I just dug this up. It’s what I handed back students after a “pre-quiz” (i.e. a quiz at the end of the unit, but before their quiz). I had forgotten that during that first year I handed back these things with class performance percentages on them.

Anyway, the way those percentages break down is interesting to me. Is it surprising that kids had so much trouble with negative exponents in numerical context, but had such less trouble with variables?

Last week I posted a short video from a tutoring session I had with a kid. We were solving equations, and he had some interesting ideas, and it was nice to have those ideas and his mental workings become explicit.

Here’s another chunk of that video:

Comment on whatever you like, but here are some prompts:

  1. Help me understand his thinking. How did he devise his test for whether his solution is correct?
  2. What does this say about what he thinks about 2/0?

Or jump in with whatever you like in the comments.

Comment on anything that you like, but here are some prompts:

  • Do you like my questioning? (Because I didn’t. Notice that moment when I pause and try to unask my question?)
  • Describe, as best you can, the way that this kid thinks about the Distributive Property.
  • What would you say next, if you were me?
  • Does this video have implications about the way that you’d teach this topic? (And, come on, don’t give me that “they need lots of practice stuff.” Of course they do. But what else?)
  • Do you prefer having videos over images on this site? (Because I have about a half hour more footage from my work with this kid…)

Looking forward to a great bunch of comments here. Don’t let me down?


Here’s how I responded:


  • The kids have a ton of confidence, even in the stuff that they haven’t formally studied in class yet.  (For this survey, Questions 1-3 had been covered formally, and Questions 4-5 had not.) To my mind, this continues to reaffirm that the most annoying mistakes aren’t the distortion of instruction; they’re the failure of instruction to override preconceptions.
  • Kids like to say that (x+7)^2 = 49, and teachers like to say that this is due to overuse of the Distributive Property. That might be true, but those teachers also have to recognize that kids said that (a+3)(a+3)=a^2 + 9 with almost the same verve and frequency. It’s hard to blame exponents or notation for that mistake, right? So where does this intuition come from?
  • A couple of kids included a 2a term in Q4 and a x^2 term in Q5. I find this interesting, but I’m not exactly sure what its significance is. Is the temptation to add a+3 and a+3 when the binomials are in the same visual position that they are for addition problems?

The idea that kids walk into our classes with these intuitions is, I think, counter to the way that most math teachers talk and think about these mistakes. I think that realizing that these mistakes are the result of deep intuitions about how math should be is important. I also think thinking about where these intuitions come from is important, because maybe we can avoid setting them in earlier years.

I hope that some of you will give this survey to your students who haven’t yet received instruction on how to multiply polynomials. The original survey can be found here.

You’ll disagree with me in the comments, right? I’m counting on you all…

matt owen


Matt submits the above, and Matt writes, “I think it’s especially interesting that this student left the mistake on the board even though she had found the correct solutions by graphing in Desmos.  I’m not really sure if she did half of forty, or sqrt 4 and then stuck a zero on it (she wasn’t sure either).”

I vote for “half of 40.” You?