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/

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.

As a kid, for sure, Douglas Hofstadter's Godel Escher Bach. I loved about it that I couldn't understand all of it but there were parts I could understand, and then as I re-read it as I got older, I could follow more and more of it. I try to make my own books work like that; you can read them with a pencil and paper and really work through the parts where it gets technical, or you can skim those parts because you're never far from getting back to stories about humans. Hofstadter's books, of course, are also really insistent on mathematics as a human activity that's inextricably wound around our other mental activities -- talking, art-making, philosophizing, etc. -- that has affected my view of what mathematics is and certainly it has affected what I choose to write about.