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.

Math for Social Good

I participated in the #ClearTheAir chat last night, discussing Episode 11 of Hasan Kwame Jeffries’ podcast Teaching Hard History, on Slavery in the Supreme Court. It was a great discussion, and I’ve learned a lot of important and neglected history from the podcast in the past few weeks. But talking about the role of math teachers in helping students to better understand our country’s racist history and build a better future, I felt a bit lost. Many folks had great ideas last night, questioning who is represented in math classes, how we understand math as a marker of intelligence, and how we create access to math classes for more students and use those classes as opportunities for empowerment. And I’m excited to do more of this work in my classroom! But today I’m teaching about polynomials and sequences & series, and when I get deep into content lots of these questions feel distant from what’s actually happening each day in class.

What is math? Some would say it is a tool for understanding the world, or a set of skills in abstract thinking, or problem solving, or content necessary to gain access to future education. I think that math education can meet those goals in lots of different ways. I’d love to imagine a new curriculum that dispels the myth that math is memorizing procedures and manipulating meaningless symbols, and builds from the ground up something that more students want to learn, that gives students agency in understanding the world and working to make change.

Complex numbers, polynomials, rational functions, most systems of equations, trigonometry, triangle congruence, circle theorems, conic sections? I could live without them. What if instead we taught a course called “Math for Social Good,” giving students skills to use math to understand injustice in the world around them? Here’s a first draft of some questions one might ask and explore:

Can math help us understand the world?

  • How can different statistical measures lead or mislead us?
  • How can different statistical representations lead or mislead us?
  • How can probability help us to understand justice and injustice?
  • What is a “fact,” and how can we make better-informed decisions based on our knowledge?

Can math make valuable predictions of the future?

  • How can linear models help us understand the world?
  • How can exponential models help us understand the world?
  • What is interest, and who does interest benefit?
  • In what ways are mathematical models used to empower or disempower people making everyday decisions?

Can math make society more democratic?

  • What is gerrymandering, and how does it influence elections?
  • Why do we elect politicians the way we do, and what are some consequences of our choices?
  • What are algorithms, and in what ways do they help or harm individuals and communities?
  • How did our country become segregated and what can we do about it?

What does it mean to do math?

  • Who has shaped the ways that we think of math today?
  • What narratives about math have been excluded from the curriculum?
  • What role does math play in a world with ubiquitous technology?
  • What math is most worth learning?

Some will say in response that this will never work, that colleges won’t accept it, that we have to get students on the road to calculus, or that the standards are what they are and no one will allow it. But it seems to me that the system we have doesn’t work very well for many students, particularly for groups that have been marginalized in the past. If we built math education from scratch, what would it look like? What might mathematics be in the future, and how might we create a mathematics that includes and empowers more learners? What is education without recognizing the shortcomings of the current system and imagining something that better prepares students to be engaged learners and informed citizens?

Reasoning through Content

I love writing because as I write, I understand things a little better. I’ve written and changed my mind about why math might be worth learning more times than I can count. Each time, I get a little sharper in my thinking. Long time readers will probably find this repetitive, but I enjoyed writing it, and hopefully there’s a bit more insight than before.

Math class has lots of purposes. Learning math keeps future doors open, allows people to better understand the world around them, helps students to experience beauty and wonder, and empowers learners by showing them what they are capable of. But the most important argument to me is that math teaches reasoning. It’s not the only way to get there, but I think math class has enormous potential to teach students to notice and generalize patterns, shift representations and see a problem from a new perspective, balance attention to the big picture with attention to the details, connect ideas that seem different on the surface, and move flexibly between abstractions. These are skills that could serve students in the future within and beyond math classrooms.

But I hesitate to argue for reasoning as my primary goal in teaching math. Reasoning is the end goal, but it’s not the means of getting there. I don’t think that reasoning is a skill that one can practice; students can’t go to class, do some hard problems, and assume they are better at reasoning. I can’t confidently say that I’ve seen my students reasoning more effectively or consistently at the end of a year. All humans can reason, and all humans struggle to reason consistently. It’s a gradual learning process, and I’m lucky if I have a small part in that as a teacher.

More than my feeling that teaching reasoning directly doesn’t work, it’s also inequitable. Reasoning always happens in context, whether we choose the context of factors and multiples, interpreting statistics, or making decisions about finances. Students who already have knowledge of that context are likely to learn, and students who don’t, won’t. Pretending that reasoning is divorced from context advantages some students while leaving others floundering.

The Road to Reasoning Goes Through Content

Reasoning is a fuzzy goal, with a long and uncertain path to reach it. But content is what I know how to teach. I don’t have much confidence that I can teach a student to notice patterns and create generalizations in a given class. I do think I can do a decent job of teaching students to solve problems with quadratic functions. So I start with content. If I do my job and give each student a foundation in the content we’re learning, they all have a chance to apply their knowledge in new ways, to stretch their ways of thinking, and to reason. And I believe that, as students practice reasoning in more contexts, it becomes a little more likely that they will be able to apply those skills somewhere new. At first it’s small jumps — learning some things about quadratics and polynomials that can then be applied to rational functions. But, over time, I think math education sets students up with skills they can transfer outside of the math classroom. I don’t claim to know exactly how to get there, but that’s the goal.

This might seem like a dull perspective. We teach math to teach reasoning, but all I want to do is focus on content? But there are so many rich questions left unanswered. What content is most useful to teach? I don’t love what I’m asked to teach right now. Functions and algebraic manipulation are great, and should be in the curriculum, but what could we cut to make room for the math of gerrymandering, statistical analysis for social good, linear and exponential modeling in the world, and more? And once we’ve decided on the content, how can I teach that content in ways that help students reason? What are the teacher moves to link students’ experiences in math class to future decisions? What experiences will help students see math as a tool that empowers them in their lives? I don’t have any more answers at the moment, but I think these are great questions, and they are even better questions to ask about the breadth of content I teach each year.

Teaching Reasoning, or Teaching Math

Why teach math? One reason I hear often is to teach reasoning. I’d love to teach students how to reason, but I’m skeptical. I have enough trouble teaching them to complete the square or graph a rational function. Is it even possible to teach someone to reason?

I wonder if the idea that math teaches some broader skill like reasoning is a kind of reaction to the experience of many teachers that it’s really hard to teach content. Lots of students leave class without the mathematical knowledge we want them to have, and we might like to believe they have. But if content isn’t the point, then it’s just fine that students forget everything they learned about logarithms and polynomials within a month or two. It’s a nice idea!

So why teach math?

Mostly, for me, to learn math. I’m biased, I really like math. And primary math is really useful! I wouldn’t mind throwing out a few things, but facility with arithmetic, fractions, decimals, proportions, the basics of geometry, and few principles of statistics and probability help people to live in the world, to work, and to fulfill their civic responsibilities.

Most high school math, I could take it or leave it. If higher education wasn’t so obsessed with calculus, high school might look very different. Who’s to say that learning about polynomials is more useful than graph theory, or projective geometry, or number theory?

Here’s the fun part. I teach math because I think math can open doors, can empower students to see what they are capable of, can inspire wonder and beauty. Those are powerful experiences for students in school, when we get them right. And there’s lots of great thinking that goes into learning math. And I think that maybe, just maybe, as students learn more math, and they think in new ways in different contexts over time, it becomes a little more likely that they can apply that thinking somewhere new, in or out of math class. I’m a skeptic here; I’m not interested in teaching a “SWBAT reason” lesson, and I don’t think I get there with very many students. But I think math has that potential. Bit by bit, over time, it becomes more likely that those ways of thinking about proportions and linear models and function transformations and circle theorems become useful outside of mathematics.

I’m happy to be humble here. I teach math because I think math is worth knowing, and because the experiences one can have in math class are worth having. But I like to dream that what I teach could help someone to live a richer life outside of math class.