The question is a really deep and really challenging one for kids. Truthfully it is probably a little over the head of my kids, but I thought I’d give it a try anyway. I’ll revisit this one (hopefully!) several times over the course of this school year – although the question confused my kids a little bit, I really like it.

Here’s my older son’s (started 7th grade today!) thoughts on Dave’s question:

Here’s my younger son’s thoughts – he’s in 5th grade. I took a little extra time at the beginning with him to work through some examples with numbers so that the abstract symbols wouldn’t be so confusing:

It is fun to hear the boys struggle to try to explain / reconcile the strange ideas in Grandi’s series. I’m also glad that they are learning to think through what’s going on rather than just believing the algebra.

For that project we actually ordered the shapes from Shapeways, but last night I learned that Henry had put a few of the prints from the book on Thingiverse!! I think many of the prints will be too complicated for our home printer, but glancing through the there were two that I thought we could do – the pseudosphere and the neat Hilbert curve creation.

Here are my older son’s reactions to and thoughts about the shapes:

Here are my younger son’s reactions and thoughts about the two shapes:

I’m so glad that Henry wrote this book – it is incredible to be able to hold these shapes in your hand and share them with kids.

and started building the cars when they arrived on Friday. Here’s what the boys thought at the beginning of the project. My younger son first:

and then my older son:

We built for about 45 min and got about 1/2 way there. The building is not that challenging, though I’d guess our 45 minutes with a 5th and 7th grader would be closer to 15 min with high school kids.

Here’s where we got to after the work on Friday night (the main tools that you need for the build are a glue gun and a soldering iron) :

After about another 30 min of building this morning we’d finished the cars. We used some spare lego pieces to help the solar panels tilt up.

Our first test out in the street didn’t go as well as I was hoping because it was a little cloudy, but at least the wheels still ran and the boys seemed pretty happy:

Later in the morning, though, we got enough sun on our porch to do a nice demonstration:

In the video he presents a neat Theorem about partitions due to Euler.

Simon Gregg, by coincidence, was looking at partitions recently, too, and has written up a nice post which includes some ideas from Rubenstein’s video:

The part that struck me in Rubenstein’s video wasn’t about partitions, though, it was about the manipulation of the infinite product. It all works out just fine, which is pretty neat, but sometimes manipulating infinite quantities produces strange results. See this famous video from Numberphile, for example:

Just as an aside, here’s a longer and more detailed explanation of the same result:

The fascinating thing to me is that Euler’s proof in Rubenstein’s video is easy to believe, but the sum in the Numberphile video is not easy to believe at all. Both are examples, I think, of what Jordan Ellenberg called “algebraic intimidation” in his book How not to be Wrong. I used Ellenberg’s idea when I talked about the -1/12 sum with my kids:

The talk I’d like to give to calculus students would start with the theorem presented in Rubenstein’s video. Once the students were comfortable with the ideas about the infinite products and the ideas about partitions, I’d move on to the idea in the Numberphile video. It would be a fun way to show students that infinite sums and products can be strange and you can sometimes stumble on really strange results.

A few years ago this Numberphile of Ed Frenkel inspired me to think more about how to share math with the public – and especially kids:

This morning I was reminded of the line around 5:41 in the video where Frenkel talks about how other fields do a better job sharing their work with the public than mathematics does:

“And I feel as though other scientists are doing a much better job; physicists, biologists. We keep talking about the solar system about the universe, about galaxies, about atoms and molecules, elementary particles and DNA.

Those concepts are no more complicated than things we do in modern mathematics. Why is it that, you know, DNA and stars and elementary particles are part of our cultural discourse but mathematical ideas are not? Well, in part because we are not doing nearly enough. We professional mathematicians are not doing nearly enough.”

What reminded me of Frenkel’s quote here was this incredible article from Natalie Wolchover (which I was listening to on Quanta magazine’s podcast):

I won’t put words in Frenkel’s mouth, but the study of neutrino interactions feels like exactly the type of thing he was talking about physicists study that is “no more complicated than things we do in modern mathematics.”

Wolchover’s article does a fantastic job of making both the problem physicists are studying and their experimental results accessible to the public. This paragraph in particular struck me as a great bit to share with high school students:

“If the seesaw is balanced, signifying perfect CP symmetry, then (accounting for differences in the production and detection rates of neutrinos and antineutrinos) the T2K scientists would have expected to detect roughly 23 electron neutrino candidates and seven electron antineutrino candidates in Kamioka, Tanaka said. Meanwhile, if CP symmetry is “maximally” violated — the seesaw tilted fully toward more neutrino oscillations and fewer antineutrino oscillations — then 27 electron neutrinos and six electron antineutrinos should have been detected. The actual numbers were even more skewed. “What we observed are 32 electron neutrino candidates and four electron antineutrino candidates,” Tanaka said.”

I love how the reader gets to see what the scientists expected in two situations (i) 23 / 7, or (ii) 27 / 6, and then what they actually found – 32 / 4. What a great example of the scientific process!

This paragraph is also a great opportunity to talk with kids about statistics. I’m sure that high school students could understand the basic statistical ideas here and have have a great discussion about the data presented in the article. In fact, this short lecture from New York master teacher Amy Hogan discusses a similar statistics problem:

So, I loved Wolchover’s article and think it is really a great model for how to communicate complicated ideas from math and science with the public. I especially love that there’s something that teachers can use in their classrooms right away. I hope that we’ll see more and more articles similar to this one that bring advanced ideas from math to the public.

My younger son has been learning a little bit about square roots over the last couple of weeks and I thought it would be fun to show him some proofs that the square root of 2 is irrational. Because this conversation was going to explore some ideas in math that are both important and pretty neat, I asked my older son to join it.

I wasn’t super happy with how this little project went – it felt a bit rushed while we were going through it. Hopefully a few of the ideas stuck.

We started by talking about the square root of 2 and what basic properties the boys already knew about it:

After that short introduction we moved on to the first proof that the square root of 2 is irrational – I think this is probably the most well-known proof. The proof is by contradiction and starts by assuming that = A / B where A and B are integers with no common factors.

The next proof is a geometric proof that I learned a few years ago from Alexander Bogomolny’s wonderful site Cut The Knot. It is proof 8”’ here:

If you like this proof, we have also explored some geometric infinite descent proofs in a slightly different setting previously inspired by a really neat post from Jim Propp:

Finally, we looked at a proof that uses continued fractions. It has been a while since I talked about continued fractions with the boys, and will probably actually revisit the topic soon. It is one of my favorite topics and always reminds me of how lucky I was to have Mr. Waterman for my math teacher in high school. He loved exploring fun and non-standard topics like continued fractions.

So, although I don’t go deeply into all of the continued fraction ideas here – hopefully there’s enough here to show you that the continued fraction for the goes on forever.

So, although this one didn’t go quite as well as I was hoping, I still loved showing the boys these ideas. We’ll explore them more deeply as we study some basic ideas in proof over the next year.

The picture on the bottom right reminds me of a really neat geometry problem that I heard MIT’s Bjorn Poonen discussing. The problem starts with the circle inscribed in the center of the picture below and then ask some questions about higher dimensional versions of the same situaiton:

I asked the boys to flip through the book and pick out two shapes each that we could then order from Shapeways. Those shapes arrived today – yay!! I’ll do a more extensive project with these shapes later, but for today I just wanted to hear their reaction to seeing the shape and holding it in their hands.

My younger son happened to be home when the package arrived so he went first:

(1) A 120-Cell

The fun thing about this shape is that we’ve played with versions of the 120-Cell before:

This shape really caught my son’s attention in Segerman’s book – I’m glad he thought it was as cool in person as it was in pictures!

(3) Double Zarf

When my older son got home we unpacked his two shapes. I’d never heard of either shape before and am really excited to explore both of them. Both shapes come from the mathematical artist Bathsheba Grossman

I asked the boys what they wanted to talk about today and got a fun response – the Koch Snowflake with squares!

Luckily we still had the Zome set out from the “tribones” project from last week – so making the first couple of iterations wasn’t that hard.

Before we started, though, I asked the boys what they thought the shape would look like:

While they were building I searched for something on line that would let us play with this particular fractal. I found these two Wolfram Demonstration projects:

Here’s what the boys had to say about the first three iterations of the square Koch Snowflake. A fun thing that happened here was that during the discussion the boys found a small mistake in the construction of their level 3 curve:

Next we moved to the computer to explore the Wolfram Demonstration projects. First up was Robert Dickau’s “Square Koch Fractal Curves.” Sorry that this video (and probably the next one, too) is so fuzzy – it looked ok in the view finder. Oh well.

Finally, we explored Tammo Jan Dijkema’s “Create Alternative Koch Snowflakes” demonstration project. This project allows you to alter the fractal. The boys had a lot of fun playing with this project. Again, sorry for the fuzz in the video.