Steven Strogatz had great series of tweets about math education earlier in the week. These two have stayed in my head since he posted them:
I know that last year Strogatz taught a college level course similar to the one he is describing in the tweets. We even used a couple of his tweets about the course material for some fun Family Math activities. For example:
Here’s a link to that set of projects:
Steven Strogatz’s circle-area exercise part 2 (with a link to part 1)
So, thinking back to projects like those got me thinking about all sorts of other ideas you could explore in an appreciation course. At first my ideas were confined to subjects that are traditionally part of pre-college math programs and were essentially just different ways to show some of the usual topics. Then I switched tracks and thought about how to share mathematical ideas that might not normally be part of a k-12 curriculum. Eventually I tried to see if I could come up with a (maybe) 3 week long exploration on a specific topic. I chose folding and thought about what sort of ideas could be shared with students.
Below are 9 ideas that came to mind along with 30 second videos showing the idea.
(1) A surprise book making idea shown to me by the mother of a friend of my older son:
(2) Exploring plane geometry through folding
We’ve done many explorations like this one in the last couple of years – folding is an incredibly fun (and surprisingly easy) way for kids to explore ideas in plane geometry without having to calculate:
Our Patty Paper geometry projects
Here’s one introductory example showing how to find the incenter of a triangle:
(3) The Fold and Cut theorem
Eric Demaine’s “fold and cut” theorem is an fantastic bit of advanced (and fairly recent) math to share with kids. Our projects exploring “fold and cut” ideas are here:
Here’s one fun fold and cut example:
(4) Exploring platonic solids with Laura Taalman’s 3d printed polyhedra nets
You can find Taalman’s post about these hinged polyhedra here:
Laura Taalman’s hinged polyhedra blog post on her Makerhome blog
And if you like the hinged polyhedra, here’s a gif of a dodecahedron folding into a cube!
Which comes from this amazing blog post:
The Golden Section, The Golden Triangle, The Regular Pentagon and the Pentagram, The Dodecahedron
[space filled in with random words to get the formatting in the blog post right 🙂 ]
(5) An amazing cube dissection made by Paula Beardell Krieg
We’ve also done some fun projects with shapes that I wouldn’t have thought to have explored with folded paper. Paula Beardell Krieg’s work with these shapes has been super fun to play with:
Our projects based on Paula Beardell Krieg’s work
(6) And Paula didn’t just stop with one cube 🙂
(7) Two more of Laura Taalman’s prints
Seemingly simple ideas about folding and bending can lead to pretty fantastic mathematical objects! These objects are another great reminder of how 3d printing can be used to make mathematical ideas accessible.
Here’s Taalman’s blog post about the Peano curve:
Laura Taalman’s peano curve 3d print
(8) Getting to some more advanced work from Erik Demaine and Joseph O’Rourke
As hinted at early with the Fold and Cut theorem, some of the mathematical ideas in folding can be extremely deep:
(9) Current research by Laura DeMarco and Kathryn Lindsey
Finally, the Quanta Magazine article linked below references current research involving folding ideas. The article also provides several ways to share the ideas with students.
Quanta Magazine’s article on DeMarco and Lindsey’s work
The two blog posts below show my attempt to understand some of the ideas in the article and share them with kids. The video shows some of the shapes we made while studying the article.
Trying to understand the DeMarco and Lindsey 3d folded fractals
Sharing Laura DeMarco’s and Kathryn Lindsey’s 3d Folded Fractals with kids
So, these are just sort of ideas that popped into my head thinking about one part of a math explorations class. Feels like you could spend three weeks on folding and expose kids to lots of fun ideas that they’d (likely) never seen before.
Mike's Math Page 
Have so much to say about this post. More than I am writing, but…
First, that book at the top, which I call the origami pamphlet, (but it has many names) , is, to me, one of the most satisfying ways to talk about math while making something. There’s embedded opportunities while making this booklet, to talk about division, fractions, addition, scaling, rotations, shapes, symmetry and more, depending on the goals of the presenter. More and more I think that a big piece of what could help kids with math is for them just to hear the language of math more often, and the origami pamphlet is just full of opportunities. on the sidebar of my blog there’s a tutorial page for making this structure, but one of my favorite sets of directions was made by Tim Winkler http://pictureengine.net/?p=7960. I especially like his image with the scissors to show where the cut needs to be made.
I watched with interest as you cut with a right handed scissors. Understanding why it’s tough for a lefty to use righty scissors is an interesting inquiry. My daughter is a lefty, as are many students that I work with, so sorting out the scissor mystery has been helpful. It has to do with the way we naturally torque the handles of the scissors so that the blades come closer together. If, when using an opposite handed scissors, the user torques the opposite from what is natural, the cutting goes much better. And then, after having this experience of torque, searching for the definition of torque in wikipedia can make one feel very clever https://en.wikipedia.org/wiki/Torque
My main thought, though, on your post and on Steven Strogatz’s tweets, is that I wish this kind of math appreciation would start with parents and teachers (especially elementary teachers) so as to encourage the use of the language of math every day in the same way that we consistently expose children to the language of literature Have you seen Kent Haines video on this? https://www.youtube.com/watch?v=OitL1ZI0azo It’s only got about 1100 views, but OMG if only every single parent and teacher watched it math education might look different. What he’s talking about lays the foundation for the math talk that you’ve been modeling with your kids, though what you add is how far you can take kids with math thinking as part of what you do daily. I have to say that it’s stunning to see the evolution, over the years, of this math talk you do with kids. (I will always be grateful to Michael Pershan for editorializing about your writing in a way that made me stop and look)
The ideas you show here remind me of something that had always been hard for me to understand, and I suspect that the same is true for the typical student. It’s that even when cool math activities, like the ones you show here, are presented and appreciated, I never understood what they had to do with math. It took me years (decades) to unravel even what patterns have to do with math. Since the idea that math is about making calculations is so deeply etched into our brains from such an early age that I think any exposure to different ways of thinking about math absolutely has to be explicitly prefaced with the disclaimer that math is NOT all about doing calculations correctly, but, instead, about inquiry and discovery.
So, yeah, I love what you are doing and thinking about here. It’s the sort of thing that I think about all the time, as I keep trying to highlight the math thinking in the visiting-artist projects that I do with kids These posts in which you put a whole bunch of ideas together are so appreciated.