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.