We should always be learning something new

Last week I started auditing a class on Hapkido, which is a Korean martial art. A friend asked if I’d be interested in attending the once-a-week class with him and I said I would. Having never done any martial art at any time in my life (I don’t even think I’ve watched any of the Karate Kid movies in their entirety), I was nervous. However, partway through the class I realized that there is clear value in learning something completely new.

I want to list a few feelings I had, as I think they reminded me of what it’s like to be a learner, as an adult or a teenager.

  • I didn’t want to make any mistakes. When the instructor demonstrated something, I wanted to do it perfectly. This notion is ridiculous because, as the instructor also pointed out, it takes thousands of repetitions before something becomes muscle memory. For as much as I preach the importance of mistakes in learning, I was shocked at how somewhere in my guts I still didn’t want to make them.
  • I didn’t want the instructor to come by me. Or at least if he did I wanted to be working on my right side (which I thought I was better at). I was afraid he’d find something I was doing wrong. Which I consciously knew would not be bad as it would get corrected and then I’d improve.
  • I compared myself to the people around me, unconsciously ranking myself. Better than that person, worse than those two, etc.
  • Frustration. I’ve never been particularly coordinated and I was consistently frustrated at knowing in my mind what I wanted my body to do, but struggling to make my body do it.

I walked off the mat at the end of class and my mind was reeling.


“Now I know why students are apprehensive to ask questions.”

“Now I understand better why a student might get uncomfortable while I hover over them watching them work out a problem.”

“I have to constantly remind myself to embrace the difficulty. That’s where growth comes from, but it’s difficult to do in practice.”

“Having an instructor that recognized we were all learning was incredibly helpful. He created an atmosphere where mistakes were not viewed as setbacks, but part of the process.”


My main takeaway was that these are feelings I need to constantly grapple with. I need to try to put myself in situations in which I’m the learner, with relative frequency. It helps me better understand where my students are coming from and I think will ultimately help me become a better teacher.

Also, here’s one more thought that has popped in my head recently and probably doesn’t need an entire blog post, but fits with the theme in this one. I’m in my fifth year teaching precalculus, AP calculus, and algebra II and I can feel myself having less empathy with my students, with people learning the concepts for the first time. The first year I taught these courses I think I had a better understanding of their struggles as I was solidifying my understanding of the concepts prior to teaching them as well. I’m not entirely sure what this means for my teaching now, but I think awareness of it is important.

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Light Bulb Moments

I often hear teachers say “I love seeing the light bulb moments in my students”. I would also put myself in this category, but as of late I’ve tried to figure why I enjoy those moments. What about a student reaching understanding brings me enjoyment?

I figured out the other day where at least part of the enjoyment comes from. I have a fairly strong understanding of high school mathematics and because of this I see connections, different methods for solving problems, and interesting patterns. For me, mathematics is truly a beautiful topic. When a student has a “light bulb moment” it’s when, I believe, they have uncovered a piece of that beauty (I know, they might not see it that way, but stay with me…). I view mathematics almost like an a mountain range. Being the math teacher I have a pretty good view of the mountains. I don’t know every peak and valley, but I know the big ones well and a fair number of the small ones. I can also see the big picture and the awesomeness of it. When a student has a light bulb moment I feel as though I’ve shown them or they’ve discovered a new part of the mountain range they’re seeing for the first time. For a moment they are experiencing the peak or the valley or some interesting nuance that they’ve never experienced before.

Mountain GIF

I created the GIF but the image is “River of Clouds” by Ragnar Jesnen

In addition there is the excitement found in solving something challenging. A minimal sense of accomplishment can be found in solving easy problems. This is part of the reason, I believe, that mundane jobs are boring. A person exclusively solves simple problems. But when a student solves a difficult problem and gets excited about it, I’m taken back to times that I solved difficult problems. There is an inherent joy in solving something difficult or challenging. That feeling intrinsically motivates students to tackle other tough problems. The only thing better than experiencing that moment for myself is sharing it with another person.

I have a daughter who is about a year and a half old. There are countless experiences that are brand new for her and her reactions to new, exciting situations is priceless. I love showing her new things, taking her different places, and sharing in the enjoyment she has in learning or experiencing something new. I find the most enjoyment when that type of experience is replicated in my classroom. Sharing in students’ new understanding and, if only slightly, shifting their view of the world and how they experience it.

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Zoii Moon GIF

Finding the moon.


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Caterpillar Gif

Checking out a caterpillar for the first time.


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Math, Play, and a bit of Desmos Love

I’m sure a lot has been written about how Desmos is an incredibly valuable tool for the math classroom. I’d like to echo those sentiments and give a few observations from my classroom related to Desmos and play in mathematics.

My thoughts for this post stem from this activity. It’s pretty math heavy but the basic idea is that students spend time making conjectures and either verifying or invalidating them. When their conjecture is invalidated, they spend time tweaking their functions in Desmos to match the actual answer. These kinds of activities are becoming commonplace in my classroom and I’ve noticed a few positive shifts in my classroom environment. First, engagement has increased. I had a couple more high fives during this activity (and a bunch more in a previous activity) and 100% engagement in class. To be fair, this is AP calculus so it is the “better” students, but I am hard pressed to find 100% engagement when I give p. 102 #1-15 from the textbook. Second, they learn better! This shouldn’t be a big surprise since we know that allowing students to explore/play with a concept before direct instruction often yields better results (See page 58 of this research and this learning model from Ramsey Musallam if you don’t believe me). My students came up with these definitions of the chain rule without me ever saying the term “chain rule”, giving any direct instruction on the chain rule, and with minimal guidance from me. Don’t get me wrong, we followed it with a formal lesson on the chain rule with several examples and they will have an assignment from the book tomorrow, but I think laying the groundwork with the activity will pay dividends later. There’s also another pay off here…

Students are more motivated to come to class. They enjoy activities in which they are allowed to, if you can believe this, play with math. I don’t think this is possible without Desmos. The power in it is that it breaks down barriers for students. It’s like giving a student a canvas, paint, and brushes and saying “here, work with this for a while, make some conjectures, tinker with it, and tell me what you come up with at the end. ” Maybe I’ve drank a bit too much of the Lockhart cool-aid, but to me this is the essence of mathematics. I don’t agree with all of Lockhart’s points, but I think the link between curiosity, play, and learning is powerful and we can leverage that linkage in ways that create powerful learning experiences.

And if I haven’t convinced you yet, check out this excerpt from one of my student’s blog posts below.

“Writing this statement in question #4 required that I fully understood what was being shown in the lab. It made me think deeper about how I came to that conclusion. I believe that looking at the graphs, analyzing relationships between them, and forming a conjecture in my own words helped me to truly learn and understand the concept rather than just being given the relationships and rules and being asked to memorize them.  By completing this lab, I have improved in my ability to  identify what the graph of a function’s derivative should look like. I feel that I have a much better understanding of the relationship between functions and their derivatives and I know that I can look back at the examples in Lab 6 if I ever need help.”

I’d love to hear your thoughts on this. What are your experiences with learning and play? How does it fit into your content area or your classroom?

The GIF(t) of Curiosity

Recently I played around with Snagit on one of the class Chromebooks and discovered how easy it was to create a GIF. I then tried to figure out how to leverage this in the classroom. What I came up with was a prompt based on the GIF below.

Secant Tangent Line gif

The prompt essentially asked students to recreate the GIF. To accomplish this they had to “get under the hood” of the mathematics. This required them to generalize (they’re used to finding secant lines at concrete points) and that was very difficult for them. We rarely ask students to generalize and when we do, it’s usually is in context of a “critical thinking” book problem that gets skipped. Worse than that, often the teacher ends up doing the problem for them at the beginning of the next class. And even if a student does try it, unless it’s an odd problem, they usually can’t see if their generalization was correct until the next day.

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