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: What I want to ask you now is, the way this is set up right now, can you do a comparison that would answer our prompt? Where we’re looking at all three DVD plans and figuring out when they’ll cost the same amount of money. Can we do that with the way that this is set up?
CECILIO DIMAS: Can we do this the way it’s set up? I’m gonna ask that if you think you can do it the way it’s set up, that you give a thumbs up under your chin, and if you don’t think we can do it the way it’s set up, show me a thumbs-down. Under your chin, please. Thumbs up or thumbs down. Yes, no. Okay. Thumbs down. Could I hear from those who said that you can’t compare the three plans? Could I hear from you, please? Charles?
STUDENT: I don’t think you can, because, like we discussed, the student got Mail Flix confused. So with that messed up, I don’t think that we’d be able to compare.
CECILIO DIMAS: Okay. Does someone want to add to that? Erin, I also noticed that your hand was up?
STUDENT: I was gonna say the same thing.
CECILIO DIMAS: Can you restate it, then?
STUDENT: Well, it’s because Mail Flix, um, because he got it all messed up. But we could not compare because the prices are double.
CECILIO DIMAS: Okay, now if we wanted, and we do want to compare all three plans, what do we need to correct? What would we need to change so that we can compare all three of these DVD plans? Go ahead and take a brief moment and chat with your shoulder partner again.
STUDENT: We could change the numbers for Mail Flix one, ‘cause it’s not the right numbers...
STUDENT: So why isn’t it the right numbers?
STUDENT: ... for Movie Buster, it’s...
CECILIO DIMAS: Okay, so what could we do, so we could compare all three plans? What kind of changes could we make? Jocelyn?
STUDENT: Probably... Mail Flix.
CECILIO DIMAS: Okay...and how would we change Mail Flix a little bit?
STUDENT: Um, probably...um... probably like...um, take off a little bit. Like make it a little cheaper and also a little bit closer to what Online Flix, Online Flix, Because Movie Buster and Online Flix are both by each rental, and Mail Flix is by month.
CECILIO DIMAS: Okay. Can someone add to what Jocelyn said? Tricia, I see your hand.
STUDENT: Um, we would leave 18 for all of them, because it’s unlimited rentals for every time you rent one.
CECILIO DIMAS: Okay, so, what would I put here? for, if we were to change this to... what would we change this label to? If we were to write a label here, what would we write, Tricia?
STUDENT: Number of rentals.
CECILIO DIMAS: Okay, and so if this was the second rental, what price would that be. What would the cost be?
CECILIO DIMAS: And for the third rental?
CECILIO DIMAS: And for the sixth rental?
CECILIO DIMAS: Okay.
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.