Category Archives: Collaboration

Adapting Lessons Part 1: Launching an Activity

As a math teacher and coach, I’ve always adapted curriculum materials. I am sure we all have—it’s part of knowing our students and wanting to provide them with access to the math in ways that make sense to them. And while we have the best of intentions, sometimes the choices we make can unintentionally take the math out of students’ hands, disconnect from what they already know, or even reflect assumptions and expectations we didn’t realize we were holding.

Adapting activities is challenging for many reasons: some decisions require extensive prep, others happen in the moment, and all of them must be balanced with the pressure to keep pace while still giving students the time and space to share the incredible ideas, experiences, and mathematical thinking they bring into the classroom. I also found it difficult at times to know exactly what to look or listen for mathematically, and I sometimes interpreted students’ responses through my own assumptions and expectations. While a high-quality curriculum can provide problems that elicit student thinking, the deeper work of examining what we notice, what we value, and how we interpret and leverage students’ ideas extends far beyond even the most comprehensive teacher guide.

While thoughtful planning ahead of time is ideal, we all know it isn’t always possible. Because so much can get in the way of planning and prep time, I think a lot about how to make quick, in-the-moment adaptations based on what students are saying and doing—and on what I still need to learn about their thinking. I often wonder: What if our adaptations did more than simply “fix” a lesson? What if they opened space for deeper student thinking, a stronger sense of community, and richer mathematical understanding?

These questions keep pushing me to look closely at the flow of a typical lesson and identify the moments where even small adaptations can make a big difference. I’ve found that there are five points in a lesson where a few quick, minor changes can have a big impact:

Launching an activity
Working in small groups
Engaging with contexts
Sharing ideas
Synthesizing the lesson.

While it would be wonderful to have a single “right” adaptation for each of these, the reality is that different classrooms and moments call for different routines or structures. In the next series of posts, we’ll take a closer look at each of these opportunities and explore practical, in-the-moment ways to adapt them.

Ultimately, the goal of these small adaptations isn’t just to adjust an activity or lesson; they’re meant to open up richer mathematical experiences for students and provide increased insight for teachers. My hope is that these structures encourage sense-making and increase access to the mathematics; provide every student opportunities to share the amazing ideas they bring; create space for questions about one’s own or others’ thinking; build a stronger mathematical community; and support teachers in reflecting on why they’re adapting an activity and how those choices best serve students.

In this post, I’ll outline a few alternate ways to launch a lesson, and in subsequent posts, I’ll explore each of the other lesson moments listed above.

Alternate Ways to Launch a Lesson

Instructional Goal: Provide students with an entry point into the activity.

When deciding which adaptation to use, it is helpful to think about what aspect(s) of the task might make it inaccessible to your students. For example:

If the context or representation of the problem might be unfamiliar to students, you could use the Tell a Story, What Questions, or Making Connections routine.
If the activity relies on a prior knowledge from earlier in the year or prior grade levels, it might make sense to use the What Questions or Making Connections routine.
If the language (including math language) adds a layer of complexity to the activity, you could use the Word Splash routine.

Try it!

There isn’t one “right” adaptation for any classroom scenario; each is an opportunity to learn more about your students. In the end, it’s not the routine itself that matters as much as what the routine affords—the ways it reveals student thinking, provides an entry point, and supports problem solving. The only way to figure out what works best is to try these structures out! Fortunately, they require little materials prep. All you need is a piece of chart paper or whiteboard and an image from the problem students will be solving.

In your next PLC or planning session, review the activities in an upcoming lesson. As you read through each problem, discuss:

What knowledge and experiences are your students bringing to the problem?
Is there an activity where students might not have access to the problem? If so, what specifically is inaccessible and why?
Which of the four routines will you use to launch the activity?*

*If you’re planning with your grade-level team, each person can try a different routine and then compare the affordances of each one. I’d love to see what you try! Share your ideas in the comments or on IG (@kgraymath)!

Part 2: Supporting student learning as they work in small groups

Wondering About Classroom Norms

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I cannot count how many times classroom norms has been a topic of my conversations in the past month. From creating and facilitating professional learning to thinking about how a curriculum can offer support in this area, I find myself obsessively thinking about ways in which norms might support both students and adults in their learning.

If you asked me a year ago about the norms in my classroom, I would have felt pretty good about how the list hung proudly on my classroom wall, was collaboratively established by students, and appeared to be in place during their math activities.

However, like the majority of my teaching life, the more I learn, the more I realize how much there is still left to learn. In this particular case, it is norms in a classroom.

I think most people would agree that establishing norms is important. Norms can encourage students to work collaboratively and productively in a classroom, elicit use of the Mathematical Practices and help students see learning mathematics as more than just doing problems on a piece of paper. But, how often do we create norms in our classroom only to complain a month or two later that students aren’t thinking about any of them when working together and we struggle with how to refocus students to keep in mind those things they said were important at the beginning of the year? I know I have been there and looking back, wonder how I could have done that better.

While I think good curriculum tasks, lesson structures, and relationships I had with students helped me a lot in encouraging students to be mindful of the norms in the classroom, I don’t think I put an equal amount of effort into maintaining norms as I did establishing them. With that, I wonder what it even looks and sounds like to maintain them?

To me, maintaining norms is about moving from a poster on a wall to a living and breathing culture in the classroom. But, what things can a teacher do to make the norms not only a list, but a part of their classroom math community?

Of course, as the journey begins on writing the IM K-5 Math curriculum, I am also wondering how a curriculum can support teachers in establishing and maintaining classroom norms in a meaningful way. Even more specifically, what could this look like in Kindergarten when we have the opportunity to influence the way students view learning mathematics?

As I think through these questions, I would love to hear how you think about norms in your math classroom. What things can we do as teachers to support students in thinking more about what it means to learn and do mathematics? How could a curriculum, especially in Kindergarten, help teachers in this process?

Two Things I Am Wondering…

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It is an interesting perspective moving out of the classroom into a coaching position. I have had more face-to-face teacher math conversations this year than ever in my career and it is wonderful. This position also lets me take a step back from the daily lesson planning and think about things I see across all of the grade levels. Most times, my thoughts are about the trajectory of mathematical ideas, however over the past couple of weeks I find myself thinking about two things I saw as norms when I taught, but now wonder more about…

1 – Is there a such thing as an addition, subtraction, multiplication or division problem?

I am sure we all can relate to the stories of students struggling with story problems. We see them be successful with Notice/Wonders and 3-Act Math tasks, however when given a story problem some “number grab” and compute without thinking about reasonableness. Why? While I think there are many factors at play here, I have another theory that has led me to question problem types. I could be completely off, but as I look through the curriculum and think about the progression in which I taught in 5th grade, I wonder if there is something to teaching “types of problems” within a unit. For example, in Unit 1, Investigation 1 could be my multiplication lessons while Investigation 2 could be my division lessons. While we don’t explicitly say, “this is how you solve a multiplication problem” and we explore various strategies to make connections between the operations, the header of the activity book pages say things such as, “Division Stories” or “Multiplication Stories.” Also, the majority of the work that week is the specific operation and applications.

From there I began to wonder, is there really a such thing as a specific operation problem? I would think that any division story could also be thought of as a multiplication problem. Do we lead students to think there are certain types of problems even if we make clear all of the strategies to find solutions? I love how CGI talks about problem types and wondering why more curriculum are not set up that way instead of keying students into operation-specific problems?

I asked some 4th graders about this exact idea. I gave them some multiplicative compare problems and asked them if they thought about each as multiplication, division or both. Then we talked about why.

2 – What makes students attach meaning to a vocabulary word? Do they need to?

Every year in 5th grade, I was confident that all of my students could find area and perimeter of rectangles. However, I was also confident that there would also be a handful of students who could find area and perimeter but didn’t know which was which. After much work with area and perimeter, they would have it by the end of the unit, but did they remember when they got to 6th grade? I am not sure.

Now, seeing all of the work they are doing with this beginning in 3rd grade, and talking to 3rd and 4th grade teachers who are seeing the same thing, I am left wondering why this is? What makes students attach meaning to vocabulary? This question is then followed by the my very next question…when do they have to?

I wonder if students should ever be given a problem where the context would not allow the students to figure out which one, area or perimeter, the problem was asking. For example, if Farmer Brown is buying fencing he would need the perimeter where if he was buying something to cover a piece of ground, it would be area. Should we ever give them just naked perimeter or area problems with no context where knowing the meaning of the word impacted their ability to solve it?

And then, after they do all of this work with both measurements, why do they forget which word is which year after year? I know the teachers do investigations with the work and use the vocabulary daily during the unit, as I did, but students still don’t hold on to it. What makes it become part of their vocabulary? Is it just too long between when they use it? Is it

These are just two things I am wondering about….

A Teacher & Mathematician Mash Up

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One of the many things I love about Twitter is the diversity of the group in which I have the opportunity to interact. Every day, Twitter provides the space for me to move outside of my classroom happenings and connect with others of varying perspectives and insights on teaching and learning. While these perspectives are so interesting to me, if I am being completely honest, they can also be quite intimidating. Not intimidating in the sense that one person’s point of view is “better” than another, but more in the sense that sometimes math conversations go to a place content-wise or philosophically that I cannot even engage. Not because I don’t feel like I don’t belong, but simply because I don’t even know what the heck to say because I don’t understand what they are talking about or it is so far removed from where I am in the classroom, I can’t relate.

The way I feel in those situations feeds my preconceived notions I have about mathematicians. Not the type of mathematicians I would call my students because they are doing great math, but mathematicians as in, that is their job title, you know, those mathematicians. I so admire the way in which they think about math, however given a choice, I would probably shy away from a conversation with them out of shear nervousness of saying something that sounded silly, or even worse, completely wrong mathematically. That was, until I started my work with Illustrative Mathematics.

Throughout my projects with all of the wonderful people at Illustrative, I have truly seen such incredible value for the perspective each and every person, whether a teacher, a math coordinator, a mathematician, or math specialist brings to the work we do in working to improve teaching and learning. From developing tasks, to facilitating professional development, the work is such an amazing collaborative effort in which I learn SO much. During this learning, my confidence in what classroom teachers bring to a math conversation grows, as does my appreciation for our different perspectives.

Most recently, a mathematician at Illustrative, Mike, and I have been working collaboratively on tasks to be reviewed for the IM site. It has been such an amazing learning experience for me. He is wonderfully thoughtful about the math, open to any ideas and/or questions I have and possibly the quickest email responder I have ever encountered:) Throughout our work together, I felt we were on the same page as far as the content of the task as well as in our thoughts about what students would do with the math of the task. I didn’t feel at all like I was “just” speaking from experience and he was talking from this “mathematician world” in which I couldn’t relate, but that we were both thinking deeply about the math and how it looks in a classroom, it was a beautiful thing.

After our first task, I thought to myself how odd it was that we thought so much alike. I was completely anticipating having these eye-opening mathematical revelations after our conversations together. However, during our second task, the revelation(s) came rolling in and the difference in our perspectives was really interesting and valuable.

The task centers around the commutative property of multiplication with fractions in the context of wrapping packages with riboon, 6 x 2/3 and 2/3 x 6. In my classroom, I am so wary of students strictly computing without making sense of problems that I make a conscience effort, probably to almost an extreme, to connect their representations to a context. For example, in the problems above, I really want students to “see” the story for each differently. I want them to see 6 group of 2/3 for 6 x 2/3 and 2/3 x 6 as 2/3 of 6 or an area model with 6 and 2/3 as the dimensions. My biggest concern as a teacher, is the students connecting the problem to the context and then noticing patterns that show commutativity. My questions primarily focus on connecting their representation and notation back to the context. Everything to me is focused on context because of my fear of them number-crunching their way through an algorithm they don’t have a contextual visualization. Did you happen to catch that I care about context in that paragraph:) I even blogged about it here: https://mathmindsblog.wordpress.com/2015/03/29/commutativity-in-fraction-multiplication/

Mike and I both agree all of this contextual work is super necessary and important. This past year, I think my students did a beautiful job seeing the commutative property come out through patterns and repeated reasoning, however, after talking more with Mike about this commutativity, I realized I missed such an important piece. A piece that would have really solidified the commutative property in their work through their representations themselves.

I wanted students to match one of those two equations to a context and develop a more appropriate context for the other, however that just shows they come out to the same answer. In my mind it doesn’t really show how they can be commutative within the same context. I had never thought of that so much until Mike emailed me this statement…

“… if someone arranged the pieces of ribbon appropriately they could argue for either equation. I think that what we are after is to match an expression with some kind of reasoning. In other words, the real question to ask the students is to explain their expression via a picture that accurately models the situation.”

This is the point where I completely wish I could reteach this lesson. I would do everything the same, but add this piece. I would love to see if students could see one representation in another for both 6 groups of 2/3 and 2/3 of 6. Have them defend their reasoning and/or find their reasoning within someone else’s work. That really would have proven to students how the commutative property looks versus just seeing I get the same answer no matter the order of the numbers. Which is kind of how I felt I left it this year.

This has been, and will continue to be, such a wonderful learning experience for me. I SO appreciate the diversity of people I have worked with at Illustrative as much as I appreciate the wonderful mix of people I get to learn from on Twitter. It is enlightening to me that as open and addicted as I am to learning, there are still so many things that I have a classroom perspective on that can be improved and extended through conversations with people who I may typically have shied away from in person. Knowing they appreciate my perspective is such a wonderfully empowering thing for me as a learner. Thank you to all involved in my journey!

MathMinds

by Kristin Gray.