Save the Least Efficient Strategy for Last

Recently, I have been thinking a lot about the 5 Practice for Orchestrating Productive Mathematics Discussions by Margaret S. Smith and  Mary Kay Stein.  In particular, I have been thinking about sequencing.  Often, I hear people say that we should sequence student work from the least efficient strategy to the most efficient strategy.  I agree that there are times that we might want to sequence work from least efficient to most efficient, but I think we should be intentional about when and why we chose those times. I also wonder about the times that it might be more useful to save the least efficient strategy for last.

Look at the image below and write down one thing you notice and one thing you wonder:

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When I asked a group of 6-12 teachers what they noticed, they said:

  • “I notice missing dots in all the corners.”
  • “I notice each image has a middle part and then dots surrounding the middle part.
  • “The middle part is a square in three of the images.”
  • “I notice the number of the dots on the outside sections increase by one each time.”
  • “I notice that the number of the dots on the top and sides tells you how many rows and columns there will be.”
  • “I notice the number of rows and columns increases by one each time. So the step number tells you how many rows of three dots you need
  • “I notice a pattern. It goes (1×1)+(1×4), (2×2)+(2×4), (3×3)+(3×4), (4×4)+(4×4).”
  • “I notice little white squares in between the block dots.”

Then, I asked them what they wonder.  They said,:

  • “I wonder what the hundredth one would look like?”
  • “I wonder what the one before the first one would look like?”
  • “I wonder what the next one would look like?”

Perfect timing. I showed them this slide:

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They all went to work.  As they worked, I monitored.  I had anticipated that, in a room full of 6-12 educators, several people would see the following expressions in the dots:

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The original version of this Illustrative Mathematics task actually gives these two expressions:

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I decided to remove the text from this task because I was hoping  to lower the floor and raise the ceiling. My goals were to get us  thinking more deeply about equivalency and the language of math. How do we use symbols to convey relationships that we see in images?  How does the structure of an expression help us represent what patterns and relationships we see? How do we know that we truly understand what another person sees and thinks? Often, I wonder if we assume we know what our students are thinking when, in fact, we are merely projecting our own thinking onto their words.

So, with those goals in mind, I set out trying to find:

Someone who quickly pulled the following expression out of the images:

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I was thinking that this is one of the more efficient expressions for finding the total number of dots in any step.  One person had already eluded to it during our notice and wonder phase. I planned on sharing it first because I wanted to use it as an anchor for equivelance.  I also wanted us to look beyond efficiency.  I wanted us to wonder if there were times when a long clunky expression with many terms might serve a purpose. Maybe? Maybe not? I didn’t know the answer, but I wanted to explore the question.

So, our first friend, Lily shared where she saw Screen Shot 2017-10-14 at 10.17.17 AM.

When Lily finished, I asked Rita, “What do you notice about the pattern you noticed earlier and the expression that Lily just shared?”

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(1×1)+(1×4)        (2×2)+(2×4)          (3×3)+(3×4)           (4×4)+(4×4)

She said “Oh. Well they are equivalent. You can see the Screen Shot 2017-10-15 at 6.54.26 AM in the first term of my expression and you can see the Screen Shot 2017-10-15 at 6.54.37 AM in the second term in my expression.”

“What does n represent in the image?”

“It is the number of dots in the bottom row.”

I asked, Lily, “Is that what n represents in your expression?”

“No. n is the step number, but it doesn’t matter because the step number is equal to the number of dots in the first row.”

Next, I asked Chris to share. When I first checked in with Chris he told me, “I am thinking about the way that Jared saw the sequence when we noticed and wondered. It sounded like he saw the three horizontal dots in the first image as the constant. I am trying to create an expression that matches his description of the pattern.”

Chris explained to the group that the expression he created was a little clunky, but he was trying to capture what Jared saw staying the same and changing.

3n + n (n+1)

Chris explained that he heard Jared referring to the three horizontal dots as the constant.

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Then, he noticed that the stage number told us how many rows of the constant we needed. The term 3n represented the array formed by n number of 3 rows.

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Then, he had some extraneous rows and columns that he needed account for.  He realized that there were always n+1 groups of n dots.

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So, he came up with the expression: 3n + n (n+1).  He asked Jared, “Is this how you were seeing it?” Jared said, “yes.”

Then I asked, is 3n + n (n+1) equivalent to Screen Shot 2017-10-14 at 10.17.17 AM?

Everyone agreed.  Someone shared that they could “see” it. “If you distribute the n, you have n squared plus n. Then, you just add the n to 3n and you have n squared plus 4n.”

Next, I asked two people to come up to the document camera, Rachael and Max. Earlier, when I was monitoring, I found Rachael describing how she saw the pattern changing and staying the same.  She said, “I can see a relationship, but I can’t find an expression to represent my thinking.”  Max asked her to describe the relationship she saw. She said, “I can move one of the dots from the top row down to the lower left corner to make a square and then just add the dots that are on the border.” Max suggested the expression  Screen Shot 2017-10-15 at 4.04.22 PM and described how he saw it in the dots:

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Rachael said, “I see where your expression is in the dots, but your expression doesn’t represent what I was seeing.”

Max and I were determined to find the expression that matched Rachael’s thinking.  We tried to rephrase what we thought she was saying.  We marked up the picture as we spoke:

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She said, “Yes! That is how I see it.”

After some false starts, questions, and dead ends, we figured it out!

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We decided n would represent the number of dots in the top row, which is also the step number. I asked Max and Caitlin to share our experience and speak to how they knew their expression was equivalent to the others. They said they were able to prove equivalence algebraically.  They shared their work.

As I sit here now, I am realizing that I can “see” the structure of Caitlin’s expression in the image. I don’t think I need to verify equivalence algebraically.  I think all the dots are accounted for.  Each term represents a group of dots.  n-1 represents the top row of dots. n+1 squared represents the lower left corner of dots (arranged as a square). Is the “action” of the dot moving represented by the subtraction symbol or is the subtraction symbol really a negative sign? I still have so many questions.  The final n represents the column of dots all the way to the right.

I wish I could say that I had made this connection during my session with the 6-12 teachers, but I didn’t. At the time, I was primarily focussed on creating a situation where a group of educators would share their thinking, listen for understanding, and change their perspectives, which is why I saved the least efficient strategy for last.






II am a K-12 math coach and I never took Calculus. I hated math in High School. I hated it even more in college. It is really challenging for me to memorize formulas, vocabulary, math facts…. anything really. I am a slow processor, but a fast talker.  I won’t remember anything unless I truly understand it which takes a looooonng time.

When I was a kid, I never knew my multiplication facts. I just couldn’t memorize them. So I taught myself some tricks. Trick #1: count really fast. If I knew 6 x 5 was 30, then I could just count up really fast to figure out 6×7.

You know what I am talking about: 6×5 is 30 so six times 7 must be 31,32,32,34,35,36, 37,38,39,40,41,42. In my head it sounded more like this

“thirtyone thirtytwo th-three,

th-four, five, th-six!” (head bobbing every so slightly- almost imperceptively.)

“th-eight,th-nine, fty, ftyone, forty two!”

Super fast. Right?

Speed matters. Right?

At some point, someone tried to teach me a trick to remember the nines, but it didn’t make sense to me so I never remembered it. Eventually, I figured out that I could use 9×9=81 to figure out 9×8 by subtracting 10 and adding one because that was the same as subtracting 9.  That was as close as I got to using the properties of operations and, at the time, I had no idea that what I was doing had anything to do with the properties of operations. I just thought it was handy that 9 was so close to ten.

Anyway, I didn’t really learn a whole lot of math as a K-12 student. Most of the understanding that I have now has been developed since I started teaching. Now, I love math. I love learning. Now, I know my math facts. I don’t need to use tricks. I don’t even need to use the properties. Using and understanding the associative and distributive properties over time eventually led me to recall multiplication facts. I own them. I can even recall them pretty quickly.  More importantly, now, I have a deep understanding of the properties of operations and the major role they play in our number system.

Every year, I stretch myself to expand my understanding of the properties of operations. I read about them, I ask questions about them, and I play with them A LOT.

This year, I decided it was time I played with Calculus. I thought about taking a Calculus class, but I don’t really want to take a class. I just want to learn; on my own, in a messy, not linear, no-time-crunch, kind of way. I just want to play with Calculus.

I am a little nervous. Sometimes, Calculus seems like a big, burly, sand throwing, dodge-ball-pelting, kind of playmate. Usually this happens when I let myself get into a space where I think I should know something. I get distracted by the little voice in my head that says, “you should know that word.  You should know that symbol.  You should. You should. You should.” Then, it isn’t fun anymore. So, I stop. I walk away. I wait for the curiosity to bubble up again.

The first day of school, my colleague/friend/mentor/math-mate Robyn told me she was trying something a little different with her AP Calculus class this year. She said she was going to give her AP Calc students a problem from Paul Forrester that she adapted. She was hoping her students would struggle a bit and, hopefully, develop an appetite for some important information. She seemed excited about starting the year with a problem. She hoped her students would ask a lot of questions as they solved this problem.  I asked if I could try it. She scribbled the problem on a piece of paper for me.

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Suppose a door that is pushed open at time t=0 and slams shut again at time t=7s. While the door is in motion, the number of degrees, d, from the closed position, is modeled by this equation. How fast is the door moving at the instant when t=1s?

One of the first things that I said was, “If  the door was slammed, why did it take seven seconds to close?” She laughed. “Okay,” she said, “maybe it didn’t slam.”

Over the next few days, whenever I had a moment- sitting in the car while my kids were at soccer practice, while I was running, driving around the school district – I would think about the door.  Finally, when I had a minute, I opened Desmos and started a graph.  You can take a look at it here, if you want. Please remember, I am sharing my messy learning with you. I am not looking for feedback, right now. I am happy to answer questions.

Seeing the graph really helped me wrap my head around the problem. I wondered a couple of things:

  • Does a negative exponent always signify exponential decay?
  • Can I call the curved part of the graph a “parabola” even though it is attached to a decaying exponential curve?  What do we call a function that is part quadratic and part exponential?
  • Does the “peak” of the “parabola” represent the moment in time when the door started closing? Is that why 100 degrees shows up twice? Once, when the door reached 100 degrees on the way out and, again, when the door reached 100 degrees on it’s way back to the door jam?
  • Can I use the formula for speed to solve this problem?  If speed is measured by distance traveled divided by time spent traveling, can I measure the difference in degrees between two moments in time and divide it by the time it took the door to travel that distance?

I decided to try this. I think I found the difference in degrees between the door’s location at 0 seconds and 7 seconds and divided it by 7.  I can’t find the scrap of paper that I worked on. I texted Robyn.

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She responded:

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I decided to call her because I couldn’t explain it in a text.  She asked me to explain what I had found. As I was explaining, I realized that my time span was too long. She reminded me that I was trying to find the speed at 1 second. I was averaging 7 seconds. She re-iterated the word, “instantaneous.”


I put the problem aside for the night. The next day, after school, I went back to Desmos.

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I added some more points to my table. What if I found the degrees at .99 of a second? That is pretty close to instantaneous, right? This is when the problem solving started to get really fun.

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I went back to the drawing board.  I started plugging in points with ridiculous amounts of nines in them. At one point, I went too far. I was using the calculator on my computer. It got tired of my nines. It started rounding.

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Screen Shot 2017-09-09 at 10.09.21 AMI walked away again. I went back to work, soccer practice, dinner making, life, but I kept thinking about the door. I couldn’t wait to find those few minutes in my day where I could open my Desmos graph and tinker around some more.

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At this point, I found a serindipitous mention on my Twitterfeed. My friend Chase Orton had shared a blog post about the importance of collaboration between HS teachers and elementary teachers.  I shared my thinking with Chase. He asked me to see if I could use a visual approach to the problem.  I thought about that for awhile.

Yesterday, I stopped by Robyn’s office and we chatted about what I had learned so far. We captured our conversation on her whiteboard:

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As we were talking, I noticed that self doubt creeped in every time Robyn used a math term to capture what I had described.

I described the change in distance divided by the change in time.

She wrote a “triangle d” symbol on the board.

That little voice started whispering, “You don’t know what that means.”

I described how I was  finding the slopes of really short lines on the graph.

She asked me what we called the line that went through a circle.

I almost panicked, but I didn’t. I was with Robyn. I don’t need to panic with Robyn. Instead, I told her about the voice. I told her I struggled with recall. I told her I wanted to understand but I couldn’t remember what it was called.

She said, “it starts with an “S”.”

I remembered a problem that I was solving with Chase last spring. “Secant?”

“Yup. What about the line that touches a point on the outside of the circle?”

I asked, “What does it start with?



“Yup.” Then, she talked for a little while about how the slopes of the little lines were related to secants and tangents.

I told her what I had noticed about place value patterns when I was solving the door problem. I noticed that I was trying to find the difference between a number super close to 1 and 1.  The delta change (am I using this correctly?) is always going to be 1 whatever the furthest place value to the right is called: one hundredth, one thousandth… millionth……one infiniti-th?

Robyn helped me articulate that I was trying to get as close to zero as possible. She introduced a new word.


I wondered if I needed a symbol to represent this distance that I can’t really pin down.

Robyn challenged me to try  this:

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She also gave me the second problem that her AP Calculus class tried. I am excited to work on both. I will let you know how it goes.

For what it is worth:

I wrote this in April and never hit publish. I have been letting it simmer. It seems like it might be worth the conversation.

I am on my way home from the #NCSM17 and #NCTM17 conference. I had the privilege of being part of a team of educators who presented at both of these conferences.  We shared a story about how we collaborated K-12 to improve our ability to teach math.

During our story, we shared some powerful #MTBoS experiences that transformed our teaching.  These experiences were so transformative because they simultaneously lived inside and outside of the boundaries of our zipcode.  We worked together in our district, but we also worked together in the Math Twitter Blogoshpere.  We reflected with each other and we reflected with educators all over the world.

I have been sitting in the Charlotte airport for 5 hours. My connecting flight doesn’t leave for two more hours. I have spent the majority of my time in the airport reflecting.

This morning, on Twitter, I shared links to our NSCM and NCTM presentation with several #MTBoS folks who were instrumental in the evolution of my (and my colleague’s) learning this year.  I engaged in a thoughtful, and somewhat provocative conversation with some twitter friends, none of whom I have ever met in person.

Take a look:

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As we talked, I wondered:

  • When and how do we establish relationships on Twitter?
  • Does humility and vulnerability impact who we engage with on Twitter?
  • Does Twitter have a culture?  Is it persavise or incidental?

Shortly after I wrapped up a two-hour reflection session with @Simon_Gregg@nomad_penguin@KentHaines@TAnnalet, and @m_pettyjohn, I read Dylan Kane’s blog post: On NCTM and MTBoS

I found Dylan’s post fascinating. It seems like Dylan is wondering if people who are new to MTBoS know the purpose of MTBos.  I have only been engaged with MTBoS since last May so I would consider myself on the “new” side of the experience. I am not sure what the collective purpose of #MTBoS is, but I have a pretty clear vision of my own purpose.  I actually tweeted it during one of the conversations that I mentioned above.

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Sometimes I do this by just engaging in a problem solving experience.  Other times, I ask for help with a teaching struggle.  I don’t actively remind myself of my purpose, but I think it is usually a motivating factor of my engagement.

I guess I wonder if it is possible for #MTBoS to have a collective purpose.  It is a community right? But it doesn’t have a structure or rules.  There are no bylaws or sign up sheets.  You can’t point to it or place it on a map. It doesn’t really live anywhere.

Yet, it most certainly feels alive.

It evolves.  Doesn’t it?

Shouldn’t it?


Coherence: A Work in Progress

Yesterday morning, I had a heated conversation with a colleague. It wasn’t nice. It wasn’t fun. At the time, I just felt angry.  Now, I feel conflicted: grateful, frustrated, and a little disappointed in myself. As I sit here, typing (and then deleting) words, phrases, & sentences, I am wrestling with the truth. Do I want to say what really happened?  Or do I want to skip the ugly emotional details and just share my refined and polished insights about when and how we should teach interpreting remainders?

I am going to try the truth.

We started our seventh grade team meeting by discussing how we were going to plan and then teach a three act lesson that we had practiced the last time we met.  It is called basketball arcade and I have taught it several times this year.  I was explaining to my colleagues that I was looking forward to us teaching it together because I really struggle with navigating the strategies that students bring to the lesson, particularly the ones that involve long division. Often, there are students who use long division to solve the problem, but they don’t know how to interpret the remainder so they get stuck. Then, I get stuck.

For example, in this particular three act task, we are trying to figure out how many basketball shots a man will score before the timer reaches zero. We know he has scored 15 shots in 6 seconds. We also know that there are 34 seconds remaining in the game. Some students divide 34 by 6 and get 5.66666666666667.  Then, they are stuck. They wonder, is 5.66666666666667 the answer? Is it shots or seconds? What do I do with 5.66666666666667?  They don’t see the relationship between shots and seconds. They can’t use that connection to manipulate the ratios. They don’t really understand how division connects to proportional reasoning or even how decimals connect to fractions.   I ask them, “what does that long decimal represent?”. They stare blankly.

I told my colleagues, “I struggle so much with this task, but I am attracted to it because it is such a challenge. I am still trying to figure out how to connect student strategies so we all increase our understanding of proportional relationships.”  I asked the other teachers if anyone had implemented the task since we met last time.

One teacher said, “I did. It was tough.  There were some crash landings.”

Another teacher asked, “Could you please tell elementary teachers to stop teaching kids to write R when they have remainders?” She sounded frustrated, accusatory.

I could feel my heart start to beat faster. My Irish blood was rushing to my head.  I felt angry.

“No. I can’t,” I said, “they have a standard that specifically mentions interpreting remainders. If there are 55 students and 16 can fit on a bus, they need to explain why they need 4 busses. Is that what you are talking about?” I probably sounded frustrated and, maybe, accusatory.

She went up to the board. I think she started to explain something about how kids don’t know how to continue dividing after they get to the remainder.  I’ll be honest. I wasn’t really listening because I felt like she was patronizing me, the elementary teachers, and her students.  I think the next thing I said was something like, “if you are talking about the long division algorithm for decimals, that is a sixth grade standard.  It is beyond elementary school.”

She said, “I don’t think you are hearing me.”

More boiling blood. Increased heart rate.

I said, “I am hearing you. That is the problem.  You said, ‘could you please tell elementary school teachers to stop teaching students to write R for remainder? That sounds arrogant to me.'”

She said I was getting defensive. I said she was getting defensive.

She left the meeting.  I took a break.

When I came back, the other teachers were discussing remainders. I said, “I am sorry I got frustrated. I am not proud of what I said. I guess I got defensive because I can’t stand it when we blame each other and our students.  What if the high school teachers came down here and said, “could you all please stop teaching students to ……. (fill in the blank)?”

They nodded. I continued, “I actually don’t know if fourth grade students should write the letter “R” or if they should even use the term remainder. It seems like an archaic convention.”

Somebody wondered, “What would they write instead?”

“I don’t know,” I said.  “Eventually, they have to learn to deal with the remainder.  They have to learn to divide it.  This concept goes beyond 4th grade. We can’t just expect 4th grade teachers to teach their standards in a vacuum.  We can’t teach middle school standards in a vacuum, either.”

I decided to put the question out on Twitter.

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As we waited for responses, we continued to wonder about the progression of learning involved in interpreting remainders.

  • What does the remainder mean, as a whole number, decimal, and fraction?
  • How are remainders related to proportional reasoning?
  • What information tells us how precise we have to be?
  • When we are precise – and divide the remainder – how do we know when to record it as a decimal and when to record it as a fraction?

We decided to do some research.  One of the teachers said we could go into her classroom that morning and collect some information. We designed an entrance ticket that would give us information about how students dealt with a remainder:

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Here is a sample of what they came up with:

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What do you notice?  I notice there are a lot of procedures written on these pages. I wonder if the students wrote the procedures because they “needed” them to solve the problem or because they were “justifying” their thinking. I also notice that most of these kids have no problem interpreting the remainder.  Almost all of them knew that they needed another bus. Although, I really appreciated the alternative solution posed by this student:

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Next, we wrote the three answers below on the board and asked students what they noticed and wondered:

3.43      4 busses         3 R 7

They said that all of the answers were correct.  They didn’t wonder much.  They were pretty quite.  I asked “what does .43 mean?”


I asked, “what does remainder 7 mean?”

They said, “it is the number of kids that can’t fit on the bus.”

I asked, “where are the 7 kids in the number 3.43?”


I asked, “Do you think 3.43 is equivalent to 3 R7?”

“They should be,” someone said, “because they are both correct answers.”

“What do other people think?”

A lot of heads nodding yes.

Hmmm. We decided to try a few more problems, without context.

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When I asked the students where they got the 4 hundredths from, they said they “dropped two zeros down”.  One student originally thought the answer was 5.4, but changed her mind. When I asked her why she changed her mind, she said, “I only dropped one zero by mistake.”

I asked them if they thought 5R1 and 5.04 were equivalent.


We tried one more problem:

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This one was a doozy. There were audible groans. The students said they needed paper. I said, “no, not yet. Let’s just talk about what we are thinking.  Does anybody have any possible solutions?”

Somebody said, “around 4”.

“Tell me more.”

“Well if 125 divided by 25 is 5, then 125 divided by 26 has to be less than 5.”


I wish I could tell you his exact answer because it was so beautifully authentic.  He described the relationship of the divisor to the quotient.

“So,” I wondered, “Does anyone have a braver estimate?  Can anyone get a solution that is closer than around 4?”

A girl in the back muttered something under her breath. I asked her if she minded sharing her thoughts.

She said, “I was thinking about 4.8, maybe.”

I said, “Tell me more about that.”

She struggled to articulate where she got 4.8 from, but she said something about multiplying 26 times 10 and then 26 times 5 and thinking the answer was between four and a half and five.  The other students seemed to think her answer made sense, but no one was able to describe a more precise justification for why 4.8 was a good estimate.

I asked the students to describe a context that might go with these numbers.  They said dividing 125 jellybeans amongst 26 people. I asked them how many jellybeans each person would get. They said, “4 and their would be 21 jellybeans left.”  I asked them if the 26 people could share those 21 jellybeans. They said it wasn’t worth it. Damn! True, but Damn!

I said, “what if it was money? Could 26 people share 21 dollars?”

They said, “yes.”

 I asked, “how?”

They started mumbling some ideas.  Someone said, “well, I think each person could get another 50 cents.”

“How do you know?”

“Because half of 26 is 13 so you would need 13 dollars to give each person 50 cents.”

“What do people think about this?”


I decided to draw a model.

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I heard some, “oh yeah!”s

I asked, “Is that it?  Can these people get anymore money?”

Somebody else said, “yes.”  Eventually, we figured out that each person could get $4.80, but we would still have 20 cents left.  They said, “we can’t split pennies.”

I asked, “was 4.8 a good estimate?”  They all agreed it was.  I asked them to take out a calculator and find the exact answer.  Everybody grabbed their computers.

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We decided to give them one more problem. We told them their work was going to help us plan their math lesson for tomorrow.

When we went back to my office to debrief, we looked at the student work and planned a gallery walk for the next day.

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We chose to include these work samples in the gallery walk:

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We brainstormed ideas for what our colleague  might do the next day:

Ask them to peruse the work and record what they notice and wonder.

As they reflect and share, monitor what they are writing and saying:

  • Do they see the equivalence of the decimal and the fraction?
  • Do they see the progression from leaving the remainder whole to dividing it further?
  • Are they thinking about why 2R1 is not equivalent to 2 1/8 or 2.125?

Ask them to discuss in small groups:

  • Is this true?  2R1 = 2.125 Why or why not?

Close the lesson by highlighting students who explained why 2R1 can’t be equivalent to 2.125. Ask students to change the statement so that it is true. 

After we finished brainstorming, we made a progression chart of all the standards from 4th – 8th grade that connect to interpreting remainders. You can see it here.

Then, we checked Twitter to see if anyone commented on our question. It turns out, people had a lot of the same questions (and, maybe, feelings) that we did.

When I went home last night, I visited and re-read the progression documents for Number and Operations in Base Ten.  Sure enough, there is a very clear answer to the question about remainder notation:

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I could be thinking, “What a waste of time! I should have gone to the progression document in the first place. I could have just had everyone read the document and then I could email it to the fourth grade teachers and we would all be on the same page.”

That is actually not what I am thinking at all. I am thinking the opposite. I am so grateful that I didn’t go to the progression document first. If I had, we wouldn’t have had to construct our own understanding of how interpreting remainders progresses through the grade levels.  We wouldn’t have experienced the messy, uncomfortable disequilibrium of our own learning.  I still need to apologize to my colleague and thank her for pushing my thinking. I am not done processing what I have learned.

I still wonder about coherence.  Does it come from a purposefully constructed sequence of lesson plans written by people well versed in the common core standards? Does it live in a document? Or is coherence more dynamic than that?  Maybe an essential part of coherence is the willingness of all of us to learn more together.

To Whom it May Concern: Learn to Love the Why.

Today, I went to my friend’s house to boil sap.  If you’re not from New England, you might not know about boiling sap.  Sap boiling is the amazing process that takes clear tree sap and transforms it into thick amber deliciousness.  This process is amazing to me because of its simplicity. You collect the sap, boil the sap, put the maple syrup in jars. That is it. Well, to be fair, our friend had to periodically throw logs on the fire throughout the day. Oh, and it takes a really long time to go from sap to syrup. At this point, you might be wondering, why would you spend your whole day at someone’s house just to watch sap boil?

Because something magical happens while the sap is boiling.  Conversations bubble up, residue gathers, extraneous details boil off, and I am left with a nugget of truth.

The conversation started when my friend and I were discussing how cool it would be to make a three act task for sap boiling.  My friend’s fourteen year old daughter, I will call her “x“, was fascinated.  She jumped right in to the conversations:

“What is a three act task?”, she wondered.

I started to explain, “It usually starts with a visual that inspires you to notice and wonder. For instance, what if we showed a short video of all the five gallon buckets that are filled with sap.”

Her mom said, “we should probably have a clip of one of the five gallon buckets attached to a tree tap so people know what they are looking at.”

Her daughter laughed, “of course they are going to know what they are looking at. We live in Maine.”

Her mother and I explained that, if we shared our three act task, people all over the country might use it.

“Oh,” She said, “That is cool.  So they would want to know how much sap you need to make syrup.  Would you need different units?”

“Tell me more.”

She went on to explain that it takes a lot of sap to make a little bit of syrup so you couldn’t measure the syrup with the 5 gallon bucket because the amount of syrup would probably be less than one gallon. She wondered:

  • Would we use millilitres?
  • Could we use fractions of a bucket as our unit?

She said, “I wish I could do this in my Algebra I class.”  I asked her what she was currently studying and she said “graphing inequalities”. I asked her if she thought sap boiling had anything to do with graphing inequalities. She said, “I don’t know. Is it like if I knew I wanted *at least* so much syrup within a certain amount of time, I could figure out how much sap I would need to boil?  Maybe the graph would show me all the possible amounts? I wish we could talk about this in my class.”

I asked her, “why can’t you? What do you do in math class?”

She said, “we do the textbook. I hate the textbook. I think we should burn the textbook.”

I asked her, “what would your teacher do without the book?”

This is what she said:

When will the milfoil overtake the lake?

What if you brought this question to class?

She will probably dismiss it.

It requires too much work on her part.

At this point, you might be tempted to think, “This kid has no idea how hard teachers work. Who does she think she is?”

Please don’t.  This kid is thinking hard about what it means to do math.  Give her a chance. Let it simmer.

She goes on to describe how she and her classmates designed their own study guide.  She says, “Do you want to hear what our study guide was – for our exams – which are like 40% of our grades?”

I said, “of course.”

She elaborated, “One day, the teacher said, ‘okay, I need everybody to go into the book and pick out a problem from each of these three chapters’. So, we did it. We didn’t know what was going on. So, she took them all, copied them, stapled them together and that was our study guide.”

I wondered, “What didn’t you like about that?”

“It didn’t help,” she said, “I threw it away. I studied out of my book… because she has been teaching us out of the book. If she had been teaching us in different ways than I wouldn’t be able to study out of the book.”

I wanted to know more, “So, you know the test is going to cover the book. Am I correct that you didn’t find the study guide helpful because you knew you were going to have to go back to the book anyway? So you couldn’t trust that your peers had picked out the right problems?”

“Well they wouldn’t have chosen the most in-depth problems, because, obviously if your teacher says pick three problems, you are going to pick three easy problems that you can do within five minutes because you don’t know that she is stapling them and that is your study guide and it only covers a very small portion of the book and the answers are in the back.”

At this point, my friend and her daughter started discussing how grateful they are for the answers in the back of the book.  My friend mentioned that having the answers is helpful when she is trying to help her daughter.

Then, x remembered problem 26.  Listen:

She has been talking to me for 15 minutes about how much she hates the textbook and the study guides, and the forced group work. I have been listening to a cacophony of frustration, relief, and longing in her words.  What could have made her remember problem 26? Was it that awful?

Actually, it was quite the opposite. Listen.

We were working together.

We were communicating.

We were problem solving.

So I asked her, “if your teacher could change three things about her math class starting tomorrow, what would they be?”

She thought for a while. Then, said this:

If she could learn to love the why, that would be good.

I asked, “It sounds like you think she gets uncomfortable when you ask why. Why is that?”

“Because when you are teaching out of the book, you don’t need it. The book gives you what you need to teach without the why.”

I was so fascinated by the clear articulation and reflection of this fourteen year old girl so I pushed, “If it doesn’t give you the why, what does it give you?”

“Formulas and specific situations. It is more helpful for me if I know why it works and how it works and I can see the inner workings of the problem because then I can apply it to different situations and different areas of math.”

After listening, I tried to rephrase what I heard.  The three things you would want your teacher to do more of are:

Talking and Discussing  

Problem solving together

Loving the why.

I said, “Those three things don’t include burning the book.”

“Well,” she said, “Then there needs to be four things.”

How should WE teach and assess fluency?

Recently, there has been a lot of discussion in the MTBoS about fluency.  I think Jamie Duncan is right. It is that time of year.  Triggered by state tests and end of the year expectations, some teachers start to panic and grasp at the paper thin promise of flash cards and algorithms. “If I just show them how to do it, they can apply it to the test.” “I don’t have time for deep understanding. The test is next week.”  I could write a whole different blog post describing my opinions about state assessments, but I am going to take a different approach.  Instead, I would like to talk about ownership and agency.

In our district, we have been working on transitioning to a standards based system of teaching and learning.  For the past six years, we have been learning to speak the language of the CCSSI for math. This year, we started to transition our report cards from letter based to standards based.  It has been a challenging, enlightening, rewarding, frustrating, and overwhelming experience.  Some of the stickiest conversations we have had revolve around teaching, assessing, and reporting whether a student is fluent with their ability to add, subtract, multiply, and divide.

I have facilitated countless professional development sessions on the importance of using and understanding the properties of operations.  I have led book studies on Number Talks and Number Sense Routines. I have modeled number talks using images and numbers. I have shared  videos from Graham Fletcher and articles by Jo Boaler. Every teacher has access to Number Talks by Sherry Parish, the Minilessons series by Catherine Twomey Fosnot and Willem Uittenbogaard, and the Mastering the Facts series by Susan O’Connell and John San Giovanni. Our district purchased multi-user licenses for a strategy based Computational Fluency Screener from K-5 Math Teaching Resources and integrated the resource into our curriculum maps.   I have organized and led K-12 summer leadership institutes using books like Building Powerful Numeracy.  My colleagues have led book studies with books like Classroom Discussions and Intentional Talk. In short,  we have spent a lot of time discussing fluency in our district.

Or have we?

Recently, I asked myself, “have we been discussing it or have I just been showing people how it should be taught and assessed?”

I thought about this for a while and I realized that I have  been trying to solve a problem without collectively identifying what the problem is.  Why was I doing this?  Because I was terrified to give up control. I thought, “I can’t ask teachers to identify expectations for teaching and assessing fluency!  What if they don’t say what I want them to say? What if they decide to use timed tests and flash cards? I have to show them all the evidence and then tell them how we should do it because I am the math coach who has all the background knowledge.”   I was scared so I just kept professionally developing everyone and avoiding tough conversations.

A few weeks ago,  I came to the realization that I was never going to get people to change the way they teach and assess fluency if I didn’t meet them where they are and give them a voice. I was at a second grade team meeting and the teachers were discussing what the words “from memory” meant in this standard:

Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers.

The conversation started to make me uncomfortable:

  • “From memory means they have their facts memorized. They may have learned the fact by using strategies, but now they have it memorized.”
  • “If we don’t time them, then how will know if they are efficient?”
  • “Is it okay if a student is asked to solve 8+7 and he/she says, “I know 7+7 is 14 plus one more is 15 or does he/she have to say “15” without using a strategy?”
  • “If they are using a strategy than they don’t know it from memory.”
  • “Isn’t it more important that they understand and use  the properties?”
  • “But eventually the strategies should lead them to knowing it from memory?”
  • “But some kids are just always going to take more time and it doesn’t mean they have less understanding.”

I wanted to just tell everyone what to do.  I wanted to say, “We aren’t going to use timed tests. “From memory” is different from “memorization”.  Let me tell you why.”  In that moment, I realized that I had been telling them why for years and, yet, they were still asking the same questions. And, do I really know the answers to these questions?

I decided to put the question out on Twitter.  Being a part of the Math Twitter Blogoshpere has helped me see the importance of being vulnerable. So, in the middle of our second grade team meeting, I projected my computer screen and asked Twitter for help. We sat and watched as the feedback poured in. You can click the picture below to see the whole thread.

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We didn’t get to see all the responses during our meeting because  we were close to the end of our meeting time. However, in that moment, I realized that whatever we decide about fluency expectations, we need to decide it together. I asked the administrative team if I could have two hours of our early release day on March 17th to facilitate a discussion about teaching and assessing fluency. They said, “sure.”

I met with the elementary interventionists and, together, we planned the two hours.  We decided that our goal was to bring everyone to the table and find out what we, as individuals, think about teaching and assessing fluency. Then, the next time we meet, we can develop a collective definition of fluency and expectations for teaching and assessing it.

One of the first things we did was ask each teacher to explain how he/she defined fluency. Then, we did something transformative. We asked teachers to read each other’s responses.

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You can read our responses by clicking on this link. If you do take time to read the responses, please take a minute to respond in the comments:

  • What do you notice?
  • What do you wonder?

After people had some time to read and reflect on their colleague’s perspectives,  we showed them  Graham Fletcher’s ignite talk about the difference between “from memory” and “memorization”. Next, we had them choose an article to read about fluency. After they read the article, they were asked to find quotes from their article that addressed teaching and/or assessing fluency.  We asked them to document those quotes in a google form.

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Finally, we asked them to reflect on some pivotal question:

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As a school group, we asked everyone to look at all the evidence we had gathered and reflect on whether we have a consistent vision of teaching and assessing fluency. We asked them to identify practices in our district that are aligned and are not aligned and record them on chart paper

We asked everyone to fill out a form with recommended next steps.  Some people said we should establish a committee of  representatives and some people said we should continue to do the work as a whole group. Some people were frustrated that some of their colleagues still thought timed tests and flash cards were acceptable. Some people wondered how we assess fluency without timed tests. Some people resented having to be a part of this conversation because they think, “I know what I am doing. Why do I have to worry about what everyone else is doing?”

I think they wonder, why do I have to be accountable for what happens in other classrooms and other buildings?  Collaboration is confusing, laborious, and rife with conflict. Collaboration often causes feelings of vulnerability and humility. Collaboration requires defending your perspective and exposing yourself to hard questions about what you believe. Collaboration takes a long time. I have asked myself the same question, “why do I have to listen to other people talk about timed tests and fact cards when I don’t agree with those practices.”  I think the answer lies in the fundamental idea that teaching is a collaborative effort.

I closed the session by saying, “I have a clear vision for what I believe it looks like to teach and assess fluency, but I realized that my vision can only take us so far.  We are a collaborative group and what we do in our individual classrooms impacts what our colleagues  do in their classrooms. We have the same students. I can’t, and won’t, stand here and define fluency for you. We have to establish fluency expectations together because we can’t hold each other accountable for a vision that is vague and misaligned.”

I think our next step is to create a vision. It might take us a little while, but it will be ours.


The Lure of a Good Rabbit Hole

A week ago, I facilitated a learning lab in a fifth grade classroom. We were trying to explore these questions:

  • What can we learn from student’s verbal explanations?
  • What can we learn from student’s written explanations?
  • How is the information we get from written explanations similar to and different from the information we get from verbal explanations?

Two nights before the fifth grade meeting, I attended a class with Kristin Gray. Kristin encouraged us to consider how we illicit and analyze student understanding. She guided us through some great activities and encouraged us to try them and think about what we learned about student thinking.

I decided to try using a Numberless Word Problem and Talking Points. I have used Numberless Word Problems before, but I had not tried asking the students to choose their own numbers.   When I shared my ideas with the 5th grade teachers, we decided we would end the lesson with a journal prompt. This would give us an opportunity to juxtapose student’s verbal and written responses.

Mrs. G, the fifth grade teacher, told us that her students were about to start investigating division of decimals. She wanted to use the learning lab as an opportunity to see what her students were bringing to the investigation.  We decided that our student objectives would be:

  • How can explaining my thinking help me learn?
  • How is dividing decimals similar to and different from dividing fractions and dividing whole numbers?

We planned on spending more than one class on these objectives.  The learning lab was our first step.

I started the lesson by introducing a numberless situation:

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This prompted immediate conversation and, understandably, the students wondered more than they noticed:

  • Is she hunting?
  • Why are you outside?
  • Are you cold?
  • Can you see?
  • Is it a full moon?
  • Are your chickens okay?
  • Are you sleeping?
  • What are you wearing?
  • What time is it?  
  • Is she awake?
  • Does she live in town or out in country?

Then, I gave them the rest of the problem, but I intentionally left out the specifics.

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This is where things started to get interesting. I asked for some possible numbers that we could use in this problem.

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One student, T,  volunteered that he would like to use one-quarter of a mile as the length of the driveway and 2 miles as the amount that I wanted to walk. I asked him how I should write “one quarter”. He said I should write one-fourth. I wrote down 1/4 mile and I asked him if he could tell me what one quarter was as a decimal. He said, “point seven five.” I said, “and how would you say that as a decimal?” He responded, “seventy-five hundredths.” I waited. Nobody said anything. After several seconds of silence, one student said, “Wait. I don’t think seventy five hundredths is one fourth because .75 times 4 is going to be more than one whole.”

I asked T, “What do you think about that?”

He said, “I am not sure.”

I told him that I was going to collect some other numbers from people and he could have some time to think about it and let us know if he wanted to change anything about the numbers he chose.  Then I called on two more students and they shared their ideas for numbers. I was just about to move onto the next part of the lesson when one of my colleagues chimed in, “hold on,” she said, “you forgot to go back to T.”


I asked T if he had any thoughts about seventy-five hundredths being equivalent to one-fourth. He said, “yes. I changed my mind. I think it should say twenty-five hundredths.”


“Because there are four quarters in a dollar and one-quarter is twenty-five cents or twenty-five hundredths.”

“Awesome. Thank you so much for sharing your thinking with us.”

At this point, I asked the students to choose any of the suggested pairs of numbers and solve the driveway problem.  Many students chose to try the first three sets of numbers and they approached the problem differently. Most students arrived at answers that made sense for these numbers, but my favorite conversation involved justifying how many times I would have to walk around a driveway that is 9 tenths of a mile long if I wanted to walk 1 and 8 tenths of a mile.

As students were checking in with each other and defending their solutions, I stumbled upon a fascinating argument bubbling up.  Is the answer 1 or 2?

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I looked down at this student’s paper and saw the number one circled. I asked, what do you think the answer is?

“The answer is 1.”

Wait. What? “Can you tell me more about that?”

“Sure. If your driveway is nine-tenths of a mile long, you will have to walk around your driveway one time in order to walk 1 and 8 tenths of a mile.”

“What do you mean  walk around?”

“You will have to walk up to the end of your driveway and then back down again to your house so that is one full loop.”

Whoa. I did not see this coming. I assumed when I told the kids that I walked around my driveway they would picture a circle.  Some groups actually asked me what my driveway looked like. This one didn’t.  Once again, my assumptions failed me.

“So,” I said “Some groups got 2 for an answer. What do you think of that?”

“Why would it be 2?  If it was 2, you would count your trip to the end of the driveway as one whole trip and that wouldn’t make any sense because why would you ever just go to the end of the driveway and stay there?  The problem says “around” so she goes up and back one time.”

Huh.  Fascinating. What do I do with this?  I decided to call everybody back together.  Both groups did a great job of arguing their points.  We decided that the solution would depend on how you interpreted what constituted a trip around the driveway. Interestingly, they all agreed on the math. It was the interpretation of the math that was nuanced and arguable.

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We spent some time discussing how many times I would have to walk around my driveway if it was .3 of a mile long and I wanted to walk 1.5 miles.  We had a similar discussion about whether I would need to make 5 trips or 3 trips.  My favorite comment in defense of 3 trips:

“Why would she just stop at the end of her driveway?  What is she just going to stay there forever?”

I was really curious to see if any of the students tried to use the numbers .99 for the length of my driveway and 1.999 for the amount that I wanted to walk.  Some did. Have I mentioned how much I love working with this class?

One student started the conversation by telling us that it couldn’t be done. Here is his work:

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Another student argued, “Sure it can be done. She just needs really small feet.”

I prompted, “Tell us more about that.”

“Well, she would only have a teeny space left after she walked up and down so she would have to have teeny feet.”

Wow. At this point, I didn’t know what to do. Should I jump down this rabbit hole?  I looked at the 6 adults in the room.  We hadn’t even gotten to the talking points yet.  I asked for advice.  Should we explore this?

Fortunately, these teachers know me really well. They know that I am mildly addicted to rabbit hole jumping. They are type As and I am type B through Z. Many of them were shaking their heads no. Mrs. G gave me an out. She said, “I wonder if I could come back to this conversation with my students.  We definitely have a lot to talk about, but I would like to see what they do with the Talking Points routine and I worry that we won’t get to it.”

Fair enough.

I introduced the Talking Points routine.  These kids are used to engaging in math discussions so they adjusted to the expectations fairly quickly.

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I wondered if they would have any disagreement about whether you learn something every time you get an answer wrong.  They didn’t. They all said making mistakes helps you learn. Mrs. G and I talked about this afterwards and she decided she was going to continue to revisit this talking point.  We wondered how we could get them thinking about how we actually learn from our mistakes – just because we make a mistake doesn’t mean we learn from it.  I was glad to see that none of the teachers pushed this thinking on the students. If they go there, I want it to be organic, not because we told them so.

The students had the gist of the routine so we moved on.  We asked them to think about this prompt and decide whether they agreed, disagreed, or were unsure about it.

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I settled down next to a group of students and watched them. Each of the students did some work on his/her whiteboard. When it was time to discuss, the first three students in the group disagreed.  I watched M carefully. I knew that, on her board, she wrote agree and her math confirmed her opinion.  As each person in her group spoke, she covered up more of her white board. By the time it was her turn, she had her white board pressed firmly against her stomach so no one could see it.  She looked at me. I told her she should share her thinking, regardless of whether it was different.  I reminded her that this routine was about learning from each other.  She and I have a good relationship.  I am so grateful for that.

She described to her classmates that there are 20 groups of one tenth in the number 2. She showed them her number line. Then, each group member went around a second time and shared whether or not they had changed their thinking.  Several students changed from saying they agreed to saying they were unsure.  They sited M’s explanation as their reason for changing their minds and explained that they were still wondering about the difference between 20 groups of one tenth and 20 tenths.

At this point, we were almost out of time. I really wanted to give the kids a chance to write. We gave them a choice for writing prompts:

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We got some really interesting responses:


So… at this point, I have a classroom full of students who are still unsure about a whole lot.  This makes me feel unsure about a whole lot.  Understanding division is hard. Understanding division of fractions and decimals is really hard.  Some students wrote about how the answer couldn’t be 20 because when we divide the answer gets smaller. Several students were representing the driveway problem with incorrect equations:

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There is not doubt in mind that there is a lot about dividing decimals that these students don’t know yet.  Sometimes, when I think about all the things my students don’t know, I get overwhelmed.  I find it more helpful to think about what they do know.  I think these students know a lot.  They know that context matters.  They know that it is okay to make mistakes. They know it takes time to truly understand something. They know that there is a relationship between multiplication and division and also a relationship between addition and division.  I am looking forward to revisiting this class next week and continuing the conversation about all of the things they still wonder.

Kristin, I will definitely use these two routines more often.  I learned so much.  When I asked the fifth grade teachers what they learned, they said:

  • These routines provide organic conversation.
  • They force the students to actually talk.
  • Small groups before large group discussion leverages the discussion – increases engagement.

and I would add…

  • They provide instant access to a myriad of potential rabbit holes.