How kids think about multiplying polynomials

The kids are confident

I recently gave my students another survey. At the time of the survey, the students had just studied exponents and were about to begin multiplying/factoring polynomials. To emphasize: they've never studied multiplying polynomials in a formal setting. The second half of this survey is what you folks call a pre-assessment or something.

This response from a student -- let's call him Mike -- is especially interesting.

Mike nails the first three questions. A conscientious kid, he even expresses modest, self-aware doubt on the Question 3, though he uses the Distributive Property like a pro in his response. In short, Mike is a student who knows how exponents and the Distributive Property work. 

So far, we haven't taken Mike out of his comfort zone. Take a look, though, at what happens when we push him out of the boundaries of what he's studied formally (this year):

The first thing to note is Mike's confidence. He expresses more confidence in the stuff he's never studied in school than the stuff that he studied a week ago. 

Two numbers, not one

Turn next to his particular responses to Questions 4 and 5. Isn't it interesting that he provides different answers to Questions 4 and 5, even though he analyzes (a+3)(a+3) as (a+3)^2? Why could that be?

Here's one take on his contrasting responses: in responding both 9a^2 and x^2 + 49 to two questions that Mike himself takes to be similar, he betrays that he doesn't see those two responses as especially different. In other words, Mike is operating on the "x" and the "7" as two independent numbers, he's doing the same thing with "a" and "3", and as a consequence there isn't that much of a difference between 9a^2 and x^2 + 49.

Debunking(?) the Distributive Property of Exponents

Let's talk, for a moment, about that x^2 + 49. It's by far the most common response to this question on the survey. It's also a very common mistake, one that we've cataloged on mathmistakes.org before -- twice, actually. Here is a sample of some of the comments that teachers left on those mistakes:

"I think the prior comments have essentially nailed it. The trouble is the “distributive property”." 

"I have to say that I don’t get why we talk about “the distributive property” of multiplication over addition, but not that of exponentiation over multiplication. If naming the distributive property is helpful in one circumstance, why wouldn’t it be helpful in the other?" 

"We don’t teach why multiplication distributes across addition, and we talk about a power to a power means you multiply, so it seems very hard to distinguish between this mistake and when you can distribute."

(Heck, I even named one of those posts "The Distributive Property of Exponents.")

But let's look at some of the other responses from this survey:

 

Suppose that the Distributive Property of Exponents theory is right. That theory has nothing to say about (a+3)(a+3). There's no reason why (a+3)(a+3) should provoke a misapplication of the Distributive Property of Multiplication. I mean, there's no visual resemblance, and it's not at all clear to me how students would be misusing the Distributive Property to arrive at a^2 + 9. (Unless they're going through (a+3)^2 along the way, which strikes me as unlikely.)

Maybe the Distributive Property explanation is helpful in some cases, but there's something else going on here. 

So, what is going on here?

Let's look at one more response, before wrapping this thing up:

Why would this student multiply the 7's together, while adding the 3's together?

I used to think that consistent mistakes were the most interesting, but now I'm pretty sure that inconsistencies are far more revealing about what students are actually thinking. They're like tensions in a story or conflicting experiments. 

I don't know how to explain these (fairly typical) responses, but here are the principles that will guide me thinking:

• I won't try to attribute to students well-thought out theories about how multiplication polynomials work. That's inconsistent with the idea that these are intuitions, pre-reflective ideas. (See previously, here.)
• I won't try to blame everything on the Distributive Property. This is bigger than that.
• I'll keep a close eye on the tendency of students to see polynomial multiplication as a series of operations on the individual numbers, rather than taking a more holistic view.

That's it for now. Go forth and comment -- here about the overall analysis, and over at mathmistakes.org for the last survey result from this post.