To John Golden’s class

Saw this blog post from John Golden earlier in the week:

John Golden’s “Math Is” post

In it he provides a link to some thoughts his students writing about what “math is.” I found the pieces to be absolutely fascinating. One description that stuck with me came from Lindsay C’s piece:

It is my understanding that mathematics is a collection of tools that we use to quantify and describe the world around us. We use mathematics very similarly to how we use language. Using language, we can identify objects, convey ideas, and argue. Math can be used in the exact same way when communicating scientific ideas, defining mathematical objects, and proving theorems. The most interesting relationship between language and mathematics is that both can be utilized to describe events and objects that do not exist in the physical universe.

Reading all of their blog posts got me thinking about what I would say “math is” if I were speaking to a room full of people who had some interest in the k-12 education side of math. Over the last two days, these are some of the ideas that came to mind.

I’m sorry this post is a little disjointed – I had a hard stop time, but wanted to try to write as much as I could anyway because the next few days are busy with work and family stuff.

(1) I think if you are interested in showing how math can be used to describe the world around you, Terry Tao’s “Cosmic Distance Ladder” public lecture at the Museum of Math should be on your “must watch” list. Tao is one of the most respected research mathematicians in the world today, but this talk is not really about theory at all. His talk shows how some amazingly simple math (in retrospect) helped us understand the world we live in. What better way to understand how math has helped us understand the world than by having Terry Tao give a few examples!!

I used three ideas from this video for three really fun projects with my kids:

Using Terry Tao’s public lecture to show math to kids

Clocks and Mars

The Speed of Light and Paralax

(2) If you like the theory side of math, Numberphile has an incredibly fun (and a bit controversial) video that shows an amazing idea in math. I was reminded about the video this morning after getting a question about it on Twitter.

and I think it is important / helpful to watch Ed Frenkel’s interview about this sum just for some context:

When I showed the original Numberphile video to my kids, my younger son (who was probably 7 at the time) was screaming “no no no” at the screen. It was fascinating to see how much emotion the (super) strange math brought out. The upside of that emotion is that the original video is a fantastic way to get kids talking about math. Several months later, after hearing Jordan Ellenberg talk about “algebraic intimidation” in “How not to be Wrong”, I talked about the series with them again. The idea for this conversation was simple – do you believe that this series sums to be -1/12?

I love my younger son’s line at the end of the last video: “This doesn’t seem right – but we proved it.”

Jordan Ellenerg’s “Algebraic Intimidation”

(3) I’ll stay with theory for a second math idea I think kids will find fascinating – John Conway’s surreal numbers. One of the neat things in math in 2015 is an amazing collection of articles and books and events involving the mathematician John Conway.

For example, Siobhan Roberts recently published an amazing biography of Conway:

This Conway-related piece written by the mathematician Jim Propp caught my attention in August:

The Life of Games

In his post, Propp explains Conway’s “surreal numbers” using a game called checker stacks. The post may seem a little difficult to understand at first, but the the exposition is so well-written that I think the main ideas will be accessible to a wide audience, and I also think several of the ideas can be explained to kids. I tried out some of the ideas with my own kids (and then later with other kids in the neighborhood, who were both amazed and engaged all the way through):

Walking down the path to the surreal numbers with kids

Walking down the path to the surreal numbers – Part 2

I’d encourage anyone looking to talk about math with kids to explore Propp’s blog post. Just play around checker stacks and see if you can find the value of some simple starting positions and then move on to the ideas about deep blue, deep red, and deep purple chips. The idea of the stack whose value is infinitely small but not zero is amazing – and double amazing when you see kids thinking and talking about it.

Remember Lindsay C’s quote from the beginning – “The most interesting relationship between language and mathematics is that both can be utilized to describe events and objects that do not exist in the physical universe.” Does this infinitely small but positive number exist in the physical universe? 🙂

(4) Finally, I sort of want to leave the world of theory because there’s some absolutely incredible mathematical work going on right now that I think kids will find fascinating. The starting point for this example is Laura Taalman’s 3D Printing Blog: Makerhome

We learned to 3D print from this blog. If the kids on our block are a representative example, kids find 3D printing to be amazing 🙂 Here are a few examples from our living room book shelves of how 3D printing work like Taalman’s can help kids see, hold, and understand math:

That shelf has sparked so many great math conversations with kids from the neighborhood who came over to play with our kids.

Here’s an example that shows one way that the ideas in 3d printing can be used to help kids understand some fairly standard k-12 math:

Using 3D printing to help kids learn algebra and 2D geometry

and here’s a project with a 3D printing example from Henry Segerman that does a little beyond normal school math:

Fine, Ed Frenkel, you convinced me

Segerman’s new project with Vi Hart, Andrea Hawksley, Emily Eifler, and Marc ten Bosch – Hypernom – goes well-beyond 3D printing. But even 4 dimensional virtual reality games can get kids talking about math – you wouldn’t expect them to correctly describe a 4 dimensional 120-cell, but what do they say?

Using Hypernom to get kids talking about math

Having brought Vi Hart into the conversation, I can’t end without mention her story about Wind and Mr. Ug which I think might be the best mathematical video ever made. I would love to see how kids of all ages react to this video:

So, I bring these examples into the “math is” conversation because these last examples show mathematical work that is happening today. When you think about what “math is” it is easy to look only in the rear view mirror, but there’s so many fun things happening in math right now – don’t lose sight of that amazing work!

An interesting problem from my older son’s math club

Earlier this week my older son attended the first meeting of his school’s math club. He brought home a collection of problems for members of the math club to work on. We are going through a few of them every morning, and one from yesterday was so neat that I turned it into a Family Math project last night.

Unfortunately my son has the packet with him as I’m writing up the project so I don’t have the exact statement of the problem in front of me, but we introduce the problem in the first video. It is an excellent counting problem for kids because it combines ideas from counting with ideas from geometry. Here’s the problem, and I’m sorry that the lighting in the video isn’t that great. It was easier to see our snap cube squares in person:

After introducing the problem we moved on to solving it. My younger son uses an interesting geometric idea to count the total number of adjacent squares in our first solution:

Now we looked for an alternate approach to solving the problem. My older son has an interesting idea for counting the total number of adjacent squares without over counting. He has a tough time explaining the process initially – and we try to spend some extra time to makes sure that both of them understand it – but his instincts were right. By the end of the video he’s able to explain why it worked.

Finally we take a totally different approach to solving the problem. Starting with a 6×6 grid might be too difficult – what happens if we start with something smaller like a 2×2 grid? Looking at easier cases when you are stuck is often a really useful problem solving strategy.

So, a great problem with lots of ideas from counting and geometry. Definitely a great problem to use for a nice conversation with kids about math.