Sunday, October 16, 2016

Differentiation in Math Class: Centers

My fourth grade class is all over the map, mathematically speaking. Some are really ready to learn fourth-grade content, while others need extra support in order to do so. Still others need time to revisit concepts from earlier grades, and a final group is ready to be pushed to fifth and sixth grade math. This is a common situation in many classrooms, and it can leave a teacher scratching her head, wondering what to do each day.

One of the ways I approach this situation is to plan some days that are based around centers. I use the term “centers” loosely. Mostly what I mean is that different students are doing different things at different times, and students usually rotate from one activity, game, or group to another. Here is a portrait of how I planned such a day last week. (Thanks to @jennalaib for inspiring me to write this blog post.)

I had several groups of students to consider in planning my math class last Friday. The majority of my plan was based on the work they had done two days earlier, when my student teacher taught a lesson on prime and composite numbers.
  • Of our 24 students, she and I determined that five of them were pretty solid on determining if a number was prime or composite.
  • Sixteen others either needed more practice (they had worked slowly on Day 1, so we weren’t sure yet how facile they were at finding factors), OR we knew they needed more support from a teacher.
  • For three of our students, determining if a number was prime or composite wasn’t a good use of their time. Of these three, one has Down’s syndrome and is learning to add and subtract numbers under 20. The other two need practice with multiplying and making arrays. The visual models of multiplication are very important for them to continue to make sense of multiplication. They can work on finding factors by making arrays, and we can talk about whether a number has many or few factors, but our focus for them is more on what multiplication means than how many factors a number has.

Based on this information, I made a center plan for our 60-minute math class. We would begin with a 10 minute number talk, then students would have slightly more than 20 minutes in each of two centers. This left a few minutes to clean up and transition after each center.

I created the following plan for each group of students:
  • I started with the 16 students who needed more work on prime and composite, since that was my biggest group and my main goal for the day. I broke them into two heterogeneous groups of 8, so that each group had only 2-3 kids who would really need more intensive help from a teacher. Each of those groups would meet with my student teacher for one 20-minute “center” to continue identifying prime and composite numbers.

    Their second center would be to meet with me for a discussion about finding factors. This was meant to be a whole-class conversation, according to our curriculum, but participation has been somewhat limited during whole-class conversations in my class so far this year. (I think this will get better as we get deeper into the year.) Plus, I wanted to be able to support different students to access my questions in different ways (questions like, “Is 3 a factor of 51? Is 6 a factor of 86?”). I thought this would be easier in small groups than in a whole class format.
  • My student who is working on counting and adding small numbers also had two centers. The first was to work one-on-one with my co-teacher (the special educator). I asked them to solve some addition and subtraction problems about making jewelry with beads and for my co-teacher to take some notes for me so I could know what strategies she was using and if these problems (with totals under 30) were the right level of difficulty for her.

    Her second center involved learning an Investigations game from the first grade curriculum called Five in a Row. This game has many versions that vary in difficulty, so I wanted her to learn the most basic version so she could play it all year, progressing from the easier version to harder ones involving adding 3 numbers or subtracting. I partnered her with a peer to learn this game on the computer. I rotate kids through working with her, usually for 20-30 minutes at a time so they don’t miss too much of their own math work. This is good pro-social time for her as well as good math practice.
  • The two boys who are working on understanding multiplication better did something similar. They also played an online game, this one from the third grade Investigations curriculum (Multiplication Compare). I had taught it to them the day before. For their second center, they worked with my co-teacher and she taught them to play How Close to 100? She took detailed notes on how they did. Her notes, indicating that the game was too hard for one of the boys, led me to create a modified version of the game for him to try this coming week.
  • Finally, the 5 students who were solid on finding prime and composite numbers had a choice of 3 multiplication / factoring games to play: a fourth grade Investigations game called MultipleTurnover; the Product Game; and Factor Find from Origo. They stayed with the game for both their centers.

So how did it all go? Mostly, it went great. Here are the highlights:
  • The room had a busy hum throughout the workshop and everyone, with the exception of one student, was working nearly the whole time.
  • My student teacher got to work with a few students who needed more practice with prime / composite and they made good progress. She took notes for me on their work in a Google doc.
  • My co-teacher took detailed notes about the three students she worked with, which helped me think about the next steps for all of them. She could focus deeply on each of them because she had them in such small groups.
  • Because the two groups of 8 were heterogeneous, both my student teacher and I were able to give students the right amount of attention. If we had had 4 or 5 students who were struggling with the content in either of those groups, it would have been overwhelming to attend to their needs. This way, we each got to check in with those who needed our attention.
  • Because these groups were heterogeneous, the level of engagement was high. My small-group conversations about factors were really rich, and I was fascinated at how quickly students took up my questions and ran with them, taking the work in whatever direction made sense to them with only a few small nudges from me. Students really pushed each other’s thinking in these conversations.

What could have been better?
  • The student who played Five in a Row had a hard time understanding the game. It would have been better if an adult had taught it to her with the peer, rather than asking the peer to introduce it on her own. After some time with an adult, the two of them could have carried on playing independently.
  •  The two boys who played Multiplication Compare played it for some time, but by the end of the 22 minutes had started playing a different (non-math) online game. (The temptations of the Internet!) Invariably during a center-based math class, students must work independently, and some are more able to do so than others.
  • The five students who got to choose a game had no time with an adult. I didn’t get to see what strategies they were using or ask them questions about their thinking. For one day, this was fine, but I have to be careful that students who have mastered the content don’t end up working independently most of the time. The next time we do centers, I’ll be sure to spend time with each of them.
  • This kind of teaching is a lot of prep. Luckily, a lot of these math games are online through our curriculum. Otherwise, I would have been pulling together tons of cards, dice, directions, etc. It’s important that my students have a repertoire of games with which they are familiar so that I can give them choices during centers and I have less prep. I have plastic boxes set up with the materials for all the non-computer games that I can quickly grab in the morning, and I have created tiny urls for the online games so they can access them easily. While some center material is new each day or two, other things are recycled and re-used, which cuts down on prep.

I don’t have three adults in the classroom every day, so on some days more kids would have to work independently. But the days when there are three of us are great days to do centers with purposefully planned small groups.

Saturday, September 17, 2016

Forging a New Identity

We've been in school for seven days. We've had seven days to start to get to know Diego, who is new to our school and to our fourth grade class. (His name and the details of his story have been changed. The traits I describe here are not true of the real student, but they give a sense of what his school experiences have been.) When his dad brought him into our class on Day 1, he said, "He's had a hard time in school." Diego turned his back to me and asked his dad to take him home.

Since then, he's been very quiet, rarely speaking during academic times. He is sometimes rude, perhaps unintentionally, when he does speak. He loves to run and jump, even in the classroom. He has broken a few of my things, sometimes on purpose. We are working hard to figure out what makes him tick, what kind of a learner he is, and what his strengths are.

Yesterday in math class, we talked about how good it is for our brains when we connect images, designs, colors, pictures, and patterns to numbers. Then we looked at pictures of arrays in real life. Students had a choice to look for arrays in the classroom, draw arrays they could think of (such as an egg carton), choose a photo of an array I had brought, or build their own arrays from tiles or wooden circles (made by Chris Danielson of Talking Math with Your Kids.) They worked on finding out the dimensions of their array and how many were in it total, then drawing a picture of it.

Here is what Diego made. His own idea, with very little input, working from right to left.

This is so awesome for our math thinking and math community. Here's why.

The image of his work is a perfect starting place for all kinds of questions for kids learning math at all different levels. I can offer many questions and prompts and let students choose what to work on.
  • What do you notice and wonder about Diego's work? (This is where I will start, with the whole class.) 

Then students can pursue any of these questions, plus those they come up with themselves:
  • How many are in each array? What kind of a multiplication equation could represent each array? 
  • Try turning the arrays on their sides to see if the products are the same. 
  • Make predictions about what would come next if you continued the pattern, then build it to check your predictions. 
  • Make predictions about how the pattern will change as it moves left, even very far to the left. (Thanks to Pierre Tranchemontagn for this idea.)

I have one student doing math at a K / 1st grade level. She can count how many squares are in each array and write the total underneath. 

Even more, though: When we highlight Diego's work, and make it the subject of our amazement, wonder, and further inquiry, we will give him an identity as a mathematical thinker. I suspect this will be new for him. And it will be an important first step as he starts his year in a new class, at a new school, with hopefully a better school experience than what he's had before. He's mostly been known in school as someone who makes trouble and needs help. How will it feel to him to be known for a novel idea that is fodder for the mathematical thinking of 23 other kids?

And to the rest of the fourth grade teachers: I know we made a plan for next week. Sorry that we'll be diverging from that plan. Something came up.

Saturday, May 21, 2016

Ten Teen

My two and a half year old daughter did several things with numbers yesterday that I had never heard her do before.

We were out on a walk, and someone walking two dogs passed us.

(That was my count, anyway.)

"Five!" Mia exclaimed.

"Where do you see five?" I asked.

"Five dogs!" she answered, pointing back over her shoulder.

Then she looked ahead to another dog that was approaching.

"Six!" she proclaimed.

Two new things here:

  1. I had never heard her use a number greater than 2 to describe the total quantity in a group of objects. She has said "two books" and "one moon," but nothing over two that would show that she understands that a bigger number can be a total quantity. (We math teachers call that cardinality.)
  2. She said "five" and then she said "six." I know this doesn't sound like a big deal. But it was the first time I had heard her count on, without starting at 1. 
Sadly, the dogs passed so quickly by that we never had a chance to see if there had been 5 dogs or 2 in that first group. (I am pretty sure I was right, though.)

Later, at dinner, she stretched her hand up in the air and started counting at 4.

"4, 5, 6, 7, 8, 9, 10, 11, 12, 16, 17, 18, 19, ten teen," she said.

"Yes!" I said. "We really should have a number called 'ten teen.'" 

"I don't think she knows about 13 and 14," my husband said.

Mia, overhearing him say "fourteen," immediately started counting at 40.

"40, 41, 42, 43, 44, 45, 46, 47, 48, 49, forty-ten!" she said happily.

"Yes," I said. "And the name for forty-ten is fifty!"

"50, 51, 52, 53, 54, 55, 56, 57, 58, 59, fifty-ten!" she continued.

"Yes," I said. "And the name for fifty-ten is sixty!"

What is so cool here is her understanding that there is a repeating pattern to our number system. She has only heard someone count above 30 about 3 times in her life, I would guess. But she has internalized something about the counting pattern.

And so we continued on to ninety-ten, at which point I told her ninety-ten was called one hundred, even though I wasn't sure if that was the right thing to say, since something big changes at 100. She's only 2 and a half, though, so I told her it was 100 without getting too complex, and we kept counting together until we got tired of the game. 

Then I dictated notes to my husband, who jotted down on the back of an envelope what had just happened while I held our wiggly ten month old with one hand and tried to finish my dinner with the other.

Wednesday, September 26, 2012


Today we were solving 8 + 3 + 2.

I asked what answers the students had found.

"12," someone said.

"15," Maggie said.

I recorded both totals on the board. "Any more answers?" I asked.

No more hands went up.

"Who will defend their total?" I asked. I pulled a name out of my cup of names. It was Jaylin.

"I know that 8+2 is 10," he explained. "And 3 more is 13."

I wrote down his solution on the board.

"Who else wants to defend their answer?" I asked. Then I saw Maggie's hand up. I called on her.

"I made a mistake," she said.

"What was your mistake?" I asked.

"I thought it was 15," she said. "But I counted wrong. It's 13."

I smiled at her.

"Wow, Maggie," I said. "You have to be really brave to say 'I made a mistake' like that."

Suddenly, Maya raised her hand.

"Yes, Maya," I said.

"I made a mistake," she said.

"What kind of mistake?" I asked.

"I thought it was 15, too," she said.

I smiled at her.

"Maya, Maggie made a mistake today. I'm not sure if you did or not, but that's okay. There will be lots of other days for making mistakes."

Tuesday, June 26, 2012

Dangers of the Pacing Guide

In the past few weeks, I have been working on a project to align our district's math curriculum with the new Common Core standards.

Since I have always taught at a pilot school, I have in many ways been sheltered from the mysterious workings of the Boston Public Schools. As part of this project, I have caught a glimpse into what teaching is like for many teachers in the district.

On the first day of the project, we were told that we would need to understand the new math standards, then think about what might help children learn them, identifying materials and resources that could help teachers introduce the material.

"You mean we don't just follow the Pacing Guide and go through the curriculum as it's written, from start to finish?!" asked one teacher.

That's exactly what we mean, the leaders replied. A collective gasp traveled the room as teachers shook their heads, wondering how they would do such a thing.

In my mind, this is what teaching is -- knowing what you want your students to think about or get better at, then figuring out how to help them do so. This, along with paying close attention to how your students make sense of the world, is the heart of the intellectual work we do.

But most of the teachers in the district aren't in the habit of doing this anymore. When it's time to teach math, they reach for a Daily Pacing Guide that tells them exactly what lesson to do on, say, November 13th. While the pacing guide has a few floating days for when your students need a little more time, there isn't much room for flexibility or you'll be (gasp!) off the pacing guide. Reading is the same but a little worse, since they use not only a Daily Pacing Guide, but also a generic, scripted curriculum.

What's the outcome of this? Teachers are forgetting how to intellectually engage with their students' thinking and their work, think deeply about what might begin to move them to the next place, and plan a lesson.

I guess this is the logical outcome of the profession becoming more scripted and "teacher-proof" in recent years. I shouldn't be surprised. What shocked me was the teachers' increasing dependence on the pacing guide. Many teachers want the pacing guide. When I suggested that it might be a disservice to professionals to ask them to blindly adhere to such a document, many looked askance. This is what their job has become. They are forgetting how to do it any other way.