This morning, I stumbled upon a video clip of two students double counting. These students were immersed in an authentic problem – how do we read two counts? I loved hearing their two voices and watching their gestures as they navigated two counts, uninterrupted by adult conversation.
This video reminded me of a question that I have been mulling over for months now: How do students construct a mathematical identity that is rooted in the context of who they are and who they want to become outside the classroom? In particular, I wonder, how does the way we teach students to construct viable arguments and critique the reasoning of others nurture or neglect their identity as a whole person?
The predominant ways that I have seen students taught to engage in this math practice have been:
sentence stems like “I disagree because…”.
orchestrated turn taking
I think these structures and norms are used with the intention of giving students equal opportunity to participate and feel like their ideas were heard. But do they work? Does each student feel like the opportunity to argue for something they believe to be true has been genuinely presented to them? Does each student feel like their true voice is heard? When I think about the word “argue”, I think about passion, frustration, and joy. I also think about family and community. What does a productive argument look like in the lived experiences of each of our students? Are there sentence stems, raised hands, and orchestrated turns taken?
I am not saying that we should abandon structures and norms in our classrooms. Structures and norms are intended to ensure that arguments are productive. I wonder if we need to reconsider how the structure and norms come to be. Who benefits from the structures and norms?
When I watched the video of these students double counting, I think I saw passion, frustration, and joy, but I don’t want to project my thoughts and feelings onto the students. I saw back and forth conversation, ideas being exchanged, and thinking being revised, all while an adult observed.
This video clip was a spring board -a push- for me to re-examine classroom norms and routines that I always used and accepted as “good practice” for all students. I am going to sit with the questions:
Where did our classroom norms and routines come from and who do they really serve?
Whose culture have I preserved when I cultivated and enacted classroom norms and routines?
I haven’t been able to write much on my blog for the last two years because I have been busy writing the 5th grade course for the Illustrative Mathematics K-5 math curriculum. I am humbled and honored to share some of the story my colleagues and I are writing.
From the start of the year, we want students to know they are capable of engaging in grade level mathematics. In the Opportunity Myth (2018), data has shown there is an opportunity gap for historically marginalized students between the grade level expectations laid out in standards and students’ opportunities to engage with this content in their math classes. Oftentimes, grade level content is withheld from students because they are perceived as being not “ready” and, therefore, they are restricted to only engage in below grade level work. If we want to close the opportunity gap, we have to do many things differently than we have done in the past, one of which is re-examine the sequence of the content we put in front of students. The Illustrative Mathematics K-5 story of fifth grade was intentionally written so all students have access to grade level content.
Typically, many 5th grade math curricula start the year with units that focus on place value concepts and whole number multiplication and division algorithms. When considering whether to start the year this way, we wondered:
What message does the mathematical content in the early units of the year send to students about what it means to ‘do math’ and what is valued in math class?
What assumptions might educators make about students based on their successes or challenges with multiplication and division algorithms at the beginning of the year?
How might these assumptions impact students’ access to grade level content throughout the course of the year?
Extend an Invitation
From the start of the year, we want to send the message that mathematics is an opportunity to be curious, collaborative, and playful. To do this, we begin by introducing students to the concept of volume. Being a new concept to fifth graders, it naturally invites students to be curious and creative, while at the same time offers teachers the opportunity to notice and build on what students do know and can do. It also diminishes, if not eliminates, the assumptions both teachers and students might make about what it looks like to be ‘good at math’.
Students begin their study of volume with many opportunities to build with unit cubes and use familiar words and phrases to explain the layered structure of rectangular prisms. Teachers listen for and display the words students use, such as “slices”, “groups”, or “layers”, to describe how they counted the cubes to determine the volume. Students connect their less formal language to multiplication expressions that represent the volume of rectangular prisms before they are introduced to more formal math vocabulary and procedures.
How do the expressions 5 x 24 and 6 x 20 represent the volume of the rectangular prism? Explain or show your reasoning.
Throughout unit 1, the measure of prism side lengths were chosen to encourage students to make sense of concepts of volume while they continue to strengthen their fluency with multiplication. Over time, students connect their informal strategies for finding the volume of rectangular prisms to generalized formulas. Later in the year, after learning the standard algorithm for multiplication in unit 4, students solve problems involving volume with larger multi-digit numbers as side length measures. This progression increases the opportunities for students to be successful with multiplication and division algorithms and decreases the likelihood that they will be perceived as not ready for new grade level topics because of their proficiency with computation. Therefore, we believe these choices will decrease the opportunity gap and increase student access to grade level content.
Put Fractions in the Forefront
Beyond Unit 1, we want to continue to maximize students’ opportunities to access grade level content. In Units 2 and 3, we introduce grade level content while providing ongoing opportunities to reinforce prior grade level understandings and fluencies, all within the work of fractions. While many curriculums organize whole number computation before fractions, this sequence has implications for extending an opportunity gap. If students are perceived as not being able to use whole number multiplication and division algorithms correctly, they may be confined to remediation cycles and denied the opportunity to engage with grade level fraction concepts and procedures. We know that understanding fraction operations is a major indicator of success with topics in high school mathematics so we want to ensure that each student has the opportunity to make sense of grade level fraction understandings and operations early in the year ( Siegler et. al., 2012, p.695).
Continue to Build on Prior Grade Level Content
In units 2 and 3, students revisit the meaning of fractions, multiplication, and division. They connect what they know about these topics to new learning.
Students begin unit 2 by interpreting a fraction as division of the numerator by the denominator. Through this work, they continually deepen their conceptual understanding of fractions and division. In the example below, students connect what they know about unit fractions and division.
Complete the table.
number of people sharing a sandwich
amount of sandwich for each person
Choose one row of the table and draw a diagram to show your reasoning for that row.
What patterns do you notice?
As the unit progresses, students extend their understanding of multiplication as they multiply a whole number by a fraction. In the example below, students leverage what they know about whole number multiplication to solve problems involving multiplication of a whole number and a unit fraction.
Find the area of the shaded region. Explain or show your reasoning.
In unit 3, students continue to use area diagrams to make sense of fraction multiplication. They build on the new understanding of multiplying a whole number by a fraction while applying prior grade understanding of area. They recognize, when multiplying fractions, the unit is the size of the piece being tiled within the unit square. They count those unit tiles by multiplying the number of tiles in each shaded row by the number of shaded rows.
For example when asked to determine the area of the shaded region below, students know that the unit square is partitioned into a array, so the size of each tile is . They then multiply to find how many tiles are shaded and determine that the area of the shaded region is. Students later apply these understandings to multiply any two fractions.
Students are offered another opportunity to revisit the major work of prior grades in the context of the major work of 5th grade when they are introduced to division of a unit fraction by a whole number and a whole number by a unit fraction. As shown in the example below, the sequence we chose offers students the opportunity to be curious and creative while deepening and extending their understanding of the meaning of division.
Sequencing fraction multiplication and division before whole number algorithms invites teachers to notice and celebrate what students do know about multiplication and division and minimizes the chances of teachers and students making assumptions about what students don’t know about fraction operations.
Make Algorithms Accessible
By the time students encounter whole number multiplication and division algorithms in unit 4, they have had significant experience building the necessary fluencies and understandings to be successful. Teachers have had significant opportunities to notice what each student knows and can do in regards to applying concepts of multiplication and division.
In telling the story of 5th grade, we chose to begin with an invitation to engage in grade level mathematics, introduce fractions early in the year, and postpone multi-digit multiplication and division algorithms until later in the year. We think these choices will:
send the message that each student is creative, curious, and capable of doing grade level math in the beginning of the year.
support teachers to notice and build on what students know and can do.
increase opportunities for students to access grade level content.
We hope these choices lead to a world where more historically marginalized students have the opportunity to know, use, and enjoy mathematics.
In the last two years, I have thought more about race then I have in my entire life. My job for the last two years has been to write math curriculum for 5th grade students all over the country. When I first got the job, I was really worried about the fact that most of my career was spent working with white children in Maine. Does it matter that I have mostly worked with white children? It seems like it would matter. What kind of contexts should I write about? What do kids in the city care about?” I think I actually said that out loud at one point: “kids in the city”. I was afraid to say the word black or brown because I didn’t know how to talk about race.
Recently, I have been asking myself, what does it mean to be white? In my everyday life, as a member of a predominantly white community, I don’t think about being white. Whiteness is what I have in common with most of the people around me.
I remember the first time I heard the term “white privilege”. I resented it. Then, about a year later, I heard a black woman share a story about a time when her son came home in a cop car. He was walking home from the store when a policeman pulled over and asked him where he lived. The policeman wasn’t convinced her son really lived where he said he did – in a predominantly white, upper-middle class neighborhood. I think about that story a lot. I think about how women of color have to worry about their sons in a way that I will never have to worry about mine because racism is still embedded in our society.
I try to notice my own thoughts and feelings about race. Often, I have to come to terms with some uncomfortable feelings. I have biased thoughts and feelings. I am learning to recognize them, sit with them, and own them. As a white person, I don’t have to make an effort to gather with other white people. My world is structured for white people gathering. I don’t worry about being excluded because of the color of my skin. I don’t worry about being killed because of the color of my skin.
Paulo Frier says, “Education either functions as an instrument which is used to facilitate integration of the younger generation into the logic of the present system and bring about conformity or it becomes the practice of freedom, the means by which men and women deal critically and creatively with reality and discover how to participate in the transformation of their world.”
As a white educator, I realize the most important thing I can do right now is cultivate the practice of freedom. It starts with me recognizing the racism in me and my world and making an effort to change it.
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 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.
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?
Last week, I was working with some second graders. We were doing a number talk and one of the students asked if he could use a hundreds chart. I said, “sure”. All of a sudden, five kids got up and headed to the envelope of available hundreds charts. I watched the onslaught for a minute, wondering how it would work out. The first two students grabbed a chart and went back to their space on the carpet. Then, two different students grabbed the envelope at the same time, each maintaining a fierce grip while glaring at the other. A third student stood quietly and watched.
“I had it first!” Elise said as she tugged the envelope towards her.
“No. I was waiting here. It’s my turn.” Joseph shoved his hand into the envelope and tried to wrestle one of the hundreds charts free while Elise simultaneously used her free hand to pinch the envelope closed.
I focussed my attention on Charlie, the boy who stood watching. “Charlie,” I said, “Thank you so much for waiting so patiently. It must be really hard to wait when you are anxious to get your hundreds chart and go back to your seat. I really appreciate how careful you are being.”
Elise and Joseph froze. Joseph dropped his hands from the envelope. Elise carefully took one of the hundreds charts out of the envelope and handed it to Joseph. I said, “Elise that was really thoughtful. I know how much you wanted that hundreds chart and you just gave it to someone else. Thank you for taking care of your friends.” Joseph passed the hundreds chart to Charlie. I continued, “Wow, Joseph. You are also being really thoughtful and taking care of your friends. Thank you.” Joseph took out a hundreds chart and handed it to Elise. Then, he took one out for himself and they all sat down.
You might be thinking, “well that only worked because Joseph and Elise don’t really have a lot of ‘behavior problems;'” Nope. Joseph and Elise have a really hard time relating to and communicating with their peers. It is not easy for them to slow down and consider how their actions are impacting the people around them. You also might be thinking, “Well, it takes forever to establish that kind of relationship with kids. You’ve probably been working one on one with them all year.” Nope. I am only in their math class two or three days a week and I just started working with them in March.
I think my response to these students “worked” because I focussed on what the students were doing “right”. I held up a mirror for them. In the mirror, they saw who they want to be. The “reward” for their behavior wasn’t a sticker, a point, or a card. The rewards were feelings: pride, purpose, and gratitude.
Later, during the Number Talk, a different student was sharing his thinking. Several students turned their bodies around to face that student. I took a moment to recognize their gesture. I said, “I notice you just turned around to face your friend. That shows them that you are really interested in what they are saying and you want to learn with them.” Simultaneously, three other students turned to face the speaker.
After our Number Talk, we worked on a Same But Different Math problem that I created. As I anticipated, there was some disagreement about which numbers belonged in the blanks. We spent the remainder of class trying to prove whether or not the same number would make these two different equations true.
Some students didn’t get to finish their work before class ended. I decided to continue the problem on Monday. For the lesson close, I asked if anyone wanted to share anything they had learned so far. Greg was anxious to share. He couldn’t hold it in anymore. He shouted, “I learned I was wrong!”.
“Wow,” I replied, “Tell us more about that.”
He continued, “I thought the answer was 22 and Riley did too, but then Riley thought maybe he was wrong, but he couldn’t figure out why he was wrong. He asked me to prove to him how I knew it was 22. When I started to try to prove it, I realized I was wrong! Riley helped me figure out I had made a mistake!”
“That is really cool,” I said. “That is what mathematicians do. They share their thinking and make mistakes. When Riley asked you to prove it, he helped you figure out what you did wrong.”
Greg grinned, “Yeah! He really helped me.”
Riley chimed in, “Its a good thing because I thought it was 22 too.”
As math educators, we talk all the time about how important it is to get students to ask questions, justify their thinking, and critique the reasoning of others. If we really want to make this happen, I think we need to pay attention to the mirrors we hold up for our students. If we use extrinsic rewards and punishments are we building agency or promoting compliance?
I’m not sharing anything new, here. I’m not claiming to be doing anything new and amazing. I just wanted to take a minute to share Friday’s math class because I’m so proud of these students. In our world at large, where adults are rarely listening to each other with the intention of understanding, I feel an urgent sense of purpose to help young people experience a respectful community of learners.
“Math is a language, sometimes intimate, often boisterous, but always layered with experience and life profoundly lived.” (apassion4jazz.net):
Last weekend I was listening to wind chimes and wondering if there are patterns to the wind. The wind reminded me of Jazz music. I don’t know much about Jazz, but I love listening to it. I love how it sounds simultaneously fluid and unpredictable. As I reflect on my time with students and teachers last week, I realize that there is a certain amount of necessary chaos in learning. The chaos often pushes my thinking and helps me connect ideas.
Last week was crazy. I was all over the place. I didn’t allocate enough travel or planning time. I double booked myself. I was late to most meetings. I left personal belongings all over the school district. I got more than one text like this one:
This was my schedule for just one of those days:
Truth: Last week wasn’t any different from other weeks. Scarier truth: I actually thrive in chaos. On Thursday morning, I was working with a small group of second grade students. The teacher I am collaborating with was working with a different small group of students. Picture me scrambling to get poster paper out of my backpack as I realize that I didn’t write the scenarios on paper ahead of time. Have a listen to our organized chaos:
Admittedly, I was feeling frustrated. I wanted to say, “just get a pencil, sit down, and listen to my story about penguins, now!” I didn’t. Instead, I tried to listen. I redirected. One of many benefits of recording myself when I teach is it forces me to be present, sort of. I spent one minute and six seconds trying to get a pencil in each student’s hand, only to tell them to put their pencils down and listen to me while I read to them about penguins.
Finally, I got them to close their eyes and listen. I said, “There are some penguins on the ice. Some are sleeping and some are awake.” Then, I asked the students to open their eyes and tell me what they saw.
Rowan said, “Three penguins sleeping. Wait. Is it three?”
Meka added. “I see ten.”
I asked, “Rowan, are you wondering, is it three? Meka, are you wondering, is it ten?”
Rowan said, “Is it okay if I say it’s three?”
I told him, “Sure. What did you imagine?”
“Three penguins on the ice. Some are sleeping and some are awake.”
“And what did you picture in your brain when you had your eyes closed? What did you see?”
“More penguins awake and less penguins sleeping?”
At this point, there was a disagreement about erasers which derailed the conversation slightly. I took it as an opportunity to build community. Have a listen.
These three students live in a small town and they go to a small school with less than 60 students. They are like a family. Even though they argue sometimes and forget to listen to each other, they really care about each other. They learn with and from each other. It is really important that I asked Rowan and Trent to listen to Meka, not me. And they did.
When I asked Trent what he pictured, he told me, “Nothing. It was just black. That’s it.” I told him, “that happens sometimes.”
I asked the students, “Is there anything else you wonder besides is there three? Is there ten?” They wondered about 100 penguins.
Next, I read the story again, but I changed the amount of penguins to 12.
This time, instead of asking them to talk about what they imagined, I asked them to write it down on paper. Then, I asked them to share their thinking. Meka volunteered first. Trent wasn’t so sure her answer made sense. Listen to them as they discuss Meka’s work:
I think this conversation is beautiful; simultaneously fluid and unpredictable. Yes, Meka convinces Trent that she also has twelve penguins and that is important, but her invitation to count with her and his respectful request to count for himself are so much more profound. The boisterous intimacy continued as Trent and Rowan described what they were seeing.
At this point in the lesson, I might have assumed that the students understood that there could be more than one way to group 12 penguins into two subgroups of sleeping and awake. They seemed amenable to each other’s thinking. However, there is a difference between being convinced that a particular approach works and generalizing an understanding of whether the approach will always work and why. I tried to push, asking, what is the same about your solutions? What is different? Listen:
After we discussed similarities and differences, I encouraged them to see if they could find all the ways we could group 12 penguins; some sleeping and some awake. Trent asked if we could write the word “exhausted” on our poster. He said, “the penguins are awake, but really really tired. They are exhausted.” So, I did. I offered the students the opportunity to use the base ten blocks or Digi-Blocks to help them solve and I went off to check in with some of the other students.
I left my phone recording and captured some wonderful bits of conversation between the students. I particularly enjoy the giggling.
Trent is thinking about a “turn around fact”. He is wondering about changing the groups so that ten penguins are sleeping and two penguins are awake. At one point, he says, “now, ten are sleeping and two are awake”, but that is not what he records on his paper. Trent is learning to take risks, but it is hard for him. I am tentative to push him too fast. I am thrilled that he is so thoughtfully engaged in learning.
Rowan and Meka identified some other combinations of 12.
It was almost time for class to end. I asked Meka and Rowan to explain how they were able to find different ways to make 12. Rowan said, “I minused some from one and I plussed it to the other.” Meka describes her hard work.
I was tempted to record Rowan’s strategy using an equation, but I wondered if my recording would be meaningful for him.
Instead, I asked him, “How would you record what you did on paper?” This is what he drew:
I wonder what to do next with Rowan, Meka, and Trent. I want to bring Rowan’s strategy to the whole class. I am hoping if we can explore Rowan’s strategy and connect it to some of the tables that other students created, it may help all the students see, explain, and use structure when solving word problems. I am so grateful for Brian Bushart because he shared a wonderful routine and penguin problem, but the heart of this lesson is not the resource. The resource is just the vehicle for students to engage in meaningful discourse. The heart of the lesson is in the language; sometimes intimate, often boisterous, that we speak as we work together to seek common understanding. Language is how we communicate. It is how we create meaning. Math is a language. Name-ing things is important, but what is more important is how we come to name them.
Recently, during a Number Talk, I wrote this problem on the board and asked students to show me a quite thumb if the problem made sense. Then, I encouraged them to try to find a solution using some of the strategies we have been working on.
26 + 10 = ________
Calvin mumbled something under his breath. Students started raising their fingers to indicate how many strategies they had used. Calvin crossed his arms and started to kick the air as he slunk down into his chair. I asked the students to whisper the answer on the count of three. I heard a chorus of “36”. I wrote the next problem on the board.
26 + 12 = ________
Calvin glared at me. He growled. He turned his back to me. Mrs. X went over to him and he whispered to her as I continued the number talk. When Mrs. X came back over to me, she whispered, “he’s mad at you because you won’t stack the numbers.” I nodded to her and continued on with the next problem.
26 + 22 = ________
Calvin glanced at me with angry eyes. Then, he looked at Mrs. X and snarled, “She’s not stacking it!” My quiet thumb dropped to my side. I took a breath, looked Calvin in the eyes, and responded, “Calvin, if you would like me to write the problem a different way, I can do that, but you need to use your words and ask me. I am happy to help you if you ask me to.” Then, I went back to thinking about 26+22. Calvin kicked the air and turned away from me. After some quiet think time, I collected solutions and asked for volunteers to defend them. Jason told us that he decomposed and added. I recorded his thinking as he spoke.
I said, “This strategy reminds me a little bit of the one Calvin likes to use. We don’t have it listed on our strategies menu, but I think maybe we should. Calvin, did you want to talk about stacking?”
Calvin turned around. His shoulders settled. He asked, “can you write the numbers so they are stacked? That is the only way I can do it.”
“Of course I can. Thank you so much for asking. I will write the numbers so they are stacked. Can you tell us how you would solve it?”
Calvin explained, “Six plus two is eight and two plus two is four. The first time I got 82, but then when Jason said 48, I figured out I was wrong.”
I rephrased, “So you added the six ones and the two ones and got eight ones. Then, you added the two tens and the two tens and got 4 tens.”
“No. It’s just four. Two plus two is 4. The answer is 48.” He beamed. “Can you write my name next to it, like you did for the other kids?”
I asked Jason if he noticed anything that was the same about his strategy and the strategy that Calvin used. Jason said, “we both decomposed the numbers. We both added the tens and the ones.” I asked Calvin if he understood what Jason meant. I motioned towards the similarities as Jason explained them.
Calvin smiled, “We both got 48.”
“Yes,” I agreed. “Calvin I am going to continue to write the problems horizontally, but I can also write them stacked, if you would like.”
“Yes,” Calvin replied.
This number talk took place about three weeks ago. I think about it a lot. I think about Calvin a lot. Usually, I get really angry when I think about Calvin. I’m not angry at Calvin. I’m angry for Calvin. Calvin wasn’t always in our district. He transferred here from somewhere else. Counting is challenging for him. His only experience with place value seem to be to “stack” numbers and use his fingers to tally up the digits in each column.
Calvin clings to stacking like a life boat. I picture him, tethered about 100 feet off shore, clinging to the lifeboat because no one taught him how to swim. I imagine, and maybe I’m wrong, that no one taught Calvin how to swim because they firmly believed that he couldn’t learn how to swim. They probably thought it was safer for Calvin to just cling to his lifeboat. I can imagine the types of conversations that happened in regards to Calvin’s potential for learning:
“Calvin is just so low.”
“Calvin just needs to be taught a procedure. He can’t think ‘like that’.”
“Poor Calvin. His life is so hard. It’s not his fault, but it makes sense that he is so far behind.”
“Calvin can’t be in class with his peers. He is just so far behind.”
“Math just isn’t Calvin’s thing.”
“Calvin should be tested.”
Who the hell are we to decide what Calvin can’t do? Maybe it isn’t Calvin’s problem. Maybe it is our problem. Maybe we need to ask ourselves, what can we do to help Calvin think deeply about mathematics?
After this number talk, I added stacking to our anchor chart of strategies. The week following this number talk, I wrote the Number Talk problems both vertically and horizontally. If I forgot, Calvin grinned and politely reminded me. Gradually, Calvin stopped asking for the numbers to be stacked. He still clings to his strategy, but he seems to be thinking about trying to count up by tens. Yesterday, we checked in on his counting skills. Look at his work and ask yourself, what CAN Calvin do?
Calvin can count forwards and backwards by tens, off the decade! He might not be able to do it all the time, but he can certainly do it. Now, we have to help him develop his place value understanding and connect it to what he knows about counting. I have been carefully observing Calvin during class lately. Here are some other things he can do:
Calvin notices patterns when we do choral counts.
Calvin always raises his hand when I ask if anyone wants to defend a solution.
Calvin takes risks. Yesterday, one of Calvin’s peers used a compensation strategy to solve 19 + 19 = ______. Joey said, “I took one from the 9 and gave it to the nine in 19 and then I had 10+10+10+8 so I got 38.” I asked if anyone else could explain what Joey meant. Calvin’s hand shot up. He had a big smile on his face. I called on him. He thought for a while, smile never disappearing. He said, “I am not really sure.” I offered, “help or time?”. He asked for help. He loves being in charge of choosing who gets to help him.
Let’s start looking at all of our students in regards to what they can do. Let’s stop finding excuses for why we can’t teach students. Calvin is far behind his peers. He didn’t have the exposure to the math practices that his peers had, but Calvin has potential. Calvin has a voice. Calvin is capable of greatness. I will admit that Calvin terrifies me because it is going to require a lot of work and reflection for me to figure out how to help him think deeply about mathematics. I have a ton of questions:
Should I give him place value blocks during the Number Talks?
Should I give him Digi-Blocks?
What if the other kids want to use them? Will it lower the cognitive demand of the Number Talks?
How do I help Calvin connect counting to place value?
I don’t know exactly how to how help Calvin. I have a lot to learn about supporting K-2 students. So, I read. I try things out. I reflect. I ask for help and feedback.
Today, I tried to help Calvin, and the rest of the class, understand Joey’s compensation strategy. I put nineteen place value blocks in each of my hands. I asked Calvin to count them to make sure I was correct. Thank goodness he did because I was one short.
Then, I asked the class if anyone could use the blocks to show us Joey’s strategy from yesterday. Ben volunteered. Ben picked up one of the cubes from my left hand and said, “Joey took one away from the nine,” he placed the cube in my right hand, “and he gave it to the nineteen.” Calvin watched carefully. He looked up at me and said, “That’s a ten! We can trade it for a stick!” I smiled and asked him if he would like get the ten stick for us. He did. When he came back, he said, “that makes three tens. So it is 38!”
I said, “your darn right it is,” and I gave him the highest of high-fives.
All of our students deserve respectful, engaging, math instruction that requires them to think deeply. How can you help make this happen?
Kindergartners simultaneously terrify and inspire me. They terrify me because they are so candid and unencumbered by humility. They won’t hesitate to look you right in the eyes, in the middle of a conversation, and say, “this is boring.” They inspire me because their sense of wonder is raw, and, also unencumbered. Five year olds wonder as naturally as they breathe. Being curious isn’t something they have to practice or strive towards. It is just what they do; breathe, sleep, eat, be curious.
Once a month, I meet with Deb Hatt, one of our building based math interventionists, and Katie Reed, a Kindergarten teacher. We co-plan and co-teach a Counting Collections routine. We are trying to learn more about how Kindergartners record their thinking and justify their reasoning. We have so many questions:
What is the difference between “how did you count?” and “how do you know your answer is correct?”
What do Kindergartners understand about the word, “prove”?
How do we honor student thinking while also nudging it forward?
What is the difference between a “nudge” and a “shove”?
During our last planning session, Mrs. Hatt and Mrs.Reed discussed a recent blog post by Heidi Fessenden. They were so appreciative of Heidi’s honest reflections about how she let go of some control in order to make space for the opportunity to learn about her students. Earlier in the year, I had shared some resources from Kassia Wedekind. We have found them incredibly helpful. Mrs.Reed encouraged us to try Kassia’s guide to conferring during our routine today. We discussed places that we might try to nudge student’s thinking.
Katie wondered, “What kind of questions should we ask that might nudge the students?” We decided we could start by just saying, “I noticed (something about how they counted). I wonder how you are going to record that?”
We knew it was still a challenge for many of the students to accurately record their count with a picture. Most of them had no problem recording the number they counted, but they were still struggling to show how they know their answer is correct. Some of them have started to record how they counted, but aren’t necessarily depicting an accurate number of objects. The last time we did this routine, I asked one of the students if she had drawn a circle for each one of the shapes she had counted. She looked at me, exasperated, and said, “No. You told me to show you how I counted, not how many I counted. This is how I counted, see?” She touched one of the shapes and then touched one of the circles she had drawn. “This (shape) is this (drawn circle).” A few students are pretty content to count the objects, write a number, and move on. Mrs. Hatt, Mrs. Reed and I have spent a lot of time thinking and talking about how we can help Kindergartners find a purpose for proving to themselves that they counted correctly.
Katie started the lesson by reminding the students about the Counting Collections routine, “We are going to do our counting collections routine today. Do you remember? We have some collections that we count and then we talk about how we counted. Today we are going to spend some time exploring how we can tell we have the right answer.”
She and Mrs. Hatt made a space for themselves on the floor, in front of the students. Katie continued, “We have some shapes that we are going to count. Mrs. Hatt is going to count them first. Then, I am going to try. We are going to see if we get the same answer because, sometimes, when we are counting, we get different answers. Has it ever happened to you, when you are counting with somebody else and somebody else gets a different answer?”
A sympathetic chorus of “yes”.
Deb shook the shapes onto the carpet and counted. She waved her finger over the pile as she quickly and haphazardly accounted for each of the shapes. “I got 21,” she told Katie, “Now, you count them.”
Katie pulled the pile of shapes into her space and began to count. She got a different answer than Mrs. Hatt did. The children were riveted. Listen.
Mrs. Reed still wasn’t sure what the answer was so she decided to check again.
At this point, Mrs. Hatt and Mrs. Reed agreed that they had 19 shapes. Deb held up the recording sheet and asked the students how she might fill it out. “It says, ‘how do you know?'” She wondered, “How do I know?”
Several students responded, “Because you counted them.”
Deb described how Katie’s arrangement really helped her see that there were 19 shapes. She drew a sketch of the group of ten shapes and the group of 9 shapes. Then, she asked the kids about the representation she drew on her paper. “Does this drawing look kind of like Mrs. Reed’s arrangement?”
“Yeah. It needs one more to make 20.”
“That’s right,” Mrs. Hatt agreed, “My drawing also needs one more to make twenty.” She asked the students if they could tell, by looking at her paper, that there were 19 shapes in her collection.
Katie introduced the visual anchor charts that we got from Heidi’s post. Deb introduced the tools; cups, ten frames, plates, and hundred charts. She reminded them,
“You are welcome to take anything that is up here. You have lots of choices.”
The kids went off to count. Katie, Deb, and I circulated and watched. After a little while, I decided to check in with Alissa. I reminded myself to take the time to notice what Alissa had done before I attempted to nudge.
I said, “I noticed that you were drawing lines on your paper, instead of drawing the shapes, is that right? Am I correct about that?”
She nodded yes. I continued, “It looks like you wrote that you have 17 shapes. I am wondering how do you know that the lines on the paper match how many shapes are in the bag?”
Alissa responded, “because I counted. I looked for the number 17 on the number chart.” She pointed to the 17 card sitting on the table. She had taken it out of the hundreds chart and used it to record the number on her recording sheet.
I decided to nudge a little bit. I asked, “What if someone came over here and said, ‘I don’t think there are seventeen shapes in that bag. Can you show me how you know there are seventeen in there?’ What if I said that? What would you show me?”
Alissa dumped the shapes out of the bag and placed them in the cup, one at a time, as she counted out loud. This time she counted 29. Then, she counted a third time, and got 19. She chuckled and said, “I counted wrong.” I wondered how we could figure out which count was the correct one. I think at this point, I transitioned from “nudging” to “shoving”. I was grasping at straws; “how can we organize?”, “can we make groups?” Alissa would humor me with a glance in my direction, then go back to counting. Listen to me fumble. Can you hear me sweating?
Did you hear what she said at the end? She said, “I need another one.” She filled up one of her ten frames. Then, she wandered across the room to the table with all the tools on it. When she got back, she picked up right where she left off. I decided maybe I should go back to noticing, instead of shoving, so I attempted to be a mirror for her. I was pretty enthusiastic. Maybe too enthusiastic?
I don’t know if I did the right thing. I wanted so badly for Alissa to convince herself she was right. I wanted her to trust her intuition. She knew it was 19. I could tell. How could I get her to show it on her paper? I knew I had to wrap it up. This conference with Alissa had lasted at least three days (or ten minutes). Ethan was waiting patiently to show me what he found out. After Alissa changed the number of items on her recording sheet, I asked her how she was going to show that she knew the answer was 19?
I asked, “What if someone came over and said ‘I had bag L and I got 17. I think it is 17.'”
She responded, “Then I would say, ‘Okay. Let’s just do another one.'” She was just going to consent to 17. Oh no! After all this work, she was just going to consent to 17! I started nudging (shoving?) again. I said, “How could you convince them? What if you said ‘no. I am pretty sure it is 19.’ and they said, ‘well show me your paper. Show me how you know it is 19.’ What would you write on your paper. How did you figure out that it was 19?”
She said, “A ten frame.”
“Okay,” I gave one last push, “so what could we put on your paper to show that? Think about that, okay Alissa. Think about how you are going to show here that you know it is 19. I am going to check in with Ethan and then come back and see what you came up with.”
I didn’t get a chance to check back in with Alissa before the closing circle. When I look at her recording sheet, I can tell she changed her thinking. She tried to erase some of her earlier lines. She still has more than 19 lines on her paper, but do those top two sets of hash marks resemble groups of five? Might that be a group of ten?
I don’t know. I have so much to learn about conferencing, but I am so grateful for my time with Alissa. She taught me a lot:
Noticing is really important. Maybe more important than nudging?
I can always go back to noticing, even after I try a nudge.
A nudge can linger.
I really enjoyed using Kassia’s recording sheet. I would like to try noticing and nudging with all my students, K-12.
Kindergartners remind me that deep learning takes time: intuition isn’t built in one day.