In my classes this year, I’ve been really concertedly trying to emphasize that students need to really understand concepts and explain ideas in written form clearly. Today I’m faced with a conundrum about how students are connecting concepts with the problems we’re doing.

On my Algebra II quiz, I asked:

Explain — using complete sentences and proper mathematical terminology — why doesn’t have a meaning [in real numbers], while does.

I was really, really, really pleased with my class’ answers. In the course of their explanations, almost students mentioned that . Literally on the same page, however, was a set of radicals that I asked students to simplify. One of them was, gasp!, . It was an oversight on my part and I will probably change if I use parts of this quiz next year. Can you see where I’m going with this?

There were a few students would could do the conceptual work — who even showed that was in their written explanation — who didn’t get the exact same question right below it correct.

Color me flabbergasted. (What is that, a pukey yellow?) It’s just so hard to figure out what was going through their heads.