Concepts and Problems

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 \sqrt{-16} doesn’t have a meaning [in real numbers], while \sqrt[3]{-8} does.

I was really, really, really pleased with my class’ answers. In the course of their explanations, almost students mentioned that \sqrt[3]{-8}=-2. Literally on the same page, however, was a set of radicals that I asked students to simplify. One of them was, gasp!, \sqrt[3]{-8}. 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 \sqrt[3]{-8} was -2 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.



  1. This kind of story makes me want to do away with assessments divorced from meaningful context altogether. It’s the deal with the devil of algorithms and algebra – we exchange meaning and context for efficiency. We would never teach one without the other, but whenever we test mechanistic skills it’s hard to avoid. (I hope I’m making sense, I’m not trying to say you’re doing something wrong, I run into the same kind of problem.)

  2. @Kate: You’re railing against mechanistic procedures instead of conceptual understanding of the procedures and why they work. Which I agree with mostly. But I guess I’m confused about your first sentence. Do you think that my question about squareroots/cuberoots is “divorced from meaningful context” or testing “mechanistic skills”? Because that’s precisely what I was trying to get away from with that question. I was hoping to get them to think about what the two symbols mean and explain it coherently. Which seems like it would be in the vein that you’d be advocating. If not, any ideas on how to introduce nth roots?

    @Jackie: I haven’t returned the quizzes yet. I’m very interested too.

  3. I suppose your first question framed the cube root of -8 in a way that led students to the right way of thinking. In the latter problem, they didn’t have your guidance. I think your test is good pedagogy: guide the students first, then see whether they can still perform without you providing context.

    This reminds me of a question my advisor asked me during my oral exams in grad school. He gave me a concrete question I couldn’t answer. Then he said “Let me ask you a harder question” and made the question more general. Still no response from me. Then he said again “Let me ask you an even harder question” and stated the problem in an abstract form that put me the right frame of mind and I could answer.

  4. I was saying your “set of radicals I asked the students to simplify” was testing mechanistic skills, not your asking them to explain the difference between the square root and cube root of a negative. Reading my comment I don’t really agree with myself. Mechanical fluency and accuracy are important, too.

    Did you ask them what they thought, like Jackie suggests? I’m interested to hear, too.

    I don’t have a secretly great way to teach nth roots. I just stick to the stuff I usually do – asking the question forward and backwards, and making them figure out as much as possible and explain it to each other.

  5. @Kate: Oh, we were totally on the same page then, Kate! Yessssss! Huzzah!

    @John: Your grad school oral exam (shudder). I hate thinking on my feet. Which doesn’t bode well for me signing on for career of “teacher” does it, now that I think about it.

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