This is from yesterday’s survey, which was discussed over at this post. What do you make of the responses, particularly the differences between  (2a+6) in the first response, and (2x+49) in the second?

This post is part of a series analyzing a bunch of survey results. For previous posts, go here and here.

Noteworthy:

• 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…

The submitter reports that this happened with several different students who went up to the board to solve proportions problems. This was the “Warm Up” exercise.

How would you react to these mistakes in class?

Thanks to Victoria for the submission!

How would you help this student?

Another thought: would this student have made this mistake at the beginning of the problem? In other words, is this mistake more likely to happen as the problem goes on than at the beginning? If so, then what does that say about problem-solving?

Thanks to Anna for the submission!

What’s the fastest way of helping these students?

Thanks to Anna for the submissions.

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?

What do you notice in this student’s work?

Thanks to Matt Owen for the mistake.

There’s a lot going on here. What about this work do you find the most interesting? How would you help this student?

Thanks to Heather for the submission!