In this clip, Margie Trainer, Stacy Emory, and Francis Dickinson share their take on re-engagement as Fran introduces us to the original task and the lesson at hand. Fran shares what he saw as he reviewed student work on the "Buttons" task. In this preparation, he identified two learners' descriptions of the 11th stage to use during the re-engagement lesson. Fran describes the general flow of the lesson as planned. You’ll also hear a description of the “what’s my rule?” number talk. Margie and Fran then discuss why this number talk was chosen. Margie asks Fran about his use of the think-pair-share strategy in his efforts to foster discourse in the classroom. Fran, Stacy, and Margie discuss the ways in which learners may struggle with different representations of the pattern in the "Buttons" task. Also, they discuss how the learners are developing their ability to generalize the pattern with algebraic representation.

5th & 6th Grade Math – Multiple Representations of Numeric Patterning*Fran Dickinson, San Carlos Charter Learning Center, San Carlos School District, San Carlos, California*

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MARGIE TRAINER: What parts of the lesson do you anticipate that your students are going to find straight forward in and be successful?

FRAN DICKINSON: I think what'll come natural to the learners is first of all the number talk. I think we'll probably go through that pretty quickly. And moving into the second phase of the lesson, looking at learner work, I think the similarities and differences will jump out at them right away. Um, I think they'll struggle a little bit with some of the multiple representations. I think...and it's going to be different for all of the different learners in the classroom. So some of them are going to struggle, I think maybe with the verbal description and how to use words to describe the pattern growing. And I think other learners will have more of a challenge with modeling how they see it growing. My hope is that the x, y...I think the in and out table will be easier for them to see the clear connection between the in and the out. But I would anticipate a little bit of a struggle with pulling that T-chart apart and looking at the mathematics behind how the learners see it growing.

MARGIE TRAINER: Will it's sounding like your focus is really on multiple representations and multiple solution strategies, and that's what you're hoping will come out of the lesson for your students today?

FRAN DICKINSON: Yes, I'm really hoping that the learners will take to heart multiple representations for modeling different functions and that we can apply this in future lessons. I've mentioned before that I don't think graphing will necessarily happen today but that is kind of the next step in this lesson, to really dig deep with the graphical representation of both of these rules. And just to kind of have a complete picture of these two different functions that Learner A and Learner B came up with; describing the same pattern and how could we have two different functions to describe one pattern.

MARGIE TRAINER: Can you talk to us a little bit about what your next steps will be after this lesson?

FRAN DICKINSON: My intention is that, um, this is the start of a yearlong study of multiple representations and being able to use multiple representations throughout the year to enhance our understanding of algebraic situations and functions. So I'm really looking forward to exploring in the future the border problem, some tables, problems, and things like that. So just kind of getting into more growth patterns as the year goes on and exploring those representations with those patterns.

STACY EMORY: One of the advantages I see or one of the strengths of this is that when you're going to expand your T-chart and let them show what they're thinking using the numbers; what's the relationship between the input and the output? And then hopefully where they're going to go with this eventually is to start to make some abstractions from their computations; what are they doing over and over in each one of these input/output pairs? And to able to generalize some sort of rule.

FRAN DICKINSON: That's a good point that I didn't really speak to is the generalization piece. I did mention in the learner work that they were fairly successful at being able to generalize. But most of them were generalizing with arithmetic, they weren't generalizing with maybe a variable, you know, making some kind of relationship between y and x. So that would be another outcome that I will be looking forward to is generating rules that look more traditional and have that y = mx + b format, so that we can start applying that to further graphing lessons, and just building our understanding of algebraic reasoning. Get ready for 7th and 8th grade.

STACY EMORY: One of the strengths of using patterns as an entry point to algebraic thinking is that the kids can start to see the same thing happening over and over, and then trying to come up with the language to describe what's happening. One of the things you said that they were going to struggle with is maybe a verbal explanation of what was going on. And to give them the ability then to move from that verbal to some sort of symbolic representation and to look to be able to describe the function symbolically, or using an algebraic expression, or algebraic equation, it's something that's going to be really powerful for them, and having such a great entry point for them to do that.

MARGIE TRAINER: And I agree wholeheartedly with the idea of making the mathematics explicit by pulling the T-chart apart and representing it numerically to let them make the connection between the input and the output. My question is...do you anticipate that AHA coming from the students as a result of this lesson or maybe further down the road, that the input and output look the same and the graphs look the same, like the way that Student A and Student B solve the pattern was different yet got to the same end result? So maybe, even though the rules, if they get to that point, look different, the rules are equivalent, is that an anticipated...?

FRAN DICKINSON: That's an anticipated outcome, so yes and no to the question. I think yes I want them to see the mathematics behind the pull-apart and that's where I'm going to focus today. I don't know that they'll see the equivalency and the graphing because that's something that will be hit or miss as to whether learners will get there today. But is definitely where I'm heading; I want them to make that leap. And um, I don't know at this point in the school year that yes, we'll get to looking at the equivalency between two different rules, but I do anticipate that eventually somewhere down the road we'll get there.

MARGIE TRAINER: Fran, thank you so much for letting us come into your class and watch this lesson.

FRAN DICKINSON: My pleasure.

This lesson is a reengagement lesson designed for learners to revisit a problem-solving task they have already experienced. My colleague, Stacy Emory, best describes reengagement by comparing it to re-teaching. Re-teaching is a teacher directed activity where we plan a different lesson to address something that is perceived to be a misconception with our students. Reengagement is a learner-centered activity wherein the original task is posed in such a way that we may expose learners to different strategies, alternate solutions, or even misconceptions. Think of the original task as a formative assessment that helps you shape the lessons that follow.

All of my 6th grade learners were able to successfully draw the 4th stage of this pattern and mostly all of them were able to correctly identify the number of white buttons in stages 5 and 6. The two exceptions to stages 5 and 6 were learners who scored the special case points for counting the black button along with the white buttons.

13 out of 28 learners incorrectly identified 33 as the number of total buttons in stage 11. These learners also went on to incorrectly list 72 as the number of total buttons in stage 24. This error, I believe, solely to be a critical reading miscue. These learners were leaving out the black button and missing the language “total number of buttons.” I base this inference on the abundance of learner work which successfully describes the growth of the white buttons.

The learners were mostly successful at generalizing the pattern in their own words or through the use of number sentences.

Some learners were describing the pattern as adding on:

“I added 4 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 + 3 [ to get stage 11.]”

Other learners were moving towards thinking multiplicatively about the +3-ness of the growth:

“ I counted by 3’s for each pattern...”

Less than half of the class chose to represent the growth with a number sentence such as:

“I [did] (11 x 3) +1 = 34 buttons [stage 11.] I added one for the black button.”

Overall, my 6th grade class performed very well on this task, which is why we can use this reengagement lesson to begin looking at multiple representations of functions.

All of the teacher tools you see at work in my classroom have accumulated over the years of observing other math teachers in video and in person. It is easy to become isolated in our profession, but there is a lot to be gained from observing others at work in their craft.