“I Don’t Get It”

Here’s something I find hard about teaching:

I want to create a classroom culture where mistakes are a normal part of learning, asking questions is encouraged, and students feel comfortable sharing their confusions. Ideally, I create opportunities to figure out what students know and don’t know, and to do something about it.

At the same time, often when students say “I don’t get it,” they’re not saying, “I would like help so I can figure this out.” Instead, they’re saying, “I don’t like this class, math never makes sense, and I’m bad at it anyway.” “I don’t get it” functions in that moment as a performance, as a way of signalling to peers that math class is a silly place and engaging is a waste of time. It’s a particular type of comment, often said quietly to someone sitting nearby or mumbled after an explanation or direction to get started with a task. Student body language is totally different from a sincere attempt to clarify something or move forward. And I understand where this comes from — many students spend years in math classes feeling unsuccessful, incompetent, and forgotten about. It’s a natural response, and not one I blame any student for.

I have a hard time responding in moments like this. I can try to circle back to a tough topic, offer another explanation, model another problem, or do something else to try to help that student learn. But engagement is typically low and the energy has been sucked out of the room, and taking a step back often sacrifices even more momentum from the lesson. I can stop class to unpack how students’ attitudes toward math influence their learning, but that often feels like the wrong time and makes me feel defensive. I can try to ignore it, but ignoring confusion is probably just going to let things fester and get worse.

I don’t feel like I have any good ways to respond in that moment. Any ideas?

Challenges in Implementing a Thinking Classroom

I’ve spent much of the last year and a half experimenting with Peter Liljedahl’s ideas on creating a Thinking Classroom. He writes:

I wanted to build, what I now call, a thinking classroom — a classroom that is not only conducive to thinking but also occasions thinking, a space that is inhabited by thinking individuals as well as individuals thinking collectively, learning together, and constructing knowledge and understanding through activity and discussion.

One way to summarize this, for me: Most students spend math class trying to avoid thinking. If that is the case in a class (and it is definitely the case for many students in mine), then making drastic structural changes that promote students engaging and actually thinking are worth trying. Below is the current 14-point framework for building Thinking Classrooms, divided into four levels.


I’ve felt largely unsuccessful implementing a Thinking Classroom. When I started, I viewed a Thinking Classroom as a space where I gave students rich tasks, put them in visibly random groups, and had them work at whiteboards. I definitely see benefits to engagement and collaboration in that environment, but it wasn’t the transformative change I was hoping for. I’ve had two insights along the way. Both of them seem obvious, but I think they’re worth sharing because I often see Peter Liljedahl’s work portrayed as synonymous with random grouping and vertical non-permanent surfaces, when there’s so much more.

First, there’s a lot more to Thinking Classrooms than the first level — tasks, groups, and surfaces. I’m working on maybe the second or third level. While tasks, groups, and surfaces are helpful, to build a successful Thinking Classroom I need to have complementary structures in place to maximize engagement, focus student thinking, and create the support necessary to sustain the work over time. One area I’m thinking a lot about now is flow. I want students to get into a space where they feel successful, and feel encouraged to continue thinking. This means finding the right tasks, and I can get students started more effectively through random groups and vertical non-permanent surfaces, but there are a lot of strategies to make sure that thinking continues. Having a few hints ready, encouraging groups to visit another group that has useful ideas, and valuing student ideas when we discuss as a class are all elements that help to create a culture where students continue thinking about problems for sustained periods of time.

Second, the principles of Thinking Classrooms each have a purpose. The purpose of vertical non-permanent surfaces is to increase student willingness to just try something and take risks, and to increase knowledge mobility between groups. But surfaces don’t do those things on their own; I need to make my goals clear to students, narrate when I see the type of thinking and collaboration I’m looking for, and recognize when a certain activity was unsuccessful and figure out why. Similarly, the purpose of hints and extensions and only answering certain types of questions is to manage flow, so that I always have tools ready to keep students engaged and doing purposeful mathematical thinking. I can’t expect a few structural changes to influence my classroom culture if I don’t follow through with making sure I meet the specific goals behind the changes. Peter Liljedahl has written extensively about Thinking Classrooms, and everything there is worth diving into.

I think there’s a ton of potential in the principles of Thinking Classrooms, and I’m excited to continue working on implementing them. At the same time, I worry that it’s easy to oversimplify, as with any idea in math education. Thinking Classrooms are far more than the first level of good tasks, visibly random groups, and vertical non-permanent surfaces. As I continue to work on my Thinking Classroom, I want to work on incorporating more and more complementary elements of the framework, while keeping in mind the broader goals and course-correcting as necessary.

Observing Students

Of all I saw and learned this past half year, one thing stands out. What goes on in the class is not what teachers think — certainly not what I had always thought. For years now I have worked with a picture in mind of what my class was like. This reality, which I felt I knew, was partly physical, partly mental or spiritual. In other words, I thought I knew, in general, what the students were doing, and also what they were thinking and feeling. I see now that my picture of reality was almost wholly false. Why didn’t I see this before?

-John Holt, How Children Fail

I don’t spend enough time observing and learning about what students experience in school. It seems silly, but I spend most of my time watching the entire class, and I tend to interpret events in my class through that lens. They understand it, or they don’t. They’re engaged. They need to let off some steam. I generalize about the whole at the expense of the individuals. At the same time, I tend to ignore the complexity of interactions between students and the social dimension of their experience in class. Here are some ways I want to work on practicing observation:

  • Listen in on collaborative work
  • Watch students  while I’m explaining an idea at the board
  • Watch students as they come into class, and as they leave
  • Watch students when I ask a question
  • Pay attention to individuals — their body language, facial expressions, who they listen to, who they don’t, what creates that sense of engagement and what causes them to turn away.

The more I observe, the more I see two things that school is very good at teaching students:

  • Adolescents learn to compare themselves to their peers, tirelessly and at every opportunity, often when I don’t realize that a comparison is possible.
  • Adolescents learn to be incredibly creative at avoiding things that are hard, and incredibly effective at “playing the game” and going through the motions of school.


Good problems are the backbone of my teaching, but I often have trouble articulating what, exactly, makes a problem “good”. This year I’m teaching calculus primarily through the Active Calculus textbook, available for free online. I previously used a Larson, Hostetler and Edwards text, and used it for practice and big-picture sequencing while filling in the rest myself. The problems weren’t very useful to me — they lacked conceptual rigor and provided more quantity than quality.

Active Calculus is the opposite. Each section has several activities designed to introduce students to new concepts and make connections between big ideas, and 3-5 challenging, conceptually focused tasks at the end. Here’s one I used recently:


Here’s what I love about this problem:

  • It offers practice with both concepts and notation, in this case the concept of the definite integral and integral notation.
  • It includes nonroutine practice, requiring students to evaluate definite integrals where the lower bound is larger than the upper bound and sketching definite integrals point by point.
  • It circles back to previous content. Integration is presented not as a one-way process to nowhere, but with an opportunity to think about the relationship between integration and differentiation.
  • It uses practice as the foundation for a discussion about a concept. Students calculate and graph several definite integrals, and can then use those graphs to unpack something counterintuitive about integration — the idea that integration produces an infinite number of possible functions while differentiation produces exactly one.

Using the Active Calculus textbook means that I have many more good problems like this one to buttress my curriculum. But more than that, it means that I spend far less time trying to write good problems, and more time planning what I will do with them. I can think about the questions I want to ask, how I want to sequence the problem, the specific goals I have for the problem and how I can make sure students reach them, and how I can structure collaboration and discussions to emphasize those goals.

This is really just an argument for great curriculum. And it’s an argument for the value of spending less time figuring out what they will put in front of students and spending more time thinking about how they will use that curriculum to meet ambitious goals of learning mathematics.

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.