The lesson study team had anticipated that students would make correct mathematical statements about Student H’s work and have some concern regarding the third table. Additionally, the team had expected that some students would present the argument that table #3 is about months and cumulative costs for that particular DVD plan. The team had agreed that any incorrect responses would be managed by asking the class if we were all in agreement by using hand signaling and ask if someone would be willing to defend a particular answer to provide further clarification. Student H had created three different horizontal tables that were all mathematically correct. However, the third table did not match the DVD plans. Additionally, Student H did not label all three of the tables, thereby allowing room for confusion and an inability to accurately respond to the original prompt. The notion that mathematical comparisons in this situation can only be made with like units is a big mathematical idea in this particular case or context. Likewise, the otion that just because there is a correct mathematical pattern doesn’t mean that the table is correct for this context. This too, is a big mathematical idea for our students.
CECILIO DIMAS: So what I’m gonna ask everyone to do right now on this green sheet, is to please make any mathematical corrections and additions onto Student H’s work. So you’re actually going to make changes onto the table, and at the bottom of the page you’ll see that there are some lines and on those lines I’d like for you to state why you made the changes that you made. Okay? So you’re gonna make changes, and then tell us why you made those changes. Go ahead and do that now, please.
CECILIO DIMAS: So boys and girls, I’d be curious to hear, what are some of the changes that you have made? And why have you made those changes? Sam?
STUDENT: Well, I changed all the multiples of 18 to 18, because the poster said unlimited rentals for 18 bucks. I put 18.
CECILIO DIMAS: Okay. So you changed these all to 18, to match the plan?
CECILIO DIMAS: Okay. What are some other changes that were made? Or additions. What did you add? Kyle?
STUDENT: For the, um, the top boxes? I labeled them as rentals, because that might have been how person H got confused with MailFlix.
CECILIO DIMAS: Okay, so instead of movies, you wrote rentals?
STUDENT: Uh, yeah. Amount of rentals.
CECILIO DIMAS: Amount of rentals, and what did you label this one right here?
CECILIO DIMAS: And.. this one here?
CECILIO DIMAS: So all three of your tables are labeled with rentals?
STUDENT: Uh, yeah, for the top, uh, column.
CECILIO DIMAS: And the bottom row, what did you label that?
STUDENT: The prices?
CECILIO DIMAS: The prices. Okay. So, just to restate what Kyle said, could I have someone restate what Kyle said? Right now, of what he added to the tables? Debra?
STUDENT: He added, you know where it says 1,2,3,4,5,6,7,8?
CECILIO DIMAS: Uh huh.
STUDENT: He put amount of rentals... (inaudible.)
CECILIO DIMAS: Okay. Could we also have kept it movies?
CECILIO DIMAS: Did you keep it movies?
CECILIO DIMAS: Did you do number of rentals as well?
CECILIO DIMAS: Okay. And what was the bottom column, or bottom row. What did you label that?
CECILIO DIMAS: Money. For all three?
STUDENT: Mmm, yeah.
CECILIO DIMAS: Okay. Amir?
STUDENT: For Mail Flix, I thought what would make it even better, was for 1 month, 1 DVD would cost $18, and so on and so on. For 2 DVDs, 3 DVDs and 4 DVDs. You just add a buck, 1 dollar.
CECILIO DIMAS: Excuse me? Could I take a peek at what you did?
STUDENT: Um, I didn’t really do it, but I think for Mail Flix, 1 DVD and 1 month, for, so, for rental of 1 month,
CECILIO DIMAS: Mmm hmm?
STUDENT: It would be, um, $18? And then when you want to get 2 DVDs? You just add 1 buck. So, yeah.
CECILIO DIMAS: So the 2 DVDs would cost how much money, then? With Mail Flix?
STUDENT: Uh, 19 dollars.
CECILIO DIMAS: Mmm. The Mail Flix plan, remember, it’s 18 dollars for unlimited rentals, so, would you pay more for 2 DVDs? With Mail Flix? It would still, I think it’d still cost you $18 there.
CECILIO DIMAS: Okay? And for 3 DVDs, it would cost you...
STUDENT: 18 dollars.
CECILIO DIMAS: Mm hm. So it’s $18 for unlimited rental, there.
This documented lesson on cost-analysis and comparison of plans depicted on tables is one of three lessons being developed around students’ misconceptions and understanding in our lesson study process this school year. This lesson is focusing on using tables to understand a cost analysis situation and will be followed by a lesson using graphs in a cost analysis situation and a lesson using algebraic equations in a different cost analysis situation. Our goal is to then have students make all three representations for a new and different cost analysis situation and discuss the merit of each representation in that particular situation. We will then give the students the Mars task, Picking Apples for our third benchmark assessment to determine the effectiveness of our lesson study lessons. The majority of my regular math classes needed three days to complete the pre-re-engagement lesson and the re-engagement lesson focusing on Students H, A, E, and J.
Through these lessons we have been better able to understand the misconceptions that some students had when comparing the tables and/or reading tables in general. Some students noticed the multiplicative relationship and completed the table based on this understanding instead of looking at the relationship between variables which led them to then struggle to interpret the data that existed within the table that they had created.