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Mathematics Reading List For High School Students?

timothy posted more than 5 years ago | from the flatland-to-hawking dept.

Education 630

Troy writes "I'm a high school math teacher who is trying to assemble an extra-credit reading list. I want to give my students (ages 16-18) the opportunity/motivation to learn about stimulating mathematical ideas that fall outside of the curriculum I'm bound to teach. I already do this somewhat with special lessons given throughout the year, but I would like my students to explore a particular concept in depth. I am looking for books that are well-written, engaging, and accessible to someone who doesn't have a lot of college-level mathematical training. I already have a handful of books on my list, but I want my students to be able to choose from a variety of topics. Many thanks for all suggestions!"

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Flatland (5, Funny)

Anonymous Coward | more than 5 years ago | (#26776809)

Sorry, my list is lacking some depth.

Re:Flatland (2, Funny)

moderatorrater (1095745) | more than 5 years ago | (#26777267)

What about "Life of Pi"? That sounds like it's got a lot of math in it.

Re:Flatland (1)

fm6 (162816) | more than 5 years ago | (#26777295)

You can always fill it out with Sphereland.

Re:Flatland (-1, Flamebait)

Anonymous Coward | more than 5 years ago | (#26777327)

Linux just isn't ready for the desktop yet. It may be ready for the web servers that the average Slashdot user uses to distribute their TRON fanzines and personal Dungeons and Dragons web-sights over the information superhighway, but the average user isn't going to spend months learning how to use a CLI and then hours compiling packages so that they can get a workable graphic interface, especially not when they already have a Windows machine that does its job perfectly well and is backed by a major corporation, as opposed to Linux which is only supported by a few unemployed nerds living in their mother's basement somewhere. The last thing I want is a level 5 dwarf (haha) providing me my OS.

How to Lie with Statistics (5, Informative)

sando101x (1058590) | more than 5 years ago | (#26776813)

How to Lie with Statistics, Darren Huff, 1954

Re:How to Lie with Statistics (1)

fm6 (162816) | more than 5 years ago | (#26777275)

That's a great book that everybody should read. But it's not about math. It's about how people misuse math. I know this because I have over a billion seconds of experience!

I'm wondering (0)

Anonymous Coward | more than 5 years ago | (#26776819)

I'm wondering what constitutes "Stimulating Math?"

Re:I'm wondering (0)

Anonymous Coward | more than 5 years ago | (#26777003)

if it's anything like The Obama/Pelosi/Reid stimulus, it probably involves phys ed, home ec, shop, art, study hall, sex ed, and maybe some math buried in there somewhere.

Prime numbers online article thing (5, Interesting)

JaxWeb (715417) | more than 5 years ago | (#26776829)

I wrote this:
http://people.pwf.cam.ac.uk/jlnw3/maths/books/prime/ [cam.ac.uk]

It was meant as an introduction to the idea of proof. Perhaps you might like it.

Flatland. (0)

Anonymous Coward | more than 5 years ago | (#26776845)

My 10th grade math teacher had me read it. It was very brain-stretching.

http://math.cowpi.com/flatland/

High school is preparation for life (3, Funny)

BadAnalogyGuy (945258) | more than 5 years ago | (#26776853)

Start with Basics... (4, Funny)

Mikkeles (698461) | more than 5 years ago | (#26776859)

Principia Mathematica. It's all there ;^)

Re:Start with Basics... (5, Funny)

fm6 (162816) | more than 5 years ago | (#26777231)

No it's not. [everything2.com] Sorry.

Re:Start with Basics... (0)

Anonymous Coward | more than 5 years ago | (#26777345)

A much better criticism would have been to discuss how dense the material is, instead of arguing against formalizing math in general.

Flatland (0)

Anonymous Coward | more than 5 years ago | (#26776889)

http://en.wikipedia.org/wiki/Flatland

Any abstract algebra text (5, Interesting)

davidwr (791652) | more than 5 years ago | (#26776891)

It's normally taught as an upper-division college class but the only real prerequisite is 2nd-year high school algebra and a mind that can think abstractly.

Students will find it different enough from trig and calculus to be fresh and knowing they can do "college math" can be a real ego-boost.

By the way, if you know any elementary or middle school teachers, many of the concepts in abstract algebra can be taught to those age groups as well. Being able to do "adult math" can be a real point of pride and inspiration at those ages.

First grade isn't too early. Anyone who can add or subtract time already has the basics for abstract algebra addition and subtraction.

Re:Any abstract algebra text (-1)

Anonymous Coward | more than 5 years ago | (#26777407)

First grade isn't too early. Anyone who can add or subtract time already has the basics for abstract algebra addition and subtraction.

You've got to be kidding. Abstract algebra is beyond the capabilities of most adults. You don't even have to get that abstract: x+y will do it.

However, algebra is a good place to start for the bright high school student. In particular, I would recommend any introductory text on Linear Algebra --- one can get a lot of mileage (computer science, physics) from that subject.

Re:Any abstract algebra text (5, Interesting)

rpillala (583965) | more than 5 years ago | (#26777451)

I teach high school and try to put in the some of the abstract algebra topics when I teach Algebra 2. Some of the students enjoy it but most of them get really pained looks and only stop me to ask if the material will be on the test. That's not a big deal all it means is I'm not presenting it right yet. It also has something to do with how they've been taught in the past. But to support your recommendation I want to share that I was able to get a review copy of the current edition of my abstract algebra book from the publisher. I think at the college level this is more common than in secondary schools. Teachers should consider this method to build their resources.

I also want to recommend Men of Mathematics by E. T. Bell. The calc kids were very interested to know about Newton and Riemann's lives. Considering that most of what we do in middle and high school is actually math history, it seemed fitting to bring some of the personalities in.

The Little Schemer (1)

pHatidic (163975) | more than 5 years ago | (#26776897)

If this book doesn't make them think that math is cool, nothing will.

Flatland (5, Informative)

Ponderoid (311576) | more than 5 years ago | (#26776899)

Flatland [wikipedia.org] by Edwin Abbott Abbott. Higher-dimensional math packaged as a parody about Victorian culture. :)

*** Ponder

Re:Flatland (0)

Anonymous Coward | more than 5 years ago | (#26777107)

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

Re:Flatland (0)

Anonymous Coward | more than 5 years ago | (#26777265)

I agree with your farther that the book is disrespectful to women. To me this showed the writer is a narrow minded type of person.

I rather read and delve into fantasy with work of a open minded writer. A good example is Douglas Hofstadter.

This was just released (4, Interesting)

rolfwind (528248) | more than 5 years ago | (#26776911)

How to Think like a Mathematician:
http://www.amazon.com/How-Think-Like-Mathematician-Undergraduate/dp/0521895464 [amazon.com]
Online here (for how much longer?):
http://www.maths.leeds.ac.uk/~khouston/httlam.html [leeds.ac.uk]

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 [amazon.com]

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 [cmu.edu]

Bourbaki! (0)

Anonymous Coward | more than 5 years ago | (#26776923)

Accessible, and it'll teach them French into the bargain. One downside: you'll have to read it too --- and pretend you understood something.

Real analysis (0)

Anonymous Coward | more than 5 years ago | (#26776925)

I believe that real analysis could be a good way to go, but then again I'm unsure what level high school mathematics is in your country. We touched on formal arguments for convergence and mean value theorem back in my high school days.

In particular, I recommend: "A Companion to Analysis" by T. W. Korner.

It's well written, but a brick. It has some great humor hidden in seveal places for the alert reader ;)

Fractals (1, Interesting)

Anonymous Coward | more than 5 years ago | (#26776941)

While I can't think of a book offhand, I learnt complex numbers and matrices through playing with both IFS and standard fractals. Advantage is that you can get visual feedback of what you're doing in just a few seconds
A few lines of BASIC or equivalent and you can be playing with them in no time.

Kids are ungreatful bastards (5, Interesting)

QuantumG (50515) | more than 5 years ago | (#26776943)

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 [amazon.com] , 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.

Get involved with FIRST (USFIRST.org) (1)

purduephotog (218304) | more than 5 years ago | (#26776951)

FIRST robotics, while not a 'reading list', would provide your math students hundreds or thousands of opportunities both in the field of mathematics but also engineering and science.

Right now I can think of a few dozen 'practical' real world problems for this years competition that I could use some students seriously grounded in math to think about and solve (radius of turn for Ackermann steering, forces on a gyro during a turn, etc) not to mention coding up and implementing algorithms.

Anyway- don't sell math short- there's money in the real world applications :)

Jason / Team Lead for 1591 Greece Gladiators

Re:Get involved with FIRST (USFIRST.org) (0)

Anonymous Coward | more than 5 years ago | (#26777453)

I am on a FIRST team and quite frankly it sucks. I highly recommend other robotics competitions like Trinity or SciOly. Maybe my negative experience has more to do with the teachers involved than with the actual competitions.

Godel Escher Bach (5, Interesting)

firmamentalfalcon (1187583) | more than 5 years ago | (#26776955)

Excellent explanations. It is completely understandable if the student puts in the time to understand it. It requires almost no outside knowledge.

I would have loved it if someone showed me this book earlier.

Re:Godel Escher Bach (1, Interesting)

Anonymous Coward | more than 5 years ago | (#26777069)

I have no mod points, but I have to second this recommendation. I first read GEB when I was twelve and I don't think I fully understood the implications of what it was saying for another decade.

Strictly speaking, this book is about the Church-Turing thesis, but it's really about what we know--what we can know--and in what ways mathematics is and isn't useful. There's a full treatment of the fundamentals of math, formal logic, and a focus on the act of interpreting what math's little symbols really mean in the real world. All of this is couched in an entertaining Lewis Carroll'ian dialog referring to Escher's artwork and Bach's music.

Re:Godel Escher Bach (1)

ThatGuyJon (1299463) | more than 5 years ago | (#26777297)

Thirding this. I first read it aged 16, so if you're worried it's inappropriate for the age group, it's not. Very inspiring.

Re:Godel Escher Bach (0)

Anonymous Coward | more than 5 years ago | (#26777485)

ha!

perhaps it's easy to have a "big picture" concept of everything in GEB, but true mastery of everything in that book takes a hell of a lot of effort

unless by "puts in the time to understand it" you mean reading every passage slowly 10 times and working out examples multiple times

Just one book (-1, Troll)

Anonymous Coward | more than 5 years ago | (#26776957)

Prepares [goatse.fr] you for anything. Especially prison niggers.

Martin Gardner's column in Scientific American (5, Informative)

Lupulack (3988) | more than 5 years ago | (#26776967)

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 [amazon.com]

"The HIgher Arithmetic" (2, Interesting)

Anonymous Coward | more than 5 years ago | (#26776973)

"The Higher Arithmetic" by Harold Davenport is a fantastic book on number theory. It explains the concept of proof in the first 10 pages without using any formal notation. All of the proofs are given in an intuitive, explanation style. Aside from being a fantastic book on Number Theory (and thus a great primer to understanding modern cryptography), it is a very good introduction to the style of thinking and argument involved in actually doing /mathematics/ (as opposed to arithmetic, which is what seems to be mostly taught in schools or the treatment of mathematics in most science and engineering fields, which tends to be algorithmic and problem focused).

Interesting math, without all the math (4, Interesting)

artor3 (1344997) | more than 5 years ago | (#26776975)

I read Prisoner's Dilemma by William Poundstone when I was that age, and found it to be a very intriguing introduction to game theory. It is fairly light on math, providing only enough to show that there are calculable solutions to situations that are otherwise difficult to reason through. It also provides some real life examples which are easy to relate to, e.g. letting one child cut a piece of food in half and the other choose the half they want in order to ensure "fair" portions.

It's a good choice for showing that there's more to math than finding the length of the hypotenuse.

My math is cool (4, Insightful)

CMonk (20789) | more than 5 years ago | (#26776979)

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 [amazon.com] 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.

My new favourite, (1)

Bromskloss (750445) | more than 5 years ago | (#26776985)

which i don't really recommend for your purposes, but want to tell everyone about anyway, is Bourbaki. Available in French and English. Have a taste [google.com] . It's very dry and concise and I love it!

Prime Obsession (2, Informative)

zackhugh (127338) | more than 5 years ago | (#26776991)

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

Save for college, it'll cost you all your beer $ (0)

Anonymous Coward | more than 5 years ago | (#26777007)

High School is not for "additional reading", young man.

And you can only hope you get your beer for free, because you aren't old enough to buy it until your last year.

Math? (1)

girlintraining (1395911) | more than 5 years ago | (#26777011)

You need to provide more information about your target audience. "16-18 year olds" is a pretty broad demographic. But let's say for the sake of discussion that you mean the average kid in the average high school math class. I can sum up your lesson ideas in a word: Practicality. For most people, mathematics is tiresome, and the majority of adults don't use it for anything more than figuring out if they got the right change at the drive-thru, not spending too much at the grocery store, and taxes (for those still doing it with pencil and paper). That's just the simple reality.

That said, if you want something engaging, give them a challenge and see what they come up with. Hands-on math, with a tangible goal. Don't make it one of those "Navigate this map to collect all the items" either, that's boring. I know that the school administrators would never approve this, but here's an idea -- why not give them a small trebucket, and throw watermelons at a designated target on a football field? Basic geometry. Assign them into teams. Or a large maze and an RC car, and they have to navigate the car to the "goal"... But without seeing the car on the track. ^_^ Hello vectors, and simple calculus. Give them a goal and let them figure out the math.

the pleasures of counting (2, Informative)

thrope (830256) | more than 5 years ago | (#26777037)

I really enjoyed this book when I was at that stage... http://books.google.co.uk/books?id=wUdtVHBr-OQC [google.co.uk] Really a book about operational research, but covers lots of maths in a really applied accessible way with examples from history (spread of cholera outbreaks, optimal fleet size to avoid submarines in WW2, enigma machine etc.) Lots of exercises, and each section is relatively self contained - so ideal for starting off the kind of short projects you are talking about. Highly recommended...

Re:the pleasures of counting (1, Interesting)

Anonymous Coward | more than 5 years ago | (#26777387)

Seconded. The author, Tom Korner, has recently published another book along similar lines: Naive Decision Making.

BetterLesson (0)

Anonymous Coward | more than 5 years ago | (#26777039)

You might want to try to get in on http://betterlesson.org private beta - they have tons of great math curricula from other math teachers - some of whom have worked to provide some of extra work.

Teachers from other States or AP teachers wont have the same standards, which might help you build your list.

First Ninnle Post! (0)

Anonymous Coward | more than 5 years ago | (#26777043)

NinnleninnleninnleninnleninnleninnleninnleninnleBATMAN!

GÃdel, Escher, Bach (1)

Stephan Schulz (948) | more than 5 years ago | (#26777049)

If you want to stretch their minds, Gödel, Escher, Bach by Douglas Hofstadter is excellent. It's much deeper than it may seems, so it needs some support from a teacher who knows what he or she is doing.

If you want to make their first year in college much easier, have them work trough Schaum's Outline of Linear Algebra by Seymour Lipschutz. It's the best introduction to LA I've ever seen, accessible, but without dumbing things down.

moving outside of 'pure' math (4, Informative)

cellocgw (617879) | more than 5 years ago | (#26777067)

Let them loose on The Feynmann Lectures on Physics. Quite readable and bound to get them interested in one branch or another of physics.

The Golden Ratio -- or some other book on the same constant -- which goes into things like sunflowers and nautilus shells IIRC.

Mathenauts: a collection of sci fi short stories in which (in most cases) the hero is a mathematician.

Quantum (1)

Ruie (30480) | more than 5 years ago | (#26777073)

First take a look at Quantum magazine (http://www.nsta.org/quantum/) it had many interesting articles, it was an attempt to port over Russian magazine "Quant" famous for many high-quality mathematics and physics articles.
Also AMS has some translated (or written anew) books from Russian mathematical tradition. The books that I read personally (and liked a lot):
  • Stories about Maxima and Minima, Vladimir Mikhailovich Tikhomirov
  • Knots : Mathematics with a Twist, A. B. Sossinsky
  • Intuitive Topology, Vol. 4, V. V. Prasolov, A. B. Sossinsky ( Translator)

I am forgetting quite a few. There should be books on number theory that talk about properties of Euler function, books about foundations of mathematics, Graph theory by Orr (american author) is very good, books on elementary group theory are quite good.
Also, find some old (1920-1930) calculus books - well written ones are a whole lot easier to read and understand than todays texts (my own favorite was written by Fikhtengoltz - not sure whether the spelling is correct).

Bringing Down the House (4, Informative)

c_forq (924234) | more than 5 years ago | (#26777077)

If you're trying to get kids interested in the possibilities of math I would suggest Bringing Down The House, about the MIT Blackjack team.

Don't let them see the movie (0)

Anonymous Coward | more than 5 years ago | (#26777417)

Read the book, but dont' let them see the movie, which sucks.

Simon Singh (4, Informative)

Ian Alexander (997430) | more than 5 years ago | (#26777081)

You might look at some of Simon Singh's stuff if you haven't already- there are some good chapters in The Code Book regarding the basics of public-key cryptography which don't require any more than a basic education in algebra.

Re:Simon Singh (2, Interesting)

Ian Alexander (997430) | more than 5 years ago | (#26777169)

Also, I was a bit of a math nerd in high school, and so I suggested to my math teacher that he try a class where you give the students a simple monoalphabetic substitution cipher, do a quick rundown on how to crack them, and then give them some time to crack it. The declaration of independence was long enough for most of the kids to have gotten most of the alphabet cracked by the end of the hour. Saved me a boring class and it was a big hit. You might think about setting some kind of similar challenge.

John Allen Paulos books (2, Interesting)

Thatmushroom (447396) | more than 5 years ago | (#26777083)

"Innumeracy" and others are very good general introductions to how math is used in the real world. The kids who are going to do an extra-credit reading list will likely be right at the target level you're going for. A lot of them are also structured so you can take in a couple small chapters at a time and move on.

When I was in high school, I enjoyed.... (1)

spinach and eggs (1472445) | more than 5 years ago | (#26777085)

- Most any of the books from the MAA, especially the New Mathematics Library (now Anneli Lax New Library?), e.g. Geometric Inequalities, Geometric Transformations, Graphs and Their Uses, An Introduction to Inequalities, Uses of Infinity, Continued Fractions, The Mathematics of Choice, etc. - Ian Stewart's books, especially Nature's Numbers. - Loren C. Larson's book, Problem Solving Through Problems. - Many of the smaller Dover books (e.g. Excursions in Geometry)

The Shape of Space (5, Interesting)

Pixie_From_Hell (768789) | more than 5 years ago | (#26777125)

I highly recommend The Shape of Space by Jeff Weeks. (He's a freelance geometer, something he can afford after winning a MacArthur Genius Grant.) I've used this book a couple of times -- once with bright high school kids and once with bright college freshman -- and even if they don't get everything, just a taste is enough.

It builds on Flatland (which someone mentioned above), but has the advantage of being more modern and not sexist. But very quickly you're learning about Klein bottles, connected sums, and all sorts of topology you typically don't see until you're well into your undergraduate (or grad!) program in math. All aimed at high school kids. Very cool stuff.

Oh, and the big punchline at the end: what is the shape of the universe? At least you'll get a good understanding of the possibilities...

Here's a taste for you from a page related to the book [geometrygames.org] .

"e": The Story of A Number (2, Interesting)

hcetSJ (672210) | more than 5 years ago | (#26777131)

by Eli Maor. ISBN: 0691141347 I read this book the summer before taking calculus, and I learned the core concepts of calculus from it (limit, derivative, integral, fundamental theorem). I still had to learn the specifics in class, but having that conceptual foundation made everything easier. The book is full of interesting historical tidbits. For instance, did you know that the inventor/discoverer of the logarithm was excommunicated from the Catholic Church? I don't remember the circumstances now--I suppose Google could help, but I know it's in this book.

Computer + Mathematics == More Understanding+Fun (1)

burni (930725) | more than 5 years ago | (#26777133)

Applied Geometry for Computer Graphics and CAD
(Springer Undergraduate Mathematics Series)

Pages: 352 Seiten
Publisher: Springer
Language: Englisch
ISBN-10: 1852338016
ISBN-13: 978-1852338015

I've read it by myself and even if it says "undergraduate" it will perfectly fit,
give it a try applied mathematics is fun either.

Anonymous Coward (0)

Anonymous Coward | more than 5 years ago | (#26777137)

I would encourage you to put Cryptological Mathematics by Robert Edward Lewand on the list
( http://www.amazon.com/Cryptological-Mathematics-Mathematical-Association-Textbooks/dp/0883857197 )

It's very well explained cryptography. You could even given give them some solvable challenges if they want something extra. Now who wants to decrypt this message and find the subject of your next examination?

Surely You're Joking, Mr. Feynman! (2, Informative)

XxtraLarGe (551297) | more than 5 years ago | (#26777163)

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

Courant-Robbins (3, Informative)

fph il quozientatore (971015) | more than 5 years ago | (#26777171)

Courant and Robbins, "What is mathematics?"

The number devil (1)

baomike (143457) | more than 5 years ago | (#26777181)

The number devil. Maybe for too young an age for high school , but maybe not.

The Number Devil
A Mathematical Adventure
Hans Magnus Enzensberger
ISBN 0-8050-5770-6

Fibinacci
Golden mean
Klien
Pascal Triangle
Sierpinski triangle

The Pattern On The Stone by Danny Hillis (1)

six11 (579) | more than 5 years ago | (#26777195)

"The Pattern On The Stone: The Simple Ideas That Make Computers Work" by Danny Hillis.

It is a masterful piece on computation, and how computers work, and uses mathematics and logic in a very down-to-reality way that I think is certainly readable by a motivated high school student.

I'd write more but my laptop battery is about to die. It's a great book!

Zero, Black Swan (1)

wraithinfinite (133900) | more than 5 years ago | (#26777197)

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 [amazon.com] 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 [amazon.com] you are depriving yourself of the fascinating history of a shockingly revolutionary idea.

At that age, I wish someone had told me about... (0)

Anonymous Coward | more than 5 years ago | (#26777199)

"A course of pure mathematics" by G. H. Hardy

It is a pure gem and a pleasure to read: unfortunately I found this book five years later. It is freely available here, as it is out of copyright:
http://www.archive.org/details/coursepuremath00hardrich [archive.org]
http://en.wikipedia.org/wiki/A_course_of_pure_mathematics [wikipedia.org] :
A Course of Pure Mathematics is a classic textbook in introductory mathematical analysis, written by G. H. Hardy. It was first published in 1908, and went through many editions.

It was intended to help reform mathematics teaching in the UK, and more specifically in the University of Cambridge, and in schools preparing pupils to study mathematics at Cambridge. As such, it was aimed directly at "scholarship level" students -- the top 10% to 20% by ability. The book contains a large number of difficult problems.

The content covers introductory calculus and the theory of infinite series. The exposition is quite leisurely, but the attention to rigour high. Hardy at the period when he wrote it had successfully implemented reforms of the Mathematical Tripos at Cambridge, making it less a test of sheer problem-solving technique. In writing his Pure Mathematics he was proposing a course of study preliminary to a French-style Cours d'Analyse, at the time a benchmark for a mathematical education leading to research in the field.

"The Code Book" by Simon Singh (1, Interesting)

Anonymous Coward | more than 5 years ago | (#26777205)

I really enjoyed

"The Code Book" by Simon Singh

From Publishers Weekly

In an enthralling tour de force of popular explication, Singh, author of the bestselling Fermat's Enigma, explores the impact of cryptographyAthe creation and cracking of coded messagesAon history and society. Some of his examples are familiar, notably the Allies' decryption of the Nazis' Enigma machine during WWII; less well-known is the crucial role of Queen Elizabeth's code breakers in deciphering Mary, Queen of Scots' incriminating missives to her fellow conspirators plotting to assassinate Elizabeth, which led to Mary's beheading in 1587. Singh celebrates a group of unsung heroes of WWII, the Navajo "code talkers," Native American Marine radio operators who, using a coded version of their native language, played a vital role in defeating the Japanese in the Pacific. He also elucidates the intimate links between codes or ciphers and the development of the telegraph, radio, computers and the Internet. As he ranges from Julius Caesar's secret military writing to coded diplomatic messages in feuding Renaissance Italy city-states, from the decipherment of the Rosetta Stone to the ingenuity of modern security experts battling cyber-criminals and cyber-terrorists, Singh clarifies the techniques and tricks of code makers and code breakers alike. He lightens the sometimes technical load with photos, political cartoons, charts, code grids and reproductions of historic documents. He closes with a fascinating look at cryptanalysts' planned and futuristic tools, including the "one-time pad," a seemingly unbreakable form of encryption. In Singh's expert hands, cryptography decodes as an awe-inspiring and mind-expanding story of scientific breakthrough and high drama. Agent, Patrick Walsh. (Oct.) FYI: The book includes a "Cipher Challenge," offering a $15,000 reward to the first person to crack that code. Copyright 1999 Reed Business Information, Inc. --This text refers to an out of print or unavailable edition of this title.

Fermat's Enigma (3, Interesting)

brechmos (679454) | more than 5 years ago | (#26777213)

I really like Fermat's Enigma by Simon Singh. Relatively easy read and I found it inspiring.

Chaos Theory (1)

RoFLKOPTr (1294290) | more than 5 years ago | (#26777215)

Chaos: Making a New Science [wikipedia.org] by James Gleick is a pretty interesting math-related read. I couldn't make it all the way through it before my brain melted, but interesting nonetheless.

Anything by Mario Livio (1)

david.emery (127135) | more than 5 years ago | (#26777223)

But particularly his book on Phi... ("The Golden Ratio")

dave

p.s. has anyone read anything by John Allen Paulos, e.g. "A Mathematician Reads the Newspapers"?

It depends on what subject you want. (2, Interesting)

JoshuaZ (1134087) | more than 5 years ago | (#26777245)

Do you want them interested in math or do you want them to know more math? Since many people have already listed more applied books I'm going to try to focus on the less applied end of things.

Books with much mathematical content I'd recommend for that age group are:

Oyestein Ore's "Number Theory and its History" which is an excellent, highly concrete introduction to number theory with a lot of interesting historical material thrown in. I read this first in 9th or 10th grade.

Sawyer's "Concrete Introduction to Abstract Algebra" is an excellent introduction to many ideas that will be necessary in higher level math classes. The material is of a level that can be understood by most high school students.

A more difficult but still good book is Adams' "The Knot Book" which is an introduction to knot theory.

All of the above do not include any understanding of calculus or any other advanced topics.

If one wants a less mathematically advanced book that is more about the stories and people I'd recommend Simon Singh's "Fermat's Enigma" which tells the story of Fermat's Last theorem and along the way sketches out the great stories of mathematicians including the tragic life of Galois, the fate of Hypatia at the hands of a mob and many other great stories, all woven into the overarching narrative the quest to prove Fermat's Last Theorem. (I'm also going to take this an opportunity to strongly disrecommend vos Savant's book on Fermat's Last Theorem which contains serious errors and other problems).

William Dunham (1)

CheshireCatCO (185193) | more than 5 years ago | (#26777277)

William Dunham's books are excellent reads. They're a mix of biography and math, usually focused on the more playful, clever parts of math. (As opposed to the tedious, but necessary bits.) He covers a lot and anyone who reads them with any attention at all would come away with a pretty good conversant knowledge of mathematics.

The World of Mathematics (1)

femtobyte (710429) | more than 5 years ago | (#26777281)

I remember a 3-volume set of books called "The World of Mathematics" from when I was around high school age. The books were a collection of short essays on mathematical topics by current and historical mathematicians. The subjects were very wide ranging and quite approachable. Those books greatly contributed to my interest in mathematics (leading me to a double-major in math and physics). I don't know if they're still in print, but they're worth tracking down.

TI-30 (0)

Anonymous Coward | more than 5 years ago | (#26777289)

The Great International Math On Keys Book [amazon.com]

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.)

Ask! (1)

fm6 (162816) | more than 5 years ago | (#26777315)

Oops, somebody screwed up. This is an Ask Slashdot that asks an interesting question that some of us are actually qualified to answer, and that can't be answered by trivial means such as googling. I don't think that's allowed!

How about some Rudy Rucker? (1)

mog007 (677810) | more than 5 years ago | (#26777323)

Master of Space and Time by Rudy Rucker. It has some math in it, and it's funny to boot.

"What is Mathematics" by Courant and Robbins (2, Interesting)

DrJimbo (594231) | more than 5 years ago | (#26777325)

I love this book. It contains a wide variety of topics and although some of it is elementary, there is plenty of depth to challenge and enchant your students.

Albert Einstein praised it as:

A lucid representation of the fundamental concepts and methods of the whole field of mathematics. It is an easy to understand introduction for the layman and helps give the mathematical sudent a general view of the basic principles and methods.

If you want to teach your students to love math, try this book. Courant was a leading mathematician of his day. He co-authored the formidable Methods of Mathematical Physics with David Hilbert. Courant's love of mathematics shines throughout What is Mathematics.

1089 and all that -- Acheson (0)

Anonymous Coward | more than 5 years ago | (#26777337)

"1089 and all that" by David Acheson seems like a perfect book for your needs.

1 2 3 ... infinity by George Gamow (2, Interesting)

_greg (130136) | more than 5 years ago | (#26777349)

Gamow's book covers some of the most interesting areas of mathematics without excessive simplification or condescension.

Another good book is

The "Language of Mathematics: Making the invisible visible" by Keith Devlin. This is an expansion of his earlier book for Scientific American Library.

Finally, consider mathematics which involves interactive projects with a computer. Turtle Geometry is a great starting place. Advanced students can tackle a professional book on computer graphics and will learn a massive amount of projective geometry and mathematical thinking while having a blast doing it.

_Greg

Purcell, Calculus. (0)

Anonymous Coward | more than 5 years ago | (#26777359)

Couldn't get enough math in HS, so went on over to the local University and enrolled before senior year. Quickly found to move up to the next level, and go for the challenge. Purcell was/is a good read, and the summer ('77) was great.

Freakanomics... (3, Informative)

lordsid (629982) | more than 5 years ago | (#26777369)

I suggest Freakanomics.

Although not really a pure math book I think you can see the relevance. I found it very enlightening to read and it provided a very interesting insight into odd things like Why Sumo Wrestlers Cheat and How much Crack Dealers really make an hour.

Ninnle for Dummies. (0)

Anonymous Coward | more than 5 years ago | (#26777371)

Should be required reading.

The Movie PI (1)

wiglebot (1348459) | more than 5 years ago | (#26777377)

PI is on You tube -- a classic film.

A History of PI (2, Insightful)

SmilingSalmon (1143805) | more than 5 years ago | (#26777389)

A History of PI [amazon.com] 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.

Gamma (1)

leopardi (456992) | more than 5 years ago | (#26777393)

Julian Havil's book, Gamma, is both a popular mathematics book and a mathematics book. It gives both history and results.
http://press.princeton.edu/titles/7494.html [princeton.edu]

Freakonomics (1)

CoolGuySteve (264277) | more than 5 years ago | (#26777397)

Unlike a lot of the posters here, I think at that age, it's more important to show students why math is important than the concepts used by upper year college students. When I started my Math/CS undergrad, the department pretty much dismissed everything I was taught in high school and started from first principles. Even things I taught myself at that time outside of school like computer graphics turned out to be irrelevant.

In relation to statistics, I think they're vastly under taught and under appreciated in the high school curriculum. As much as engineers and scientists like to scoff at the lax rigor that's employed sometimes, statistics are essential to the social sciences. We need good psychologists, good economists, good politicians, and insightful voters, and statistics is how we get there.

Also, every time some USian I work with spits out that asinine Mark Twain quote about statistic or says "14% of all people can tell you they're made up", I just want to hit them. It seems like rhetoric has totally destroyed data in this country's discourse.

Anyways, the most interesting book I've read when considering this aspect is Freaknomics. It shows how data analysis can be used to explain everyday phenomena in society in laymen's terms. It's pulp, but it's interesting. There might also be others with a similar bent.

Div, Grad, Curl and All That (1)

Fished (574624) | more than 5 years ago | (#26777405)

The book that made me understand multi-variable calc... well, kind of:

Div, Grad, Curl and All That

Of course, that would only be suitable for students with at least some calc.

Also, how about A Beautiful Mind (the book, not the movie.)

Serge Lang - Math Talks for Undergraduates (0)

Anonymous Coward | more than 5 years ago | (#26777413)

Serge Lang's "Math Talks for Undergraduates".

A Pathway Into Number Theory (3, Interesting)

dtmos (447842) | more than 5 years ago | (#26777423)

A Pathway Into Number Theory [amazon.com] , 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.

Continued Fractions (0)

Anonymous Coward | more than 5 years ago | (#26777435)

I've used Continued Fraction by C. D. Olds successfully with kids as young as 14.

Alge

The Code Book, by Simon Singh. (0)

Anonymous Coward | more than 5 years ago | (#26777437)

http://www.simonsingh.net/The_Code_Book.html

It's a well-written history of cryptology, with explanations of the algorithms.

Books for students interested in competitions (1)

justanothermathnerd (902876) | more than 5 years ago | (#26777447)

Many of the books suggested here are really more about the history of mathematics with a small dose of mathematical explanation added. Some books in this category are better than others, but nearly all of them provide such a shallow explanation of the underlying mathematics that students really can't learn much mathematics from them, even if they do pick up some interesting biographical and historical information. It's sad that the publishers have churned out so many of them in recent years.

Several posters have also mentioned books in the "introduction to proof based mathematics" genre. This is certainly an important topic, but many of these books are a bit too advanced for most high school students.

Another important category that I haven't seen mentioned are books on problem solving techniques for mathematics competitions. In this category, I'd strongly recommend "The Art Of Problem Solving" by Richard Rusczyk.

The Book of Numbers (1)

Chih (1284150) | more than 5 years ago | (#26777457)

http://www.amazon.com/Book-Numbers-John-H-Conway/dp/038797993X/ref=pd_bbs_2?ie=UTF8&s=books&qid=1234135581&sr=8-2 [amazon.com] 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.

George Polya and Imre Lakatos (0)

Anonymous Coward | more than 5 years ago | (#26777467)

Two of the top names in mathematical education, these guys come at math with an eye that sees that it doesn't need to be impossible, and that there are things you can learn and teach without being an expert.

FailzorBs (-1, Redundant)

Anonymous Coward | more than 5 years ago | (#26777481)

on my Penti0m Pro

In Code (0)

Anonymous Coward | more than 5 years ago | (#26777493)

In Code by Sarah Flannery is perfect for high school age kids with an aptitude for math but lack the exposure to the field of number theory, plus it's a cute autobiography of a female mathematician growing up in math.

Lady Luck (1)

dgun (1056422) | more than 5 years ago | (#26777503)

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

Lady Luck: The Theory of Probability [amazon.com]

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

How about just plain reading math? (1)

ciaohound (118419) | more than 5 years ago | (#26777507)

Encourage kids to read math and consider the subject worth studying. I sure wish someone had clued me into that in high school! How I went on to become a high school math teacher is still a bit of a mystery to me, as my high school math teachers were pretty uninspiring. They never encouraged us to read the textbook, let alone any outside texts. Perhaps that was because high school texts tend to be pretty uninspired -- I'm a fan of Key Curriculum Press texts, and I encourage math teachers to check them out, in particular Discovering Geometry by Michael Serra.

Discrete mathematics? (1)

that_itch_kid (1155313) | more than 5 years ago | (#26777513)

How about discrete maths/combinatorics? The intro material is not overly difficult, but I find it a very interesting branch (As a CompSci student). Set theory, graph theory, logic theory, advanced probability. Enumeration, generating functions...

Number theory? It's a bit more advanced, but some kids should grasp it.

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