I stumbled on Christopher Wolfram’s agent based virus model last week totally by accident. The model is here:
I’d played around with it a bit over the last two days and decided to share some of the ideas with my kids this morning.
We started with the basic idea of networks and graphs:
Now we stepped away from Wolfram’s mode for a second to look at several of the different kinds of graph structures he was studying. The boys had some pretty interesting things to say about the different types of graphs:
Next we looked at one of the results in Wolfram’s project that I thought was particularly fascinating – how a seemingly small change in assumptions can cause a virus to change from hardly spreading at all to spreading across the entire network:
Finally, my older son (in 10th grade) had looked through Wolfram’s presentation yesterday and I asked him to show some of the ideas that had caught his eye:
I love Wolfram’s post – both for showing how mathematical modeling can showing you interesting ideas about the spread of a virus and for showing the power of Mathematica to make these models accessible to everyone. This was a really fun project to share with the boys.
Over the last month I’ve seen several great twitter threads on ellipses. Sort of a strange coincidence, I guess. Today I finally got around to sharing them with the boys.
We started with this tweet from Lucas Vieira. I’m having a fight with WordPress on the embedding of this one – the tweet we are looking at is the “ballistic ellipse” one at the bottom.
Here’s what the boys thought of the ideas in the animation:
Next we moved on to a post from Greg Egan that was inspired by Lucas’s post:
The kids had a tough time explaining what they were seeing here, so we talked about this picture for a little bit longer than usual:
Now we moved on to Jacopo Bertolotti’s Physics Factlet #216 on 1/r^2 orbits. This animation helped me make sense of a point about General Relativity that I’d heard, but never really understood.
The boys thought the animation was fascinating:
Finally, since Jacopo share his Mathematica code, we took a look at the program. The boys were surprised by how short it was. After looking at the code for a bit we changed some of the parameters and got a fun surprise:
I love that so many people share their amazing work on Twitter. Looking at these animations was a fun way to share a bit of math and physics with the boys this morning!
Yesterday we played around with some introductory ideas in Stephen Wolfram’s Physics Project. Today we moved on to looking at the 3d examples in the project. We were entirely on the computer today looking at shapes, so this project was more about experiencing the ideas rather than diving into the details. Still the kids had a great time.
Here’s how the introduction to the 3d shapes went:
Now I had the boys each try to produce one of the 3d shapes using a rule they made up. This part of the project turned out to be a bit harder because of our lack of familiarity with how the underlying details work. Still, though, they produced some neat shapes and talked about them here:
Despite the difficulties today, I’m excited to play with this project a bit more. The math ideas here are something that I think the kids aren’t going to see anywhere else, and I love the lessons about building complex shapes from simple rules.
I learned about Stephen Wolfram’s Physics Project last week from a Steven Strogatz tweet:
It looked like something that would be really interesting for kids to see, so I spent a bit of time diving in. This morning I took a shot at introducing some of the basic ideas to them and they were fascinated by what they saw.
I started by talking about directed graphs – the introductory ideas are definitely accessible to kids (sorry this video is a bit out of focus – I forgot to check the focus before we started today):
Now (off camera) we constructed the next step in the sequence of graphs Wolfram is studying. My younger son explains our work below and my older son shows a slightly different way of thinking about it that he thought might be more illuminating.
Now we went to Mathematica to explore a bit. Wolfram has made his code available, so exploring his ideas at home is really easy as long as you have the latest (as of April 2020) version of Mathematica. Just to be clear, I’m using their code and don’t yet fully understand how it all works. I was able to understand it well enough to play around, though. It was fascinating to see how the graphs changed when the boys changed the rules a bit.
But the most important thing I think is in this video is just how interested the kids were in these amazing shapes.
Finally we looked at a selection of graphs made from random rules. Again the boys were fascinated by these shapes and seemed to really enjoy thinking about them.
This project was incredibly fun – hoping to find other ways to share these ideas with the boys.
I saw an interesting paper tweeted out by Nassim Taleb yesterday:
Working through some of the ideas made for a welcome distraction yesterday afternoon. When I finished up I thought that there was an important idea in the paper that kids could understand even if the math behind the result was beyond them. That result is here:
So, I started today’s Family Math project by explaining this result in a way that hopefully kids can understand:
Next we went to a short program I wrote in Mathematica to explore this result – here’s how I introduced that program to the boys:
Now I had my younger son vary the number of initial trials we had – my program had been picking 20. We looked to see if we picked “n” numbers that the chance that a new selection would be bigger than the maximum of our n numbers was roughly 1/n.
Finally, I had my older son vary the tail parameter of the Pareto distribution. There’s a little surprise that came from my not updating the program correctly – an accidental good experiment / programming lesson for the boys – but eventually we were able to find that our experiment matched Nassim’s result:
It is fun to try to explain results like this to kids. Again, there’s no chance that they can follow the underlying math, but they can certainly see the ideas from the computer experiment.
The overall idea of Nassim’s paper is also an important statistical point for kids (and everyone!) to understand -> what you see and what you don’t see are both important!
Yesterday I saw an amazing twitter thread from Zeno Rogue which shows a bunch of non-Euclidean geometres:
Today I thought it would be fun to have the kids go through this thread and see what their thoughts were on each of them. My younger son (in 8th grade) went first. Here are his thoughts on the first 5:
Now the next 5 and then further discussion of his favorite one:
Now my older son (in 10th grade) reacted. Here’s the first 5:
Here’s the final 5 plus some extra discussion on his favorite one:
It is so fun to share advanced ideas in my with kids. I love the way that Zeno Rogue presented all of these non-Euclidean geometries – it gives kids a great opportunity to see and react to some ideas that they’ve probably never even gotten near before!
Last week Grant Sanderson published a fantastic video showing some simple models of how a virus can spread through a population.
All of the common pandemic models are pretty complex and have tremendous uncertainty in their parameters, but Grant’s video does an incredible job of showing their strengths and weaknesses.
Today I watched the video again, but this time with my kids. I asked them to take some notes and then we talked about what they thought was interesting. It is always fascinating to hear what kids take away from math / science content.
Here’s what my younger son (in 8th grade) had to say:
Here’s what my older son (in 10th grade) had to say:
With so much terrible news about the corona virus lately, I thought it would be good to talk through some of the numbers and models with them. One thing I thought would be particularly interesting for them to see is why the virus didn’t show up on some of the flu tracking maps, yet.
We started by looking at some of the flu maps from the CDC so they could see how those maps work. Those maps are here:
Here’s what the boys had to say:
Next we moved to this interesting flu tracking map which uses internet connected thermometers. The interesting thing about this map is that it indicates that the flu-like systems are declining rapidly right now:
Here’s what they boys had to say looking at Miami, New York, and Boston on this map:
The next chart we looked at tracked movement in the US using cell phone data. This map allows us to see how the lock downs around the US are working. The map is here:
Finally, we looked at a new model making predictions about the spread of the virus in the US. Here’s that website:
Again, we looked at Florida, New York, and Massachusetts:
I saw an amazing resource for looking at the spread of the Corona virus today – Alison Lynn Hill is a researcher specializing in mathematical biology at Harvard:
I thought that Hill’s program would be a terrific one for kids to use to see how a researcher studies the spread of a virus. We loaded the program on they boy’s computers and they played around with it for about 15 min.
Here’s what my younger son (in 8th grade) had to say – he was particularly surprised by how many variables there were:
Here’s what my older son (in 10th grade) had to say – he was interested in how the curves changed when he played with the transmission rates.
I’m grateful to Hill for sharing her incredible program, and really think it could help kids see (and understand a little) the modeling the modeling involved in studying pandemics.
I’ve been looking for fun ways to review calculus topics with my older son and found a great post from the Calculus VII blog:
Higher order reflections
Today I had my son read through the post and then we discussed the ideas. His initial thoughts are in the video below – he understood most of the post and also had a couple of good questions:
After we talked through the post we went to Mathematica to take a look at some of the example “reflections” which preserve the 1st and 2nd derivatives:
One of my son’s questions in the first video was why the blog post was using functions like f(x/2) and f(x/4) to make reflections. I’d mentioned that these were essentially arbitrary choices. Below we saw what would happen if we used f(x/3) instead:
We finished up by going back to Mathematica to see what these new “reflections” would look like: