“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
When students learn in a communicative classroom, their written work becomes another place through which they are communicating their mathematical thoughts and ideas. Two-column proofs are the typical format that is used in high school geometry. However, do two-column proofs exist anywhere outside the secondary school curriculum? The answer is no. So, if we want to bring the activities of the classroom closer to the profession of mathematicians, then we ought to ask them to work in a medium and in a format that people who use proof actually use. Mathematicians use the language of mathematics--algebra and logic--and they use their spoken language. Thus, my students have learned to write their proofs with both prose and algebra.
The following are examples of student-generated proofs: