Unlocked: How to Teach Math

I wanted everyone to read this previous post for paid subscribers: Teaching math well will lead to improvement, says an instructional coach—but that’s no small job

Hey there, readers! Here’s a post that was previously for paid subscribers only, now free for everyone. It’s a long conversation about math instructional practices, what ‘guided inquiry’ learning looks like (and why so many charter schools are seeing success with it), and why teaching math well is so…hard. Looking forward to hearing your thoughts! If you’re interested in getting more out of The Bell Ringer with posts like this one—all the paid posts, access to the full archive, upcoming interviews and talks with some of the science of learning’s most important people—you can become a paid subscriber here.

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 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.

The least surprising thing I've ever heard is that a one-size-fits-all inquiry curriculum isn't working—when you think about what you watched in that fifth grade video, she is so intentional about who her students are. 

Schools are often getting this curriculum that says it's an inquiry-based curriculum. They open it up, they do the lesson that's on the page, and there's no asking, ‘Do my students have the foundational knowledge for this? How did yesterday's lesson go?’ Because the entire thing is dependent on yesterday's lesson. So if that didn't go great, it's gonna go even worse today. 

The Bell Ringer: This was great to watch, but I’m also wondering if It's really hard to teach math well when you do it this way. The teachers have to understand the concepts of the math. Is there some massive teacher training project that needs to happen? 

Neily Boyd: The first challenge is the level of teacher content knowledge that you have to have about the concept and about how the concept builds. It’s extremely hard, because most people who are currently teaching math were not taught math that way, and so you're being asked to teach a concept and at best, what you remember is the procedure for how to do it.

The Bell Ringer: So what are the obstacles to coming into schools and training teachers in this way?

Neily Boyd: The biggest challenge for a teacher is starting where your students are, because that’s not always the same place where your scripted curriculum starts. 

You have to start where your student knowledge starts. And so if the curriculum is on step four of this concept, let’s say calculating fractions of a set, but your students need to be on step one of that concept, you can't just put the task in front of them as it's written in the curriculum. 

But lots of teachers are getting opposite advice to that right now. The advice they're getting is to teach the grade level standards at the grade level the way the curriculum is written. And I think the spirit behind that is good—we don’t want to set the bar too low for kids. 

But if you start too many steps ahead of where their knowledge is, math is so hierarchical, it won't happen. If you don't activate the last thing they truly understand and then build from there, you're not going to get anywhere.  

That raises the second challenge. As a teacher, I have to understand this concept well enough to say, even though the sixth grade portion of this concept is only this tiny piece—I've got to know what it looked like in fifth grade and fourth grade, so that if fourth grade is really where they're starting, I can start it there. This is not easy to do. 

The Bell Ringer: A fourth grade math teacher I spoke with recently said the kids who have come to her this year are way behind in math. She’s a good, experienced teacher, but she was saying, ‘Now I have to figure out, how do I keep the fourth grade curriculum moving and fill in all the gaps at the same time?’ 

Neily Boyd: When the gaps are as significant as they are, then you need teacher expertise in essentially planning and adapting their own lesson. You've got a set of materials from your scripted curriculum to start with, but you're not guaranteed that's the right fit, so a lot of teacher planning is still required. 

I'm very firm in my belief that, with the right training and coaching, and for students with the right teacher, this is the strongest method of teaching math I have ever seen. It's just that, if one of these challenges is in place and doesn't get addressed, the entire thing falls apart. 

This is what we do in education reform. We say, the problem is this one thing and now everybody has to change to this one thing. We didn't put any systems in place. We didn't change our professional development.

The Bell Ringer: Emily Hanford said she’s worried about this happening right now in reading.  States and districts have said, we have to ban 3-cueing, the evidence-based curriculum is the thing that needs fixing. But without the teachers understanding the science, it's not going to work right exactly, because kids are not copy/paste versions of each other. 

Neily Boyd: You still have to have the right knowledge of how to adapt. There is no version of teaching where it follows exactly the same process every single time. The more hands-off a teacher becomes, the more expertise it requires. I've never said that out loud before, but I think it’s really important to note. There's an inverse relationship between how teacher-directed the math lesson is, and how much teacher expertise you need to have. 

The Bell Ringer: Just talked with a parent the other day who had moved her kids to a different school, and she was just so relieved they were showing them how to do math problems. According to this mom, what her kids were doing before in some classes was watching a video and then doing a worksheet. So that's not inquiry teaching, and that's not direct instruction—that's actually not anything, there's no teaching.

Neily Boyd: So when several charter schools started doing this guided inquiry method of math instruction—not only Brooke, but Achievement First, DSST in Denver was doing something similar—people like me at other charter schools took notice, because it was working. But you can watch how, over the progression of a decade and as its spread to schools without robust teaching coaching structures, it has often become such a watered-down, ineffective version of inquiry. 

Teachers have to have enough judgment to translate the curriculum for themselves, depending on their students. And sometimes what happens is they might say, to be honest, ‘I don't know how to teach this, so I'm going to put this video on and kids will watch this,’ like the classroom you referenced.

This is a thing I genuinely struggle with all the time. I'm not sure that a first-year teacher should even be attempting inquiry teaching like this if they don’t have an instructional coach working with them on it daily. I don’t know how to balance what I have come to believe over the years is the right way to teach math with how teacher skill-set dependent it is. At what point does it stop being the right way to teach kids? 

The Bell Ringer: So this leads (as it often does) back to how teachers get trained. Shouldn't they be doing more of this kind of work in education school? 

Neily Boyd: Every teacher who teaches math should leave their teacher training program being able to say, in the grade I’m teaching, I teach the concept this way—even if that’s not a super-efficient way to solve this problem, but it’s essential to understand because in two years, they’ll have to do it in a more efficient way, and if they don’t understand it this less-efficient way now, they won’t understand it the other way later on. 

I have to believe that's not happening in teacher ed programs, because this isn’t how most math teachers talk about their curriculum. It’s essential for teachers to understand that while the strategy they teach in their grade level might be objectively inefficient, it’s been intentionally put in their grade level to build conceptual understanding for when students learn a standard algorithm in future years so that students will have this greater understanding of the pattern of the numbers. When teachers are able to talk about math this way, it’s typically something they’ve been trained on by a coach in their school or knowledge they’ve gained through resources they’ve acquired on their own.

Often teachers only see the little slice of math they’re teaching and they aren’t always getting training on the math concept of progressions, how concepts build from year to year.

The Bell Ringer: What happens in reality is more like: You were a reading teacher last year? Oh, sorry, you teach math this year. When would some teachers even get time to build expertise on anything this complex? 

Neily Boyd: Yes–and there are other things that happen. Like ideally a teacher who's taught 7th grade and then moves down to 5th grade might say, ‘Oh, I now see what’s going wrong, because in 7th grade they needed this.’ But the opposite happens without training. Without support, what that teacher might say is, ‘In 7th grade, they didn't know how to do this thing I needed them to do, so I'm going to teach them the trick faster in 5th grade.’ Instead of seeing why 5th graders have to draw pictures of fraction models over and over again—which is so they can do the algorithm in 6th grade, and by 7th grade it all makes sense—they’re skipping over important pieces. It becomes this messy, messy hodgepodge. 

I feel terrible for teachers in all of this. Teachers and students are both caught in this whiplash of ‘Do this curriculum, teach it this way,’ but nobody's actually giving them solid training or solid information. 

And then teachers will open the curriculum—the number of times I hear teachers say, ‘This is not right for my kids, my kids aren't ready for this’ is a lot. Their instinct is: I have this lesson I'm supposed to put in front of them, but they're not ready for it. And then we train teachers out of that instinct, right? By saying, just follow the scripting. 

What they need is for someone to say, ‘That's actually exactly right. What information are you pulling that's making it feel like your students aren't ready for this? Let's talk about how we could get them to this point, but on a different route, because they aren't going to follow this scripted pathway there.’ No one's teaching teachers how to do that. 

It's unfortunate because as a teacher, your instinct that this material I'm putting in front of my students isn't quite right, is a really important teaching skill. 

The Bell Ringer: Is this going to get worse because of the push for high quality instructional materials? Has there been an overreaction to the idea that we have to teach everyone at grade level? 

Neily Boyd: You can't just say, okay, fine, my students didn't solidify any of their math skills from second grade through fifth grade, but now I'm just going to put this sixth grade lesson in front of you, and it's going to go great. That is not real. And the number of times that I hear people higher up in curriculum-making decisions talk about it as if that’s all you have to do—put this sixth grade lesson in front of this sixth grade student, and everything will be fine. And I'm thinking, what? 

The Bell Ringer: Right, the students don’t have the prerequisites. We have to talk about the knowledge needed to do this more complex math. What is that background knowledge? What do students need to know by the time they get to fifth grade, that they are able to do fluently, so they can start to do this more conceptual stuff? 

Neily Boyd: A big one is understanding place value and our base 10 number system, and this pattern that everything in numbers and place value is built on repeating groups of 10. Like most concepts, you can fake it without full place value understanding when numbers are small, but especially when you start adding in numbers less than one, and all the decimal place values, if kids don't have strong place value knowledge, their ability to visualize numbers really suffers. 

Place value is more than ‘this is the tens place,’ etc. It’s really understanding, I have a 2 in the tens place means I have 2 tens, which is 10 times more than a 2 in the ones place, and 100 times more than a 2 in the tenths place. The entire number system is predictable, but kids don't really always understand that.

The Bell Ringer: If you don't have a good understanding of place value, what happens to you in fifth grade?

Neily Boyd: I think it hurts students’ ability to visualize a number and be able to look at a problem and say, ‘is that a reasonable answer?’ If I'm multiplying by a fourth, I need to understand that that's the same as dividing by four. But then I need to use my place value knowledge to understand, what's a reasonable answer if I'm dividing 250 by 4? It's not reasonable that my answer would still be three digits. When you get into more complex calculations, and they are just trying to hold all the steps in their working memory, knowing place value helps a lot. 

Fifth grade is a time where the number of procedures you've been taught, you can't hold all of that in your head, in your working memory anymore, right? If it hasn't connected to some larger understanding about our number system, you lose things. 

And then we haven't even talked about fractions. Fractions are just another version of decimals, but kids don't often understand that either. Multiplying by 0.25, multiplying by 1/4 and dividing by four are all going to give you the same answer. 

The Bell Ringer: This is getting back to my idea that really what we're talking about is students having a flexibility with numbers. And of course, obviously your long term memory and background knowledge plays a big role here, because without those addition and subtraction and multiplication facts on lockdown, this kind of mathematical thinking is going to be a stretch, right? 

Neily Boyd: Not only do you need to have the facts on lockdown, you also need to have all of the patterns that exist between those facts. 

When it comes to fact fluency, I always feel like conversations are focused on the last step, which is the kids have automated retrieval. But I feel like we are skipping over this middle piece, which is, to me, the most important piece—the reasoning strategies, or the decomposition strategies. It’s the piece between counting and automaticity. 

What I see happen in US schools, we go from counting strategies—which is to add 5 plus 7, put 7 in your brain and count 5 more, 8, 9, 10, 11, 12, —or you count all. Either way, you're counting to solve. You're not using another easier fact to solve. But often we go straight from, ‘I can figure out 5 plus 7 by counting all or counting on,’ to, ‘Now I should have this memorized and automatic.’

But what research shows (take a look here and here) is that middle phase—I've started to build some automaticity with really easy facts, like the plus 1 facts or doubles facts, and I can use those to get to the answer for the facts I have not memorized. It's faster than counting. The middle phase, then, is how you get to the automaticity, in a stronger way.

The Bell Ringer: In the book How Do We Learn? A Scientific Approach to Learning and Teaching, by Hector Ruiz Martin, who’s the director of the International Science Teaching Foundation, he writes about how background knowledge has to be meaningful. It has to be connected to other knowledge in order to be useful. 

Neily Boyd: Exactly. If a kid can tell me that 6 times 8 is 48, but he can't tell me that that's 8 more than 5 times 8, how useful is that when you're trying to find common denominators? Kids have to know the relationship between those two denominators. 

Simplifying fractions is another one where kids look at 12 and 36, for example, and they might be able to rattle off all of their facts, right? But if they don't think about the relationship between 12 and 36 they can't generate what’s the largest factor of both 12 and 36.

So here's what I think happens in lower grades—I've never read research on this, but I know it's true. The concepts in kindergarten, first grade, sometimes even into second grade, are simple enough, and our working memories are strong enough, typically, to just hold it all in our working memory from just pure rote memorization. 

Teachers who don't know the full scope, they only know what it is important for kids to know in kindergarten, or first grade, they look at what students are doing and say, they're getting the problems right. They have what they need. Then those students get into third grade, fourth grade, fifth grade, and they have to move between different representations of numbers, or they have to pull all of the factors that are in common between different numbers, or now you're adding more procedures, and their working memories are getting overloaded, and all of a sudden you realize everything that happened in kindergarten and first grade wasn't as strong. 

The Bell Ringer: Something similar happens in reading. It might be easier to memorize words in kindergarten and first grade, they’re two or three letters long. Then by third and fourth grade, the words get harder, students are no longer able to memorize all these words you're reading. You have to be able to sound out the words to know how to read. 

Neily Boyd: Unlike letter sounds, math facts work slightly differently—addition or subtraction facts aren’t the smallest unit of foundational skills you can have. There’s actually a step below that. In math, single digit addition, subtraction, multiplication and division facts are not the smallest skill. Addition and subtraction facts are the next step after you understand number relationships. That's the piece I feel that’s missing from the conversation about math facts right now. Two plus 2 equals 4, but it’s also 1 more than 2 plus one and 1 less than 3 plus 2, for example.Two plus 2 equals 4 is not a single, isolated piece of information. 

The Bell Ringer: What I want to understand from your point of view as an instructional coach, is how do we start to fix math teaching so more students can learn math? 

Neily Boyd: If I was a superintendent of a school district, what would I do right now? I can't change what's happening in teacher ed programs and I can't change the teacher pipeline that I have. I would focus our math resources on getting pre-K, K, first and second grade right. But we would do guided inquiry like Brooke, and I would train every person touching it, every principal who's observing it. I would make the problem so small that I could fully wrap my arms around it, and I would say: this is what we're going after. In order to get math in the upper grades right, we have to get foundational math right first.

The Bell Ringer: Tell me what would happen if you had your way. How would we be setting kids up differently than we are doing right now? 

Neily Boyd: I think we would be truly teaching them both the concepts and the procedures. And I think right now we're just like whiplashing back and forth between the two, and neither one is happening. 

It's hard because nobody wants to say that, right? Nobody wants to say this problem is so big that we can't immediately solve it across 13 years of schooling. 

If we're going to do it, we actually have to start talking about a smaller problem. The smaller problem is foundational skills. We start with kindergarteners and work for 3 years with those kids, and the model grows with them. And in 15 years we will… 

* [I crack up laughing here] 

I know! But I think that's what people are unwilling to do. People are unwilling to say, in 15 years, we will have solved this problem. That's a shocking amount of time to say it's going to take to fix something. 

But the alternative is, we just keep teaching math ineffectively, and we just keep swinging wildly back and forth every decade or so between models of teaching math that haven’t led to results.