# Productive Struggle? Count K In.

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.”

“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.

# “Wait….What?”

About two weeks ago, which is 100 years in Twitter time, I saw this tweet by Elizabeth Raskin:

I thought, “hmm. -4.8 plus seven jumps towards the positive side of things equals….-3.8, -2.8, -1.8, -.8, 1.8, 2.8.  Yup. Makes sense to me.

So… why is this her favorite mistake?  Hold on. Let me try that again:

-4.8 plus seven jumps towards the positive side of things equals….-3.8, -2.8, -1.8, -.8, 1.8, 2.8. Yup. Makes sense to me…..Wait. What?

Then, I did the same thing AGAIN. I won’t bore you with the details.

At this point I felt stuck and confused. I also felt curious and determined. Why is this her favorite mistake? What am I missing? I decided to try to solve the problem on my own, without looking at the student’s work.  I wondered how many hops would it take to get back to zero?  +4.8.  Okay. How many hops do I still need to make to cover the total of 7 hops? 2.2.  So….. the answer is 2.2  Oh!!!! I get it.  This IS a cool mistake.

I responded to Elizabeth:

I wondered how I could use this problem with elementary teachers. This semester, two of our elementary schools are participating in learning rounds which focus on the NCTM teaching practice, Support Productive Struggle in Learning Math. In mixed grade level teams, we visit 2-3 classrooms and look for evidence of this practice. We record evidence of “look for’s” that we think we see:

Finally, during the debriefing, we try to synthesize our observations to increase our understanding of the practice.

To prepare us for each learning round, I facilitate a professional development session that takes place during a staff meeting prior to the observations. Elizabeth’s problem pushed my thinking about productive struggle. I decided to use it as my entry point to explore this teaching practice with the staff.

I knew I was going to be working with staff in two different buildings, but I decided to plan the same general session and adapt it to the needs of the staff. I thought I would learn a ton from the first session that would impact how I facilitated the second session (and I did). In the interest of blog efficiency, I have combined the experiences.

As the teachers settled into our staff meeting, I explained that our learning target would be to identify characteristics of productive struggle. I shared our guiding questions for this series of learning rounds:

• What is the difference between productive struggle and unproductive struggle?
• How do developmental stages and prior knowledge impact whether a struggle is productive?

I told the story about Elizabeth’s Tweet. I showed them a poster with the problem on it. The elementary teachers who have been in our district for at least three years are used to doing math together, but that doesn’t mean it is easy or comfortable for all of them to take math risks in front of their peers. Sadly,  I knew that there would be at least a few teachers whose heart rates would increase as they experienced genuine panic about solving a math problem. Fortunately, our elementary schools are small. This building has seven k-5 teachers. They depend on each other for support.  I encouraged them to work together if they wanted to. I told them it was okay to struggle. I shared that it took me several tries to figure it out. I asked them to try to solve the problem in several different ways so they could truly understand the student’s mistake.

The teachers dove right into the problem.  One group (the kindergarten teacher and the second grade teacher) saw it right away.  Here is their justification:

Some other teachers experienced similar disequilibrium to mine:

• “Can we change it to 7 – 4.8?”
• “Why am I getting 3.6?”
• “If I start at 5, do I have to add .2 or subtract .2 when I get to 2?”

Then I showed the teachers this problem:

I asked them to show multiple ways to arrive at the solution. Here is an example of the strategies they used (Incidentally, it is the work of the same two teachers whom I referenced in the first problem):

Then, I asked, “What is the same about these problems?  What is different?”

“They both use a numberline, but one deals with crossing zero and one deals with crossing a decade.”

“They both have to do with place value patterns.”

Tell me more.

“Well, crossing a hundred is challenging because the patterns in the ten place change -now you have a hundreds place.”

Me:  And what about the first problem?

“The pattern in the tenths place changes AND it is even more difficult because of the transition from negative to positive.”

Me: Can you see that on the number line?

“Yes!” (Points to change from -.8 to +.2)

“Both problems have to do with decomposing.”

“You can use compensation for both…. wait. Can you? How do you use compensation with negative numbers?”

“Well. If you add 1 jump of -.2 to -4.8, you will land on -5.  So….Wait. Is that constant difference?”

“Keep going. If we add -.2 to the 7, we will get 6.8. Then we would have -5 + 6.8. That doesn’t work because the answer is 1.8.”

“What if you add +.2 to 7. Then, we would have -5 + 7.2.  Yes!! That works. -5 +5 is 0 plus 2.2 is 2.2. But why do we have to make it positive?”

At this point, I was so excited about all the math that these K-5 teachers were doing.   I was also stressed out because we had about 15 minutes left in our staff meeting and we had yet to identify characteristics of productive struggle. Should I just tell them all the rules for adding and subtracting positive and negative numbers?  Give them a link to a Kahn Academy video?  Maybe assign them 42 practice problems?  I decided to go with being honest.

“You are doing some awesome thinking.  It seems like you are engaged in productive struggle. I am too. I am also trying to figure out how the rules for adding and subtracting positive and negative numbers impact the discovery you just made.  I need to explore it more and I encourage you too, as well. Maybe we can revisit the same problem next month and share what we learned. For now, I would love to hear what you think it looks like and sounds like when someone is engaged in struggle.”

“It looks like us trying to solve that negative number problem.”

So, what were we doing and saying that tells you we were engaged in struggle?

• making mistakes
• talking through our thinking
• saying bits and pieces of information that are leading up to a solution
• crossing things out
• trying once to see if your answer makes sense, deciding it doesn’t, and trying again
• saying, “wait. what?”

I asked if there was anything that they see in their classrooms that wasn’t on the list. They agreed that they see a lot of the same evidence of struggle in their classrooms. They added these:

• student sharing the wrong answer, but is totally convinced he is right
• students arguing
• “I don’t get it”
• students destroying his/her paper

This brought us back to one of our guiding questions,  What is the difference between productive struggle and unproductive struggle?  I asked the teachers to place some of their evidence on a continuum:

Then, I asked, “How do we keep the struggle productive?”

(Thoughtful silence as the clock ticked closer to 4:00.)

Slowly, they came up with some ideas:

• You have to have a culture where it is okay to disagree
• ..and mistakes are valued
• You have to anticipate who will know what and how you will navigate confusion
• You have to know when it is time to take a break or move on
• You have to ask the right questions
• It is hard.  It is really hard… to balance pushing their thinking without giving them answers and/or confusing them to the point of frustration.

Me: Who is it hard for?

“The student… and, well, me.”

Me: Who struggles more?

“Good question. It depends.”

(More thoughtful silence and clock ticking.)

Me: “This is a huge question. I don’t think we can answer it in a day. We can come back to it each time we meet and discuss how our thinking is evolving. Thanks for taking a risk with me today.  I can’t wait to be a part of your lessons tomorrow on learning rounds. I always learn so much from all of you.”

And learn I did, from each of the 11 classrooms that I got to observe. I wish I had time to write a blog about each and every one of them, especially my new hero, Mrs. Chalmers, who took a huge risk and offered her kindergarten students a 7 foot long piece of yarn on which to place the numbers 1-10.  She navigated their struggle (and her own) with deliberate thought and humble presence.  Thank you Mrs. Chalmers.

# Counting groups or “grouping” counts?

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?

# Dial the math down.

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: