# Conversations on the Rifle Range 23: The Quadratic Formula Ultimatum, and the Substrate of Understanding

Barry Garelick:

t took me about three weeks to learn all the names of my students. Identifiable patterns of behavior took me a little longer. For example, Cindy, who is in one of the two algebra classes, tended to stop me in the midst of explaining a new procedure and say: “Wait, wait, I’m confused, I don’t understand.”

Over time it got so I could anticipate when this would occur. When showing the factoring of 9×2 –16, for example, I paused after writing down (3x + 4). As expected, I heard: “Wait. I’m so confused. Where did the 3x + 4 come from?” I knew it was Cindy.

“I just don’t understand why it works that way,” she said. I had started the lesson by having students multiply (x-y)(x+y) and other similar problems showing how the middle value drops out. It is not unusual for students to have difficulty extending the pattern of the x2– y2 form to one like 9×2–16. I explained how it worked. Students got it but Cindy persisted. Once she understood something, she got it, but until she did it was painful—particularly when she would get frozen and could not move on until she understood, which was the case here. Students who manage to get it groan when this happens. Someone told Cindy “Because it works out that way; just follow the rule and figure it out later.”