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:

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Then it went here:

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and here:

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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?”

screen-shot-2017-02-18-at-7-58-01-amHere is my response to Simon:

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Here is Telanna’s (Lana’s) response:

Screen Shot 2017-02-18 at 8.12.16 AM.pngSimon told us that this question was inspired by a quote from Borges:screen-shot-2017-02-18-at-8-04-15-am

He said he wasn’t necessarily thinking about this quote metaphysically.  I couldn’t help but think about it metaphysically.

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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.

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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:

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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:

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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?

“… Like a tree extending its roots”

 

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Yesterday morning, Dan, Bill, and I were discussing this pattern as we planned the lesson we were about to collaboratively teach to Bill’s 6th grade math class.

Bill: “What I wonder is what happens if I go below zero? Can you go in both directions.”

Dan: “As a teacher, I wonder about the word wonder. If you have kids wondering at all, you are in a good place and it is a hopeful statement. There are kids who won’t wonder and the goal is to get them to that place.What about the kids who says, “why are we doing this?”

Bill:  “And I would say that since I have been using this language this year, because I didn’t use it at all last year, they are really into it because it is so accessible to all of them.”

Me: “And there is no wrong answer. Actually, if somebody said, ‘I wonder why I have to do this’ I would probably say that is a great question. I hope you can answer it by the end of the class.”

Dan: “Bill, you have me curious right away.  I want to know the body of the language you are talking about.”

Bill: “Those two questions. What do you notice? What do you wonder?  I love them because, for kids who might struggle with getting an answer, I’ve got them hooked! At least for the beginning.”

Sarah: “So the point of starting with this is that the answers are already there. I am not asking them for answers. I am asking them what they wonder about those answers.”

At this point, I showed them the next slide:

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Bill started the conversation, “To arrive at the same number for two different expressions, you need to increase by the same number….”

Dan: “……The increase and decrease must be equal.”

Bill: “Will it work with a decrease? I wonder.”

Sarah: “so, to arrive at the same number in what…”

At this point we realized that we didn’t have the vocabulary we needed to state our claim. We all remembered the answer being called the “difference”, but we wondered, “is the first number in a subtraction problem the subtrahend or the minuend?” We looked it up online.  We decided we would use these words throughout the rest of our discussion and with the students. We wondered if we might remember them in the future because they came out of the problem solving. We needed them. We weren’t just asked to look them up along with 50 others.

I restated our claim,”okay so what you are saying is to arrive at the same number for the difference, you need to increase the minuend and subtrahend by the same amount.”

We all agreed.

The next step was to find evidence to support our claim. We all tried different problems. We explored going below zero.  We felt pretty confident that our claim worked.

Then, I showed them the next slide.

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Dan described how he was “seeing” it: “Eleven is a higher elevation so base camp is seven and the farther you climb up, the greater the distance to base camp. So if it is 11,000 foot peak, it is 4,000 feet down to the 7,000 foot base camp.”

I drew what he was describing on the whiteboard.

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This is where the conversation got really interesting. I was picturing a different representation in my head so now I had to figure out how to understand Dan’s thinking.

I asked, “Where are our numbers in your representation?”

“Well,” Dan said, “It is not the same because the difference is ever increasing.”

I wondered, “So where is that in our numbers?”

“It is the difference between the minuend and the subtrahend… it is increasing all the time… No. No it is not by definition because it is 7…. because they are increasing by the same amount….”

“You originally said base camp represents 7. Where would 8 and 1 be?”

“8 would be here… and the  distance from here to here is only 1.” Dan recorded his thinking as he spoke.  Below is my rendering of what was on the whiteboard.

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I still couldn’t see it. I asked, “Where is our claim?”

Dan said, “I am not sure the picture translates to our claim.”

I asked, “Could we change it to represent our claim?  Could we use your context and make it represent our claim?”

Dan thought aloud, “Difference and distance really do mean different things. So I look at that map and it makes sense to me and I look at (our claim) and I can make it work, but difference and distance mean different things.”

He paused.

Then continued, “The range or the difference say would be three or 4 and the camp for the night is at ten or eleven. So base camp is 7. The range is, the distance traveled, is 3 and the camp for the night is at ten. Now, how do we get 7 to stay the same?”

He added, “It stays the same because it is not…. if you looked at it as what is it going to take to get back to base camp…it will be a different amount, but base camp is always the same so if you go up farther, you have to come back farther. If you go out 27, you have to come back 27, but 27 is going to be a measure of, in this case, elevation from sea level which is the distance you’ve gone plus the 7,000 feet of base camp.”

I was still struggling to see it.”So, I need your help now because what your saying makes sense to me.”

“But I am not translating it well to the claim.”

I decided to share my thinking. Maybe I could adjust my representation to accomodate for Dan’s context. I shared, “The representation that I was picturing is a slide.  When you originally started talking about base camp, I was thinking of that (slide) happening on my representation, but now I can’t see it. I pictured the slide being the same distance traveled by people starting and ending at different locations on the mountain.”

I drew this on the board:

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We all thought for a while.

“Well…base camp is not going to move.  So you can go up as far and come back as far as you want, but you are always going to have the same difference from sea level.”

I still wondered, “So what would sea level be over here.” I pointed to the original equations.

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“It is the difference between 7 and 0, but I am doing that knowing the visual – zero is not represented in that chain of problems.”

Bill added, “You gotta start at sea level. You gotta start at the ocean. I think you’ve gotta start here (sea level) and go to 8 and come down to here (seven). That is 8 minus 1. ”

“Yes,” Dan said, “If you want to go to the base camp, why climb 9,000 feet?”

Bill  agreed, “So 7 minus zero is equivalent to 9 minus 2. So why go 9 and then down 2 when you can just go to seven.”

Wow. I was really struggling to wrap my head around how Dan’s visual represented our claim.  I kept asking myself, “where do I see this in that?”, but I was struggling.

At this point, we had to move on. We were supossed to be teaching this lesson in 20 minutes and we hadn’t even gotten to the visual pattern part of the lesson – the main part of the lesson. I decided I was going to let this idea simmer and come back to it when I had some time to think.

Fortunately, we have a snow day today. I woke up early and  reread our conversation from yesterday. I thought about what Dan was trying to say. I tried to adapt my representation to show  what Dan was describing. This is what I came up with:

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I still wonder, does the context of sea level and base camp represent our claim?  Are  constant distance and constant difference related?  If yes, how? How are they similar? How are they different?

I don’t know the answers to these questions, but I am so grateful that Dan and Bill and I explored them before we went into the classroom. We didn’t get to the representation part of the lesson with the sixth graders, but they had some interesting questions of their own.

Many of them noticed the constant difference pattern and were able to articulate it as a claim.  They also found evidence to support the claim. They, like us, were prompted to think deeply and wonder about relationships.

  • Can you go below zero?
  • Does the same thing work with addition?

And my personal favorite:

  • Why doesn’t it work if you multiply the subtrahend and minuend by the same number. If multiplication is repeated addition, shouldn’t it work?

One student actually provided and supported an answer to this question. What do you think it is?  Go ahead.  Extend your roots.

 

Modeling problems

This week, I attended the High School Math team’s common planning time.  We were trying to select some common assessment tasks for this standard:

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We started by looking at the tasks from Illustrative Math.  The first task we chose was:

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As soon as this problem came on the screen,  I started whispering excitedly to my colleague Robyn.

“Robyn! It is a visual pattern!  This is great!”

Robyn smiled. Robyn and I are in a K-12 learning group. Recently, we have been discussing the value of using visual representations in math class.

Robyn and I both went to work solving the Illustrated Math problem.  I could “see” the first expression right away.

Here is a rendering of what was drawn on my paper:

I thought, “So n must be the number of dots on the bottom row which also corresponds with the step number.”

Then I started thinking of the second expression:

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I couldn’t “see” this expression in the first image, but I could see it in the subsequent steps.

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I checked in with Robyn. “Is the square in the middle of the image?”

She said, “I think so.”

I went back to thinking of the first step. I realized the first step was a little trippy because the square I was seeing in the subsequent steps was actually made out of circles.  That is why I can’t see it in the first step. It isn’t there. In the first step, n is the number 1 so 1 dot squared is still one dot. Weird. I was about to check in with Robyn again, but I missed my chance because it was time to discuss the problem as a whole group.

Somebody said, “Time’s up. What do we think?”

One teacher said, “I don’t see it. I haven’t done any dot problems.”

Another one said, “I am voting down this problem.”

And finally, “This is over the top. I would have to spend a lot of time to teach this and it would take away from what we have to do.”

I couldn’t say what I wanted to so I wrote it on my paper:

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Not enough time?  Over the top?

In my head, I was thinking “this is what we have to teach.  This is where we have to spend our time.”

As a learner, I was feeling really frustrated inside.  When I took Algebra I in High School, it was all procedures.  I never understood one bit of it because procedural recall isn’t my strength.   If I don’t understand something, I won’t remember it.  I hated Algebra, but I loved Geometry.  Geometry made sense to me. I could see it.  I remember thinking “Why can’t Algebra be more like Geometry?”  Back then, I thought Algebra and Geometry were two completely different subjects that had nothing to do with each other.  Now, I realize that Geometry made more sense to me because I could “see” it. I wonder if Geometry and Algebra are more intertwined than I ever realized. I would love to take these classes again, but with an integrated approach.

So…. I chose not to say anything.

One of the teachers said, “I think I can see it, but I don’t know how a student would explain this. How would you answer this question?”

Robyn spoke up.  She asked, “How would you explain it?”

He started  to explain where he saw n squared in the image.

Robyn kept asking questions to draw out his thinking.

She asked, “Why?”, “Can you explain where +2 is?”,  “What does the (n+2) squared represent?”

Robyn’s questions helped me understand my colleagues thinking.

Finally, someone said, “I don’t know how we would expect students to write all that.”

I said, “I didn’t write it. I drew it.”  I held up my rough sketch.

Silence.

I think Robyn said, “If you do these types of problems with kids on a regular basis, they get good at seeing the expressions and explaining their thinking.”

Silence.

“I would not use this or teach this because we have so much to do. I am not going to waste time teaching this when I barely have time to teach everything else. This is “over the top”. It is nice to have over the top, but I don’t have time for it.”

I didn’t say anything.

“Well,”  someone else said, “It is time to vote. We have other problems to look at.”

We lost.  3 to 2.

We analyzed the other Illustrative Math problems and we chose to use  The Physics Professor and Mixing Fertilizer.  These are both great tasks. Why do I still feel like we are jipping our kids because we left out the Dots tasks?  It is one task.  Leaving out one task can’t have that much of an impact on our Algebra curriculum?

Or can it?

Maybe it isn’t about the task.  Maybe it is about what the task represents.  The reason we are choosing common assessment tasks is to deepen and calibrate our understanding of the standards.  Unfortunately, I think calibrate has become somewhat of a loaded word in our district. When I say “calibrate our instruction”, I think some teachers hear “stifle, homogenize, anesthetize” our instruction.  I think, sometimes, my definition of calibrate gets lost in translation.

To me, calibration is an ongoing process. You can read more about my definition of calibration here, but I think the dual purpose of calibration is professional growth and equitable math experiences for students. At the heart of calibration is transparent and collaborative reflection.   Calibration means continuously and collaboratively asking, what do we want our students to learn?  Why do we want them to learn it?  How do we want them to learn it?

I guess the reason I can’t seem to “let go” of the Dots task is because it represents a crucial answer to the questions I just asked.  It represents the integration of visually representing Algebra.  I feel pretty strongly that all students should be able to “see” Algebra.  Earlier, I refered to myself as a “visual learner”.  I often wonder about the value of this term. Should there be a certain kind of learner who sees things visually or should visual learning be an expectation for all students?  Is there value in being able to move from the abstract to the visual and back to the abstract?  As a learner,  I realize now that it is really important for me to able to move fluidly from the visual image  to the expression and back to the visual image. Isn’t this the essence of modeling with mathematics?

Dan Meyer describes modeling  as “the process of turning the world into math and then turning math back into the world.”

I think I can see how the other tasks we chose would offer opportunities for modeling with mathematics, but I still want the Dots task.  I can’t seem to let go of it. Maybe it is personal.

 

 

 

 

 

 

 

 

 

Mistrust

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It wasn’t long after I asked students to share their thinking about the problem above that one of the students started commenting on her classmate’s solutions.

“That is wrong.  You are wrong.”

I was recording possible solutions on the board while she was holding court behind me.

I wrote :

10 1/2

11 1/2

10 1/3

“Those are wrong.  The right answer is 10 1/2. Those other answers don’t even make sense.”

This voice had been the most prominent voice since I walked into the 6th grade classroom. It was loud and disruptive.  It overpowered the other voices and it demanded attention. It was accompanied by evasive eyes, sneering, and whispers to a friend.  I had been trying not to pay attention to it.  I was waiting for her to contribute something authentic so I could recognize her as a contributing member of our community, but it wasn’t happening.

I asked the class, “does anyone want to defend any of these answers?”

One girl came up and drew a number line  on the board. She said she thought the answer was 10 and one half. She circled each of the ten groups of two-thirds. She circled the one-third cup at the end of the number line and told us she thought it represented one half of a meal.

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One boy, Max,  raised his hand.  He said, “I don’t think it matters if you say 10 1/2 or 10 1/3. You can call it one half or you can call it one-third. Maybe they are the same thing?”

“What?!” She interrupted, “That doesn’t even make sense. That is wrong.” Her voice was getting louder and less respectful.

At this point, I had to say something.  This one voice was starting to supersede all of the other voices in the classroom.  It was eroding the fabric of trust faster than I could establish it.

I raised my voice. I said, “We are sharing our math thinking.  You are being disrespectful.  I need you to listen to your classmate’s ideas without putting them down.”

Silence.

I made eye contact.

More silence.

She looked away.

I felt bad.  I didn’t want to call negative attention to her. I think she felt embarrassed because I called her out in front of her peers. Did I lose her? I wanted to build her trust, but  the time and space that I was offering to her was being used to damage the relationships I was trying to build with the other students. I don’t think it was intentional.  I think it was coming from a place of mistrust.  What did she mistrust?

Me.

The math problem I proposed.

The model her classmate was drawing.

This girl does not have an easy life.  I don’t need to go into the details. We all know this girl.  She has every reason not to trust me, her classmates, or the math we are doing.

What does she trust?

I think, in this particular instance, the one thing she might trust is the algorithm. That is all that was on her paper. In the beginning of class, I asked the students to draw a picture of the situation described above. She didn’t. Maybe she couldn’t.

Before I arrived in this math class, these students were taught the algorithm for dividing fractions. Their math teacher asked me to help him teach these students how to model fractional division. He said they all know the algorithm, but they are struggling with the modeling part.  My goal was to teach them how to model a fractional division situation  so they could explain and interpret the units they were working with.

As I sit here, I wonder if my goal should have been to get them to trust themselves as mathematicians.

At this point, we had three different answers on the board.  It was up to the students to decide which answer made the most sense.

I asked Max if he would come up to the board and show us his thinking.  He did. He drew seven circles. He partitioned each circle into three parts.  He colored in groups of two-thirds.  He counted 10 groups of two thirds and then said,

“See.  There  are ten.  And, then, there is one-third of a cup left so I think the answer might be 10 and one-third, not 10 and one half.”

“I got 10 and one half,”  she said. “I don’t get it. This math class is making me feel retarded.” Her voice was softer than it was before.  She was talking to her teacher now- my colleague.

I felt uncomfortable again. I don’t like it when people use the “r” word. I don’t think she meant to be disrespectful when she said it. I think she meant to be self deprecating. I don’t like self deprecation either.  I was trying to think of what I should say to her – should I call her out on using the “r” word?  Would that make things worse? Before I could say anything, my colleague spoke:

“Don’t use that word.  Can you think of another word?”

“Fine. This math class is making me feel stupid.”

Ouch.  That is just about the worst thing I could hear in a math class.  She sat down next to her teacher.

I wondered, what should I do?

I looked right at her. “Did I hear you say that you are not sure why your classmate is getting 10 and one third for an answer because you got 10 and one half?”

“Yes.”

“THAT is a really interesting question. Let’s talk more about that. Can anyone answer that question?”

A different boy volunteered to come up to the board and share his thinking. His writing is in the dark blue at the bottom of the picture. He drew rectangles.

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As he presented, we discussed our thinking.  I asked him, “What does the one half represent in the story and the picture?”

“One half of a meal.”

“Some people are saying one half is the same as one-third because 1/3 is half of 2/3. What do you think about that?”

Other students chimed in, “Yes. But the question asked about meals and you are going to have one half of a meal, not one-third of a meal.”

“Oh.” I said, “So one-third is one half of two-thirds but the answer to this question is 10 and one half because that is how many meals we will be able to feed the dog.”

Some nods of agreement.

The girl was quiet for the rest of class.  When I left class that day, I worried that I should have done something different.  I decided that, no matter what, when I went back to class the next day, I was going to try to help her believe in herself as a mathematician.

The next day, we were using virtual Pattern Blocks to model how many sixths were in one-third.  I introduced the Illuminations website to the students and encouraged them to explore. Then, I asked them to use the Pattern Blocks to show how many sixths were in one third.

I immediately went over to my friend.  This is what was on her screen:

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My first instinct was to say, “You can’t use the squares.” or “Don’t you want to use the triangles?”  My second instinct – my growing intuition – reminded me to listen.

I asked, “What are you thinking?”

She said, “There are six of them.”

“Do you think you could use the squares to show one sixth.”

She repeated, “There are six.”

I was thinking, “don’t shut down. Please don’t shut down.”  What can I say to get her to trust herself?  I wondered, “So… what is one sixth?”

She pointed to one of the squares.

“Okay. So there are 6 squares and you are saying one of them would be one sixth. What was the other part of the question that I asked?”

She looked at the board.  “How many sixths are in one-third?”

“So where is one-third in your picture?”

“I don’t know.” She gestured towards three of the blocks. “Forget it. I don’t know how to do it.”

“Yes you do.  Can I try something?  Do you mind if I move your blocks around a little bit?”

“Sure.”screen-shot-2017-01-10-at-4-22-24-pm

“Show me one sixth again.”

She pointed to one of the squares.

I asked, “How do you know that is one sixth?”

She said, “Because it is one out of 6 parts.”

“What if we wanted to show one out of three equal parts of the same rectangle.”

She thought about it for a while.  Then, she slowly traced the line that marked off one of the thirds of the rectangle.

I wanted to jump up and down and make a really big deal out of her success, but something inside of me told me not too.

Then, I asked her, “Can you see how many sixths are in one-third?”

“One?  No. Two!  There are two.”

“Are you sure?”

“Yes.”

“I am going to go check in with some other kids. Why don’t you share your thinking with Jamie.  How do you feel about sharing what you found with the class?”

“No.”

“I understand. I only ask because you taught me something. I hadn’t thought to use the squares to answer this question. I thought you could only use the triangles and the rhombus.  I am wondering if other people thought the same thing. They might learn something from your strategy.”

“I don’t think I want to share.”

“Okay. If you decide to change your mind, let me know.  You can bring a friend if you want.”

The next pair of students that I checked in with had this on their screen:

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Wow. I was starting to wonder if anyone was going to use the ol’ triangle and rhombus combination.

It was time to share our thinking. I decided to check in one more time with my new math friend.

“Do you want to share your thinking?”

“Okay. But can I bring Jamie?”

“Absolutely.”

She and Jamie projected her computer screen up on the white board.  She started to explain her thinking. Two students kept whispering. She tried to talk over them.  Another two students started a side conversation.  She looked down at the ground and said to her shoes, “no one is listening to me.”

“Hey!” I raised my voice again.  “She didn’t want to come up here. I asked her to share her thinking and she originally said, ‘no’.  Then, she changed her mind. She is taking a huge risk right now and you are being disrespectful.  Stop talking and listen.”

Silence.

“Well. I thought I needed six squares so that I could show one sixth.”

“And how many sixths are in one-third?”

“One. No.. two.”

“One or two?”

“Two. It is two. See them?”  She pointed. “One. Two. Two sixths.”

“Does that make sense to you?”

“Yes.”  Was that a smile I saw?  I can’t remember for sure, but what I do remember is that she went to her seat in the back of the room, picked up her belongings, and moved to an open seat in the second row.

She moved closer to me.

 

 

 

 

 

Upside Down and Super Fast

Recently, my colleague, Dan, asked me to help him help his students understand how to model division of fraction situations.

When I walked into Dan’s classroom, he was asking his students a question about decimals.  He wrote this number on the board:

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Dan asked his students to divide this number by 100.  He said this was a short routine they had been doing during transition time for the last couple of days. Dan said they enjoyed being able to quickly do a problem that looked really hard at first glance.

I asked, “Is the answer going to be bigger or smaller?”

Many of the students said “smaller because you are dividing.”

I asked, “can you think of a time when you would use a number like this?”

One student said, “When you are counting sand.”

Me: “Why?”

“Because grains of sand are sooooooo small.”

Another student said, “I think I know what the answer would be, but I don’t know how to say it.  How would you say that number?”

Me, thinking out loud: “Hmmm. Let’s see, tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, ten millionths??  I am not really sure. Maybe fifteen thousand eight hundred ninety-two ten millionths??? I need to ask my math friends for some help with this.”

I told the kids about the Math Twitter Blogosphere while I typed my question on twitter. I am so glad I asked for help. I hadn’t even considered scientific notation until I started reading some of the twitter feedback.  I wondered “if scientific notation is the aspirin, is 0.0015892 the headache?” I haven’t had time to dive into the twitter feed. I am looking forward to revisiting it. I would also like to revisit the comment above about sand.  I want to ask, “tell me more about how 0.0015892 could represent sand.”

At first, I thought, this student was wrong.  He is thinking that really small numbers represent really small objects.  0.0015892 can’t represent sand. Then I realized maybe I don’t know enough about what, exactly, this student was thinking. Can 0.0015892 represent sand?  What if .0015892 represented the size of a grain of sand in relation to the rock that it was once a part of?  Could a grain of sand be .0015892/10,000,000 of a rock? How do I get the students to consider the  relationship of a unit this size?

My colleague Dan and I are pretty similar.  We tend to live “outside the box.”  In fact, at times, we humbly admit that we need help getting back in the box.  The boundaries of the box are often elusive to us.  We don’t even know how we wandered outside of them.  We were probably following something shiny. For example, when I arrived to teach in Dan’s classroom yesterday, I asked him to video tape the lesson for me.  This morning, I anxiously sat down to watch myself teach and reflect on the lesson.  This is what I found:

I love this video.  At first, I didn’t love it. I thought, “Dan!  What the heck did you do?” Then, I laughed.  Next, I tried to edit the video. Can I slow it down to “normal pace”? Then, I laughed some more.  There is no sound.  I can’t read lips. What is the point of slowing it down? Finally, I settled into leaving it the way it was.  I wondered, what can I learn from this super fast, upside down video?

At first, I thought, “bummer.  All I see is me in front of the class the whole time.  Was I being the ‘sage on the stage’?”

Then, I looked a little closer and noticed something interesting.  There are a lot of students bobbing their heads and turning around.  There are a lot of students moving their mouths. This part of the lesson is when I was doing a Number Talk from the book Making Number Talks Matter. I am trying to record student thinking on the board.

You can’t see what I am recording because my Google Slide presentation for the next part of the lesson is taking up most of the space on the white board.  Here is the perfect example of me not even being slightly aware of the boundaries of the box.  As I watch myself now, I am thinking (kind of yelling), “TURN OFF YOUR SLIDESHOW!”  “Hello??? You are monopolizing most of your board space with a presentation that you aren’t even using right now!”

See. I can learn stuff from watching a super fast upside down lesson.

I remember this part of the lesson.  In fact, If I blow up a screenshot from the video and rotate it I can see what I did.

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See. Doesn’t that help you?

What do you mean, “no”.  It helps me.

Down on the bottom, I see the number 99 and I am remembering the great conversation we had about whether 50/99 was more than one half or less than one half.   Some students said less than one half.  Some said more than one half.  I highly suggest you take a minute to figure it out for yourself before you move on.  I asked the students if anyone wanted to defend either of the solutions.  One boy said he wanted to defend less than one half.

He said “half of 100 is 50 and 99 is less than 100 so 50/99 would be less than one half.”

I remember thinking really carefully about how I should record what he said.  I chose to use words.  Immediately, one of the other students said, “but half of 99 is 48.5”. At this point, I chose to just stand there and wait.  Then, I said, “can you tell us more?”

“Well, half of 99 is 48.5 and 50 is more than 48.5 so 50/99 has to be more than 1/2”

Then, the first boy chimed back in.

“I was wrong. It is more than one half, but it is 49.5, not 48.5.”

Me: “What do you mean?”

“half of 99 is 49.5, not 48.5.”

other boy:  “oh, yeah.”

Me: “So, why did you change your mind?”

“because 49.5  would be half and 50 is more than 49.5.”

At this point, I am noticing that no one has actually referenced the units. They aren’t talking about the fraction in it’s entirety. The students are referring to the relationship of 1/2 to the numerator or to the denominator, but no one has actually said, “49.5 ninety ninths or 50 one hundredths”  You can’t see this because the video is upside down and super fast. I remember it because I was trying really hard to listen “to” and not “for” answers. I was also trying really hard to cultivate my students’ math intuition.

I remember thinking, “how do I get them to think about the unit without telling them to think about the unit.”

Finally, I said, “how many hundredths would be equal to one half?”

Silence.

“one?”

“ten?”

“two?”

“fifty?”

Wow. This is most definitely NOT what I anticipated for answers.  I thought I would hear a loud chorus of “fifty.”

So. I recorded those answers on the board, somewhere up in the top corner, smooshed in, next to the magnetic marble tracks. (Get back in the box Sarah!!).

I asked, “which of these answers is correct?”

Several students said “fifty.”

Me:  “Why?”

“because fifty is half of one hundred.”

“fifty cents if half of one dollar.”

A lot of “oh yeahs….”

“Does anyone still think it is something other than 50?”

Silence and heads nodding “no”.

I asked the students, “Does anyone want to share why they changed their minds?  Some of you originally thought the answer would be 1/100 or 2/100 or 10/100.  Can you tell us why you changed your mind?”

Silence.

Okay. What if I try this, “Does anyone want to share why you think someone might have gotten one of these other answers the first time (1/100, 2/100, 10/100)?”

One boy said, “I could see why someone might say 2/100 because 2 x 50 is 100 and 50 is half of 100.”

Several students nodded in agreement.

I silently did a giant happy dance.  I think I am doing it!! I think I am helping my students develop their math intuition.  Wow. I need to ask those kinds of questions a lot more.

At this point, we had to move on.  This was a nice segue to the division of fractions exploration that I had planned for today. (I could write a whole other blog post about the second part of the lesson.)

We definitely need to offer these students more opportunities to explore and discuss the relative magnitude of numbers. We also need to cultivate more opportunities for them to explore and communicate the units they are working with.  At first glance, this post may seem like a series of upside down and super fast examples that are unrelated, but I actually think they are the beginning of a journey towards deeper understanding of when and why the unit matters.

 

Resolution: A firm decision to do something.

A few weeks ago, I read Tracy Zager’s post about intuition. I also took the opportunity to go through her slides from her #CMC North presentation and push myself to think more deeply.  In an effort to understand the role of intuition in math thinking, I recorded my entire thinking process – all of my mistakes, questions, and partial understandings in this post. I kind of feel like I am about to expose myself.  I am making myself publicly vulnerable. This is not easy for me, but I am doing it in the interest of becoming a better math teacher and coach. Tracy has got me thinking that my job is to help students and teachers refine their math intuition. Therefore, I have to explore my own.  Here are my candid thoughts.  The picture on the left is a screenshot from Tracy’s slideshow (which can be found on her blog. Screenshots of my actual thoughts are on the right.

 

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At this point in the exercise, things got really interesting.  I started to wonder:screen-shot-2016-12-07-at-1-44-18-pmI followed my intuition both times.  When I listened to it, it helped me. Why did I get a closer estimate on the smaller pumpkin?

Was it because I have more experience carrying objects that are closer to 5 pounds?  Does it have to do with my question about weight distribution?  Is it because heavy objects all feel so “heavy” – there is less clarification for me about “how heavy”?

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Reuben Hersh says intuition is a “trace in our mind left by our experiences”.  Was I less intuitive about my larger pumpkin guess because I have less experience with carrying things that are over 5 pounds?  Would a professional delivery person have had better intuition?

I think it is about those little voices, Tracy.  Most of my childhood schooling was spent listening to the voices of my teachers. I didn’t spend any time developing my intuition. I think I started listening to those little math voices when I first started teaching. I was in a one room school house ten miles off the coast of Maine. I was the new teacher and my students were in Kindergarten, third grade, and sixth grade. I had no idea what I was doing. I will forever be indebted to Thom Buescher, a G/T consultant from Camden who was hired  to help me figure out how to teach such a broad spectrum of learners. He suggested that the school board purchase the K-5 math program, Investigations in Number, Data, and Space and the 6-8 math program, Connected Math. So, they did. And I started opening up the communication lines with those little math voices who I hadn’t heard from in a very long time. I took those resources home and I did math, math and more math. I loved it.  I learned so much math the 6 years that I taught on Monhegan, but I only scratched the surface. I left Monhegan in 2007. Since then, the amount of math voices in my head has grown exponentially. 

However, I am finding, there are still a lot of communication lines with weak connections. Back to Tracy’s slideshow:

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Now, I want to take a minute to remind you that you are looking at naked thinking. I am sharing screenshots of my  thought process. You are seeing all the nuances of how  I approached the problems in this slideshow. You are “seeing” me revise my thinking in my head. These notes have not been edited. While I took notes, I was intentionally trying to follow my own intuition and write down everything I thought, in an effort to see if I could find the communication “weak spots” in my own intuition. Watch what happens.

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As I re-read these notes, I am fascinated.  I have always been a “rusher”.  I think too fast and it is not a surprise to me that I “saw” 4×4.  I wonder how thinking fast impacts my intuition. As a student, I wasn’t given the time to hone my intuition.  As an adult, I am realizing how important it is.  The good news is – somewhere in my head- there is a “slow down voice”.  Reread my notes above.  Can you hear it?  I think I would like to turn up the volume of that voice.

So, I continued to think about the other estimates on the slide.

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So my “gut” knows why 6,700 isn’t reasonable, but I am really struggling to find the precise words to explain it.  Fascinating. When my intuition is strong, I struggle to explain it.

Try.

If I had 6,700 groups of 47, clearly I would have more than 6,739 because 6,700 x 47 is equivalent to 47 groups of 6,700 and 6,739 is barely over 1 group.  

Ha!! Did you see that? I think I just refined my intuition by having to articulate it. What did I refine my intuition of? Perhaps the relationship between multiplication and division? Maybe the magnitude of numbers?

150 is unreasonable because 150 x 100 is 15,000. (150 x 10 is 1,500)

Wait a minute. I don’t want to just move up a place value – don’t rely on old tricks – refine your intuition, Sarah!

(15 x 10) x (10 x 10) = 15 x (10 x 10) x 10

15 x (100 x 10) = 15 x 1,000

15 x 1,000 = 15,000 

Yes!

So if 150 x 100 = 15,000 than 150 is unreasonable because 150 x 50 = 7,500 because half of 100 is 50 so I need to cut 15,000 in half. 

So it has to be less than 150 and greater than  120. Aha! So the most reasonable estimate on the slide is 130.

Ha! I just got to the part of the slides that talks about externalizing my thinking and I have been doing it the whole time. 

Intuition is developed. Wow. 

Tracy goes on to to say “My intuitions are based on my knowledge and my experience. The more I have, the more robust my intuitions are likely to be.” I am thinking it is really important to connect this with the earlier quote from Erich Wittmann: 

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So, as a learner, the key words are My. Me. I. If I want to hone my intuition, then the knowledge and experiences have to be mine, all mine.  What does that mean?  I think it means I have to make the mistakes, acknowledge the mistakes, articulate how I would correct the mistakes, and reflect on what I learned from the mistakes.

So is it just about mistakes?

No.  I think it is also about connections.  If I want to hone my intuition, I have to connect my experiences.  I have to wonder how everything fits together. You can ask me questions to prompt me to wonder, but it has to be me who does the wondering and it has to be me who makes the discovery.

In their book, Making Number Talks Matter, Ruth Parker and Cathy Humphreys say some provocative things about whole group sharing.  They say “Students are used to listening to their teachers’ explanations, and saying something once won’t hurt – as long as you don’t expect the students to understand just because you have explained it.  They need to make sense of ideas for themselves.”

They also go on to say:

“Currently, deciding the order of sharing based on the sophistication of strategies, from least to most, is a popular notion. Using this method, it is argued, allows the underlying mathematical concept to build. We have valiantly tried this approach at one time or another but generally take a more organic approach….preselecting an order for sharing or scaffolding toward our “best explanation” can take the agency right out of the hands of students and make processing time almost algorithmic”

Wow.  I recently posted this quote on Twitter because it really hit me in “the gut”. It spoke to me.  It was one of those little voices.  It connected a lot of other recent wonderings:

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The Twitter post prompted a conversation about when and how to sequence student work. It prompted a great conversation about purpose and intention. As a facilitator,  I wonder, is it possible to sequence work AND hone intuition?  Is there a difference between organic sequencing and planned sequencing?  Am I facilitating opportunities for students to connect ideas or am I sharing ways that I connect ideas? When I solve problems as I anticipate student thinking,  am I taking the time to connect my ideas or am I just rattling off a bunch of different ways that students might use to solve the problem? Am I honing my own intuition or am I “planning my presentation”? Yikes. This is tough.

How do I balance honing and planning?  Is there a connection between honing my intuition as a math learner and honing my intuition as a math teacher?  Does the former inevitably impact the later?

In her slide show, Tracy prompted me to look through a series of questions that promote intuition, choose one from each category, and incorporate it into my teaching. My goal for 2017 is to take Tracy’s advice, but make a slight modification:

  • I will incorporate the following questions into my teaching AND learning:
    • Forget about the question for a second. What’s going on in this situation?
    • Let’s refresh our memories about what each of these numbers represents.
    • Slow down. We want to follow your thinking. Can you tell us your reasons for approaching it that way?
    • How did thinking about your experience with __________ help you here?
    • What tipped you off that something wasn’t right?
    •  It feels counterintuitive, doesn’t it? Say more about that.
    • What’s making you doubt?
    • Did anyone change your mind today? How?

My favorite part of this goal is that I haven’t even read Tracy’s book, yet!! I am waiting for it to arrive in the mail. I am just finishing up Making Number Talks Matter.  I am really looking forward to continuing my “gut checks”.

Enough about me. Let them talk about eggs!

In my last post, I shared an experience about trying to explore the properties of operations with K-12 teachers and students in order to understand the progression of how these properties are taught and learned.  Since then, so much exciting learning has happened – a lot of it has been mine.

One of the fourth grade teachers in our K-12 Professional Learning Group approached me to talk about her experience exploring some complicated expressions with her students.    She said she showed her students this expression, (7×8) + (8×3) and asked them what they thought the solution was.  You can read more about her experience on her blog, but essentially, this is how a few of her students approached the problem:

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Chrissy was thrilled that her students were decomposing 7×8, but she was hoping they would see that you could recompose the factors into 8×10.  When she came to me, she said, “I got the students to see how you could combine 7×8 + 8×3 to make 10×8 and we talked about how “seeing” 10×8 would be so helpful because they know 10×8 is 80.”  Then, she shared something provocative.

She said, “one of the students, Emerson, said, ‘you can’t do that. You have to do what is in the parenthesis first.'”.

Wow.

“What do I do with that?” she asked me.  “What should I do next?”

“Great question. I am not sure.” I wondered what some of my colleagues on Twitter would recommend as a next step.

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Wow. We got so much instantaneous feedback. It was awesome. We were still left with a lot of questions:

  • Is there such a thing as the “reverse distributive property”?
  • Is factoring out the 8 an example of “using the distributive property” or is it something else?
  • How do we get these students to shift their perspective about decomposing factors so they can see the potential of recomposing factors? How can we get them to connect decomposing and recomposing?
  • How do we get them to see and use parentheses as a tool instead of a rule?
  • Is it possible to help them figure out that this “regrouping” only works when the expressions in the parentheses share a common factor?
  • Will they, can they, figure out why?

We asked our Twitter friends for some advice about how to pursue these questions, our questions, and still honor Emerson‘s original disequilibrium.

David Weese actually started a planning doc so that multiple people could contribute to the brainstorming. You can see it here.

Christopher (@Trianglemancsd) suggested using a picture of Eggs!

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We decided to ask the students, “what are all the different ways we could figure out the total number of eggs in the picture? Don’t tell me the answer. Just tell me all the ways you could find the answer.” Chrissy and I  didn’t have a ton of time to connect in person. We were connecting via Twitter, David’s planning doc, and short conversations in the hallway en route to and from the bathroom. This was all happening 24 hours before the last day prior to Thanksgiving vacation. I anticipated what the students might say. You can read about this in the planning doc that is linked above.

I emailed Chrissy and begged her to let me come in to math the day before Thanksgiving break to try this lesson with her kids. I told her I completely understood if she wanted to tell me to shut up and go on vacation already. She didn’t. She was equally as curious to see what her kids would do with this picture of eggs.  We didn’t have a ton of time to hone our plans, but  so we decided to jump in because we really wanted to revisit the topic before too much time had passed . Keep in mind, what you see and read below grew out of unrefined plans.  It is bumpy.  Chrissy and I are unsure at times.  I am sharing it with you so I can learn from the experience and so we can learn together.

We showed the students the picture of eggs and asked them to record ways that they could organize them to find out how many.

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This is what they did:

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Chrissy and I started a discussion by asking Chase to share his thinking.  The expressions below were written before Emerson asks her question.  You can’t see the video of Chase, but you can hear him. The black lines indicate what he was drawing on the smart board. The pause in the middle of the video is because the white boards kept falling off the ledge. Hence, my high pitched reminder of why I had suggested NOT putting the white boards on the ledge. #reallifeclassroom

Have a listen.

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As I listen to him now, I wonder if I missed an opportunity. I got caught up in the fact that Emerson noticed one of Chase’s expressions didn’t match the image.  I didn’t catch the fact that he says, “and I would add those together.”   I wish I would have asked him what he meant by that.

At one silent point during the audio, Chase changed one of the expressions on his whiteboard from (2×4) to (2×3) and handed it back to me:

 

We only had 15 minutes before lunch.

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“Emerson, it is because he is adding….”

Let her finish, Sarah!

It is so hard to listen to myself in this video. I want to jump through the recording and put my own hand over my mouth. I wish I hadn’t cut Emerson and her classmate off. The more I listen to and/or watch myself teach, the less I want to talk in a classroom. (This is a good thing!) I talk so much less than I did when I started as a math coach, and yet, I still think I talk too much.

At this point in the lesson, a few boys came in. They weren’t present at the beginning of the lesson so we tried to catch them up by explaining what we were doing.

Have a listen. This conversation is where we start to really wonder about the role of those parentheses.

 

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As I listen to this clip, I wish I had prompted Gavin to tell me more about what he was thinking.   I was so focussed on getting these students to discover what I wanted them to discover that I missed a golden opportunity with Gavin.  I wish I had asked Gavin, “Why do you want to take the two out of the second set of parentheses?” (Tracy Zager, you’ve got me thinking about asking better questions to encourage relational thinking.)

I wish I could rewind and pause.  I was listening for answers instead of to my students. (Where did I read about this last weekend? On Twitter? I think it came out of the #CMCmath North conference? Was it Zak Champagne who said it? Whoever said it, it has really stuck with me.)

I wonder what would happen if I showed Gavin the video above, paused it, and asked him why he wanted to “take the two out”?

At this point, it was time for lunch. How many times have you been in the middle of some deep, messy thinking and the bell rings?  I wish I could have spent the whole day with Gavin and Emerson.  Interestingly, they hung around and chatted with me while everyone else got ready for lunch.  We continued our conversation for a few minutes.

As I reflect on this lesson, I think I  was rushing. I was so desperate to get Emerson and Gavin to figure IT out before the bell rang. 

What was I thinking? The IT is huge! Gavin and Emerson are wrestling with some giant ideas about the distributive property. They are wondering about the limits of parenthesis. They are trying to figure out when to add partial products and when to multiply them. They are manipulating expressions to match a context so that the math makes sense. This kind of learning doesn’t happen in 15 minutes or less. 

I was so worried about losing the opportunity to connect big ideas that I hurried right past several opportunities to connect big ideas.

It has taken me weeks, literally, to write this blog post. It takes me a loooooong time to process experiences because I have to understand all of “it” before I can make sense of parts of “it”.  See Pam Harris’s post about the three groups of people. I am a classic “c”.

I am so grateful for taking the time to process this post.  I used to think all of Gavin and Emerson’s thinking would disappear over Thanksgiving vacation.  I thought I had to be the super hero math teacher lady who swooped in and helped them organize their thinking in a neat and tidy pre-lunch math chat. 

Now, I realize that Emerson and Gavin are doing some serious thinking. I need to let them be the hero’s of their own math stories.(Thanks Dan Meyer for planting this seed.) It takes a long time to construct the understandings that they are wrestling with. They haven’t even started their multiplication and division unit this year. 

Wow. I need to say that again.

They haven’t even started their multiplication and division unit this year.

I wonder what’s on the menu for that unit?

How about eggs?  

Lots and lots of eggs.