As a K-12 math coach, I am all over the place. We have about 50 teachers in our district who teach math in some form. I haven’t found a way to divide my time equally among those 50 teachers and make a difference. My superintendent tells me to work with teachers who want to work with me. So I do.
In the beginning of the year, I had two high school math teachers ask me if I would do Number Talks with their classes. I was thrilled. First, I gave them both a copy of the book, Making Number Talks Matter by Cathy Humphreys and Ruth Parker. (If you don’t have this book, go get it.) Then, I added both classes to my calendar and told them I would be there, once a week, for the rest of the year.
I didn’t have a pre-planning meeting with them. I didn’t write down formal goals. I probably could have. Should have? The reality is I don’t always make the time to follow elaborate coaching protocols. I am not saying that I shouldn’t. I probably should, but I don’t. My number one priority is getting into a classroom, as quickly and regularly as I can. My number two priority is getting invited back. All the formal protocols in the world aren’t (necessarily) going to help me build relationships, BUT building relationships might help me use coaching protocols more meaningfully.
Ms. S and Ms. K, were really excited about me doing Number Talks in Algebra II and Transitions to Algebra. They talked about wanting to build their student’s number sense and get them to have more meaningful math conversations. Those sounded like great goals to me. My unofficial goal for the first few Number Talks was to cultivate a space where the guiding principles for Number Talks could bubble up. In chapter three of Making Number Talks Matter, Cathy and Ruth introduce ten guiding principles:
- All students have mathematical ideas worth listening to and our job as teachers is to help students learn to develop and express these ideas clearly.
- Through our questions, we seek to understand student’s thinking.
- We encourage students to explain their thinking conceptually rather than procedurally.
- Mistakes provide opportunities to look at ideas that might not otherwise be considered.
- While efficiency is a goal, we recognize that whether or not a strategy is efficient lies in the thinking and understanding of each individual learner.
- We seek to create a learning environment where all students feel safe sharing their mathematical ideas.
- One of our most important goals is to help students develop social and mathematical agency.
- Mathematical understandings develop over time.
- Confusion and struggle are natural, necessary, and even desirable parts of learning mathematics.
The first couple of Number Talks I did in both classes were kind of a hodgepodge. I was trying a bunch of things, looking for the “sweet spot” of just enough disequilibrium to prompt some spontaneous questions, revisions, and “wait… what?” moments. I was less interested in the content of the Number Talk. I was cultivating the process. I started with dots and subtraction.
I saw all kinds of strategies from counting dots one at a time to subitizing. Most kids said they used the algorithm for the subtraction problems. A few subtracted too many and adjusted. I thought one student used constant difference, but it turned it I was just projecting my thinking onto her strategy. Many of the kids seemed to feel comfortable sharing their ideas. Some were really open about changing their thinking.
The next time I came in, I used only dots and a Number Talk Image. I wondered if the images would “nudge students beyond the algorithms.”
The highlight of this class was when I heard one student say, “I used ________’s method. I added dots in the corners, multiplied, and then subtracted the dots that I added.” I hadn’t officially named the strategies. I hadn’t even “officially” encouraged them to try each other’s strategies. They just thought “R’s strategy was cool.”
Cathy and Ruth talk about the importance of “helping students develop social and mathematical agency”. They say, “Students with a sense of agency recognize that they are an important part of an intellectual community in the classroom; that they have worthwhile ideas to contribute, and that they learn from considering, and building on, the ideas of others.”
For the next Number Talk, I decided to try some multiplication. I know, I am all over the place. Remember, I am just poking around right now. I am trying to see what these students are willing to share, what they know, and what they are not sure of.
A lot of the strategies used during this Number Talk were based in addition and many students struggled to figure out why their partially correct approaches were not working. There was a whole lot of talk happening, which is why it went way beyond 15 minutes. It might have even lasted 30 minutes. Don’t call the Number Talk police, yet! These kids – all of them – were so present and invested.
When it came time to discuss 35 x 24, they were talking over me and each other.
They were arguing.
There was one girl, who always sits outside of the group, by the wall. She hadn’t said anything in a while – maybe not ever. I heard her whisper, “I don’t think you can do that.”
I barely heard her because two other students were going back and forth about whether you should multiply 35 x 2 or 35 x 20.
I had to raise my voice a little, ” Hold on! You need to be respectful of each other’s ideas. I want to hear all your thoughts, but you need to be respectful and I can’t hear what _____ is saying.” They listened! I swear to you that they actually listened to me, and there is no doubt in my mind that the only reason they listened to me was because I said I couldn’t hear their classmate.
It was silent. I was super nervous, but I asked anyway, “Do you mind saying more? You don’t have to, but I would love to hear your thoughts.”
She said, “I am just not sure if you can do that. I don’t know the answer, but I am not sure you can just add the four like that.”
“Do you think we should multiply 35 by 20 or 2?”
“I think 20 because it is twenty four, but I don’t think we can add the 4 after that. It doesn’t seem right.”
“Yeah,” the boy said – the one who originally suggested an answer of 704 – “I don’t think it makes sense either, but I am not sure what else to do with it.”
It was silent. I thought about drawing an area model, but I just couldn’t bring myself to do it. I didn’t think it would mean anything to them. I am not saying I will never show them an area model, but I just couldn’t commit to it at that moment. Instead, I waited.
A different girl said, “I got 840, but I think it is wrong.”
I asked, “how did you get it?”
“I did it on my whiteboard. 4 times 5 is 20, carry the 2, 4 times 3 is twelve, plus 2 is 14, 2 times 5 is ten, carry the 1, 2 times 3 is 6 plus 1 is 7. 140 plus 70 is 840.”
I was kind of bummed. Where did she get a whiteboard? It must have been in her desk.
At this point, I remember feeling completely overwhelmed and exhilarated at the same time. How can I get them to see the groups of 35? I decided to show them what they already knew. We went back over the other problems in the string.
“We know that 35 x 20 is 700. We talked about how multiplication can mean ‘groups of’. We can see the twenty groups of 35. How many more groups of 35 do we need?”
Someone said, “I think we need to do 35 times 4, not just plus 4.”
I asked, “what do other people think?” I really wanted to scream, “YES! You are so right! We definitely need to multiply the 35 by 4!” I didn’t.
“I don’t know,” somebody said. “It seems like we could add the 4, but then it doesn’t.”
I was exhausted. They were exhausted. I didn’t know what to do. Should I leave them hanging? Should I draw an area model? Should I add 35 twenty-four times?
I think I wrote the partial products on the board and asked, “Why does this makes sense?”
I think at least one or two kids were able to articulate that we needed to multiply 35 times 4. I am pretty sure one of them referenced 35 x 20 being 700 so 35 x 24 had to more than 704. There were definitely more than a few kids who were still unsure, but they looked like they would take our word for it.
One boy said, “that was so hard. My brain hurts.”
“Can we do one more?”
I laughed. I said, “No. I want you to want me to come back. Everybody looks pretty fried right now.”
On my way out, I said, “I think multiplication is a good place for us to spend some time. Do you mind if I bring another multiplication string next time?”
They nodded. I think one of them even thanked me for coming. Their teacher, Ms. S, said, “that was so cool. I am learning so much from these Number Talks. I didn’t learn how to do math like this when I was in school.”
I have read several books about math coaching and I have found them helpful. However, my favorite way to think about math coaching is through the lens of a teacher. Often, when I read books about teaching math, I replace the word “students” with “teachers” and I find provocative advice for myself:
“Teachers with a sense of agency recognize that they are an important part of an intellectual community in the classroom; that they have worthwhile ideas to contribute, and that they learn from considering, and building on, the ideas of others.”
So, at this point, I have done about four Number Talks in each high school math class. Tonight, I emailed Ms. S and Ms. K. I asked them,
- What have you noticed?
- What do you wonder?
I hope they mention the guiding principles, in their own words, of course.