Thanks to Tina, we’ve got this great example of a tiny little error that crops up during complex numbers. Here’s my take on it: there’s no way that this kid would make this mistake if their problem was just “Simplify the square root of negative 4.” When the skill is laid forth is such a direct way, it’s very clear what the student is supposed to do. But when the skill is embedded in a much more complex problem, the student “handled” the negative root by realizing that this was a context that deserved a complex number. Happy and satisfied that they noticed and “handled” every aspect of the problem, the student moved on.
I like calling these sorts of mistakes “local maxima” mistakes, and I think they’re fairly common. To me, the importance of these sorts of mistakes is that they reveal the problem with testing any skill in isolation of others. I’m <i>absolutely sure</i> that this student could simplify the square root of negative four if plainly asked to. But that didn’t mean that this student was able to use that skill in this context, when there are many more things to juggle.
To me, this means that you can’t really assess any individual skill in that sort of isolation. Instead, I’d prefer an assessment system that gives students a bunch of chances to use a skill — unprimed — in the context of a fairly difficult problem. If the student can simplify negative radicals in 3-4 more involved problems, then I’m pretty confident that this kid has that skill down.
This is a straight up student-empathy question: what was this kid’s thought process like?
Thanks to Tina for the submission!
Editor’s Note: I categorized this as Grade 7 – Geometry in the CCSS, but I’m not really sure if that’s right. Where does this belong?
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!
This is just a note to make something official that I’ve been doing unofficially for a few weeks.
I’m now posting 3 mistakes a week instead of 5. I think that’s more keeping with people’s ability to process this stuff and comment intelligently. Bonus: it gives me a bit more time, a bit less pressure.
Think this is the wrong move? Let it rip in the comments.
Thanks all for all your help and support and things.
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!
Let’s take for granted that these students don’t have conceptual understanding of the Pythagorean Theorem, because if they did, then they wouldn’t make these mistakes. (I actually think that we need to be more careful with the ways that we toss around phrases like “conceptual understanding” but whatever.)
What do these mistakes reveal about how these kids think about right triangles and the Pythagorean Theorem in the absence of conceptual understanding? Why does this ever make sense to the student?
Thanks to Michael Fenton for the submission!
These are some “Always, Sometimes, Never” questions. Like, “Is it always, sometimes or never true that a rhombus is a parallelogram.”
What’s the fastest way to help these students?
(Thanks for the submission, Tina C!)
I’d love to hear anything that you’re thinking about this in the comments.
I wonder: should we be giving kids explicit guides for these sorts of algorithms at all? Algorithms ought to be patterns in thought. Does a mistake like this begin to make a case against these sort of printed aids?
Thanks to Professor Triangleman for the submission.
What strikes me about this piece of student work is how clean and predictable their mistake is.
Is this sort of mistake the rule or the exception? Does a mistake like this reflect the fact that many/most student errors are due to coherent mental models, or is it the rarer exception in a world dominated by stormy minds that fling ideas at math less predictably?
Thanks again to Dionn!
I’ve been reading some constructivist stuff lately, so…
What resources does this student have for refining their understanding of fractions?
(Thanks for the submission, Dionn!)