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.