Where We've Been: We've just started to wet our feet with formal two-column and paragraph proofs. I have demonstrated the proofs that are the focus of this lesson, but students have not written any proofs on their own.
Where We're Going: By the end of this unit, students will be writing formal two-column and paragraph proofs on their own without any aids or scaffolds.
So in this section of the lesson, I just want students to remember the two-column proofs they've seen and to get into the right frame of mind for writing proofs. To do this, I just set aside five minutes of quiet time and instruct students to study the examples of two-column proofs I've given them as notes.
This section of the lesson is student-centered. Each student receives a copy of the Developing Proof Templates, Developing Two Column Proof Cutouts, and a pair of scissors. I tell students to cut out the statements and reasons only for the proof on which they are working so that they don't lose any steps.
I explain that they may or may not use all of the statements and reasons. I also remind them that, for efficiency's sake, multiple statements can be linked to the same statement. Students are not allowed to use their notes for this exercise
And they're off...
As students get started I circulate to make sure everyone is on task and that no one is stuck. When students ask me, "Is this right?", I respond, "Explain to me why you think this is correct." As I listen, I respond in several ways depending on the situation:
If there is a substantial error, I think of an open ended question that will force the student to confront their misconception or mistake and then I tend to walk away so that I don't end up answering my own question.
If the student has the pieces arranged correctly but isn't effectively communicating their understanding of the "why" behind the arrangement, I ask more probing questions as to why certain steps are placed as they are. As I listen to their answers, I repeat with correction/clarification/elaboration as needed.
If the student has the pieces arranged correctly, and can convince me that they understand the proof and the arrangement of the steps, I focus on refining their academic language. I also try to think of higher order questions that get at their understanding of the more subtle aspects of the proofs.
When I'm convinced the student has learned what they need to learn, I give them approval to number and paste the steps of the proof and then begin work on the next proof.
In this section, I just want to make sure that students get another chance to think about the proofs they've assembled. I divide the students into groups of four. Each person in the group takes responsibility for explaining one of the four proofs. In their explanations, I direct students to talk about two things:
1. why each statement-reason pair belongs together; and
2. how the steps are arranged in logical sequence
Students will be required in a later lesson to write these proofs without the benefit of the cutouts. During that lesson, I will also provide more thorough explanations of the proofs. At this point, though, I just want students to begin taking ownership of the proof thought process.