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?
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.
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?
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.
I’ve spent a lot of time at math teaching conferences over the past few years. It’s left me a bit cynical about professional learning. Of all the sessions I’ve been to, only a handful have changed my practice. And many times I’ve been the person up front presenting. What have I been doing with all that time and energy?
This summer I learned about the idea of pedagogical judgment via Darryl Yong at PCMI. Ilana Horn and Sara Campbell write in their paper:
Teachers need frameworks to work nimbly in the commonplace situation where students do not respond to activities as expected. For this reason, our learning goal was for novices to develop pedagogical judgment. Perennial puzzles of teaching – whom to call on, how much time to spend on a topic, whether to proceed with a lesson as written or attend to an unanticipated student misunderstanding – often have indeterminate answers and rely on teachers’ pedagogical judgment built on their situated knowledge of their particular teaching context. Pedagogical judgment is at the very heart of ambitious teaching practices.
[A] hallmark of sophisticated pedagogical judgment is ecological thinking about the classroom. That is, teachers accomplished in ambitious instruction reason about situations in ways that keep in mind the interconnectedness among things like classroom climate, teaching moves, student participation, mathematical activities, and student learning.
Horn & Campbell, 155
Here’s a tension I’d love to explore. In professional learning spaces, presenters try to get some idea across to teachers. But teachers aren’t blank slates; we come in with beliefs about learning, knowledge of our students, and knowledge of our context. How can professional learning engage productively with the ideas that teachers bring with them?
I think of pedagogical judgment as a core part of what makes the teaching profession unique. Nothing anyone shares at a conference works everywhere. Teachers need to adapt ideas to their context and the practicalities of their classroom. If we see teachers and classrooms as interchangeable, we share ideas in ways that assume teachers can apply them regardless of context. But if we respect the knowledge that teachers bring to professional learning spaces and build learning from that knowledge, we recognize how critical it is that teachers apply sound judgment in bringing new practices into their classrooms.
Are there professional learning spaces that value pedagogical judgment? What do they look like?
I’m trying to spend more time playing with math. Because it’s fun, because I learn about learning, and because I want to practice the discipline I teach. Here are a few problems I’m playing with right now. No spoilers please, but feel free to play along!
The two diagrams above are different pictures of “Borromean Rings.” They are each a link of three components, which basically means three loops of string arranged together. The Borromean Rings have the property that the components cannot be separated as pictured, but removing just one of the components means that the other two can be separated. A link with this property is called “Brunnian.” Can you find a Brunnian link of four components? Of more components? (problem from Colin Adams’ The Knot Book)
This is the most recent problem from Play With Your Math and I have already had a ton of fun with it. It’s neat because the number of possible mountain ranges grows quickly, so it is very hard to count each possible mountain range individually for large numbers. I think it is possible to organize my work to find patterns that help me to make better predictions. I thought I had a breakthrough yesterday and then realized it didn’t work. So fun!
The Fibonacci numbers are really cool. They go like this: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55,… where each number is the sum of the two numbers before it. They seem to have a neat property. 21 is the 8th Fibonacci number. 8 is divisible by 2 and 4, and 21 is divisible by the 2nd and 4th Fibonacci numbers, 1 and 3. Is this property true in general? Why? What other divisibility properties do Fibonacci numbers have? (problem inspired by this book, which was based on the PCMI 2012 math course)
I played a neat game recently. In a group of at least three people, each person randomly chooses two other people. Your goal is to stay equidistant from (though not necessarily at the midpoint of) your two people. Some positions act as an “equilibrium.” For instance, with three people, an equilateral triangle is an equilibrium position. What do some equilibrium positions look like for larger numbers of players? What is the probability that equilibrium is possible with randomly chosen people? What about a different version where each person chooses a “hero” and a “villain” and tries to keep their hero between them and their villain?
I’m getting sick of teaching precalculus. Don’t get me wrong, there are some gems in there. I love teaching probability and bringing in Ben Orlin’s excellent The Bear in the Moonlight series of fables. Teaching sequences and series is a ton of fun; I get introduce students to some fascinating and thought-provoking perspectives on infinity.
But too much of the curriculum is there for the same reason: “you’ll need this in calculus.” Nothing deadens my soul like hearing that students needs to learn something for the sake of learning something else in the future, on and on forever.
Juxtapose with the big event in the mathematical news this week:
My favorite response:
The first has become a bit of a trope. Problems like it come along every few months, with the apparent goal of making people feel dumb and reinforcing the idea that math is this inscrutable language with arbitrary rules that don’t make sense. The second isn’t a thing; while language can be ambiguous, we understand that meaning comes from context and don’t feel stupid if we aren’t sure what the writer is saying.
Back to precalculus. One negative experience I have had more times than I can count teaching this course for the last four years goes something like this. Let’s say I’m teaching logarithms. Most students have seen them before, and most students have forgotten what they are and why they might be important. We take our time making sense of the idea of a logarithm, first informally (credit to Kate Nowak) and then with more precision. It seems a bit arbitrary that students have to learn this, but it makes sense. Then we move into the other log rules and it all goes to shit.
Sure, if I was a great teacher we could take our time and do this all right — there’s plenty to make sense of here. But precalculus is a race. There’s so much to cram in that spending the time to do log rules right gives short shrift to something else. More likely I skim a few key ideas and move onto the next thing.
The problem is the curriculum. It’s not designed for students to understand a coherent body of mathematics. The goal is to be able to push symbols around and remember some disembodied rules that might be useful in the future. And students often come out the other end with their worst ideas about math confirmed. Math is about manipulating letters and numbers in weird ways, doing what you’re told, and getting on with your life.
Much of high school math works like this. It’s a race to calculus, with everyone’s pet topic shoved in just in case. And it leaves teachers with a choice. Take the time to do things right and skip out on some topics, or sprint through everything and pray.
Right now, the central goal of high school math is to prepare students for calculus. What if, instead, our goal was for students to believe that math makes sense? What wouldn’t we teach? What would we add? What would math class look and feel like?