Math teachers disagree about mistakes. Just check out the comments on this blog post from Dan Meyer. Some folks argue that students don’t make mistakes, they just sometimes answer a different question than the question the teacher was asking. Others argue that mistakes entrench unhelpful ideas in students’ memories, and that mistakes are the enemy of learning.

Here’s a bit of thinking I’ve seen a few times in my precalc class recently, in a unit on graphing sine and cosines functions. We’re early in the unit, and a student is writing an equation for a sine function, maybe like this one:

She writes the period as pi over 2, reasoning that the period is how long a function takes to repeat, and the y value is the same at both 0 and pi over 2.

One path forward is to label this thinking a mistake and explain the “correct” way of finding the period of a sinusoidal function. But that path serves to invalidate the student’s knowledge, and pass up on an opportunity to help them connect what they already know to what I would like them to know. A second path is to help the student see the valuable thinking they did, offer another example, and link their thinking to the missing pieces. For instance, using the same reasoning but focusing on either the maxima or minima of the function, one gets a different answer. In this situation, if I have an instinct to label student thinking a mistake, I miss an opportunity to build off of what they know and help them see themselves as effective mathematicians. If I choose to see their strengths and find the question they are actually answering, I have an opportunity to validate their mathematical thinking and expand what they know, rather than trying to replace something I’m labeling as a misconception with the “correct” way of doing it.

But there are actually two different places I see this mistake. One is early in the unit, when students are first trying problems and applying their knowledge. The other is a few days later, practicing the concept to consolidate their understanding and solidifying their fluency. And I think those two places require different approaches. When I’m introducing a topic, I want to find every way I can to build off of students’ prior knowledge, to help students feel successful with a new concept, and to give students a sense of agency in their ability to make sense of mathematical ideas. These are important opportunities to look at mistakes as opportunities for learning and draw out the valuable ideas even when student thinking falls short of where I want them to be.

During practice, I fall on the other side. Practice is where students make associations and consolidate their understanding. Practicing the wrong way just leads to confusion and frustration. This doesn’t mean I want to shame students making mistakes during practice; I still want to connect what they know to where I want them to get. But I’ve spent my entire life confusing affect and effect; I’ve used them wrong as often as I’ve used them right, and even when I look up the distinction I forget it in a morass of incorrect usage. This confusion be really frustrating for students; I don’t want them to feel like I let them practice something the wrong way and confused them when I had the chance to correct them earlier. Even where the “mistake” is a matter of arbitrary convention, when students feel wrong they often get frustrated and disengage with math class. I’d love to celebrate mistakes and help students see mistakes as an essential step on the path to understanding, but I also want to make sure that those mistakes are authentic opportunities to build understanding and not undermining future success for students who already struggle to feel successful in math class.

There are two psychological phenomena relevant to mistakes. The first is the generation effect. Having someone guess an answer before learning it improves the quality of learning. Even if the learner generates an incorrect answer, as long as they get quick feedback on that answer they are likely to learn more than if they were just told the information to begin with. From this perspective, asking students to try to figure things out and then giving corrective feedback as necessary is an effective teaching strategy. Those mistakes are valuable; they’re both authentic to the practice of mathematics and improve the quality of learning with immediate feedback.

Second, the new theory of disuse suggests that we don’t ever really forget things, they just become less and less accessible. Even if I can’t remember my friend’s phone number from twenty years ago, if I am reminded of it I will retain it much better than if I had never learned it. Anything students learn incorrectly can never be replaced, we can just try to make the new learning more readily accessible. Mistakes from this perspective are permanently damaging. I think of my struggle to remember the difference between affect and effect. I have used them incorrectly so many times that I struggle to remember their correct usages; the wrong usage is embedded just as well as the correct one. Making a mistake once will be much less accessible once students have had a chance to practice the concept, but making a mistake over and over again can start a cycle of confusion and frustration.

I think I have a responsibility to help students see the ways they can be successful in math class, and to see mistakes as opportunities for learning. But all that nice talk about how much we can learn from mistakes feels limited to the first case, where it’s early in the learning process and there’s lots of time to practice and consolidate ideas. When students are practicing concepts incorrectly, I still want to help them feel like their thinking is valuable, but I also need to be proactive in catching and changing their practice before they start a cycle of negative feelings and confusion. And in lots of situations, this is a blurry line! There’s not always an easy answer.

Moral of the story: I think mistakes are great, but there are important boundary conditions that help me understand when mistakes are helpful and when they might be counterproductive.

Popsicle Sticks and Accountability

When I started teaching, I was told that cold calling was important because taking hands selects only confident students, and I might think a class understands a concept when the silent majority are all confused.

I think there are lots of other ways to avoid selection bias. I can give exit tickets or mid-class hinge questions, circulate to look at student work or listen in on student conversations, use whiteboards to quickly see answers to a particular question, and more broadly cultivate an environment where students check their own understanding and feel empowered to speak up and ask questions. Cold calling is a bit of a blunt instrument; it samples randomly, but only one student at a time, and can still misrepresent where the class is.

But my core issue with cold calling was the shame certain students — and it was always the same students — felt when they were called on and didn’t know what to say. Some students bring negative experiences to math class, and putting them on the spot is likely to entrench negative feelings toward math and threaten the social safety of mathematical risk-taking and idea-sharing.

At the same time, I want to create a sense of accountability in my class. I don’t mean accountability in the sense of punishing students when they don’t participate. For me, accountability is creating an environment where students know I care about their learning, I will make sure that every student is set up to succeed, and I follow through to see if they’ve learned and do something about it if they haven’t.

In Ilana Horn’s book Motivated, she describes three norms of participation that can help to create a sense of accountability:

  • Everyone participates
  • Listening matters
  • The focus is on mathematical ideas

I’ve started to use popsicle sticks to cold call students again, but rather than trying to see which students know or don’t know a certain answer, my focus is on creating a sense of accountability and reinforcing these norms. I like popsicle sticks because they are visibly random — students don’t feel like I’m picking on them or trying to catch them not paying attention. And I use them only in a few specific places where there is no right or wrong answer, but instead ask students to share a mathematical idea with the class:

  • After students attempt a problem in groups, or reflect on an idea and share with partners, I call on students asking, “How did your group approach the problem?” or “What is something useful that you or your partner shared?”
  • After looking at a mathematical prompt, for instance a Connecting Representations routine, but with pencils down and before solving, I ask students, “What did you notice that might be mathematically important?”
  • After reflecting on a situation and making estimates of answers to a problem, I ask students for their estimates.

In each of these cases, I’m not looking for a right or wrong answer, but for students to share ideas and approaches. Every student is expected to participate. And in particular when I ask students what they talked about with their partner or group, the message is that they are expected to listen to each other’s ideas.

More broadly, while popsicle sticks are something my students groan about at times, I’ve found that using them in a few specific places helps to create an environment where students know they are expected to participate and that I care what every student thinks, without shaming students for not knowing answers.

On “Low” Students

Here’s a version of an exchange I’ve seen on Twitter more and more often:

Person 1: “Help! I’m struggling to engage my low class with rich tasks. How can I get these students to be willing to try harder?”

Person 2: “Respectfully, I wonder about your use of the word “low.” What makes those students low? What assumptions might you have about those students? How might expectations play a role in the challenge you’re describing?”

These are really thought-provoking conversations! How we talk about students matters — do we talk about students in ways that reinforce stereotypes, or in ways that remind us to seek out every student’s brilliance?

Reading these conversations reminded me of something in Rochelle Gutiérrez’s article “Strategies for Creative Insubordination in Mathematics Teaching.” Gutiérrez references that many of the teachers she works with teach Black, Latin@, historically looted, and/or emergent bilingual students, with this footnote:

Thinking again of the term “low” students, I love Gutiérrez’s term “historically looted.” In other contexts, someone might describe those same students as of low socioeconomic status, underprivileged, lower-class, or other euphemisms for poverty. All of those terms are fundamentally passive; they say that this group of students is lacking access to economic resources, but do not ask why. They are also loaded with assumptions about “those students” and “those schools” that many people bring to conversations about education.

“Historically looted” chooses a different emphasis. If these students lack economic resources, it is not because of any fault or deficit of their own, but because they live in a country that has chosen to extract resources from some at the expense of others, and to allow poverty to exist despite the fantastic wealth of the ruling class.

Here’s the thing. No matter what language we use, it’s loaded with assumptions. We can choose language that is passive, that lets those assumptions go unchecked. Or we can make the assumption that students’ challenges are because of a system that has chosen not to support them. Martin Luther King wrote in his Letter form a Birmingham Jail:

So the question is not whether we will be extremists, but what kind of extremists we will be. Will we be extremists for hate or for love? Will we be extremists for the preservation of injustice or for the extension of justice?

It might be radical to call a group of students “historically looted.” But maybe being radical is a way to start conversations, question assumptions, and begin to see students from a new perspective.

So how might one describe those formerly referred to as “low” students? I’m not sure. Students whose brilliance we haven’t yet learned to see? Students whose talents aren’t valued in our education system? Students who we have taught not to take risks in school? Students who haven’t received the quality of education they deserve?

Any of these phrases and many others check assumptions at the door, and just might start a conversation that helps to reset expectations and see the challenges of teaching from a new perspective. And as I talk about students in new ways, I find myself acting differently, and acting myself into new ways of being and seeing students and schools.

Reflections on Discovery

I’ve gone back and forth on this more times than I can count.

I think that pushing students to figure out large portions of the high school math curriculum for themselves, even with some structured guidance, doesn’t work very well in practice. Discovery can increase inequities as students who have strong background knowledge succeed and those who don’t struggle. It can exacerbate feelings of frustration toward math when students feel unsuccessful over and over again. It’s easy for a few students (or a lot of students) to fall through the cracks and miss key ideas. And asking students to figure things out for themselves takes a ton of time, and I think there are often better ways to spend my time. If discovery edges out time for students to practice and apply what they’ve learned, all that time they spent exploring becomes pretty worthless as what they discovered floats away without reinforcement.

However, I do value discovery in other ways. I think every student should experience mathematical discovery at least a few times a year, and some topics lend themselves to this well enough that my reservations can be put aside. I really love the Binomial Theorem, and once students have solid background with calculating combinations and multiplying polynomials, a structured exploration of the intersections between combinations, binomial expansions, and Pascal’s triangle can be a ton of fun. Figuring out new ideas is an essential part of the practice of mathematics, and that’s an experience I want all students to have.

I also think it’s often helpful to have students try to figure something out to see how well they apply their prior knowledge and identify where I might need to provide some extra support. I like to begin a unit on arithmetic series by telling students about how Gauss would finish his work early in elementary school. One day his teacher, to keep him busy, asked Gauss to add up all the numbers from 1 to 100. The teacher was amazed when Gauss found the sum in seconds. How did he do it? With this prompt, if some students can figure out how to sum an arithmetic series, awesome! I can try to spread those ideas through discussion and group work. More likely, I’ll lead some summarization and explicit connection and move into a practice or extension activity. I’ve also learned it’s important not to do this every day; if I’m always asking students to figure something out but I’ll explain it to them after 10 minutes whether they get it or not, they figure out the game and are much less likely to engage.

There’s some fun middle ground in a lot of situations as well. Recently I was beginning a unit on function transformations in Algebra II. I started by asking students to sketch rough graphs of a bunch of quadratics in vertex form — something they were pretty rusty on, but able to remind themselves of in small groups. After spending time playing with quadratics, we summarized the rules for the different types of transformations. This was a great transition into more abstract function transformations, making the connection between their prior knowledge and our next unit explicit. I was the one introducing new ideas, but students were still exploring to start the lesson and taking some ownership of their learning.

It’s easy to treat a discovery lesson as some big monolithic thing, but my choices depend on the content, students, and my broader goals and the time I have. Here are my core principles:

  • Every student should experience mathematical discovery at least a few times a year to participate authentically in the practice of mathematics
  • Beginning a lesson by asking students what they already know about a topic is a great way to get a sense of where they need support and to activate background knowledge
  • Making concepts and connections explicit is an important practice to prevent exacerbating inequities that already exist
  • Everything depends on context
  • The central activity of math class should be students doing math; if students are spending all their time trying to figure out new things, they’re not spending enough time applying what they already know

Research and the Messiness of Classroom Teaching

I’m interested right now in learning more about what it means to hold an asset orientation as a teacher. My working understanding is that when teachers hold an asset orientation, they frame everyday interactions by looking for strengths, valuing student thinking and assigning competence. There’s a ton to read on the topic; I tweeted and got a great reading list from many folks in response, and a recent issue of Teachers College Record focused largely on asset-oriented and similar approaches to teaching and teacher education.

Asset-oriented teaching is a fascinating body of research to study, and I’ve been interested in learning more about research in education for a long time. I think I made a mistake for a while thinking that an asset orientation isn’t “real” research. It’s not a variable a researcher can control in a lab. But teachers make hundreds of decisions a day; an asset orientation helps to understand where those decisions come from, and the impact they have on students. I see an asset orientation as a problem frame, as Horn, Kane, & Wilson write:

This approach contrasts with much of what I see shared as evidence-based teaching. I see folks sharing about spaced practice, or interleaving, or dual coding. And these seem really valuable! But they are typically focused on inputs: what does the teacher do, how is the curriculum designed, what questions are asked. An asset orientation is focused on how teachers respond: what to do when a student is stuck, or shares a misconception, or has a conflict with another student.

I’ve found Daniel Willingham’s comparison to architecture helpful in the past:

Architects, like teachers, usually have multiple goals they try to satisfy simultaneously. Safety is nonnegotiable, but architects may also be thinking to a greater or lesser extent about energy efficiency, aesthetics, functionality, and so on. In the same way, some goals for teachers are nonnegotiable — teaching kids to read, for example — but after that, the goals are likely to vary with the context. In addition, architects make use of scientific knowledge, notably principles of physics, and materials science. But this knowledge is certainly not prescriptive. It doesn’t tell the architect what a building must look like. Rather, it sets boundary conditions for construction to ensure that the building will not fall down, that the floors can support sufficient weight, and so on.

In the same way, basic scientific knowledge about how kids learn, about how they interact, about how they respond to discipline — this knowledge ought to be seen as a boundary condition for teachers and parents, meaning that this knowledge sets boundaries that, if crossed, increase the probability of bad outcomes. Within these broad boundaries, parents and teachers pursue their goals.

-When Can You Trust the Experts? (p. 221)

But there’s an issue with the comparison between teaching and architecture. When an architect designs a building, odds are pretty good the building will stand and serve its basic purpose. Yet in my class, every year, I fail. Students leave math class with profound negative feelings toward math. They can’t graph a polynomial, they forget the meaning of expected value, or they reverse the rules for function transformations. Education isn’t like medicine, where once someone is diagnosed with strep throat a doctor can be fairly confident in the success of the treatment. Teachers need to respond, in the moment, every day, when things go wrong — and not for one patient, but for a classroom of students. I’m interested in research that accounts for the uniqueness of humans, for the idiosyncrasies of young people, for the vagaries of learning and memory. Yet research often gravitates toward what is easiest to measure, or what can be distilled into a randomized controlled trial. I see this research as valuable, but also as a bit sterile and sometimes distant from the realities of classrooms.

What does a vision of research look like that’s responsive to the experiences, struggles, and individualities of all the humans in a classroom? What does research look like that honors the messiness of young humans, of learning, of everything that’s happening in classrooms each day, the in-the-moment decisions, the emotions, and the identities of the individuals involved?

I have a lot more to learn, but I think the shift I’m making right now is framing my problem differently. My problem is that I want to learn more about research on teaching and learning. But in trying to meet that goal, I have a natural bias for easy and concrete solutions. I wonder if diving into research that offers fewer easy solutions and more tensions to navigate and is more authentic to the discipline of teaching and the challenges of classrooms. So my problem is not to find solutions, but to find harder questions that get at the messiness of classroom teaching.

Making Things Explicit

I teach students to do polynomial division using the box method, which I originally learned about from Anna Vance. I’ve found students typically really enjoy the box method once they get the hang of it, and it reinforces their understanding of multiplication at the same time. But the initial stages of teaching the topic have always been hard and result in a lot of student confusion and frustration. Here are a series of puzzles I designed to try to improve the transition for students:

For those curious, here is a doc with the images and a template for making more, and I create the smudges by uploading images to Google Keep.

Pretty clever, right? I found students work their way through the puzzles without too much guidance, applying what they know about multiplication and gradually working up to problems where they’re doing polynomial division. Then, I say “hey, you don’t even realize it, but you’re doing polynomial division! Isn’t that cool!” Then I give them some problems for practice, and they’ve magically learned a new topic.

Only it didn’t work out so smoothly. Students could work through the puzzles, but the transition from puzzles to a division problem stumped them. The connection was totally unclear. Some students given a polynomial division problem just multiplied the two polynomials and built bad habits before I realized what was happening. I ended up having to completely reteach the topic, and to work through some well-justified frustration along the way.

So I took the same puzzles, but used a different approach. After each problem, we paused and summarized what was happening. What patterns are we seeing, and why are they there? Why are like terms on diagonals? How is that useful? Which expressions are being multiplied? What is the corresponding division sentence that must be true? How would you write that division sentence? How does that reasoning apply to this next problem? What’s a different way to write this problem?

Then, on the transition to polynomial division problems without the scaffolding of the puzzle, I modeled what the first steps of the setup look like. Suddenly life was great, polynomial division was easy, and the lesson breezed by.

I think there’s a really important bit of learning in the difference between these two approaches. I love finding ways to reframe content as a puzzle. It gets students curious and it reflects the discipline of mathematics. But there’s also something significant happening cognitively when students see a topic as a puzzle, rather than starting a lesson with SWBAT polynomial division. Starting with an explanation creates all kinds of potential confusion. Students bring in their associations with polynomials, with division, with negative experiences learning a new and abstract concept. Framing the topic as a puzzle helps students to zoom in on the nitty-gritty, get some muscle memory with the basic elements of the procedure, activate background knowledge, and build confidence. But that transition from a fun puzzle to the abstractions I want students to take with them needs to be explicit in the lesson; if I leave those connections to chance, students’ learning will never get linked to the contexts I want them to apply it in, living in their minds as a fun but isolated puzzle.

A similar topic came up at PCMI last summer. We were talking about Smudged Math, and one participant shared frustration that it often seems like teachers can find a million ways to help students to do algebra but feel like they’re not doing algebra, without ever actually linking that learning to algebra. I think that’s dead on. It’s often fashionable in math teaching to talk about how students figured something out entirely on their own, or to tell students they’ve been applying a particular concept without realizing it. I see these as fantastic pedagogical opportunities, but only when we make clear exactly how what students are doing connects to new and more abstract math we also want them to do. As teachers we see the bigger picture, but students, and any novices learning a topic, struggle to step back and see the forest for the trees, getting lost in the specifics of a particular question or task. I think the crucial pedagogical move here is to make those connections explicit for students rather than leaving them to chance. Otherwise, it’s just clever packaging for a lesson students enjoy but don’t learn anything they can apply in the future.

Little Things

I used to have a value in my teaching that students should figure out as much of the math as possible on their own. “Never say anything a kid can say” or something like that.

That’s not a value of mine anymore. Instead, my value is that students should spend as much time as possible doing math. Doing math might look like figuring out something new, but it might also look like applying what they already know in new ways to practice, consolidate, and extend their understanding. I’m unafraid to tell students things if it means more time for doing math.

The issue with telling students things is it works best in small chunks. The longer I’m talking and the more new ideas I’m introducing, the harder it is for students to follow along.

In class yesterday I was working on conic sections in Precalc, and I wanted to introduce graphing circles. First I wanted students to get their heads around simple graphs of circles centered at the origin, like

Then I wanted them to translate those circles around the plane, graphing equations like

Students have seen similar transformations before, but likely in a different form, working with equations written as a function of x. The way x is transformed might be familiar, but doing the same thing for y is new. I could do some explaining to fill in the gap, but that can lead to a mix of blank stares and questions that lead us down confusing rabbit holes. This is tricky; I’m building off of students’ prior knowledge, but it’s a totally different structure than what they’re used to, so those connections are unlikely to be clear. And there’s nothing worse than telling students they should know something that feels confusing and counterintuitive to them. How can I bridge that gap?

I started class by giving students a set of parabolas to graph in groups:

There were a few false starts, but the ideas came back quickly. After graphing a few of these, we were able to discuss and summarize the core property of these transformations: subtracting something from one variable translates the function in the positive direction, and adding translates the function in the negative direction. Counterintuitive, but consistent with what students have seen before — and suddenly this rule holds for translations in both the x direction and the y direction.

With this knowledge explicit, translating circles is a cinch. I can introduce the standard form of a circle and get students practicing quickly, and the transformations feel like something that builds off of what they already know, rather than a mysterious new idea. My favorite part is that the introduction to transformations with parabolas probably cut the time of my explanation by half, if not more. I got to start class with students doing math, and we had more time to build off of that knowledge and solve harder problems with circles at the end of class as well.