Moving Toward Approximations of Practice in Teacher Professional Development: Learning to Summarize a Problem-Based Lesson

Abstract

This chapter describes a set of activities in a professional development intervention with in-service geometry teachers that engaged them in role-playing a lesson summary. The activities included discussion of animations of classroom instruction with examples of how to lead a summary of a problem-based lesson. The animations were representations of teaching that supported the development of shared knowledge about how to summarize a lesson. In addition, the teachers decomposed the practice of summarizing prior to their engagement in a role-play of a summary. This role-play constituted an approximation of practice that enabled the teachers to envision how a summary would unfold in real time and required the teacher leading the summary to make tactical decisions. Observations of a teacher leading a summary in his classroom provided evidence of teacher learning. Overall, the professional development included a sequence of activities that validated teachers’ knowledge and provided them with the capabilities for engaging in an authentic approximation of practice in a safe environment.

Appendix

Story 1
TEACHER:	Okay, you all did a great job with the problem. I was pretty happy with what I saw as I was walking around. I want us to take some time for us to share some of the things we discovered and wrap up the lesson. I’m going to give you two minutes to determine in your group what you want to share with the class. I can call anybody in the group so make sure that you are ready. Talk to your group mates and see what you want to share.
TEACHER:	All right, let’s start the discussion. Where are my notes? Oh, okay, here. Let’s start with group 1. Your strategy was very similar with what I saw in other groups. So, if group 1’s solution is like your group’s, keep a mental note about it and add your ideas at the end. Conrad, can you please share the work from group 1?
CONRAD:	Sure.
	[CONRAD GOES TO THE BOARD]
TEACHER:	You can use the projected map and write there.
CONRAD:	So, like, we found the middle first.
TEACHER:	What do you mean by the middle?
CONRAD:	Right. So we looked at the two schools and found the midpoint.
TEACHER:	The midpoint of what?
CONRAD:	Like the midpoint of this segment.
TEACHER:	I see. Remember that we want to use correct vocab. It isn’t the midpoint of two locations, it’s the midpoint of a segment. Keep going.
CONRAD:	That was our first point. Then we found the other two possible places like above and below the midpoint.
	[MARKS DIAGRAM]
TEACHER:	Does anybody have a question for Conrad? [Silence.] What do you think that I’m going to ask? Mariana?
MARIANA:	How did you pick the other points?
TEACHER:	Yes, that’s the question I would ask. Remember in Geometry we always want to know how, not just what. You should be able to say HOW you found the points so that others can follow the same method. Can you tell us how you decided to pick the other points?
CONRAD:	Well, I think we just looked a little bit on each side and that’s it.
TEACHER:	Okay, thanks. That was a pretty common strategy. Now, how did you know that those points were fair? What did your group have as an idea of what’s fair?
CONRAD:	It’s fair because it’s very close to the midpoint, and we knew that the midpoint was fair.
TEACHER:	Can anybody add to this? Maybe another group with a similar approach?
MARIANA:	Well, we were thinking like if this was a real map and there was something there like a gas station or something, then moving a little bit would be alright because it’s the closest you can move away from the midpoint.
TEACHER:	So you were thinking about what other actual buildings would be there, and you want to get away from those buildings. I see your point. Now, how do you know that those points are actually “fair”? I was talking to group 5 and they had a strategy to check if the points are fair to both schools. Audrey, do you want to talk about your strategy? Thanks Conrad. You can go back to your seat.
	[CONRAD GOES BACK TO SEAT]
AUDREY:	Well, we did it like that group, only we started measuring so that it would be the same to both schools.
TEACHER:	Can you come up to the board and show what you measured?
	[AUDREY GOES TO BOARD]
AUDREY:	So, here it’s obviously the same distance away from both schools.
TEACHER:	The midpoint.
AUDREY:	Yes, the midpoint. But then if we moved the ruler a little bit, then we have like this is the same distance from one school to this point and from the other school to that point. And we found a bunch of other points with that strategy.
TEACHER:	Okay, that’s interesting. And I have to say that when you are measuring you get to be more precise and you can be sure you have the right location. So you are looking at these points (adding markings, and this is the same as this, this is the same as this, and so forth…). Everyone agrees with this? Do you see the strategy?
STUDENTS:	STUDENTS MUMBLE, “OH YEAH”, “I KIND OF SEE THAT”, “YOU’RE RIGHT”, ETC.
	[AUDREY GOES BACK TO SEAT]
TEACHER:	Take some time and talk to your group now and see if you notice something interesting about this line. What’s special about this line?
STUDENTS:	TALKING IN GROUPS. “I DON’T SEE ANYTHING, “WHAT’S INTERSTING?” “IT CROSSES AT THE MIDDLE.”
TEACHER:	Okay, what do you notice about the line? Eric?
ERIC:	It’s like going through the middle.
TEACHER:	Yes, going through the middle. What do we call that?
ERIC:	Bisecting?
TEACHER:	A bisector. What else?
MARIANA:	It looks perpendicular.
TEACHER:	It looks perpendicular? How do we know?
MARIANA:	We can measure.
TEACHER:	Did anybody measure?
ERIC:	We did. It’s 90.
TEACHER:	Yes, so, it’s a line that goes through the midpoint, so it’s a bisector, and it’s also perpendicular to the segment. We call it the perpendicular bisector. Two words, because it has two characteristics—it bisects a segment and it’s perpendicular.
TEACHER:	Ah…let’s do this, since we are running out of time. I am giving you a couple of problems for homework. Here, you need to determine what kind of line you have from some possibilities—a perpendicular bisector, a median, or an altitude. Review your notes and see what’s similar or different. Now, you need to be careful, because in some cases, it can be more than one thing, and that’s okay.

Story 2
TEACHER:	Okay, everyone, let’s get started with presentations of the work and wrapping up what you did. You all did a great job. I asked some people to come up and present. Let’s start with Leah. Can you come up and show what your group did? Remember that we want to show not just the location of the points but how we knew that the points are fair.
	[LEAH COMES TO BOARD]
LEAH:	We started with the midpoint because that’s fair to both schools. But then we were thinking about the other two points and so we made sort of a square, well, like a rhombus. All the sides equal, so, yeah, a rhombus.
TEACHER:	And how is that fair?
LEAH:	It’s fair because you are moving the same distance to the schools, it doesn’t matter where you are. Like this and this and this and this…is the same, they are all the same.
TEACHER	That’s an interesting strategy and very unique. Thanks Leah. Let’s leave it up there for now. You can go back to your seat.
	[LEAH GOES BACK TO SEAT]
TEACHER:	So, let me ask everyone, do you think that their solution is fair? And why?
JACKSON:	Well, I think it is, because you are moving the same distance away. We didn’t do it that way though.
TEACHER:	Yes, Jackson, hold on for a minute. I know that your group did something different, but I want to stay with this for a while. How do we know that this is fair? That’s the question that I want you to answer, and what is fair for them? Think about it for a little bit. They say that if you move the same distance and make a rhombus, then you get “fair locations.” How do I know that’s fair?
NATALIA:	We can measure.
TEACHER:	We can measure what?
CONRAD:	Like the midpoint of this segment.
NATALIA:	We can measure the distances and they all will be the same.
TEACHER:	That’s right. If we measure from-let me put some labels, Catalina High School to X and Tyrian High School to X is the same as Catalina to Y, and Tyrian to Y. Now, let’s consider group 3. Can you show your solution? Elijah?
ELIJAH:	We actually have something that doesn’t look like a rhombus.
TEACHER:	Exactly, that’s why I want you to show that. So what does that look like?
ELIJAH:	It’s more like a kite.
TEACHER:	A kite. And when we have something like a kite, you said, what things are equal.
ELIJAH:	These two are equal and these other two are equal.
TEACHER:	Okay, so let’s say I have Catalina to point, let me call it A and Tyrian to A is the same. But then Tyrian to B and Catalina to B are the same. Do you see where this is going? Not all the four distances are equal, but each pair is equidistant-the same distance.
TEACHER:	We are running out of time. I want you to think about this—are these locations that make a kite, fair? And also, are these locations that make a dart fair? [Teacher shows a dart.] You should provide an answer in your journal and then we’ll talk more about the wrap up to this lesson tomorrow. I’m sorry that we are running out of time, but it’s better to look at this closely tomorrow. Thanks all for your hard work.

Story 3
TEACHER:	Okay, let’s start talking about what you did in the problem. In this class it’s important that we share our mistakes so that we learn from them. Group 5 agreed to come up and talk about how they corrected their mistake and show their solution. Trey can you come to the board?
	[TREY GOES TO BOARD]
TREY:	So, we thought first that we needed to pick an area, so we did like two circles here and here and then we said, any points here would make it.
TEACHER:	And how did you realize that this was a mistake?
TREY:	Because you told us.
TEACHER:	Okay, but what was the problem of that answer.
TREY:	Well, that the points are like all over the place and it’s like too many points.
TEACHER:	Actually, I don’t think that having too many points is a problem. Let’s look at this again. So let’s say that I choose this point that is in the area, would it be fair to both schools?
STUDENTS:	“NO.” “NOT EVEN CLOSE.” “MAYBE?”
TEACHER:	Why not?
RAQUEL:	It’s obviously closer to Catalina.
TEACHER:	Yes, and Tyrian High School would complain, right? So, just by being in one area doesn’t mean that it’s fair. Okay, so Trey, what did you do to correct the mistake?
TREY:	We looked at the intersecting points and they are fair because it’s same, same, same, same.
TEACHER:	Same, same…, you mean, equidistant?
TREY:	Yeah, that.
TEACHER:	Okay, so what made you look at the intersection points?
TREY:	It was Elijah’s idea, it’s like the radius is the same.
TEACHER:	Radii—congruent radii. That’s a good point. And let’s draw it here. Let’s go back a little bit. What do we know about these circles? How did you construct them?
TREY:	They are the same—congruent—like they have the same radius.
TEACHER:	So then we have this is a radius, this is the radius, and the same. That’s a good point. How did you choose the radius?
TREY:	I don’t know.
TEACHER:	Let me ask you another way, what if you choose another radius like let me draw this with a bigger one? And you can go back, thanks. I don’t want to put you on the spot, this is for the whole class. Yes, Taylor.
	[TREY GOES BACK TO SEAT]
TAYLOR:	I wanted to say that we didn’t do it that way. We didn’t even use a compass.
TEACHER:	Yes, I know, and I’ll get to your method soon. Let’s finish with group 5. So, if I have another pair of congruent circles, would these points be equidistant? Let’s try it.
TEACHER:	Why are these equidistant?
JAMAL:	Same radius. The same. As long as the two circles are the same you are set.
TEACHER:	Yes, exactly. Now I want to go to group 3. Taylor, what was your method?
TAYLOR:	We measured. So it would be the same and the same here. And then the same and the same here…
TEACHER:	That sounds very good. Now, how similar is this to the circle method?
TAYLOR:	Oh, I guess you can make circles too—I see.
TEACHER:	Can you say it again?
TAYLOR:	We didn’t use a compass, but you can actually do it with a compass, like this.
TEACHER:	Can anybody summarize the main idea here? Why are these new locations fair?
RAQUEL:	They are fair because you are the same like the same, like moving the same from each school.
TEACHER:	Okay, so let’s see if there’s a pattern here. What if I choose these two points, or these two points, or these two points? What do I see with all the points?
JAMAL:	They are in a line.
TEACHER:	Yes, they are in a line—and that line is very important. So it doesn’t matter what radius you choose, the two circles will always intersect on that line.
RAQUEL:	What if you choose something too small?
TEACHER:	Yes, I mean as long as it actually makes two intersecting circles. That’s a good point. If you choose a radius that is too small it’s not going to even intersect.
TEACHER:	So, alright, this is actually a theorem and for homework you need to summarize in your own words the relationship. Look closely at the diagram and try to figure what’s the theorem. We want to say something about that line in relation to the segment.