The idea was “new to me” and I thought it would be fun to share with the boys when I got back from the conference. The set up of the problem is definitely something that kids can understand:
You start with 500 people who each have $100. At each step of the simulation each person gives $1 to another person selected at random. How does the distribution of the money evolve over time?
Here’s how I introduced the problem to the boys – the really fun thing is they both guessed that there would eventually be a few people with lots of money and people with little amounts of money. They way my younger son reasoned through how the distribution would evolve was really fun to hear.
Now we went to the NetLogo website and ran the simulation:
At the end of the last video the simulation had run about 4,000 steps. We let it keep running to get a better sense of what would happen, and while it was running in the background I explained a modification to the problem that we’d look at next.
The modification is at each step a player will give away 1% of their wealth rather than $1. Here’s what the boys thought would happen in this new game:
Before jumping in to the next simulation we went back and looked at the prior simulation which had now run roughly 10,000 steps. After we discussed what had happened with the prior simulation I showed the boys how to modify the code to produce the new simulation:
Finally, we wrapped up the project by looking at the modified simulation:
This project was really fun to share with the boys. After we finished they played with a few more built in examples – flocking birds and one with bacteria spreading. I hope to share more of these examples with the boys in the next few weeks!
This school year my younger son is working through Art of Problem Solving’s Precalculus book. This week we started chapter 4 -> Applications to Geometry. The first section is on right triangle trig.
As we started working through the example problems, I was struck by technique my son was using to work through the problems. Once he finished, we went back through one of the problems on camera to share the technique.
Here’s the problem and how he got started:
Next he dove into the solution. His technique is to first draw a right triangle with a hypotenuse of 1 and then scale up:
Finally – here’s how he solved the equation he found in the last video (with some not so great calculator help from me . . . )
I’ve had this blog post kicking around in my mind for a month or so. During that month it has morphed from not quite started to not quite finished. But I’m leaving on a work trip next week and decided to push publish today.
There are so many great math and science ideas that researchers are sharing publicly these days. These ideas have led to dozens of really fun projects that I’ve been able to share with my kids. So, for anyone looking for neat math and science ideas to share with middle school through high school students, I’ve put together a bunch of projects below that were really memorable. All of these projects were inspired by work that experts were sharing publicly – and I’m incredibly grateful that they all took the time to share their ideas.
So, this one is definitely not one of my better written posts, but don’t let my poor writing get in the way of these really fun math and science ideas to share with kids.
(1) Heather Macbeth’s talk about developable surfaces
In the winter of 2018 I saw Heather Macbeth – now a math professor at Fordham – give a amazing lecture about developable surfaces at MIT. The mathematical properties of these shapes are incredible all by themselves, but what made the talk especially neat was that Macbeth shared ways to make these shapes on your own! These surfaces are a great way for kids to be surprised by the shapes you can make from simply twisting pieces of paper!
Earlier today I attended an absolutely fantastic public lecture given by Heather Macbeth of MIT. The topic was "developable surfaces" -> surfaces that can be made from a sheet of paper. Can't wait to share this topic with kids: https://t.co/SjSbGfWxrq#math#mathchat
Richard Green is a math professor at the University of Colorado. He used to share his ideas on Google+, so unfortunately I’m not sure if his original posts still exists or not, but at least we go through one of his a bit one of his number theory posts in the first video in the project below. We used several of Green’s posts for projects – this one is about a neat question in number theory about representing integers in different bases.
Alissa Crans is a math professor at Loyola Marymount University and gave a great talk at the Joint Math meetings two years ago. The ideas in Crans’s talk inspired me to share some basic group theory with kids
The image of the black hole is one of the most exciting scientific achievements that I can remember getting to see! Since doing the project below with my younger son I’ve learned a bit more and been to a few more talks. It feels like the more I learn about the project the more amazed I am that the team was able to accomplish what they did. Actually, at this point I can’t believe that anyone thought it was even possible in the first place – ha!
When the image was published I was able to find ideas about the image shared by three physics professors, Chanda Prescod-Weinstein from the University of New Hampshire, Katie Mack from North Carolina State, and Leo Stein at Ole Miss. There was also a nice Ted talk by Katie Bouman, one of the computer scientists involved in the project who was a post doc at MIT at the time and is now at Caltech.
All of that information inspired me try to share some ideas about the image with my younger son.
(5) Laura DeMarco and Kathryn Lindsey’s folded fractals
An article in Quanta Magazine introduced me to the absolutely fascinating work of Laura DeMarco and Kathryn Lindsey on folded / folding fractals. DeMarco is a math professor at Northwestern and Lindsey is a professor at Boston College. This topic is probably as far out of reach for kids as the black hole image is (!), but the article gave several neat ways to play around with the ideas. Allie helped us hold one of the shapes in our hand by sewing one of the examples in the article:
Sharing statistical ideas from Nassim Taleb with the boys has been an absolute joy since explaining the ideas helps me understand them better. Taleb is terrific at explaining the ideas in ways that are easy to understand and then backing up the explanations with deep mathematical explanations if you want to dig deeper.
(7) Ole Peters ideas about ergodicity and economics
I’m a gigantic fan of the work that Ole Peters is doing at the London Mathematical Laboratory. That work, I think, is going to cause many people to see financial risk in a totally new light. As with Taleb’s work, I love sharing Peters’s work with my kids since explaining it helps me understand the work better.
(8) Chanda Prescod-Weinstein and Bruce Macintosh sharing a neat idea from astronomy
This example is one that I felt especially lucky to see – thanks to a retweet from Chanda Prescod-Weinstein – as it shows how concepts from high school math courses are used to solve problems in physics and astronomy. It also shows that ideas in physics that initially might not seem super hard can be pretty tricky.
The start of Bruce Macintosh’s twitter thread that inspired our project is here – Macintosh is an astronomy professor at Stamford:
Thinking about brightness of a moving object in magnitudes is complicated, so here’s a brief tutorial with some calculations at the end. Imagine a telescope with one arcsecond pixels, excellent seeing, and some idealization, and a 4th magnitude satellite.
(10) A neat mathematical game shared by Jordan Ellenberg
Ellenberg is a math professor at the University of Wisconsin. We’ve used several of the ideas that he’s shared on his blog and in his book How not to be Wrong for projects. We used his post about a neat math game with a quarter circle just yesterday – it was sort of the motivation to get going and write up this collection of projects.
Katherine Johnson was a mathematician at NASA, is a Presidential Medal of Freedom winner, and was one of the inspirations for Margot Lee Shetterly’s Hidden Figures. After seeing the movie I was interested in finding ways to share some of her work with the boys. Fortunately the NASA technical papers are available online, and I used ideas from one of her papers for two projects with my older son:
(12) Federico Ardila’s incredible Numberphile video
This video featuring Federico Ardila, a math professor at San Francisco State University, completely blew me away:
It is an absolutely amazing video to share with kids, and I’ve used it combined with a Zometool set for several “hands on” math lectures for high school students. I was 47 years old and had a math PhD when I learned that the rows of Pascal’s triangle are hiding in n-dimensional cubes!
Larry Guth is a math professor at MIT. I think it was in 2015 when I saw him give a public lecture on what he called the “no rectangles” problem. The idea is to put X’s in boxes in an NxN grid without any 4 of the X’s forming the corners of a rectangle. Solving for the most boxes that can be filled in without forming a rectangle is an incredibly difficult problem in general, but the smaller cases (like filling in boxes in the 3×3 and 4×4 grids) are great to share with kids.
I’ve shared this problem with kids as young as 2nd graders – it is so fun to see kids work through it! The two projects below show my younger son working through the 3×3 and 4×4 cases – you’ll see how this problem brights out lots of different mathematical ideas in kids.
O’Neil is a PhD mathematician who is doing work outside of academia studying problems created by / exacerbated by computer algorithms. Her book Weapons of Math Destruction is a must read. After seeing her speak at Harvard last year I did a project with the kids showing how even seemingly small biases can have large impacts.
Arndt is the head of the Climate Monitoring Branch at NOAA’s National Centers for Environmental Information. He is active on twitter sharing terrific data visualizations that show how temperatures are changing all over the world. This public work he is doing is something that kids can understand and appreciate and I think is a great way for kids to learn about both climate change and also statistics.
Finally, Moon Duchin is a math professor at Tufts University who has been doing fascinating work studying gerrymandering. She also has led several conferences on math and gerrymandering. One of the really great resources that has come out of Duchin’s work is a set of materials to help k-12 educators talk about gerrymandering with their students. A link to those recoursec is in the blog post below. In that blog post I go through one of the exercises Duchin’s team came up with to help kids understand gerrymandering better.
Today I wanted to explore that idea a bit more and also include my younger son. So, I thought it would be fun to see if we could find a way to see what the 5d permutohedron looks like by looking at slices of it in 4d.
I started by reviewing the 3d permutohedron and how it is embedded in 2 dimensions. It was nice to go back to the beginning here – especially so that we could explore how slicing with lower dimensional slices works.
Next we tried the same “visualization by slicing” idea with our 4d permutohedron embedded in 3 dimensions:
Finally, and sorry this one is long, we got to the heart of today’s project. Here we’ll be using some code I wrote in Mathematica to view 3d slices of the 5d permutohedron emedded in 4d space. It is close to a miracle that I was able to get these visualizations to work correctly – maybe the extra hour this morning helped! It was super fun to hear the boys talk about what they saw with these shapes:
It looked like something that the boys might enjoy playing with, so we gave it a shot this morning. Thankfully Ellenberg included some code in his blog post which made it really easy to implement this game in Mathematica. We’ll get to the computer simulations in the last video, but I started out by just explaining the game:
Now we tried to solve the game for a circle of radius 3. I started out with this extra small version of the game to make sure that the kids understood the rules and also to be sure that they knew what “winning position” and “losing position” meant:
Now we moved on to a circle of radius 6. This example was a little harder, but it really helped both kids get to the finish line on understanding how the game worked. I definitely think anyone exploring this game with kids should run through a few small examples first since there are a few potential areas of confusion that probably aren’t obvious to adults.
Here’s how this next case went:
Finally, we moved to the computer program. Roughly speaking the first 3 min of this video are me explaining the code, and the last 5 are playing around with different cases. It was fun to see the kids describe the different patterns they were seeing.
Also, my older son wanted to explore the ratio of winning points to losing points inside the circle. We’ll either tackle that off line or maybe for a project tomorrow.
It is always super fun to be able to share ideas that are both accessible to kids interesting to professional mathematicians. I really like this game since it gives kids a nice opportunity to think through a pretty complex problem and also because the game is so easy to play with on a computer.
Thanks to Jordan Ellenberg for writing up his thoughts (and sharing his code!).