Using Steven Strogatz’s Infinite Powers with a 7th grader

My copy of Steven Strogatz’s new book arrived a few weeks ago:

The book is terrific and the math explanations are so accessible that I thought it would be fun to ask my younger son to read the first chapter and get his reactions.

Here’s what he thought and a short list if things that he found interesting:

After that quick introduction we walked through the three things that caught his eye – the first was the proof that the area of a circle is \pi r^2:

Next up was the “riddle of the wall”:

Finally, we talked through a few of the Zeno’s Paradox examples discussed in chapter 1:

I think you can see in the video that Strogatz’s writing is both accessible and interesting to kids. I definitely think that many of the ideas in Infinite Powers will be fun for kids to explore!

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What do kids see when they see ideas from advanced math?

Saw a really neat tweet from Steven Strogatz tonight:

I thought it would be fun to share it with the boys and just listen to how the described what they saw.

Here’s what my older (8th grade) son said:

Here’s what my younger son said when he saw the video:

It is fun to see ideas from advanced mathematics through the eyes of kids 🙂

Playing with 3d printed versions of shapes theorized by Hermann Schwarz

Saw a neat tweet earlier today about 3d printing, math, and engineering:

I recognized some of the shapes in the article as ones that we’d played with before:

The grey shape displayed in the article is a “made thicker for 3d printing” version of the surface \cos(x) + \cos(y) + \cos(z) = 0. I thought it would be fun to print that shape today and use it for a little project with the kids tonight. Here’s the Mathematica code and what the print looks like in the Preform software:

8 hours later the print finished and I asked the boys to describe that shape plus the gyroid. It is always fascinating to hear what kids see in unusual shapes. My younger son went first:

Here’s what my older son had to say (and he’s starting to study trig, so we could go a tiny bit deeper into the math behind the shape I printed today):

Next we watched the video about the shapes made by Rice University:

After watching the video I asked the boys to talk about some of the things they learned:

Of course, mostly they didn’t want to talk about the shapes – they wanted to stand on them! So much for an 8 hour print and 45 min of trying to clean out the supports . . .

Here’s how the standing went:

Definitely a fun project and a fun way to show kids a current application of both theoretical math and 3d printing!

Having kids play with ” swarmalators”

Saw a couple of neat tweets on a new paper in Nature by Kevin P. O’Keeffe, Hyunsuk Hong, and Steven Strogatz:

It looked like playing with the “swarmalator” program would be a really fun way for kids to experience ideas from current math research even though the math underneath these results is a bit out of reach.

So, this morning we just played. Here’s how I introduced the ideas of the program – the two most important ones are (i) the strength of attraction of similar colors, and (ii) the strength of the desire for neighbors to have the same color (and both of these “strengths” can be negative):

After that short introduction I had my younger son (in 6th grade) play with the program to see what he found:

Next I let my older son (in 8th grade) play:

Finally, to talk about the ideas a bit more we went through 4 of the 5 examples at the bottom of the web page of the program we were using. I had the kids try to guess what was going to happen before we set the coordinates. Here are the first two examples:

Here are the last two examples – in this video the boys are getting the hang of how the program works and have several pretty neat things to say about what they are seeing (and what they expect to see):

We played with the program for about 20 min more after we turned off the camera. This program is definitely fun to play with and it was really fun to hear what the boys were guessing the various different states of the program would look like. Even with just two parameters, the kids really had to think hard to talk about the expected behavior. I think that lots of kids will really love playing around with this program.

Thinking about a math appreciation class

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:

Tweet9

Tweet10

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:

OUr Fold and Cut projects

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!

dodecahedron fold

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.

Sharing Kelsey Houston-Edwards’s topology video with kids

Kelsey Houston-Edwards’s latest video is terrific:

This one is particularly easy to share with kids because there are several puzzles where she asks you to stop and think about the solution. I began the picture frame puzzle as the starting point for our project today.

The puzzle goes roughly like this:

A common way to hang a picture is to use two nails in a wall and run the wire around those two nails. Assuming the nails / wall are strong enough, if you remove one of the nails the picture will still hang. Is there a way to hang a picture with two nails so that if you remove either of the nails the picture will fall?

We took a shot at this puzzle using yarn and snap cubes. It was a good challenge for the boys:

In the last video we got the picture to fall once, but the boys weren’t quite clear what happened – but now they at least knew it was possible! Here we explored the idea more carefully:

Next we finished watching the video and then discussed what we saw (as I publish this post the video preview isn’t embedding properly, but is really just audio anyway):

Finally we looked at two sets of shapes that appeared in the video that we’ve looked at before. The first is a 3d print of Henry Segerman’s “Topology Joke” and the 2nd is a set of “rollers” that we’d made after seeing a Steven Strogatz tweet. The tweet and the roller project are here:

3d printing and rollers

Another fun project from Kelsey Houston-Edwards’s amazing math series. Sorry to be brief on this project, but I had to get this one out quick because of a bunch of activities going on today.

Using 3d printing in the college classroom

I saw a couple of tweets from Steven Strogatz yesterday that got me thinking about how you might use 3d printing in the college classroom:

The last tweet, in particular, made me think that having the 3d print versions of the two shapes would be useful. Before I get too far in to this post, though, I had to throw this post together pretty quickly to be able to fit in an hour of shovelling prior to heading to work! Sorry if it isn’t the most well-written or well-argued post. The main takeaway I want is that I think there are many great uses for 3d printing in the college math classroom.

The topic of 3d printing and calculus is one that I’ve thought about briefly before – see these old posts:

3d Printing and Calculus Concepts for kids

Using 3d printing to explore some basic ideas from calculus

Here are the shapes from the first post linked above – I think they would help students understand ideas like Riemann sums and volume by slicing:

Here are the two 3d shapes from the second Strogatz tweet from yesterday. Unfortunately we lost power in the middle of the night before the print project was complete, but you’ll get the idea. One of the things that comes through immediately in the prints is the difference in size of the two shapes:

Finally, an important shape from advanced algebra – a cube inside of a dodecahedron. This shape appears (and plays an important role) in Mike Artin’s Algebra book:

Dodecahedron

I found it hard as a student to understand the shape solely from the picture. Holding the shape in my hand, though, makes it much easier to see what is going on (I have made the cube slightly larger to highlight it):

So, while I’m sure it is true that learning to draw some of these shapes by hand is useful, I also think that 3d printing can be an important tool to help students see, understand, and experience the same shapes.