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. 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 know how, 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’re used to, but better
- Problems: then a miracle occurs
- Projects: what they’ll remember in 20 years
- Active Caring: the essential ingredient

This blog originally appeared on Emergent Math.