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

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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
  • asking questions
  • 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.


Power or Influence?

Recently, I watched an NCTM shadowcon talk by Robert Kaplinsky. I can’t stop thinking about it.  His words echo in my thoughts.  I am a district math coach. I have no administrative power.  I only have the opportunity to influence.  Sometimes, my influence is positive. Sometimes my influence is negative. Often, the difference between the two is how honest I am with myself about my intentions and how intentionally I reflect.  

Last week, I led a learning round in an elementary school. I watched an exceptional lesson in a third grade classroom. What made the lesson exceptional was how little the teacher said.  For ten minutes, 7 different students participated in a student generated investigation about how many lines you need to draw to show fourths on a number line.

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When we walked in, we heard:

Student #1: “You have to draw more than three lines because if you only drew three lines, you would have thirds.”

Student 2: “I agree. If you didn’t have the end line, you would have thirds.”

Student 3: “You don’t count all the lines.  You don’t count the 1.”

Student 4:  “I see both sides of the story.  What does the zero stand for?”

Pause. A long, silent pause.

Student 5:  “If you count the 1 line, you have to count the zero line.”

Student 6: “I disagree with you… about not counting the 1 line.  If you didn’t count the 1 line, you would only have three fourths. You would only have three pieces.  It really wouldn’t make sense without the end line.”

Student 7: “Yeah!  If you don’t have the zero line and the 1 line, the numbers would go on forever.  The zero and the one are like the start and the stop.”

Several Students:  “I agree with (student 7).  A number line goes on forever.  When you make fractions on a number line, it is kind of like you are showing the pieces. You need the zero line and the 1 line to show  the “piece ” of the whole number line.”

Another long, silent, pause.

Mrs.Watkins:  “I think I heard you say a couple of things.  When showing fractions on a number line, it is really like showing a piece of the whole number line – a line segment.  We need to draw the zero line because it tells us where to start. We need to draw the 4/4 or 1 whole line because it tells us where the whole ends. You taught me something today. I could be more specific when I am using number lines to show fractions.  I could call the fraction pieces line segments.”

These students presented, questioned, and defended their own and each other’s reasoning.  The teacher listened.  

I was participating in this learning rounds with an interventionist from another building. She didn’t know these students at all.  Afterwards, I asked her, “Would you be able to guess which students received “gifted and talented” services?”  She said she would have no idea. Then, I asked her, “Would you be able to guess which students receive interventions?”  She said she would have no idea.  I pushed.  I asked her, based on the thinking she just observed,  choose a student who you think sounded like a typical “gifted and talented” student.  She chose a student who receives math interventions. 

After I left this classroom, I couldn’t stop thinking about these students and their teacher. Their words were echoing in my thoughts.

During the debriefing session of our learning rounds, I asked my peers who observed after we did, “What happened next?”  I was hooked. I only saw ten minutes of this math lesson. What else did these students do and say after I left?

They shared with us that, after we left, the teacher presented the students with a problem about cupcakes.  “There are 3 cupcakes and  4 kids.  How should they share?  I wonder if they should each get half?” She sent the students off to work in small groups to come up with a solution.

As the observing teachers circulated, they noticed the level of engagement in math talk. One student said to her partner, “no offense to Mrs. Watkins, but I think her estimate is off. I am pretty sure each kid could get more than half a cupcake.”

Later that day, I checked in with Mrs. Watkins.  I asked her what the kids came up with for solutions.  She showed me the white board below and said, “this was really interesting.”

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“Is carot a type of cupcake?”, I asked.

“No. It is the name of the kid. They chose their own names.”

Of course. Carot, Tomato, Charlie, and Joe.  Just your typical group of nine year old names.

She told me that these students presented to the class that the solution could either be 3/4 or 1/4 or 3/12.  She endured yet another silent pause as I processed.

“1/4?”, I asked. She was being really patient with me.

“1/4 or 3/12 of all three cupcakes. These students explained to us that the answer depended on what you considered to be the whole.”

Again, she waited for me to catch up.  I felt humbled.  I wouldn’t have thought to consider either of those answers. She might not have thought of those answers herself, but she was open and quiet enough to allow her students to consider them.

Yesterday, I went back into third grade.  I can’t stay away.  Mrs. Watkins has inspired me. She’s got me thinking more deeply about fractions.  I shared with Mrs. Watkins that, since I left her, I saw this message on twitter:

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She and I had a great conversation about how and why we label fractions on a number line. She told me that one of the first lessons she does with her students is create fraction strips.  She has the students label each interval as a unit fraction.  She doesn’t introduce labeling the hash marks on a number line until later.  The lesson that I described above was her first introduction of labeling the hash marks. Again, I learned from her – the progression matters.

Mrs. Watkins has been teaching third grade for 30 years.  She often talks to me about how much her math instruction has changed since she first started teaching.  When I first started teaching, I worked across the hall from her.  She is quiet.  I think she would consider herself an introvert.  This year was the first year we completed learning rounds in her building.  Learning rounds are opportunities for teachers in a building to observe short segments of each other teaching with the intention of learning together to improve math instruction. Some of the teachers that Mrs. Watkins works with said, “We have worked together for 16 years and today was the first time I saw her teach.”  I am so grateful that we are all able to see Mrs. Watkins teach.  Thank you, Mrs. Watkins.