# What is Algebra? (continued)

Last Thursday evening, I met with the K-12 collaborative group to discuss the most recent task we had taught. Our K-12 group meets once a month. We choose a low-floor/high ceiling task, do the math together, plan how we will adapt the task for our students, teach the task, and then reflect on the lesson together. The task we were debriefing is from the Georgia Math Curriculum. The task is based on a visual pattern that looks like this:

Here are my thoughts about my lesson:

You can take a look at everyone else’s reflections here. We were able to introduce the task at every grade level, K-12, including a Special Education classroom and a Life Skills classroom.

Next, we shared our thoughts.  Our conversation reminded me of the conversation I was having with third graders in Cassie’s classroom on Wednesday. Jaime started the conversation by sharing a question that one of her third grade students asked, “Can you ever have a problem with two unknowns?” Listen to where the conversations leads:

In my last post, I described how several third grade students were wondering about Algebra.  What is it?  Do we do it or is it something that lives at the middle and high school?  Is it just a bunch of expressions with letters or boxes or smiley faces in them? Talking with my colleagues about our lessons left me wondering about how I would define Algebra.  Maybe define is the wrong word.  I don’t want to look it up. I want to own it.  I want it to be alive for me, like it was when I was talking with my colleagues.  I want it to be alive for my students.

# What is Algebra?

Last Wednesday morning, I texted Cassie and asked her if I could do a number talk in her classroom. I wanted to see if I could use number talk images to create spontaneous true/false statements out of student thinking. Cassie welcomed me into her classroom.

When I got there, Cassie  asked me about two of the questions on our benchmark assessment.  She had given it to her students and they hadn’t done well on these two questions. She asked me if we could talk about how she might revisit some of the concepts that the questions were supposed to assess. These are the questions:

I have seen this assessment many times. I helped create it. I think these particular questions are actually  Smarter Balanced released items. They are supposed to assess 3.OA.5 and  3.OA.6.

While I stood there, in Cassie’s room, waiting for her students to come back from music,  I listened to Cassie explain her concerns. She wondered:

• I haven’t presented letters as unknowns. Do third grade students need to use letters to represent unknowns?
• Are my students struggling with the language “the product of 7 and 9”?

I wondered:

• What are these questions actually assessing?
• Are these good assessment questions?
• Does the language have to be so cumbersome?
• Do these question offer an opportunity to engage in the math practices?
• Should we be assessing these standards in isolation?
• How do our benchmark assessments help and/or hinder our instruction?

She asked me if we could adapt my Number Talk to probe her student’s understanding of unknowns, products, expressions, and equivalence? I thought that was a fantastic idea.

Quickly, we brainstormed how we could adapt the lesson. First, we looked at the standards and the assessment and formed a guiding question. We asked ourselves, what do we really want to know about student thinking?  This is what we came up with.

Then, we came up with some more specific questions that we hoped would get the students talking.

• What are some math symbols that we use?
• What do the symbols mean?
• What are some words that we are using in math?
• What do the words mean?

We started class by handing out sticky notes and asking students to share their thinking about the following question:

In my quick sweep of the poster, I noticed that no one mentioned any symbols for “unknowns”. I decided I was going to ask about unknowns.  I started the conversation by listening for these answers.

I was NOT listening for this answer:

I was going to dismiss the smiley face because it wasn’t the answer I was looking for.  Boy was I wrong.

Watch what happened:

When I watch this video, I think I can see myself listening for understanding.  It happens when the talking shifts from me to them.  They start wondering, debating, conjecturing:

• “You can use anything for an unknown, except for a number.”
• “Actually, you shouldn’t do letters for an unknown.”
• “You can do letters.”

I am listening for understanding, but, in the moment, I don’t know what to do with their thinking.  I don’t even know which poster to write it on.  All I know is that I should try to follow it and write it down somewhere.  I have watched this six minute video clip at least five times.  Every time I watch it, I notice something else. Then, I wonder something else.  Here are just a few examples:

I decided to tell the students that we would come back to their ideas and questions. I wondered if the students would make connections between their questions about Algebra and our number talk.  I wasn’t sure how to answer their questions. When Cassie and I planned the number talk,  we tried to anticipate how the students would approach equivalence.  We chose to use images of eggs. You can see our plans here. We started with a dozen eggs. When I projected the image, kids just started sharing what they noticed and wondered:

“Why are there shiny ones?”

“Why is there a bigger one?”

“Why is there green and red ones?”

“Those aren’t green. They are brown.”

Then, someone noticed that it was an array. This prompted other students to bring the numbers to the table.

One student said, “Yeah. One plus one equals two. Two by six. You are just using the two since the two is broken up, you are just adding the one and the one for the two.”

When I heard him, I thought he might be trying to explain the distributive property. Was he thinking about two groups of six? I was going to use the opportunity to connect what he was saying to the expression (1 x 6) + (1 x 6), but another student joined in the conversation and it went in a slightly different direction.

When I watch the video, I wonder what would have happened if I had asked the original student to record his own thinking.  It is so challenging to record someone else’s thinking. Was I really capturing what he was thinking when I represented the decomposition of 2?

As I listened to the students share their thinking, I was trying to find someone who did not have to compute both sides of the equation. Was there anyone who was seeing that both sides were equal – using the properties – without having to do all the arithmetic?  After I listened in on a few conversations, I noticed that Cassie was having a very interesting conversation with two boys.  Listen in:

Two by six.

We showed the students another image and discussed their thinking:

Then, we showed them this:

We asked them what they noticed and wondered. Without missing a beat, many of them started talking about how there were probably 3 cartons of eggs under the grey rectangle. We asked them to use numbers, symbols, or words to convince us there were 3 cartons. They did not have a problem with this:

Most students used multiplication to justify their thinking. One student used addition.  As a whole, the students seemed to have a pretty good understanding of all the ways you could decompose 48. Most of them were more comfortable with adding partial products (distributive property) than multiplying partial products (associative property). One student tried to use the associative property but I think she lost track of how many times she had actually multiplied 12. She is thinking about “groups of groups”, but her understanding is incomplete. It would be helpful to ask her to find the expressions in her picture.  She will probably realize that she has too many groups of 2 x 12.

Finally, we revisited our original question about whether the vocabulary was hindering the students on the assessment. We asked them:

Is 2+2+2+2+2+2 equivalent to the product of 6 and 2?

Many of the students said yes and justified their thinking:

One student changed his thinking:

Several students asked some great questions:

We talked about the definition of product and how it was similar to, but different from the definition of sum. We learned that we needed to revisit what a “difference” is. One of the questions that came up in our discussion of this vocabulary was  “Can two statements be equivalent if they have different operations in them?” We explored this question, but I am not convinced that the students have convinced themselves of the answer.  How can we explore this more deeply?  Why is it important for students to understand that expressions with different symbols can be equivalent? Several students also wondered why this isn’t true:

One student tried to explain why she thought it was true by using smaller numbers and an image. She said, “see.  If I have 6 and I divide it into groups of 3, I have two groups.  This picture shows both expressions.”

This was a pretty tricky claim to navigate.  Cassie and I weren’t sure where to go with it and we would love some advice if you have any. I wish I had done a better job of closing the lesson. I wish I had revisited our guiding question and asked the students to write their thinking in a journal.  I think we still have a lot of wonders about Algebra. I think we can dig deeper here.  Grab a shovel. Help us out.

# God, Love, and Counting Birds

Do yourself a favor.  Close your eyes.  Picture walking down a street.  When you look up, you see some birds on a wire.  Pause for a minute and get a clear image of what you see. Try to describe it in words – maybe jot down some notes – but DON’T try to draw a picture. Just use words.  It would be wonderful if you could share your thoughts,  in the comments, when you are done reading the blog.

I spent a good portion of Valentine’s Day engaged in a fascinating conversation on Twitter. It was one of those conversations that had a lot of twists and turns.   The conversation started like this:

Then it went here:

and here:

Simon has blogged about a lot of fascinating math explorations.  When I mentioned that I wanted to explore geometric representations in other base systems, Lana told me that Simon had written a blog post about that.  “Of course he did”, I said.  Then I wondered, “Is there a question that Simon hasn’t written about yet?”

Here is my response to Simon:

Here is Telanna’s (Lana’s) response:

Simon told us that this question was inspired by a quote from Borges:

If, in my minds eye, I see more birds than I can subitize, can I ever truly count them in their original form? Can I capture them? Or will they always be a “clump” somewhere between 10 and 15?  When I try to count them, do I change them?  By assigning them a number, do I bring them into existence?

I decided to ask my husband what he saw.

What does our mind do with indefinite numbers? Can we ever truly communicate what we see in our mind’s eye?

Lana and I decided we wanted to explore this with children.  She shared some beautiful pictures that her students had drawn when she asked them about the birds. These were my favorites:

Lana got a lot of definitive answers from her students. I wondered if the pictures were what the students originally saw or were they manifestations of what they orginally saw. Did the question, ‘how many?’ change what they saw?

I decided to ask my own children to answer the question.  Based on what Lana found, I changed my approach.  I interviewed my kids separately and I devised  a follow up question. We don’t often see groups of small birds sitting on telephone wires in rural Maine.  When I think of birds on a wire, I think of a lone hawk or an owl.  I wondered what my kids would say if I asked them about turkey vultures. We often see large groups of turkey vultures circling in the sky.

First, I asked my 11-year-old daughter. Here is what she said:

Then, I asked my 7-year-old son. Keep in mind, this is unedited.  Therefore, you get to experience my home at breakfast time.  Yes, my son is taking this opportunity to talk about bird poop “on the record”. Yes, my husband is yelling at the dog in the background. This is my life and I love it. Here is what my son said:

He continued talking about the turkey vultures and he said something that I found fascinating. I asked him if he saw ten the first time or if he “made” ten so that he could count them more easily.  He told me he made ten so there could be an even number for his circle. I asked him if he could draw a picture for me.  He tried, but he got really frustrated:

He was about to give up because he couldn’t make the picture represent what he was seeing in his mind. I gave him some pennies. Watch what he did:

He miscounts the pennies in the circle.  I don’t think that is important right now.  He is grappling with bigger questions. He is trying to communicate bigger ideas.  He is trying to translate what he sees so that I can understand it. It is time for me to listen. I love this video.  I have watched it many times.  I will watch it many more.

Some nights, after dinner, My husband and I walk around our driveway. We live in the woods. Our driveway is long.  We look up at the stars.  Sometimes we walk together and sometimes one of us stops to look up at something and the other one keeps walking. Later, we meet back up again.  Last night, he was showing me how the big dipper and the little dipper are related. It is hard to have a conversation about stars because they are so far away.  It is challenging to decipher each other’s perspectives.  I was trying to find the little dipper.

“Is that it or is that the seven sisters?”

“That is Orion’s belt.”

“Where are the seven sisters?”

“Over there. They are also called the Pleiades.”

“They look like a little dipper.”

“They are smaller than the little dipper.”

“Like a tiny dipper? Is the little dipper the one that looks like it is pouring water on the trees over there?”

“I guess it could look like that.  It depends on how you look at it. The handle on the little dipper points to the North Star.”

“I think I see it.”  We kept walking.

I was watching Orion’s belt.  Each time I walked a loop, I would look up at Orion.  After the third time, I saw what I thought was his bow.  I asked my husband, “Am I supposed to be able to see Orion’s bow? I think I can see it in that line of curved stars.  Can that be his bow if his belt is so much smaller?” My husband just listened.  “I’m not sure what you mean,” he said, and we kept walking.

“Did you see that?!,”  He asked

“No.  What?”

“A shooting star.”

I looked up, but it was too late. I missed it.

He stopped and stared for a while.  He said, “I am going to make a wish.”

Over the last year, I have thought a lot about the overlaps between my relationships with students, my relationships with teachers, and my relationship with the people I love. I have been trying be a better listener.  Recently, I noticed that I have gotten better at listening to students.  I still need to work on being a better listener to teachers and loved ones.  I wonder if love lives in the space where we try to understand each other?  What would happen if every single  person in the world decided to simultaneously listen to every other person?  Would there be a profound silence?  What would follow the silence? Who would speak first?  What would that person say?  Would we listen?