HOT v. the most appropriate

Every now and then I’m confronted with the idea of asking higher order thinking (HOT) questions. A lot of teachers are. In fact, there seems to be an unhealthy infatuation with HOT questions in education.

Don’t get me wrong, HOT questions certainly have their place. They serve to connect ideas, broaden perspectives, and deepen understanding of the content being learned. I strive for HOT questions. Even if it’s indirect, the heart of any good lesson, I believe, revolves around them. They’re a must.

That said, there’s much, much more to questioning than HOT questioning. In fact, by placing so much focus on HOT questions, we can lose sight of how questions build off one another and their dependency on current levels of student understanding. What is the goal of the question? How will it lead to the next? HOT questions can be too demanding and, consequently, create a gulf between what is currently understood and what’s expected. Any question is entirely dependent on our students – nothing else. They must meet them where they are.

What I strive for is not necessarily asking more HOT questions, but finding the most appropriate questions given the context. “What” and “when” questions should not be frowned upon if they are frequently used during a lesson. Instead, we should be critical of the sequence of any and all questions we ask and how this sequence impacts students’ abilities to answer HOT questions.


Teacher = engineer


Up until recently, I never thought of what I do everyday as engineering. I always felt that engineers were those highly intelligent, creative people who mastermind the things we encounter every day. Those solution-oriented folks that use mathematics, science, and their own insight to solve problems that impact a vast array of human needs. They were the engineers.

But what greater, more important engineer is there than a teacher? Don’t we create solutions that enable other humans to make meaning? Don’t we create moments of debate and wonder? Don’t we design learning? Don’t we make every other profession possible?

I think it’s fair to say that the average teacher doesn’t use differential equations or Ohms law or advanced mechanics to reach their kids. I get it. But the classroom is a complex system in which three powerful forces – content, management, and pedagogy – all interact in dynamic ways. We teachers attempt to make sense of these three forces and their relation to one another. We are critical of every moment – every thought – since each one has a momentous impact on the next. We use constraints and limitations, from learning styles to broken copy machines, to construct magical moments that alter lives.

And just like engineers, we fail. A lot. No matter how seasoned the teacher, expected learning doesn’t always happen. In fact, everything we do is trial and error. Good teachers know that the complexity of our work causes us to fail early and often.

Do I hope to imitate an actual engineer? No. Am I going to add “Learning Engineer” to my resume? Of course not. Besides, I hate titles. They clutter the real work that needs to be done.

But I know that there’s a gold standard when the term “engineer” is used. It symbolizes serious can-do thinking. What I aim for is to view teaching through the sophisticated lens of an engineer. To advocate for teachers as problem solvers whose success is contingent upon high levels of critical thinking, analysis, and creativity. To remind us that our work is inspired by discovering high leverage solutions for our classrooms – solutions that directly address a multitude of human needs.

Maybe not in the traditional sense, but, yes, teachers are engineers.



The day after


I don’t like review days before exams. I’d much rather spend that day after an exam analyzing mistakes and relearning. I find this to be crucial in promoting a growth mindset in my students. My struggle has been how to structure these post-exam days. Here’s a formative assessment idea that I’ve used a few times this year.

The day after an exam, I set up the room in 3-5 stations. Each serves as a place to study a particular concept that was on the exam.

My bell ringer asks students to check their exam performance on the bulletin board in the back of the room. It lets them know for which concepts they earned proficiency. I also email the kids their performance immediately after assessing the exams, but many don’t check.

I hand back the exams and they move to a concept that they need help with based on their performance. If they have earned credit for every concept on the exam then I ask them to float and help others. At each station they use notes, each other, and the feedback I provided on the exam to analyze and learn from their mistakes. I also have practice problems at each station so they can make sure they understand the concept. I float around the room and help. Of course, the SBG data allows me to sit with students who need me most.

After a student feels they have successfully relearned a concept, and I usually check in to confirm, they can retake that concept. The retakes are in folders in the corner – students grab one and do it anywhere in the room. They submit it and begin working on another concept, if necessary. It doesn’t matter how many concepts a student retakes during the period, but it usually works out to be 1-2.

Before I did this activity I was concerned that since the stations would be full of students that struggled on a concept that they would all sit together and get no where. This hasn’t been the case. The kids are diligent to relearn. This may be because they like retaking exams and earning proficiency during class time, as I usually make them come after school to do this. It helps that the relearning is targeted and individualized to each student. Plus, it’s all formative. They go wherever they feel they need to. They assess themselves, but use one another in the process.

It can look and feel chaotic. But that’s the point. Improvement is messy. It’s also amazing – especially when it happens amongst your students.

The asymptote as more than an invisible line 


Suppose you ate Chinese food that was left on the kitchen counter for four days. You didn’t know, and now you’re sick. You go to the doc and get some medicine. You’re diligent and take the prescribed dosage each day, but what if you don’t? What if you take the meds one day and stop? 

With help from Lois and Max that was the problem I posed to my students recently during my exponential functions unit.

What initiated it was asymptotic behavior. I no longer want this powerful constraint to be “an invisible line that a graph approaches infinitely.” In a world of NYS Regents exams, I want that line to have meaning. Note: my students have not graphed exponential functions in my class (it’s part of the algebra 1 curriculum), but they have explored function transformations.

When the kids walk in, I have three containers in the front of the class. Thank you science department for the borrow.

From left to right: waste water, body water, replacement water

The bell ringer opens things up with an exponential functions exploration. I want them to relate the value of b with either growth or decay. This will help later with constructing a model for our medicine.


We discuss that over time the medicine gets diluted and you must consume it everyday to maintain its potency. Otherwise, you’re never going to feel better. I mention that we can model this situation with mathematics and represent it with the containers of water.

Then I put 5 drops of food coloring into the “body” container and mix it up. I originally wanted to use 5ml, but realized in preparation that this was way too much to see a difference in color for our trial. To represent the medicine leaving the body each day, I tell them that I will remove 1 cup (10%) of the body water and replace it with 1 cup of clean water. Before we start, I have them estimate, based on our fictional simulation, how much medicine would remain in their body after 2 months if they only consume one dose of meds.


We start diluting. I do it 7 times (to represent a week) and they calculate the amount of drops remaining in the container after each day. They graph. The decreasing nature of the graph is noted and I ask them to find an exponential model for this data.

Screen Shot 2016-03-28 at 7.49.52 PM

They use the model extrapolate 2 months, with one week intervals, and graph the data.


At this point a few students start to get it. Bam. The asymptote is born.


Here’s the handout.


  • Pretty cool activity. Somewhat teacher centered, but highly engaging. The design was such that the intervals of estimation, graphing, and modeling helped to put their focus on the mathematics and not me.
  • My focus was asymptotes, but it also served as an introduction to exponential functions in algebra 2. I’m not sure the exponential relationship between the variables was really understood. To that end, I’m not convinced that this was the best activity for the job – at least how it’s designed right now. I should tweak things.
  • I was pleased with the bell ringer. There was some good debate around b being fraction and whether that’s why y2 was decreasing.
  • My goal after the demo was to go into graphing exponentials with transformations, so I included a series of questions that are typically found on NYS Regents exams. It tanked. I didn’t make a strong connection between the Desmos activity and this one. The 2nd class did slightly better because I bridged the two concepts a little better on day 2 (with an improved bell ringer), but for the goal I set out here, I still need to adjust my planning to better relate things like f(x) = 2^x and f(x+2) = 2^(x+2). They knew what transformation f(x+2) meant, just not how to interpret it as an exponential function.
  • Neither class answered the last writing prompt (stop & jot), mainly because I wasn’t feeling it during the lesson. Still thinking about how to wrap things up.
  • I started this on one of our shortened days, so I’m not sure if we could have done this in one day. Probably not.
  • I didn’t realize it until now, but this whole thing has a 3-act feel to it. I like!