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7 comments on “Routines, Lessons, Problems, and Projects: the DNA of your math classroom”

Routines, Lessons, Problems, and Projects: the DNA of your math classroom

This blog post introduces a new mini-series from Emergent Math: Routines, Lessons, Problems, and Projects.

all four icon

In my time in math classrooms – my own and others’ – I’ve developed a rough taxonomy of activities. Think of these as the Four Elements of a math class: the “Earth, Air, Fire, Water” of math as it were. Or perhaps think of these as the Nucleic Acid sequence (GATC) that creates the “DNA” of your math classroom. Or the Salt, Fat, Acid, Heat of a class. 51dtoe1qufl-_sx260_Speaking of which, the author of Salt, Fat, Acid, Heat, Samin Nosrat, suggests “if you can master just four basic elements … you can use that to guide you and you can make anything delicious.” While I’m certainly not the first to think about teaching-as-cooking, I’m compelled by the way Nosrat distills cooking into four essential elements. I’d similarly posit if you can master these four elements of math instruction – Routines, Lessons, Problems, and Projects – and apply them in appropriate doses at appropriate moments, you can craft lessons and an entire course year for maximum effectiveness and engagement.

Let’s define our terms – after which we’ll criticize them.

Routines – Routines are well-understood structures that encourage discourse, sensemaking, and equity in the classroom. A teacher may have many different types of routines in her toolbelt and utilizes them daily.

Lessons – Lessons include any activity that involves transmitting or practicing content knowledge. Lessons can vary from whole class lectures to hands-on manipulative activities.

Problems – Problems are complex tasks, not immediately solvable without further knowhow, research or decoding of the prompt. Problems can take anywhere from one class period to three or four class periods.

Projects – Projects apply mathematical knowhow to an in-depth, authentic experience. A project occurs over the course of two to four weeks. Ideally, projects are outward facing, community based, and/or personally relevant.

These definitions may not be perfect. I’d encourage you to come up with better (or at least more personalized) definitions and toss ’em in the comments. I reserve the right to change these definitions throughout this mini-series.

To be sure, these four elements often blur and lean on each other: you might teach a lesson within a project. You may employ a routine while debriefing a problem. Many times I’ve been facilitating one of Andrew’s Estimation180’s as a routine and it wound up leading to a full on investigation (which we’ll call a “lesson,” I suppose). Is Which One Doesn’t Belong? a routine or a lesson? Or maybe it’s a problem. It honestly probably depends on how you facilitate it.

Most of the time, however, you’ll be able to walk into a classroom and identify which one of these four things are occurring. If students are engaged in some sort of protocol, they’re in a routine. If the teacher is standing at the front of the class demonstrating something, we’re looking at a lesson. If students are engaged in a complex task, we’re probably in a problem. And if students are creating something over the course of days or weeks, we’re probably in a project.

But why bother with such distinctions?

Perhaps I’m overly interested in taxonomy, but I find it helpful to sort things into categories (perhaps it’s a character flaw).

The real answer to the question of “why bother with such distinctions” is that I was trying to describe the difference between a “traditional” math classroom and a more “dynamic” one. Both of these terms are meaningless, even if they do connote what I’m trying to convey: traditional = bad; dynamic = good. Traditional classes are ones where teachers are lecturing most of the time. Dynamic classrooms are ones where kids are working in groups most of the time. But even that’s not a sufficient clarification: good classrooms employ all kinds of activities, including lectures, including packets.

So it began as an attempt to describe the ideal classroom juxtaposed against a teacher-centric one. A teacher-centric classroom might employ lessons 85% of the time, while a dynamic classroom might employ lessons 55% of the time (I’m making these numbers up entirely).

Then I began to find it challenging to talk about Projects vs. Problems. In my work I’m often asked to describe an ideal classroom: wall-to-wall Project Based Learning (PBL) or Problem-Based Learning (PrBL) or a mixture of both? And how often ought we actually teach in a PBL or PrBL learning environment? How does an Algebra 1 class differ from an AP Stats course?

I’m not going to answer these questions for you, but I hope that this framework will equip you with the vocabulary to design your best math class.

And just like halfway through my adolescence, they discovered a fifth taste (“umami”) we can’t discuss these four elements without the thing that binds classes together: active caring. Perhaps it’s backwards, but we’ll conclude this mini-series with a discussion about active caring and how it’s essential. The best routines, lessons, problem, and projects in the world are moot to a classroom without caring. I suppose it’s a bit too on-the-nose to make a Captain Planet reference with the fifth planeteer’s power being “heart” but that works well as a metaphor if we’re looking for a fifth, I suppose.

One last metaphor: you know those sound boards they have to mix songs? Those ones with a million knobs? And in every movie about a band there’s always a really cool scene where the band is killing this one song and the sound engineer slowly pushes those levers up while bobbing his head and looking at the producer all knowingly? That’s kind of what we’re doing here: playing with the knobs and seeing what it sounds like. We want to get better at each of these instruments individually, and put them together to make beautiful music. Or food. Or genes.

Coming up in this mini-series:

  • Routines: the driving beat of your class
  • Lessons: the stuff we envision, only better
  • Problems: then a miracle occurs
  • Projects: what they’ll remember in 20 years
  • Active Caring: the essential ingredient
10 comments on “Working while sad”

Working while sad

Until recently, I would have classified myself as a “happy” person. Now I’m not so sure.

Every day when I or my wife picks up my son at school there’s a 50/50 chance he’s in the counselor’s or principal’s office because he hates himself for something he did or didn’t do. When something – anything – negative happens, it’s a flip of the coin. Sometimes he’s able to slough it off. And sometimes, he goes into a complete and unstoppable downward spiral. He says he’s the “worst person in the world” or the “dumbest person in the world.” Neither of those things are true, nor is he receiving that message from anyone at home or at school (who have gone above and beyond trying to make emotionally safe accommodations).

So all day I’m on edge about 3:08pm, when his class lets out. Will I see my son bounding out with joy, ready for a rollicking afternoon of fun and games? Or will I see that grimace on his teacher’s face when we make eye contact which tells me everything I need to know about how the next few hours will be?

I check my inbox constantly, anxiously just waiting for that email to show up with the subject matter that simply states his name or something foreboding like “Today…” with my wife, his teachers, his counselor, and his principal all cc’ed. Once that email hits, or once I see his school on the caller ID, the rest of the day is over. It’s time to go pick him up early because he won’t be rejoining the class and he’s unsafe at that point. (I just checked it again.)

It’s not easy to enjoy things when your brain is occupied with such concerns. It’s very difficult to work in a profession that requires social interactions. It’s hard to do much of anything – go out to lunch, exercise – when a significant part of your brain is wondering “Is my son wanting to hurt himself right now?”

When people ask how he’s doing, my answer is “good,” because there’s a good chance that yes, at this very moment, he’s “good.” So it’s technically, possibly not a lie! But he’s not good. He struggles with mental illness in a way that we are all unprepared for. That I am unprepared for.

Thankfully, by dint of never seeking medical attention for myself, I have a fair amount of money stored away in an HSA, which I will be using to attend to my own mental health as I start therapy this month. Even after just two sessions, I feel better equipped to manage my own emotions and responses to challenging situations. Even just talking openly and getting acknowledgement of how goddamn hard life can be has been helpful. And hopefully with hard work it’ll get better.

So I guess I should end this blog post with a Point of some sort. So it’s this: consider whether talking through your anxiety / stress / struggles might help. Really consider it. If you have HSA dollars, use ’em. If you have free counseling sessions associated with your work (as my wife did at her previous job) use ’em. Or seek out a therapist that works on a sliding scale if the price point is challenging (which it truly is! Side note: my insurance will pay through the nose for medication and zilch for therapy, which is both dumb and Another Story).

Don’t try to go through things alone. Don’t bottle things up. Talk to your school counselor. Talk to a therapist. Talk to a pastor. These people are great at what they do. They’ll help you feel better about what you do too.


0 comments on “When 1/25 ≠ 2/50: team teaching”

When 1/25 ≠ 2/50: team teaching


My son attends an “open concept” school, a term that belittles the potential for such learning space. Before he started attending that school, I had heard of “open concept” as a fad that passed through schools in the 1970’s and fell out of fashion due to their unwieldiness. I had an image of two hundred students corralled in a gym-like room with their teachers trying to shout over the hundreds voices reverberating off the walls.

First off, that image is woefully misrepresentative, at least at my son’s school. Each “pod” has two grade levels in it. And even each pod has enough physical distance and visual blocks between the grade levels that there’s never really an issue of noise. In fact, the first thing that struck me when I was touring the school a few years ago was how quiet it felt. The students in the “open concept” school were much better at regulating their voices and being aware of their peers needs than in a smaller classroom with fewer students.

But that’s not the biggest boon offered by this open concept – as realized by my son’s school. The biggest boon is that teaching is a team approach at this elementary school. Each grade has 50 students with two professional instructors. While each student technically assigned to a home teacher, the day is fluid.

When you have two teachers teaching 50 kids, rather than one teacher teaching 25, it opens up endless possibilities for small group workshops, differentiation, and enrichment. One teacher can work with a handful of students while the other teacher can facilitate the rest of the grade. If one teacher is passionate about, say, Science and the other Social Studies, they can utilize their particular teaching strengths or passions. The two teacher divide and conquer certain subjects and certain concepts. By having the same room, their planning time is more natural and organic.

Even more than the logistical, technical, and pedagogical advantages of a team teaching approach for elementary school is the assurance that there is nearly always an adult in the room who knows every student on a deep level (and vice versa). Substitute teachers were always difficult for my son to handle: they don’t know the rules, they’re not following the schedule, and so on. Now, even when one teacher has a substitute, with rare exception he can make eye contact with the other teacher that knows him well and how he struggles in certain environments. If one teacher needs to go to an IEP meeting, the class doesn’t get put in “time out” or “baby-sitting mode.” If a kid is having a melt-down one teacher can take him or her aside without pausing the entire class.

I realize it’s not possible for schools to employ team-teaching. The numbers have to work out kind of nicely, with the number of teachers-per-grade being even. The physical space needs to be amenable to such a work space. The teachers require a level of professionalism and trust that isn’t as necessary when everyone is siloed. But it works at my son’s school and it works for my son. Every day he knows there will be someone in the class who knows him, and he never goes a day without seeing friends from previous years.

2 comments on “Stop Thief!, The Fugitive and introducing equations of circles”

Stop Thief!, The Fugitive and introducing equations of circles

When I was a kid, we had this super high-tech board game called Stop Thief!. The gist was this: someone committed a crime somewhere on the game board, which was rife with jewelry displays, unattended cash registers and safes. Your job as the detective was to identify where the thief was. The location of the thief was tracked by a phone looking device that calls to mind those old Radio Shack commercials with car phones. After each turn, the invisible thief would move some number of spaces away from the crime scene. The phone made these noises indicating where he could be – opening a door, climbing through a window, breaking glass. Based on these clues and the number of turns that elapse, you’d try to identify where he was.


Fast-forward a few years. We all remember this scene from The Fugitive:

These are the artifacts that were going through my head as I designed this lesson, linking the pythagorean theorem and equations of circles. In it, students must overlay a circle to establish a “perimeter” (side note: shouldn’t Tommy Lee Jones have used the term “circumference?”).

While this task only starts from the origin, you could quickly modify it to have other starting points, which would allow students to explore what the equation of a circle looks like when you center it wound non-origin points. I’d expect that to occur the next day or later in the lesson as part of the debrief.

Feel free to tweak it to make it better. The desmos graph is linked below, along with a couple word handouts.


(Note: a version of this task will appear in my forthcoming book from Stenhouse Publishers, Necessary Conditions.)

The set-up: a crime has been committed and it’s up to the students to establish a perimeter based on how much time has elapsed. After using the pythagorean theorem a few times to identify buildings the thief could be hiding in.

PrBL - RunningFromTheLaw-01

Given the time that’s passed and typical footspeed, the criminal could be anywhere up to 5 kilometers from the crime scene.

Which of the buildings above could he be in?

[Desmos Graph]


Additional resources:

Running From the Law

Running from the Law Student Handout

0 comments on “Necessary conditions: understanding groupwork with a three-legged pedagogical framework”

Necessary conditions: understanding groupwork with a three-legged pedagogical framework

At some point this year (2018), I’ll have a book for you to read from Stenhouse that proposes a framework for effective math classrooms. These are the three broad ingredients that create a successful math classroom as well as how a student experiences math. They are:

  • Academic Safety – the social/emotional state of a student and her self-regard as a mathematician
  • Quality Tasks – the thing that students are doing, working on, and/or creating
  • Effective Facilitation – the short- and long-term moves that promote mathematical thinking and sensemaking

Every successful math classroom I’ve been in has each of these three hallmarks in spades. In fact no successful classroom I’ve been in hasn’t had each of these hallmarks working for it. They’re our necessary conditions for great classrooms.

They work independently and in concert and can be the lens through which we can better understand classroom issues. Let’s take a common issue of unproductive or inequitable groupwork. Effective strategies will tackle one, two or all three of these elements. Let’s use this framework to better understand the issue, before we jump into the solution.

Fig 1-2

Is the issue one of Academic Safety?

Students may not be engaged in groupwork if they self-identify as a “non-math person.” It’s possible they’re only living up to the social academic status they’ve been given. How do students see themselves as mathematicians? Do they see themselves as mathematicians? Do their peers see one another as mathematicians? If so, how so? Are they publicly acknowledging the mathematical smartness of their peers?

Is the issue one of Quality Tasks?

I’ve been in classrooms where the issue around groupwork began with the fact that students weren’t being assigned groupworthy tasks. If you’re going to require an assignment occur in a group setting, the task ought to require (or at least be enhanced by) groupwork. Tasks are often developed for individuals but assigned to groups.

Is the issue one of Effective Facilitation?

How was the groupwork time introduced? Did you just assign the task and say “go” or did you have a structure in place? Do students have specific and understandable roles or is it the onus of the student to figure out where they fit into the groupwork dynamic? What norms are present in your classroom (and no, not the norms that are on the wall, but the ones that are actually present)?

Once we start to answer some of these questions, we might be able to better identify potential solutions. Maybe the classroom needs defined group roles. Maybe a norm of “same problem, same time” needs to be enacted. Maybe tasks need to be developed to push students deeper into the math content. Or maybe it’ll generate additional questions or additional need for understanding the issue at hand.

And as I mentioned, it’s possible (probable) that issues will bleed between our three pedagogical elements. Certain tasks can reinforce messages about mathematical self-regard. Unstructured groupwork can reinforce issues of academic status. It’s messy work, this teaching. Hopefully this framework will help you better understand the complex dynamic of a classroom ecosystem.




1 comment on “Counting Idling Cars”

Counting Idling Cars

pickupI’m sitting in my car, waiting to pick up my son from school. It’s too cold to wait outside  this time of year so I keep the heat on, the engine running, and continue listening to the Dunc’d On Basketball Podcast, the nerdiest podcast about basketball out there. I’m also quite anti-social, so I prefer to sit in the car, rather than, like, talk to people.

The driver of the car in front of me is doing the same (presumably, minus the podcast listening), ditto for the car behind of me. Maybe they’re reading “The Pickup Line,” an e-mag specifically for parents who sit in the car, waiting to pick up their kid from school. It occurs to me: boy there are a lot of cars idling in front of the school right now. I’d guess about 40. But y’know, someone should really count these up.

I get typically get to the school about 10 minutes before the release bell rings and I’m sort of in the middle of the pack of idling cars. I’d guess it’s about the average of when most cars arrive, again, most of which are idling. While I don’t conduct this environment-destroying practice all year long – when the weather is nice I’ll get out and check my phone, rather than talk to other parents – I practice it for maybe half the school year. That’s about 80 days or so.

80 days x 10 minutes of idling per day. Boy, 800 minutes of idling seems like a lot doesn’t it? And if there are indeed 40 cars at my son’s school, averaging a similar amount of idling time, we’re looking at 32,000 minutes of car idling. That’s over eight days of just idling.

We have a train that goes through town and we have signs encouraging us to turn off our car, rather than sit there idling, while we wait for the train to pass through. And I sometimes follow that instruction! I should probably follow it more often and more aggressively. But what about idling in the school pickup line? Or along the side of the school for us anti-socialites?

How much gas are we wasting?

How much Carbon Monoxide are we putting in the air?

How much gas waste / CO is the entire town/state/country contributing?

Would it be better to just switch off the car and start it later?

Boy, oh boy, someone oughta do the research on this…

What about at your school? How much gas is wasted in a day, week, or school year? Could students do the research? Could they create an awareness campaign for reducing gas waste (and presumably promoting cleaner air at their school)? Seems like something a bunch of go-getter students could handle.


1 comment on “Why we teach the “other stuff””

Why we teach the “other stuff”


“I don’t know what to say.”

“I don’t know how to talk to him.”


I’m sitting in a coffee shop with my back facing a mentor and her mentee, a college student who is apparently struggling through her semester. I can hear them clearly, even though I’m trying not to eavesdrop. The mentor is pleading with her mentee to email one of her professors to get help with an assignment, or even figure out when office hours are. “I don’t even know what to say!” the student response. The mentor patiently describes what questions to ask and how to start off the email, to no avail. “I don’t know how to talk to him!” she keeps responding. Pretty soon the student starts doing that thing that teenagers do where they start laughing when they’re really uncomfortable . It’s an unintentional defense mechanism employed by nearly all adolescents. She’s embarrassed by her inability to perform a seemingly simple task, so she starts blushing and laughing.

“I don’t want to talk to someone else.”

“That sounds like so many words.”

Later on, the mentor is trying to get the mentee to call the registrar’s office to find some information about something or other. Again, the student giggles that she doesn’t know how to talk over the phone. The mentor clearly cares deeply for her mentee. The mentee is clearly embarrassed at how nervous she is communicating to professionals.


Teachers sometimes scoff when we implore them to teach those other things. Things like communication skills, groupwork skills, self-reflection, these are “soft skills” that don’t appear in our scope and sequence. The state test doesn’t address them, and we have too much content already to cover to worry about these phony-baloney skills. I’m a math teacher, I teach math, that’s what I do. That’s my responsibility.

I can’t speak to the mentee’s instruction, but I’ve seen it frequently enough. Listening to her, she was paralyzed when it came to communicating with other adults, or at all. This is something she did not learn, let alone practice, while in High School. And now it’s preventing her from succeeding at the post-secondary level. Her paralysis wasn’t that she didn’t know her content well enough, it was that she didn’t have the ability to find out how to further her content knowledge. Whether that was technically the purview of her secondary math teacher or not, the responsibility to prepare students for post-secondary life falls on each of her teachers and the system in which they teach.

Does that mean that it is your responsibility – as, say, a math teacher – to teach students how to make phone calls? Or email professionals? Or create a study group? Or manage their time? Or teach all of those non-math skills?

Yes. Yes, it is.

The reason I taught math was because I love math. The reason I taught math the way I did was because I wanted students to grow as communicators and problem-solvers. Thankfully, I taught in a school in which we were unified that these were indeed skills we wanted our students to have when they graduated. I now coach in a model that aspires to make that happen for all students.

In our classrooms, we have students call and set up meetings with mentors in the business or academic communities. We have students stand and defend their work. We assess students on eye contact as well as content knowledge. And we teach them how to do it, not just tell them to and let them flounder. We do these, not in isolation, but as an entire school. In my class, this included these activities, as well as complex math problems that required collaboration, presentation of new and novel ideas, and practice and structures to handle it when these proved exceptionally difficult.



I’m heartened by these mentor-mentee relationships. Whether they were established via a specific program or they grew organically, there’s genuine care there. As exasperated as the mentor was, she showed active, authentic caring. I’m glad the mentee has that support system in place now that she’s in college. She’ll need the skills her mentor offers, as she clearly didn’t receive them in High School.