I find myself always pressed for time in class. Doing inquiry-based work, and going at the pace of the kids’ understanding, means that inevitably there are stretches of time when I’m feeling like “ARGH! We need to go faster! We have to be further along to cover the content!” Every teacher I know feels this, and in my office we talk about this, so that makes me feel better. I’m not alone. I also know in the back of my mind that somehow, each year, we manage to get things done. That helps.
That being said, this past week I had one of the very few times that one class was about half a day in front of the other and I didn’t feel the need to forge forward as acutely as I usually do. And in that short exciting window, I saw a student’s shirt which I thought was mathematically beautiful…
… which of course I told to the student. “Math?” He didn’t see it, nor did others in the class. It was an 90 minute class, and when having kids work on some problems together, my mind started thinking… I want to show kids how I see this shirt. The glasses I use to see the world are different from theirs. We take a break in the middle of class for kids to get water or quickly grab a snack, and so 5 minutes before the break I stop everyone. I have them look up. And I say something like: “I know many of you don’t see math in [Stu]’s shirt, but I think if you start looking at the world with math in mind, something like that shirt will pop out as beautiful mathematics. So grant me 5 minutes where I’m on stage, you’re the audience, and I live give you my thought process for creating a version of that shirt on desmos. It may not work, but I think it will be neat to try.”
So they are sitting watching me sit at a laptop. I start by graphing . They ooh. Maybe they don’t. But in my mind’s recreation of the event, I hear them ooh. I explain that this isn’t a random thing I created. I ask who is taking statistics. I mention the “normal distribution.” A few nod knowingly, and for those who don’t, I say “this isn’t genius, this is me seeing [Stu]’s shirt and recognizing one of the most famous equation shapes in the world.” Then I graph the reflection over the x-axis. They understood that. But then I said I need more lines.
So I say: “I am now going to try to make the same curve with a higher peak. I think I should do a slight vertical stretch.”
But then I note that it isn’t just that each curve gets slightly stretched, but also the width of the bump gets slightly widened too. I go to the board and explain how I’m going to do a horizontal stretch too, and write up how I’m going to alter the x-variable in the equation to do that.
I flipped that over the x-axis and then manually entered a bunch more equations that did the same thing — slightly higher peaks, slightly wider bumps. Kids asked me to add in the two circles in the middle, so I did. It looked meh because I only had 6 or 8 curves. I sent them on break and promised them I’d get it to look a bit better when they returned. And that I would do this with just one equation.
During break, I whipped this up using lists.
When they returned, I explained how the list worked in Desmos — so one equation actually plotted a bunch of equations.
I didn’t know what to make of doing this. I wanted them to see how I saw things, how I thought about things. That math is in lots of places if you just look for it. That playing with math can be fun and what they already know mathematically are quite powerful tools. If for just a second one of the kids was like “Oh, yeah, wait, math is pretty neat,” I’d be happy. It might have happened because the next day I was talking with a science teacher who was telling me that my kids who she also taught were talking about it in her class.
Also, you know, I always find that when I deviate from my plans for something I’m excited about, I always feel so good about doing what I’m doing with my life. I have to keep this in mind and try to go off the beaten path more…
PS. Of course when I saw the shirt, I didn’t initially “see” the normal distribution. I saw fluid flow around a cylinder:
But I forgot everything mathematical I know about that. :) So normal distribution it is!
OMG, Sam. This is such a cool moment of modeling how a mathematician sees things.
Your talent for seeing and expressing this visual image through the language of math is fascinating. I don’t have a head for math and have never been very good at it, but I appreciate how some people can bring it to life. Your students are fortunate to have you as their instructor.
Great post! You are a great teacher. I enjoy you posts on maths and I love your enthusiasm.
That is so sweet. I don’t think I’m a great teacher (yet), but I’m getting better!!!