Our first math theme on geometry ended on Friday. Cathy taught students about polygons, Gabe taught quadrilaterals and I taught angles and triangles. We each did a three day series of lessons and our classes rotated through each series.
One of the key things we worked on was measuring and drawing angles using a protractor. Angle measurements are surprisingly abstract. One of the things that we've learned through our lesson study process (which is part of our professional development) is that listening to students as they work through a math concept can provide invaluable information on how to best teach that math concept. I had the luxury of teaching this set of lessons three times and each time I honed my language and approach based on the thinking I was seeing.
Some students wanted to measure the length of the rays to measure the angle. Others would place the protractor on the paper but not so that it aligned with the angle. Others weren't sure which numbers to read (or what they meant). I worked hard to find language that clearly communicated just what a degree was and what we were measuring - the more I listened to the students, the better I got at anticipating mis-conceptions and addressing them.
But even after a careful introduction, student understanding is often shaky. On the third day of our lesson, I asked them to build shapes with angles and lines using the logo programming language. We used very simple commands to order a turtle about the screen.
As students experimented to make a triangle, hexagon and octagon - they began to really understand the concept of degrees. Their initial guesses as to how much to turn the turtle were often far off -- even though they had all shown "mastery" on the angle measuring worksheet the day before. They needed to use the measurements in order to really get them. The more they worked with degrees, the better their estimates became.
This, of course, is the crux of progressive education. Students learn things deeply when they understand the utility of what they are learning -- when they feel a need for the concept. While the foundation lessons of using a protractor and learning different angle types were necessary background, it wasn't until they were ordering a turtle around that they truly saw got what these numbers meant.