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 notion 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: Today we’re going to look at some different tables, that you’re going to have in your packet, that we’re also going to have up here on the board. And as we look at each one of these tables, I’d like for us to analyze, again, when will all three of the DVD plans cost the same amount of money? So that’s going to be the prompt that we continue to focus on. When will the three DVD plans cost the same amount of money? We’re going to look at similarities between the tables that we were studying, and we’re going to look at the differences between the tables that we’re going to be analyzing. So I would like for you to flip to the very second page, which I think is going to be green, And here we’re going to be looking at student H’s table. I would like for you to think about...does this representation make mathematical sense? Why or why not? Does it make mathematical sense? Why or why not. I’m also going to be asking you to think about, whether or not these three tables that we have here match the DVD plans that we have been working with for Movie Buster, Online Flix, and Mail Flix. So does it make mathematical sense, and do they match the plans? So I’d like for you to talk to your shoulder partner about those two ideas, right now please.
STUDENT: Online flix? It says right here, ...
STUDENT: I don’t think it makes sense because... Mail Flix is going by a month,
STUDENT: but the other ones are obviously going by each movie, because Movie Busters
STUDENT: looks like, you know, how much it goes per movie, so it’s 3 dollars per rental, then 6, then 9, then 12,
STUDENT: Then Mail flix is going by months, ‘cause this is supposed to be unlimited, and it’s 18, 36, 54.
STUDENT: These two are going by... by movie rentals, and this one is going by the month. That’s why.
STUDENT: it makes sense.
CECILIO DIMAS: So, boys and girls, can I have your attention back, please? Um, let’s go ahead and talk about the very first question. Do these three tables make mathematicalsense? Let’s go ahead and share out. Sam?
CECILIO DIMAS: Okay, why don’t they make sense?
STUDENT: Because the last one is supposed to be, um, all 18, since it says unlimited rentals,
CECILIO DIMAS: Okay. So, Sam, you’re saying that it doesn’t make mathematical sense because it doesn’t match the plan? Okay. Could I get someone to agree or disagree with Sam’s idea? Debra?
STUDENT: I agree.
CECILIO DIMAS: You agree? That it doesn’t make mathematical sense? Okay. Is there someone who sees that it makes mathematical sense? And could you tell us why it makes mathematical sense? Seneca?
STUDENT: Um, I think that it does make sense, because, like, if you add 18 plus 18 it would equal, um, 36, but it wouldn’t make sense if it were for, like, the real price. But it does make mathematical sense.
CECILIO DIMAS: Okay. Kyle, you also had your hand up?
CECILIO DIMAS: Okay.
STUDENT: It does make sense because it follows a pattern, like 18 plus 18 is 36, but it doesn’t match the Mail Flix plan.
CECILIO DIMAS: Okay. So Sam and Debra and Seneca and Kyle, I think you all agree that it doesn’t match the plan, and then we looked at it, and we can see that they, that we are counting by 18. So that there is some mathematical sense there. Okay? Why are we organizing this table? What’s the purpose? What’s the mathematical purpose of creating these three tables? Why are we doing this? Go ahead and take a few moments to talk to your shoulder partner about why we are creating these three tables. What’s the purpose?
STUDENT: So we can see which one costs less. We’re also putting these up to see if the plans are correct.
STUDENT: See if it makes mathematical sense.
PRINCIPAL: What was the question he asked? Because I didn’t hear.
STUDENT: Why are we doing this?
STUDENT: I’m nervous
PRINCIPAL: Oh, don’t be.
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.