Lessons: the stuff we envision, only better

June 28, 2018

Geoff Krall

When you think of a math lesson, you probably conjure up an image of a teacher in front of the classroom demonstrating mathematical concepts. While that certainly qualifies as a lesson, I’d like to broaden your mental image. Consider a “lesson” any facilitated activity where students are building or practicing their content knowledge. In addition to our imagined lecture, let’s also consider activities such as card sorts, investigations, practice time, and other structured times in the classroom.

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

Lessons probably make up the bulk of your course. Students walk in to your room, you teach them some stuff, the day ends. That’s a lesson. How you teach offers endless possibilities. Let’s look at some of these possibilities.

The Lecture

There’s nothing inherently wrong with a lecture. I’d suggest it’s not always the best way to engage students. But oftentimes it is the most efficient way to transmit information, provided you are lecturing effectively. How does one lecture effectively? Despite being perhaps the most oft-used instructional approach, little time if any is spent in pre-service teacher programs in how to do lecture well.

Things to consider:

  • How will you ensure all students are engaged throughout the lecture, not just an eager few?
  • What’s the shortest amount of time you could possibly do the talking? Go with that. And maybe subtract a few more minutes.
  • Are you incorporating visual elements into your lecture?

When you’re lecturing, you want to stop and prompt discussion often, perhaps every 3 minutes or so. Rather than asking a question and waiting for a student to raise a hand, consider utilizing some of our general discussion routines from the previous post. The more you can make your lecture feel like a conversation the more successful the lecture will be.

When you’re lecturing, try to get students in the mode where they’re talking to one another rather than to you. See this blog post on various lecture models.

Some additional tips for lecturing:

  • Start your lecture with pizazz. Bring in a recent news article that pertains the the topic. Start with a memorable or funny quote. Post a picture or diagram and ask a question about it. For example, launch a lecture on horizontal asymptotes with the following graph and the prompt “Do you think these lines will ever intersect? Turn and talk to your neighbor and explain your reasoning.”

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Create a hook that will grab students’ attention. A picture plus the Notice and Wonderprotocol works extremely well.

  • Question authentically, not putatively. Questioning to get to deeper understanding is a skill that takes years to hone. It’s important to get genuinely curious about students’ ideas. As much as possible try to avoid the punitive, I-bet-he’s-not-listening questioning. Of course we want students to be paying attention, but we don’t need to “gotcha” students by asking them to derive the quadratic equation on the spot when we’re actually trying to make them feel foolish for zoning out during our boring lecture on the quadratic equation.
  • Talk slower. Every human talks 30-40% (not precise calculations) faster in front of audiences than they do in normal conversation. I’m not sure why, but it just is. Slow down. You need natural pauses and a good cadence, otherwise your words will morph into that of Charlie Brown’s teacher. I found this potentially effective technique:

Mark a paragraph / in this manner / into the shortest possible phrases. / First, / whisper it / with energetic lips, / breathing / at all the breath marks. / Then. / speak it / in the same way. / Do this / with a different paragraph / everyday. / Keep your hand / on your abdomen / to make sure / it moves out / when you breathe in / and moves in / when you speak.

Before you whisper each phrase, take a full bellyful of air and then pour all the air into that one phrase. Keep your throat open, and don’t grind your vocal chords. Lift your whisper over your throat. Pause between phrases. Relax. Then, take another full breath and whisper the next phrase. Whisper as if you were trying to reach the back of the room.

The Investigation

As a fan of the Discovering Mathematics series of math textbooks, investigations were a staple in my classroom. These lessons involve an intentionally structured activity that reveals some new mathematical truth.

Using tools or manipulatives

As an example, here is an activity on Triangle Inequality and dried spaghetti:

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Kids use their hands and dried spaghetti to determine the triangle inequality theorem: the sum of two sides of a triangle must be greater or equal than the third side.

Discovering Geometry was big into patty paper activities. These were excellent, cheap ways to get kids using their hands to make discoveries.

Using a highly scaffolded series of questions

This was the mode for my Running from the Law lesson.

In this activity students (much like in the spaghetti activity) identify mathematical concepts through purposeful questions. In Running from the Law, it was the connection between the distance formula and Pythagorean’s Theorem.

The questions are carefully ordered to point out possible discoveries hidden in the mathematical weeds. In some ways these activities mimic a quality activity debrief.

Using technology

The desmos team – and many of their contributors via the activity creator – use the clean interface to construct lessons that allow for students to construct their own understanding through carefully designed activities.

Desmos’ Central Park is a great example of this.

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Students begin by interfacing with a challenge, notably without any discernable mathematics. Throughout the activity, students are prompted to identify what information would be helpful to solve the challenge. Eventually, we build enough knowhow to write expressions that help us out. Each slide presents an additional prompt intended to get students to think mathematically about the scenario.

The Card Sort

Another general type of lesson is The Card Sort. Teachers provide students materials that need to be matched up or ordered in a specific way to make the puzzle work. The most common type of card sort is matching.  Students match two or more like items, typically in the form of paper or card cuttouts.



A twist on the matching card sorts I quite like is that of “dominoes.” It’s like card sorts in that there are cuttouts and students are asked to arrange them in the matching order. But in this case each cuttout has two “things” on it and they match with another “thing” from another card. The result is a circular matching activity:

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I like it because it offers an immediate check: the “dominoes” should circle completely around and there shoulndn’t be any gaps.

Things to consider:

  • Card sorts take a little time to build. It’s helpful if you have a template. Here’s one: Card Sort Template
  • Card sorts take significant time to cut out and put into plastic baggies. However, if you do it once – and have students place them back in the baggies at the end of the period – you’ll have them forever. I’ve had some card sorts in baggies for almost ten years now.

The Practice Problem(s)

Some classes and class days incorporate a lot of practice problems, packets even. That’s ok. We can work with that. A packet of a few high-quality problems can be an effective means of deepening understanding. I’ll go ahead and re-emphasize it for ya: a few high-quality problems. Now that we have that out of the way, we can hone in on effective means of teaching on a day – or a time of day – with a lot of student practice. I’ll offer two strategies that make the Practice Problem lesson an effective one.

Same problem, same time

Assuming students are progressing through practice problems in groups (which I recommend), make this a norm in your class: “same problem, same time.” This means that group members cannot proceed to the next problem or next page until all their group members are ready and have demonstrated understanding. Every group and every group member ought to be on the same problem so they may discuss it when it becomes challenging. You should never have a student call you over to ask about a problem that they’re working on and their groupmates aren’t (either because they left him in the dust or vice versa).

Participation Quiz

What are the norms of groupwork you want to see in a given problem work time? Make those public and identify when those moments are happening – or not happening. This can easily be achieved through a document camera or anything that’ll project a document.

In this case, the teacher identifed “plusses” and “deltas.” Or, positive behaviors or phrases students are exhibiting and behaviors that need to be changed.

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In this case, plusses include “OH I GET IT NOW!”, all heads in, paper in middle, “how do we solve this” and other markers of persistent problem solving. The deltas include “crosstalk” and “phone out”. At the end, you can debrief with the class with this document: how did we do today? What do we need to focus on for tomorrow? What ought we celebrate?

Note that the teacher has maybe five “plusses” for each “delta.”


That’s four lesson “types,” which is certainly not exhaustive. This exercise through the DNA of our classroom is not meant to be exhaustive or definitive. But it is meant to give us some common vocabulary. And, as with routines these activities are malleable, and even interchangeable. You may wish to employ specific sharing Routines throughout your Lesson. You may wish to follow up a Lecture with a Card Sort (is that a Lesson followed by another Lesson?).

What other lesson types or structures ought we add to our list?

What else ya got?

  • I have this facilitation one-pager from Necessary Conditions (Krall 2018). That might give you a nice menu of teaching techniques.

Read more at Emergent Math.

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