When I woke up this morning, I went right to Twitter to see if I could find video footage of the keynotes from the California Math Conference. I watched Fawn Nguyen and I  jotted down the things that made me think, “Yes! I want to do that!”.

Then, I watched Dan Meyer. He challenged me to turn a question into a story and put students at the center of it as the hero. I thought about that for a while. How do I draw a student into the narrative so she is the heroine – the owner of the learning? How do I give her the power?

Later in the day, I was exploring the geometry progression with a colleague.  We were playing with triangles and parallelograms in order to understand the first and second grade geometry standards better.  By accident, I created a small parallelogram and a large parallelogram.  I wondered if they were proportionate.  As we worked, I kept generating more questions about the situation. My favorite question was, is the area of  proportionate figures related to the scale factor?

I remembered Dan’s keynote. I wanted to make myself the hero of this story. I didn’t want someone to tell me the answer to this question. I wanted to figure it out.  My friend, who has taught high school math, was eager to help me make connections. I told her I wanted to “own” my understanding. I needed some think time. She respected  my wishes.

While we were exploring, another colleague approached me.

“How long have you have been a math coach?”

“5 years.”

“Oh. I noticed you were really excited about learning math. What is your background?”

“I taught K-8 in a one room school-house off the coast of Maine for 6 years. Then, I taught fourth grade for four 4 years.”

“What grades do you coach?”

“Well I started as a 3-8 math coach, then a K-5, now I am K-12. We have a really small district and my role has kind of changed and evolved over the years. We also have math specialists in all the buildings, including one in the high school.”

“I only ask because sometimes it is hard for generalists to become math coaches because they don’t have the content background. You work with high school teachers?”

“Well…. yes. We work together. I ask them questions about HS standards.  Sometimes they ask me questions about the K-8 standards. My role is to be a conduit.”

My cheeks felt like they were getting red.  I wondered if other people were listening. I quickly started packing up my things. I wanted to hide all the pieces of scrap paper with my notes on them.

I nodded and smiled.  I said, “I just really enjoy learning math.”

Ouch. I tried not to feel it, but it seeped into me before I could stop it.

Shame.

I don’t even like typing that word. It is the worst feeling and to feel it about a subject that I have finally grown to trust again…… It just sucks.

I wanted to shake it off. I wanted to forget about it. This person doesn’t know anything about me. Later, I thought about all of the things I could have said to this person. I thought about all the qualifications that make me a good fit for my K-12 coaching position.  I almost started listing them here, but my qualifications aren’t what this story is about. This story is about how shame got in the way of me becoming the hero of my own math story.

If I wasn’t still feeling ashamed, I would probably describe my favorite question  in more detail. I would stay up until 2:00 am pouring over my notes from today, in an attempt to articulate my current understanding of how area is and isn’t related to the scale factor of proportionate figures.  I would share all the mistakes I had made. I would ask more questions. I would try to connect my thinking to all the standards.

I don’t really feel like doing any of that right now.  The shame has gotten in the way.

When I was grappling with my questions about proportionate parallelograms,  I felt empowered.

How did those words,

“You work with high school teachers?”

take me from feeling empowered to feeling ashamed in a matter of seconds?

Was it because I felt vulnerable?  I thought vulnerability was a good thing.  When I am vulnerable, I am open, unassuming, ready to learn.  When I was exploring my favorite question with my colleague, my vulnerability pushed my thinking and fed my curiosity.   It allowed me to take a risk, to explore something that I wasn’t totally sure of.

Suddenly, when confronted with a question about who I work with, vulnerability was not so good. It meant I was weak.  I didn’t have the “right” experience.

It ignited shame.

I am not sure what I want to do with this experience.  I am not even sure why I am sharing it. I guess maybe I hope that if I acknowledge the shame, I will take away its power.   I can’t help but wonder, would this person have asked the same question to a high school teacher who told him that she was coaching elementary teachers?

I used to think that being vulnerable was my strength. It was at the heart of the relationships I have built over the last five years.  It was where I tried to start every day. It was what allowed me to feel so comfortable, and even excited, about saying,

“I don’t know.”

Now, I am not sure.

Unfortunately, vulnerability doesn’t have a switch.

Fortunately, I will learn from this. I will think about the teachers and student I work with.  I will continue to value and protect the space where being vulnerable can be empowering.  I will encounter more situations like the one I just described.

Maybe, after some time, I can get myself to a place where I can say,

“Yes. I work with High School Teachers. And they work with me.”

Recently, I have been fascinated by the resurfacing of the counting and cardinality standards in third grade.  Third grade is when students start exploring multiplication and division. They have just spent three years grappling with how to count, add, and subtract whole numbers.

They started the journey in Kindergarten wrestling with the fact that one digit could represent multiple items.  Minds were blown by this concept.  Just when they started to apply it, they came across the number 10.  What ?!? two digits – one and zero – can represent ten items.  That number “1” now means “1” ten?!? Craziness.  Eventually, these same students developed deep understandings of some important units in our number system: one, ten, 100, 1,000, 1/2.  They learned how to take these units apart and put them back together. They used them to represent and understand their world.

Now, they enter third grade and we start presenting situations where 1 can mean a group of any number of items AND you can have multiple groups of those items.  Enter the parallel progression.

In kindergarten, students learn to count a group of objects, they pair each word said with one object.  When these same students start to learn multiplication, sometimes, they revert back to counting by ones, especially if the objects they are counting are presented in an array that is composed of numbers that make skip counting challenging.  

Watch Jack:

When students were in Kindergarten, they learned that the last number name said in counting tells the number of objects counted. When students work with arrays, they may take some time to construct the understanding that a 7×7 array that is subdivided contains the same amount of squares as a 7×7 array that is not subdivided. They wonder, do they really both have 49 squares in them?

Take a look:

I asked this student which one of these boxes of truffles was easiest to determine a total. His answer is a reflective and clear statement about why this progression is challenging

I am fascinated by how students progress from using and understanding units of 1 “item” to using and understanding units of “1 group”.

When Jack and I were talking about how the two subdivided arrays were different, he commented on the different numbers he used to do the subdividing. I asked him, “What does that 3 mean? ” He struggled to come up with the words to describe it. He tried “three slots of truffles in a line at the top”, but he recognized that his words weren’t capturing his meaning.  I decided to focus on the middle array since that was the one he proclaimed was easiest because he could skip count.  He could “see” and use  the groups in the middle array.

Listen to him. You can hear him revising his thinking about “items” and “groups”.

After students are comfortable using skip counting, they usually start thinking about and using repeated addition.  This is when they start grouping their groups.  Listen to Ellen describe how she knew the amount of truffles in these arrays:

I have talked a lot about how I think the Counting and Cardinality progression resurfaces in third grade. Now, let’s look at the Operations and Algebraic Thinking progression as it applies to third grade. If you haven’t read the progressions documents that are on Bill McCallum’s website, you really should. I reread them all the time.  They really help me understand the standards.

Here are some quotes from the third grade section of the OA progression.  See if you recognize any of these in the videos you just watched:

• “In the Array situations, the roles of the factors do not differ. One factor tells the number of rows in the array, and the other factor tells the number of columns in the situation. But rows and columns depend on the orientation of the array. If an array is rotated 90 degrees, the rows become columns and the columns become rows. This is useful for seeing the commutative property for multiplication”

The progressions describe three levels of representing and solving multiplication and division problems:

•  “Level 1 is making and counting all of the quantities involved in a multiplication or division situation.”
•  “Level 2 is repeated counting on by a given number, such as for 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30. The count-bys give the running total. The number of 3s said is tracked with fingers or a visual or physical (e.g., head bobs) pattern.”
• “Level 3 methods use the associative property or the distributive property to compose and decompose. These compositions and decompositions may be additive (as for addition and subtraction) or multiplicative.”

I want to understand more about how children use what they have constructed about counting, addition, and subtraction and apply it to make meaning of  multiplication and division situations. Bill McCallum says, “These skills and understandings are crucial; students will rely on them for years to come as they learn to multiply and divide with multi-digit whole number and to add, subtract, multiply and divide with fractions and with decimals.”

I wonder if we give them enough time to really explore the connections between these concepts.  I also wonder if we give them enough time to defend and question the ideas they are developing.

I also wonder:

• Would these students have used the same strategies if I showed them this picture:
• How could I help Erin see her addition statement, “I know 5+5 =10 and 10+10 =20” as a multiplication statement, “2 groups of 2 groups of 5”.  Is Erin thinking intuitively about associativity?
• Are certain properties more intuitive to certain kids or is it about what learning opportunities they are exposed to?

I wonder.  I wonder. I wonder. What do you wonder?

Listen. This is the sound of sense making.

I taped this audio recording at the end of the day.   The three boys were waiting to picked up. They came over and said, “What are you doing Mrs. Caban?” I told them I was reading a blog post about decimal quick images (thanks Kristin Gray).  They asked me if they could do some more problems like we did this morning. So, I showed them the one above.  It was so cool to listen to their voices collectively buzz as they wrestled with unit conversion.

I started the morning with these boys in their fifth grade math class.  Mrs. Gordon and I have been co-planning and co-teaching this week. We are introducing operations with decimals and we decided we want to anchor the unit in the student’s prior knowledge about decimal fractions.

Yesterday, Mrs. Gordon did count around the circle. First, they counted by tenths. We had planned that she would intentionally ask each student how he/she wanted Mrs. Gordon to record what they said – at least in the beginning. We were curious about how many of them were actually picturing decimal fractions and how many were picturing decimals. We knew that how we recorded what a student said would influence how the subsequent students in the counting sequence responded. We also decided she would record in rows of 10.  Some interesting things happened.

The boy who started the circle told Mrs.Gordon to write “one over ten”.  The tenth person said, “ten tenths or 1 whole.”  Mrs. Gordon said the students seemed to enjoy the challenge of changing from mixed numbers to decimals to improper fractions and back to decimals again.  They noticed that the halfway point of each line was always composed of 5 tenths.

Then they counted by hundredths.

Today, when I went in, we wanted to see if we could get the kids to establish some relationships between tenths and hundredths.  We knew we wanted to try to switch the unit as we counted – start with tenths, move to hundredths, move back to tenths. We decided to record the counting sequence on a series of blank number lines that were segmented into ten sections.

When we got to 1 and 1 tenths, I asked everyone to pause.  I told them we were going to switch to counting by hundredths. I asked Liz if she wanted any suggestions from her classmates or if she wanted to try it herself.

Liz said, “Well. I think it will be 1 and 1 tenth and 1 hundredth.”

We wondered, “What do we call that?”

“Well,” said Liz “I think there are ten hundredths in 1 tenth so that would be like having 1 and ten hundredths plus another hundredth which is 1 and 11 hundredths?”

I wish I had written 1.10 underneath 1.1 on the recording sheet,  as Liz was speaking, so she could see the ten hundredths that she had so elegantly described.  We continued to count by hundredths until we got to one and 17 hundredths. Then, I switched us back to tenths. There were audible gasps.

“Whoa.”

At this point, we could tell that the transition back to tenths would be tricky for a good number of the kids. There were some who thought they could just add one tenth to the tenths place, but they struggled to convince some of their peers.  One student was stuck.  I asked him, “Do you think you could use a number line?”

He drew a number line and figured out that he could decompose the tenths into ten hundredths.  Then, he broke up the ten hundredths into 3 hundredths – to get the nearest tenth and then added 7 hundredths to arrive at 1.27.  Yay! We were really hoping someone would consider decomposing.  He still had to count be one hundredths to arrive at his solution, but he was able to see the .3 and .7 chunks after he counted. I decided to add his representation to our chart. You can see it above.

I wish that I used a bigger number line so everyone could see the hundredths. I also wish we grouped the kids in smaller circles so everyone could engage in more counting. So… we made three big number lines to use later in the week.

I have been thinking a lot about productive struggle.  What does it really mean to struggle productively? This is what I found when I Googled the words:

: achieving or producing a significant amount or result.

That seems pretty straight forward.  How about “struggle”?

Yikes! I would like to think we can avoid violence in math class.  Could the purpose of productive struggle in math class be to “strive to achieve or attain something significant in the face of difficulty or resistance”?

This week, I  facilitated Math Learning Labs with our elementary school teachers. It might have been my favorite week of school so far. Learning labs are a time for us to learn and grow together. Last week, I wrote about my experience with the Kindergarten teachers. Today, I get to share what happened with the third grade teachers.

Once a week, since early September, I have been co-teaching with Mrs. Watkins.  We have been trying to help her third grade students build stamina for problem solving. You can read more about that here. Since the grade level meeting was going to take place in Mrs. Watkin’s building (we live in a small rural district so each elementary school takes a turn hosting a meeting), I wanted the learning lab to meaningful for her. We discussed what kind of experience would be beneficial for the teachers AND also for Mrs. Watkin’s students. We decided that we would like to give the students an opportunity to justify their thinking and the teachers an opportunity to support productive struggle.

Here is what we came up with for an agenda:

We started by reading  two blog posts about the Hundreds Face Challenge, one written by Malke Rosenfeld and one written by me.

Then, I gave the teachers a collection of statements about productive struggle. I got most of these statements from NCTM’s recommended teaching practice, Support Productive Struggle in Mathematics. I added some of the statements that Mrs. Watkins and I had discussed with her students.

I asked the teachers to arrange the statements so they showed a progression or “road map” through productive struggle. This is what they did:

Then, we jumped right into building our own Hundreds Faces.

Some people started by counting out a collection of rods that equaled 100 and then building and adjusting as they went.

Others, started a design and kept track as they went:

Everyone was able to build and justify a hundreds face:

After we built our hundreds faces, we shared our thinking and reflected about how our approaches were similar and different. I shared how I had figured out that if I could rearrange my Cuisinaire rods into a 10 x 10 square  than I could prove that I used the equivalent of 100. I wondered, would you always be able to make a 10 x 10 square out of any hundreds face?  Could you make a 25 x 4 rectangle?  They wondered these things too. We agreed to explore these questions further during our next unit.

Next month, we will be using Muffle’s Truffle, one of the mini units in Cathy Fosnot’s Context for Learning Math series, to kick off our third grade exploration of  the relationship between arrays, addition, multiplication, squares, and rectangles. This will be a perfect time to explore our questions further.

As we revisited  our maps of productive struggle. I asked, “did you experience any of these statements while you were building your hundreds face?” The teachers responded with many connections:

“We all solved problems.”

“We had to find our own mistakes.”

“We had to prove and justify our thinking.”

“We asked a lot of questions.”

Then, they started to have a really interesting conversation about whether the math leverages the creativity or vice versa.

Have a listen:

After this conversation, I asked the teachers to each make their own individual map of productive struggle. We didn’t have a lot of time, but I was hoping they would transfer whatever meaning they had constructed to a sketch.  We each thought of our journey through productive struggle differently:

After we shared our maps, we brainstormed some questions that we could have in our back pockets as we interviewed the third graders about their hundreds faces. Here are some that we came up with:

• How do you know if this is equal to 100?
• Can you show me?
• Is there another way we can figure if it is 100?
• What does that (having too many or too few) mean?
• What do you need to do next?

Finally, we went into third grade to see whether or not the students could justify their hundreds faces.  As we circulated the room, the kids were eager to share all their strategies with us. Unfortunately, most of the video footage from this experience is full of so much math talk that it is hard to discern who is saying what.  Fortunately, there is a TON of wonderful math talk happening in this classroom. Here is a sample of an exchange between a teacher and student:

S:  “Right now I am at 81.”

T:  “and you have to get to how many?”

S: “100”

T:  “so how many more do you need?”

Below, you can see some sample faces with their justifications. Can you find the math in the pictures?

Right before we were wrapping up with the kids, our district curriculum coordinator managed to capture her conversation with a student who was navigating his way through some big ideas about area, geometry, and multiplication.  Take a look:

If the purpose of productive struggle in math class is to “strive to achieve or attain something significant in the face of difficulty or resistance”, was Marshall engaged in productive struggle?   What did he produce that was significant?  I noticed that he produced a pretty significant statement regarding the classification of the shape that he created:

“It’s a rectangle!”

I wonder which was more significant; the statement he made or the resistance that got him to a place where he could claim it? Think about the questions/statements that Nancy contributed to her conversation with Marshall:

• What is this?  Can you tell me about what you made here?
• How do you know it is a square?
• Rephrase:  “So, anything with four edges and four ends…”
• How do you know that those are equal edges?
• Prove it to me.
• Oh, you are measuring with your hands. What if I said show me with the rods?

Now, think about what would have happened if Nancy had never asked the first question. What if Marshall left class thinking he had built a square?  Teachers often admit to me that their biggest concern about allowing students to experience disequilibrium is that the student might leave the classroom confused about something.  What if they leave thinking the wrong answer is actually the right answer?  What if they leave my classroom not knowing all the answers?

I usually respond that the difference between the struggle being productive and unproductive is the teacher’s level of awareness. If you know your students have partially formed understandings then you can revisit and explore these with carefully planned questions and problems. But, how do we know what our students don’t know?

We ask.  Then, listen. I mean really listen.

One recent morning, my son, daughter, husband and I were trying to count the flock of turkeys in our front yard.

“I got 19.”

“I see 20… no 21, there is one in the ditch!”

“Wait. Three more just came out of the woods.”

“No. I already counted them.”

“Somebody take a picture!”

My son took a picture and we used it to help us organize our thinking. It turns out there were 24 turkeys in our yard.  It took us a little while to prove it.  We had to work together and we needed some tools to help us organize our thinking.

This week, I had the pleasure of joining the kindergarten teachers for a Learning Lab. Recently, we have been exploring how we can create more opportunities for students to “play” with the math they are learning.  We have also been discussing what productive struggle looks like in Kindergarten. We decided to create a giant ten frame, present it to the students, and see what they noticed and wondered. We brought different materials to the meeting: clear shower curtains, felt, tablecloths, duct tape, and Sharpies.  Each teacher created their own giant ten frame.  Then, we brainstormed questions we wanted to ask Kindergarten students and we predicted what they might say and do.

We wondered:

• Will they recognize it as a ten frame?
• Will they use it to problem solve?
• Can we engage them in productive struggle? If yes, how will we know it is productive? How long will it last before it becomes unproductive?

When the students first came in, they sat around the ten frame and we asked them what do you notice?

• “It is Memory.”
• “It is a ten frame.”
• “It is five frame.”
• “a 6 frame”
• “a 7 frame”
• “an 8 frame”
• “a 9 frame”
• “a 17 frame”
• “a hund… a hund… an infinity frame!”
• “It’s soft!”

Okay, so now that you have been sufficiently introduced to the lesson, I will tell you that teaching Kindergarten students terrifies me.  I have read books and taken classes about teaching Kindergarten students math.  I am very familiar with the Kindergarten Common Core math standards. I have repeatedly read the Common Core progression for counting and cardinality.

All of this goes out the window when I walk into a classroom of 5 year olds.  They are so alive!  They are growing right in front of me – constantly trying to make sense of their world and how they fit into it.  How do I get from the books, classes, and standards to the actual little people standing in front of me (without any of us (including me) ending up under the table in tears)?

Fortunately, I work with an amazing group of Kindergarten teachers who I am constantly learning with and from.  At our grade level meetings, we have cultivated a space where it is okay to take risks because we support each other.  So here is me knee-deep in my own risk taking:

It might look like I am nodding and smiling, but really, I am thinking, “11? 62? 100? 73? 9 frame? infinity frame?  two five frames?  What am I supposed to do with all this????  Do these kids see ten or not? Help me!!” At this point, I am experiencing productive struggle. I am trying not to panic. I decide to move forward with the exploration.

I asked them, “What could we do with (this big frame)? Do you have any ideas?” Several kids wanted to play memory.

One girls said, “I think we could all sit on one square.”

I asked, “Do you want to sit on a square?”

“Yes.”

Then, all the students started immediately and simultaneously scooching towards the ten frame.

Me: (heart rate increasing) “Hold on. Hold on. Let’s go one at a time. I pointed to the student who came up with the idea, “You came up with the idea, why don’t you find a square to sit on.”

Okay. That went well.  So, I pointed to one other student and said, “go ahead.”  Then this happened:

“Hold on.  Hold on. Hold on. Back up. Back up. Wait one second.” I wondered, how do Kindergarten teachers do it?  How do they  build space for productive struggle without completely losing every kid to distraction or impulsivity?  I decided to ask for help. I turned to one of my brilliant Kindergarten teaching colleagues and asked her what she wanted to know.  Watch what happened next:

I found this fascinating.  What is going on here? I thought some of the students were counting the empty spaces on the ten frame, but then when I asked them to explain, they didn’t really have much to say about it. Were they changing their thinking because other students started counting students?   In the end, I followed Mark’s lead and tried to steer everybody towards figuring out how many kids were on the ten frame and how many kids were off the ten frame?

What happened next was really interesting.  One of the girls said, “Hey. Can we get in a line so we can count?” My favorite part of the next clip is an exchange between Christy and  I. I think we are both wondering how long this will last before everyone falls apart. What do you think? Are we still struggling productively?

So, after everybody was lined up, I figured this was it. They would count who was in the line and clearly see it was 12.  Well that is not what happened.  And we, the adults, were starting to get squirmy. We REALLY wanted to help.  It was hard for us to watch these little people struggle.  Even Mark was trying to figure out why on earth did his peers keep getting 11 for an answer when he got 12?  Check it out.

Here is where I say a big giant thank you to Pam.  I went into this lesson with the intention of trying to figure out what it looked like for kindergarteners to struggle productively.  I learned pretty quickly that I couldn’t keep the struggle productive by myself. I knew I didn’t want to feed the kids the answers, but I didn’t have enough experience to know how to scaffold the struggle.  Pam came up with a scaffold that allowed the struggle to linger a little longer and still stay productive.  Thank you Pam for supporting MY productive struggle. As you can see, maintaining a neutral voice is not really my “thing.”  I got pretty darn excited when  Evvy figured out that Henry was forgetting to count himself. Sometimes, you gotta throw neutrality out the window and celebrate a lightbulb moment.

At this point, I wondered whether I should just end on a happy celebration. Yay!  We all know how many kids are not on the ten frame. It is 12. Great struggle. Go ahead and have a snack.  After all, we have been struggling for a solid 15 minutes, right. But….. I couldn’t help myself.  It was like I was addicted to pursuing the struggle. Could we go a little further?   Could we just push a little bit more? Sure. Why not?

Now I have to say a big thank you to Christy.  She knew when to jump in with a good ole’ “prove it!” We had talked about it enough.  We had the ten frame right in front of us. Let’s just get on with it.  And so we did.

After the kids left, the teachers and I sat down to debrief.  I asked them, what did they notice? They said:

• “The students were practicing  so much: 1-1 correspondence, subitizing, solving for unknowns, adding and subtracting!”  
• “They were involved in high levels of analysis and using a lot of language.”
• “The lesson moved from conceptual to procedural.”
• “My favorite part was when they noticed that they all wouldn’t fit on the ten frame.”
• “I really enjoyed the opportunity to watch my own students. There were some surprises. Some students who don’t usually talk were speaking up.  Some students showed me they had a better understanding of 1-1 correspondence than I had seen in the past.”

Then, I asked the teachers, what would you do next?

• “I would bring it out again.”   
• “Could you bring out number cards – show me this number in the ten frame?”
• “After we have done it physically, I might add objects to manipulate. Roll the dice and put the objects in.”
• “I like the idea of exploring.”

And finally, I asked the teachers, what did they learn?

• “You taught me to let go a little bit. They are learning.  They are the ones doing it. “
• “They are engaged in real life problem solving.  They will transfer it to make a decision down the road. “

And what did I learn?

I learned that counting is hard and so is teaching it in a rich and meaningful way. I also learned that Kindergarten students are beautiful little people who want to move, talk, organize themselves and each other, articulate their opinions, share,  explore, and count. I am just a little bit less terrified about supporting them.

I also learned that the kindergarten teachers in our district our thoughtful, reflective, observant, curious, knowledgeable, and so very patient.

Recently, I read a post by Kristin Gray titled What is it About these Questions?  It got me thinking about our 4th grade students know about area and multiplication.  We have four small elementary schools in our district.  About every 6 weeks, we hold monthly elementary grade level meetings.  During that time, we do learning lab in one of the classrooms per grade level. The purpose of the learning labs is to collaboratively and plan a lesson that allows us to learn from each other and our students.  We usually pose a guiding question to help us identify what it is we want to learn about.

After I read Kristin’s post, I thought it would be interesting and informative to use  explore these questions in our upcoming fourth grade learning lab.  I shared her post with the fourth grade teacher with whom I will be planning the learning lab.

As you can see, Deb thought this was a great idea! Deb and I decided that we would give her students the same exact questions that Kristin used, analyze them together, and decided what we still wondered. Then, we would think about how we could structure the learning lab to explore what we still wondered.

Since Deb and I will not have a lot of time to plan when we meet, I thought it would be helpful if we could process what we notice virtually – through Twitter and  blogging.

I am going to get the ball rolling by sharing my thoughts about what I noticed. The first thing I did was sort the student work into some groups. The first group represents students that got all three problems correct:

A

B

C

D

E

F

The first thing I notice is that even though all of these students got all three answers correct, I think they demonstrate very different understandings of multiplication and area. Student A decomposed all three problems to find partial products.  Student B used partial products to solve 19×7, but his explanations for the other two problems makes me wonder whether he added or multiplied to find his answers. Student C’s answers are really interesting to me because this student decomposed the rectangle but used addition to arrive at a solution. I wonder what student C and student A would say if we asked them to compare and contrast their strategies.  Student D got me thinking about my own understanding of doubling and halfing. He found 9 groups of 14 plus 7 more.  Do you see it?

What does student F know about multiplication and area? He didn’t show any work for the first problem, used partial products to answer the second one, and appears to have counted individual squares to answer the third problem.

I want to ask all of these students if they can use the array to show me where their decomposed problems are.  Students A, B, C, and E seem to be referencing the array in their strategies. Can student D see the 9 x7?  Does he understand what that partial product represents in terms of the array?

Even though these students got these answers correct, I still have a lot of math I would like to explore with them and I am not sure it would suit them best to put them in a group together just because they all got the answers correct.

The students below all got two of the answers correct.  I would like to ask students H and K why they both got different answers for 7×9.  I would like to show the whole class student I’s work and ask them what they think he is doing. Where is his work for 7 x 19 in the array?  I don’t even know if I would show the answer he arrived at. I think I would just show them some of his beautiful number arrays.  I think he might have a natural affinity for the associative property

G

H

I

J

K

The following students did not get any of the problems correct and  I don’t think I would put them in a group together. I need to ask student L more about how he arrived at his answers. I would like to get L and M to look to the array to support them. I wonder about putting N, M, and J in a group and asking them to compare and contrast all the different ways they decomposed 19 x 7.  Can they use different colors to show where the partial products are?  Would this help them revise some of their thinking?

L

M

N

I think it would be really interesting to ask all of the students, how could we adapt the 7 x 19 array so it shows 19 x 4?  What about 21 x 8?  Maybe we could start the class with this problem.  What do you think?

Full disclosure:  I am a district math coach. I work with K-12 teachers in 6 different buildings.  I am always seeing something shiny.  I am never bored. This blog is the space where I take all those shiny things and try to create something coherent and, perhaps, beautiful. It is my chance to make sense of all the bits and pieces. It takes me a while to pull it all together.  Brevity is not my strength. This post still feels clunky. Consider yourself warned.

On Wednesday, I went back into Mrs. R’s third grade class.  She and I have been collaborating to help her students build stamina for persevering.  I originally wrote about this here.

After last week’s lesson, I thought a lot about where we should go next with these students.  I kept trying to make it about the math.  Should I do a mini-lesson about equivalence?  Should I juxtapose student’s strategies to encourage more efficient strategies?

Something didn’t feel right.  It still felt forced.  My original goal was to encourage perseverance – to build stamina.  I have learned from experience that stamina is directly related to engagement. There were still quite a few students who built a #hundredface that was not equivalent to 100. I wondered, do they care that their face might not equal 100?  Are they engaged enough to want to make adjustments to it?  THIS is what interested me.

As I continued to let my third grade plans simmer, I began to plan the Elementary Learning Rounds that I was going to facilitate.   This year, our learning rounds are focussing on the NCTM teaching practice, Supporting Productive Struggle in Learning Math.  As I planned the debriefing meeting for the learning rounds, I wondered:  When is struggle productive?  What does it look like?  How does it become unproductive? What do we do when struggle becomes unproductive?   How does struggle relate to engagement?

How does struggle impact our work with the hundred face challenge? At this point in my planning, I felt stuck.  So, I decided to work on a hundred face.  I wanted to build an owl. Owls are my favorite animal.  I connect with owls.  I took out a bunch of Cuisinaire rods and started to build.  I looked at pictures of owls.  My first owl was equivalent to 122 Cuisinaire units.  Too big. But I loved it!  I didn’t want to change it. Enter struggle.  Should I start over?  Should I build something else?  No. I am going to start tweaking it.  What do I know about Cuisinaire rods?  Well I am going to need less rods AND less units.  I can’t just replace the smaller rods with larger rods because that might not change the total value or, worse, it might increase the total value. Hmm.  I have 4 rods that are equivalent to 7 so that is a total of 28.  If I take those rods away I will be down 28 (-28).  If I replace them with 3 rods that are equivalent to 6 then I would have +18.  So -28 +18 = -10.  I would be down 10 units so I am still over because 122-10  = 112.  I kept going with this line of thinking.  It was a struggle, but it felt productive because my owl was changing in a beautiful way AND I was making connections to integers.  Could I do this activity with some of the middle school classes? The math was leveraging my creativity.  The creativity was leveraging the math. Here is where I landed:

Now what? I am psyched because I made a cool owl.  What does this have to do with the third graders with whom I am supossed to work in one hour? Should I share my work with them? How? I guess I could rebuild my owl and put it under the doc camera. I wish I didn’t have to put the cubes away.  Maybe I don’t.  I could cut out paper Cuisinaire rods and build a replica of my owl. That would be cool, but I wouldn’t have the shadows or the cool wood grain background. Let’s just see what it looks like:

This is kind of cool. I wonder if the third graders would be interested in making their hundred faces more permanent?  It will take me a long time to cut out all those pieces. I am not just handing those out wily nily. If they want to make a paper copy of their hundred face, they will need to prove it is equivalent to 100. Hmm. Am I making the math more relevant?

Enter the voice of Dan Meyer.  On Twitter this week, I was following a series of conversations about whether it was possible to create a math lesson about Bottle Flipping that was truly relevant. Does the relevance leverage the math? I read Dan’s blog post and all the comments.   I watched his NCTM talk again.  I thought about this:

• “We need to focus on the rot not the coat of paint.”
• “Teachers are so eager to get to the answer that we do not devote sufficient time to developing the question.”
• “How mathy does the room feel right now?”

I decided I was going to share my idea with the third graders and see what they thought.  I started the lesson by sharing the story of my owl.  They were intrigued. They were actually pretty excited to make their own paper hundred faces.  I told them about the constraint: You have to prove, without a doubt, that your face is equivalent to 100.  I am not wasting little paper pieces. Then, I took them on a field trip to the entrance hallway in our school.  I told them I thought we should hang our faces on clothesline across the ceiling tiles so that everyone will see them when we come into the school.  They loved this idea.  They commented that we better make sure our math is correct if we are going to be displaying it.

When we got back to the classroom, every student got to work:

Please note that these faces are NOT all equivalent to 100 because the work is NOT done. These students need more time to struggle.  Many of them have moved on to the justification portion of the task. Several have already figured out that they will need to make adjustments.  One boy spent almost twenty minutes working out the math for his face:

Originally, he did not have 4 brown rods in the center and he only had one brown rod for each wing, but he added the rods in the center because the didn’t have “enough” to equal 100. He had some calculation errors when he determined what 8+8+8+8+8+8 was. I believe he was off by 8. However, he was correct about needing more units to reach 100. After he added more brown rods, he realized that he had too many.  This child spent the entire class revising his thinking.  He still has more revising to do.  However, in the last 5 minutes of the class,  amidst groans of frustration, he said, “I don’t even have to do the addition. I already know it is too many.”  Mrs. R said, “Tell me how you know that.” He said, because 70 plus 60 is more than 100 so 76 + 64 will definitely be more than 100.    His teacher said, “Wow. If you already know that then you know a lot.  Next week, we can start with what you have and see if we can keep adjusting it.”  He asked me to take a picture of his eagle so he could remember where he left off.

These students are in a beautiful place.  They are learning that their discomfort is leveraging their creativity and their mathematical understanding.  This takes time. I think, next week, we can dial-up the math a little by focussing on precision.  My next question is how do I ensure that I am dialing it up thoughtfully and intentionally?

For the last few weeks, I have visited a third grade classroom every afternoon. I love this time because I get to work with a teacher for whom I have a tremendous amount of respect. I wrote about her here.

This teacher has often told me how much she loves to teach math. She has been teaching for thirty years and she is always growing and learning.  She often comments about how much her math instruction has changed over the last ten years.  She has become an expert at honoring student thinking, valuing mistakes as learning opportunities, and cultivating perseverance.  She told me in the beginning of the year that she was worried about math this year.  She has a group of students who have so many social and emotional struggles in their lives that she is not sure how she will be able to get them to build their stamina for problem solving. She invited me into her classroom and I was honored to accept.

Admitedly, I was nervous.  I am great at “talking the talk” when it comes to teaching kids how to persevere and communicate their reasoning, but “walking the walk” is really hard. This teacher “walks the walk” so well and she does it with patience and love.  I knew I would be learning from her, but I was not sure that I would be bringing anything new to the table.

Awhile ago, I had read about the Hundreds Face challenge and thought, “That is so cool. I am going to try that with kids.”  Then, I filed the idea with the thousands of other amazing ideas that I have seen on Twitter and WordPress in the short 6 months that I have been spiraling through the MathTwitterBlogoshpere. The beauty of the #MTBOS is that, yesterday,  I found myself reading Malke Rosenfeld’s post, #HundredFace Round 2 and thinking, “Oh yeah. I remember that. That is so cool. I am going to try that TODAY with the third grade class that is struggling to persevere.”

I was still nervous.  What if they didn’t get it?  What if they gave up?  What if they didn’t want to make faces?  Should I let them build other things?  If I did, what would the constraint be?  I decided to ask Malke for her thoughts.  Here is another beautiful thing about the #MTBOS – you can ask far away people for help.  At 10:31, I posted some questions on Malke’s blog.  At 11:56, Malke responded with some suggestions and a few more questions. At 12:30, I responded with my new ideas. At 1:15, I taught the lesson.

Stop for a second and think about what I just said.  Now, think about this:  I have never met Malke. This was the first time that we ever communicated.

When I got to third grade, I told the third grade teacher my idea.  She was thrilled.  “That is a great idea!” she said. Then, she took out her giant tote of Cuisinaire rods. As the kids came in and got settled, I caught her up to speed about what Malke and I had discussed. We anticipated what students would do and what we would look for to share with the whole group.

First, we took a page from Malke’s book and told the kids to explore with the rods.  We smiled as we saw several kids building stairs and then noticing that they could compose rods out of other rods. They told us that 10 whites equals 1 orange. Here are some of the other equivalent compositions that they built:

Next, I told them about the challenge. Malke, the teacher, and I had all agreed that starting with 50, instead of 100, would offer scaffolding without lowering the cognitive demand of the task.

All of the kids started building faces.  Most of them were thinking about how to make the face equivalent to 50 white cubes. At this point, the third grade teacher and I agreed that our priority was to encourage students to stick with the task for as long as they could.  It was okay if some of the faces didn’t equal 50. We could come back to the faces next week and explore the math more deeply.  Here are some of the 50 faces:

The one on the right is one of my favorites. When I first approached the student, I asked him how he knew that the face was worth 50? He wrote, “blues are 9 so I need 5 ones.” I don’t know if you can tell from his picture, but he used 4 blues to outline the face, 1 blue for the mouth, and then he tacked on a white cube to the end of the mouth.

Some of the kids were struggling with making a face that was equal to 50 and some of the kids were done and ready for more.  The teacher and I conferenced and decided to offer a choice for the next challenge:

• Choice 1:  Build a face that is equivalent to 100!
• Choice 2:  Design your own face and tell us how much it is worth.

We weren’t sure if the second choice would offer enough of a constraint, but we think it did. The students who chose to design their own face were constrained by justifying the value of the face they had created.  The additional constraint of defining a value was beyond their zone.

Here is a sample of two students who designed their own face and told us the value.

And here are some of the #Hundredfaces:

There are still some faces that we need to revisit next week. I love these faces, even if the math is a little off:

The best part of this lesson was at the end. We asked the students what they learned today. Here is what they said:

• “Math can be fun!”
• “Never give up! You’ll never figure your way through it and you won’t learn if you give up.”
• “We made a bunch of mistakes today so our brains must have grown.”
• “Keep going – it’s a challenge.” (This one is from the boy who looked up at me half way through the lesson and said, with a big grin on his face, “This is so much harder than I thought it would be.”

And my favorite one…

“I learned to keep trying. I was trying to make a face with only 50 and I kept getting over 50 ,and over 50, and 0ver 50. I just kept trying the 50 face. Then, I got 50. Now, I am trying to build the 100 face.”

Thanks Malke.

This weekend, when I looked at my calendar, I realized that I had double booked myself for Monday. I was supposed to spend the whole day with the seventh grade math team AND I was also supposed to teach a 6th grade math class.  I thought about canceling the math class, but I really didn’t want to do that to the kids or the teacher I have been working with.  I decided I would just let the 7th grade math team know that I had to leave for an hour to teach 6th grade math.  I knew they would understand.

Then, halfway through my Sunday morning planning session, my kids started fighting, the dog ran away, and I realized that no one had eaten anything yet.  THAT never happens. Fast forward to the ride to school on Monday and I am panicking because I only have half a lesson planned. I also only have half the 7th grade math meeting planned.  One half plus one half equals a whole lot of random unfiltered blah bit e blah from me as I try to fake my way through the day.  Then, I had an idea.

What if I invite the 7th grade math team to join me as I teach the half planned 6th grade math lesson? Then, they could give me feedback about whether my lesson was only half a failure or a complete failure.

Fixed mindset voice in my head: Seriously, Sarah?  Why would you invite teachers to watch you teach a lesson you are so completely unsure about?

Growth mindset voice in my head: They could help me.  We could do it together.

Fixed mindset voice in my head:  And if it doesn’t work?  If you make a bunch a math mistakes, or worse yet, teaching mistakes?

Growth mindset voice in my head: I will just tell them up front that I am not sure about the lesson, I need their help, I only want specific feedback, and I am nervous about making mistakes in front of them.  Maybe I will ask them if they see me making a mistake, could they frame it as a question?

Fixed mindset voice in my head:  This is NOT a good idea.

Growth-ish mindset voice in my head: The worst thing that can happen is that I end up under the table crying.

Fixed mindset voice in my head:  Actually, the worst thing would be if the kids are totally confused because I didn’t get to take the time I needed to try all the math and anticipate what students will do.

But, what if I just say that to my peers.? What if am transparent?  Won’t the lesson inevitably be better if I have my peers help me analyze my half-plans before I go in to class, assist me as I navigate the nuances of students grappling with new concepts, and debrief with me afterward?

Yes.  The lesson will most definetly go better if I collaborate with my colleagues.

So I did.

When I got to the seventh grade meeting, I explained the situation.  I invited the 7th grade math teachers to join me in 6th grade math and then I waited.

Would they even want to do this? Some of them only teach 7th grade math?  What was I thinking?!??!? Why would they want to teach with me?

Silence.

“That sounds fun.”

“I would love to do that.”

Wait. What?

I continued to explain why I wanted their help and that I was really nervous about this experience.  I have had many elementary school teachers observe and teach with me, but being observed in a middle school math classroom was new territory for me.  I listed off all of my criteria:

• I only want feedback about posing purposeful questions
• I want you to use this “look for” sheet that I made and I want you to write specific evidence.
• I want your help.  I might ask you for suggestions in the middle of the lesson.
• If I make a mistake, please don’t point it out in a really obvious way. Ask me a question that prompts me to check my reasoning.

AND

• I need your help planning the lesson before we go in.

They were all in.

I shared how I wanted to introduce the tape diagram as a tool for creating equivalent ratios.  We had been using tables, interlocking cubes, and some equations.  I said, “I don’t want to force the tape diagram, but I want to share it as a possible strategy.  I don’t want everyone to have to use it.” I shared the Bubble Juice recipe that I had created as a context for us to work with:

Recipe for Bubble Juice

Makes 6 cups

4 cups juice

2 cups bubbly water

Then, we brainstormed:

• What is a tape diagram?
• How does a tape diagram keep track of the unit?
• How does a tape diagram relate to a table?
• How does a tape diagram relate to the interlocking cube model?

We didn’t have much time to plan, but it was enough for me to establish a feeling of trust: we were in this together and the focus was on learning from and with each other and the students.

When we first started the lesson, all the visiting teachers sat in the back of the room. That didn’t last long.  When I asked the students to create a ratio from my recipe, some of them were struggling.  I took a teacher time out.  I asked my peers, “Can you help me?  I don’t want to do the thinking for the students, but my questions aren’t helping.” One of my peers joined me.  One by one, my colleagues got up and integrated themselves into the class.

The rest of the class followed a similar rhythm.  I would pose a question to the students and/or my peers and we would navigate the learning trajectory together:

Me to my peers: “The recipe was the warm up problem.  I was going to move on to a problem about running laps. Do you think I should stick with the recipe context?”

Peers:  “Yes!  Do you want to make sure they have found all the different ratios in the recipe?”

Me to my peers: “Yes, but I don’t want them to just name them.”

Peers:  “Well. You have some listed on the board.  What if we  label the ones on the board with letters and ask the students to write down a letter that is an example of a part:part ratio?”

Me: “Yes! They can work with their table partners to come to consensus.”

Students: (having already heard the directions) “There is going to be more than one answer. Should we find them all?”

And this is how the rest of the class went.  All of us learning together.  In fact, when it came time to introduce the tape diagram, I handed off the marker to one of my peers.

“Sherri, would you mind explaining how you use a tape diagram to create equivalent ratios?”  She did.

Then, we asked the students, “How many cups of bubbly water and cranberry juice would you need to make 60 cups of bubble juice?”  Again, we circulated.

Toward the end of the lesson, I called a teacher team huddle.  I said, “One of the things I am trying to work on this year is the closing of the lesson. I want to take the last 5-10 minutes to encourage kids to connect what we did today.  I want them to walk away thinking and making connections. What should I ask them?”

• “What about, ‘How would you be able to find the amounts of bubbly water and cranberry juice needed to make any # of total cups of juice?'”
• “Or we could ask them why an answer is incorrect. ‘So and so said you would need 30 cups of cranberry juice and 30 cups of bubbly water, but that isn’t correct.  Why wouldn’t 30 and 30 work'”

We decided to ask both.  One of the students closed the class by sharing,”It can’t be 30 cups of cranberry juice and 30 cups of bubbly water because that would be too fizzy.  You have to keep the amounts of cranberry juice to bubbly water so they match the “juicey-ness” of the original recipe.”

I couldn’t have said it better myself.

So, it turns out, the lesson was 100 times better with my peers in the room.  Kids benefited. I benefited. My peers benefited.  Collectively, we magnified the learning.

The best part?  When we got back to our 7th grade meeting, after we debriefed, everyone agreed that we need to do this at all of our meetings.