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Why I love lesson study

I love lesson study.  I recently finished another cycle and I was thinking back to when I first learned about it. I remember talking with Sadie about my frustration of doing workshops and having some teachers say, “That sounds great and all, but I can’t do that with my kids.” Sadie pushed me to use lesson study as a way to help teachers see what their kids CAN do.

Adapting lesson study for the urban school I work with took some work. I spoke about the structures and systems we used at NCTM with an assistant principal, and dear friend, who has been a key player in making lesson study part of the school’s culture. Over the past two years, we’ve made modifications to lesson study that helped make it a sustainable initiative for the school.

Here’s the main reason I love lesson study: It is a vehicle for studying problems of practice. Many questions teachers and administrators have about teaching and learning can be explored in lesson study. Curious about how to adapt a new curriculum for your students? Let’s do lesson study. Want to explore how to do more problem-based lessons with your students? Let’s do lesson study. Want to explore how to help struggling students? Lesson study. It’s become such a part of the culture that that teachers will say, “I want to try out this idea I heard out, can we do a lesson study on it?”

I also love it because it’s collaborative. Too often, teachers are left on their own to solve problems of teaching and learning. Sure, they are helped accountable and given “feedback” during observations, but schools rarely provide them with the tools and support to examine the problems they are most interested in studying. Lesson study creates a risk free way to experiment with new ideas.

The same goes for administrators. Having assistant principals and principals involved in lesson study has been so helpful, even if they never taught math. They offer a different perspective to the group and they learn more about teaching and learning math.

It’s similar to what I love about research so it’s not a huge surprise that it’s the work I am so drawn to it.  It’s also been the most successful PD I’ve been involved in. Given that I believe we learn by doing, it makes sense to me that we learn more when we are engaged in doing the work of teaching and learning together.

Want to read more about lesson study?

Check out the lessons study group at Mills College.

Check out the lesson study group at Teachers College: 

Read “A Lesson is Like a Soflty Flowing River: How Research Lessons improve Japanese Education

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.

Fair Share Task:

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?

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?

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.

2. Problems of the week from the Math Forum

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?

Welcome back


I’m back. It’s been a while.

I took a break this summer–from blogging, from twitter, from workshops. I did a little writing, some for my dissertation and some for academic papers. I went to Vancouver and gave a talk at a conference. I visited Puerto Rico and drank pina coladas by the pool. I spent some time in the mountains in Utah, and at the beach in Cape Cod. It’s been a great couple of months.

And now it’s back to school. I always have mixed feelings when summer ends. It’s by far my favorite time of year. For many reasons. The beach and the pina coladas being a major one. Another is that it gives me the time and space to reflect on the past school year. What went well? What didn’t? What are my goals for this year? How do I do things better? Do I still want to do what I am doing?

Here’s what I realized this summer. The thing I love most about what I do is building relationships–with students and with teachers. It was hard for me to do that well last year because I was bouncing around to different schools all the time. So this year, I’ll only be working at three schools. I’ll be at each school at least once a week supporting the math teams. I’m excited to really be part of the teams instead of popping in for a visit here and there.

I also reflected on I want to learn more about this year. I narrowed it down to two (for now):

  • Meaningful Feedback.  I had the chance to finally meet Michael Pershan and pick his brain about math mistakes over coffee. We talked about a lot about meaningful student feedback. What does it look like? How does written feedback differ from verbal feedback? What types of tasks or questions promote learning from one’s mistakes?
  • Guided Math Groups and/or Centers. Most classrooms I work with contain children with a wide range of abilities. While there are a lot of great ways to use open ended tasks that students of all abilities can access, we also need to work with students at their level in small groups. I’m trying to figure out the best ways to make that work in both elementary school and middle school.

I’m excited to get back on twitter and read what all of you have been up to. It’s good to be back.

Multiplying Fractions with Meaning

When I do workshops with parents, I often get complaints about why students don’t just memorize things.  It’s often followed by, “I learned math that way and I’m fine!”

I then ask them to solve a multiplication of fractions problems, say 4/5 x 2/3.  Inevitably, one person will tell me to draw “butterfly wings” and cross multiply. Another will tell me to find a common denominator, multiply the numerators and leave the denominators the same. Eventually, someone will say that you multiply the top and then multiply the bottom.  Don’t even ask about what happens when I ask them to explain why that procedure works.

I tell this story because most of us learned multiplication of fractions without any meaning. As a result, if we forget the memorized procedure, we don’t know how to reason about it.

I’ve been using the following type of problem to help learners begin to develop meaning for what it means to multiply a fraction by a fraction.

Ms. Placa made a tray of brownies.  She put icing on two-thirds of the pan.  She then put sprinkles on four-fifths of the brownies that had icing on them.  What fraction of the pan of brownies have sprinkles and icing on them?

Before students begin, I have them estimate whether the answer is going to be bigger or smaller than two-thirds. This leads to an interesting discussion and will help addresses a major misconception later on that students have about multiplication always making things bigger.

Then students draw pictures.  Here’s one possible sequence of drawings and student thinking:

1.  I’ll draw a pan and shade two-thirds with icing.


2.  Now I’ll cut the iced brownies into fifths and put sprinkles on four-fifths of those.


3. Hmmmm, I know I have 8 brownies with sprinkles and icing on them, but what size are they?  I can’t tell because all the brownies in the pan aren’t the same size. Oh, I have to make some more cuts to have equal sized brownies.   


Now I know that the brownies are fifteenths and eight of them have sprinkles and icing on them so eight-fifteenths of the pan are brownies with sprinkles and icing. 

Eventually we can get to writing number sentences and to imagining what would happen with larger numbers. We can then start to generalize what rule would work for multiplying any two fractions. But I think starting with a picture and context provides a nice foundation for starting to think about multiplying fractions.

What are your thoughts?  How do you usually teach students to multiply fractions?

Want to know more?  The study below gives a more detailed progression of how this type of thinking was fostered and some of the background knowledge it requires.

Mack, N. K. (2001). Building on informal knowledge through instruction in a complex content domain: Partitioning, units, and understanding multiplication of fractions. Journal for Research in Mathematics Education, 32(3), 267.

The “Math Wars” still rage on

What amazes me about working in schools is that when I walk into one I am immediately transported back to being a student.  The clothes may be different and the hairstyles may have changed, but in the over twenty-five years that has passed since I was a student in an elementary school the model for teaching math still looks very similar in a lot of classrooms.

The teacher stands at the board (maybe now it’s a smart board), models an algorithm and the students practice it at their desks.  Now I’m not saying that this happens in every school.  I’ve had the chance to visit classrooms that use very different approaches and I like to think that when I was a teacher, I pushed for an inquiry-based way of teaching as much as I was able to at the time.

But it’s perplexing to me that 25 years of research haven’t been able to change the traditional model in a lot of schools.  Imagine if the hospital I went to 25 years ago still used the same methods today.

However, after reading the recent op-ed piece in the New York Times that argues against math education reform, I see why it is so hard to enact change.   There are still many people who don’t buy into new methods of teaching math, hence the so called “math wars.”   The fact that this debate between traditional and reform methods of teaching still wages on is shocking to me because we know that our traditional ways of teaching are failing lots of students.

After reading the piece in the Times, you should check out Professor Keith Devlin’s thoughtful paragraph- by-paragraph critique of the article.   It makes a number of good points that I won’t rehash here.

The thing that bothered me most about the op-ed was that there was no connection to what we know about how students learn.  There is no attention given to what the research says about student learning.   There has been plenty of work (see the references at the end for some of it) that demonstrates that students learning math in reform-based classrooms outperform students in traditional classes and these students report stronger motivation and interest in math.  What’s more, the reform approach has been shown to be successful with students of diverse backgrounds.

It just doesn’t make sense to me to keep doing what we’ve been doing and expecting different results.

What do you think?

Want to know more? Check out some of the research below.

Boaler, J. (2002). Experiencing school mathematics: Traditional and reform approaches to teaching and their impact on student thinking. Mahwah, NJ: Lawrence Erlbaum Associates, Inc.

Knapp, M. S., Adelman, N. E., Marder, C., McCollum, H., Needels, M. C., Padillia, C., Shields, P. M., Turnbull, B. J., & Zucker, A. A. (1995). Teaching for meaning in high poverty schools. New York: Teachers’ College Press.

Silver, E. A., & Stein, M. K. (1996). The QUASAR project: The revolution of the possible in mathematical instructional reform in urban middle schools. Urban Education, 30, 476–521.

Van Haneghan, J. P., Pruet, S. A., & Bamberger, H. J. (2004). Mathematics reform in a minority community: Student outcomes. Journal of Education for Students Placed at Risk, 9(2), 189–211.

What Predicts Success? A Look at Grit.

When Angela Duckworth was a NYC math teacher she noticed something that I think will sound familiar to most teachers: it wasn’t always her most talented students that performed best in her class.

She had a hunch that there was some quality that made those students successful and that doing well in school and life was about more than learning quickly and easily. She left the classroom and set out to research this hunch. After studying what made kids and adults successful in a variety of settings, she found a quality that was a significant predictor of success. She called this grit— a passion and perseverance for long-term goals

Grit was a predictor of which cadets would be successful at West Point, which salespeople would earn the most money, which students would graduate high school, which kids would succeed at the National Spelling Bee and which new teachers in tough neighborhoods would succeed.

I think grit is particularly important in succeeding in math class. I recently worked with a group of fourth graders. When I first began working with them, many students quickly gave up when presented with novel problems that didn’t allow them to follow step-by-step procedures they previously learned. It wasn’t because they didn’t have the ability to solve the problems. It was because they expected to be able to solve math tasks quickly and easily and when they couldn’t, they gave up.

Over time, I tried to create a culture where doing mathematics was not about who completed the most problems in the shortest amount of time. I encouraged them to use their resources to solve the problems and talked to them about how mathematicians often spend a long time solving one problem. After several months, students slowly became more comfortable when presented with challenging problems and the classroom culture began to shift.

The jury is still out on how to best foster grit. What we do know is that Duckworth has found that talent doesn’t make you gritty. In fact, grit is unrelated or inversely related to measures of talent. So how do we best foster grit in our students? I’d love to hear your ideas.

Want to know more?
Find your Grit score and other resources