the Common Core Standards for Mathematical Practice reflect the NCTM Process Standards, it is predictable that there would be some fierce resistance to the latest efforts to make changes in mathematics curricula. However, the most recent reform efforts carry a ton of political baggage unprecedented in nature such that there is enormous opposition from political and educational conservatives as well as many who consider themselves to be political and/or educational progressives. And unfortunately, while I think much of the criticism of the overall national standards initiative is valid, the result has, over the last few years, degenerated into hysterical reactions to problems and worksheets like the ones on display in this article.
It’s great that you have done a lot here to unpack and explain some of the thinking behind how the teacher has graded these problems, but I’m skeptical that anything much is lost by interpreting 5 x 3 as three groups of five versus the “correct” interpretation of five groups of three. Note that this is not a word problem; there is no context in which it is meaningful to claim that there is a material difference between the student’s answer and the “correct” one. Multiplication (and addition) are indeed commutative, and it doesn’t take a teacher explicitly teaching that to students for it to be a property of operations on the real numbers or any of its major subsets. And the inverse operations of subtraction and division will not be commutative in any of those sets regardless of what teachers have taught or any set of curriculum standards state. Kids will figure that out for themselves in many cases long before the official words “commutative,” “associative,” “distributive,” et al. are mentioned to them, and it’s a bit asinine, in my view, to penalize students for offering three addends of five giving the correct result for 5 x 3. If one is going to argue the doubtful proposition that multiplication IS repeated addition, then I can’t see how in the abstract there is any meaningful difference between what the student answered and what the teacher rather rigidly required for credit. This strikes me as the sort of absurd nit-picking that can quickly make students despise mathematics as a bunch of arbitrary rules to be memorized regardless of any sense-making.
Were this a class for elementary teachers, I could see justification for insisting on a particular interpretation. I have much more difficulty seeing a reasonable justification for penalizing primary grade students for thinking perfectly correctly about how to COMPUTE the result of 5 x 3 as an addition problem. Make it a word problem in which there are three bags with five oranges each and that’s a different thing. But this was not that.
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Thanks, Michael! Great (and thorough) points! I agree with most of what you say here.
I will have to look up Devlin’s arguments against thinking of multiplication as repeated addition!
I agree that nothing is written here is unique to “Common Core math” (whatever that actually is)–in fact, that is part of my point: just because someone (rightly) disapproves of the grading of these problems, they shouldn’t immediately form an opinion about Common Core!
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