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Mathematics Reading List For High School Students?
Troy writes "I'm a high school math teacher who is trying to assemble an extra-credit reading list. I want to give my students (ages 16-18) the opportunity/motivation to learn about stimulating mathematical ideas that fall outside of the curriculum I'm bound to teach. I already do this somewhat with special lessons given throughout the year, but I would like my students to explore a particular concept in depth. I am looking for books that are well-written, engaging, and accessible to someone who doesn't have a lot of college-level mathematical training. I already have a handful of books on my list, but I want my students to be able to choose from a variety of topics. Many thanks for all suggestions!"
I wrote this:
http://people.pwf.cam.ac.uk/jlnw3/maths/books/prime/
It was meant as an introduction to the idea of proof. Perhaps you might like it.
- Jax
It's normally taught as an upper-division college class but the only real prerequisite is 2nd-year high school algebra and a mind that can think abstractly.
Students will find it different enough from trig and calculus to be fresh and knowing they can do "college math" can be a real ego-boost.
By the way, if you know any elementary or middle school teachers, many of the concepts in abstract algebra can be taught to those age groups as well. Being able to do "adult math" can be a real point of pride and inspiration at those ages.
First grade isn't too early. Anyone who can add or subtract time already has the basics for abstract algebra addition and subtraction.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
How to Think like a Mathematician:
http://www.amazon.com/How-Think-Like-Mathematician-Undergraduate/dp/0521895464
Online here (for how much longer?):
http://www.maths.leeds.ac.uk/~khouston/httlam.html
I bought this in the discount bin for $1 somewhere, I think it's (Playthinks) really good to develop logic and just try a little bit of every mathematical discipline:
http://www.amazon.com/Big-Book-Brain-Games-Mathematics/dp/0761134662
This isn't pure math, but lisp, but since Lisp is inspired by lambda calculus, perhaps it'll inspire more programming (shrugs):
http://www.cs.cmu.edu/~dst/LispBook/index.html
Even the dullest high school student has a memory that makes us adults seem slow. There is exactly one way to motivate teenagers: tell them they are not "ready", although telling them they are "not allowed" has a similar effect. With that in mind I recommend you give one or two of them a copy of All the Mathematics You Missed But Need to Know for Graduate School, and suggest they pass it onto someone else if they find it "too hard". It's a great book that gives a quick skim over all the different fields of mathematics that a graduate student in mathematics is expected to know. A typical college student will read this book, shake their head and decide that maybe graduate school isn't for them. A typical high school student, even one not interested in math, will read this book and decide that mathematics is awesome and maybe they should pay attention in class, because if they can't grasp differential linear equations then they're never going to understand Lebesgue integration and infinite Fourier series.
How we know is more important than what we know.
Excellent explanations. It is completely understandable if the student puts in the time to understand it. It requires almost no outside knowledge.
I would have loved it if someone showed me this book earlier.
I read Prisoner's Dilemma by William Poundstone when I was that age, and found it to be a very intriguing introduction to game theory. It is fairly light on math, providing only enough to show that there are calculable solutions to situations that are otherwise difficult to reason through. It also provides some real life examples which are easy to relate to, e.g. letting one child cut a piece of food in half and the other choose the half they want in order to ensure "fair" portions.
It's a good choice for showing that there's more to math than finding the length of the hypotenuse.
I highly recommend The Shape of Space by Jeff Weeks. (He's a freelance geometer, something he can afford after winning a MacArthur Genius Grant.) I've used this book a couple of times -- once with bright high school kids and once with bright college freshman -- and even if they don't get everything, just a taste is enough.
It builds on Flatland (which someone mentioned above), but has the advantage of being more modern and not sexist. But very quickly you're learning about Klein bottles, connected sums, and all sorts of topology you typically don't see until you're well into your undergraduate (or grad!) program in math. All aimed at high school kids. Very cool stuff.
Oh, and the big punchline at the end: what is the shape of the universe? At least you'll get a good understanding of the possibilities...
Here's a taste for you from a page related to the book.
I really like Fermat's Enigma by Simon Singh. Relatively easy read and I found it inspiring.
A Pathway Into Number Theory, by R. P. Burn.
It's the most unique math book I've ever read. There is no prose in the book per se; rather, the book is a series of small tasks and questions (usually starting by identifying patterns in tables of numbers) that, as the title suggests, gently lead the reader into Number Theory. All the major topics of a first course (the fundamental theorem of arithmetic, quadratic residues and forms, etc.) are there; the beauty of the book is that each task is such a small step from the previous one that the reader is led painlessly to a mastery of each concept. (Just don't skip steps!) This feature makes is suitable for advanced high school students looking for "stimulating mathematical ideas."
It's a wonderful book, on a wonderful subject. I have often wished for books written in this format on other mathematics subjects.
An English teacher of mine lent me "Godel, Escher, Bach" in eighth grade (I suspect he taught English by necessity, not choice!), and I found it one of the most fascinating pieces of reading I'd come across in my life. Frankly, it still holds up, if you ask me - even though I don't agree with a lot of what Hofstadter says, almost everything he writes is worth reading because it brings up so many thoughts. After practically every page I would find myself feverishly jotting down my own notes and going on my own tangents, often to discover that Hofstadter would pursue exactly those ideas in the next few pages. Quite a fun read.
That simple act of lending probably had more of an impact on my future intellectual path than almost anything else in school. Gotta remember to send a thank you to that teacher one of these days.
1. A Long Way From Euclid
Constance Reid
A survey of math from the ancient Greeks on.
Very accessible.
I spent months reading it in 6th grade.
2. Innumeracy: Mathematical Illiteracy and Its Consequences
John Allen Paulos
Lots of cool stuff on probability, estimation, and application of math to current events.
You laugh and mod parent funny, but I actually picked up the book on a whim because I wanted non-fiction. What I got was a kid in a rowboat with a tiger. Back on topic though, I really liked "The Code Book" by Simon Singh, and it has a significant amount of number theory and statistics that is light enough for someone without too much background to pick up.
Empires grow and crumble, and the Turtle Moves. Gods come and go, and still the Turtle Moves. The Turtle Moves.