Jordan-Ellenberg

Mean Girls

The obvious answer for "what to add" is a course in data analysis and statistics, but it needs to be said that to some extent this has already been added! Statistics is taught in a substantial proportion of high schools now, and the stat AP is more popular than the calc BC AP and almost as popular as calc AB. And having two kids on their way through K-12, I can tell you that ideas about data, visualization, and statistical analysis are now present throughout the curriculum, starting in kindergarten. BUT: I do think it would be good for a year to be devoted to that topic and for it to be marked as part of the sequence, not as an elective.

But what to take out? I'm going to cheat a little, because I don't think any existing course is fully expendable. I think I would make both pre-calculus and calculus shorter by reducing the emphasis on games with trig identities in pre-calc and the emphasis on integration tricks in calculus. (But again I'm still cheating because the integration tricks are in calc BC which is really not part of the 6th-12th sequence per se!)

OK serious answer: after all these years, the class of the field remains the NOVA episode "The Proof," about Wiles and Taylor's proof of Fermat's so-called Last Theorem. It captures the emotional side of doing research in mathematics as no other documentary has done. https://www.pbs.org/wgbh/nova/proof/

I've heard great things about the movie about Maryam Mirzakhani but I haven't seen it yet! http://www.zalafilms.com/secrets/

And I can't omit the NOVA episode I'm in and where through the miracle of special effects they make it look like I can shoot a three-pointer over my shoulder. https://www.pbs.org/wgbh/nova/video/prediction-by-the-numbers/

1. I don't know if it's more political! Of course there is the second on gerrymandering, where the subject itself is the interaction of mathematics and politics; so unavoidably that part is political (though I strive to be nonpartisan.) I wrote the book during what was (is?) obviously a very politically strained time in the U.S. and I actually found it kind of a respite to mostly stay at arms length from that stuff in this book.
2. Oh I had no idea I was going to write any of that in advance. Like a lot of mathematicians I suddenly got very interested in pandemic modeling and mathematical epidemiology in early 2020. But because everything in math is connected, it all ended up very much in tune with the stuff about random walks and networks I already knew I wanted to write about, the use of differential equations to study the natural world brought it in contact with Poincare, etc. I didn't have to take anything out because I don't write to a strict word limit. I suppose if I told Penguin I wanted to do 800 pages they might start to get itchy but by and large their philosophy is that I should write the book I want to write and that determines the length.
3. I didn't mention it because I don't know about it! Part of talking about the book, once it's out, is finding out about all the research tunnels I didn't go down and stuff that could have been in the book but isn't. So tell me!
4. There's no non-flawed system. My inclination is certainly to see IRV as an improvement over what we do now, though I'll be watching carefully what happens in Maine and NYC and other places that have taken it on to see if that changes my assessment. I don't have a good answer as to why IRV is getting traction while, e.g. approval voting isn't. It may just be psychological -- when people vote they like to be able to put their favorite candidate first. IRV feels like less of an emotional departure from the system Americans are used to, in that respect.

And as a researcher, Joe Silverman's book The Arithmetic of Elliptic Curves (a lot of number theorists would say the same thing) which just absolutely sells the hell out of the idea that the road to understanding the most classical, simple-to-state problems inevitably runs through abstractions developed only in the 20th century, and that the path doesn't have to be brambly and terrible but is in fact really fun to follow.