Anticipating student responses 


A colleague and I recently planned a lesson on developing a need for factoring when solving quadratic equations. It was totally unexpected. He walked into my room after school and we started talking. Two hours later, we had the framework of a lesson.

Two teachers. Two hours. One lesson (sort of).

What struck me was that the bulk of the time was spent anticipating student responses to questions we wanted to ask during the course of the lesson. We went back and forth about the roles of the distributive and zero product properties and how students might interpret these ideas in context. How could we use their responses to bridge an understanding of solving linear equations to solving quadratic equations? We didn’t want to shape how they answered, but simply craft questions that would naturally guide them to worthwhile discussions and new understandings. Whatever question we toyed with throughout the two hours, it always came back to the same criteria.

How might the kids answer? How will their response draw them nearer to the goal? How can their thinking help the lesson tell a story? 

Subconsciously, I think I do this. Just not enough. This experience connected well with the principles of my current book as well as providing me a nice reminder to plan the critical points of a lesson so that they pivot on student thinking.



Entry ticket

Yesterday my students were introduced to logarithms. Today, to follow up, I used an “entry ticket” requiring students to evaluate basic logarithms. It’s pretty simple and requires no prep. It’s one of the few times during the year it can work because of the minimal computation involved (another one is the unit circle).

Here’s the deal. I stood outside my classroom before class and had students line up single file along the wall. I waited for the late bell to ring and asked the first person in line to evaluate a given logarithm. For example:

If he answers correctly, he’s allowed to enter the classroom. If he doesn’t, then to the back of the line he goes! The person next in line steps up and the process continues until everyone has gained entry. For 27 kids, it took about 20 minutes.

I’ve done this several times and I’m still surprised on how wildly successful, and effective, it is. The kids love it! They always embrace the challenge with smiles and good energy. Even random kids passing by my classroom and other staff members root them on or cackle when students are sent to the back.

Other details:

  • I made up the problems on the spot, so planning was minimal.
  • I used a whiteboard to present the problems.
  • Kids that were absent yesterday had to learn from someone else in line that was present for the lesson. Interdependency!
  • After students were in the classroom, they played log war.



Summation notation, but way more

I’ve been rethinking all of my lessons this year. My hope has been to get my students to reason more. To think independently. To not be sponges. I’d like to think it’s been working. Here’s a recent lesson on summation notation that showcases this shift.

To open things up, I gave them this.

Super accessible and relevant to summation notation. In the past, I would have chosen a bell ringer that was closely connected to a prior lesson (i.e. review) than the current one. I wanted to provide remediation. I’ve learned this year that a relevant bell ringer is pivotal to any lesson.

Here’s what came next.

Again, very accessible. Last year Jennifer Preissel mentioned the “Stop & Jot” idea as a simple way of getting kids to write and reflect more during a lesson. Here, I gave them five minutes to express, on their own, what they wondered and noticed about the expression. After, they shared with their groups and we discussed as a class. By including “left side” and “right side,” I wanted to focus student responses. There were comments like “the +2 happens in every parenthesis” and “the number next to the +2 is going up by one.” Their observations led us to the brink of directly relating sigma notation to its expanded sum. In the past, I would jump right into defining sigma, the upper and lower limits, argument, etc. There would have been no exploring or thinking on their own.

Next, I ask them to move on to another example with the hope of finding a relationship.

It worked like magic. They see the same pattern from the Stop & Jot and they start to generalize. They have no idea what the “E thing” is, but it’s beginning to settle in how the left and right sides relate to one another. They discuss all of this in their groups. I float around. Observing. Listening. In the past, I would show them how to find this sum and answer their questions. Again, no self-exploration and making meaning of what they see.

Now they are to dissect and interpret.

This lacks clarity. Some students knew to write their interpretation next to the arrows, but many did not. As a checkpoint, we came back together and discussed.

Next: remove the right side.

Things are flowing now. The scaffolds are working. They know the relationship and successfully express the sum. In the past: The students would probably be completing this problem, but instead of using their own insight to drive the work, they’d be following what I said was the correct procedure.

Finish it off.

We come back together one more time to debrief and to address any questions the groups haven’t already. To bring things full circle, I mention the task from the bell ringer. “Ohhh!”

Lastly, on the next page, the proper names are reveled.

We then have just enough time for an exit slip.

This lesson is heavy on notation and I didn’t want to bog them down with symbols. The goal was to find meaning first, then discuss representation. It succeeded. What I miss out on is working in reverse. Namely, using sigma notation to represent a given sum.

What I love most about this lesson has little to do with summation notation. It’s much bigger. It stems from the approach. Bottom up. Using their own insights to help them find meaning. Doing less and allowing them to put the pieces of the puzzle together. This lesson is a microcosm of how I try to teach nowadays, which is much different than in the past. It symbolizes my growth as a teacher, as a learner.

Here is the handout.



Thank you hint tokens.

A few weeks ago I stumbled across the idea of a hint token. Think of it as a get out jail free card, but for the classroom. While working on a task, groups can trade one for a hint from me.

Loving this idea, I immediately went to implement it. This, I thought, would be a great way to give students more ownership over their learning and hopefully learn to rely more on one another. The first time around we were studying sequences and I gave each group two hint tokens in the form of Jolly Ranchers (thanks Sam).

What happened was something unexpected: no tokens were used.

They may have simply wanted to eat the candy afterwards, I’m not sure. I wouldn’t doubt it. That said, what was most impressive was how they worked interdependently to solve the problems. I was essentially ignored.

Afterwards, I realized how empowered I was. The kids need me far, far less than I think. Understanding something and feeling something are very different phenomena. I’ve always known that my students should need me less, but I now know how that feels. It’s incredible. I even communicated this to the kids and saw the realization in their faces. They felt the same way.

This experience has had a dramatic affect on my teaching. What’s ironic about this is that you’d think I would move to incorporate hint tokens every day. The thing is, I’m not. Instead, I ensure that students have the opportunity to own their learning and sit longer in each other’s thoughts. It usually consists of 10 minutes of focused, small-group discussion and productive struggle during every lesson (I have 42 minutes class periods). During this time, I provide no any assistance of any kind.

As a result, it’s common for me to pull up next to a group, watch and listen. Before, they would be inclined to ask me something simply because they could. Now, they forget I’m even there. I silently assess their thinking the entire time – which reminds me of a live version of video-based PD.

It’s a win for everyone. They purposefully and interdependently think through a problem, which spurs engagement and ownership, and I get valuable insight into their thinking that serves as a driving force for the rest of the lesson…and beyond.

Thank you hint tokens. Thank you for facilitating this change in the culture of my classroom.

I’m left thinking that this shift may be directly related the class chemistry I’ve developed this year – which has cultivated a willingness to learn and explore amongst the kids. In other words, the tokens could have simply been what I needed use in order to realize the new path that learning is taking in my class. I don’t know. Maybe next year I will need the tokens. We’ll see.

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