Wednesday, February 14, 2018

Who Talks? "Numberless" Graphs in Grade 2

A class of second graders is having their first Hands Down Conversation (HDC). They are discussing what happens when you add 10 to a number, and what will happen if we count by 10 but start at 8. I'm sitting outside their circle, taking notes on everything they say. This is how the conversation ends.

David: It would be 0, 4,5, 8, then it would be 18, 8 at the end, 28, 8 at the end, 38, 8 at the end, 48, 8 at the end, 58, 8 at the end, and it would keep going.

Eddie: I get what you’re saying, that there would be an 8 at the end of everything, but we kind of know that already.

David: Not everybody.

Theo: Not everybody knows that. Babies don’t know that. (He giggles).

There is a low hiss of voices around the circle. Some students don't like this way of talking.

I call a timeout to reflect on how our first hands down conversation has gone.

This class is not an easy one to facilitate. There are 22 students. 14 are boys. 8 are girls. More than half have an abundance of energy which makes it hard for them to sit still. Of those 14 boys, a majority (about 8) are quick-thinking, verbal, and impulsive, which means they often call out answers or comments before everyone has had time to think. Without careful management, this means that other students are completely excluded from the conversation.

(This is our hands down conversation anchor chart. It comes straight from the work of Kassia Wedekind and Christy Thompson, who have been working together on facilitating hands down conversations in literacy and math.)

The second week

After the second HDC, I list the names of the students who spoke on the board. We count how many spoke (7 of the 22 in the class). What do you notice? I ask. No one notices gender, so I ask how many boys spoke (5) and how many girls (2). We talk about some of the reasons so few children spoke. The same people talked over and over again. Some people tried to speak but their voices were quiet and they were drowned out by louder voices.

The third week

Before our next hands-down conversation, we work on a list of ideas for hearing more voices. Their ideas are spectacular.

Gender comes up a number of times. Girls suggest that boys should listen more and wait longer before speaking more than once. Each time the girls or I mention "boys," there is a murmur across the rug. The boys don't like it. I make a split-second decision not to write "boys" on our chart. The truth is that there is one girl who also spoke often and assertively, drowning out other voices, so this is not just a boy problem.

But I do tell them that men often talk more than women. I tell them that when grown-ups have meetings, the men talk more than the women, even when there are fewer men in the group. I say, "boys and men have to practice listening more and speaking less."

The next day, we have another hands down conversation. First we re-read our chart. We talk about what to do if two people speak at the same time (one person can say "go ahead" to the other). I ask the most talkative and impulsive boy to sit outside the circle with the teacher and a class list, and keep track of who speaks and how many times. That way he has an important job and won't overpower others so much. And I ask a quiet girl who rarely speaks to launch the conversation.

They do better. More students speak. More girls speak. And more boys speak than the week before. Some boys still talk a lot. But I can see others working really hard to make space for more voices. Some speak only once or twice. One stops himself a number of times and offers the floor to other kids. One girl tries asking another girl, "Would you like to speak?" At the end of the conversation we tally up the data: how many students spoke: 11. 7 boys and 4 girls.

This week

Before our next hands down conversation, I wanted to show them the data of how their conversations had gone over the past two weeks. I wanted to present the data to them visually so they could make sense of it, see the positive trend of more participation, then hopefully increase that participation. I decided to do so with a variation of numberless graphs, a routine Kassia and Brian Bushart (among others) have been working on. My graphs aren't numberless, but they have few numbers, few words, and I shared them from simplest to most complex.

After I projected each graph, I simply asked "What do you understand?" and "What do you wonder?" They discussed the graphs for more than 30 minutes. The turn and talks were loud and animated. Best of all, nearly everyone spoke in the large group, even students who rarely participate or who, when they do speak, are often confused.

Here is the first graph:

And here are their observations:

Here is the second graph.

And their observations:

Finally, the third graph.

Their observations:

Some of my favorite observations:
  • The girls and boys combined to make everyone.
  • The blue bars show who spoke on Week 1
  • The reds are taller than the blues (because Week 1, only a few boys and girls talked)
  • Girls still talked less than boys Week 2
  • The reds are the same on slide 2 and slide 1 (because the total number of students stays the same!)
  • The green is bigger than the blue (because more people talked Week 2 than the first week)

How many kids spoke anyway?

We didn't talk much about the actual numbers of students who spoke, but we did spend some time discussing whether there are 7 or 8 girls in the class. Besides the kids who just looked around the rug and counted, there was some discussion of how you could use a graph with only a few numbers on it to figure out how many each bar represented. One student described how he imagined these numbers in between the 6 and the 12.


I know most adults who look at these graphs will be interested in talking about the dynamics of our hands-down conversations. What do these data mean? How can we change them? How did the class talk about these questions and decide to move forward?

But what I loved about today's conversation what how easily the second graders stuck to low-inference observations. Only once did someone make a suggestion to the class, and it was after the comment that "the red has to be bigger than the blue." We talked about when that might not be true: when would the red and blue (or green) be equal? Only if every single student participated in the hands-down conversation.

Looking at data this way, and starting with what we understand, opened the door for everyone to participate. It was the perfect low-floor, high-ceiling task. Jayvaughn, a student who raises his hand to participate all the time but who is often confused and hard to follow, observed that "the reds are taller than the blues." And in fact (as I told the class), this is how grown-ups make sense of graphs like these. We look for trends overall. Figuring out exactly how many girls spoke or how many boys are in the class is less important than the trends, than observations like the fact that each time the reds were taller than the blues. After Jayvaughn made that observation, I asked if someone else could explain what his observation meant in terms of real life. Tyrell explained that this was the case because "week 1, only a few boys and girls talked."

I could go on and on about each of their observations and why I like them all so much, but I'll stop myself here. What I most was reminded of was how much depth and understanding can come out of just talking about what we see. This is why we spend so much time noticing and wondering (in math and in all subjects). When we slow down and just observe, we make the most sense for ourselves.

A Note on Gender Identity (added 2/24/18)

One thing I thought about but didn't include in this post originally was the gender binary implied in this whole data analysis. This is a class where everyone identifies, so far, as either being a girl or a boy -- but this analysis I encouraged them to do did not question that gender binary in any way. I am not sure how to hone my students' ability to pay attention to the gender dynamics of who speaks in class without reinforcing the gender binary, but I'm sure there are good ways. I'd love suggestions.

Saturday, January 6, 2018

Counting Collections: One Nearly-Perfect Answer to Inclusion

I have a new job this year, and with a new job comes a lot of learning -- which is fun and hard.

The biggest challenge I face is that I am tasked with creating inclusion opportunities for several students who are in a substantially separate classroom for students with Autism Spectrum Disorder. Although I have taught students with autism before, I have never worked with students who have this level of disability. I have been making a LOT of mistakes and, most of the time, learning from those mistakes.

Sometimes I bring a few of the students from the ASD classroom upstairs to the second grade general education classroom. This is hard, though, because it is a small, crowded, relatively noisy room. I feel a little claustrophobic and distracted with the extras kiddos in there, so I can only imagine how they feel.

Other times I bring between 4 and 10 students from both classrooms (the ASD room and the general ed room) into the Learning Center, where we either do the same lesson they are doing in the classroom, or a variation on it, or a variety of centers.

But the best part of my week is the 50 minutes on Mondays when I bring four students from the second grade general education class downstairs to the ASD classroom. They pair up with four students with autism for Counting Collections.

I bring different students down most weeks, so that everyone has a chance to go, but some students go more often than others -- because I want them to build off what they did the week before, because I think they need the counting practice, or because they are really, really good at being partners with their special education peers.

(A note on language: I wish that we had names for our classrooms that were, say, animal names, instead of calling them "second grade general ed" and "the ASD classroom," but we don't. Soon I'll introduce you to a few of the students and call them by name because it is both unwieldy and somewhat distasteful to call them the "general ed students" and the "special ed students.")

A few weeks ago, I paired Ricky up with Elena to be counting partners. Ricky is a second grader with ASD. He is working on matching quantities to numerals and counting quantities under 30. He is a quiet guy, but he does communicate verbally and understands a lot of what is said to him. Elena is in the general education second grade class. I had noticed her having some hesitation when counting under 100 (pausing at a new decade, for example), and she needed practice with making groups of ten and understanding place value.

Ricky and Elena chose a collection of links. There were 46 links in the collection. The first time I checked in with them, they were lining up the links and counting by ones. I saw Elena doing most of the counting, so I asked her to stop and let Ricky count, telling her she could help him if he got stuck or made a mistake. He paused for several seconds at each new decade after 20 and sometimes needed help remembering what came next as he counted.

The first time they counted, they got 39 links. I commented on how they had organized the links, and asked them to recount so I could see how they counted. This time they got 46.

"How can you be sure it's 46 and not 39?" I asked. They both looked at me for awhile. Then Elena started to count again. She counted one link twice and got 47.

"Is there a way you could organize them so they would be easier to count accurately?" I asked.

I have a variety of tools they can use for organizing: ten frames, cups, and plates. They chose cups, and started to put two links in each cup.

When I came back, they had 2 in each cup and had put the cups into groups of 4 cups (8 links), with one group of 3 cups (6 links).

"It's 46," Elena said.

"Did Ricky help count?" I asked.

"Sometimes," she answered.

"Show me how you figured out it's 46," I said.

She counted them by ones. She got to 30, then counted one more group of 8 (by ones), miscounted, and got 39. "So 30 plus 8 is 39?" I asked.

She thought for a few seconds and said, "No! It's 38." She recounted them to be sure, then continued on to 46.

I suggested they label each group with the total they had after counting that group. Elena re-counted and I asked Ricky to make the labels. He used a hundreds chart to find the numbers to write. He could find numbers like 38 and 46 but didn't know how to write them on his own. He wrote carefully, clearly, and proudly.

By that point, time was up. Between Elena and Ricky and the other pairs working that day, we had a variety of grouping strategies. Several had grouped by tens, and one pair had made groups of four. Here are a few photos of their work.

I decided the next week to share some of this work with the whole group. One of my goals was to encourage students who had not grouped by tens to think about the advantages of doing so.

The next week, I started class by telling them I wanted to tell them a story. I had a slideshow of Ricky and Elena's process, and how they had gone from counting all in one big line to grouping in cups and labeling their groups. At the end I included a few other pictures, including the recording sheet showing the groups of ten. Many of the same students were back again (like Elena, who I partnered with Ricky again), but a few were different. Some students were interested in my "story" and the pictures. Others didn't seem to be paying much attention.

I asked which way of counting they thought would be more efficient and accurate: counting by 2s, 4s, or 10s. A few said they thought ten was quickest and therefore they were less likely to make mistakes. Seeing the recording sheet also gave them, I hoped, ideas for how to record their counting.

I showed them the collections of objects they could choose to count and the visual directions (below), reminded them of the options for recording sheets, and sent them off with the partners to start. (Thanks to Chelsea Schneider for ideas for recording sheets, and to Pierre Tranchemontagn for the photo of a simple recording strategy for my visual.) That day, a few teachers from another district were visiting, so I benefited from their extra eyes and ears -- they got to see things going on that I missed.

Elena and Ricky chose to count links again (they like them!), and they had a bigger collection. Here is how they grouped them:


And here is Elena's recording of her counting:

Ricky was less involved this time -- he kept hiding links and playing with them, while Elena patiently (or sometimes not-so-patiently) coaxed him back into counting.

After counting, I asked Elena how she would group her collection net time. She said she would try by 4s. So she's not sold on 10s yet.

Lily had come from the general education class as well that day, and she worked with Nick. Lily and Nick get along really well and love working together. Lily suggested they count their collection by twos. Nick didn't know how to count by twos. I came upon them to find her teaching him to count by twos using a hundreds chart. He circled the numbers as they counted together. I didn't watch them for long, but our visitors commented that they could see Lily deepening her understanding of skip counting as she explained it to Nick.

One more piece of work from that day (or it may have been the next week, I'm not sure). This shows the potential of Counting Collections to push students wherever they are in their understanding of number and operations.

What I've noticed and learned:

1. Choice

It is good to offer a lot of choices. I try to bring some familiar and some new objects to count each week. I have larger collections and smaller ones. There are three recording sheets to choose from (the two linked above, plus a sheet with 3 ten frames on it). Partners don't have to choose the same recording sheet. I suggest that the pairs decide together what they would like to count, or I encourage the student with autism to make the choice. They struggle more with attending and being engaged, so I want them to choose an object to count that they enjoy.

2. Collection size

One of my questions when I started Counting Collections with heterogeneous pairs was what size the collections should be. Most of the students with autism are working on counting up to 30 or just above 30. Most of the other students would do well to count larger collections, up to 100 or above 100. I asked on Twitter for advice, and the consensus was to ask each pair to count two collections, a smaller one and a larger one. In practice, I have found that some pairs don't get to two collections in one period, but if they count a collection between 40-60, it seems to be a good challenge for both members of the pair (as in the case of Elena and Ricky).

On one branch of that Twitter thread, Kristin Gray and I discussed recording, and agreed that if students counted more than one collection, they could choose just one to record on paper, and attempt to explain the larger collection verbally if it was a challenge.

In reality, I need an even more flexible definition of recording. On the first day Elena and Ricky worked together, Ricky's recording was figuring out how to write the numbers for each group as they counted. On another day, he was able to show how they counted a much smaller group on paper (pardon the water spill):

One student in the class communicates using a communication device, mostly one word at a time. For him, recording means finding the total on a hundreds chart. With a significant amount of help, he can draw a small collection on paper. Figuring out the best way to make Counting Collections work for him is a challenge I am still working on.

3. Everyone is learning.

My primary goal for this time is that both members of each pair are learning. I have worried from time to time that the students with autism might not get much out of the work, if their partners do too much for them. But I have seen partners working together well -- helping each other count, sorting and organizing together, working together to record. With some suggestions from me, they often both have a challenge that pushes their thinking.

Our visitors were most impressed at how much the general education students were learning. One assumption about heterogeneous pairs like this is that the general education students will help the special education students and not be challenged themselves. (I hear this thought often as my school discusses increasing the amount of inclusion we will do in the future.) Because of the low-floor, high-ceiling nature of the task and the many choices involved, most of the time this does not seem to be a problem. One student deepens her understanding of counting by twos, another gains fluency with counting, others build their understanding of our system of tens, and another works on multiplication. It is really exciting to watch.

4. Don't micromanage.

The first week or two, I was obsessed with making sure everyone was productive at all times. I couldn't engage in deep listening or conferring because I was trying to get Chima to stop rolling shapes across the table and dropping them on the floor, or I wanted to be sure everyone was accurately recording their counts on paper. I rushed from pair to pair and never stayed in one place for long.

I soon realized I was spending my time talking about following directions and not math. I stopped trying to make sure everyone was moving forward all the time. Instead, I confer with pairs about their counting (Kassia Omohundro Wedekind has shared some great resources on conferring during Counting Collections) and let go of needing everyone to be productive at all moments. It turns out many of the students are working and thinking whether I'm on top of them or not. And if others aren't, I'm okay with that. It's more important that I have real conversations about the math.

5. Let partners do the work

Related to the last take-away is that fact that the heterogeneous partners can do a lot to keep each other moving. Sometimes they go in unexpected directions. Sometimes one of the pair ends up doing most of the work. But often they nudge and support each other beautifully. This frees me up to do the kind of conferring I want to do.
The last time I led this group, I tried Choral Counting with them. I'm eager to gradually add that in as part of our routine and see where it takes us. Then I'll be thinking of what other low-floor, high-ceiling tasks can meet all of these learners where they are. Send me your ideas!

Sunday, December 3, 2017

8 1/2 by 12 Inch Paper: Cardinal vs Ordinal Counting in Third Grade

One of the advantages of my new job this year is that I sometimes (not always, not even most of the time, but sometimes) have the luxury of not feeling rushed and not being in charge of the whole class, and I can spend a long time watching just one or two students as they work through something.

That is what I did on Friday, when I worked with two third graders who were tasked with finding the perimeter of a piece of paper (an 8 1/2 x 11 piece of paper, for those of us in the know).

When I first came upon Sonia and Freya, they had carefully marked the inch lines from their ruler along the length of their piece of paper, and labeled the marks from 1 to 12.

It's twelve inches long, they told me confidently.

Interestingly, they had already measured the other dimension, and told me (correctly) that it was 8 1/2 inches long.

See how they did the two sides differently?

I asked them to show me how they counted each side. What ensued was fascinating. Each time they counted, they got a different measurement. Sometimes they counted the inches themselves, by putting their fingers in the middle of the inch (cardinal counting). When they did that, and got 11 inches for the length, they would say something like, "Oh! It's 11 inches long!"

I would then say, "Oh, okay. So it's 11 inches long. Show me how you can use the ruler and get 11 inches." And they would line the ruler up with the 1 at the edge of the paper, count using the numbers on the ruler (ordinal counting, but thinking of 1 as the beginning of the first inch, instead of the end of the first inch), and get 12. "No," they would say, "we were right after all. It's 12 inches long."

"Where is inch Number One?" I asked. They pointed to that little tiny tick mark at the edge of the paper.

When I asked them to show me how they got 8 1/2 inches for the width, they did this:

Look at that! When they started from the "wrong" end of the ruler, they got the right measurement, because they didn't use the numbers on the ruler at all. They counted the inches themselves (see that pencil pointing in the middle of the inch, rather than at the number?) and didn't let the ordinal numbers on the ruler confuse them.

When I asked them to turn the ruler around and re-measure this side, they got 9 1/2 inches. At some point, I grabbed color tiles and, once I had them confirm that each tile was 1 inch long, we lined them up along the edge of the paper and counted them. 8 1/2 inches. Then we added the ruler to see how it compared to the color tiles.

Again, they lined the 1 up with the edge of the paper, but they counted correctly because they were counting tiles.

This whole conversation went on for at least 45 minutes, so I don't have a detailed record of what I asked and what they said. What I can tell you is that they measured the dimensions of that sheet of paper over and over and over again. Sometimes they got 11 inches. Sometimes they got 12 inches. Each time, they were convinced they were right, until I would remind them that the last time they had reached a different total, and ask them to show me again how they knew for sure. Then they would get the other answer.

After many, many counts, they concluded that the paper was 8 1/2 inches long by 11 inches wide, and added up all the sides to find the perimeter.

Their next step was to find the perimeter of something else in the room. They chose a copy of Strega Nona which, coincidentally, was also 8 1/2 inches by 11 inches. And... once again they placed the 1 on the ruler at the end of the book and told me the book was 9 1/2 inches wide.

Again we lined up the color tiles and counted them (8 1/2), then added the ruler.

I can't remember what led to them lining the ruler up correctly this time. What I do remember is that Freya continued to insist the book was 9 1/2 inches long. (Can you see in the picture how her tiles continue past the edge of the book? She attempted to make her tiles equal what she thought the measurement should be.) I remember touching that first tile on the right of the picture and asking her, "What number tile is this?" "0," she answered. (Which would have resulted in a measurement of 7 1/2 inches if she had followed this line of thinking!)

"Zero?" I asked. "If you were counting the tiles, you would touch this tile and say zero?" (A leading question if there ever was one.) "Oh, no," she answered. "It's number one." Eventually, they talked each other into the correct measurement and seemed convinced.

At times, Sonia and Freya seemed to see the ruler as some kind of magical tool that gave them an answer but that didn't have to make sense. It was an object that was separate from their own counting of the tiles. I see this often among the students I work with: small comments like "Well, you can't ..." or "I have to..." as they name rules that either someone has taught them or they have made up for themselves. I have taken to saying, over and over again, "Math is supposed to make sense. There aren't rules that you have to follow just because. Whatever you do in math, you should do because it makes sense."

This is the same thing I say to beginning readers: "Whatever you read is supposed to make sense. If it doesn't make sense, you need to go back and re-read it until it makes sense." It's an idea that is not intuitive to beginning readers. Like these beginning measurers, they think that people magically know what the words on a page say, and that the words on the page are disconnected from each other and don't have to make sense.

I have written before about the importance of letting students take their time to do something over and over and over again before they trust it. (To me, trusting it means noticing the pattern or procedure, believing that it always works every time, and then beginning to use it automatically.) This year I am relearning that as some of my second graders repeatedly count groups of 10 by ones instead of counting them by tens. They have a healthy skepticism about numbers. Does this always work?

Sonia and Freya needed to do just that -- measure over and over again, and have someone (me) there with them to point out that their measurements were different each time, and ask how they could be sure the most recent measurement was correct. I was surprised, but also fascinated, that it was this challenging for them. It made me wonder:
  • How does their understanding of the numbers on the ruler, and the difference between cardinal and ordinal numbers, connect to their understanding of the number line, which they have been using to add and subtract numbers in the hundreds? 
  • Do they see the ruler as something completely different from a number line? 
  • What questions would I need to ask to figure out if they think about the number line in the same way they think about the ruler?
It is so tempting to tell students how to do something. "You have to line the very end of the ruler up with the end of the paper," I've said to students in the past. And then they learn that as a rule, until they come to a test question like this, or find themselves in a real life situation where they can't start at the end of the ruler. When I have the time, I remind myself not to jump in -- to watch as they say "one" at the edge of the paper -- to wait, and then to ask questions. Moments of dissonance like these will help solidify their understanding, which is wobbly now, more than if I told them a rule. Meanwhile, watching their thinking and their questioning is such a treat.

Saturday, June 17, 2017

Trusting the Pattern

There was recently a long Twittersation about multiplying by multiples of 10. It's something my students have been working on a lot. I have not told them that you "add a zero" in order to multiply by a multiple of 10 (I wouldn't dream of it!), but I would not be surprised if other people have told them that. Also, when we've worked on number strings like this:

6 x 10 = 60
6 x 100 = 600
6 x 1,000 = 6,000
6 x 10,000 = 60,000

My students themselves have said that you "add a zero" each time. We have talked about the fact that adding a zero means 60 + 0, which is not 600, and they have made representations of these kinds of problems as arrays in order to see the increasing magnitude (ten times bigger) as one of the factors is multiplied by 10.

After these conversations and explorations, I've been looking to see what my students do when multiplying 4 digit numbers by 1 digit numbers. Last week, when I took their work home to look at, I found this (as one step of a longer problem):

Many Tweeps wondered what this student, who I'll call Shayla, was thinking about, so the next day I asked her. I took notes, then promptly lost my notes, then asked her to repeat her explanation. Each time I beckoned her over she rolled her eyes, but she smiled too. She liked knowing a bunch of teachers were wondering about her thinking.

"Well," she said, "2 x 7 is 14, and 14 x 10 is 140, which is the same as 20 x 7. And 140 x 10 is 1,400, which is the same as 200 x 7. And 1,400 x 10 is 14,000, which is the same as 2,000 x 7, which is what I figuring out."

"How did you decide how many times to multiply by 10?" I asked.

"Umm, because there were three..." she trailed off, her finger waving above the zeros in 2,000. "I don't know what to call these. The two was three..."

"Sometimes we call them 'places,'" I suggested. "You know how we say this is the ones place, and this is the tens place, and this is the hundreds place. Does that sound right?"

"Yes," she said. "There were three places here before the 2. So I multiplied by 10 one time for each place."

"And why did you put three checks there?" I asked, because some of us had wondered if she put them there to keep track of the "places."

"Because Aliyah and I got different answers, so we were checking it over and over again to figure out where we made a mistake," she explained in a tone that said this is so obvious.

Here is the whole of her work on that problem:

[Here is the slideshow about refugees arriving in Europe via the Mediterranean Sea, which connects to our social studies unit on immigration.]

Here's what's interesting to me about this: once again, I thought that if students saw this pattern one or two times, they would internalize it and trust it, and not need to go through all those steps of repeatedly multiplying by 10. Some of my students don't need to go through these steps, but there are maybe 5 or 6 who are doing this each time they have to multiply a 4-digit number. Here are a few examples:

They don't trust the pattern yet. This is just like first graders who need to count groups of 10 by ones over and over again until they finally trust that a group of 10 is always a group of 10. They need to go through that process enough times, and they need to be given that time.

The other night, when I found myself solving problems in base 2 through the Exploding Dots project, I found that I didn't trust the pattern either. I needed to walk through each step for each problem. If someone had told me I had to skip the steps I wanted to go through and following a quicker procedure, I could have followed the procedure, but I wouldn't have understood it deeply.

The next day, by the way, here's what Shayla did:

She's starting to be able to skip a few steps. Trusting the pattern. But in 8 days, I won't be her teacher anymore, and I shudder to think how quickly another teacher will tell her, as she faces bigger factors, that she can just "add zeros" for each place. 

Thursday, April 27, 2017

Just Right Conjectures

We are winding up our fractions unit, having ended with adding, subtracting, and multiplying fractions. Today my student teacher Alex led class, and she asked students to think about how operations with fractions are the same or different from operations with whole numbers.

This is a pretty wide and deep question. I watched curiously to see what would emerge.

Students turned and talked with a partner. The two boys next to me sat together quietly. They looked at the board, around the room, and fidgeted. When I asked what they thought, they responded with, "What's the question?"

Around us, pairs were talking animatedly, but I couldn't hear what they were saying, although I caught words like "multiplication" and "denominator." At least they seemed to be talking about math.

"Okay," Alex said, "What did you come up with?"

Vanessa started us off. She had several false starts, and other students kept breaking in to tell her what she was trying to say. Alex quieted them, and I sat poised with the marker, ready to write whatever Vanessa said.

"When you add a fraction to a fraction," she said, "you add the numerators, and the denominator stays the same. Well, if the denominators are the same, that's what happens."

That's not a very sophisticated conjecture, was my first, uncontrolled mental response. Or it sounds like a procedural trick someone taught her. But I knew it wasn't. Vanessa was making sense of fractions for herself, stating a pattern she had noticed. I thought back to the first day we had added fractions, when she had carefully added the numerators and the denominators like this: 2/4 + 1/4 = 3/8, until I asked her to draw a model.

Vanessa continued. "Also, when you add a fraction to a mixed number, you get a mixed number that is greater than the fraction you started with."

Again, my immediate mental reaction was to be unimpressed. I kept my mouth shut and wrote.

Having finished, Vanessa turned around to see who wanted to respond. She called on Amit.

Amit shared a conjecture that some students had started talking about a few days earlier. "When you multiply a fraction by a whole number, your answer is more than the fraction and less than the whole number. But when you multiply two whole numbers, the answer is more than both the numbers."

As I wrote what he said, I asked if I could change the word "more" to "greater." I paused when I got to the word "answer."

"Is there another word we could use for answer, when we are multiplying?" I asked.

"Product," several voices chimed, so I used that instead. I paused again before writing "both the numbers."

"Do you remember that those numbers you are multiplying together are called factors?" I asked. The word factor is familiar enough to our class that they nodded, and I made the substitution.

We didn't dig into this idea more. Amit called on another student, who shared another seemingly simple conjecture: "If you add a whole number to a fraction, the answer will be a mixed number made up of the whole number and the fraction."

I asked if anyone knew what the "answer" was called when you add two numbers. "Product!" Amit said. "No," Jonelle corrected him. "Sum." I wrote sum and went back to add it to the second conjecture on our list as well.

Aliyah shared next. "If you subtract a fraction from a whole number," she began, "the answer will be a fraction."

I mentally paused as I began to write her idea down. Was this true?

I took a second to introduce the word "difference" (mentally chiding myself for never making a chart of these terms for students to refer to).

"But if you subtract a whole number from a fraction," she continued, "you can't really do that."

There was a thoughtful moment of silence, then hands started to wave and voices started to rise.

"Go ahead, Aliyah," I said, "Call on someone."

"You CAN subtract a whole number from a fraction," Joseph said. "If I have 12/8, I can subtract 1."

"Oh, I meant if you were using a fraction that was LESS than 8/8," Aliyah clarified.

"Less than 1," Sandy added.

"So you're saying you can't subtract a whole number from a fraction less than 1," I repeated back.

I wracked my brain for exceptions. On the surface it made sense. But was I going to mess up those middle school teachers if I agreed that you couldn't do it? Should I bring up negative numbers? That was NOT what we were trying to learn about today.

I kept my mouth shut and wrote.

"No!" Sean exclaimed excitedly. "If you subtract a whole number from a fraction less than 1, you get a negative number!"

"Yes! That's right!" several other kids clamored.

"What? What is a negative number?" Skye asked.

Alex jumped in and drew a quick number line on the board. She started to explain, but students excitedly took over her explanation. (They think negative numbers are SO neat!) She wrote in several whole numbers greater than and less than zero and asked what the kids noticed. "It's like a mirror!" someone said.

Alex then added fractions between the whole numbers, greater than and less than zero. Many students started talking at once with questions and observations.

We had to make a choice. I looked at Alex. We stopped them. Back to fractions.

"How should we word this, then?" I asked, and I took suggestions from the class about what to cross out and change. I drew a small number line to illustrate the conjecture. And our time was up -- in fact, math had gone twenty minutes over.

I loved that some students could generalize like this. And I wondered about others, like the two boys I sat with during the turn and talk, who didn't talk at all during this conversation, and mostly fidgeted, heads down.

I spent some time thinking about my mental reactions to these ideas, the fact that my mind kept wanting more sophistication out of these ten year olds. Looking back at the list, I could see that these weren't simple ideas. They were the ideas they were beginning to really solidify for themselves about fractions. They were Just Right Conjectures, for them.

Unsure of what should be my next step (especially considering we are at the end of our time for fractions and have so much to move on to before the end of the school year), I got some advice from Kristin Gray and Jamie Garner.

I'm excited to see what they come up with.

I'll be the teacher in the corner with her mouth shut, writing.