Having my younger son explore a fantastic website shared by Bill Hanage explaining how social distancing changes network connections

Yesterday I saw a tweet from Bill Hanage linking to a really interesting website:

From just a little bit of time on the site, I thought having my younger son read and play with some of the ideas would make a great project. So I asked him to spend 20 min reading and exploring, and then we talked.

Here are his initial thoughts:

One of the things he thought was interesting was the idea of 3 and 6 degrees of separation when you have a few connections and how much the network changes when you just add one connection (on average) per person:

Another thing he thought was interesting was the companion site that allowed you to modify connections in the network. Here he looked at the size of the largest group when you made the change from connections with only essential works to again adding 1 connection on average for everyone:

I really like how the ideas of network connections are explained on this site. Their work makes a fairly complex idea accessible to everyone – including kids. Thanks to Bill Hanage for sharing this site!

Talking conditional expectation with the boys thanks to examples from Alex Kontorovich and Gil Kalai

I saw a neat twitter thread from Alex Kontorovich yesterday:

It reminded me of a fascinating conditional expectation problem on Gil Kalai’s blog from 3 years ago:

Here’s the original blog post where you can fill in your guess in the poll (and see the solution, too!):


I thought that talking through these problems would make a nice project for the boys today, so we started in on Alex’s problem. The nice thing right from the start is that the boys had different guesses at what the expected value of one die would be when the combined roll was 8:

Now that we had a good discussion of the case where the sum was 8, we looked at a few other cases to get a sense of whether or not the intuition we developed from that discussion was correct:

Next I introduced the problem on Gil Kalai’s blog – again the boys had different guesses for the answer:

I had the boys write computer programs off screen to see if we could find the answer to the problem on Kalai’s blog via simulation. The interesting thing was that the boys approached the problem in two different ways.

First, my younger son started looking at dice roll sequences and he stopped when he found a 6 and always started over when he saw an odd number. He found the expected length of the sequence of rolls was roughly 1.5:

My older son looked at dice roll sequences and he stopped when he found a 6 but instead of starting over when he found an odd number, he just ignored the odd number. He found the expected length of the sequences looking at them this way was 3:

This turned out to be a great project. I’m glad that the boys had different ideas that we got to talk through. These conditional expectation puzzles can be tricky and subtle, but they are always fun!

Talking through Felix Salmon’s dice problem with my older son

At the end of the Slate Money Podcast this week there’s a discussion of a dice game that Felix Salmon asked about in last week’s podcast. Here’s the podcast from this week:


The game is easy to explain. You roll a fair 6-side die N times, where N is any number you pick. You also choose an amount of money to bet – say X. If you never roll a 6 in your N rolls, you win 2^N times your money back. If any of your rolls are a 6, you get $0 back. Salmon’s questions are -> (1) How much money would you bet if you could play this game once, and (2) how many rolls would you select?

I thought this game would be fun to talk through with my older son. Here I explain the game and he talks about a few of the ideas he thinks will be important for answering Salmon’s questions. He has some interesting ideas about “high risk” and “low risk” strategies. We also talk through a few simple cases:

In the last video my son was calculating the probability of winning the game in N rolls by calculating the probability of not losing. That’s, unfortunately, a fairly complicated way to approach the problem so I wanted to talk a little more so he could see that a direct calculation of the probability of winning wasn’t actually too hard. We talked through that calculation here. We also find that if you roll 4 times you have roughly a 50/50 chance of winning the game.

Before we played the game he wanted to calculate the expected value for your winning in this game. Here we do that calculation and find the surprising answer. We then play the game. He decided to bet $100 and roll three times, and . . .

This was a fun problem to talk through with my son, and I’m excited to talk through it with my younger son tomorrow to see if he reaches a different conclusion. It had never occurred to me to talk through this or any version of the St. Petersburg Paradox with the boys before, so thanks to Felix Salmon for sharing this problem.

Using Christopher Wolfram’s virus program to show kids some ideas about how a virus spreads through a network

Yesterday we looked at a very simple model of how a virus spreads through a network – the assumption was that everyone infects everyone they are connected to. In that (obviously simplified model) the structure of the network affects the structure of the spread:


Today we are looking at another model – still simplified, but not as much as yesterday. This model (and the code we used today) was created by Christopher Wolfram and is here:


In Christopher Wolfram’s model, we use Mathematica to make a network and then study how the virus spread through the network by varying the average number of connections per day that people in the network have. The surprise (that we discuss mostly in the last video) is here the different network structures seem to behave in nearly identical ways. So the result today is very different than yesterday’s result.

I introduced today’s idea by asking the boys to think about how to build a more realistic model of how a virus spreads. The first network we looked at was a simple 2d grid:

Now we looked at a 3d grid:

Next up was a Delaunay triangulation:

Now we looked at a pure random graph network:

For the last two we looked at two graph networks that look a lot like connections in the “real world.” First up was a Watts-Strogatz graph:

Finally we looked at a Barabasi Albert Graph. This graph looks like the pure random graph we looked at, but you can see in the video that the degree distribution is really different. At the end of this video the boys talk about some of the surprises in this project and what they learned.

I think Christopher Wolfram’s program is one of the best I’ve seen for helping students understand some of the difficulties in modeling how a virus spreads. It seems like a big surprise that all of these networks seem to behave the same way, but understanding why it maybe isn’t a huge surprise helps kids see some of the key ideas in these simple models.

Using an idea from Stephen Wolfram to show kids how a virus can spread through different kinds of networks

This week I watch an interesting live coding video from Stephen Wolfram:

Right at the beginning of this video Wolfram shows how to use some simple Mathematica commands to make a simple model of how a virus spreads through a network. I thought it would be fun to share this idea with the boys for several common networks.

I introduced the idea on a 2d grid:

Then we moved to a 3d grid:

Then we moved to a type of network called a Delaunay triangulation:

Now we moved away from these relatively simple graph networks and looked at a completely random one:

With these examples out of the way, we moved to two types of networks that more more commonly used to model a network of human interactions. The first was a Watts-Strogatz network:

Finally we looked at a Barabasi-Albert graph:

This was a really fun project and I was really excited to hear how the boys thought about the different types of networks. The math to properly describe what’s going on in these networks is over my head but I am really happy that Mathematica makes it so easy to explore.

Finally, the idea for looking at these 6 different graphs comes from Christopher Wolfram’s fantastic agent based modeling example. In that program he dives into these different networks much more deeply than we do here – this program is definitely worth checking out if you’ve not see it already:


Having my older son work through Christopher Wolfram’s agent based virus modeling program

A few months ago I saw an amazing program from Christopher Wolfram showing how agent based modeling works for modeling the spread of a virus:


We did a project with Wolfram’s idea back in May:


Since the boys have been learning more about programming in Mathematica this summer, I thought it would be fun to review Wolfram’s program again. My older son spent the week looking through the notebook. Tonight we talked about some of the things he thought were interesting.

The first thing that caught his eye was how the average number of interactions per time step affects the spread:

The second thing that caught his attention was how Wolfram was able to model how the virus spread across different kinds of graph networks:

Finally, he thought the “network of networks” model was really interesting and Wolfram’s graph of how the number of connections between the individual networks changed how the infection spread, in particular, caught his eye.

I think that Wolfram’s work here is one of the best examples I’ve seen that makes virus modeling accessible to students. I also really love that there are many different areas to explore further in Wolfram’s work. Definitely interesting for my son to play around with this program a bit more.

Extending our talk about additive and multiplicative random processes to an example I saw from Ole Peters

Yesterday we had a nice discussion about additive and multiplicative random processes:


Today I wanted to extend that conversation to an example I first learned from a talk by Ole Peters.

I introduced the additive version of the game first and asked the boys what they thought would happen:

After the boys thought about the coin flipping game where you bet $100 each time we moved to the same game where you bet your entire net worth each time/. This one is a little harder to think through, but they boys has some good intuition:

Now we moved to looking at the two games in Mathematica. Here’s how the additive game plays out:

Finally the boys got to see the surprise in the multiplicative game – average wealth increases, but eventually any individual players ends up losing all of their money:

This game is really fun to think through and also a nice example to share with kids to illustrate additive and multiplicative games. Happy that yesterday’s detour into multiplicative processes led us to this conversation today!

A nice surprise walking the boys through terrific introductory stats visualization from Nassim Taleb

I saw an interesting tweet earlier in the week from Atrin Assa:

I thought the idea would make a neat project for the boys today, and it turned out to be even more interesting that I expected.

I started by just having the boys look at draws from a uniform distribution. You never really get a chance to go back and see ideas like this for the first time, so I’m always really interested to hear how kids describe what they are seeing:

Next we looked at the average of two draws from a uniform distribution. The boys had different thoughts about what this would look like before we saw the outcome – that was fun to hear. My younger son had some interesting intuition based on a dice game he’d played previously:

At the end of the last video my older son wanted to check what would happen if we looked at the geometric mean rather than the arithmetic mean. This idea wasn’t what I was planning to study, but it seemed like a great idea so we tried it out. They both had interesting guesses at what this new distribution would look like:

We wrapped up today looking at the arithmetic mean and geometric mean of three draws. With the arithmetic mean they started to see the normal distribution appearing. With the geometric mean they didn’t recognize the shape – my guess is that they’d never seen a distribution like this before:

This was a really fun project. Nassim’s intuition about how to explain important ideas from probably and statistics is incredible. I love working through his ideas with my kids.

Having the kids talk about some corona virus data presentations

100+ days into the pandemic and I’ve found several sites producing data and data presentations that are helping me track the spread of the corona virus. There is also, obviously, lots of bad information. For our math project today I wanted to share a few visualizations with the boys to (hopefully) help them understand the pandemic better – especially in the US.

We started by looking at Apple’s mobility tracking site:


This site is terrific for seeing how people from all over the world have changed their travel behavior. Here’s what the boys had to say:

Next we looked at the site: https://covidtracking.com/

I learned about this site relatively recently. It does a great job collecting and presenting data in the US. Here’s what the boys thought of the various presentations:

During the conversation in the last video my younger son said that he was surprised to learn that deaths in the US from the corona virus had been declining until recently. To help him understand why that was happening we looked at the data presentation on the FT’s website:


Finally, we looked at two presentations that I made this morning playing around with the data mapping tools in Mathematica. I’m still very much a novice when it comes to making these presentations, but I still thought it would be interesting to hear how the boys interpreted these presentations:

Talking through different kinds of corona virus visualizations with kids

Yesterday Jordan Ellenberg posted a really good thread about a different viral twitter thread about the corona virus:

Inspired by that thread, I decided that we’d talk through several different corona virus visualizations. The pandemic has hit different parts of the US (and different parts of the world) so differently, so I was really interested to hear what the boys thought of the various graphs and presentations.

First we looked at a county by county comparison of the pandemic in Massachusetts and Texas. We looked at cases and deaths per 100,000 people from January through June in each county:

Next we looked at a more traditional data presentation with graphs of total cases by state

Now for a different perspective on the cases in each state, we looked at the graphs of cases over time weighted by population. I think the difference in the total cases graphs and the population weighted graphs are easy for adults to understand, but the differences were a little harder for the kids to interpret:

Finally we looked at a data presentation that I think I’d never seen before the pandemic. Mathematica calls this presentation a Matrix Plot – I don’t know what these plots are usually used for.. These plots were hard for me to understand when I first saw them, but they made a bit more sense to the kids this morning:

I think that showing kids data about the corona virus helps them get a better understanding of what’s going on. Talking through different kinds of presentations is an important exercise, too, as kids will often see ideas in these presentations that are different from what you were expecting them to see.