How to Teach Math

Teaching math well will lead to improvement, says an instructional coach—but that’s no small job

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In this letter:

• What a ‘guided inquiry’ approach to math looks like in classrooms

• Why high-quality instructional materials for math could backfire

• Why teacher training often doesn’t prepare teachers for teaching math

• The missing info in conversations about math fact fluency

“I think right now we're just whiplashing back and forth between the concepts and the procedures, and neither one is happening.” 

Over the past few weeks, I’ve been having a series of conversations with Neily Boyd, an independent math consultant and instructional coach based here in Nashville. I interviewed her earlier this spring about her Instagram handle Counting with Kids—but that’s only part of her work on math teaching and learning. She has taken her knowledge as former director of STEM for the KIPP Nashville charter schools, where she built a math program that outperformed both the district and state math achievement, and is now helping other schools build their math programs in a similar way. 

I wanted to talk with her because she’s in a unique position to explain what’s happening in math classrooms—she’s sort of at ground zero between the research, the curriculum, the schools and the teachers, since she works with all of them to help train and coach teachers. Her insight and outlook is based on where the rubber meets the road: how teachers look at the curriculum they’ve been handed and try to figure out how to teach it to the students they have. 

This is a piece of one of our conversations, in which we talk about the ‘guided inquiry’ method of instruction. She strongly believes, a lot like mathematician and JUMP Math founder John Mighton, that it’s the way to teach math so students can build a strong conceptual and procedural foundation that go hand-in-hand as they progress. But, as you’ll see in this conversation, doing this kind of instruction well is often not an easy thing for teachers to do. It requires a lot of skill and judgment, and we often don’t set teachers up so they can easily access the tools they need.

We talk in this conversation about some math teaching at the Brooke Charter School, a network of four charter schools in Boston, and Chelsea, Massachusetts, because Boyd thinks they are doing a particularly excellent job teaching math. (Check out the math scores for their mostly low-income students, they are consistently way above the state average. Student scores also grow over time, unlike most school districts). Luckily for us, they have a pretty robust video library of how they are teaching math at different grade levels, so we can take a look at some examples of what they’re doing. 

I’ve linked the videos within the interview so you can watch them and fully understand what we’re talking about.

I think you are going to get so much out of this interview on what it’s really like teaching math, and teaching teachers to teach it well. It’s a long but worthwhile read. Let’s get started. 

The Bell Ringer: So let’s start with this fifth grade math class video from Brooke, they are learning a skill called ‘fraction of a set,’ and the teacher is using what they are calling “productive struggle” to try to get them to figure out the next level of problem. 

[Watch the video here.] 

First the teacher reviewed a problem from the day before, where the division of a set worked out evenly, to ensure all students understood the necessary prior knowledge. But then she moves on to a slightly harder problem, dividing fractions where it doesn’t work out evenly, and she wants them to reason their way to the answer. But importantly, she doesn’t just send them off on a quest—It looks to me like she’s doing what John Mighton would probably call guided inquiry. And this kind of “productive struggle” that is so heavily guided by the teacher looks like what the science of learning folks might call “effortful thinking.” 

Neily Boyd: So many conversations on productive struggle just stop before they get anywhere, because people are not using the same definition for the same term. I’ve been in classes when the students didn’t have the prior knowledge to access the task in front of them and the teacher said, “I need to let them do this, this is productive struggle.” And I in that case I would say, ‘No, this is not productive struggle, I would call it unproductive struggle, this isn’t going to produce anything.’ I often talk to teachers about, what does a struggle need to actually be productive? It needs a path forward. 

With the fifth grade lesson in the video, look at the level of intentionality that you see there. Their warm up problem was one that activated the knowledge from yesterday. It works out evenly, right?

The Bell Ringer: Rosenshine would love that. 

Neily Boyd: [laughs] Exactly right. 

The Bell Ringer: There are two things I thought about as I was watching. First of all, she has carefully planned out every single step of teaching this. Number two, the kids are fluent with how to divide quickly in their head and all that stuff. 

Neily Boyd: The teacher makes sure they've gone over it as a class, that everybody understands this warm up problem first. Now that's right there in your brain, all we do is add one new step to it— but you have to have enough procedural fluency with calculations, with fractions, to start to make sense of the problem. 

Did you see how fast students went from one-fourth to .25? One kid said ‘You split it into four, because I need four groups. We take this one whole chocolate bar, we split it into four, so then each part is .25.’

It’s important to note that a lot of the work in the lesson can't be done if kids are struggling to divide right in their head, or they don't understand how all the numbers fit together. 

The other piece here is that Brooke has their own curriculum, they have built their own inquiry-based curriculum, following exactly this process. When I was at KIPP, I met with teachers in every grade level for the upcoming week, we would look at every ‘opening task’ much like what you just watched in that fifth grade lesson. We would pull it up, and teachers would sometimes say, ‘I'm worried that my students are going to get stuck on this—we're trying to get them here, but I don't think anybody's going to give this answer which we need to get here.’ And so we would—in real time, in that meeting—edit the task so students could reach it. Sometimes it would be written as one problem, but sometimes we’d split the problem into parts,  and scaffold them through it a little more. Sometimes we’d decide the task was totally wrong for this group of kids and we’d need to give them something different. Our goal was to make sure the opening task was accessible to students so that we could successfully guide the conversation to the key takeaway in the lesson.

And I don’t think that’s the common practice in most schools using scripted inquiry curricula. Scripted inquiry curriculum should be an oxymoron. How can I write a script for somebody else in a classroom I've never seen?

But many of the inquiry curricula that schools are using is one-size-fits-all inquiry and that's not a thing.