I have, broadly speaking, two goals for my students. The first is that they leave my class knowing a whole bunch of new stuff about math. The second is that they leave my class with a greater appreciation of the joy that can be found in experiencing mathematics.

Last year, I focused most of the energy around my second goal on discovery activities. There’s something special about figuring out a new piece of mathematics, and I wanted to share that with my students. It was successful — for some of them. But it wasn’t successful for others. And it wasn’t successful day after day after day.

Looking back, I was putting all of my eggs in one basket. Discovery activities work well for some topics, and don’t work for others. I was also banking on students appreciating the joy of discovery in mathematics, and I think that was ultimately a product of my own enjoyment of discovery — but it just didn’t click for many students.

I’ve found it useful to separate the *learning *of math from the *appreciation* of it. In getting students excited and invested in doing math, I found much more success in my former eighth grade classes through three-act tasks and problem-based lessons than the vast majority of my discovery activities.

Now I teach high school, mostly precalc and calculus. I’m a little thin on three-acts, just based on the standards I teach, but I’m finding new ways to help students enjoy mathematics. Check these out:

This is sin(xy)=cos(xy) (it’s fun to zoom out, and if you’re teaching calculus, to ask students to implicitly differentiate and see what happens).

This is the Weierstrass function, which is continuous everywhere but differentiable nowhere (more info here). Try increasing “c” in the linked graph — or, if you’re really adventurous, ask it to graph the derivative of f(x) and watch Desmos freak out.

This is a fun polar function that I think also illustrates a lot of useful properties of polar roses and looks awesome with sliders (click through the link to see the slider version)

This is the Euler spiral. I dare you to analyze this with some BC Calc students. What questions can you answer? What questions can’t you answer?

Or even just take this humble rational function and zoom way in on the point where x=3. Suddenly moves removable discontinuities from boring to fascinating.

I make no claim that these specific examples work any particular magic on teenage brains. I’ve found plenty more that have worked in my classroom (and my students have as well as they’ve become more comfortable exploring). I’m sure different strategies would work in different classrooms. But thinking about the level of engagement I’ve found in some pretty simple graphs has reinforced what are for me two truths of teaching.

- Students are almost always more willing to see the beauty and wonder in an image than in an equation.
- That wonder is often best situated
*after*students have learned about and gained expertise with a topic, and have the capacity to engage with all of the relevant complexity.

As a corollary, I’ve found Desmos to be a particularly effective place to create that sense of wonder. Patrick Honner describes Desmos as a “mathematical makerspace” — it lowers the threshold to experiment and can reveal unexpected and powerful mathematics through what seems like a simple tool.

Most importantly to me, there’s something powerful about seeing the wonder in my students’ eyes when they see a piece of mathematics that sparks their curiosity and prompts them to ask new questions. It’s especially powerful to be able to do so in a way that works for all of my students, and doesn’t put a huge strain on my class time. It’s gravy. The substance of my class is that students walk out the door every day with some new knowledge about mathematics. But to be faithful to the discipline of mathematics, and to help me sleep at night hoping I am stoking the flames of future mathematicians, I am finding more and more minutes of my class to spend exploring ideas both fascinating and beautiful, finding wonder in mathematics and learning at the same time.