The groups are instructed to display a proof of one of the quadrilaterals on a poster. The plan is to follow up the lesson with presentations by the groups. The groups will use the posters to help communicate their findings with a formal justification.

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

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CATHY HUMPHRYES: So I have to check the clock and we have six minutes until the bell rings. So here is what, how you can decide what...I would like to have your posters up even if they are not finished, so feel free to put them on top of any other poster in the room or anywhere on a white board. Excuse me guys. And you also need to...I do want to have these papers again. So individual work that you did yesterday, staple together and clap together by the table. I need these cards to be turned in and everything needs to be cleaned up. Alright, so you could do that at your leisure, whatever you need to do to get it done.

Thank you for your good work today. So um, as soon as your table – if you were not tardy, you will be excused as soon as your table is clean and your chairs are pushed in. See you tomorrow.

COMMENTARY BY CATHY HUMPHREYS: Closure is a word that has always bothered me. Think about it: what would it mean for this lesson to have closure? Clearly the students need more time to finish the posters and analyze each other’s work. But once they have done that, does closure mean that we have “finished” with the myriad pieces that make up understanding how to construct a mathematical proof? Does closure mean that all of the groups have presented their proofs? So often, closure means that we have done as much as we dare take time for on a particular concept and move along, whether the students are with us or not.

After the cameras had left the room that Thursday morning, and the students returned to class the following Monday, the process of finishing and analyzing continued. Actually, it continued until the end of the year.

Some people have asked me how I could afford to spend all of this time on one standard when there are so many to cover. Truth be told, I never cover all of the chapters of a book. To me, understanding one thing well far surpasses knowing a dozen things at a superficial level.

Is this an irresponsible decision? Do I jeopardize my students’ “opportunity to learn?” I can only say two things: first, that at NCSM last spring I heard Zal Usiskin talk about how lists of content have eclipsed the opportunity to learn to reason, investigate, conjecture… all of the mathematical habits of mind that comprise real mathematical thinking. Second: 93% of the students in my two classes were Proficient or Advanced on the California Standards Test; 18% of these students increased their proficiency level from the previous year, while only 3 students (5%) dropped a level. Clearly, a focus on reasoning did not hurt the students’ mathematical performance.

COMMENTARY BY COACH DAVID FOSTER: Most of the groups were able to complete the posters. A few of the groups still needed to complete their proofs. Of the posters completed, the students were successful in making rigorous arguments and valid proofs. The students that did not complete the poster would do so in the next period.