I never thought they would have trouble with the definition of the word "angle".

Apparently the textbook writer's didn't think so either because a definition for angle is nowhere to be found. So, I began to probe and ask for students to tell me what an angle is. I knew I would not get a rigorous mathematical definition, but I expected some understanding of the concept. A girl suggested, "In a shape, it's the corner." A decent beginning. I hoped for others to build upon that idea.

Nothing.

Complete silence.

I realized some of them had ideas but weren't going to share. Others had no clue. Still others could care less. They expected me to give up and tell them the answer. I refused. I told them:

"I know what you're doing. You think I'm going to break down and tell you the definition. Well, I'm prepared to wait you out."I made each table discuss and come up with some definition. Wrong answers were OK, even good as a starting point for learning. But, every table had to have some reasonable definition. Then I would randomly call on one member of the group to share.

This worked pretty well. We had some interesting ideas and I tried to use Socratic questioning to get at student's underlying thinking and assumptions. I had limited success. Eventually I dragged them kicking and screaming to the idea that an angle was a measure of rotation in the same way length was a measure of distance.

The point of all this is not the students lack of understanding of what an angle is. It is my lack of ability to draw out what I am sure they already intuitively know... an angle measures rotation. I need so much more practice at Socratic questioning. I need to plan ahead and script a discussion so I can anticipate likely misconceptions and difficult concepts.

I was excited about the overall effectiveness of the discussion despite it's unimpressive final results. As I tell my students, initial failure is good if it moves us towards understanding and learning.

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