Why Instructional Routines?

I’ve been digging into the book Routines for Reasoning, by Grace Kelemanik, Amy Lucenta and Susan Janssen Creighton.


The book focuses on instructional routines — activities that use a consistent structure to reach a range of mathematical goals. Here are three quotes from the book that have framed my thinking:

 It is significant to realize that the most creative environments in our society are not the ever-changing ones. The artist’s studio, the researcher’s laboratory, the scholar’s library are each kept deliberately simple so as to support the complexities of the work in progress. They are deliberately kept predictable so the unpredictable can happen.

-Lucy Calkins, Lessons From a Child

Mathematical reasoning requires students to be attentive to both the content and to one another. Listening to students’ ideas and building from them to a rigorous mathematical curriculum requires extraordinary concentration on the part of the teacher. New instructional routines were needed to make it possible to manage thirty students in one room while they were reasoning and critiquing the reasoning of others, as well as to make it possible for students to construct new kinds of relationships with their classmates, in which it would be safe to say what they think and appropriate to raise questions about someone else’s thinking. Instructional routines could enable both students and teacher to focus on the mathematics rather than on who is supposed to be doing what when. And they could change what we think it means to learn in math class.

-Magdalene Lampert, Routines for Reasoning, Foreword

When people are first learning to drive, they are faced with a million small details to attend to: when and how to adjust mirrors, how to operate headlights, how to operate wipers, how to operate the radio or music, finding money for tolls at an upcoming toll booth — and all this on top of the crucial skills of steering, accelerating, braking, and paying attention to the movements of other drivers around them. As drivers become more familiar with their vehicle and the act of driving, many of these small, repeated actions become automatic and require little attention or thought, allowing drivers to focus most of their attention (we hope!) on their own movement and the movement of other drivers around them. Instructional routines serve the same function: they make more predictable the design and flow of the learning experience: “What is it that I’m supposed to be doing?,” “What question will I be asked next?,” or “How will things work today in the lesson?” The predictable structure lets students pay less attention to those questions and more attention to the way in which they and their classmates are thinking about a particular math task.

-Grace Kelemanik, Amy Lucenta, & Susan Janssen Creighton, Routines for Reasoning

I’m excited to apply these ideas in my classroom because instructional routines have the potential to:

  • focus student thinking on learning, rather than classroom procedures or the structure of the lesson
  • provide opportunities for repeated, focused practice exercising the mathematical practices
  • provide effective support for struggling students to access challenging mathematics
  • scaffold discourse for students to more effectively share ideas

Routines for Reasoning complements many ideas from David Wees, Kaitlin Ruggiero, and Jasper DeAntonio’s morning session at Twitter Math Camp in July. That session inspired me to begin implementing an instructional routine called Contemplate then Calculate and to experiment with other routines. Reading this book and diving more deeply into what routines can look like and their various benefits for students has me thinking about how I can take other elements of my class and adapt them, using elements of instructional routines to do better by my students. More to come on both of those goals.

Teachers as Architects


I think architecture serves as a better comparison field than medicine does. Architects, like teachers, usually have multiple goals they try to satisfy simultaneously. Safety is nonnegotiable, but architects may also be thinking to a greater or lesser extent about energy efficiency, aesthetics, functionality, and so on. In the same way, some goals for teachers are nonnegotiable — teaching kids to read, for example — but after that, the goals are likely to vary with the context. In addition, architects make use of scientific knowledge, notably principles of physics, and materials science. But this knowledge is certainly not prescriptive. It doesn’t tell the architect what a building must look like. Rather, it sets boundary conditions for construction to ensure that the building will not fall down, that the floors can support sufficient weight, and so on.

In the same way, basic scientific knowledge about how kids learn, about how they interact, about how they respond to discipline — this knowledge ought to be seen as a boundary condition for teachers and parents, meaning that this knowledge sets boundaries that, if crossed, increase the probability of bad outcomes. Within these broad boundaries, parents and teachers pursue their goals.

-Daniel Willingham, When Can You Trust the Experts? (p. 221)

I really like this comparison, especially as someone who has compared teaching with medicine in the past. It’s making me think about what my “boundary conditions” are for learning. What is the scientific knowledge that I use to inform my teaching? Here are six principles that I think about with some frequency, and have found useful in my classroom:

  • Practice is essential to building flexible, durable knowledge. This practice should be spaced over time and interleave topics whenever possible.
  • The teacher’s job is not to make learning easy. While learning should not be so difficult that students are likely to give up, activities that are hard can lead to more powerful learning because the brain is more active in retrieving information and making connections.
  • A growth mindset is a major factor in learning. Shifting students’ mindsets  is difficult, but influencing these beliefs has a great deal of influence on future learning.
  • Problem solving and critical thinking cannot be taught independently of content. While these are critical goals, a strong base of knowledge across a range of content areas is an essential prerequisite for problem solving and critical thinking.
  • Treating students like experts is an inefficient way to lead them to expertise. Experts and novices think in qualitatively different ways, and expertise requires a great deal of carefully structured deliberate practice that is often distinct from the activities that experts excel in.
  • Incentives and feedback play a complicated role in students learning. Grading and other normative comparisons can reduce intrinsic motivation, inhibit future learning, and reduce the effectiveness of feedback.

None of these are prescriptive, but I find all of them useful. While they don’t offer specific guidance on how to teach, they can be particularly useful when I find out that students haven’t learned something. If I teach a topic, assess students, and learn that they don’t know what I thought they would know, I can use these principles to try to diagnose where something went wrong.

I’m curious if I’m off base with anything here, or if other principles belong on this list. For instance ,there’s a great deal of literature on feedback, explicit instruction vs constructivism, formative assessment, and more, but much of that research is conflicting or difficult to distill into concrete boundary conditions. I’m also curious about the complementary knowledge necessary for teaching that is not based on scientific principles, and how that all of that knowledge can fit together.

Defining Sense-Making

One goal I have in my teaching is to support students in developing productive beliefs about what mathematics is and how they perceive their relationship with mathematics.

One phrase I’ve found myself using more and more with respect to student beliefs about mathematics is sense-making, and it seems like a useful exercise to try to define sense-making and articulate what I do to promote it in my students.

sense-making (n): a belief that mathematics is a logical system where new knowledge is consistent with and connected to previous knowledge; a disposition to search out that logic when learning something new

I don’t have any easy solutions for making this happen for my students. I do have a few things I try to do, and most important for me is spacing these activities over a unit to give students multiple at-bats with this type of thinking.

One useful method is what Ben Blum-Smith calls jamming — “posing a mathematical task in which the underlying concepts are essential, but the procedure cannot be used”. This Illustrative Mathematics task is a nice example. Kids tend to see function notation as “plug that number in for x and see what pops out”.
Screenshot 2016-11-07 at 6.54.30 PM.png
Here they have to connect that knowledge to what they know about functions and equality, and do so in a way that is often unfamiliar for students.

My second go to is just asking students to explain why something is true. For a function y=(x+a)(x+b), why are the x-intercepts at (-a,0) and (-b,0). In my experience, even kids who have discovered that principle on their own are likely to forget it and have trouble explaining it in the future. Spacing practice with this type of thinking helps to mitigate that forgetting.

Finally, I explicitly ask students to draw connections between problems. Figure out the odds of rolling three dice where the product of the three numbers is odd. Now figure out the odds of rolling three dice where the product of the three numbers is even. How can you use the first problem to help you solve the second problem?

These approaches often feel mundane to me. They don’t result in my most spectacularly engaging teaching. But they are short tasks, they lead to fruitful discussion, and they very often reveal to me that my students understand less than I think they understand. Then I can go back, dig into their understandings, and try to support their sense-making. No shortcuts, just hard work worth doing.

Growth Mindset: From Rhetoric to Action

I presented yesterday at NCTM Phoenix. It was titled Growth Mindset: From Rhetoric to Action. The ideas come largely from this post from last year and the thoughtful responses in the comments. My slides are here, and here is some fascinating further reading on academic tenacity.

My thesis is that promoting a growth mindset is hard, and is particularly hard for a subset of students who are most disaffected and have had the most negative experiences with mathematics. In my experience, most of the interventions that are commonly talked about — praising effort rather than ability, encouraging students to try new strategies when they are struggling, creating space for collaborative work — are ineffective for these students.

I see a student’s mindset as a function of two variables:

If I am telling a student to have a growth mindset, but those words don’t match the experiences that student is having in my class, they’re unlikely to think of themselves differently.

I do think there are a few things we have control over that can influence this function. We can

  • Carefully define what success looks like in math class
  • Build relationships so that students are willing to take risks
  • Pay particular attention to students who have a history of failure
  • Have scaffolds and supports ready to move struggling students toward success

None of these ideas are groundbreaking. I asked the group at the start of the presentation to think about and discuss some possible negative consequences of growth mindset, and folks in the room named pretty much everything I shared. If there’s one idea I hope people took away, it’s that this work is both extremely hard and absolutely worth doing.

I got to spend some time with Henri Picciotto, and he said something smart at dinner last night. “When you grade, you help one child at a time. When you plan, you help all kids. Spend your time accordingly.” I’d love to add a corollary — in the same way that our time planning lessons is valuable, we can spend extraordinarily valuable time planning for how we create a classroom that makes students feel like they belong and that they can be mathematical thinkers. It’s relatively easy to spend some time puzzling over how I want to introduce polynomial division next week. It’s much harder to spend some time trying to figure out how I can define success in my class so that every student is able to feel like they can be a math person.


Student Feedback

I think #observeme is a really cool thing. I’ve enjoyed following the hashtag and seeing great things happening in a ton of classrooms. My school is working to increase professional collaboration and observation, and I’ve had several colleagues visit my class so far this year. Thing is, I haven’t received much helpful feedback. I know I’m not alone with this challenge. Other teachers often say they weren’t there for long enough to give helpful feedback, or just never follow up.

I don’t mean to criticize #observeme. It’s powerful just to have other teachers in my classroom and get these conversations started. I’m excited that this is a priority for my school and we are working on common language and goals to focus these observations. I’m sure that over time I’ll receive more useful feedback from my peers. The goal of #observeme isn’t just to get me some useful feedback; it’s also to create a professional community that values observation and continual learning.

But if my goal is to get some useful feedback on my teaching that I can use to get a little better tomorrow, I think I’ve found a different way to do that. Here’s a survey I asked students to fill out via a Google Form:


It’s adapted from the ideas in Dylan Wiliam’s book Embedding Formative Assessment. I tried to make the language more accessible for students, and briefly framed what these ideas mean to clarify what I was looking for from students. The survey was anonymous, and only took students a few minutes. Here are a few of the responses I received:

Even though we have been given a sheet that tells us what we are being assessed on for synthesis tasks it can still be a bit confusing what precisely we are being assessed on for a specific synthesis task. Maybe this is a misunderstanding that I have but it would be helpful just to revisit that topic quickly again.

I think that it is really hard to give peer to peer feedback. Coming up with a strategy to help facilitate that would be something to look at. I want to know how these problem solving skills can help me in the future. So every one in a while just vocalizing what that looks like would be an area to improve on. I think also when giving feed back on synthesis tasks, it can be nice to have a very brief verbal explanation as well as what is written on the page.

I feel that even when we do collaborative work it still is very independent. I think finding a better way of holding students accountable for their group work is an area to improve on.

These are pretty insightful comments into my classroom, from the people who are likely to best understand what is happening. There were plenty more, affirming things I do well, reminding me of weaknesses I need to work on, and providing insights into areas of my teaching I hadn’t been thinking about. For a few minutes of my time mocking up this survey and a few minutes of their time filling it out, I’ve got a lot of great feedback to work with.

How My Teaching Has Changed

I think I’m a better teacher than I was a few years ago. I look pretty different on the surface. I’m more relaxed in the classroom, I speak more clearly and confidently, I can plan classes more quickly, and I’m generally less stressed about the day to day responsibilities of teaching.

But most of that is incremental and only tangentially connected to student learning. Here are some more substantive ways my teaching has changed that I think have actually made a difference.

I ask myself, pretty incessantly, what my goals are for a lesson and how an activity I have planned meets those goals. I don’t do activities that don’t meet my goals, and I’m likely to think in terms of these goals when I decide to allow more time for an activity or cut it short.

If students don’t know something, my old response would be to jump in, offer an explanation of what I thought they were missing, and try to push my knowledge into their brains on the spot. Now, I’m much more likely to step back and realize that the best response is often to circle back to that topic later in class, the next day, or the next week, once I’ve had time to think through student misconceptions and figure out next steps that are more likely to make a difference.

Student Thinking 
When I first started teaching, a lot of my goals were around getting students to say right answers. I spent class time asking questions that were implicitly seeking validation of my teaching by trying to lead students to say clever things. I’m much less interested in that now, in comparison with students thinking smart things. I’ve become much more comfortable with wait time, and I’m less concerned with that perfect series of leading questions to get kids to say some right answers than with a smaller number of questions that kids think about for more time, talk about in partners or groups, then share with the class. Maybe they don’t share that perfect answer. That’s fine. It’s about the thinking.

When I give a task, I almost always give students time, individually, in partners, or in small groups, to work through a problem or task. Then, I try to start any full-class sharing with a few students chosen in advance to move discourse toward my goals for the task. I still take hands, but far less of the time, and I always have a voice in the back of my head reminding me that when I take hands, I’m hearing from an unrepresentative subset of the class.

Discovery, Follow-Up

Lots of really thoughtful people got me thinking in their responses to yesterday’s post. Check the comments and Twitter for a sample. Three are really sticking with me.

Anna Blinstein:

I think that my only addition or caveat to your post, Dylan, is to push back a bit on the goals of math class. If the primary or only goal is remembering/applying mathematical content knowledge, then your post makes complete and total sense – we should probably use discovery sparingly; it is helpful as a motivator (basically, the intellectual need and wonder categories you listed) and maybe helps some students remember some ideas some of the time. But, if one’s goal is to teach students to think like mathematicians, then I don’t know of a better way than having them engage in the process of doing math consistently and frequently while also seeing models of what this might look like and getting feedback on their efforts and ideas. I don’t think that anyone would argue that a discovery approach is the most efficient method of transferring knowledge, but for me at least, that’s not the primary goal.

Dan Anderson:


(my answer)


Avery Pickford:


These all hit me pretty hard, and I’m questioning a bunch of what I wrote yesterday. I’m going to try and reframe my argument and see where this goes. Call me out if I’m totally off track here. There are two issues I see. Does discovery lead to better learning of content, and does discovery learning teach students to think like mathematicians and acquire future knowledge for themselves? The second question is the one I want to tackle.

I think the answer is yes. First, an example.

Introducing the Unit Circle

I recently introduced the unit circle. Students started by constructing right isosceles and equilateral triangles to derive the values of sine and cosine at a few angles. I chose a discovery approach for this lesson because it had students practicing relevant triangle geometry, seemed manageable, and the much more didactic alternative I was imagining sounded boring.  The activity felt successful; every group found the value they were looking for, we followed up with a brief discussion, and then groups derived the rest of the unit circle.

I was surprised when, three days later, many students failed to produce a similar proof on an assessment. I don’t mean to argue that discovery never works, just that in this instance it came up short, and that doesn’t feel like an isolated example to me.

Contrast: after that assessment, I gave students this image:

I told them that this was a new unit circle, with angles at 15, 75, 105 and etc degrees. I gave them the values of sine and cosine at 15 degrees and asked them to figure out as much additional information as they could.

I would argue that this activity also asks kids to think like mathematicians. It’s much more subtle, in that there is no grand reveal of the unit circle at the end. But it’s still valuable thinking. And, more importantly for me, I find that more students have a chance to access a task like this, because they all have similar background knowledge at this point. This task also involves acquiring knowledge, and can be metacognitive in thinking about strategies that are useful for acquiring knowledge. And if I have to pick between a discovery activity introducing the unit circle and a demanding task asking students to reason with what they’ve learned, I prefer this one. Not by a ton, but I do.

Discovery often meets ambitious goals of mathematics learning. I think that, for many students, my unit circle exploration did that. But it didn’t land for everyone. And, based on my experience with that activity and similar activities, it’s too often the same few kids who are left behind.

I don’t believe that discovery has a monopoly on teaching students to think like mathematicians, and I don’t believe that every lessons has to involve discovery in order to teach kids how to acquire knowledge themselves. I’m at a 3.5 on Dan’s scale, which I take to mean somewhere around a third of my lessons involve some element of discovery. I would like to nudge that number upward a bit — but only if it means I’m doing discovery well, for every student. I would rather focus a small amount of energy on these goals but do that well than try to do it every day and possibly undermine those same, important messages.

I hear the argument that it’s important to teach students to think like mathematicians and acquire knowledge for themselves. But I want to avoid being dogmatic about using discovery as a means to do so. Instead, I want to focus on doing discovery well when I use it. Maybe I’m just coming up short in my knowledge and skill with that particular pedagogy; I’d love to hear folks sound off in the comments with the practices that have proven useful for them with these types of lessons. But in the meantime, I’m going to try and meet all of the above goals with as many different means as I can.