Recently, I have been thinking about how coaching isn’t something you do “to someone”. It is something you do with someone. I used to think coaching was about structures and protocols, but now I think the heart of coaching is in the conversations I have with my colleagues, the learning we do together. My colleagues push me. They ask me questions that I don’t know the answers to. I used to think I had to know the answers, but I realize now that the most important learning happens when we co-construct the answers together.

This blog post is co-written by Deb Hatt and I. This year, I am working with the math specialists in our district to design and implement a collaborative intervention model.

I am a math specialist at one of our elementary schools. Sarah and I have worked together for a long time. During Sarah’s last year in the classroom, we co-taught fourth grade. I work closely with the third grade teacher in my building. We meet weekly to plan together, and we co-teach daily.  I also work with some of the third grade students during the school wide intervention block for grades 3-5. This morning I showed Sarah this representation of a 3rd grade student’s work.

Take a minute to notice and wonder. What do you think this student is doing? 

Here is what Deb and I noticed:

• The student is flexibly regrouping based on what he needs.
• He knows how to keep track of his regroupings using expanded form.
• He might prefer subtracting multiples of tens.
• His regroupings are based on the properties of operations, as opposed to the place based structure of our number system.

Here is what we wonder:

• How would this student use the same strategy without expanded form?
• Will this strategy become cumbersome with larger numbers?
• If he tries to use this strategy without expanded form, will he forget about the true value of all of the digits he is subtracting?
• How do we honor student agency while simultaneously introducing other perspectives and strategies that highlight the useful structures of our number system?

Deb and I talked for awhile about what to do next. Neither of us knew the answer. I asked Deb what she was thinking about doing.

I talked to Carolyn about this strategy at length.  I really wanted her opinion as an experienced 3rd grade teacher.  We both thought it was fascinating, but had some qualms about what comes next.  We see a lot of place value understanding in Sam’s work in general, but we worried that might be lost if he tried to make his strategy look more “compact”, like the algorithm.  Would he remember that the 6 was worth 6 tens when he was presented with 38-6?  We thought it would be interesting to explore further with him.  

Now I’m thinking that I want to encourage him to continue using this strategy, but also use a strategy based on place value.  

I said, “Tell me more about why you want him to use the strategy based on place value.”

“We emphasize these strategies based on place value and I am wondering why,” I said to Sarah.  “I feel this strong urge to teach him place value strategies, but I’m asking myself why?”  I felt like using place value strategies helped me so much as a mathematician, but in that moment,  I was struggling to see why Sam would want or need to use them if this strategy is working so well for him.  

“I am wondering the same thing,” I said, “ it seems like we should teach him to use the place value structure, but I don’t know if I can say why. Let’s play it out. Why does he need to know place value strategies?”

“Well, I really want him to use numbers flexibly,” I started. “I don’t want him to only think about adding and subtracting what he needs.  We have this wonderful system of tens. The ten structure is so helpful when we think about multiplication, division, exponents. I want him to have flexibility with numbers, not just be confined to adding and subtracting small bits.”

I paused and then continued, “I think the part that gets to me is that I almost shut it down. We were talking about using the specific subtraction split strategy.  I had asked the group to try it, allowing that they could use another more familiar strategy to check their work once they had attempted this one.  When he was showing me his work, I saw that he had the correct answer. But then I saw how he had solved it and said, ‘Hold on. What did you do here?’  

He said, ‘I just took what I needed.’”

“At this point, I almost said, ‘Well, we’re trying this new strategy today.  Can you show me how you would solve it using subtraction split?’ I stopped myself. Instead, I said, ‘Tell me what you did here.”  I thought I understood pieces of it, but I didn’t know exactly what he had done. After he had explained his thinking, I took it and said, “This is really cool!  I am going to have to think more about it.”  I really needed time, without other kids in the room, to really think about how he had solved the problem.

I think you showed a lot of respect for Sam and his work.

“That is my big thing this year,” I responded, “ I teach these kids and they are so worried about math. I have fourth and fifth grade students who have gotten a message, somewhere, that they are not good at math and their ideas aren’t as good as other people’s ideas. I don’t want to perpetuate that at all.”  I paused. “I was so close to screwing it up.”

“But you didn’t screw it up. I so appreciate your honesty, Deb. I have felt the same way so many times. I see myself in you. Don’t be so hard on yourself. It is what you told me the last time we reflected together.” 

Deb and I continued to reflect on how we have changed as teachers and learners.  We used to think teaching and learning math was all about decision making.  We were always thinking about which step to do next, following a prescription.  We never used our intuition.  Now, we are re-training ourselves to build and use our intuition.  We look towards ourselves, our students, and each other to figure out what to do next.  The decision making process is much more complex.

As we were wrapping up our meeting, I said, “I’m very glad I caught myself and listened to Sam. I am thinking a lot about how I introduce strategies. There are some strategies that lend themselves to students discovering them on their own. I have found that subtraction split isn’t necessarily that type of strategy. But is a strategy that I have found helps many students when they learn it. I think I would like to reframe the way I introduce strategies. I think I need to present new strategies as more of an invitation, instead of a prescription.”  I told Sarah that I was so grateful that I was able to take the time to think about Sam’s strategy on my own and reflect with both her and Carolyn about our next steps.  I’m learning so much about being a more reflective, responsive teacher from both of them.  

Later this afternoon, I was talking with Abby, one of our other school based math specialists. I was relaying the conversation that Deb and I had. Abby agreed that Sam’s strategy was unique. I asked her what she would do next. She wondered if we could compare and contrast Sam’s work with a student who used the place value structure. We could ask him what is the same?  What is different? Both strategies are decomposing and finding equivalent ways to represent the subtrahend.  The only difference is that Sam is only taking what he needs. He is taking smaller amounts.

She also wondered whether Sam was actually using the place value structure, but in a different way. I hadn’t thought about that before. She said her gut says this kid is thinking flexibly about numbers. It might not be a far leap for him to find the commonalities between his strategies and the place value strategy.  

I am so grateful for Deb and Abby. They care deeply about the students they work with. They show so much respect for student thinking. They push me to question what I think I know about teaching and learning. I just love this strategy that Deb shared with me. It is like a little nugget of truth. What if we all just took what we needed?