I look forward to learning with you.

The day before a teaching observation I send out an email confirming that I have the correct details and reminding the teacher that I will need an action plan beforehand to focus my attention. I try to end each of these emails with the statement provided at the beginning of this post. It serves as a reminder that the observation will be a learning opportunity for both of us and not just a dog-and-pony show.

Sometimes, I take for granted my role as a learner in these experiences. I was reminded of this during a recent observation. The teacher wanted me to focus on whether or not she was providing adequate support to students as they were learning how to multiply and factor polynomials. This was in an eighth-grade class and they were halfway through the unit - just wrapping up multiplication of polynomials.

The lesson went well and the students were engaged in what Fisher and Frey call Collaborative work. This entailed a few items that provided students practice multiplying polynomials and a worksheet called Polynomial Puzzler that the teacher had modified from a lesson found on

*Illuminations*. The teacher went over the instructions,and demonstrated using the first puzzle.Fill in the empty spaces to complete the puzzle. In any row, the two left spaces should multiply to equal the right-hand space. In any column, the two top spaces should multiply to equal the bottom space,

As I looked through the entire worksheet, however, I identified an area I thought would give the students trouble; there were places where the four entries in the upper left were missing entries. This meant the students would need to factor some polynomials in order to complete the puzzle. The teacher had not provided the necessary support for student success. I made a note to talk about this during the debriefing.

Sure enough, after students had completed the first puzzle and multiplied what they could in the second puzzle, many started to ask, "What goes here?"

The teacher responded with something like, "That's a great question. I guess I didn't give you enough support." She pointed to the upper right corner and said, "We want to find out what goes here. In other words, what times this {pointing at (-15

*x*+3)} will equal this {pointing to the lower right corner}? Okay?"
I thought to myself, "No. Not okay. They need to know how to factor." But the students seemed satisfied and went on about their work. And the surprising thing was that they were okay. In fact, they were better than okay. They were amazing.

The students at the table nearest me began looking back at the work they had already done and sharing what they noticed. "Look. -4

*x*+10 divided by 2 is this one {pointing at -2*x*+5}," one of the girls exclaimed with more enthusiasm than I usually see in math class. The group then began talking about dividing the polynomials to find the missing entries. Sometimes they tried using guess-and-check to identify what was missing. The main point for me was that they did not just give up.
They seemed to be embracing the struggle that so many students in math class are determined to avoid. And they weren't alone. I heard several ahas from students seated in different groups around the classroom. I do not know why this class acted so differently than others I have seen. The cooperating teacher is a former GVSU graduate, so I would like to think that the learning environment had something to do with it. Or maybe it was the fact that it was a puzzle and not homeWORK.

After the lesson, I asked the teacher if she had anticipated any problems. She had thought that the instruction might be confusing. When I pressed about the factoring, she acknowledge that it could have been a problem but that she thought it was actually good that they did not know exactly how to do the puzzle our way because then they would obsess about doing it right. "Besides," she said, "I just wanted it to foreshadow factoring. Now they're ready for the next lesson."

I love my job. I learn something everyday. Even on those days when I think I'm the teacher.

I love the idea of figuring out what factoring really is ... this beats the pants off of giving them "long division with polynomials" and having them not even recognize the connection to long division which they didn't really get either, and then move on to something a little easier but equally comprehensible...

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