Cathy Humphreys introduces the Properties of Quadrilaterals task. The lesson is introduced by explaining the students’ roles and responsibilities to carry out the investigation. The teacher models how to gather information for the tinkering stage of the investigative process. She also poses the math problem in the context of a kite making company. The kite making company uses sticks to make kites. The investigation involves how two sticks will be selected and positioned to determine the shape of a kite. She creates a purpose for exploring the key factors (length, intersection point, and angle position) that define the shape of a quadrilateral. She introduces manipulatives (different and like length strips with holes and a brad) and demonstrates how one quadrilateral might be determined by the arrangement of the diagonals. She demonstrates how the stick might be used to draw a rhombus.

9th & 10th Grade Math - Properties of Quadrilaterals*Cathy Humphreys, Fremont High School, Fremont Union High School District, Sunnyvale, California*

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CATHY HUMPHREYS: Okay, so as soon as I put these on your paper, why don’t you decide right now who is going to be your resource manager.

STUDENTS: I will.

CATHY HUMPHREYS: Okay, resource manager, resource manager. I actually didn’t ask the resource manager to open the package yet, but that’s okay.

STUDENT: She told me to do it.

CATHY HUMPHREYS: Um, she told you to do it? Um, I am just trying to think now if you should have, while you are fiddling around by yourself, if you should have the scratch paper with you. Um, I think maybe, because otherwise you are just going to be messing around and I haven't thought about that. So how about I walk around and put a stack of scratch paper on every table. What you are going to be doing is you are going to fiddle around at first; you are going to tinker, but by yourself without talking to anybody. Why would I have you do this? Ori, why do you think?

STUDENT: So that we can get information without the thoughts of anyone else and then after that we can share what we think; and then maybe come to some conclusions.

CATHY HUMPHREYS: Okay, that’s really good. It's a good way of saying it because, um, if you all start together you might all go down one path. But if there are four of you thinking about things in a different way then the more chances there are of different ideas coming out. So I will bring scratch paper to everybody and meanwhile now resource managers you can take your...and everyone can take the brad out and just start seeing what happens with some of these sticks. And let’s say five minutes to eight o’clock. Facilitators would you please watch the clock. Team captains would you please make sure no one talks. Oh you know what, I forgot. Ariel thank you, thank you, thank you, thank you; I forgot to ask a really important question. And what I meant to do is have you first talk and make sure you understand what it is you are supposed to do and what the question is. So will you talk in your group for just a minute and then when you are done talking, wait.

STUDENT: ...find a variation of the strips for (inaudible). So I guess right now we'll just tinker through and find out what type of strips we can make using...

STUDENT: What the different holes are?

STUDENT: Mm hmm.

STUDENT: So that's it. Wait. What if you want to make a rhombus? I don't know. We'll see, we'll find out.

CATHY HUMPHREYS: Show me that you are listening to one thing. As I was walking around I heard something and it made me realize that I needed to say something else. A lot of you have had experience with geometry before and may have studied this before. So if there is something that you know to be true, I hope that you will hold back um, the information that you have; use that on your own. But just like I don’t want to be telling you answers, you also don’t want to be telling each other answers because that takes away the chance to fiddle around with yourself. Eventually you are all going to want to come to a conclusion and your information that you knew ahead of time is going to be really important. But at this stage everybody needs to be able to think about it in as open a way as possible. All right? Um, okay, any questions? All right, let’s have until about three minutes after eight. Silent individual work.

COMMENTARY BY CATHY HUMPHREYS: I have posed this problem as a “real-world” situation, which it clearly is not. The idea of the kite factory motivates a context for the investigation but it is a fanciful context, indeed. There are many truly real-world applications of mathematics (e.g., the mathematics of global warming). These applications are especially important for students to experience in order to understand the relevance and importance of mathematics in their lives.

I also want to emphasize the importance of the process of private individual thinking prior to the group addressing a task. This, as Ori said, ensures individual ideas and different ways of thinking that enrich the group’s approach to the problem.

As I closely watched video of the small-group discussions (an opportunity which a teacher rarely has), I noticed that although the students had individually made some conjectures, there was no emphasis made on recording these; and this also obstructed examination of exactly what the person had said. When I do this lesson again I think I will have one student at each group record for the group: 1) “What have you found out so far?” and 2) “What are you wondering about?” This way they will have a better starting point for their group work.

Finally, I would like to comment about the structure of the group work during this task. The idea of these roles comes from Elizabeth Cohen’s work on Complex Instruction. I have, however, not employed these roles in the true spirit and intent of Complex Instruction, in which the roles are assigned so that all students have the opportunity to perform each of the roles. Assigning the roles can help negate status issues that may arise from certain students always selecting a specific or preferred role.

COMMENTARY BY COACH DAVID FOSTER: Cathy states explicitly how students are responsible for investigating, working with others, making sense of the mathematics and sharing their findings. She describes the thinking the students will experience in the investigation, reminding them of the process they have both experienced in the past and how it is depicted in the process on their poster. She uses the notion of tinkering to help the students understand that they will have to work at making sense of the situation before they come to understanding it completely. Cathy poses the math problem, with a purpose that student relate to and can understand. She introduces the manipulative before having the students begin their investigation. She gives two sets of manipulatives to each group of four to foster collaborative exploration.