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: If you think you’re ready and you have a watertight proof um, you can start your poster. Alright? Do you have any...let’s see. You did your givens right? So what are the criteria for a square?
STUDENT: Perpendicular diagonals…
CATHY HUMPHREYS: Okay, what else?
STUDENT: The diagonals are congruent.
CATHY HUMPHREYS: Okay. Anything else?
STUDENT: They intersect at the midpoint.
CATHY HUMPHREYS: Okay, they intersect at the midpoint. Okay, good. How do you prove...so how would you...what do need in order to prove that it’s a square?
STUDENT: That these two sides are parallel so we need to find the alternate interior angles.
CATHY HUMPHREYS: Okay, so you have to prove these two are parallel. What else do you have to prove for a square?
STUDENT: That this is ninety degrees.
CATHY HUMPHREYS: Okay and there’s one other thing.
STUDENT: All the sides are congruent.
CATHY HUMPHREYS: All the sides are congruent. Okay, so you have three things to show. Make sure before you finish your prove that you show all three things okay?
STUDENT: Um, this line has to be bisected or yeah. Is that what you call it? This line has to be bisected?
CATHY HUMPHREYS: Uh-huh.
STUDENT: But this line is – can’t be?
CATHY HUMPHREYS: Oh okay. Um, it isn’t on there. What you would say is um, you could say exactly one diagonal is bisected.
STUDENT: So if the diagonals are not congruent and are perpendicular and exactly one is bisected then it is a kite?
CATHY HUMPHREYS: Uh-huh. Now I want to ask you a question. Could they be the same length?
STUDENT: Which one?
CATHY HUMPHREYS: Could the diagonals be the same length if it’s a kite?
STUDENT: The diagonals can’t be the same length. They can be the same length actually.
CATHY HUMPHREYS: Would you test that before you put that statement in?
STUDENT: No, we did test that. We said that the diagonals can be made from a long and a short stick or two long sticks. So I read that wrong.
CATHY HUMPHREYS: Okay. Alright. Good. Okay, so does everyone have something important to do? Alright. I wonder if people should...if you should kind of start your poster now. Like maybe somebody could start writing if the one you chose and all that.
STUDENT: So what’s that rule if you line an angle, I mean a segment – two pair of segments that the same congruent segment? It’s something…
STUDENT: Oh, is it the perpendicular thing?
STUDENT: No, it’s the division property, I think.
STUDENT: Oh, division property...oh yeah, the division property!
STUDENT: No, no. Okay, just call it the division property.
STUDENT: Put six?
STUDENT: Not right now. We just have to find this.
STUDENT: Wait because step six... Do you want to write it out?
STUDENT: I’d rather write it out and not actually list. I guess we could just list it. But then you know what, she’ll probably still make us write it all out. I mean still I’d rather get...
STUDENT: No, not there.
STUDENT: Up there, they didn't (inaudible).
STUDENT: Just put the numbers?
STUDENT: Okay, then just put the numbers.
TEACHER: You know what you might say? That’s correct up there but it’s a little incomplete. If you would say...because all...because what are you trying to prove?
STUDENT: A rhombus.
TEACHER: But how do you know what a rhombus is?
STUDENT: Define what a rhombus is.
TEACHER: By the definition of a rhombus so…
STUDENT: So write the definition of a rhombus?
TEACHER: You say definition and then you might put in the step numbers that show it.
TEACHER: The reason could be the definition of a rhombus.
STUDENT: Okay, cool.
STUDENT: These two sides have to be congruent…
CATHY HUMPHREYS: Okay, Rose, I’m going to interrupt you because you are telling me the definition of an isosceles trapezoid. What I’m asking you is what are the criteria for the diagonals? What’s your...what’s going to be your "if" statement?
STUDENT: If the two diagonals are the same length then they intercept um…
CATHY HUMPHREYS: Do you feel like you’re on the spot? I’m sorry, I didn’t mean to put you on the spot. I want to make sure everybody in your group is together on what are – what are the things that you need. What are the exact attributes you need for the diagonals in order to guarantee that it’s an isosceles trapezoid; and what is the definition of an isosceles trapezoid? Those are two separate things you need. So stick together and Rose, you can demand your right to understand what they are talking about.
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.