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.

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.

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.

What insects need to live

One of the objectives of our second grade insects curriculum is that students should learn that insects need 4 things to live.

You know them, right? Quick! Name them!

I'm sure you got it right, but in case you didn't, here are the 4 things: food, water, air, and space.

In the past few years, we've started writing learning targets for our lessons, so that both the teachers and the students know what they are supposed to learn. One of our learning targets is this: "I can list what insects need to live." If you can list those 4 things, you get it right. If you can't, you have missed your target.

(We write more interesting learning targets than that one. But it is easiest to write and assess objectives that involve remembering facts, and hardest to write and assess objectives that involve thinking and analysis. This is one danger of objectives.)

A few weeks ago, in an inquiry group for science teachers, we took a deeper look at some student work having to do with this learning target. We looked at one student's observational drawing of a milkweed bug habitat, and we looked closely at partial transcripts of science discussions from two classrooms.

We noticed, when using the Collaborative Assessment Conference to look at the drawing, that while the student had painstakingly labeled, with arrows, the insects' food, water, and air holes, when she wrote the word "space," she drew an arrow pointing to the word "space" itself. Her label was pointing at itself.

This prompted us to think about how abstract "space" is. Who decided insects need space, anyway? What does that mean? Do they need a space to live in? Do they need just enough space for their bodies so they don't get squished? We began to eye our list of four needs with some suspicion.

In the science talk transcripts, the students dug deeply into the idea of what insects need to live. That's not what the discussions were intended to be about -- the teachers had asked where insects live. But as students shared ideas about where they live, they naturally started to talk about what they need to survive. They talked about food, and that insects live in places where they can get the kind of food they need. They talked about protection -- insects live under logs because it is dark and safe and hard to find them.

Then one student said that insects need each other to survive.

We, the teachers, thought hard about that. It made us wonder: What does "live" mean? Does it mean that an individual insect lives? Or does it mean that a species survives? If it's the latter, they most certainly do need each other. 

The second graders, though, weren't thinking about reproduction. They were thinking about safety. They were pretty sure that some insects protect each other. If that was the case, didn't those insects need each other to survive? (If you're not sure about this, check out this video of fire ants making a raft so they can survive a flood in the jungle.)

[The students in one class designed an experiment to see if insects need each other to live. They took one mealworm and put it, alone, in a habitat with food, water, air, and space. It died. The mealworms die easily, so this is hardly incontrovertible evidence, but the second graders were pretty convinced.]

The more we thought about it, the sillier this learning target seemed to us. Do the curriculum writers have any idea of the diversity of insects on earth? Those insects need very different kinds of things to live in very different places. What is really interesting about insects is how they live in certain places so they can get what they need to live -- an idea the students began discussing almost immediately. This seems like a Big Idea about insects (and all living things) that could lead to all kinds of thinking and analysis, instead of just memorizing four things that insects supposedly need to live. 

This is one of the many stories that make me think we should always word our objectives as questions, not as answers. What do insects need to live? There are many answers, and we could investigate them all year.

What I Learned at Harvard

It's been a year since I last wrote a blog post -- a year in which I haven't been teaching, but have been learning instead. My year of graduate school is at an end, and so I'd like to share the biggest thing I learned at Harvard:

We only really learn the things we figure out ourselves.

This actually seems so obvious to me that I'm a little embarrassed to write it down, but from looking at the world around me, it appears it's not so obvious.

This big take-away can be phrased in other ways. You can't make anyone learn anything they're not ready to learn. People don't learn anything if you tell it to them. You can't really teach anybody anything -- you can only guide them in exploring ideas.

It seems to be a more or less controversial statement depending on how you say it. And I'm not sure all those statements really mean the same thing, or are always true. But I do know a few things.

I know that the person who does the talking, the one who explains, is the one who learns. "Learners talk and teachers listen," as my wise Professor Duckworth wrote on my journal this year, and she's right. If I hear someone explain something that makes sense to me, I kind of understand it, but I don't really own it until I explain it to someone else -- or, better yet, to several other people. Even then, it's very possible I won't remember it a few weeks later. If I really want to learn it, permanently, I have to experience it, struggle with it, and figure it out myself.

Here's another part of this that I cannot believe I never figured out in all my years of teaching, and no one ever told me (or maybe they told me, but I didn't learn it): students only learn what they DO.

In other words, as I've heard said at least one hundred times this year, "Task predicts performance." The idea is well explained here, but what it means is that we only learn to do what we practice doing. If we practice a procedure until we have memorized it, we are only learning to memorize a procedure. If we work with a team to solve a construction problem with blocks, we are learning about constructing buildings, and learning to solve problems in teams. If we sit and listen to a teacher talk, we are learning to sit and listen to a teacher talk, but nothing more. We only learn to do the things we do.

The obvious problem here is that when we think of what it means to be a student, we think of students sitting and listening to the teacher. And when we envision a teacher, we envision someone standing in front of students, talking. According to what I've learned this year, in this scenario the person doing the learning is the teacher, because the teacher is the one talking and thinking. The students are learning to sit and listen.

This was brought home to me a few weeks ago. I'm part of a group that has Harvard professors come and speak with us for one hour on Friday afternoons. It's a chance for us to hear about different professors' work, even those whose classes we didn't get to take.

Until last Friday, the visits were almost indistinguishable. The professor would announce, "I'm just going to talk for 20 or 30 minutes, and then we can have a conversation." They would open up a powerpoint presentation with 40-60 slides, talk for 50 minutes, then entertain 2 or 3 questions before leaving.

A few weeks ago, Steve Seidel, a truly great teacher, came to speak with us. He brought 15 slides, and only got through 7. He told us about some thinking he'd been doing lately, with some quotes from Frederick Douglass that had pushed his thinking. He asked what the quotes made us think. Then he told us he had to give a talk soon, and would we think about the topic of his talk, and maybe connect it to the Douglass quotes, and tell him our thoughts. He took careful notes of each person's ideas, and thanked us for helping him plan his talk.

It was the polar opposite of the other Fridays, and I was struck by how much more useful it was. Unlike the other talks, I can still remember what we talked about, and I suspect I'll remember it for some time. Most importantly, Steve came to listen, think together, and to learn from us, and the result was that we thought and learned, too.

All of this thinking about how people really learn -- by doing and talking, not by listening -- and how to teach -- by listening -- has me trapped in a new quandary. All around me, I see people "teaching" by telling people things -- Harvard professors, elementary school teachers, and myself, on a pretty regular basis. But I don't think people are learning much this way.

Teachers, policy-makers, administrators, and academics need to learn that this isn't how people learn. And my instinct is to tell them this, to say, "You know, no one's gonna learn that if you just tell it to them."

You can see my problem, of course. No one will learn this from me telling them, because that's not how people learn. Teachers won't change their practice, and school leaders won't change their priorities, because I (or some other little pipsqueak) come along and says that people don't learn this way. People have to learn it for themselves, when they're ready to learn it. They'll be ready to learn it when they experience it, or closely watch their students and observe it. And there's no way to make that happen quickly.