Following up on the work of Serra and De Villiers, and in the spirit of recent discussions about the success Bloom's Taxonomy has had in penetrating classrooms, I present the Hexagon of Proof.
The idea is that the reasons that are needed for proof can be developed through a variety of contexts that kids are more familiar with, such as arguing with each other over something controversial. Once a reason have been exposed, though, the reason can have a life of its own in a proof.
Here's how this might look in class:
Disagreeing - Hook the kids into a disagreement. "Draw two triangles. Measure their angles. Do you think that all triangles have that many degrees?"
Debating - In the face of disagreement, ask kids to defend their views. "Some folks here aren't sure that all triangles will have 180 degrees. Why do you think they do?"
Convincing - Give kids a chance to win over their peers. "And what do you respond to your skeptics?"
Explaining - In the face of agreement, explain why something is true. "So, why do all triangles have 180 degrees?"
Teaching - Explain something whose explanation you don't yourself require. This might involve pretending, as in "We know that all triangles have 180 degrees. How could we teach this to a 4th grader who doesn't know this?"
Proving - Use the traditional language and structure of mathematics to prove that something is true, whether or not its controversial or needs explanation. "Prove that all triangles have 180 degrees."