A window into my classroom. This isn’t what we do every day (tomorrow, we’re playing War (the card game) with irrational numbers). Other days we do 3-act tasks or variations on them, or play with Desmos, or build bungee cords for stuffed animals that don’t reinforce negative stereotypes about women. But most days — the large majority of days — look like this.

**Warm up:** Warm up is a few question on one topic. I write them all at once for a week at a time. Most review a specific, previous topic. Some are formative assessment for a future topic. Some are quiz or test review. Some are problem solving, meant to create argument. If I’m giving a quiz, that’s the warm-up. They’re never graded (except for quizzes); I allow enough time for most students to finish, but never all — takes too much time, and is why I never grade it.

**Number talk:** Put a mental math question on the board. Students solve mentally, and can feel free to estimate to find a close answer. First, I take hands for all answers students have. Then, I pull popsicle sticks for random students to share. Then I take hands for more methods. I scribe these methods on the board. I often have students turn and talk to analyze which methods they like best. At the end, students write on a weekly sheet which method they preferred and why.

**Warm up review:** I almost always put student work up on the board. It just becomes normal, and isn’t a big deal — everyone’s work will go up at some point. I ask directed questions to focus on key parts of the warm up I know will be difficult. I allow students to ask questions to the student whose work it is to clarify, and probe for understanding wherever possible.

**Intro to new material:** I introduce new material with a few questions for students to explore on their own. They’re meant to start easy, and lead toward the key understanding for the day. Students work on their own, after I have framed what they’re doing and what they’re looking for. Some students will get it, some won’t, and that’s fine — this part of class is as important for building habits of mind as it is for learning content.

**Discussion:** Usually starts with focused partner discussion about what students were just working on. Then, I’ll often show different approaches students took to analyze them, or we’ll discuss our way through a new topic. If students aren’t getting it on their own, I am 100% unafraid to teach them explicitly how to do it.

**Notes:** Students record key ideas. These are short and sweet, and everyone records.

**Group practice: **Several questions on the topic that we’ll do as a group. These solidify basic skills, apply the concept in a new context, stretch it further, or synthesize it with other topics. Sometimes students work alone and then share out, sometimes they work in partners, sometimes we solve as a class.

**More practice:** More questions on the day’s topic, progressing from simple to complex, that students solve on their own. If students finish early, I keep challenge packets of hard problems pulled from the internet for them to work on for extra credit. This is prime time for me to pull up a chair with students who are struggling.

**Exit slip:** A few short questions, taking less than 5 minutes, for students to complete. I look at them every day, and grade them same days, to figure out what students learned and where we will go next.

**Homework** is mixed review of key topics for the year, checked every day for completion, occasionally for accuracy. 90% of students should spend less than 20 minutes on it, half spend less than 10.

That’s a typical day. It’s not every day. Sometimes I tell students what to do; some days we spend immersed in one task; some days are practice days, some spend more time on one part and skip another.

I think it’s important to note that, while I am interested in and influenced by people who label their teaching as “Problem Based Learning” and “Inquiry” (I’ve been called such a teacher in the comments on this blog), I don’t always think of myself that way. I believe practice is important; I believe having daily learning goals is critical; I will always give direct instruction if I don’t feel like I have a good inquiry activity for a topic. And even on inquiry days, students are rarely spending more than 15 minutes discovering and discussing new math. I believe that spending whole classes discovering and discussing leaves no time to move learning into long-term memory. Students should explore and discuss to build understanding, but there needs to be codification of knowledge and practice — of the day’s topic and prior topics — in order for students to move from novice to expert mathematicians.

Dan KearneyInteresting. So what drives your “new content/topics” that you introduce (via direct instruction?) Is there some sort of “master plan/schedule”?

dkane47Post authorI’m definitely still figuring out how to do that, but I try to go backwards from big ideas — what students should be able to understand and do by the end of a unit or topic. For instance, this unit is on square roots and exponents, and starts with approximating imperfect squares, then simplifying square roots, then cube roots, then rational and irrational numbers. Then we move through exponent rules and to more complicated applications of exponents. Then, new unit on scientific notation, unit conversions and quantities.

I just try to break down those topics into small chunks so that each day’s understanding builds on prior knowledge — for instance, one day we would approximate square roots using whole numbers, and apply that in a few different ways. Then, the next day we would move on to decimal approximations.