Building Better Lessons

Last fall (2017) I was invited to participate in a research project that was to involve video-recording a class I teach. (side note: is it still called "videotape" even though there is no tape involved? what's the current word?) I was to choose a Formative Assessment Lesson (FAL) from the TRU library at mathshell. The purpose was to help build a library of lesson videos that teacher training programs could use.

On the day of recording, I was surprised that I wasn't particularly nervous. I had participated in the Measures of Effective Teaching project years ago, and I suppose I was used to the idea of being recorded. So the day of my lesson came and went, and aside for some nice anchor charts that we generated, I didn't think much about it.

Come the new semester, and I joined a Professional Learning Team (PLT) at Math for America titled Building Better Lessons in Mathematics. In this PLT we were going to watch videos of teachers using the TRU FAL, and see what we could learn from them. We had two guiding questions in mind: 1) What are the different ways that students appear to understand and misunderstand specific mathematical ideas? and 2) Imagine we could go back in time to this part of the lesson and put ourselves in the teacher's shoes. Are there questions we might ask or moves we might make to help students think more deeply about these ideas?

This past week, we watched portions of the recording of my own lesson. I remembered some of the dialog, but only vaguely. Fortunately, one of the researchers provided a transcript, which spurred my recollection. Following is the transcript of a portion of my class in which I interact with a grouping of three students. The part of the task they are grappling with card T3 asking them to transform (x,y) to (-x,-y).

S1: He said you have to translate the x-axis to the negative x-axis, right?
S2: Oh, I see, we're reflecting this.
S1: No, that's a translation.
S2: Let me see [inaudible] negative, for this point, this point is negative two ... negative two, three. If you use this method then ...
S1: You have to [inaudible] the y axis.
S2: He [the teacher, me] messed up. He messed up something.
S1: You have to move this figure, to her, you know what I mean?
S2: Yeah, I know what you mean. He mister, I think you made a mistake.
T: Me? Show me. Teach me.
S2: So, for the example her you have x and y, however this one already starts out a negative.
T: Yes, isn't that x value negative two?
S2: Yeah.
T: and isn't the y value three?
S2: Yeah, so wouldn't this be the opposites?
T: What's the negative of negative two?
S1: It's negative four.
S2: Negative? ... Negative two?
S1: Negative negative two, that's going to be negative four.
T: The negative of negative two.  What if I have the negative of negative three? What's that going to be? [writing on paper -(-3)]
S2: Three?
T: Negative negative four? [writing on paper -(-4)]
S2: Four.
T: What's negative of negative six thousand five hundred thirty one? [writing on paper -(-6531)]
S1: Six thousand five hundred thirty one?
S2: Oh. So ...
T: So this [pointing at transformation rule] I don't know what the value is, but whatever it is, I take the negative of it.
S2: So I'm just adding?
T: Same thing with the y, I'm just taking the negative of whatever value it is.
S2: So, flip this all the way over here [pointing from quadrant 2 to quadrant 4] because negative negative makes a positive.
T: Yes. What's really going on, when I'm doing this, I'm moving these points, but I'm moving EVERY point on the plane.
S2: Yeah.
T: I'm just using this [pointing to the polygon in the preimage] ... to keep .. so my eye can focus.

The first thing that jumped to my mind is, I completely missed the error that S1 was making at first. As a result, I was answering a different question. I think now that he was seeing an x-value of -2, and he interpretted x to -x meaning that he takes the x-value (negative 2) and subtracts the absolute value (2) from it, to get negative 4. That is also why he was thinking it was a translation.

Fortunately S1 was in sync with my questioning, which was intent on clarifying the meaning of the notation -x. I think by emphasizing the language "negative of negative" he saw through the orthography of how we write the operation algebraically.

I was happy with my initial reaction to the question "Hey, mister, I think you made a mistake." I try to encourage my students not to be afraid of errors. As Jo Boaler might say, mistakes are when our brain grows. One of the other teachers in my study group also made note of that line, saying he wants to use it in his classroom.

I was not so happy with how quickly to responded during our interaction. I even feel I may have cut off the students. I think I may have already formed my idea of how I was going to explain things, and as a result I wasn't listening fully to our interaction.

Ultimately the students got to describing the move as a flip, which is appropriate for a reflection (the particular transformation was a point-reflection through the origin.

Despite feeling some discomfort in being recorded (who among us is 100% confident of their teaching practice?), I am convinced that the benefit of being able to critique myself is valuable. In the process, if I can provide some materials that help other teacher reflect on their own practice, all the better.