Cecilio Dimas had asked Danielle, Tricia, Charles, and Alex to participate in a post-lesson student interview. In the post lesson interview, he asks students to share what their previous experience had been with tables, key learning, confusion experienced as well as what they did to achieve clarity, the role of zero, their perspective on making corrections to incorrect student work, and assumptions.
CECILIO DIMAS: So good afternoon boys and girls. I would like to ask you a few questions now that we have gone through our re-engagement lesson of the DVD Plans. And one of the first questions I would like to ask all of you is, what is your experience with tables? What have you done with tables in the past?
STUDENT: I would add up everything together and then see what the total amount is.
CECILIO DIMAS: Okay. Anyone else want to add in?
STUDENT: Uh, you can use tables to organize things.
CECILIO DIMAS: Okay.
STUDENT: And compare.
CECILIO DIMAS: Okay. Um, you don't have to answer if you don't want to. Okay, another question I have regarding table is, how much have you done in the past with comparing information within tables?
STUDENT: Well, I remember starting in 3rd grade, pretty sure, and then we've done it every year since then.
CECILIO DIMAS: Okay. What were some of the key learnings that you had today?
STUDENT: How to compare all the prices with the table.
CECILIO DIMAS: Okay. Danielle.
STUDENT: Understanding, like, what the difficulties are and what mistakes you can make.
CECILIO DIMAS: Okay. Someone else with anything to add in? Charles.
STUDENT: Like the mistakes, it helps us see if somebody made mistakes so we could correct it. So that's, yeah. Showing us the prices, lots of stuff.
STUDENT: Because if someone made the same mistake, they would learn, um, how to fix it.
CECILIO DIMAS: Okay. Was there any point during the lesson today that you experienced confusion? And if you did, I'd like for you to state what that confusion is and what you did to...kind of get yourself out of that place of confusion?
STUDENT: I got confused with, I think the last one we did, and it's because with the 12 and the 25. So I wasn't sure, like, because the plan wasn't fixed, so I wasn't sure if they were trying to do 12, 24, so 12, 13, plus 13. I wasn't sure which way they wanted it.
CECILIO DIMAS: Okay. So what did you do to help yourself with that confusion?
STUDENT: I guess I kind of waited for somebody else to answer it.
CECILIO DIMAS: Okay. Someone else want to add in?
STUDENT: I got confused too on the same thing he did. And I thought you just always added 13 to everything and I thought the problem was correct, you just had to take away the zeros but...13 remains the same and you just add the DVDs.
CECILIO DIMAS: With table A, with Student A's table, we saw zero and I would like to know what your thoughts are about zero.
STUDENT: Um, I think zero is important to include in the table because it shows that even when you don't rent any DVDs, you still have to pay a certain amount or nothing. Depending on which plan.
STUDENT: Yeah, I agree with that because somebody, that's probably how somebody got confused. They didn't look at it clearly, so if you know it's 12 per month or 18 per month, that'll help.
CECILIO DIMAS: With Student H, we had labels missing. When you looked at those three tables, did you assume that the labels that were on the first table were also in place for the 2nd and 3rd table?
CECILIO DIMAS: Okay and when you made that assumption, that you looked up the numbers, then how did you read the last company, where it was multiples of 18? How did you read that then?
STUDENT: Well, I can help, I guess. What do you mean? How do we see it?
CECILIO DIMAS: Yeah.
STUDENT: Well, yeah. It looked like it was just 18 plus 18, plus 18 over and over again. So it wasn't following the plan because it was unlimited movies, so. Yeah.
STUDENT: But it didn't make mathematical sense.
CECILIO DIMAS: Okay, so talking about mathematical sense, you bring up right now, Kyle, do you see the difference between something making mathematical sense and it matching the plan?
CECILIO DIMAS: Could we talk about that a little bit?
STUDENT: Um, so sometimes it made mathematical sense even though it didn't match the plan and probably because the student saw a different pattern, maybe.
CECILIO DIMAS: So when we're looking at that, it reminds me of things we've done in the past with our warm-ups, where we look at two different problems, where one is correct and one is incorrect. And we're still trying to figure out the logic behind that. Did this kind of feel like a lesson like that?
STUDENT: And I have a question. You know, like the 12, the 12 one, when it's 12 + 13 + 13, would that make mathematical sense or no?
CECILIO DIMAS: Well, you're constantly adding on 13, so you're adding 13, starting at 12. So we're working with building 13, so yeah, there's some sense there. And we talked about earlier that the 13 came from the $12 flat fee, plus $1 for the additional rental.
CECILIO DIMAS: Was it helpful to you to make corrections to the tables?
CECILIO DIMAS: And in what way was that helpful to you?
STUDENT: Because you get to understand what kind of problems, like even if you think it's right and it's wrong, you understand, like you could've make that mistake yourself too.
CECILIO DIMAS: Okay. And did...even though we didn't quite finish, um, did the lesson leave any questions for you right now? Are you still wondering something that you would like to address?
STUDENT: No, I think I understood it pretty well.
STUDENT: Yeah, I think it all made sense.
CECILIO DIMAS: Okay. Thank you.
Kyle brought up an interesting idea regarding the mathematical sense of the student work regardless of it accurate depiction of the three DVD price plans. Charles sought clarification about one of the student errors. I didn’t anticipate receiving this question from him, because his work and comments during the lesson appeared to show understanding. This interview process reiterated the importance of multiple measures and providing opportunity for further clarification and affirmation of ideas and questions. I found the opportunity to interview a sample of students after a lesson to absolutely insightful. I can only imagine the significant impact of including students in the process of reflection would have on classroom instruction.