Explain Less

I’ve developed an instinct against explaining things to students. It’s an instinct I’ve developed for practical reasons. The more I talk, the less students listen. The longer I talk without a break, the more tenuous my hold on what students understand. Students get confused when I spend too much time sharing how I think, and too little time understanding how they think.

I have no philosophical issue with telling students things. In short bursts, when students feel a need, there’s nothing better than a concise explanation that lets them solve a new problem. Then, I ask students to solve the problem, see how it goes, and consider whether there’s something else I need to address or if they are ready to extend their thinking a step further.

Something I’ve found myself doing is finding more and more ways to explain less. Asking students to figure things out for themselves often doesn’t work, and when it does it tends to privilege students with strong backgrounds and positive past experiences in math class. Explaining less might mean breaking a lesson into little chunks, so that I’m only talking for a minute at a time in between having students solve problems, warm calling students after each problem to summarize a key idea for the class. It might mean using worked examples to have students generate an explanation themselves, and listening in to notice their thinking and build from it. It might mean presenting a problem as a puzzle drawing on prior knowledge, figuring out where students are with that prior knowledge, and building my explanation from there.

I read this great post about teaching Computer Science yesterday and couldn’t stop thinking about the elegance of the ways this teacher found to explain less. My favorite:

Here’s an example — the second thing I ever show students, right after print("hello world") , is this right here:

name = "Tamara"
print("Hello" + name)

And then I ask one simple question:

Don’t answer out loud — just think. What will happen when you run this program?

Don’t answer. Just think. What will happen?

Turn and ask your neighbor for their prediction.

Literally every student intuits that this program is going to greet Tamara.

And then after that, we run the program, find out that it prints  HelloTamara  without a space, and we also do our first round of debugging. High fives all around!

I can imagine myself as a novice computer science teacher trying to explain what this example will do. I imagine it as a complete mess. There will be a time, later, to formalize student knowledge of variables, strings, and more. In the meantime, students learned something , and they’re primed to learn more. This is a beautiful example. It’s not too complex, builds from intuition, and packs in surprise. I’d love to find more ways to do this in a math classroom.

Here are three things that I think matter in finding moments to explain less. First, you can’t force it. I never want to become dogmatic and refuse to explain things. Explanations are valuable, and they’re especially valuable when they’re used at the right time and place. Second, I don’t want to let perfect be the enemy of good. My worst explanations are when I try to explain something with mathematical precision and address every possible case. But that precision can create confusion. Sometimes students are ready for an informal understanding of a concept, but struggle to engage with too much complexity at once. Finally, finding ways to explain less trusts students more, and sends a message about what they can do. If every lesson begins with explanation, students learn that knowledge always moves from the teacher to the student. I don’t expect students to derive centuries of math on their own, but there are plenty of opportunities for students to extrapolate form their knowledge to something new. And every time we do that, students have the opportunity to trust themselves and their ideas a little more.

Math Recess & the Challenges of Play

I recently finished reading Math Recess by Sunil Singh and Dr. Christopher Brownell. I enjoyed it! It’s full of mathematical stories, games, puzzles, problems, and more. I’m still exploring and learning from many of their problems. I love playing with math, and Math Recess makes an argument for rethinking math education on a foundation of play and joy.

It’s a compelling argument, and it’s particularly compelling to me. I love math and I’ve had lots of positive experiences exploring math. But I’ve tried in fits and starts to incorporate more play into math class, but I’ve failed a lot more often than I’ve succeeded. The biggest challenge has been my own perspective. Math that looks fun to me often doesn’t look fun to my students. Playful math involves taking risks and experimenting, but I struggle to create a culture where every student is willing to do so. Some students come into class with assumptions about who they are and what they are capable of based on their past experiences. These assumptions can foreclose taking risks, and turn what I see as playful math into another frustrating day of math class.

Here’s one example. At my school we take students on extended backpacking trips in Colorado and Utah and try to help students connect with the natural world. I designed an assignment on the Fibonacci sequence to connect math class and students’ backpacking trip. Students read an excerpt from A Mathematical Nature Walk by John Adam about how the Fibonacci sequence shows up in nature. Students also worked together on a short set of problems about the Fibonacci sequence, exploring some of its interesting properties and surprising places where it shows up. I put a ton of effort into the assignment, but it was a mixed bag. Some students enjoyed it, but those were mostly students who were already successful in math class. Others struggled. They got frustrated quickly, tried to rush through the problems as fast as possible, and didn’t have much of anything positive to say about it.

I’ve used the assignment a few more times since then, and I’ve made some changes that help more students access it. But it’s still not what I want it to be. There are a few different conclusions I could draw here. I could blame the students. But the students have good reasons to be skeptical of the assignment. They’ve spent years frustrated with math class and receiving messages that they’re not math people. If exploring and playing is what math is actually about, you would think we would do it more often. I could blame other teachers. But teachers are doing the best they can with what they have, and another teacher might criticize me for the same reasons. I could blame myself. But if we want to see more play in math classes, we need to find ways for it to work for every teacher, not just those who happen to have special knowledge or skills. It’s not as simple as dropping a few new problems or games into the curriculum. Singh and Brownell acknowledge this — they are arguing to redesign math class from the ground up. But from where I’m standing, I need incremental steps, and incorporating play bit by bit can be a frustrating road.

So what does a path toward more play in math class look like?

First, I want to keep expanding my knowledge of mathematical games, puzzles, and problems. I need to both collect more examples, and better understand how to use them in ways that encourage students to take risks and try something new.

Second, I want to find more ways to find moments of play while we are doing the regular everyday math stuff of learning new content. Knowing lots of different versions of Nim is a ton of fun, but Nim is hard to connect to required content. I’d love to add to my toolkit new ways to make learning the typical curriculum fun and playful. I wrote a while back about an attempt to do that with polynomial division, but it took a lot of refining and careful facilitation to get it to work. How can I find more examples of playful ways to introduce new topics?

And finally, there’s a foundation of culture and knowledge students need to just be willing to engage with mathematical play. If a student has never felt successful in math class before, why should they expect to do so now?

So here’s my review of Math Recess: Read it! If you love math, you will come out the other side with fun new ideas and experiences with playful math. Enjoy entertaining the possibilities of a math class designed around play from the ground up.

But at the same time, I found myself wanting to better understand the foundation I need to build for students to be willing to engage with mathematical play, and to find more ways to infuse play into the math I’m already teaching. It’s fun to reimagine math education from the ground up, but the practical challenges I face every day require more incremental solutions. I’d love to learn more about what those small steps might look like.

On Writing

I wrote a thing over at Edutopia. Read it if you’re interested. My original title was “Small Steps Toward Rehumanizing Mathematics.” I wanted to write it because I have found myself struggling with the tension between ambitious thinking about rehumanizing math class, and the slow crawl of making changes in practice. The more I read and talk with people about equity in math education, the more I realize how little I know. But if my lack of knowledge paralyzes me from growing as a teacher, I end up in a vicious cycle. Even if the steps are small, they’re worth taking. I hope I captured a bit of that in the piece.

I’ve also been trying to branch out in my writing. I love my blog, but blogging has also felt stagnant recently. I’ll keep writing here, but I want to challenge myself to write elsewhere as well. Along the way to writing this piece I had half a dozen other pitches turned away, at Edutopia and a few other publications. I’ve learned a lot about formal education writing and I’ve become sharper in how I adapt my writing to an audience, space, and style. I’ve also struggled with ceding control to an editor. I get the kind of granular feedback I never get on here, which I am grateful for. But I also have to live with things like having someone else decide the title for my piece. All decent tradeoffs, but it takes getting used to.

I’ve been writing about teaching since the beginning of my first year in the classroom. I was lucky to start teaching in what felt like the golden age of math education blogging. It was fantastic for me. It’s still fantastic for me, even as most of the folks whose blogs I started my career reading have moved on.

I’ve always written for myself. Writing challenges me to explore new ideas, and pushes me to think more clearly about the problems of teaching. And for every post I write I have three more that don’t become full pieces, yet still teach me things about teaching. If I had to give any advice to other aspiring teacher-writers, it would be to write for yourself, write to learn about your teaching, about whatever interests you and nothing else. That was how I grew the habit of exploring new teaching ideas through writing, and became someone who walks out of every dud of a lesson synthesizing my learnings for a potential blog post. Writing isn’t the only way to think hard about teaching, but it’s been a good one for me.

Writing beyond this blog means being a little bit less selfish. I learned a lot writing this last piece, but it’s for other teachers, not me. And that’s a whole new interesting challenge, putting myself in the shoes of other folks, hypothesizing about their everyday trials, considering what might feel useful. It also means feeling frustrated and useless in my writing. When this last piece went live on Edutopia, I hated it. I still kindof hate it. Trying to write for other folks, with the pressure and constraints that come from a real publication, is frustrating. It doesn’t quite feel like my voice. But a little farther out I feel a little better about it.

I’m not sure where all this goes. Do I want to start writing more for other publications? My poor batting average pitching pieces so far suggests that could be a frustrating road. I also want to make sure I’m doing it so I can challenge myself and grow, not to see my name in lights. And I still love my blog, and I want to keep finding time to write here.

Assumptions & Language

Last night I was flipping through my copy of Euler: The Master of Us All. It’s a book about Leonhard Euler’s mathematical accomplishments. It’s interesting! I’d recommend it, despite the pretentious title. So Euler is playing with infinite series, which Euler loves to do, and the author inserts this bit of commentary: “By this time the reader must have noticed a number of symbolic manipulations that require careful handling.” That put me off a little bit. I hadn’t noticed, actually, Mr. William Dunham.

But this type of language, making assumption about one’s audience, is common in writing about mathematics. Here’s another one from a book I was reading about abstract algebra:

“The theorem we have just proved has several obvious but important corollaries:”

Obvious to who?

I find myself falling into this language in class. “It is simple to…” “You’ll notice that…”

This language reflects an ugly part of the culture of mathematics. For a long time, math has acted as a gatekeeper, labeling some students as “smart” and others as “not smart.” We tell ourselves that math is sequential and missing one day can cause a student to fall behind for a year. The way we talk about math reinforces these stories, and they function to recreate patterns of who has been successful learning math.

Here’s another fun quote I stumbled across last night:

One of the stories we tell ourselves about math is that, once you fall behind, it’s hard to catch up. For instance, yesterday I was teaching about rational functions. It’s easy to play with this chain of logic. I assume that first students need to understand fractions, variables, the order of operations, polynomials, intercepts, asymptotes, limits, and more. We could spend weeks searching for misconceptions in students’ prior knowledge, assuming they won’t be able to access the content until they’re fluent with every little piece. But is this always true? Is it possible to drive a car without being able to build an engine? What would mathematics look like if we chose to ask a similar question: Is it possible to engage in mathematical thinking without understanding what we assume is prerequisite knowledge? How might we restructure math class to make it more likely that every student can engage with key mathematical ideas every day?

Diagnostic Questions

Things are always more complicated than I want them to be.

Here’s something that happens to me all the time. I introduce a concept through some activity or discussion. It seems like students understand it. I give them a few problems to check their understanding. Suddenly it’s a disaster, everyone is confused, and we have to circle back and clean up the mess. Now I’m all for seeing mistakes as learning opportunities. But too often students feel frustrated and that frustration leads to spiraling and entrenching negative feelings about math class. Definitely worth avoiding.

This fall I’m experimenting with diagnostic questions. I use them right before I have students try to apply a concept on their own. Here’s one I used in a class on graphing sine functions:

We had spent some time talking about how to find the period of sine functions. Which is a hard concept! And I thought they had it. Not so fast. I asked this question, and half the students answered B. It led to a great discussion. I had students chat with the person next to them, and most pairs reminded themselves of the formula for calculating period after talking with a partner. We talked briefly as a group and did another example together. I sent students off to practice feeling like I had done something productive, surfacing how students thought about period before letting them flail on their own.

One logistical note. I do what Dylan Wiliam recommends in his book Embedding Formative Assessment. There are lots of ways I could collect student answers, from clickers to cups to moving around the room. Dylan Wiliam’s thought is that students rarely forget to bring their fingers to class, and fingers don’t need an internet connection. One finger for A, two for B, three for C, and four for D. It’s been hard to get every student to raise their hand. I’m uneasy pressuring students to answer if they’re guessing, but I don’t want it to be too appealing to opt out either. Students only need to flash their answer for a moment; I try to reduce opportunities to look at each others’ answers and engage in social posturing.

So here’s my dilemma. When students are split between two answers, my next move as a teacher seems pretty straightforward: have students discuss, in pairs and then as a full class, which of those two answers makes more sense. But in a different class, working on writing exponential functions, I asked this diagnostic question:

This time, all but two students answered B, and the other two answered D. My instinct here was to pat myself on the back. Go me!

But what do I say to the class?

One option might be to say, “awesome, almost everyone got it right! Nice job!”

What message does that send to the two students who picked D?

Instead, I did the same thing as when the class was more evenly split. “I’m seeing some disagreement between B and D. Chat with the person next to you about which answer you think makes more sense.”

There’s a lot of complexity here. Coming in, my thinking was pretty straightforward. I wanted a better way to figure out whether I should move on, or if students needed more time as a whole class. I figured I should ask a quick question, and based on their answers decide whether to stop and discuss or move on.

But it will be pretty rare that every student gets a question right. And it always seems useful to take a moment and discuss a question like this. I now look at these more as discussion starters than diagnostic questions. The information I get about who answered what is definitely useful. But so is listening in on a quick partner discussion.

And even asking a quick question to gauge student thinking feels tricky. I like multiple choice here because it helps make the questions accessible and efficient. But trying to do it quickly can undermine the culture I want to create where speed isn’t the most important thing in math class. My goal is to figure out how students are thinking about one piece of a concept, and it feels hard to linger on a question for too long. But I really don’t want students to feel rushed — and it would probably be the same students every time who feel rushed, building a negative association with these types of questions.

I’ve found it useful to take a step back. One goal is to better understand what students know before they jump into independent work, to see if we need to spend more time talking as a full group. But an equally important goal is for students to avoid reinforcing negative narratives students might have about their ability as mathematicians. And there are all sorts of things here that are in tension with that goal. An implicit value on speed. Social risk in sharing answers so that every student can see. Comparing oneself to others. Surfacing ideas that might single out one student who feels like they are on the spot.

I think that the benefits outweigh the drawbacks here. There are also plenty of liabilities to throwing away diagnostic questions. And the issues above are ones I can manage through class culture. And that’s teaching. Something I thought would be simple actually has a lot more layers than I initially thought. And there’s a lot of useful stuff here. In both of these instances, I helped to avoid the phenomenon I wanted to avoid: sending students off for some independent practice when they still have very different conceptions of some mathematical idea. Avoiding that is worthwhile, but will take more nuance and subtlety than I first anticipated.

Pedagogical Judgment and The Thing

I thought Standards-Based Grading was The Thing. I tried it at my last school and I convinced myself it would be revolutionary. Now I’m at another school, still using SBG. I’m still waiting for the revolution. Sure, some things are a little better, but it hasn’t changed my teaching in the way I hoped. Then I thought doing a clever warmup each day was The Thing. I did it for two years. When I stopped, nothing changed except I had a few more minutes each class to teach. I thought Smudged Math was The Thing. I thought Desmos was The Thing. I thought spaced practice was The Thing. No dice.

But there’s no The Thing that will, alone, make me an amazing teacher. And that’s something I’ve come to love about teaching. If it were that simple we’d have robots do it.

Maybe if there’s a Thing it’s having a dozen different tools for building relationships with students so no one falls through the cracks. And constantly finding new ways to be curious about and explore student thinking and respond to that thinking in the moment. And understanding the prejudices of our country and our schools, and how I can mitigate the impacts of that prejudice and empower every student in my class. And finding a way for every student to recognize the ways they are mathematically smart and helping my class to value those smartnesses. And recognizing that humans are complex animals that are impossible to predict or fully understand and trying to do so could be the project of a lifetime, making halting and incremental progress and still being surprised every day.

What I like about that last list is the role of pedagogical judgment. In the past, I might find some clever idea on the internet that I can try the next day or next week. A rubric for assessments, a fun activity to teach polynomial division, or a new group work structure. And these things are important! They’re parts of my toolbox, and I can’t teach without them. But pedagogical judgment is taking that toolbox and figuring out which tool is right for this job, in this moment, with this content and these students.

There’s no how-to or quick trick for pedagogical judgment. It’s something I practice over time. And if we see teachers as professionals, we need the concrete tools in their toolbox. But it is just as important that we develop pedagogical judgment to use those tools to adapt to the needs of the students in front of us.

Here’s something I want to explore more. Pedagogical judgment can seem distant from classroom teaching. It involves all these abstractions that can feel like they don’t connect to the decisions I make in the classroom each day. I’m curious how I can make those links clear and coherent. Where are the moments? What goes into the decisions? How can I practice pedagogical judgment in a way that develops skills I can use tomorrow, and next week, and next year?

Anti-Deficit Narratives

I just read a great article in the Journal for Research in Mathematics Education, “Anti-Deficit Narratives: Engaging the Politics of Research on Mathematical Sense-Making” by Aditya Adiredja. I need to spend more time with the article, but I have two takeaways so far.

First, Adiredja shares the idea of “deficit master-narratives.”

Deficit master-narratives are socially circulated and reified stories in society that suppress morally relevant details about a person or group with the impact of disrespecting or misrepresenting such a person or group (403).

Deficit master-narratives impact who we perceived as mathematically smart. Whether I like it or not, these master-narratives bring the prejudices of the world into my math class. Adiredja distinguishes master-narratives from stereotypes in that master-narratives act as scripts that play out in everyday life, while stereotypes might only live in someone’s mind. In math class, the master-narrative is that only a narrow subset of students are likely to be mathematically smart. this script plays out in the ways that students look to others for help, the ideas they respect, and the voices they listen to.

I’ve seen conversations about asset-oriented teaching or strengths-based teaching become more prominent in the last few years. Deficit master-narratives are a useful way to understand the necessity for asset-oriented pedagogies. The opposite of holding an asset orientation is not being neutral; without an explicit asset orientation, deficit master-narratives creep into my classroom and undermine student learning.

Second, Adiredja breaks down what a deficit perspective looks like in practice:

I argue that deficit perspectives are generally supported by principles that overprivilege (a) formal knowledge, (b) consistency in understanding, (c) coherent or formal mathematical language, and (d) immediate change in understanding (413).

I find Adiredja’s perspective useful in being specific about the behaviors that lead to a deficit orientation. Sometimes in conversations about ambitious equity-oriented pedagogies like holding an asset orientation or rehumanizing mathematics, the ideas feel really big and broad and impossible to tackle. In some ways they are; that’s an essential part of a project that aims to reimagine mathematics education. But at the same time, Adiredja points to actions within my sphere of influence that I can take today. And there’s a bit of urgency. Without deliberate action, deficit master-narratives continue to undermine learning. Reimagining math class isn’t necessarily about tearing everything down and starting from scratch. It can begin with simple actions that undermine the systems that perpetuate inequities. A better understanding of deficit master-narratives and perspectives feels like an important step.