Category Archives: Uncategorized

Getting Better

When I think about my growth as a teacher, it’s easy to focus on concrete changes in my practice — standards-based grading, visibly random groupings, a problem-based approach to introducing new content. For an observer walking into my classroom today, comparing with my classroom a few years ago, those might be the more visible changes But I think that the biggest drivers of my improvement in the classroom have been incremental changes to some of the fundamental elements of my teaching.


Of course I have to have goals, and everyone has an opinion about what they should look like – EUs, SWBAT, learning intentions, or something else. But the form matters a lot less than spending time thinking carefully about what, exactly, I want students to learn that day, and finding the best ways to get there. And then coming back to a lesson I’ve taught before and felt good about at the time, realizing that an activity is actually not as tightly connected to my goal as it could be, and adapting or replacing it. Beyond tying my teaching carefully to my goals, I’m constantly expanding the scope of my goals for students. Content, mathematical habits of mind, skills of collaboration and communication with their peers, and positive beliefs and dispositions toward math are a few of those, but I’m always expanding and re-conceptualizing what I want students to get out of my class, and changing how I teach accordingly.


The problems I put in front of students are the heart of my teaching.  No matter how well I facilitate a problem, no matter how much students practice, if they aren’t doing worthwhile mathematics they aren’t learning very much. I am constantly finding new, more conceptually rich problems to better probe and push student thinking. I model problems to share a strategy or make a connection explicit, use problems as anchors while exploring new topics to link what students are learning with what they already know, and assemble challenging, varied practice that keeps students thinking and applying what they know in different ways while keeping the math accessible and scaffolding success as much as possible. As I assemble and write better and better problems, I expand my vision of what I want for students, and I provide them new opportunities to think deeply about the math we’re learning.


I remember, during student teaching in college, having to fill out a box for “differentiation” in a lesson plan. I wrote that I would walk around the room while students practiced several problems to see if they had questions. At the time, I thought that differentiation in math class was easy. I just give students some problems, see if they can do them, and then answer questions. I’m in awe looking back at the complexity I didn’t appreciate in teaching a range of students, how many other great ways there are to probe what students know, and how hard it is to actually uncover student understanding and adjust a lesson on the fly. I am always developing new strategies to figure out what students know and don’t know and becoming more adept at changing plans, circling back to a tough concept, or leading a discussion targeted at a particular preconception. But most of all, I am always getting better at looking at a piece of student work and, in a split second, figuring out what they know and don’t know and how I can respond in a way that moves their thinking forward while valuing them as an individual.

I feel like I write something like this every year or two, but I also think it’s one of the more important insights I’ve had into my teaching. Getting better at teaching isn’t always a flashy new pedagogy, it’s just as much the small things that add up over time, the small things that I try to practice deliberately every day but always fall a little short of where I want to be. But this stuff is powerful, and it’s iterative. As I set more ambitious goals, I write and seek out richer tasks to move students toward those goals, and I see student thinking in new ways that teaches me to adjust my teaching in new ways. I find new ways to probe what students know and don’t know, realize that I haven’t actually reached the goals I set for students, and design a new task to fill in that gap. This is teaching, for me. This is the little stuff that I love, day after day, week after week. This is what makes teaching such a great profession to spend a career getting better at.

What I Can Learn From Direct Instruction

Did you hear the one about a curriculum with fifty years of research that actually demonstrates its effectiveness? There’s a new meta-analysis in the peer-reviewed journal the Review of Educational Research that looks at over five hundred articles, dissertations, and research studies and documents a half-century of “strong positive results” for a curriculum regardless of school, setting, grade, student poverty status, race, and ethnicity, and across subjects and grades.

Ready for the punchline? That curriculum is called “Direct Instruction.”

To clarify, Robert Pondiscio was writing above about capital D capital I Direct Instruction. Direct Instruction is a curriculum developed in the 1960s that is scripted, relies on explicit instruction, sequences lessons deliberately so that each day builds directly off of the last, and designs with the explicit assumption that “that every child can learn and any teacher can succeed with an effective curriculum and solid instructional delivery techniques.”

I don’t think teachers should embrace Direct Instruction in large numbers. I don’t think it’s the most ambitious curriculum out there. I do think that there are lessons to be learned and conversations worth having about why there is such consistent evidence that Direct Instruction has been successful.

  • Sequencing matters. Direct Instruction is designed carefully to build off of previous ideas, and for teachers putting together their own curriculum sequencing is often an afterthought.
  • Attitudes matter. The idea that every student can succeed is baked into the heart of Direct Instruction. It’s easy, when teaching is hard, to lower expectations for certain students. DI counters that.
  • Motivation matters. When students don’t feel competent, they tend to disengage. Direct Instruction is designed to help students feel successful in small pieces before they move forward to the next idea.
  • Curriculum matters. Curriculum frees up teacher energy to do things beyond figuring out “what do I need to teach tomorrow?” That energy can be spent working with students who need extra support, planning more purposeful questions, or just showing up to work a with a bit more energy and enthusiasm for the day.
  • Finally, you’ve got to do the basics well. I don’t think that Direct Instruction is the best product around, but I think execution matters more than ideology. In other words, teaching an ok curriculum faithfully is better than trying something ambitious with poor fidelity. Direct Instruction is well-designed to scale to large numbers of teachers and students.

It’s easy to let a discussion of Direct Instruction devolve into ideological arguments — “it’s just increasing test scores without supporting conceptual understanding”, “it demeans teachers by scripting lessons.” But the evidence is clear that, for decades, Direct Instruction has had positive effects on the learning of some of the most disadvantaged math students. I think it’s important to recognize what can be learned from those successes, and use them to build the next ambitious curriculum.

Meritocracy or Aristocracy?

Ever since reading Rochelle Gutierrez’s article “Embracing the Inherent Tensions in Teaching Mathematics From an Equity Stance”, I see more and more hard questions in education in terms of the tension they create. In many cases there are two or more valid perspectives on a tough question, and exploring the inherent tensions — the importance of context, the impossibility of managing what is best for every student simultaneously, the contradictions inherent in many teacher choices — is a better approach than trying to come down on one side or the other. From my perspective, grading presents exactly that type of tension.

I see the pitfalls of grading every day. Grades create incentives for students to perform rather than to learn and to focus on individual tasks rather than big ideas. Grades perpetuate status issues, as some students perceive themselves as “smart” while others perceive themselves as “dumb”, perceptions that they often carry with them for their entire lives. Grades encourage measurement of what is easy to measure, and discount what is hard. Grades waste time that could better be spent focusing on learning. There may be better and worse ways to grade, but the constraints that schools put on most teachers are not well-designed for learning.

Then I read a recent blog post by Doug Lemov arguing that eliminating grades would bring back aristocracy:

Among other reasons there’s the fact that there will always be scarcity, and that means not everyone will get the best opportunities. (Everyone wants their kids to go to top universities, not everyone can. Sorry.) So you have to have some way to sort it all out. 

Meritocracy is the best way to do that, and meritocracy requires valuation.

When there is no grounds to judge, the elites will win all the perquisites. This is to say that when meritocracy disappears, aristocracy returns.

There is the tension. Whether I like it or not, grades serve the function of sorting and ranking students for their future pursuits, and that sorting and ranking will continue regardless of my decisions as a teacher. I’ve had too many students from well-off backgrounds better able to advocate for themselves and figure out the system, or have their parents advocate for them. And I’ve had too many students who have fewer resources to draw on, unable to receive those same advantages. Would eliminating grades exacerbate those inequities, so that education will become one more place where the rich get richer and the poor get poorer?

I have no answers. But I do know that navigating this question requires navigating the tension between the damage grades can do to a learning environment with the damage eliminating grades would do to equitable student outcomes.

Race and Language


Warning: The videos in this post use the n-word and other derogatory epithets. 

Like any teacher, race impacts my classroom and my students. One challenging topic is when the n-word comes up, either in class or elsewhere in school. As a White teacher, I don’t pretend to know the experiences of Black students. I do take it as my responsibility to educate myself about the experiences of others to the extent that I can, and to learn about the history and the contexts of words that I hear students use. I don’t want to argue for or against the use of any particular word, and I have been obligated by the schools I have worked for to respond in certain ways in these situations. I do want to have as much knowledge as I can, and as many resources to offer as possible, to prepare myself to engage in those conversations productively.

Here are two videos that have helped me to better understand how words impact students in my classroom, and the context in which the n-word may be used. They offer different perspectives, but together help me to be a little bit more competent in those situations. The first is Ta-Nehisi Coates answering an audience question about the meaning of the n-word, and other derogatory words, in specific contexts. The second is a slam poem by Julian Curry, exploring the history of the n-word and his experience juxtaposing that history with its common usage.

Professional Knowledge

Saphier imagines a situation where a teacher tells you about a successful classroom practice that’s different from the one you’ve been using. If you believe there’s generally a right and a wrong way to teach something (the effectiveness or “best practices” paradigm), your reaction may be that this colleague is trying to show you up or is patronizing you. But if you view teaching as a vast repertoire of practices that need to be matched to individual classroom situations, you’ll have a different reaction: Hmmm, that’s an interesting alternative. I might want to try it. “That view of professional knowledge not only accepts the legitimacy of different ways of doing things,” says Saphier, “but also encourages debate and professional problem solving.”

Kim Marshall (the Marshall Memo) summarizing Jon Saphier’s article in The Learning Professional, “The Equitable Classroom: Today’s Diverse Student Body Needs Culturally Profiicient Teachers”

I wonder if the perspective that Saphier describes is a distinguishing feature of the teaching profession. He argues that a “best practices” perspective is disconnected with the realities of teaching. A more productive perspective is looking at teaching as a broad repertoire of skills matched to the vagaries of classroom situations. I don’t know much about anything besides teaching, but that seems like a defining feature of the experience of teachers, and one that is less present in other respected professions like medicine and law.

Another point Saphier makes is that, once teachers move away from treating teaching as a set of best practices to be learned, they need to constantly ask themselves what’s working and what’s not, and use assessment evidence to back up their decisions. I find this fascinating. I think formative assessment, broadly speaking, is one of the most important things I can do to improve student learning. Saphier argues that formative assessment can also be an important tool for teacher learning.

I’m curious about the experiences that cause teachers to shift perspectives on skillful teaching, and the school cultures that help to facilitate this process.

Student Engagement

My work on this paper began over 10 years ago with my research on the AHA! experience and the profound effects that these experiences have on students’ beliefs and self-efficacy about mathematics (Liljedahl, 2005). That research showed that even one AHA! experience, on the heels of extended efforts at solving a problem or trying to learn some mathematics, was able to transform the way a student felt about mathematics as well as his or her ability to do mathematics.

-Peter Liljedahl, Building Thinking Classrooms: Conditions for Problem Solving

A convergent observation across psych applications is that behavior change->belief change is easier to accomplish than belief change->behavior change.

-Brian Nosek on Twitter, references here

I often feel confused by student engagement. What does it look like? What creates it? What sustains it? How can I better understand the subtleties of how it plays out in different classroom situations? Engagement feels like a black box. Some inputs go in, engagement may or may not come out, and I don’t have a great understanding of what happens inside the box.

Despite that pessimism, here are two things I think I understand:

  1. Convincing students to be more engaged is really hard
  2. Facilitating situations where students feel successful helps improve that student’s engagement in the future

Students often come to class with deeply ingrained beliefs about who they are and what they are capable of. Convincing them they can reason mathematically is an uphill battle because it goes against the weight of their experience — by the time I teach them, over a decade of it — in math classrooms. If instead I can provide new experiences where students feel successful, where they feel their voice and their ideas are valued, where they have those AHA! moments, I can create a new foundation to build off of and positive momentum for the future.

Where I get confused again is how to facilitate those experiences for every student in class, and to sustain those experiences over time, while meeting the demands of the curriculum. I have some strategies, but when I look honestly at my students and their growth over time it feels like I make progress far more unpredictably than I would like.

“Natural” Teachers

“What explains America’s love affair with the untrained, the unschooled, the uninitiated?” – Peg Cagle

Peg Cagle’s ignite talk above masterfully takes down the trope of the “natural teacher”, arguing that painting some teachers as natural is unrealistic and demeans the teaching profession. I want to expand on her argument and unpack what people often seem to mean when they describe a teacher as a natural.

Walking into a classroom, an average member of the public might describe a new teacher as natural based on surface characteristics that actually aren’t essential to helping students learn. Speaking confidently, explaining ideas clearly, having some content knowledge, and getting students to like you are all things that many humans get good at outside of teaching contexts, and at first glance someone with those skills might appear to be a natural teacher. That’s not to say those skills aren’t important, but if a teacher stops growing there, they are falling short of the potential of impactful teaching.

The heart of teaching is much more subtle, much harder to learn, and much more counterintuitive. Where else but classrooms do people ask questions not to learn the answers, but to provoke thinking in others? Where else do people try to figure out how someone thinks about an idea that they don’t yet understand? Where else do people design an experience with scaffolds for someone who is struggling? Where else do people have to manage a room by both maintaining a thread of instruction and paying attention to the motivation, engagement, and understanding twenty-five individuals? Where else do people design experiences where, multiple times in an hour, they need to react on the fly to something a participant said or understood and possibly change course? These are skills that new teachers are extraordinarily unlikely to find “natural”, and they are only a small subset of the skills that make great teachers great.

Teaching is not natural. And, for many of the uninitiated, seeing someone as a natural means focusing on surface-level features of teaching, cutting away the complexities and challenges that lie under the surface and simplifying the profession down to the lowest common denominator that those outside of it can understand.