“The essence of metaphor is understanding and experiencing one kind of thing in terms of another.”
-George Lakoff and Mark Johnson, Metaphors We Live By
Geogebra and Google Docs
While class discussions have been very productive for enhancing discovery and searching for deeper understanding of the content, I found that students would agree with each other's claims, even when they were misusing their mathematical vocabulary. For instance, a student would say, "Point A is parallel to line BE," and another student would respond with, "yeah, you're right, and that is why the angles are congruent," instead of picking up on the flaw of the statement--that points do not have extension, and cannot be parallel. While they are getting accustomed to the conversational/discovery approach, they are often insecure about articulating their mathematical thoughts. Students empathize with each other, and are more inclined to be agreeable than critical.
To address this issue, I constructed an environment where students participated in discussions in writing. In these sessions, students were not allowed to talk to each other out loud. Rather, they wrote to each other in a Google Doc. I wrote applets in Geogebra that had questions that they were asked to discuss. They worked in groups of three and talked through their discoveries--much like they do in class around the table. What I found, though, is that students were more focused on the precision of their language when they communicated in writing than when they were communicating face to face.
Here is the link to my geogebra book: http://tube.geogebra.org/student/b151520