# Summer PD: The Fair Share Task

For those of you who think I spend the summer sipping pina coladas on the beach, you’re right—I do. Don’t be jealous. However, I also spend some of the summer going to and running professional development.

I had the chance to work with some amazing teachers this week on the fair share task I talked about last time. Here’s how it went.

Warm Up : Today’s Number: Write or draw ¾ in as many ways as you can.

I started with this for two reasons. One was because I wanted to start a conversation about what types of tasks help engage students at the start of class. I introduced High Yield Routines as one place to get ideas from.

The other reason was that it allowed me to assess what the group’s understanding of fractions was before we went on to the fair share task. I charted out the responses the group shared out:

Pretty impressive, no? We had lots to talk about in terms of the math and in terms of using this type of activity with kids.

After some mandatory discussions about the Danielson framework, we got to work on the task. We started with reading the problem together and making a prediction. The predictions gave me an idea of whether the group understood the task. I also was able to clarify some parts of the task, like the sandwiches being equal sizes. However, next time I think I’d change this to a notice/wonder protocol.

Teachers then worked on the task in groups of 2 or 3. Some struggled with how to start the task. I encouraged using pictures. I also had some linking cubes and paper and scissors for those who wanted to use them.

I walked around as everyone worked. An interesting misconception came up when trying to figure out how much each student ate:

One group said that when sharing 3 subs among 4 people, each person would get 3/12 of a sub.  I tried to recreate the picture they drew below:

After the group explained why it was 3/12, I asked them to make a new picture that showed the reverse in order to prove their answer works. If four people each ate 3/12 of a sub, how much did they eat all together?  I left them to work this out and when I came back, they changed their answer.

This misconception led to an interesting conversation about the whole. I posed the question: each student did eat three-twelfths, but 3/12 of what? A discussion about the whole had come up earlier during the warm up when someone said they we could represent three-fourths with money and use 3 quarters. Another teacher brought up the idea that when we talk about money, 3 quarters could be 3 wholes or three-fourths depending whether the whole was a quarter or a dollar. This connected nicely to our conversation here about whether one sandwich or three sandwiches were the whole.

I love the idea of using the chart as a way to prove. Here’s an example:

Gallery walk: The plan was to do a gallery walk and have everyone walk around and note whether the other groups solved the task in a similar or different way and why. Because of time and because we didn’t have a large group, we shared out the charts. An interesting conversation came up here as a group shared one of their mistakes and how they realized their strategy of how they drew the sandwiches needed to change. It was so helpful to have their thinking visible.

I shared how I often tell students they don’t have to use pencil in class. And if they do use a pencil, they should NOT use an eraser. This is always a controversial statement but the reason is that I want to see what the student is thinking at all points of the problem solving process. Plus, it encourages us to all make mistakes.

I’m still looking through the reflections that were filled out by the teachers, but I wanted to capture my reflection first. I loved the task and I think it allowed different learners to access it. Of course, I ran out of time for the rest of the activities I had planned, but I think it was worth it.

Here are some questions I’m wresting with right now:

• What’s the best way to group teachers? One participant felt uncomfortable with her math skills and being in a group of “stronger” math students didn’t help this. I had to come over several times and check in and at one point redirect the other members of the group so that her and I could chat about the math without the other group members telling her what to do. There are also teachers who do not want to work in groups and would rather work alone. I’m not sure how to address these grouping issues. Thoughts?
• How do I help teachers generalize from this task? What did we learn about engaging students in problem solving tasks? How do we transfer what we learned from this experience of doing math to teaching math?

# PD: A Math Task for Teachers

I’m planning a PD for a group of elementary math teachers that I’ve never met before. This makes things difficult because I don’t know anything about what they know or what their experiences are.

I was told that one of the sessions should be about engaging students in doing mathematics–which is what I think all PD sessions should be about:).  I’m trying to use a math task to engage teachers in doing math before we talk about how to engage students. My goal is to use the task to build some content knowledge around fractions and to have a shared learning experience that we can generalize from. Here’s the task I’m planning on using during the session:

I used the guidelines I posted last time to help me select and modify this task.  Here’s what I was thinking as I planned:

1. The task starts by asking for a prediction or estimate of the final answer:  I added this in because I wanted to get a quick read of the group and to make sure that everyone understood the task. I’m debating about using notice/wonder instead.

2. Knowing the formula is not enough to complete the task: It’s not enough to know the procedure to find equivalent fractions to complete all parts of the task.

3. I can provoke an interesting discussion based on a common misconception. The numbers in the task allow me to pose the following misconception: “The first three situations are equivalent since there is always 1 sandwich fewer than the number of people.”  I’m hoping this will lead to an interesting conversation about why this thinking doesn’t work and what you might do with students that think it does.

4. They have to prove WHY something is true. I like the idea of having them create a poster with convincing evidence that supports their solution. The discussion that comes out of comparing different posters will allow us to discuss what it means to convince or prove in math.  I’m going to encourage using diagrams and I also plan on handing out connecting cubes that they can use to work through the problem.

5. It can be used with students with minor tweaks. The task will need to be modified based on the level of students it’s used with, but I think it could be used across grades 3 to 5.

After we complete the task as a group, I’m planning on using these prompts as a reflection:

• Doing mathematics: Write a about the activity from the perspective of a learner. Think about the learning processes. What helped you as a learner? What helped you sort out the mathematics
• Teaching mathematics: Write about the activity from the perspective of a teacher. How is this activity different than other lessons on fractions? What do you like about it? What are some concerns?

The comments you all left last time were so helpful. You pushed me to think about now about how I will decide what warm up task I’ll use, what type of reflection prompts might be helpful and to keep in mind that some teachers that might be uncomfortable with the content.

I’m excited to hear your feedback on this task and any suggestions on how you might change things before I actually try it out.

# Professional Development: Doing mathematics

This may sound like a no-brainer, but I find it’s really important to engage elementary and middle school math teachers in doing mathematics during professional development. The experience of doing math in a different way than the way it was learned is critical before we talk about how to teach it in a different way than we learned.

It’s hard for me to figure out which tasks to use. I try to find activities that allow teachers to explore a particular mathematical concept in a different way than they may have learned it when they were students. For example, I recently worked with a group of teachers who knew the formula for surface area and volume of a cylinder but never had a chance to unpack why or how it works.

My goal is that through experiencing math this way, teachers will see a benefit to this way of learning–that when we have the experience of seeing why a formula works or how it works, we have a different experience, which leads to a different type of understanding.

I’ve been trying to think about how I select and modify tasks I use with teachers. It’s similar to how I select tasks for student in some ways and different in others. Here’s what I have so far:

1. The task starts by asking for a prediction or estimate of the final answer: This gives everyone an entry point, builds on what they already know, and it allows me to assess where the group is. It’s not always necessary but it helps.
2. Knowing the formula is not enough to complete the task: Because teachers have often memorized formulas or procedures, I need to make sure the task can’t be answered solely by knowing the procedure. I want a task that creates a need to unpack the formula or procedure they already know or apply it in a new way.
3. I can provoke an interesting discussion based on a common misconception. Sometimes someone in the group has the misconception. Sometimes I will bring it up. Either way, this type of discussion allows me to have a conversation about how important it is to anticipate misconception.
4. They have to prove WHY something is true. This allows me to set norms about what it means to convince or prove in math. I can create the shared understanding that we don’t just want to prove something works but that we want to explain why it works.
5. It can be used with students with minor tweaks. At the end of the day, I want teachers to walk away with something they can try out in their classes

I’ll share some of the tasks I’ve modified next time, but I’m curious what else you would add to this list. How do you choose tasks to use in professional development?

# How do we evaluate our coaching work?

The school year is wrapping up for me. Like most of you, the end of the year is a time of reflection for me. I’m thinking about the professional development and coaching work and I did this year and what revisions I want to make for next year.

In doing this, I started thinking about how I could assess myself. If I were working with students, I would give a final exam or project that allows students to demonstrate what they have learned all year.

But the last thing I want to do is add more work onto teachers’ already packed schedules. Plus I’m not sure what a final exam or project looks like for teachers. At the same time, I need their feedback in order to improve my practice for next year.

So I started thinking about what data I already collected that I can look at. Here’s what I have:

• My notes from visiting classrooms throughout the year
• My notes from my meetings with teachers individually and in teams
• Their observations from principals and other feedback principals gave me
• Emails from teachers
• Students work they’ve shared
• Exit tickets from PD
• Surveys
• Mid-year reflections

This gives me some rich data about how their thinking and teaching has changed throughout the year. But I also wanted to send out an anonymous survey just to get some direct feedback on my work with them.

I’m working on a draft of the questions I’m going to send out next week. Probably through survey monkey unless any of you have a better platform you’ve used.

Here are the questions from my initial brainstorming.

General questions:

• In what ways do you feel that you grew as a mathematics teacher this year?
• How have your beliefs on student learning changed? On teaching mathematics?
• What types of support did you receive from Nicora this year?
• In what ways did the professional development and coaching offered this year by Nicora impact your teaching and/or your students?
• What types of support would be helpful during the days Nicora is with us next year?
• What topics would you like additional professional development on?
• If someone else wanted Nicora to work with their school…what would you say in the way of recommendation? Why?

Questions specific to lesson study

• What is one word you would use to describe lesson study?
• What have you learned or what have you thought about differently as a result of our lesson study work? Try to be as specific as possible.
• What have you tried out in your classroom because of our work in lesson study (outside of the lesson we planned together)?

I’d love your feedback and any other questions you use!

# Professional Development: Why does it go wrong?

I sat in one of the worst PDs ever this week.

It didn’t build on what those of us in the room knew. It didn’t engage us in conversations or activities that were relevant. It shared products and not processes. At times, it treated us like we were no different that the children we teach. I’m sure many of you have sat in similar ones.

Most likely, the people giving the workshop had good intentions. Some of them were probably effective at teaching children math. But they were completely ineffective at teaching teachers.

I’ve come to realize that we don’t do a good job of providing the people who give PD with the right tools to facilitate teacher learning. They have to make it up as they go.

When I started doing professional development, I had no idea what I was doing. So I started reading a bit of the research out there about what makes for good PD.   This classic by Ball and Cohen was a good start.

Using what we already know about effective PD as guidance, I started making lesson plans for each PD session I did.

Over the years, I developed a list of questions that help guide my planning:

• What is the objective of the workshop?
• What should teachers know or be able to do at the end that they didn’t know before?
• What is the motivation for teachers to be interested in this topic?
• What prior knowledge and experiences do the group of teachers I am working with bring to the sessions?
• How can I build on these experiences?
• What is the best task sequence that meets the teachers where they are and helps them develop new understandings?
• What activities facilitate teacher learning?
• How do I engage teachers in productive struggle so that they construct their own understanding of the topic?
• How will I know if participants met the objective? What assessments will I use throughout?
• How will I differentiate the lesson for different learners? What interventions will I use? What enrichment will I provide?

This doesn’t look all that different than the questions I ask when I teach kids math. However, the answers are.

I’m still working on what theories to use to help me answer these questions. I’m spending some time looking through the research to help me with this.

Even if we don’t have all the answers, the effective PDs I go to are a result of someone carefully thinking through a lot of these questions.   The ineffective ones could be improved a great deal by thinking more carefully about them.

What do you think? How do you plan PD?

# Hooking non-math people on math

A strange thing has been happening in my workshops. People are getting really excited about solving problems. Now this isn’t really a surprise when I’m talking about workshops with math teachers. I would hope they would be excited to do some math.

No, I’m talking about the non-math teachers—those that teach something boring like social studies (I kid…kind of) or PE or even the guidance counselors that somehow wind up in my meetings.

It turns out that even those people that claim to hate math are tinkering away at puzzles, shooing people away until they find the answer, and then lighting up when they think they found a solution. Sometimes people who don’t have to be in my meetings are asking to sit in or for copies of the problem so they can work on them in the other room.

It’s very strange. Especially when the adults I see working hard on the problems tell me over and over how much they hate math or how they have never been good at it. But there they are trying to solve problems, writing equations, constructing arguments and critiquing the arguments of their peers.

It seems that when I present the task as a puzzle and not a math task, it draws them in. Yes, some get frustrated for a bit. But quickly it becomes a mission and they keep going back to it. They persevere. They use multiple strategies. They engage in mathematical reasoning despite the fact that they claim to hate math and are bad at it.

One of the things we wind up talking about after doing these activities is how we translate this to our classrooms. How do we create this same excitement with our kids in our classroom?

The push-back is there—that teachers can’t do “fun” problems all the time, they don’t have enough time or that their kids don’t even know their multiplication facts. But when we dig deeper, it turns out that math teachers aren’t necessarily good marketers.

We don’t always know how to hook kids. We don’t highlight the sexy parts of math. We don’t draw them in with an intriguing puzzle and then make them beg for the tools to solve the problem with. Dan has talked about this a lot, but I think it bears repeating. We need to be better at selling our subject.

In an attempt to do that, here are some of the tasks that have created the most energy in my sessions:

The puzzles from the transition to algebra program

Fawn’s Noah’s Ark problem

Writing equations for geometric patterns:

# Lesson Study

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I’ve been hearing and reading about lesson study for a long time, but I never had the experience to be part of one.  Luckily my work this year involves getting lesson study off the ground at a handful of schools.  It’s been an amazing experience. In fact, I think it is the best professional development I’ve been a part of. For those of you who don’t know what lesson study is, here’s how the Teachers College Lesson Study Research Group describes it:

“Lesson study is a professional development process that Japanese teachers engage in to systematically examine their practice, with the goal of becoming more effective. This examination centers on teachers working collaboratively on a small number of “study lessons”. Working on these study lessons involves planning, teaching, observing, and critiquing the lessons. To provide focus and direction to this work, the teachers select an overarching goal and related research question that they want to explore. This research question then serves to guide their work on all the study lessons.”

I think part of the reason I love lesson study is because it aligns with my experience doing research, but it has a more immediate impact than the research that I am involved with in my academic life.

For me, setting the research question the team wants to study is the most interesting and important part of the cycle. It allows the group to really focus on an aspect of teaching and learning that they want to improve.

One school I work with is trying to shift to a culture of problem-solving.  They want to move away from the idea that in math class the teacher shows students how to do a procedure and students blindly follow it.  Instead, they want to build problem solving skills and perseverance in students so that students can build on what they know to solve problems.  As many of you know, this is a difficult change for students (and sometimes teachers) to make.

Lesson study allows us to struggle with this challenge together.  We can look at questions like: “How do we best allow students to productively struggle?” or “Which types of tasks build problem-solving skills?”  Having multiple brains to look at these questions in the context of a specific lesson allows us to build on the collective wisdom of the group. It also allows for the experience to be shared so that it isn’t about a particular teacher, but rather about a particular lesson.

Lesson study is some of the most interesting work I’ve done as an instructional coach.I can’t wait to share more in the weeks to come. I’d also love to hear about your experiences with lesson study.

# Rules that Expire

Why can’t we just teach them the rule? At what point do we stop this “exploration” and just give them the rule? When I was in school, the teacher just taught us the rule–we didn’t need to understand it.

I’ve talked before about why teaching math as a bunch of rules to memorize is a problem–even if you teach the rules in a really creative way or with a catchy song. If students don’t have any understanding of why rules work, they begin to think of math as this mysterious thing that doesn’t make sense. They mix rules up and constantly need a teacher or answer key to tell them if they picked the right rule. They lose the ability to make sense of things on their own. I often talk with students who can reason about a problem and get the right answer, but get it wrong because they are following a rule that they incorrectly memorized.

I recently read a great article “13 rules that expire” that talks about another problem—the fact that many of the rules we teach kids in elementary school don’t work anymore when students move to middle and high school math. For example, when I taught middle school, my students were often confused about multiplication of fractions. They kept telling me that “multiplication makes numbers bigger.”

I recommend taking the time to read the article. It’s is an important reminder to be precise with language when working with young students. It’s also a good read for middle and high school teachers. It gives some insight into the struggles students face when they try to reconcile the rules they may have learned with new experiences that break those rules.

I’m interested to hear what you think of it.

# Three things I’m loving this week.

September is off to a great start.  Here are some highlights from my week:

1. This activity:

I love that this task is accessible and challenging to different level learners.  Students who didn’t know their multiplication facts used drawing and square tiles to help them figure out the dimensions. Higher level students were able to tackle part C.

After watching Annie Fetter’s webinar about Math Forum’s Problems of the Week, I am hooked. Not only is it a great source of interesting tasks, the teacher resources for engaging students in solving problems are awesome!

3. Mystery number

Kids love solving puzzles.  At the start of the week, each student in the class picked a favorite number. Every day when the students entered the class, they listed different facts about their number. The day I visited they were working on factors. A student would share the factors for his or her mystery number and the class would have to guess the mystery number. I’ve never seen kids so excited about factors.

Unrelated to work, I am loving training in this fall weather for my next half-marathon. I am also loving that is pumpkin spice latte season (even though I’m a little sad to see iced coffee season leave)  What about you? What are you loving this week?

# Setting Routines that Build Number Sense

This was the first full week of school in New York City! As a result, I’ve spent a lot of time talking with teachers about routines. Veteran teachers know the importance of setting routines and new teachers quickly learn that without routines, the best planned lessons quickly fall apart.

While routines for classroom management are super important, they aren’t what I want to talk about. I’m more interested in setting routines that help develop mathematical thinking– especially number sense.

Counting circles are a perfect example of such a routine. Once you set the routine, the counting circle can be used in many different ways and in many different grades to develop number sense. I learned all about them from the one and only Sadie. Definitely read her post to learn more. I also highly recommend reading more about them in Number Sense Routines even if you don’t teach K-3.

The other book I’ve been loving is High Yield Routines. It’s an easy read and I got a lot of great ideas from it. Here’s one I’ve been playing with.

Today’s Number  Students are given a carefully chosen number and asked to list everything they know about it. Students then generate different representations of the number–drawings, equations, examples. Through discussion about the different representations, student can be pushed to think about different ways to decompose and recompose today’s number.

What I love about this is that it can be easily adapted to different grades. Today’s Number can be a whole number, a fraction, decimal, a negative number, an irrational number and so on… In addition, students can access the task at whatever level they are at. Some may have only one representation, others may have many that they can begin to compare and contrast.

For me, this would work well as a Do Now. Maybe I would do it once a week or maybe more at the beginning or the year. It can also be used as part of your homework and the discussion can take place at the start of class. I also think it might be neat to share with parents as an activity they can do at home with their kids. However you use it, I think it’s a great routine to start at the beginning of the year and track how students’ number sense builds throughout the year.

I’d love to know what you think.

What routines do you use that help students build number sense?