Last week, I sat down to mark a quiz on radicals and exponents that I had given my grade 10 class. I knew that students had found the quiz difficult (there were gasps and shrugs from each student at some point during the quiz), but I was still surprised by the results. Problems that required skills that I had demonstrated accurately in class were answered incorrectly, and students made errors that suggested misconceptions. What was most discouraging was that both my students and I knew that their performance on the quiz did not reflect the significant amount of work they had done to prepare.

As I mulled over the results, and contemplated what to do next, two thoughts emerged:

- Students saw questions that looked a bit different from what they had seen in their homework and they panicked. As soon as the sinking feeling of
*I don’t know how to do this*crept in, they weren’t able to employ the skills that they had used successfully several times before. - Students had learned how to use specific skills to solve certain types of problems, but didn’t know how those skills could be applied together. For example, they knew how to convert a fractional exponent to a radical, and they knew how to rewrite a number with a negative exponent as a fraction with a positive exponent, but they didn’t know how to deal with an exponent that was a negative fraction.

With these two observations, I was reminded of my greatest weakness as a math teacher: as someone who is predisposed to think quantitatively, logically and to make connections between math skills, I often forget that many (possibly even most) students have a different cognitive disposition and need to learn how to think mathematically. In other words, the strategies that have largely contributed to my success in math are strategies that are intuitive to me. Being aware of my (subconscious) intuition, and making my strategies explicit/visible, is essential in helping students to develop those same strategies.

So, rather than asking my students to do corrections on their quiz to see if they can improve their score (which is essentially asking them to do the same thing but expecting a different result), I have attempted to map out what the skills and content *look like* in my brain (radicals-and-exponents-concept-map). My hope is that by making my thinking more visible, students will be able to articulate and deepen their own thinking in order to enhance their understanding.