• 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?

I’m trying very hard to get people as excited by exponent mistakes as I am. I just think that they’re really cool and interesting. I gave kids this survey today in class to 9th graders who have never seen negative or rational exponents before, just to see what they’d do.

survey 1

The results did not disappoint.  The mistakes they made will find their way to the site soon enough, but for now, drop by Rational Expressions and let me know what you think of the experiment and its results.

Here’s one result of the survey to whet your appetite:


If you decide to give your students the exponents survey, or make a survey of your own on exponents or any subject, I would sure love to see it. Send it my way, if you will.