Slashdot Mirror

Here are some books I've enjoyed reading and would highly recommend to interested students: The man who loved only numbers, by Paul Hoffman The Wild Numbers, by Philibert Schogt Chaos, by James Gleick - this one is a classic and is a fantastic introduction to the field of chaos, fractals, etc. The Penguin Dictionary of Curious and Interesting Numbers, by David Wells - I'm reading this at present, it's easy to pick up at random and just read, and quite fascinating. How about How to Solve it, by Polya? Enigma, by Robert Harris - you can recommend the film too. Very very very good indeed. A Beautiful Mind, by Silvia Nasar, you can find it here http://www.amazon.com/Beautiful-Mind-Mathematical-Genius-Laureate/dp/0743224574 Who got Einstein's office? by Ed Regis http://www.amazon.com/s/ref=nb_ss_b?url=search-alias%3Dstripbooks&field-keywords=einstein's+office&x=0&y=0 Simon Singh's books - I know they've already been recommended but I just wanted to add my vote, they're brilliant A history of mathematics by Boyer & Merzbach http://www.amazon.com/History-Mathematics-Carl-B-Boyer/dp/0471543977 There's a few more for you.

sorry--it was sheldon ross

http://www.amazon.com/dp/0125980639

another thought i had was complex variables. not the a+bi bullshit, but doing a good treatment of the complex exponential function (seeing all of those cryptic trig formulas just drop out is a potentially life-changing experience) and analytic function theory (cauchy-reimann equations, taylor series, Cauchy formula). i first took such a course concurrently with calculus in high school and it's not too much of a stretch. the stuff all seems really fancy, but is conceptually no more difficult than calculus. if students give you guff about "imaginary" numbers not being real, well...negative numbers are real either, in some sense. The square root of -2 is just as natural as the square root of 2.

Try "The Haskell Road to Logic, Maths, and Programming". http://www.amazon.com/Haskell-Logic-Maths-Programming-Computing/dp/0954300696/

It's an intro set theory and logic book disguised as an intro to Haskell programming. It's the perfect thing for a motivated high schooler to dive into. (And the Haskell syntax is much cleaner than Lisp for representing math problems, so new programmers won't be so hung up on language difficulties and will learn some programming on the side while focusing on the mathematics.)

No question, these are the books you want. From the publisher's description: Product Description "aha! Gotcha" and "aha! Insight" are here combined as a single volume. The aha! books, as they are referred to by fans of the author Martin Gardner, contain 144 wonderful puzzles from the reigning king of recreational mathematics. In this combined volume, you will find puzzles ranging over geometry, logic, probability, statistics, number,time, combinatorics, and word play. Gardner calls these puzzles aha! problems. He explains that aha! problems "seem difficult, and indeed are difficult if you go about trying to solve them in traditional ways. But if you can free your mind from standard problem solving techniques, you may be receptive to an aha! reaction that leads immediately to a solution. Don't be discouraged if, at first, you have difficulty with these problems. Try your best to solve each one before you read the answer. After a while you will begin to catch the spirit of offbeat , nonlinear thinking, and you may be surprised to find your aha! ability improving." Studies show that persons who possess a high aha! ability are all intelligent to a moderate level, but beyond that level there seems to be no correlation between high intelligence and aha! thinking. So dig into some of the puzzles in this book, and prepare yourself for an aha! experience. Book Description Previously published separately, the two books aha! Gotcha and aha! Insight are here combined as a single volume. The aha! books, as they are referred to by fans of the author Martin Gardner, contain 144 wonderful puzzles from the reigning king of recreational mathematics. See http://www.amazon.com/Aha-Insight-Gotcha-Spectrum/dp/0883855518/ref=sr_1_2?ie=UTF8&s=books&qid=1234183387&sr=1-2, also available separately.

This was a recent series on BBC TV. Also a level 1 course at the OU which would be good for your brighter high school students. See: http://www.open2.net/storyofmaths/ http://www3.open.ac.uk/courses/bin/p12.dll?C01TM190 http://www.amazon.com/Story-Mathematics-Anne-Rooney/dp/1841939404

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.

by Ian Stewart

highly readable non-technical tour of things like congruences, axiomatics, abstract algebra, topology and other elements of "real" mathematics, although as he rightly points out, he doesn't do much with analysis, because you really can't do much with analysis that isn't technical in nature.

http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewart/dp/0486284247

I concur with your Feynmann recommendation, but would in addition propose his delightful proof of Kepler's hypothesis on planetary orbits, done entirely with geometry -- no calculus needed!

http://www.amazon.com/Feynmans-Lost-Lecture-Motion-Planets/dp/0393039188

Although not strictly speaking a book on mathematics, this is the story of Andrew Wiles trials and tribulations in solving Fermat's Last Theorem and is an incredibly well written account. You can find it on Amazon here: http://www.amazon.com/Fermats-Enigma-Greatest-Mathematical-Problem/dp/0385493622/ref=pd_bbs_3?ie=UTF8&s=books&qid=1234170288&sr=8-3.

I mean, there's two ways you can go. There's high school level problem solving texts and then there's college level math.

If they're looking for interesting high school level math problems, then I'd look into Art of Problem Solving series (the two texts at the bottom). There's also Problem Solving Strategies by Engel, which is also an excellent book. If they're looking into some accessible "higher level" (meaning, proof-based) math, I'd suggest number theory. Niven has an excellent treatment of introductory number theory.

As far as introductory college level math goes, you could always hand them a multivariable calculus, linear algebra or differential equations book. As far as recommendations go, Axler has a pretty good treatment of linear algebra if you already have some background, Apostol's series on calculus (vol. 1 and vol. 2) has a more theoretical approach to single and multi variable calculus (it's more advanced than high school calculus, but not quite an undergraduate analysis text), and Birkhoff/Rota's differential equations book has an advanced approach to differential equations probably not for high schoolers. If you don't think they'll be scared by rigor, you could always refer them to Little Rudin, which is pretty much the standard for an undergraduate real analysis class. If you're looking for an analysis book that isn't ridiculously overpriced, I've heard about this one but I've never actually read it. For a pretty readable treatment of algebra (a.k.a. modern algebra or abstract algebra), see Artin. For a more theoretical approach (less recommended for high schoolers I guess), see MacLane/Birkhoff.

I mean, there's two ways you can go. There's high school level problem solving texts and then there's college level math.

If they're looking for interesting high school level math problems, then I'd look into Art of Problem Solving series (the two texts at the bottom). There's also Problem Solving Strategies by Engel, which is also an excellent book. If they're looking into some accessible "higher level" (meaning, proof-based) math, I'd suggest number theory. Niven has an excellent treatment of introductory number theory.

As far as introductory college level math goes, you could always hand them a multivariable calculus, linear algebra or differential equations book. As far as recommendations go, Axler has a pretty good treatment of linear algebra if you already have some background, Apostol's series on calculus (vol. 1 and vol. 2) has a more theoretical approach to single and multi variable calculus (it's more advanced than high school calculus, but not quite an undergraduate analysis text), and Birkhoff/Rota's differential equations book has an advanced approach to differential equations probably not for high schoolers. If you don't think they'll be scared by rigor, you could always refer them to Little Rudin, which is pretty much the standard for an undergraduate real analysis class. If you're looking for an analysis book that isn't ridiculously overpriced, I've heard about this one but I've never actually read it. For a pretty readable treatment of algebra (a.k.a. modern algebra or abstract algebra), see Artin. For a more theoretical approach (less recommended for high schoolers I guess), see MacLane/Birkhoff.

I mean, there's two ways you can go. There's high school level problem solving texts and then there's college level math.

If they're looking for interesting high school level math problems, then I'd look into Art of Problem Solving series (the two texts at the bottom). There's also Problem Solving Strategies by Engel, which is also an excellent book. If they're looking into some accessible "higher level" (meaning, proof-based) math, I'd suggest number theory. Niven has an excellent treatment of introductory number theory.

As far as introductory college level math goes, you could always hand them a multivariable calculus, linear algebra or differential equations book. As far as recommendations go, Axler has a pretty good treatment of linear algebra if you already have some background, Apostol's series on calculus (vol. 1 and vol. 2) has a more theoretical approach to single and multi variable calculus (it's more advanced than high school calculus, but not quite an undergraduate analysis text), and Birkhoff/Rota's differential equations book has an advanced approach to differential equations probably not for high schoolers. If you don't think they'll be scared by rigor, you could always refer them to Little Rudin, which is pretty much the standard for an undergraduate real analysis class. If you're looking for an analysis book that isn't ridiculously overpriced, I've heard about this one but I've never actually read it. For a pretty readable treatment of algebra (a.k.a. modern algebra or abstract algebra), see Artin. For a more theoretical approach (less recommended for high schoolers I guess), see MacLane/Birkhoff.

I mean, there's two ways you can go. There's high school level problem solving texts and then there's college level math.

If they're looking for interesting high school level math problems, then I'd look into Art of Problem Solving series (the two texts at the bottom). There's also Problem Solving Strategies by Engel, which is also an excellent book. If they're looking into some accessible "higher level" (meaning, proof-based) math, I'd suggest number theory. Niven has an excellent treatment of introductory number theory.

As far as introductory college level math goes, you could always hand them a multivariable calculus, linear algebra or differential equations book. As far as recommendations go, Axler has a pretty good treatment of linear algebra if you already have some background, Apostol's series on calculus (vol. 1 and vol. 2) has a more theoretical approach to single and multi variable calculus (it's more advanced than high school calculus, but not quite an undergraduate analysis text), and Birkhoff/Rota's differential equations book has an advanced approach to differential equations probably not for high schoolers. If you don't think they'll be scared by rigor, you could always refer them to Little Rudin, which is pretty much the standard for an undergraduate real analysis class. If you're looking for an analysis book that isn't ridiculously overpriced, I've heard about this one but I've never actually read it. For a pretty readable treatment of algebra (a.k.a. modern algebra or abstract algebra), see Artin. For a more theoretical approach (less recommended for high schoolers I guess), see MacLane/Birkhoff.

I mean, there's two ways you can go. There's high school level problem solving texts and then there's college level math.

If they're looking for interesting high school level math problems, then I'd look into Art of Problem Solving series (the two texts at the bottom). There's also Problem Solving Strategies by Engel, which is also an excellent book. If they're looking into some accessible "higher level" (meaning, proof-based) math, I'd suggest number theory. Niven has an excellent treatment of introductory number theory.

As far as introductory college level math goes, you could always hand them a multivariable calculus, linear algebra or differential equations book. As far as recommendations go, Axler has a pretty good treatment of linear algebra if you already have some background, Apostol's series on calculus (vol. 1 and vol. 2) has a more theoretical approach to single and multi variable calculus (it's more advanced than high school calculus, but not quite an undergraduate analysis text), and Birkhoff/Rota's differential equations book has an advanced approach to differential equations probably not for high schoolers. If you don't think they'll be scared by rigor, you could always refer them to Little Rudin, which is pretty much the standard for an undergraduate real analysis class. If you're looking for an analysis book that isn't ridiculously overpriced, I've heard about this one but I've never actually read it. For a pretty readable treatment of algebra (a.k.a. modern algebra or abstract algebra), see Artin. For a more theoretical approach (less recommended for high schoolers I guess), see MacLane/Birkhoff.

I mean, there's two ways you can go. There's high school level problem solving texts and then there's college level math.

If they're looking for interesting high school level math problems, then I'd look into Art of Problem Solving series (the two texts at the bottom). There's also Problem Solving Strategies by Engel, which is also an excellent book. If they're looking into some accessible "higher level" (meaning, proof-based) math, I'd suggest number theory. Niven has an excellent treatment of introductory number theory.

As far as introductory college level math goes, you could always hand them a multivariable calculus, linear algebra or differential equations book. As far as recommendations go, Axler has a pretty good treatment of linear algebra if you already have some background, Apostol's series on calculus (vol. 1 and vol. 2) has a more theoretical approach to single and multi variable calculus (it's more advanced than high school calculus, but not quite an undergraduate analysis text), and Birkhoff/Rota's differential equations book has an advanced approach to differential equations probably not for high schoolers. If you don't think they'll be scared by rigor, you could always refer them to Little Rudin, which is pretty much the standard for an undergraduate real analysis class. If you're looking for an analysis book that isn't ridiculously overpriced, I've heard about this one but I've never actually read it. For a pretty readable treatment of algebra (a.k.a. modern algebra or abstract algebra), see Artin. For a more theoretical approach (less recommended for high schoolers I guess), see MacLane/Birkhoff.

I mean, there's two ways you can go. There's high school level problem solving texts and then there's college level math.

If they're looking for interesting high school level math problems, then I'd look into Art of Problem Solving series (the two texts at the bottom). There's also Problem Solving Strategies by Engel, which is also an excellent book. If they're looking into some accessible "higher level" (meaning, proof-based) math, I'd suggest number theory. Niven has an excellent treatment of introductory number theory.

As far as introductory college level math goes, you could always hand them a multivariable calculus, linear algebra or differential equations book. As far as recommendations go, Axler has a pretty good treatment of linear algebra if you already have some background, Apostol's series on calculus (vol. 1 and vol. 2) has a more theoretical approach to single and multi variable calculus (it's more advanced than high school calculus, but not quite an undergraduate analysis text), and Birkhoff/Rota's differential equations book has an advanced approach to differential equations probably not for high schoolers. If you don't think they'll be scared by rigor, you could always refer them to Little Rudin, which is pretty much the standard for an undergraduate real analysis class. If you're looking for an analysis book that isn't ridiculously overpriced, I've heard about this one but I've never actually read it. For a pretty readable treatment of algebra (a.k.a. modern algebra or abstract algebra), see Artin. For a more theoretical approach (less recommended for high schoolers I guess), see MacLane/Birkhoff.

I mean, there's two ways you can go. There's high school level problem solving texts and then there's college level math.

If they're looking for interesting high school level math problems, then I'd look into Art of Problem Solving series (the two texts at the bottom). There's also Problem Solving Strategies by Engel, which is also an excellent book. If they're looking into some accessible "higher level" (meaning, proof-based) math, I'd suggest number theory. Niven has an excellent treatment of introductory number theory.

As far as introductory college level math goes, you could always hand them a multivariable calculus, linear algebra or differential equations book. As far as recommendations go, Axler has a pretty good treatment of linear algebra if you already have some background, Apostol's series on calculus (vol. 1 and vol. 2) has a more theoretical approach to single and multi variable calculus (it's more advanced than high school calculus, but not quite an undergraduate analysis text), and Birkhoff/Rota's differential equations book has an advanced approach to differential equations probably not for high schoolers. If you don't think they'll be scared by rigor, you could always refer them to Little Rudin, which is pretty much the standard for an undergraduate real analysis class. If you're looking for an analysis book that isn't ridiculously overpriced, I've heard about this one but I've never actually read it. For a pretty readable treatment of algebra (a.k.a. modern algebra or abstract algebra), see Artin. For a more theoretical approach (less recommended for high schoolers I guess), see MacLane/Birkhoff.

I mean, there's two ways you can go. There's high school level problem solving texts and then there's college level math.

If they're looking for interesting high school level math problems, then I'd look into Art of Problem Solving series (the two texts at the bottom). There's also Problem Solving Strategies by Engel, which is also an excellent book. If they're looking into some accessible "higher level" (meaning, proof-based) math, I'd suggest number theory. Niven has an excellent treatment of introductory number theory.

As far as introductory college level math goes, you could always hand them a multivariable calculus, linear algebra or differential equations book. As far as recommendations go, Axler has a pretty good treatment of linear algebra if you already have some background, Apostol's series on calculus (vol. 1 and vol. 2) has a more theoretical approach to single and multi variable calculus (it's more advanced than high school calculus, but not quite an undergraduate analysis text), and Birkhoff/Rota's differential equations book has an advanced approach to differential equations probably not for high schoolers. If you don't think they'll be scared by rigor, you could always refer them to Little Rudin, which is pretty much the standard for an undergraduate real analysis class. If you're looking for an analysis book that isn't ridiculously overpriced, I've heard about this one but I've never actually read it. For a pretty readable treatment of algebra (a.k.a. modern algebra or abstract algebra), see Artin. For a more theoretical approach (less recommended for high schoolers I guess), see MacLane/Birkhoff.

You might try Ian Stewart's Concepts of Modern Mathematics. Quoting from the end of the book:

The reader who has persevered this far must by now be a cultivator of mathematics, even if he was not at the start of the endeavour. He will therefore appreciate that, while it may be ancient and venerable, it is far from complete; that not all of it is dry; and that its reasoning has not always been either unambiguous or irrefutable - nor is it yet.

Which really captures what the book is about. It's an extremely accessible introduction to abstract algebra, topology, probability, and several other topics. It does a great job of presenting the overall structure of mathematics, and giving just enough of an idea of what's going on to make you want to learn more, without being dry, boring, or bogged down in details. I found it quite an inspiring book, and several friends that I lent it to found the same. Judging from the Amazon reviews, we weren't the only ones. All that, plus it's available as a low-cost Dover book :-)

A fantastic book that helped put me on the path to math grad school was William Dunham's "Journey Through Genius". Every chapter builds up to an important proof, explaining the historical context and necessary mathematical ideas along the way. By the end of the book, it hits the idea of different kinds of "infinity", and why there are "more" real numbers than integers! That's a really deep, fun and important theorem! I really can't recommend this book highly enough for a motivated high-schooler.

http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X/ref=sr_1_1?ie=UTF8&s=books&qid=1234163005&sr=1-1

To show a student a completely different kind of math than they're used to, you might want to let them check out some topology. "The Knot Book" by Colin Adams is a tractable introduction to knot theory, which isn't a bad way to get some exposure to topology.

http://www.amazon.com/Knot-Book-Colin-Adams/dp/0821836781/ref=sr_1_1?ie=UTF8&s=books&qid=1234163179&sr=1-1

A fantastic book that helped put me on the path to math grad school was William Dunham's "Journey Through Genius". Every chapter builds up to an important proof, explaining the historical context and necessary mathematical ideas along the way. By the end of the book, it hits the idea of different kinds of "infinity", and why there are "more" real numbers than integers! That's a really deep, fun and important theorem! I really can't recommend this book highly enough for a motivated high-schooler.

http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X/ref=sr_1_1?ie=UTF8&s=books&qid=1234163005&sr=1-1

To show a student a completely different kind of math than they're used to, you might want to let them check out some topology. "The Knot Book" by Colin Adams is a tractable introduction to knot theory, which isn't a bad way to get some exposure to topology.

http://www.amazon.com/Knot-Book-Colin-Adams/dp/0821836781/ref=sr_1_1?ie=UTF8&s=books&qid=1234163179&sr=1-1

The Manga Guide to Statistics is pretty accessible for high school kids. http://www.amazon.com/Manga-Guide-Statistics-Shin-Takahashi/dp/1593271891

A couple more I forgot to add:

http://www.amazon.com/Godel-Escher-Bach-Eternal-Golden/dp/0465026567
Godel, Escher, Bach: An Eternal Golden Braid
by Douglas R. Hofstadter
The big one - worth triple points.

http://www.amazon.com/Cracking-Math-Test-Graduate-Prep/dp/0375762671
Cracking the GRE Math Test, 2nd Edition
by Steve Leduc

This book is about the GRE subject exam, not the general math test. This test is intended only for college senior math majors.

This book is not listed here as a test prep book but as the only book I have ever seen that clearly explains a wide range of true higher mathematics. High school students should be able to progress more in understanding the essence of undergraduate math for math majors by reading this book than any other they could read.

A couple more I forgot to add:

http://www.amazon.com/Godel-Escher-Bach-Eternal-Golden/dp/0465026567
Godel, Escher, Bach: An Eternal Golden Braid
by Douglas R. Hofstadter
The big one - worth triple points.

http://www.amazon.com/Cracking-Math-Test-Graduate-Prep/dp/0375762671
Cracking the GRE Math Test, 2nd Edition
by Steve Leduc

This book is about the GRE subject exam, not the general math test. This test is intended only for college senior math majors.

This book is not listed here as a test prep book but as the only book I have ever seen that clearly explains a wide range of true higher mathematics. High school students should be able to progress more in understanding the essence of undergraduate math for math majors by reading this book than any other they could read.

I strongly recommend An Imaginary Tale: The Story of i [the square root of minus one]. I received a copy of it when I was in high school and had a very hard time putting it down. The book takes the approach of teaching the history of complex numbers - what problems needed to be solved, how complex numbers were discovered, and how they solved these problems. The book is fantastic as it is a good, fun read, while simultaneously being extremely educational. It requires no prior understanding of complex numbers to read, and should, therefore, be easily accessible to high school students. At the same time, it goes more in depth into complex numbers than I have encountered anywhere else in my academic career (including a BS in Math). I strongly recommend this book to anyone who wants to know more about complex numbers or anyone who wants a fun, educational, math read.

You can always fill it out with Sphereland.

Good book. Everyone should get credit for reading anything Rudy Rucker has written. More high weirdness than math, though.
___
Here's a bunch of really good stuff:

Mathematics for the Million by Lancelot Hogben
http://www.amazon.com/Mathematics-Million-Lancelot-Thomas-Hogben/dp/0393063615
Review
"It makes alive the contents and elements of Mathematics" -- Albert Einstein"

http://www.amazon.com/Infinity-Beyond-Lillian-R-Lieber/dp/1589880366/
Infinity: Beyond the Beyond the Beyond (Paperback)
by Lillian R. Lieber (Author), Barry Mazur (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Einstein-Theory-Relativity-Fourth-Dimension/dp/1589880447/
The Einstein Theory of Relativity: A Trip to the Fourth Dimension (Paperback)
by Lillian R. Lieber (Author), David Derbes (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Quantity-Real-Imaginary-History-Algebra/dp/0452288533/
Unknown Quantity: A Real and Imaginary History of Algebra (Paperback)
by John Derbyshire

http://www.amazon.com/Fractal-Geometry-Nature-Benoit-Mandelbrot/dp/0716711869 The Fractal Geometry of Nature
by Benoit B. Mandelbrot

http://www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0140092501
Chaos: Making a New Science
by James Gleick

Rather than just reading a book, installing the following software and working through the following tutorials should be worth beaucoup extra credit:

Geometric Algebra (GA) is one of the most exciting developments in Mathematical education and Mathematical Physics. It presents in a unified mathematical language vectors, complex numbers, quaternions, spinors, and more.

GA handles rotations easily (because it includes the quaternion algebra) and also provides a mathematical description for projective geometry. Because of this, GA is being used more and more by Computer Science (virtual reality modeling, simulation, computer vision) and Robotic Engineers (arm/body movements). ...

Geometric Algebra is also called Clifford Algebra.

Geometric algebra software GAViewer for all major OSes: http://geometricalgebra.org/gaviewer_download.html

http://www.science.uva.nl/ga/files/GABLE15plus.pdf

In this tutorial we give an introduction to geometric algebra, using our GAViewer software. In the geometric algebra for 3-dimensional Euclidean space, we graphically demonstrate the ideas of the geometric product, the outer product, and the inner product, and the geometric operators that may be formed from them. We give several demonstrations of computations you can do using the geometric algebra, including projection and rejection, orthogonalization, interpolation of rotations, and intersection of linear o set spaces such as lines and planes. We emphasize the importance of blades as representations of subspaces, and the use of meet and join to manipulate them. We end with Euclidean geometry of 2-dimensional space as represented in the 3-dimensional homogeneous model.

http://www.science.uva.nl/ga/tutorials/CGA/

This tutorial introduces the conformal model of 3D Euclidean geometry, to date the most

You can always fill it out with Sphereland.

Good book. Everyone should get credit for reading anything Rudy Rucker has written. More high weirdness than math, though.
___
Here's a bunch of really good stuff:

Mathematics for the Million by Lancelot Hogben
http://www.amazon.com/Mathematics-Million-Lancelot-Thomas-Hogben/dp/0393063615
Review
"It makes alive the contents and elements of Mathematics" -- Albert Einstein"

http://www.amazon.com/Infinity-Beyond-Lillian-R-Lieber/dp/1589880366/
Infinity: Beyond the Beyond the Beyond (Paperback)
by Lillian R. Lieber (Author), Barry Mazur (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Einstein-Theory-Relativity-Fourth-Dimension/dp/1589880447/
The Einstein Theory of Relativity: A Trip to the Fourth Dimension (Paperback)
by Lillian R. Lieber (Author), David Derbes (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Quantity-Real-Imaginary-History-Algebra/dp/0452288533/
Unknown Quantity: A Real and Imaginary History of Algebra (Paperback)
by John Derbyshire

http://www.amazon.com/Fractal-Geometry-Nature-Benoit-Mandelbrot/dp/0716711869 The Fractal Geometry of Nature
by Benoit B. Mandelbrot

http://www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0140092501
Chaos: Making a New Science
by James Gleick

Rather than just reading a book, installing the following software and working through the following tutorials should be worth beaucoup extra credit:

Geometric Algebra (GA) is one of the most exciting developments in Mathematical education and Mathematical Physics. It presents in a unified mathematical language vectors, complex numbers, quaternions, spinors, and more.

GA handles rotations easily (because it includes the quaternion algebra) and also provides a mathematical description for projective geometry. Because of this, GA is being used more and more by Computer Science (virtual reality modeling, simulation, computer vision) and Robotic Engineers (arm/body movements). ...

Geometric Algebra is also called Clifford Algebra.

Geometric algebra software GAViewer for all major OSes: http://geometricalgebra.org/gaviewer_download.html

http://www.science.uva.nl/ga/files/GABLE15plus.pdf

In this tutorial we give an introduction to geometric algebra, using our GAViewer software. In the geometric algebra for 3-dimensional Euclidean space, we graphically demonstrate the ideas of the geometric product, the outer product, and the inner product, and the geometric operators that may be formed from them. We give several demonstrations of computations you can do using the geometric algebra, including projection and rejection, orthogonalization, interpolation of rotations, and intersection of linear o set spaces such as lines and planes. We emphasize the importance of blades as representations of subspaces, and the use of meet and join to manipulate them. We end with Euclidean geometry of 2-dimensional space as represented in the 3-dimensional homogeneous model.

http://www.science.uva.nl/ga/tutorials/CGA/

This tutorial introduces the conformal model of 3D Euclidean geometry, to date the most

You can always fill it out with Sphereland.

Good book. Everyone should get credit for reading anything Rudy Rucker has written. More high weirdness than math, though.
___
Here's a bunch of really good stuff:

Mathematics for the Million by Lancelot Hogben
http://www.amazon.com/Mathematics-Million-Lancelot-Thomas-Hogben/dp/0393063615
Review
"It makes alive the contents and elements of Mathematics" -- Albert Einstein"

http://www.amazon.com/Infinity-Beyond-Lillian-R-Lieber/dp/1589880366/
Infinity: Beyond the Beyond the Beyond (Paperback)
by Lillian R. Lieber (Author), Barry Mazur (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Einstein-Theory-Relativity-Fourth-Dimension/dp/1589880447/
The Einstein Theory of Relativity: A Trip to the Fourth Dimension (Paperback)
by Lillian R. Lieber (Author), David Derbes (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Quantity-Real-Imaginary-History-Algebra/dp/0452288533/
Unknown Quantity: A Real and Imaginary History of Algebra (Paperback)
by John Derbyshire

http://www.amazon.com/Fractal-Geometry-Nature-Benoit-Mandelbrot/dp/0716711869 The Fractal Geometry of Nature
by Benoit B. Mandelbrot

http://www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0140092501
Chaos: Making a New Science
by James Gleick

Rather than just reading a book, installing the following software and working through the following tutorials should be worth beaucoup extra credit:

Geometric Algebra (GA) is one of the most exciting developments in Mathematical education and Mathematical Physics. It presents in a unified mathematical language vectors, complex numbers, quaternions, spinors, and more.

GA handles rotations easily (because it includes the quaternion algebra) and also provides a mathematical description for projective geometry. Because of this, GA is being used more and more by Computer Science (virtual reality modeling, simulation, computer vision) and Robotic Engineers (arm/body movements). ...

Geometric Algebra is also called Clifford Algebra.

Geometric algebra software GAViewer for all major OSes: http://geometricalgebra.org/gaviewer_download.html

http://www.science.uva.nl/ga/files/GABLE15plus.pdf

In this tutorial we give an introduction to geometric algebra, using our GAViewer software. In the geometric algebra for 3-dimensional Euclidean space, we graphically demonstrate the ideas of the geometric product, the outer product, and the inner product, and the geometric operators that may be formed from them. We give several demonstrations of computations you can do using the geometric algebra, including projection and rejection, orthogonalization, interpolation of rotations, and intersection of linear o set spaces such as lines and planes. We emphasize the importance of blades as representations of subspaces, and the use of meet and join to manipulate them. We end with Euclidean geometry of 2-dimensional space as represented in the 3-dimensional homogeneous model.

http://www.science.uva.nl/ga/tutorials/CGA/

This tutorial introduces the conformal model of 3D Euclidean geometry, to date the most

You can always fill it out with Sphereland.

Good book. Everyone should get credit for reading anything Rudy Rucker has written. More high weirdness than math, though.
___
Here's a bunch of really good stuff:

Mathematics for the Million by Lancelot Hogben
http://www.amazon.com/Mathematics-Million-Lancelot-Thomas-Hogben/dp/0393063615
Review
"It makes alive the contents and elements of Mathematics" -- Albert Einstein"

http://www.amazon.com/Infinity-Beyond-Lillian-R-Lieber/dp/1589880366/
Infinity: Beyond the Beyond the Beyond (Paperback)
by Lillian R. Lieber (Author), Barry Mazur (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Einstein-Theory-Relativity-Fourth-Dimension/dp/1589880447/
The Einstein Theory of Relativity: A Trip to the Fourth Dimension (Paperback)
by Lillian R. Lieber (Author), David Derbes (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Quantity-Real-Imaginary-History-Algebra/dp/0452288533/
Unknown Quantity: A Real and Imaginary History of Algebra (Paperback)
by John Derbyshire

http://www.amazon.com/Fractal-Geometry-Nature-Benoit-Mandelbrot/dp/0716711869 The Fractal Geometry of Nature
by Benoit B. Mandelbrot

http://www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0140092501
Chaos: Making a New Science
by James Gleick

Rather than just reading a book, installing the following software and working through the following tutorials should be worth beaucoup extra credit:

Geometric Algebra (GA) is one of the most exciting developments in Mathematical education and Mathematical Physics. It presents in a unified mathematical language vectors, complex numbers, quaternions, spinors, and more.

GA handles rotations easily (because it includes the quaternion algebra) and also provides a mathematical description for projective geometry. Because of this, GA is being used more and more by Computer Science (virtual reality modeling, simulation, computer vision) and Robotic Engineers (arm/body movements). ...

Geometric Algebra is also called Clifford Algebra.

Geometric algebra software GAViewer for all major OSes: http://geometricalgebra.org/gaviewer_download.html

http://www.science.uva.nl/ga/files/GABLE15plus.pdf

In this tutorial we give an introduction to geometric algebra, using our GAViewer software. In the geometric algebra for 3-dimensional Euclidean space, we graphically demonstrate the ideas of the geometric product, the outer product, and the inner product, and the geometric operators that may be formed from them. We give several demonstrations of computations you can do using the geometric algebra, including projection and rejection, orthogonalization, interpolation of rotations, and intersection of linear o set spaces such as lines and planes. We emphasize the importance of blades as representations of subspaces, and the use of meet and join to manipulate them. We end with Euclidean geometry of 2-dimensional space as represented in the 3-dimensional homogeneous model.

http://www.science.uva.nl/ga/tutorials/CGA/

This tutorial introduces the conformal model of 3D Euclidean geometry, to date the most

You can always fill it out with Sphereland.

Good book. Everyone should get credit for reading anything Rudy Rucker has written. More high weirdness than math, though.
___
Here's a bunch of really good stuff:

Mathematics for the Million by Lancelot Hogben
http://www.amazon.com/Mathematics-Million-Lancelot-Thomas-Hogben/dp/0393063615
Review
"It makes alive the contents and elements of Mathematics" -- Albert Einstein"

http://www.amazon.com/Infinity-Beyond-Lillian-R-Lieber/dp/1589880366/
Infinity: Beyond the Beyond the Beyond (Paperback)
by Lillian R. Lieber (Author), Barry Mazur (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Einstein-Theory-Relativity-Fourth-Dimension/dp/1589880447/
The Einstein Theory of Relativity: A Trip to the Fourth Dimension (Paperback)
by Lillian R. Lieber (Author), David Derbes (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Quantity-Real-Imaginary-History-Algebra/dp/0452288533/
Unknown Quantity: A Real and Imaginary History of Algebra (Paperback)
by John Derbyshire

http://www.amazon.com/Fractal-Geometry-Nature-Benoit-Mandelbrot/dp/0716711869 The Fractal Geometry of Nature
by Benoit B. Mandelbrot

http://www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0140092501
Chaos: Making a New Science
by James Gleick

Rather than just reading a book, installing the following software and working through the following tutorials should be worth beaucoup extra credit:

Geometric Algebra (GA) is one of the most exciting developments in Mathematical education and Mathematical Physics. It presents in a unified mathematical language vectors, complex numbers, quaternions, spinors, and more.

GA handles rotations easily (because it includes the quaternion algebra) and also provides a mathematical description for projective geometry. Because of this, GA is being used more and more by Computer Science (virtual reality modeling, simulation, computer vision) and Robotic Engineers (arm/body movements). ...

Geometric Algebra is also called Clifford Algebra.

Geometric algebra software GAViewer for all major OSes: http://geometricalgebra.org/gaviewer_download.html

http://www.science.uva.nl/ga/files/GABLE15plus.pdf

In this tutorial we give an introduction to geometric algebra, using our GAViewer software. In the geometric algebra for 3-dimensional Euclidean space, we graphically demonstrate the ideas of the geometric product, the outer product, and the inner product, and the geometric operators that may be formed from them. We give several demonstrations of computations you can do using the geometric algebra, including projection and rejection, orthogonalization, interpolation of rotations, and intersection of linear o set spaces such as lines and planes. We emphasize the importance of blades as representations of subspaces, and the use of meet and join to manipulate them. We end with Euclidean geometry of 2-dimensional space as represented in the 3-dimensional homogeneous model.

http://www.science.uva.nl/ga/tutorials/CGA/

This tutorial introduces the conformal model of 3D Euclidean geometry, to date the most

You can always fill it out with Sphereland.

Good book. Everyone should get credit for reading anything Rudy Rucker has written. More high weirdness than math, though.
___
Here's a bunch of really good stuff:

Mathematics for the Million by Lancelot Hogben
http://www.amazon.com/Mathematics-Million-Lancelot-Thomas-Hogben/dp/0393063615
Review
"It makes alive the contents and elements of Mathematics" -- Albert Einstein"

http://www.amazon.com/Infinity-Beyond-Lillian-R-Lieber/dp/1589880366/
Infinity: Beyond the Beyond the Beyond (Paperback)
by Lillian R. Lieber (Author), Barry Mazur (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Einstein-Theory-Relativity-Fourth-Dimension/dp/1589880447/
The Einstein Theory of Relativity: A Trip to the Fourth Dimension (Paperback)
by Lillian R. Lieber (Author), David Derbes (Foreword), Hugh Gray Lieber (Illustrator)

http://www.amazon.com/Quantity-Real-Imaginary-History-Algebra/dp/0452288533/
Unknown Quantity: A Real and Imaginary History of Algebra (Paperback)
by John Derbyshire

http://www.amazon.com/Fractal-Geometry-Nature-Benoit-Mandelbrot/dp/0716711869 The Fractal Geometry of Nature
by Benoit B. Mandelbrot

http://www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0140092501
Chaos: Making a New Science
by James Gleick

Rather than just reading a book, installing the following software and working through the following tutorials should be worth beaucoup extra credit:

Geometric Algebra (GA) is one of the most exciting developments in Mathematical education and Mathematical Physics. It presents in a unified mathematical language vectors, complex numbers, quaternions, spinors, and more.

GA handles rotations easily (because it includes the quaternion algebra) and also provides a mathematical description for projective geometry. Because of this, GA is being used more and more by Computer Science (virtual reality modeling, simulation, computer vision) and Robotic Engineers (arm/body movements). ...

Geometric Algebra is also called Clifford Algebra.

Geometric algebra software GAViewer for all major OSes: http://geometricalgebra.org/gaviewer_download.html

http://www.science.uva.nl/ga/files/GABLE15plus.pdf

In this tutorial we give an introduction to geometric algebra, using our GAViewer software. In the geometric algebra for 3-dimensional Euclidean space, we graphically demonstrate the ideas of the geometric product, the outer product, and the inner product, and the geometric operators that may be formed from them. We give several demonstrations of computations you can do using the geometric algebra, including projection and rejection, orthogonalization, interpolation of rotations, and intersection of linear o set spaces such as lines and planes. We emphasize the importance of blades as representations of subspaces, and the use of meet and join to manipulate them. We end with Euclidean geometry of 2-dimensional space as represented in the 3-dimensional homogeneous model.

http://www.science.uva.nl/ga/tutorials/CGA/

This tutorial introduces the conformal model of 3D Euclidean geometry, to date the most

I would highly recommend Journey through Genius by William Dunham.

http://www.amazon.com/Journey-through-Genius-Theorems-Mathematics/dp/014014739X

It explains pretty nicely 12 mathematical theorems, from a proof of the Pythagorean theorem to Cantor's non-denumerability of the continuum. I used this book senior year of high school (just a few years ago) and it introduced me to some really interesting theorems. I especially liked learning that there were different sizes of infinity.

An Introduction to Number Theory is a fantastic book that assumes no familiarity with number theory. I used it to teach a high school number theory course with great results. Starting from essentially no prerequisites, it reaches important topics like the Chinese Remainder Theorem, and quadratic fields, as well as fun topics like Magic Squares and Continued Fractions. Perhaps the best part is the opening chapter on why you need proofs. He shows this by giving a half dozen examples of results that are "obviously" true (many of which were believed true for hundreds of years) that turn out to be false.

...or underestimate the prerequisite mathematics knowledge of 16-18 year old students.

My suggestions: (I do not endorse amazon.com in any way; I am only using their site as a reference)

Tufte, Edward R. The Visual Display of Quantitative Information, 2nd ed.

This is a classic text, brilliant for its simple thesis that data needs to be thoughtfully organized and presented in such a way as to obtain maximum impact, and therefore insight.

Hawking, Stephen. God Created the Integers: The Mathematical Breakthroughs that Changed History.

This voluminous, expansive text is somewhat on the advanced side of mathematics. However, not only is it unafraid to delve into the pure mathematics of many of the most significant discoveries and treatises of mathematical discourse, it also provides substantial historical context. Caveat: There unfortunately appears to be numerous errata for this text.

Various authors. The Contest Problem Book.

This is an entire series of books that focus on competition mathematics at the high school level. With a variety of difficulty levels, this series should provide a solid foundation for any students who love the problem-solving process, and would like to further develop their proficiency in mathematics below the calculus level. If that's not hard enough, try:

Various authors. International Mathematical Olympiads.

Again, a series of contest books, though at the Olympiad level. These are challenging enough for ANY student. But since we're still not at calculus yet, we have:

Various authors. The William Lowell Putnam Mathematics Competition.

Undoubtedly, these contain some of the most difficult math problems ever presented in contest form to students who have yet to receive their undergraduate degree. Now let's bring things back down a few notches:

Wenninger, Magnus J. Polyhedron Models.

This is a wonderful book filled with detailed diagrams, photographs, and instructions on how to build the uniform polyhedra and some notable stellations out of paper. It is a bit dated, but it provides a window into the beautiful mathematics of polyhedral geometry, while practically inviting the reader to build some of the models described. The sophisticated student may even wish to use the information contained therein to program and draw their own templates by computer.

Please note that several of these titles have related titles that you should search for.

There are more books I could recommend, but I think that this list so far does a fairly good job at touching upon areas that are at once very mathematical while not making it appear too course-like. The problem with some of the suggestions I've seen so far is that they are really geared toward a college-level understanding of mathematics, and the reading level is such that the student would presumably have to be a lot more self-motivated. Another problem I see is the suggestion of books that are not very mathematical at all, or have a "pop math" feel which I am admittedly biased against. Furthermore, not every student will be drawn into

...or underestimate the prerequisite mathematics knowledge of 16-18 year old students.

My suggestions: (I do not endorse amazon.com in any way; I am only using their site as a reference)

Tufte, Edward R. The Visual Display of Quantitative Information, 2nd ed.

This is a classic text, brilliant for its simple thesis that data needs to be thoughtfully organized and presented in such a way as to obtain maximum impact, and therefore insight.

Hawking, Stephen. God Created the Integers: The Mathematical Breakthroughs that Changed History.

This voluminous, expansive text is somewhat on the advanced side of mathematics. However, not only is it unafraid to delve into the pure mathematics of many of the most significant discoveries and treatises of mathematical discourse, it also provides substantial historical context. Caveat: There unfortunately appears to be numerous errata for this text.

Various authors. The Contest Problem Book.

This is an entire series of books that focus on competition mathematics at the high school level. With a variety of difficulty levels, this series should provide a solid foundation for any students who love the problem-solving process, and would like to further develop their proficiency in mathematics below the calculus level. If that's not hard enough, try:

Various authors. International Mathematical Olympiads.

Again, a series of contest books, though at the Olympiad level. These are challenging enough for ANY student. But since we're still not at calculus yet, we have:

Various authors. The William Lowell Putnam Mathematics Competition.

Undoubtedly, these contain some of the most difficult math problems ever presented in contest form to students who have yet to receive their undergraduate degree. Now let's bring things back down a few notches:

Wenninger, Magnus J. Polyhedron Models.

This is a wonderful book filled with detailed diagrams, photographs, and instructions on how to build the uniform polyhedra and some notable stellations out of paper. It is a bit dated, but it provides a window into the beautiful mathematics of polyhedral geometry, while practically inviting the reader to build some of the models described. The sophisticated student may even wish to use the information contained therein to program and draw their own templates by computer.

Please note that several of these titles have related titles that you should search for.

There are more books I could recommend, but I think that this list so far does a fairly good job at touching upon areas that are at once very mathematical while not making it appear too course-like. The problem with some of the suggestions I've seen so far is that they are really geared toward a college-level understanding of mathematics, and the reading level is such that the student would presumably have to be a lot more self-motivated. Another problem I see is the suggestion of books that are not very mathematical at all, or have a "pop math" feel which I am admittedly biased against. Furthermore, not every student will be drawn into

...or underestimate the prerequisite mathematics knowledge of 16-18 year old students.

My suggestions: (I do not endorse amazon.com in any way; I am only using their site as a reference)

Tufte, Edward R. The Visual Display of Quantitative Information, 2nd ed.

This is a classic text, brilliant for its simple thesis that data needs to be thoughtfully organized and presented in such a way as to obtain maximum impact, and therefore insight.

Hawking, Stephen. God Created the Integers: The Mathematical Breakthroughs that Changed History.

This voluminous, expansive text is somewhat on the advanced side of mathematics. However, not only is it unafraid to delve into the pure mathematics of many of the most significant discoveries and treatises of mathematical discourse, it also provides substantial historical context. Caveat: There unfortunately appears to be numerous errata for this text.

Various authors. The Contest Problem Book.

This is an entire series of books that focus on competition mathematics at the high school level. With a variety of difficulty levels, this series should provide a solid foundation for any students who love the problem-solving process, and would like to further develop their proficiency in mathematics below the calculus level. If that's not hard enough, try:

Various authors. International Mathematical Olympiads.

Again, a series of contest books, though at the Olympiad level. These are challenging enough for ANY student. But since we're still not at calculus yet, we have:

Various authors. The William Lowell Putnam Mathematics Competition.

Undoubtedly, these contain some of the most difficult math problems ever presented in contest form to students who have yet to receive their undergraduate degree. Now let's bring things back down a few notches:

Wenninger, Magnus J. Polyhedron Models.

This is a wonderful book filled with detailed diagrams, photographs, and instructions on how to build the uniform polyhedra and some notable stellations out of paper. It is a bit dated, but it provides a window into the beautiful mathematics of polyhedral geometry, while practically inviting the reader to build some of the models described. The sophisticated student may even wish to use the information contained therein to program and draw their own templates by computer.

Please note that several of these titles have related titles that you should search for.

There are more books I could recommend, but I think that this list so far does a fairly good job at touching upon areas that are at once very mathematical while not making it appear too course-like. The problem with some of the suggestions I've seen so far is that they are really geared toward a college-level understanding of mathematics, and the reading level is such that the student would presumably have to be a lot more self-motivated. Another problem I see is the suggestion of books that are not very mathematical at all, or have a "pop math" feel which I am admittedly biased against. Furthermore, not every student will be drawn into

...or underestimate the prerequisite mathematics knowledge of 16-18 year old students.

My suggestions: (I do not endorse amazon.com in any way; I am only using their site as a reference)

Tufte, Edward R. The Visual Display of Quantitative Information, 2nd ed.

This is a classic text, brilliant for its simple thesis that data needs to be thoughtfully organized and presented in such a way as to obtain maximum impact, and therefore insight.

Hawking, Stephen. God Created the Integers: The Mathematical Breakthroughs that Changed History.

This voluminous, expansive text is somewhat on the advanced side of mathematics. However, not only is it unafraid to delve into the pure mathematics of many of the most significant discoveries and treatises of mathematical discourse, it also provides substantial historical context. Caveat: There unfortunately appears to be numerous errata for this text.

Various authors. The Contest Problem Book.

This is an entire series of books that focus on competition mathematics at the high school level. With a variety of difficulty levels, this series should provide a solid foundation for any students who love the problem-solving process, and would like to further develop their proficiency in mathematics below the calculus level. If that's not hard enough, try:

Various authors. International Mathematical Olympiads.

Again, a series of contest books, though at the Olympiad level. These are challenging enough for ANY student. But since we're still not at calculus yet, we have:

Various authors. The William Lowell Putnam Mathematics Competition.

Undoubtedly, these contain some of the most difficult math problems ever presented in contest form to students who have yet to receive their undergraduate degree. Now let's bring things back down a few notches:

Wenninger, Magnus J. Polyhedron Models.

This is a wonderful book filled with detailed diagrams, photographs, and instructions on how to build the uniform polyhedra and some notable stellations out of paper. It is a bit dated, but it provides a window into the beautiful mathematics of polyhedral geometry, while practically inviting the reader to build some of the models described. The sophisticated student may even wish to use the information contained therein to program and draw their own templates by computer.

Please note that several of these titles have related titles that you should search for.

There are more books I could recommend, but I think that this list so far does a fairly good job at touching upon areas that are at once very mathematical while not making it appear too course-like. The problem with some of the suggestions I've seen so far is that they are really geared toward a college-level understanding of mathematics, and the reading level is such that the student would presumably have to be a lot more self-motivated. Another problem I see is the suggestion of books that are not very mathematical at all, or have a "pop math" feel which I am admittedly biased against. Furthermore, not every student will be drawn into

...or underestimate the prerequisite mathematics knowledge of 16-18 year old students.

My suggestions: (I do not endorse amazon.com in any way; I am only using their site as a reference)

Tufte, Edward R. The Visual Display of Quantitative Information, 2nd ed.

This is a classic text, brilliant for its simple thesis that data needs to be thoughtfully organized and presented in such a way as to obtain maximum impact, and therefore insight.

Hawking, Stephen. God Created the Integers: The Mathematical Breakthroughs that Changed History.

This voluminous, expansive text is somewhat on the advanced side of mathematics. However, not only is it unafraid to delve into the pure mathematics of many of the most significant discoveries and treatises of mathematical discourse, it also provides substantial historical context. Caveat: There unfortunately appears to be numerous errata for this text.

Various authors. The Contest Problem Book.

This is an entire series of books that focus on competition mathematics at the high school level. With a variety of difficulty levels, this series should provide a solid foundation for any students who love the problem-solving process, and would like to further develop their proficiency in mathematics below the calculus level. If that's not hard enough, try:

Various authors. International Mathematical Olympiads.

Again, a series of contest books, though at the Olympiad level. These are challenging enough for ANY student. But since we're still not at calculus yet, we have:

Various authors. The William Lowell Putnam Mathematics Competition.

Undoubtedly, these contain some of the most difficult math problems ever presented in contest form to students who have yet to receive their undergraduate degree. Now let's bring things back down a few notches:

Wenninger, Magnus J. Polyhedron Models.

This is a wonderful book filled with detailed diagrams, photographs, and instructions on how to build the uniform polyhedra and some notable stellations out of paper. It is a bit dated, but it provides a window into the beautiful mathematics of polyhedral geometry, while practically inviting the reader to build some of the models described. The sophisticated student may even wish to use the information contained therein to program and draw their own templates by computer.

Please note that several of these titles have related titles that you should search for.

There are more books I could recommend, but I think that this list so far does a fairly good job at touching upon areas that are at once very mathematical while not making it appear too course-like. The problem with some of the suggestions I've seen so far is that they are really geared toward a college-level understanding of mathematics, and the reading level is such that the student would presumably have to be a lot more self-motivated. Another problem I see is the suggestion of books that are not very mathematical at all, or have a "pop math" feel which I am admittedly biased against. Furthermore, not every student will be drawn into

...or underestimate the prerequisite mathematics knowledge of 16-18 year old students.

My suggestions: (I do not endorse amazon.com in any way; I am only using their site as a reference)

Tufte, Edward R. The Visual Display of Quantitative Information, 2nd ed.

This is a classic text, brilliant for its simple thesis that data needs to be thoughtfully organized and presented in such a way as to obtain maximum impact, and therefore insight.

Hawking, Stephen. God Created the Integers: The Mathematical Breakthroughs that Changed History.

This voluminous, expansive text is somewhat on the advanced side of mathematics. However, not only is it unafraid to delve into the pure mathematics of many of the most significant discoveries and treatises of mathematical discourse, it also provides substantial historical context. Caveat: There unfortunately appears to be numerous errata for this text.

Various authors. The Contest Problem Book.

This is an entire series of books that focus on competition mathematics at the high school level. With a variety of difficulty levels, this series should provide a solid foundation for any students who love the problem-solving process, and would like to further develop their proficiency in mathematics below the calculus level. If that's not hard enough, try:

Various authors. International Mathematical Olympiads.

Again, a series of contest books, though at the Olympiad level. These are challenging enough for ANY student. But since we're still not at calculus yet, we have:

Various authors. The William Lowell Putnam Mathematics Competition.

Undoubtedly, these contain some of the most difficult math problems ever presented in contest form to students who have yet to receive their undergraduate degree. Now let's bring things back down a few notches:

Wenninger, Magnus J. Polyhedron Models.

This is a wonderful book filled with detailed diagrams, photographs, and instructions on how to build the uniform polyhedra and some notable stellations out of paper. It is a bit dated, but it provides a window into the beautiful mathematics of polyhedral geometry, while practically inviting the reader to build some of the models described. The sophisticated student may even wish to use the information contained therein to program and draw their own templates by computer.

Please note that several of these titles have related titles that you should search for.

There are more books I could recommend, but I think that this list so far does a fairly good job at touching upon areas that are at once very mathematical while not making it appear too course-like. The problem with some of the suggestions I've seen so far is that they are really geared toward a college-level understanding of mathematics, and the reading level is such that the student would presumably have to be a lot more self-motivated. Another problem I see is the suggestion of books that are not very mathematical at all, or have a "pop math" feel which I am admittedly biased against. Furthermore, not every student will be drawn into

And in similar vein I found The Number Devil to be very good.

I'd recommend ET Bell's "Men of Mathematics". A collection of biographies of prominent mathematicians from classical times to the early 20th Century with a light description of their work. First published in 1937, it is more than readable. Including an interesting 1937 perspective of the work of George Boole.

Men of Mathematics(Amazon)

Godel, Escher, Bach: An Eternal Golden Braid has already been mentioned, but bears repeating. Godel's Proof by Nagel and Newman makes a good companion to it. Finally, How to Solve It by G. Polya will help make up for the deficiencies in modern mathematics textbooks. I know I wish my mathematics instructor in high school had given me Polya.

Godel, Escher, Bach: An Eternal Golden Braid has already been mentioned, but bears repeating. Godel's Proof by Nagel and Newman makes a good companion to it. Finally, How to Solve It by G. Polya will help make up for the deficiencies in modern mathematics textbooks. I know I wish my mathematics instructor in high school had given me Polya.

Godel, Escher, Bach: An Eternal Golden Braid has already been mentioned, but bears repeating. Godel's Proof by Nagel and Newman makes a good companion to it. Finally, How to Solve It by G. Polya will help make up for the deficiencies in modern mathematics textbooks. I know I wish my mathematics instructor in high school had given me Polya.

James Gleick, "Chaos". Best darn book about math and science I ever read. Not just numbers, it will change the way you look at the seeming randomness of life and give it new meaning. http://www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0140092501

n Code: A Mathematical Adventure
by Sarah Flannery

Autobiographical book by an Irish girl about how she learned cryptography, number theory, etc. and won competitions in high school for her work.

http://www.amazon.com/Code-Mathematical-Adventure-Sarah-Flannery/dp/1861972717/ref=ed_oe_p

Why Math by R.D. Driver is an excellent book which is accessible to anyone with basic arithmetic skills. This book really drives home a deep appreciation of the power of Math!

It actually may work better for kids who are younger, or for a certain type of student, but it's a fairytale where the main character solves a series of math puzzles (solutions fully explained in text) while running all over an arabian like land. There's even a princess.

The Man Who Counted by Malba Tahan (aka Julio de Melo e Sousa>=)

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

I can't reinforce the GEB reccomendation enough. Not only will it make you lol it will also teach you more math than two undergrad years at the state univ I attend. Metamagical Themas (also by Hofstadter) is a terrific choice as well.

mod parent +1 Well Read

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