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.
CATHY HUMPHREYS: Okay, if you can’t explain it, you don’t understand. So make her explain it Drew. What would be helpful is if you work in the center of the table so that everyone could see what you’re doing and can contribute to it. Okay and Tanana what were you writing about?
STUDENT: I labeled ABCD and then I put – I was going to put AEB is congruent to AEB because they're side angle side.
CATHY HUMPHREYS: Okay.
STUDENT: Or it could be…
CATHY HUMPHREYS: But you’re doing rhombus right?
STUDENT: No, we did a kite.
CATHY HUMPHREYS: Oh, you did a kite. Okay, so what do you have to prove…lets back up here for the kite? What do you have to prove?
STUDENT: We have to prove that there are two sets of adjacent sides that are congruent.
CATHY HUMPHREYS: Okay, good. Now what are the criteria that you have for the diagonals?
STUDENT: Um, they have to be perpendicular.
STUDENT: One has to be bisected or they intersect.
CATHY HUMPHREYS: Okay, anything else? What about the lengths?
STUDENT: The generic one is used like twice as long as…like the generic leg.
CATHY HUMPHREYS: Okay, I want you to stop thinking about kite as a real thing in life.
STUDENT: So to make a kite, do the diagonals have to be the same length? Do they have to be different lengths? Did you experiment with that? Okay, why don’t you fiddle around with that because you have the three things; you have to consider the angle and you’ve got the intersection but you don’t have the lengths part yet. So as soon as you’ve got that you can write your conjecture. Okay?
STUDENT: So M is congruent to...okay.
STUDENT: And then if those two are congruent then…
STUDENT: So we have to use our flex property for MQ MQ is congruent to QM. And then for this one we're going to have to say QO is congruent to MQ. And since this angle is congruent, we have side angle side for every single one of them; and then we use the CPTCP right?
STUDENT: Yeah, what’s the title?
STUDENT: Yeah, just say Proofing Points.
CATHY HUMPHREYS: I don’t care about decorations.
STUDENT: Oh yeah.
CATHY HUMPHREYS: I just…okay good! I just want to see the math.
STUDENT: So um, when we're writing our conjecture, does it have to be like...when you’re making a shape or can it be related to the problem with the sticks and fastening the holes and stuff?
CATHY HUMPHREYS: As opposed to…?
STUDENT: Like saying diagonals or sticks.
CATHY HUMPHREYS: Oh no, diagonals is better. The sticks thing was the sort of a launching idea and diagonals is the correct mathematical term.
STUDENT: So you can’t really say that you’re trying to fasten it at the same number of hole…
CATHY HUMPHREYS: Oh right! And how would you say it if these weren’t holes? What would you say?
STUDENT: Same distance from the end point.
CATHY HUMPHREYS: Great! Same distance from the end point. That’s really good!
STUDENT: Okay then and intersect at the same distance from the end point – from one end point.
STUDENT: They intersect at the same distance from the end point.
STUDENT: We’ve kind of already proved that it’s a kite so I think we did something wrong.
CATHY HUMPHREYS: You kind of are ready to prove that it’s a kite or you cannot prove that it’s a kite?
STUDENT: Uh, we wrote that two sets – two adjacent sides are congruent. We wrote that all quadrilaterals would be congruent...
CATHY HUMPHREYS: All what would be?
STUDENT: Sorry, I said that completely wrong. Um, the two diagonals that intersect would be perpendicular and just saying that automatically makes it a kite because there is no way that it couldn’t be a kite.
CATHY HUMPHREYS: Okay, so that’s what you need to prove; that there is no way it couldn’t be a kite. That’s what you have to show – step by step by step. That’s what proving means.
STUDENT: Oh…it’s just like... I kind of get it more now but it just seems like I’m saying what a kite is and then saying it’s a kite. I’m giving it the exact properties of a kite and then saying it’s a kite but then saying all the properties of a kite are given.
CATHY HUMPHREYS: Are they given?
STUDENT: That’s how we wrote it.
CATHY HUMPHREYS: So if the properties are given the perpendicular diagonals, that doesn’t guarantee that it’s a kite. You have to show why that makes it a kite.
STUDENT: Oh, okay.
CATHY HUMPHREYS: You have to show why and you might have to use congruent triangles or something like that.
CATHY HUMPHREYS: But in other words, it seems “obvious” to you but um, it’s actually not unless you can use mathematical reasons why those sides – like, do you know why those side ̶ these two sides are congruent? Why the adjacent…why?
STUDENT: Because the...I don’t know what it’s called. The opposite side of the right angle...do you know what that’s called?
CATHY HUMPHREYS: Well, you can have lots of right angles Ryan – the hypotenuse.
STUDENT: Where are the giant rulers?
STUDENT: What's the name of...?
CATHY HUMPHREYS: The hypotenuse.
STUDENT: Yeah, above the hypotenuse...
CATHY HUMPHREYS: Just because these are opposite right angles does not mean that they are congruent.
STUDENT: Okay, I get it now.
CATHY HUMPHREYS: What can you do to help get this expedited?
STUDENT: The list.
CATHY HUMPHREYS: The list?
STUDENT: The two (inaudible).
CATHY HUMPHREYS: Oh! I wonder if one person could write on the statement and the other could write the reason. Could that help? Could you both...could two people write at once?
STUDENT: Could somebody do my hair at the same time?
STUDENT: We are starting to write the proof.
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.