# PCMI 2022 Post 2: 3D printing

I was so grateful to the neat ideas that I got on twitter about 3d printing, which I included in my last post. (If you want to check the tweet and the replies, they are here.) Some of the responses that really stood out to me are here:

(1) Have students design (using their knowledge of some core functions and transformations) bubble wands using Desmos. Ashley Tewes wrote a moving blogpost about it here (and how she tied it in with empathy and a larger audience than just the students). And just look at how fun and beautiful these are!

And in a similar vein, Martin Joyce has kids use Desmos functions to create objects involving their own names! And @dandersod showed how to 3D print a polar graph from Desmos to be an ornament, which I then did on our school’s 3D printers:

… and I was going to have my kids do our polar graph contest and have the winner’s graph get 3D printed (but the designs were too intricate for that, sadly).

(2) Kids can design their own tesselations (learning the ideas behind how various constructions can build tesselations) and then create 3D printed cookie cutters for them — so they can create “cookie tiles” that tesselate! Or penrose tiles! I initially found a neat blogpost which I’ve lost, but here’s a tweet that showcases it!

(2) Mike Lawler has almost a hundred posts where he and his two sons (who do math together for fun) have used 3D printing. And they are all pretty dang fabulous — an amazing resource. He even chose his favorite ten 3D printed projects here if you don’t want to scroll through all of his posts. The last one he listed in his ten was a model that illustrates Archimedes’ method for deriving the volume of a sphere (without calculus)! I remember learning this in high school and was blown away (so unexpected! so beautiful!), but in all my years teaching, I had never seen this particular manipulative. You can see and download the manipulative here, but I’ll throw down a screenshot of it:

(3) When I taught Multivariable Calculus, we had talked about mappings and coordinate systems, and so one year a student 3D printed this stereographic projection (among other things) and then wrote a paper which analyzed how this all worked:

And I remember showing my multivariable calculus students, in another year, a bunch of optical illusions made by Kokichi Sugihara. They blew my mind, and the kids were smitten. One read some papers on the math behind how you can design these and wrote up a cogent explanation of how this worked using a neat analysis of vector-valued functions.

And goodness knows 3D printing is so cool for surfaces in multivariable calculus, and so much in regular calculus.

But I have to say: after doing a lot of sleuthing, getting things sent to me by others, and just trying to wrack my brain, I’m honestly pretty disappointed with what I think I can do with it in the classroom. It might just be me, but all these schools a decade or so ago were like “WE NEED THESE 3D PRINTERS BECAUSE THEY ARE GOING TO REVOLUTIONIZE STEM EDUCATION.” Maybe so. But after doing an initial foray into them, my current thoughts are: pfft. Maybe I’ll change my mind, but right now: pfft.

Right now, for me, I see the value in 3D printing in two main domains:

MANIPULATIVES: So as I noted, in my last post, there are tons of cool manipulatives a teacher can find and 3D print to illustrate an idea. Like the Archimedes’ proof for the volume of a sphere, or the optical illusion, or creating penrose tiles or printing many of the 15 pentagons that tile (so kids can fit them together and play!), kids will learn. They may be captivated. But kids are learning just from the manipulative, not from the process of 3D printing. That’s just the point of the manipulative — and the 3D printing is one way of getting the manipulative. So great. It isn’t the process of 3D printing that drives student understanding, it is just the manipulative that the teacher finds to illustrate the idea, that happens to be a 3D manipulative. And that’s cool. There’s some value. But in the same value that you can open any math teacher catalog and find lots of hands-on things for kids to play with. This is just a 3D printer printing them, instead of ordering them.

OBJECTS TO SPARK JOY, BUT DON’T HEIGHTEN MATHEMATICAL UNDERSTANDING: Then there are things that I think kids would love doing with the 3D printer in a math class… building bubble wands by using Desmos and function transforemations… developing cookie cutters by learning about transformations… creating polar ornaments by designing creative and beautiful polar graphs. Kids will be able to hold their creations, feel an ownership of mathematics, be proud! So I think there’s a lot to be said for these types of activities. I want to do them! But at the same point, I also truly feel like all the conceptual mathematical learning is happening before the 3D print. The 3D print doesn’t do anything to build on that understanding. What does printing the polar graph ornament from the 2D Desmos polar graph actually teach kids in terms of math? Nothing. I’d argue a kid who printed their bubble wand and a kid who didn’t probably learned the same things. Yes, these things are dang cool, so there’s something to be said for that, but I would argue they don’t build student understanding.

I posited in my last post that there might be a third domain where 3D printing is powerful: where the act of kids actually doing the building in tinkercad or whatever software builds conceptual mathematical understanding. This has been my unicorn, the thing I’ve been really trying to think about or find in the past few days. Because if I’m going to have kids spend time learning new software and troubleshooting finicky 3D printers, there better be a big learning payoff. But at least for Geometry, Algebra 2, and Precalculus, I have yet to anything that really fits the bill.

So for now, as a teacher, I say “hey, 3D printing is cool, but overall, pfft.”

(You might feel differently about this and that’s cool. And I might change my mind. But since I’ve been sending a lot of time trying to think about this and look stuff up, I have just felt a lot of disappointment when I was hoping there was a lot of untapped promise.)