On Perspective

I think I would identify myself as a social justice educator. I hesitate because it’s an ambitious title; there’s a lot that goes into educating toward social justice, and I’m not very good at most of it. I’m not sure I even know what most of it looks like.

But one piece I’ve thought a lot about is understanding the perspectives of those with different backgrounds from me.

I’m reading Carol Anderson’s White Rage as part of the #ClearTheAir chats, and it’s all the history I never learned but need to know. The erosion of progress during Reconstruction, the forces working against the Great Migration, the violent and defiant opposition to Brown v Board, the reversals of gains from the Civil Rights era, and finally, “How to Unelect a Black President.” This is the history that has created the world we live in today, the intentional discrimination and hostility toward the black community that has changed form but not impact over the last 150 years.

We often tell this history as a narrative of progress, as major victories that have moved our society forward. I think we tell those stories because they are comfortable, and because they fit into the narrative we want to tell ourselves about our country. What would it look like to value discomfort and truth-telling over progress and American exceptionalism when we talk about history? Whose stories do we tell? Whose narratives do we hear?

I’ve thought about this tweet often in the last few months. Which authors do I read? Whose stories do I hear?

I think about this in other contexts as well. Who do I retweet and amplify? Who do I reference and quote in my writing? Which students do I gravitate toward?

I am far from perfect; it would take me a very long time to retweet as many people of color as I have white people, or to link to as many white education writers as people of color on this blog. But White Rage is one example of a place where I’ve been pushed to expand my horizons. And White Rage is giving me new insights into the world around me, and how I can exist as an educator in it.

And here’s something I’ve noticed. As I’ve been more intentional about whose ideas I’m reading and sharing and referencing and amplifying, I’ve learned a lot that I wouldn’t have otherwise. I’m exposed to more culture, richer conversations, perspectives that push me to think in new ways, and ideas I wouldn’t have found without that active effort. I see the world around me through a new lens, and I am better for it.

I’ve read a lot recently about why diversity matters, in different contexts and from different perspectives. There’s not just one reason to seek out diverse perspectives. Take a sample here, here, here, and here. These are thoughtful folks and great arguments.

For me, one important driving force has been my personal experience that being intentional in who I read, reference, and amplify is leading me to a richer life than the comfortable, familiar perspectives I’m likely to learn from without that effort.

Active Caring and Patient Caring

I recently finished Geoff Krall’s book Necessary Conditions. It’s great! You should read it too. Here’s one of my takeaways:

Geoff writes about “active caring” and shares this great graphic in the book:

Active caring demands a two-way relationship independent of the student’s academic dispositions. Students who don’t demonstrate a preternatural appreciation for the subject receive the same level of personal and cultural care as those who do.

I think this distinction is really neat, and Geoff has great ideas for how active caring can play out with large class sizes and busy schedules. Check out his blog post on the topic.

But there’s another type of caring I think connects with active caring, and can be equally important. I’m going to call it “patient caring.” In the world, in particular when I read news about politics and prominent folks, there’s this assumption that humans are static, that the traits someone has today are the traits they’ll have tomorrow and next year and in ten years. I think this plays out in education; humans have a disposition to forget that other humans grow and change, yet that growth is a core part of our job as educators. And that growth doesn’t happen over time; it takes time and deliberate effort. I don’t have Geoff’s great visual designer, but here is my list:

Patient Caring

  • Teacher plans to revisit norms and routines throughout the year, knowing that classes need reminders and to hear expectations explicitly multiple times.
  • Teacher knows that students forget things they learned last week and last month and last year and plans proactively to reteach concepts and revisit key ideas over time.
  • Teacher recognizes when a lesson is going poorly, and finds ways to reframe expectations and adjust the goals to help students feel successful.
  • Teacher gives time for students to think and wrestle with ideas on their own terms, even when class is running behind.
  • Teacher listens to students rather than for right answers, and works to begin where students are rather than where one might wish they would be. (See Max Ray-Riek for listening to vs listening for.)

These are small changes in my attitudes and responses in the moment, and there are lots of times that the practicalities of classrooms get in the way of patient caring. But, at its core, I think patient caring is recognizing that students are works in progress, that learning is messy, and that things often don’t happen on my schedule. My response in these moments should be patience, and flexibility, and the perspective to consider the broader goals of math class.

Valuing Calculation

G was a sixth grader in my first full-time student-teaching stint in summer school. She didn’t conform well to typical classroom norms. She struggled to sit still for extended periods of time or raise her hand when she had something to say. And G had never felt successful in math class. I remember her saying, “I can do addition, but I haaate subtraction.” She would quickly become frustrated with any math that wasn’t easy for her, or just scribble down the first thing that came to mind. But she was good at addition; she could add multi-digit numbers incredibly quickly and accurately, and she loved doing so. She would have been happy to hang out and solve addition problems all day if I left her to it. Unfortunately, it was my job to teach her subtraction.

I thought of G reading Michael Pershan’s recent post on ancient Greek mathematics. He explores the two ways of practicing mathematics in ancient Greece: the theoretical, abstract, and logic-oriented branch, and the practical, social, procedural branch. We often associate Greece with the culture of theoretical mathematics, yet there was a large and thriving group largely interested in solving practical problems and finding new ways to calculate, measure, and count. Michael writes:

Looking at the ancient Greek example makes me think that we really ought to respect practical mathematics — which by definition is mathematics that is not concerned with the “why.”

And yet there is so often disdain among some teachers for “mindless” calculation or “thoughtless” problem-solving. That seems unfair to me.

Here’s a quote from a recent piece of mine that seems to argue the opposite perspective:

Lani Horn writes that “Schooling favors one type of mathematical competence: quick and accurate calculation” (Motivated, p. 61). Horn argues that we can value broader mathematical competencies — making astute connections, seeing and describing patterns, developing clear representations, being systematic, extending ideas, and more.

I’ve been interested in finding ways to broaden what we think of as school mathematics, finding new ways for students to recognize their mathematical competence. But the key word there is broaden: it seems silly to find new ways of being mathematically smart if at the same time I devalue others.

I was a pretty inept teacher when I met G, and I missed opportunities to build from her competencies toward new ways of mathematical thinking. But if I only saw mathematical competence as making connections, seeing and describing patterns, and the rest of Lani’s list, I would have missed the most important asset G brought to class, and the one she identified as her greatest strength: quick and accurate calculation. I don’t see any reason to diminish G’s self-efficacy by telling her that math is really about understanding why or explaining ideas or making connections, and not calculating things. She didn’t see those other competencies as her strengths. And students in math class desperately want to feel smart and successful in their learning; G needed to find ways to build from the skills she had toward those I wanted her to learn.

There is a tricky balance to walk here. Quick and accurate calculation is the dominant competence that is valued, implicitly and explicitly, in many classrooms. Given its prevalence, I see value in putting extra effort toward seeking out different ways of being mathematically smart that broaden students’ conceptions of what it means to practice mathematics and helps more students feel valued in math classrooms. But its prevalence also means that there are lots of students, like G, who come to class with strong skills in computation. And if I choose to tell them that those skills aren’t “real math” and are less important than other skills I prefer, I am reframing those strengths as deficits exactly as I am trying to find broader ways to see students’ strengths in math class.

Learning versus Performance

I hate using turn-by-turn directions. I often refuse to turn them on even in unfamiliar places. I look at a map and try to find major landmarks or make inferences based on what I know about an area. And I often get a little lost, but getting lost helps me learn where I am and I can find my way around more easily the next time.

I think this is a useful illustration of the difference between learning and performance. If I am interested in performance — getting where I’m going on time — I would want to use turn-by-turn directions. If I am interested in learning — being able to navigate around an area in the future — then looking at a map and thinking about where I’m headed is likely to be more helpful. As Dan Willingham writes, “memory is the residue of thought.” If I’m thinking “ok now I turn left here,” I learn less than if I’m thinking, “ok after I pass I-25 I need to keep an eye out for the exit, get off, and turn north.” That thinking might mean I trade a short-term drop in performance for durable learning I can use in the future.

Nicholas Soderstrom and Robert Bjork write about this in their review of research on learning versus performance:

…learners often mistakenly conflate short-term performance with long-term learning, ostensibly thinking, “If it’s helping me now, it will help me later.”

…instructors and students need to appreciate the distinction between learning and performance and understand that expediting acquisition performance today does not necessarily translate into the type of learning that will be evident tomorrow. On the contrary, conditions that slow or induce more errors during instruction often lead to better long-term learning outcomes, and thus instructors and students, however disinclined to do so, should consider abandoning the path of least resistance with respect to their teaching and study strategies.

I think of learning and performance when teaching topics like end behavior of polynomials and rational functions. End behavior is determined by a small number of principles — larger powers get larger much faster than smaller powers, even powers are always positive, and fractions behave in certain ways. But students often want quick, easy rules, like “If the degree is even and the leading coefficient is positive, the end behavior is positive in both directions.” The latter rule will improve performance in the short term, but misses an opportunity to connect the problem to the students’ prior knowledge in ways that can be applied in different contexts in the future.

Elizabeth and Robert Bjork call these “desirable difficulties,” difficulties that reduce performance but create thinking that improves learning. But here’s the catch. I sometimes get lost trying to navigate without directions, but I’m an adult with half-decent patience and self-regulation skills. And I still sometimes drive people I’m with crazy. Lots of students have a much more fragile relationship with math class. In a lab experiment, desirable difficulties sound like a great idea. But outside of psychological studies, students are largely motivated by their perceptions of success in school. Desirable difficulties might cause students to give up or convince themselves they’re destined not to understand a concept, erasing any benefits to learning. Students want to feel successful in school, and deliberately reducing performance undermines those perceptions and can prevent them from recognizing the valuable ideas they bring to class.

I think this is a great example of the complex relationship between research and practice. I have been dogmatic in the past pushing students to think in certain ways or struggle through problems, swinging the pendulum too far in one direction. But desirable difficulties don’t offer easy and straightforward solutions for the classroom, and neither do other areas of research.

Instead, I think it’s worth thinking about the tension between learning and performance, and the balancing act between desirable difficulties and unproductive struggle. And more broadly, whenever I encounter an idea that seems simple on the surface, it’s worth probing for the tensions and contradictions that come up whenever principles of teaching make contact with the practicalities of classrooms.

The Limits of Language

And so I’ve been asking myself lately, is equity even a useful term for me anymore? And I’ve kindof come up with the decision that it’s not. Not because it no longer works for me personally; when I think of that word equity I know what I’m talking about. But equity is this word that has been used so often now, it’s like that word diversity, that it kindof doesn’t mean anything anymore, or at least it’s not clear what it means. I liken it to the organic and natural movement of the 1970s. Back in the 60s and 70s when somebody said that they had something that was natural or that was organic, there was a certain standard we held ourselves to, right? Now, Campbell’s soup is organic, you’ve got strawberries that are grown in fields that the organic ones are grown right next to the non-organic ones, it’s the same distributor, so you have to ask yourself, what is this thing we even say when we say equity? Because historically what has happened with that word, and how it has gotten taken up, I think that it fails to promote dialogue and visioning, because when we’re in a meeting, and somebody says “we really need to address equity in our math department” or “in this committee when we’re selecting a new person to come into our field we need to be thinking about equity issues.” Well we don’t tend to stop ourselves and ask, “what do you mean by equity?” We tend to all in our heads go, “yea, I agree, I want to address equity too,” but there’s many different versions out there.

-Rochelle Gutierrez

The answers to why we should be racially diverse often hover most around “because I know it’s the right thing to do”, which just doesn’t cut it. I wanted to ask:

Why is it the right thing to do?
Why do you want me here?
What do you miss by my absence?

All met with silence.

-Marian Dingle

First, go watch Rochelle Gutierrez’s talk and read Marian Dingle’s blog post. They’re fantastic, and I learned a ton from both of them. I want to pick up on one idea I took from both pieces.

Language is tricky. When I use a word, I might mean a certain thing, but I don’t know if those I’m engaging with understand exactly what I mean.

The trickiness is compounded in equity work. First, because equity work is messy, especially in social media-oriented spaces that incentivize catchy sound bites and quick fixes. And second, because disrupting the status quo is uncomfortable, and humans don’t like being uncomfortable.

The word “equity” is a great example. When I first understood the distinction between equity and equality, that idea was really useful. Equality has a connotation of giving everyone the same thing, while equity means giving every individual what they need to thrive. Equity helped me to think more critically about what I want for my students. But there’s a lot more complexity under the surface. Similarly, diversity is something most folks in education seem to believe is a good thing. But does our collective belief in diversity mask hard questions that might force us to confront the places we’re coming up short, and think critically about why we want diversity? These are the questions Rochelle and Marian are asking, and they each pushed me to reconsider the ways I use language, and the ways that I might be able to use language to better say what I mean.

So certain words have useful meanings, but as they become more and more common that meaning becomes normalized, becomes comfortable, and stops dialogue and reflection rather than encouraging it.

I think this happens in the language we use around teaching math as well. I want to teach so that students develop “conceptual understanding” of math content. It’s the difference between teaching how to solve a single problem and teaching how that problem represents a larger concept, is connected to other problems, and how that knowledge might be applied in new contexts in the future. A nuanced perspective on conceptual understanding helps me to better make sense of times when students struggle to transfer their knowledge to new problems and inform what I emphasize when introducing a concept.

But when I talk to other teachers, conceptual understanding often ends conversation rather than starting it. Everyone agrees conceptual understanding is important. But when you say conceptual understanding, do you mean the same things I do? How, exactly, does one teach students in ways that promote conceptual understanding? In what ways has that idea changed your teaching practice? The language alone often serves to create agreement and consensus, rather than conversation and further thinking.

I see something similar with “misconceptions.” Students aren’t blank slates; they bring their ideas and prior knowledge to class, and sometimes the ideas they bring are counterproductive for what I want them to learn. Anticipating those misconceptions and addressing them is much more productive than pretending that each student is an empty vessel that I’m pouring knowledge into.

But some folks argue that labeling student thinking as a “misconception” is counterproductive. They argue that students are trying to make sense of new ideas based on what they know, and their ideas are really just conceptions — “misconception” is a judgment based on our perspective as teachers. I find this really useful too! Not that I think misconceptions are evil or don’t exist, but thinking of student conceptions instead of misconceptions helps me to seek out the valuable thinking that students bring and build off of it, rather than see their ideas as broken or unusable.

The language I use shapes the way I act. If I am in the habit of labeling student ideas as misconceptions, I am likely to emphasize correcting them over valuing their thinking, and to miss opportunities to see potential bridges between where students are and where I want them to get. If I push myself to think more about conceptions, I get in the habit of assuming students bring valuable ideas to class, and creating space for students to recognize their value as mathematical thinkers. This isn’t just semantics; language influences how I see the world, and changing my language helps me to see and act in new ways.

Some people would argue that the policing of political correctness has run amok, that liberals only want to tell people what they can and cannot say, and blame and shame those who don’t conform. That argument misses the point; the point is not to change language, but to use language as a lever for action. Being more thoughtful and specific and concrete in the language I use is a generative experience; it leads me to new ways of thinking, being, and doing. Rochelle articulates the ways that “equity” has become inadequate and offers “rehumanizing mathematics” as a framework for thinking more deeply about equity in math education. Her talk helped me to find new places to work on my practice, and a new lens through which to look at my classroom. Marian asks hard questions about the ways that “diversity” promotes agreement without pushing people to clarify why they want to be in diverse spaces. Her post helped me to interrogate my motives, to better understand why I believe what I believe, and to locate the places I need to do more internal work. I’m grateful for both perspectives. Both took ideas that might seem common-sense and comfortable, and unpacked the messiness and the places where folks are falling short. Both seek to hold folks accountable for following through in practice. And both will help me to find new places where my language is inadequate to a task, and push myself to find new language that holds me accountable for more productive action.

Complicated Problems and Complex Problems

Matt Enlow shared this problem on Twitter this morning:

I had a ton of fun playing with it! It’s one of those problems that takes ideas that I think I understand — in this case, properties of equations and exponentiation — and turns them on their head, forcing me to think in new ways and helping me to better understand math I learned a long time ago. Play with it! There are more solutions than I thought at first. If you’d like a hint, check out the replies on Twitter.

My first instinct when I see something like this is to ask, “How can I engage my students with this problem?” I love math, and I love problems, and I want my students to experience the joy of solving problems. For a long time I would seek out problems like this one, problems I loved, to share with students. But many of those experiences were counterproductive, and I’d like to try to explain why. First, here’s another problem that I recently saw on Twitter and enjoyed playing with:

Give it a shot!

Interlude: Complicated vs Complex

Atul Gawande writes in The Checklist Manifesto about the difference between complicated and complex. Sending a rocket to the moon is complicated. There are lots of little things that have to be figured out and designed and built and work right and lots of people who have to collaborate to put the pieces together. But once we get one rocket to the moon successfully, we can pretty well follow those steps and get another to the moon, and another.

On the other hand, raising a child is complex. There are lots of moving pieces, and lots of nuance and judgment, and raising one child does not mean that raising the next suddenly becomes a task of copying what was done before.

Working with something complicated involves coordinating lots of little things that have to be done right and add up to one big thing. Working with complexity involves much more judgment, subtlety, and responsiveness.

Back to Problems 

One reason to give students problems is to teach content. That’s important! But it’s not what I’m interested in here. The problems I give students also send messages about what it means to do mathematics. I worry that the first problem, with the factoring and exponentiation and all of the subtleties embedded in it, sends a message that practicing mathematics is complicated. It sends a message that math involves learning lots of little things and then piecing them together in unusual and contrived ways to figure out new things, but to be successful you have to remember all those little pieces and put them together in just the right way. I think problems like these play out in inequitable ways; students who already have strong skills and a disposition toward making sense of and persevering on a math problem are likely to get some positive reinforcement, and students already disaffected feel confused and left out of the conversation.

I think the dragon problem sends a different message. It invites experimentation and sense-making, and it can be represented lots of different ways, all from a very simple prompt. I think it sends a message that practicing mathematics is complex. Math isn’t easy; it takes originality, depth of thought, and a willingness to try new ideas and take risks. And it has value precisely because it’s not easy, and working through something hard can feel gratifying and fun. But that’s a very different message about the nature of mathematics, and why someone might want to pursue it in the future.

I love both of these problems, and the first problem was still fun for me. I still find it elegant and thought-provoking. I want to design some sequences of problems that get at similar ideas, where students can engage with the idea of exponentiation and the properties of equations. Those might serve a really useful purpose in helping to illuminate deep mathematical concepts that I often hurry past in the high school curriculum. But I only have so much time to engage students with problem solving for the sake of problem solving. For the purpose of helping students see themselves as potential mathematicians and illuminating the depth of what it means to practice mathematics, I think complex, inviting problems are where I want to focus my effort.

National Board Certification

Well, I did it. I’m a National Board Certified Teacher. Here’s my favorite of many tweets celebrating after the score release last weekend:

I don’t think that being National Board Certified makes me a great teacher. I can give you dozens of ways I’m not. I do think it reflects that I care about the teaching profession, and that I’m working to get better. Board Certification is premised on five core propositions, and I think that these came through in my portfolio — but notice that these speak more to teachers’ growth than their expertise.

  • Teachers are committed to students and their learning.
  • Teachers know the subjects they teacher and how to teach those subjects to students.
  • Teachers are responsible for managing and monitoring student learning.
  • Teachers think systematically about their practice and learn from experience.
  • Teachers are members of learning communities.

Some things were frustrating about the certification process. The feedback on my portfolio was hard to understand and not very helpful. The guidelines and rubrics were complicated and took forever to sort through. For Component 1, I had to drive two hours to the nearest testing center in Denver and sit in a cubicle staring at math on a computer for three hours; not fun. For Component 2, I had to figure out how to assess student learning at the beginning of a unit, use that assessment to differentiate and give feedback effectively within some uncomfortably prescriptive guidelines, and assess progress again at the end of the unit. For Component 3, I had to struggle to get intelligible audio and video of my teaching, throw out lots of bad clips, and then write something articulate about my teaching. For Component 4, I had to gather information from colleagues and students’ families about their learning, show evidence of how I design assessment systems based on student needs, and demonstrate that I’m learning outside of my school to meet those needs. This last one was a mess; it was hard to sort through exactly what I needed to do for each step and how the different pieces fit together. For the three portfolio components, I had to do a ton of pre-work planning when and where I was going to gather evidence and be prepared when things didn’t work out the first time. Then, I had to piece together what I wanted to communicate in my portfolio, and then actually write the thing. Luckily I like writing about teaching, but it was exhausting. 

I think part of the value of Board Certification is that it is a ton of work. It takes time, it costs money, it’s complicated. Teachers can’t start the certification process until they’ve been in the classroom for at least three years, and the credential isn’t worth much outside of schools. It’s not something people are likely to do if they’re on their way out of the profession. And all of the work is embedded in teaching; it’s not like writing papers for a master’s degree because so much of what I did was analyzing my actual teaching practice and talking about where I was working to improve.

Some folks say that the certification process is one of the best professional learning opportunities for practicing teachers. I don’t think this is true, but I feel incredibly lucky to have the  MTBoS as a space to share ideas on teaching, hear from others, and push my thinking forward. The NBCT community won’t replace that. But the MTBoS community is different. For one, it’s not all teachers. Lots of people I connect with work in curriculum, technology, instructional leadership, PD, and more. And that’s great! I went to NCTM in Seattle two weeks ago, and those were lots of the people I was hanging out with, and lots of the folks who read this blog. Hi! I appreciate you. The MTBoS is the best place I’ve found for engaging intellectually with teaching math, and I wouldn’t be the teacher I am today without it. National Board Certification dug into the practicalities of classroom teaching in a different way. It was messy and imperfect, but so is the reality of schools and teaching. I have no illusion that being Board Certified will influence my career the way the MTBoS has. But it serves as a symbol of my commitment to the classroom, and my commitment to improving my teaching in the classroom. 

My advice to other teachers: if you’re committed to teaching and your school or district is willing to support you financially, take a look at Board Certification. Be careful taking on too many components at a time. Learn to love writing. Know that the first lesson you want to videotape or assessment you want to use work samples from might not work out. Plan the logistics early. Know that it will be frustrating, the rubrics and criteria will be obtuse, and the portfolio will feel like a mountain of paperwork at times. Find someone you trust to look over your work. It’s less about being a brilliant teacher than it is showing off what you already do well, and being willing to reflect on where you want to improve. And you might fail — I didn’t pass by much, and better teachers than me have failed but pushed through.

Board Certification doesn’t decide what good teaching looks like. But it does serve as a marker of commitment to the hard intellectual work and richness of teaching, and a step toward a profession that defines and regulates excellence and receives the respect it deserves.