Slashdot Mirror

I highly recommend the following two:

A Tour of the Calculus by David Berlinksi
This is a remarkably literate survey of the topic of the calculus. It does a wonderful job of connecting the real world with the calculus. The author just doesn't show calculus applications, but that calculus is omnipresent and defines everything we see and do. Your students will never watch someone on a diving board the same way again.

http://www.amazon.com/Tour-Calculus-David-Berlinski/dp/0679747885

The Universal Computer: The Road from Leibniz to Turing by Martin Davis

I enjoyed this history of computation from its very earliest origins. I recommended it to young students because it enlightens math's fascinating history and that math has a higher order than just longer word problems.

http://www.amazon.com/Universal-Computer-Road-Leibniz-Turing/dp/0393047857
   

I'd highly recommend Tesla: Man out of Time, by Margaret Cheney. It is very approachable and engaging, and will give them the following:

a) appreciation of an underappreciated scientific genius

b) understanding and awe of the power of resonance

Not necessarily what you're looking for, but a good book involving math nonetheless: Uncle Petros and Goldbach's Conjecture

It's not exactly a new book, so some of the unsolved problems listed in the book may now be solved. In any case, it's one of the few I know that help a younger student go into more depth in an area where there's still active research going on. It's a difficult subject where many of the theorems can be proved without resorting to higher mathematics.

I'd imagine that there are probably similar texts for some areas of number theory and game theory, but nothing springs to mind. Non-Euclidean geometry may also be an option if the students have already taken geometry, and there were some text books that I found at least partially accessible in high school.

The Mathematical Tourist is even more out-of-date by this time. Since it's really a survey of many areas, it doesn't really meet your need, but you may find it useful yourself for looking into other areas that may be accessible to your students.

Finally, contact your local mathematics and math education departments. The math education folks may have some good suggestions. Many mathematics departments also do some sort of outreach to high school students, so there may also be some faculty there who could offer ideas.

It's not exactly a new book, so some of the unsolved problems listed in the book may now be solved. In any case, it's one of the few I know that help a younger student go into more depth in an area where there's still active research going on. It's a difficult subject where many of the theorems can be proved without resorting to higher mathematics.

I'd imagine that there are probably similar texts for some areas of number theory and game theory, but nothing springs to mind. Non-Euclidean geometry may also be an option if the students have already taken geometry, and there were some text books that I found at least partially accessible in high school.

The Mathematical Tourist is even more out-of-date by this time. Since it's really a survey of many areas, it doesn't really meet your need, but you may find it useful yourself for looking into other areas that may be accessible to your students.

Finally, contact your local mathematics and math education departments. The math education folks may have some good suggestions. Many mathematics departments also do some sort of outreach to high school students, so there may also be some faculty there who could offer ideas.

I'm a HS Math teacher myself and I once read a book called 'A History Of Zero'. It was pretty fascinating. It didn't deal with a lot of higher math, but had some really interesting stuff about the number which is zero. Check it out here: http://www.amazon.com/Nothing-that-Natural-History-Zero/dp/0195142373

More of a mechanical, physics based look at the world. Focuses a lot on biological comparison to things we've made. But there are some good chapters that focus on math in the real world without actually going into the math. (e.g., conical progressions in seashells)

Paperback's only $12 at Amazon: http://www.amazon.com/Cats-Paws-Catapults-Mechanical-Worlds/dp/0393319903

I can't remember the name of it, but I had a Diff. Eq. prof in college who had us read a 'history of Diff eq' sort of book as extra credit. lots of newton, galileo, celestial body tracking, etc. again, no hard math, but a lot of the who behind it all. Interesting read, especially if any of your HS math students are also rather good on the English lit. side.

Complex numbers are important in so many aspects of math and physics, and despite the name they are not so complex. This book has a lot to teach even those who think they know complex numbers well, since most of us never learn much about the geometry of these numbers. And for those new to the subject, this is an endlessly stimulating introduction. available here at amazon: http://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469 also see the author's page about the book here: http://www.usfca.edu/vca/ Also, I'll throw in the Feynman Lectures on Computation, since it is a nice introduction to the physics of computing; plus it's hard to go wrong with anything by Feynman.

by Charles Seife

Mathematics Made Difficult by Carl Linderholm. I read this book in high school and when I didn't get something, I took the time to look it. I just went to see if I could find a copy and for some reason their priced at over $100. Pretty good for a book written in 1972.

http://www.amazon.com/Mathematics-made-difficult-Carl-Linderholm/dp/0529045524/ref=sr_1_1?ie=UTF8&s=books&qid=1234144871&sr=8-1

Cryptonomicon by Niel Stephenson.

I've used the example of mapping the US vs Europe by placing lights on the tops of people's heads and mapping when they go up and down (off of sidewalks) to explain statistical analysis to MBAs way more successfully than anything else.

This latter-day sequel is good: Flatterland: Like Flatland, Only More So.

I initially found this book when I was researching for a philosophy paper. I really find the combination of mathematics and philosophy to be exciting, mind expanding, etc etc. It provides a bit of math history beginning with pythagorous and his bafflement over the 2^(1/2) if I recall correctly. The GEB is an excellent book but as everyone is saying, you probably won't be inspiring any students that aren't already on the geeky side to begin with. Most students would probably flip through it and say "WTF?!" http://www.amazon.com/Mystery-Aleph-Mathematics-Kabbalah-Infinity/dp/0743422996/ref=sr_1_6?ie=UTF8&s=books&qid=1234143978&sr=1-6

This book is very accessible and has some very interesting things and relationships about numbers. http://www.amazon.com/Mathematical-Mysteries-Beauty-Magic-Numbers/dp/0738202592/ref=sr_1_1?ie=UTF8&s=books&qid=1234144461&sr=1-1

My father wouldn't let me read this because it's somewhat anti-feminist.

"Somewhat"? In Flatland, the social status of men is proportional to their number of sides (triangles are the lowest class, and priests are nearly circles); women are even lower, being straight lines. Women are not allowed to walk in public spaces without swaying and emitting noises, so that men do not accidentally get impaled on them. They have to enter their houses by the back door. They are considered "wholly devoid of brain-power", driven by emotion and instinct and lacking memory, and they receive no education.

But it's social satire, not a reflection of the author's views. He was "a firm believer in equality of educational opportunity, across social classes and in particular for women", and the book is attempting to highlight a Victorian mindset that was still prevalent at that time. The women in the book act in far more complex ways than their men give them credit for. The author even says "To my readers in Spaceland the condition of our Women may seem truly deplorable, and indeed it is" - he's not happy with how they're treated, and readers in Spaceland will hopefully see that it's caused by the absurd class system holding them back, though the narrator can't avoid falling back into the prejudices of his society.

The book makes more sense when you understand the context. The Annotated Flatland is quite interesting, providing some background on the author and mathematics and the society of the time.

("more sense" doesn't mean it actually does make sense - it all still seems a bit muddled to me, with a random mixture of physical differences and social differences between people, and strange science (like Lamarckian evolution where the actions of a parent affect the number of sides (hence social status) not of themselves but of their offspring), and sections that I don't understand the point of (like the whole thing about colour being discovered and then banned - it makes sense within Flatland but is it meant to be satirising anything in real life?). Much of it is probably because the world has changed so drastically in 125 years that I just can't understand where the author was coming from. But it's an interesting book despite (or perhaps because of) that.)

Anything by Martin Gartner, specifically http://www.amazon.com/Aha-Insight-Gotcha-Spectrum/dp/0883855518

Combinatorics is a field of math that's easily accessible, but runs very deep and is fundamental to many other areas of mathematics. All you need to start is a good sense of logic.

I TAed an introductory combinatorics university course, and this textbook was decent:

http://www.amazon.com/Applied-Combinatorics-Alan-Tucker/dp/0471735078/ref=pd_bbs_sr_4?ie=UTF8&s=books&qid=1234141953&sr=8-4

You can explore things like counting (how many ways to pack n balls into m boxes, considering the balls distinct or not, and similarly for the boxes), probability, graph theory, design theory, and all kinds of fun stuff. Sudoku, for example, is combinatorially very interesting, and a good way to motivate anyone into the field.

Archimedes Revenge by Paul Hoffman
http://www.amazon.com/Archimedes-Revenge-Joys-Perils-Mathematics/dp/0393327752/

A collection of short essay on various topics. I particularly enjoyed the section on game theory and how it applies to voting systems.

Martin Gardner's written a lot of amazing stuff. You could pick up a copy of The Colossal Book of Short Puzzles and Problems and then work through puzzles from it in groups.

Also, this could be interesting, if a bit different: Unit Polyhedron Origami by Tomoko Fuse. Basically, unit origami is about building large shapes by making many small modules and combining them. It can be quite fascinating from a geometry perspective: given a square piece of paper and no other tools, the book will show you how to construct an equilateral triangle, or a regular pentagon, or a regular hexagon. In fact, not only construct them, but with pockets and tabs so you can join them together.

Martin Gardner's written a lot of amazing stuff. You could pick up a copy of The Colossal Book of Short Puzzles and Problems and then work through puzzles from it in groups.

Also, this could be interesting, if a bit different: Unit Polyhedron Origami by Tomoko Fuse. Basically, unit origami is about building large shapes by making many small modules and combining them. It can be quite fascinating from a geometry perspective: given a square piece of paper and no other tools, the book will show you how to construct an equilateral triangle, or a regular pentagon, or a regular hexagon. In fact, not only construct them, but with pockets and tabs so you can join them together.

Seconds on Innumeracy.

Also, I can't say it goes "in depth" on a topic, but "Aha! Insight" is very fun and engaging while teaching some tricky and useful math.

I love this book. Even if they don't understand any of it, it has enough problems and examples to make you think. http://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234140945&sr=8-1

You should DEFINITELY advise them the *great* book from Rozsa Péter (the hungarian mathematician who discovered - despite all common thoughts - the Ackermann function). "Playing with Infinity" is not well know but it was a big success in my case and for all the persons I advised it to. It is an incredibly pedagogic and fun book, definitely recommended for high school but to my mind advisable also to all maths enthusiasts, researchers included : if you don't learn anything on the technical side, you'll surely learn a lot about pedagogy and have a great time! It starts from the very beginning (how many sheeps) on a very practical point of view to elaborate concepts in various fields of mathematics, even the most complicated and abstract ones like topology and number theory. Please read it and spread the word, I assure you that is is worth it. http://www.amazon.com/Playing-Infinity-Mathematical-Explorations-Excursions/dp/0486232654/ref=sr_1_5?ie=UTF8&s=books&qid=1234139922&sr=8-5 PS: look also at her photo on Wikipedia. How could such a beautiful and sweet old lady not write beautiful things?

I agree sort of. I actually did a major in math and I focused primarily on algebraic geometry. I have to admit that math didn't really get interesting in college until upper division math courses. The problem is these courses are extremely rigorous. I remember abstract algebra being very difficult to learn when I was used to my previous college level calculus courses which were basically memorization and solving equations. Abstract algebra on the other hand was taught by proof. Groups, rings, fields, homomorphisms, isomorphisms, and Galois Theory are all very interesting, but I think this might be tough to teach to high school kids.

I think perhaps a better subject to teach would be topology. I realize this is probably a more rigorous class than abstract algebra, but I think you can skip some of the details and present it to them in an easily understandable way. Also, the pre-requisites are fairly minimal if you don't advance to algebraic topology, you really only need a decent background in set theory. I think for an average high school student it'd be hard to grasp the idea of what a homomorphism is, or an automorphism. These are largely shown through proofs. However, you can show what a homeomorphism is visually by using say, a rubber band, or a piece of clay. I think at the high school level you really only need to impart the idea behind the math and perhaps get them interested.

Also, if you skip metric spaces you can bypass the analysis prerequisite. I think you could easily teach them what a topological space is, the fundamental idea behind homeomorphisms, closure, compactness, connectedness, path-connectedness*, and the separation axioms.

This is the book I used in my topology class, although I think it'd serve better as a reference to the teacher than the students.

They might not understand the prototypical example of a topological space which is connected but not path-connected though.

While in high school I read "Mazes for the Mind: Computers and the Unexpected" by Clifford A. Pickover. It ties math to a wide variety of topics, and should be entertaining and mostly accessible even if a little of the math goes over their heads. And it's full of pretty pictures!

Apparently he has written a number of books since then. I haven't read any of them, so I wouldn't know which to recommend.

(As mentioned a number of times already, I'd recommend Godel, Escher, Bach as well.)

Seconding "The World of Mathematics" by James R. Newman. If you have any students who are bright and curious, these four volumes provide endless opportunities to explore. They are well written and are still in print as paperbacks, but kinda pricey. For your classroom, I'd suggest a used set of the hardback printing, available used in good condition for cheap.

Hawking's God Created the Integers really shows off the beautify of some of the most seminal developments in mathematics over the millennia. Working through the proof for Gödel's Incompleteness Theorem was rewarding.

All of the Feynman stories are available in _Classic Feynman_ http://www.amazon.com/Classic-Feynman-Adventures-Curious-Character/dp/0393061329/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234139461&sr=8-1 I have my students read this and it's incredibly inspirational to them.

I can heartily recommend "The Enjoyment of Mathematics" to gifted students:

http://www.amazon.com/Enjoyment-Mathematics-Selections-Mathematical-Recreations/dp/0486262421

I loved this book when I was in high school. All it requires is algebra and plane geometry. It covers many interesting topics and is extremely readable. A few parts are out of date (4-color theorem and Fermat's Last Theorem are now solved), but the subtracts nothing from its value.

Expose them to Clifford Algebra before they start screwing up their brains with linear algrebra.

Jan Gullberg's book Mathematics: From the Birth of Numbers is a great read! It covers, well, a hell of a lot: number theory, trig, fractals, matrices, calc, probability, diff eq, combinatorics, symbolic logic, etc. It includes anecdotes and historical notes, and does a very good job of explaining many different things.

Amazon has a couple of reviews.

... 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".

Seconded. I'm in grad school and I have that right next to my desk. I wish I would've found that before I started grad school.

If you don't have Journey through Genius by William Dunham, you should. It is a GREAT book that shows beautiful mathematics while telling interesting historical stories.

I read it in high school, and it helped me develop a love for mathematics.

It remains on my bookshelf today.

I was thinking something similar, but wasn't sure if it would be outside of the scope of what the OP wanted.

In particular, I really like this: Linear Algebra Done Right.

It's an abstract linear algebra book that does a great job of motivating topics *without* resorting to determinants, which are weird to get your head around.

Anyway, getting through it would give students some good insights into the mathematical process, I think.

I was thinking along the lines of Abstract Algebra too, but I considered that it might just be too upper-division for the average high-school student.

But I do feel that any high school student motivated enough can at least tackle some of the basics. The right book is important too. The 900-page textbook I am using for my graduate level course is probably not the best idea.

A text book that isn't too dense should be fine. I think problems might arise with constructing new groups, though, like modding out by the kernel of some homomorphism, Field of Fractions, etc... Students will need a grasp on Set Theory for that.

Last March, I introduced Dihedral groups to my brother's 3 older kids, 11 to 13 at the time, and they were perfectly capable of filling out a Cayley table for the groups of symmetries of the equilateral triangle and square and had some fun with it. It wasn't difficult for them to notice that each element appeared in any given row or column exactly once, aside from the headings.

But perhaps the book I might recommend is my textbook for Sets and Logic:

Mathematical Reasoning: Writing and Proof by Ted Sundstrom
http://www.amazon.com/Mathematical-Reasoning-Writing-Proof-2nd/dp/0131877186/ref=sr_1_1?ie=UTF8&s=books&qid=1234136896&sr=8-1

It would be great for teaching students that writing and mathematics are not necessarily two different animals.

The book How to Solve It by G. Polya is a
classic must read. While it was given to me by one of math
professors in undergraduate school it should not be over the heads of
advanced high school students.

It's an old book, but I like it.

Lady Luck: The Theory of Probability

The author does a great job making the subject easy to understand for non-math people like myself.

http://www.amazon.com/Book-Numbers-John-H-Conway/dp/038797993X/ref=pd_bbs_2?ie=UTF8&s=books&qid=1234135581&sr=8-2 I'm suprised this hasn't been mentioned yet. It is a full-color introduction to many areas of mathematics, perfect for the age group specified and not deep enough to get dull.

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.

A History of PI by Petr Beckmann is a great book for that age group. It has lots of historical information about PI and its calculation by various historical figures and cultures. The writing style is engaging and even moving. Another plus for that age group - it's less than 200 pages long.

I second a previous poster's suggestion of Simon Singh's The Code Book.

The Great International Math On Keys Book

Okay, I'm joking. But what's the modern version of this book?

I got this with my TI-30 in 1976 and went through the whole thing because it was cool to have things to do with the calculator. I was in no way a gifted or dedicated student. I was just a bright and bored kid, and didn't get great grades.

What have we got like this today that uses the existing software on our computers? (Do our computer now all have good software like that great graphing calculator that was shipped with PPC Macs? I've no idea what's on XP/Vista these days.)

If what you're looking for is just readable books that bring forth a new perspective on maths, then I personally recommend Nassim Nicholas Taleb's Black Swan: The Impact of the Highly Improbable http://www.amazon.com/Black-Swan-Impact-Highly-Improbable/dp/1400063515/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234133817&sr=8-1 This book is a highly engaging, readable introduction to thinking about the limitations of statistical probabilities. Also, if anyone has not read Zero: The Biography of a Dangerous Idea by Charles Seife http://www.amazon.com/Zero-Biography-Dangerous-Charles-Seife/dp/0140296476/ref=sr_1_1?ie=UTF8&s=books&qid=1234134095&sr=1-1 you are depriving yourself of the fascinating history of a shockingly revolutionary idea.

If what you're looking for is just readable books that bring forth a new perspective on maths, then I personally recommend Nassim Nicholas Taleb's Black Swan: The Impact of the Highly Improbable http://www.amazon.com/Black-Swan-Impact-Highly-Improbable/dp/1400063515/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234133817&sr=8-1 This book is a highly engaging, readable introduction to thinking about the limitations of statistical probabilities. Also, if anyone has not read Zero: The Biography of a Dangerous Idea by Charles Seife http://www.amazon.com/Zero-Biography-Dangerous-Charles-Seife/dp/0140296476/ref=sr_1_1?ie=UTF8&s=books&qid=1234134095&sr=1-1 you are depriving yourself of the fascinating history of a shockingly revolutionary idea.

Not strictly mathematics, but Richard Feynman's "autobiography" might be a good one for inspiring your kids to show what they can do with their math knowledge.

Prime Obsession: A well-written history of the still-unproven Riemann Hypothesis. Maybe one of your students will solve it over summer break!

http://www.amazon.com/Godel-Escher-Bach-Eternal-Golden/dp/0465026567/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234132982&sr=8-1 Godel, Escher, Bach: An Eternal Golden Braid Very interesting book and should get students of that age excited about math and science IF they are predisposed to that sort of thing.

was full of the sort of stuff that's fascinating to inquiring minds. I read one of his collections many moons ago and was enthralled! Not common to find a math book that could be called a "page turner"

Link is to a CD-ROM of all his books
http://www.amazon.com/Martin-Gardners-Mathematical-Games-Gardner/dp/0883855453

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 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

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