The students work through their understandings of congruent triangles, the triangle postulates, parallel lines, transversals, and other geometric properties to apply those to create proofs for the quadrilaterals. Making sure the proofs are rigorous (including all steps necessary) is a challenge of any geometry class. Students struggle with how thorough and precise a proof needs to be. The students move between group work and whole class interaction throughout the lesson. In small groups, the students discuss and debate proof arguments. At selected times, the teacher pulls the class together to share findings, ideas, or sample justifications. After sharing ideas or arguments with the entire class, studentsthen return to working in their small groups. The teacher illustrates how to use selected otation in a proof as students work through the logic and reasoning. Instead of a more traditional approach to teaching mathematics, where the teacher presents mathematical notation and format up front, in this class the sharing of how to communicate, using mathematical symbols, occurs when learning situation arises. Once the group has worked through the reasoning of the proof, the teacher checks in with the group and instructs them to begin designing a poster that will display the proof they had created. Groups are instructed to design a poster that contains a drawing of the figure, the conjecture of what is to be proved, a list of the given from the conjecture, and what needs to be proved. The students can use a two column or a flow chart format of the proof.
STUDENT: Well, do the same thing but with different letters.
STUDENT: Yeah, just do the same thing except switch out the step right before it. So instead of A B C D, it’s A B...B C A D.
STUDENT: Now we don’t have a proof for that one over here.
STUDENT: I know but this is the same thing.
STUDENT: Just write the same thing except change that. Yeah, change the transversal. Are we going to rewrite it?
STUDENT: Yeah, we’ll rewrite it. So the definition is here and basic is picture here; this is nicer.
STUDENT: Yeah and bigger.
STUDENT: So that's basic definition and conjecture…
STUDENT: I don’t think I need this anymore.
STUDENT: Do like a rhombus.
STUDENT: Go like that and like this?
STUDENT: It doesn’t really matter.
STUDENT: So write rhombus.
STUDENT: A big rhombus?
STUDENT: Yeah, so three space.
STUDENT: In capital letters or just...
STUDENT: And then we start the definition. The diagonals…start here. So "if the diagonals are at different lengths" and then comma and start there; and "they bisect each other at their midpoints."
STUDENT: I’m going to underline the “if then.”
STUDENT: Hey Sage, after this we’re done right?
STUDENT: Yeah, I think so.
STUDENT: "Perpendicular to each other" and the next one. Then "a quadrilateral is a rhombus." Okay, so next do you guys want to do the conjecture first or make a diagram?
STUDENT: I think we should do it like she did.
STUDENT: Go with a diagram.
STUDENT: I think we should kind of do it like Ms. Humphreys did on the board – how it has the definition.
STUDENT: Of what?
STUDENT: Oh yeah, a parallelogram and then the diagram.
STUDENT: Okay, we can do that.
STUDENT: And put the proof over there.
STUDENT: Where is the definition paper?
STUDENT: It’s right there. Yeah, put that over here and then we put the proof over there.
STUDENT: We have to summarize it – parallelogram is triangle…
TEACHER: You only have to...a parallelogram you can stop right there. That’s all you have to do.
STUDENT: Oh, so a rhombus is a parallelogram in which two adjacent sides are equal; that’s the definition.
STUDENT: Are we doing it here?
STUDENT: Use a different color, use a different color.
STUDENT: Do you have a bigger ruler?
STUDENT: Oh yeah, I do – my foot long.
STUDENT: Yeah, foot long.
STUDENT: Don’t, don’t say anything. Don’t get off topic.
STUDENT: Do you want me to make a star?
STUDENT: Do you want me to make a star?
STUDENT: Do them all over here because we won’t have too much space over there.
STUDENT: I just hope it doesn’t get marker on it. This was in the box and I’m going to…
STUDENT: Are you writing it all out? So next...it’s sticking.
STUDENT: Let me help you with that. It stuck to the table.
STUDENT: Where do you write the paragraph proof?
STUDENT: We’re going to write it right over there.
STUDENT: I think we should write the proof on the back but this sticks.
TEACHER: You can always get a second paper.
STUDENT: Just write it?
STUDENT: So, let’s sketch it out first.
STUDENT: How do you sketch it?
STUDENT: Just sketch out what Sage did. Just sketch it out exactly like what Sage did.
STUDENT: Here, here…
STUDENT: No, I’ll just mark it.
STUDENT: Like stab through it.
STUDENT: I am.
STUDENT: Use Sage’s pencil.
STUDENT: It’s a natural pencil.
STUDENT: Leave a mark and everything.
STUDENT: I don’t think that’s a bad thing.
STUDENT: No, we’ll use it as…
STUDENT: There’s a hole here; there’s one there, there’s one there and that one there. You know, let’s just use the dots.
STUDENT: These ones are awesome too.
STUDENT: I’ll get a second paper.
STUDENT: I’m going to outline.
COMMENTARY BY CATHY HUMPHREYS: Although a wonderful conversation is happening here between Omer and Jerry, I was struck by Julie’s absence in this conversation. Even though she was the one who had the idea to use alternate interior angles as well as the idea of proving triangles congruent, once Omer and Jerry understood her idea they were off and running on their own again. Their eye contact and body language was for a group of two, not three. This raises many issues for me –questions, for example, about the gender equity in this class (even though it is my class!). I know that the boys were not intentionally selfish or mean-spirited, but the result was that Julie was excluded from most of the group work. This clip highlights the necessity of building better group skills. It also made me want to teach interrupting as a skill to be used when necessary. I heard Madeleine Albright talk about how difficult it was at first for her as Secretary of State because women generally do not interrupt, while men do so freely. I think not only Julie, but Omer and Jerry as well would have benefited from a little interrupting so that she would become an intrinsic part of the group.
This clip also makes me wonder: would assigning the group roles and structuring the group work around those roles have mitigated what happened here?
Another thought I have is that perhaps this task would have been better as a partner task; when people are trying to wrap their heads around a logical path, four people’s inputs can make it harder. With partners, there would be more for each person to do and less ideas to form into one coherent argument; this would ensure more equal participation and hence more learning.
COMMENTARY BY COACH DAVID FOSTER: Students developed a deeper understanding of congruent triangles and parallel lines in applying that prior knowledge to this new situation of proving certain quadrilaterals are formed from the formation of diagonals. Students would work through their fleeting understanding of triangular postulates (SAS, ASA, etc.) to use those geometric building blocks to make their proofs. Cathy used students’ partial proofs to improve the students’ understanding and skills with making rigorous arguments. She stopped the group work and had the class hear from a student as she explained how to make the proof complete and valid. Students sharing their work and ideas with the entire class are an effective instructional technique to deepen students’ knowledge and improve their work products. Cathy introduces conventions and notation as the students explore mathematical ideas. This technique of waiting for “just in time” information is more conducive to learning. Too often teachers give students a lot of up front information that students often forget or never use. This “just in time” methodology is more aligned to how students learn. As she moved from group to group, Cathy checks to make sure the students’ reasoning were sound and that there were no holes in their logic. Occasionally she presses on students to explain their thinking or check out an error or faulty assumption. She then would advise the group to start to design a poster that would contain the proof of the quadrilaterals they selected. Most of the groups selected a two-column proof instead of the flow chart format.