I have been thinking a lot about this word: agency.  I have been attending sessions at NCTM and NCSM all week.  Many speakers pushed my thinking about what it means for students to be agents of their own learning. During Cathy Humphrey’s session at NCSM, she shared a slide that described what student agency might look like.  It really resonated with me.

This week,  I had the great privilege of presenting at the National Conference of Supervisors of Mathematics with two teachers, Deb Hatt and Carolyn Watkins, who are trying to make our district vision of math support a reality.

This was Deb’s first year in the Math Interventionist position. She believes in our mission statement, but she wasn’t sure how to put it into action.  She and Carolyn, who teaches third grade math, didn’t start out the year collaborating.  In September, Deb was taking a group of 3rd grade students out of Carolyn’s math class to provide interventions.

Deb and Carolyn quickly realized that pulling students out of their math class was not effective or equitable.  They decided to change their approach.  Deb started joining Carolyn and her students in math class. She and Carolyn started planning together once a week. Historically, teaching has been something we have done in isolation, behind closed doors. Deb and Carolyn decided to make their teaching visible. They took a giant risk. They were vulnerable and honest.  Their collaborative journey was not predictable or neat. Like most meaningful learning experiences, it was unvarnished and gritty; full of questions, partially formed ideas, and mistakes.

At first, their planning was focussed on supporting students who needed intervention, but, over time, the line between “Deb’s students” and “Carolyn’s students” blurred.  This seemed to be working well for most students, but there was one student who needed more. Carolyn and Deb decided that Deb would work with Jayden daily, before her math block, to support her as she continued to build her understanding of multiplication and division. This is Jayden. She is working on her anchor chart of known or derived math facts.

Even though Deb sees Jayden outside of math class, Jayden is also Carolyn’s student.  In the clips below you will see Carolyn working with a heterogeneous group of students. Jayden is one of these students. Take a minute to think about the problem these students are working on before you listen to the clips.

As you listen to these students working, notice that their thinking is fluid as they try to make connections between the different shapes and their values.

 

At this point,  you might be wondering, where is Deb? She’s there. Do you see her in the background? She’s working with other students.

After these students figured out that the triangle is worth one-third, Carolyn noticed that Jayden was thinking about the blue rhombuses.

Carolyn said, “Oh wait a minute. Let’s take a look at what Jayden just did.” Carolyn  intentionally positioned Jayden as a significant contributor to this community of mathematicians. Both Willow and Madison built on Jayden’s thinking and figured out that the blue rhombus is worth two-thirds.

Jayden was not convinced so Carolyn conferenced with her to find out what she knew.  Too often in education, we use a deficit model to describe student understanding. When we do this, we end up drowning in our own assumptions, lowering our expectations of what students can do, and creating an inequitable learning environment that hinders student agency.  As you watch the following clip, look for evidence of what Jayden DOES know about fractions.

 

After Deb, Carolyn, and I watched the clip above, we talked about next steps.  We thought Jayden would be able to identify the value of the blue rhombus if she worked on it for a little bit longer. We agreed Deb would work with Jayden on this problem during her next 1-1 session with Jayden. Take a look at a clip from that session.

 

In the previous videos, you saw Carolyn and Deb help Jayden clarify her understanding of fractions as numbers.  They amplify her understanding of fractions by expanding her opportunities to talk about what she knows, ask questions, make connections, and revise her thinking. Deb and Carolyn held up a mirror for Jayden and she saw herself as a mathematician.

While planning for our presentation, we anticipated that participants would ask us, how do you know that collaboration is working? Janet Delmar, principal at the school where Deb and Carolyn work, said,   “The students in this classroom feel respected by their peers and teachers. They know their contributions in math class are valued and important. We are seeing evidence of their learning in their conversations with their peers and teachers. They enjoy mathematics and see themselves as mathematicians.”

As I listened to her, I wondered, would people be satisfied with this answer? I said, “What if the participants ask, but what about the data?”

Janet responded,  “If students hate math and don’t see themselves as mathematicians, then who cares about the data.  We will continue to look at the NWEA, the MEA, and our district Common Assessment tasks, but we don’t need test data to show us that collaborative teaching is good for kids.”

As I reflect on what I think I know about student agency, I’m wondering about teacher agency.  When teachers feel ownership of their own learning, they are more likely to “offer their thoughts, attend and respond to each other’s ideas, and generate shared meaning or understanding through their joint efforts”  Admittedly, as a math coach, there are times when I find myself overly focussed on what teachers aren’t doing. Sometimes, I use a deficit model of coaching. This experience has taught me the importance of cultivating teacher agency and paying attention to all the challenging but essential work that teachers are doing.

During NCTM’s Shadowcon18, Javier Garcia asked, “What will it take to make our math classes more about mathematics and less about status?” What a great question! I think it will take a concerted effort by all of us to stop using standardized test data to sort teachers and students into fixed categorical bins. It will take us emphatically committing to a strength based model of learning, teaching, and coaching.

 

Recently, I have been helping a third grade teacher learn how to use Number Talks to  develop computational fluency and deepen student understanding of multiplication and division.  The first time we met, we planned a Number Talk using MiniLessons for Early Multiplication and Division by Willem Uttenbogaard and Catherine Twomey Fosnot.  We rehearsed the Number Talk.  As we discussed the math, we anticipated what students might say, how Rachel might record student thinking, and what questions she might ask to connect student thinking. We read the teacher’s guide and noticed some of our ideas in the description. We discussed how Rachel could follow the same structure we just used to plan subsequent Number Talks.

Last Thursday, Rachel and I met again to discuss how the number talk went. She told me she was so glad I showed her how to use her Smartboard for Number Talks.  She described how helpful it was for her students to be able to describe their own thinking and see each other’s thinking. She pulled up the Smartboard Notebook she had created for the last Number Talk so we could discuss her student’s thinking. Unfortunately, during this Number Talk, the markers on her Smartboard weren’t working so she had to record student thinking with numbers, symbols, and words.

At this point, Rachel asked me an awesome question.  She wondered, “my students started doing this thing where they find a fact that doesn’t work evenly and then add the extra squares. Should I encourage them to do that? Will it distract them for what they should be doing?” I asked her to tell me more. She showed me some examples. She said the students were thinking about division. They were noticing that you couldn’t divide 25 in half. They tried to divide it into three groups. One of the students said, “You can do it any way you want. You just have to add the extras.”

Rachel looked at me, waiting for me to tell her what to do next.

She said, “Should I let them do this?”

I responded, “I don’t know. Let’s find out.”

Then, I got really excited.  “Oh my gosh, are they thinking about division? Are they thinking about remainders, but they don’t know it.  This is so cool. Do they wonder if this will always work? Do they wonder how many ways there are? I’m wondering how many ways there are! Are there a certain number of ways? Does it depend on the array? What does this have to do with factors? There must be a pattern. Is there a pattern?  Can we do the math? Do you want to explore this right now? I feel like we have to explore this right now to answer your question? Are you okay if we try this right now?”

Rachel smiled. I started scribbling things down on scrap paper.

It didn’t take long before we realized we needed some kind of system for organizing our thinking.

At this point, we started talking about the commutative property.  We discussed the difference between 3 groups of 8 and 8 groups of 3. We wondered, “are we only using arrays or can we use “groups”?” If we are only using arrays than we can’t see any 3 by 8 arrays in a 5×5 array. We can see seven 3 by 1 arrays, but then we have 4 leftovers. We can’t make anymore 3 by 1 arrays. If we are using groups, we can see 8 groups of 3 plus one leftover and we can also see 3 groups of 8 plus one left over. We were in this funny place. We were mixing array language with “equal groups of” language, trying to figure out how the constraints of contextualizing the community property impacted our problem.  I thought about this some more this weekend. I tried to anticipate what students might do when they confront this situation.

There is the potential for this conversation to get messy fast. We could all end up on the fast track down a major rabbit hole. I think it is worth the risk.  As I explored this problem, I was forced to articulate the difference between an area representation of multiplication and a set representation of multiplication.  These third grade students are about to start their unit on area and perimeter. This seems like a meaningful mess.  It might be worthwhile to pause during the exploration, examine a few of the decompositions and discuss; What is the same? What is different?

When Rachel and I were exploring this problem, we didn’t define the constraints. We were in the initial stages of our problem solving. We were messing around with  mathematics. We were thinking about arrays, but using “groups of” language. Towards the end of our exploration, Rachel stumbled upon a conjecture. She said, “I think we have found all the ways, but I can’t really explain why.” I asked her to tell me what she was thinking, even if it was still fuzzy. I wrote down what she said:

She said she wasn’t sure how to describe what needed to be bigger, but she just knew that we could make more groups with the leftovers. I knew what she meant, but I couldn’t find the precise words for it, either. I tried to use numbers and symbols to record what was happening:

 

Then, Rachel said, “I think I can explain why we can’t find any more ways. If the number of leftovers is larger than the group size, you will always be able to make more groups.”

“Yes. That makes sense to me.”

While Rachel and I were working, every so often, we would zoom out and discuss the implications of Rachel exploring this problem with her students. She wondered, would it confuse them? Would it dissuade them from using efficient strategies? I kept trying to find connections to the third grade standards.  I was reminded of an excerpt from Bill McCallum’s progressions document:

I saw a lot of connections to fourth grade standards, particularly the one about interpreting remainders. Rachel and I both worried, “is it okay if these students start talking about remainders if they are only in third grade?”

I have been thinking about this 4th grade standard a lot this year.  Marilyn Burns, Kristin Gray, and Jody Guarino have all pushed my thinking on this standard. I started a post about a lesson I did with a fourth grade class earlier in the year. I haven’t finished it, but I’m posting it in draft form because it is really connected to this post. How is this fourth grade standard connected to work in third grade? It should be, right? Mathematics is a system of interconnected ideas. Remainders don’t just drop out of the sky in fourth grade? They shouldn’t, right? What might we explore in third grade that would connect to a deeper exploration of remainders in fourth grade? I’m thinking about this cluster:

I’m not totally sure if/how these standards might connect to the problem that Rachel is  going to explore with her students, but I’m wondering about it.

I shared some of these thoughts with Rachel. Together, we decided this problem would be an opportunity to discuss the difference between arrays and groups, connect multiplication and division, look for structure, persevere, and organize thinking. Should Rachel create a space for her students to explore this problem?

Of course she should! They created it!  It is their problem!

We talked about how she would present the problem and support the student exploration. We revisited how the question originally came about and how she would phrase the question so it was clear and still captured the student’s voice.

Rachel took all of our work with her so she could share it with her students after they tackled the problem. She thought they would be really excited to see that we worked so hard on a problem that they created.  I’m still thinking about ways that Rachel can use this problem as a spring board to deepen understanding of grade level standards.  She plans on using it with her students this coming Monday. I would love to hear your thoughts about questions Rachel could ask, connections she might make, next steps for her students.  Try this problem out! Get messy! Let us know what you find out.

 

A well placed Notice and Wonder routine can make all the difference when you are trying to elicit and use evidence of student thinking. Take a look at the picture below. What do you notice?  What do you wonder?

We asked a group of fourth and fifth grade students these questions and here is what they said:

This Notice and Wonder routine was inspired by many recent conversations about interpreting remainders.  As far as I understand it, the term “remainder” is a convention. There isn’t really anything “math-y” about it, yet it pops up in very “math-y” places.  It is in the Common Core Progressions document:

Paul Lockhart says, “The general problem then becomes how to efficiently determine the division of the total, as well as the number of leftovers, if, any.  Incidentally, the number of leftovers is usually called the remainder (from Latin remanere “to stay back”).  He also says, “The thing about verbs is that whenever we have one, we always seems to get two.  If I lock the door, then at some point I will need to unlock it. Tie a piece of string, and sooner or later someone will want to untie it.  Actions that can be done almost always need to be undone. And this is especially true in mathematics, where symmetry is so highly prized and where the imaginary nature of the place allows us the freedom to reverse our actions so easily.”

There are situations where we will have to deal with remainders.  We will have to interpret them. We often teach kids they have three choices when dealing with a remainder:

• Use it as a decimal or a fraction.
• Ignore it.
• Round it.

The problem is: how do we record it the remainder, especially when we don’t have a context? In the past, we have used the letter “r” to denote a remainder.  For example, we might write 13 / 4 = 3 R1. This method of recording can lead to problems. 13/4 is not equivalent to 3 R1 because “r” isn’t an operation. Ideally, we would love for students to create an equivalent expression as a way to represent the remainder.  Any of the following would work, if we were talking about money or cookies – something “soft” as Lockhart would say.

• 13/4 = 3.25
• 13/4 = 3 1/4

But how do we get students to write equations about remainders that can’t be split into smaller pieces. There are contexts where we have to round a remainder or possibly ignore it, but how do we represent them with equations?  We could write this:

• 13 = 4 x 3+1

This works, but how do we create a context that lends itself to writing this equation in the context of division? How do we create a need for this equation?

One day, last week, I was planning with a colleague and telling her about my quest for a division context that prompted students to naturally think about writing the remainder as an expression of multiplication and addition.

She said, “Well, you need to think of a situation where there are pairs – like shoes.”

At this point, I remembered a picture I had taken a few weeks earlier. It was a picture of a big pile of footwear on my mudroom floor. We decided to show it to her class of 4th and 5th grade students that afternoon.  The picture shows all pairs and then there is just one lonely sandal. We hoped this picture would elicit equations that might prompt a discussion about recording remainders with mathematically accurate expressions.

Then, we asked them to write as many equations as they could think of to represent this pile of footwear. Here are some of the equations they came up with:

 

 

 

 

 

We asked the students to explain how the equations related to the picture.  This is where things started to get messy, in a really meaningful way.

Someone wondered, “Where is 4 x 2 + 9=17?  Where did the 9 come from?”

“It is the boots.”

“No, it is the shoes.”

“It doesn’t matter.”

“Yes it does! It can’t be the boots!”

A really important argument ensued about what the numbers in the equation represented.  One student finally convinced everyone that the 9 had to represent the shoes because there are 4 pair of boots in the picture.  The shoes are the category that have the extra- the remainder. We have to deal with the remainder in the context of the shoes. We had a great conversation about what to call these units that we were dealing with. Do boots count as a sub category of shoes? Can we call the total shoes or do we have to call it footwear?