Mathematics Reading List for High School Students

By Mark Trodden | February 8, 2009 7:58 pm

Via Slashdot, I came across the following question

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!”

There are some good suggestions in the comments, and some not so good ones. Surely our wise and mathematically sophisticated readers will be able to help. Add what you can there, and in the comments here if you like.

CATEGORIZED UNDER: Advice, Mathematics
  • Joshua

    Hey,

    I’d suggest “Introduction to Mathematical Reasoning” by Peter Ecceles. It’s a textbook that helps bridge the gap between high school and college level mathematics. It’s a great book that teaches student how to write up arguments in a logically rigorous manner; another book in the same vein would be “Tools of the Trade” by Paul Sally Jr.

    Also, I’d suggest anything by HMS Coexter. Another book, also focusing on geometry/topology, would be “Intuitive Topology” by Prasolov. It’s a book used for students at Moscow Math School 57, and the first couple of sections are pretty readable–just playing around with knots and links. I think it’s a nice book, but it might be a little “unstructured” for students.

    Also, there are two classics I’d recommend: The Moscow Puzzles (Kordemsky) and Mathematical Circles: Russian Experience (Fomin). Both books are excellent. They cover a wide array of mathematical topics disguised as brain teasers, puzzles and riddles.

  • Brian

    1. After algebra II, the high school math track moves toward calculus. Fine. You need calculus for any kind of engineering or science. But during the Fifteenth Century, mathematics moved in the direction of more advanced algebra. People figured out how to solve cubic and then quartic equations. When no one could crack general fifth degree equations, Abel and Galois investigated the roots of such equations and determined the impossibility of solution by radicals. I don’t know of any general book, but I bet a good student could trace this development by Googling.

    2. Diophantine Equations.

    3. There are some good books of hard-to-prove geometry theorems.

    4. Fermat’s Last Theorem – among other things this book makes students aware that there are unsolved problems.

  • Quarters
  • Brian

    Oh yeah. The Code Book, by Simon Singh. Maybe it’s not all strictly math, but it is a good read about codes and ciphers.

  • ts

    The list may vary depending on the goal. The age group (16-18) doesn’t really mean much without knowing their goals and at what level they are being taught. Are they the kind who would be satisfied by merely passing remedial type math courses? Or are they future engineers and scientists? Few high school students get intrigued by pure maths, though they actually get intrigued by something like physics when the right buttons get pushed. For many it’s easier to “get it” when there are intuitive contexts, just like dinosaurs, astronomy, and such are a great tool to lure the scientifically challenged into learn some science.

  • jay

    George Polya’s “How to Solve It” remains one of my all time favorites. Accessible to students with a rudimentary knowledge of geometry, it nevertheless retains its power for undergraduate and graduates students alike.

    Also, Imre Lakatos’s “Proofs and Refutations” was a “must have” according to one of my CS professors..

  • http://localseasoning.blogspot.com/ Theo

    I highly recommend The Symmetry of Things, by John H. Conway, Heidi Burgiel, and Chaim Goodman-Strauss. The first section does lots of real math (classifies compact surfaces with boundary, and therefore orbifolds with nonnegative curvature), but is completely accessible to a math-inclined reader who doesn’t know anything.

  • http://rigtriv.wordpress.com Charles Siegel

    I’m going to second the recommendation of Coxeter. His Regular Polytopes was suggested to me by my high school geometry teacher, and I got a LOT out of it. I’d also suggest Famous Problems in Geometry and how to Solve them http://www.amazon.com/Famous-Problems-Geometry-explaining-science/dp/0486242978/ref=pd_bbs_sr_2?ie=UTF8&s=books&qid=1234149208&sr=8-2

  • hypatia cade

    Flatland, How to Solve it.
    The lady drinking tea (about stats) by David Salsburg
    Passionate Minds and E=MC2 (both by David Boadanis)
    Feynman’s books on physics

  • CoffeeCupContrails

    S.L. Loney: Plane Trigonometry Part I
    S.L. Loney: Co-ordinate Geometry
    Hall & Knight: Higher Algebra

    These were classics that I found very helpful as a high-school student. Classics that are over a century old and still in publication. There was one more book for calculus that engineering students in the US use in their freshman years, but I forget which one – had a russian author.

  • http://astrohacker.com/ Ryan Dickherber

    Wikipedia.

    I know that sounds ridiculous, but I learned a lot about set theory on Wikipedia that helped immensely in the courses I later took. There are many great books, but Wikipedia is an excellent way to browse a ton of mathematics freely and easily.

  • Brian

    In my post at 8:33 PM, I mistakenly referred to the book Fermat’s Enigma, by Simon Singh, as “Fermat’s Last Theorem.”

  • james

    I have to agree with “Flatland” – it’s an interesting blend of philosophy and mathematics. Regardless of the subtext of the book, it’s a mind opening story.

  • Patrick Dennis

    “Fundamentals of Mathematics” by Moses Richardson (MacMillan, Various editions 1939-1966, now out of print. There is also a 1973 edition co-authored with – I think – his son.) I have the 1966 edition. It is a survey, yet one not only of astounding breadth, but also of great depth. Richardson maintains throughout the book the spirit of mathematical rigor. He begins with logical systems, then moves through the customary progression from counting numbers through to complex numbers, arithmetic, algebra (including group theory), functions,calculus, probability, even non-euclidian geometry and transfinites. All this in about 550 very well-written pages! It is a book that for forty years I have been able to pick up, ever confident that I would come upon an interesting passage or chapter.

  • Soto

    I always like learning the history of the mathematical concept. I find that this helps with understanding. A book that does this well is “Zero: The Biography of a Dangerous Idea” by Charles Seife. Numbers, series, and integrals are among the many mathematical topics covered in the history of zero.

  • Jonathan Moore

    Sticking to just one, I’ll recommend “Forever Undecided: A Puzzle Guide to Godel”, by Ray Smullyan. It’s fun, challenging, and introduces some serious but sexy mathematics.

    Also, all of Joshua’s suggestions sound good. Respectfully, I am going to recommend against some of the others: (i) Polya; better to try solving some problems (ii) the Singh books ; basically pop science (iii) Gardner; generally good, but the excellent Moscow Puzzles is very much better.

  • Mark

    Most useful mathematics book:
    Advanced Mathematics for Engineers and Scientists. Yes, it’s a Schaum’s Outline, it’s easily the most useful mathematics textbook I ever had and I wish I had a copy earlier in life.

    Most inspiring sciences book:
    A Short History of Nearly Everything by Bill Bryson, this book puts the sciences into perspective. It tells a story of the knowledge of the earth and the way that knowledge was attained.

  • Angus Bohanon

    Coincidences, Chaos, and All That Math Jazz. Fantastic book, very accessible and actually fun to read. Won’t actually teach anything, it’s more a way to grasp concepts like infinity, dimensions beyond the third, fractals, etc. It’s perfect for high school.

  • http://arcmathblog.blogspot.com/ Mr B

    I have a manuscript posted on my math blog that some of my students are reading to get better acquainted with basic calculus concepts. It’s written for people who have a passing acquaintance with algebra and a dash of trig. Folks are welcome to take a look.

    A Stroll through Calculus: A guide for the merely curious

  • http://rightshift.info Dileep

    “Differential and Integral Calculus” – Richard Courant
    Most of my undergraduate peers (in my country) understand calculus as a series of derivative and integral formula for standard functions. They can’t appreciate what a limit or divergence means. A good book on Calculus must build stuff from the ground up.

    That apart, “One, Two Three . . . Infinity” – by George Gamov was a fun general read.
    And as someone already mentioned, Feynman’s Lectures are a must.

  • http://whenindoubtdo.blogspot.com/ Eugene

    They could have a lot of fun reading Neal Stephenson’s Cryptonomicon.

  • senderista

    Why not linear algebra? It’s not exactly traditional high school material, but it doesn’t really have any prerequisites (beyond high school algebra and complex numbers), and it’s full of easy-to-visualize examples. Elementary group theory would also be appropriate, I would think.

    I recommend “Linear Algebra Done Right” by Sheldon Axler, although it’s clearly intended for math rather than science students.

  • http://diracseashore.wordpress.com/ moshe

    David Foster Wallace on set theory and the foundations of mathematics:

    http://www.amazon.com/Everything-More-Compact-Infinity-Discoveries/dp/0393326292

  • Pseudonym

    I think that the book that I wish I’d read as a teenager (assuming it had been written then) is “Conceptual Mathematics”.

  • Anon

    While in school, someone had gifted me Courant & Robbins ‘What is Mathematics?’ That book was a revelation.
    But, If I were to go back in time, I’d give myself Weyl’s ‘Symmetry’.

  • http://www.phys.uconn.edu/~yerubandi Kishan Yerubandi

    The Art of Problem Solving – volumes 1 and 2 – by Richard Rusczyk and Sandor Lehoczky

    it’s written explicitly for high schoolers

    http://www.artofproblemsolving.com/Books/AoPS_B_Texts_ProbSolve.php

  • Harold

    I absolutely loved Morris Kline’s book: Calculus: An Intuitive and Physical Approach

    http://www.amazon.com/Calculus-Intuitive-Physical-Approach-Second/dp/0486404536/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234162943&sr=8-1

    Morris Kline was a math professor and a critic of how math was taught.

  • jester

    What about the following books?
    J. Weeks, The Shape of Space
    T. Needham, Visual Complex Analysis

  • Bruce Rout

    Flatland is excellent but there is also a set of four books called from one to infinity that is a collection of all the major papers of mathematicians throughout history. A 15-year-old girl who hated math said it was the best book she had ever read. Ever!

  • http://tristram.squarespace.com Tristram Brelstaff

    From Here to Infinity by Ian Stewart.
    The Knot Book by Colin C Adams.

  • Fermi-Walker Public Transport

    How about “Number Theory in Science and Communication: With Applications to Cryptography,
    Physics, Digital Information, Computing and Self-Similarity” by Manfred Schroeder.
    Minimal background needed and as the title suggests, it has an applied orientation.

  • trailblazer

    I think that some basic differential geometry/vector calculus/complex analysis will be suitable-for tensor calculus I think the best book is Borisenko and Tarapov-old russian book-one of the best in the subject(a good book is also Introduction to Geometry by Coxeter). Probability theory is also a very good add in to the curriculum-for instance the book by Kolmogorov-not the classic Foundations of …, but a small, really interesting, high school level book , which goes all the way from dice to the central limit theorem-it is really interesting to learn the subject from the master’s book.

  • randomeda

    Flatland -Edwin Abbot

  • http://www.pieter-kok.staff.shef.ac.uk/ Pieter Kok

    Chaos by James Gleich.

  • David Derbes

    I’m a high school physics teacher. There has been an explosion of extremely enticing math books for the general reader (I really think Hawking’s “Brief History” was the start of this.) Many are published by Princeton, some by Johns Hopkins. The three big names are Ian Stewart, Paul J. Nahin and Eli Maor. Another reader above suggested abstract algebra; Stewart’s “Why Beauty is Truth” goes a long way toward addressing group theory for high school students. Barry Mazur and Simon Singh have also published good books for high school students and the general public. For logic, it’s very hard to beat Raymond Smullyan’s books. My favorite is his first, “What is the Name of This Book?” which gets to Gödel’s theorem via jokes and riddles.

    For classics, there are these: “Calculus Made Easy” by Silvanus P. Thompson, or the revised version with Martin Gardner, and Michael Spivak’s largely unknown “The Hitchhiker’s Guide to Calculus”. Courant and Robbins, “What is Mathematics?” is another classic, as is Hilbert and Cohn-Vossen’s “Geometry and the Imagination”. These are a little dry, as is Weyl’s “Symmetry”.

    I think the suggestion of the Feynman Lectures is not really a good one for most high school students. On the other hand, Feynman’s “Character of Physical Law”, though not strictly mathematical, is an excellent choice. Better yet in my opinion is “Feynman’s Lost Lecture” which is actually mathematics (that the inverse square law leads to elliptical orbits, done with very little more than Euclidean geometry.)

    Finally, some of the quirky books of Lillian and Hugh Lieber have recently been reprinted. These are “The Education of T. C. Mits”, “Infinity” and “The Einstein Theory of Relativity”. (Disclaimer: I LaTeXed and helped to edit the last, though I have no financial interest in the book’s success.)

  • Thomas

    Roger Penrose – “The Road to Reality”
    a breathtakingly beautiful survey of mathematics and physics

    Sheldon Axler – “Linear Algebra Done Right”
    sharp, engaging introduction to mathematical reasoning and proofs, through linear algebra

    Harold Abelson & Gerald Jay Sussman – “Structure and Interpretation of Computer Programs”
    ingenious introduction to, well, I’m not really sure… “programming” would be a vast understatement. Its arc spans from Ackermann’s function, to symbolic computation of polynomials, to an interpreter for the Scheme programming language (which the book is written in, by the way)

    Richard Feynman – the lectures, volume 1
    no introduction needed :)

  • http://blueollie.wordpress.com/ ollie

    I’d suggest NO EXTRA CREDIT at all. What happens is that students get used to this option and then show up at college and expect a “way out” of learning the assigned material. :)

    But as far as reading, Abott’s “flatland” and (forget the author) “How to Lie with Statistics” are both readable and good.

    The Shape of Space by Weeks will probably be too much for them, but has lots of cool pictures.

  • Nova Terata

    Neil Stephenson’s Anathem would be particularly relevant to students

  • Kyle G

    For an introduction to advanced mathematical ideas that should interest, rather than frighten, high schoolers, I think Burger & Starbird’s Coincidences, Chaos, and All That Math Jazz: Making Light of Weighty Ideas is a good choice. It’s heavy on the concepts but light on the complicated math. I’ve had Starbird as a professor before, and he’s extremely talented at teaching non-math majors about weighty mathematical concepts.

    I’ve seen him teach topology to run-of-the-mill liberal arts majors with success.

  • Count Iblis

    I have yet to find a suitable math book for high school students. The problem is that in math teaching, little is done to present the material so that it looks very exciting. What you need to do is to present some easy to understand spectacular stuff that will motivate the students to spend a lot of time studying math.

    E.g., you could teach modular arithmetic and then immediately show the power of this method. You can give an elementary proof of Fermat’s little theorem (a^(p-1) = 1 Mod p), Chinese Remainder theorem, Euler’s generalization of Fermat’s little theorem etc.

    Calculus: Why not explain Taylor’s expansion intuitively (just fit a polynomial by requiring that the derivatives match). Then a spectacular result would be to use it to give Euler’s intuitive nonrigorous argument that zeta(2) = pi^2/6. Also, use that to prove that the probability that two large integers have no common factor is 6/pi^2.

    If students are exposed to math in this way, then they will like it a lot more and be prepared to go through the tedious rigorous proofs.

  • http://www.soulphysics.org Bryan

    Richard Courant’s What is Mathematics?.

    This is the mathematician’s classic. It’s a serious introduction to the field. It’s also readable in highschool. and has the amazing propensity to make you want to do graduate work in math, before you even get to college. (That’s what happened to me, anyway.)

  • Mike

    I teach mathematics at a highly selective liberal arts college in the Midwest, and I have assigned as supplementary readings a number of excellent books written for a general audience that do a wonderful job of conveying the joy of mathematical discovery. I’ve seen two of them in the preceding comments, but I’ll repeat them here to add another vote for their selection.

    Chaos, by James Gleick
    The Shape of Space, by Jeffrey Weeks
    Sync, by Steven Strogatz
    Godel, Escher, and Bach, by Douglas Hofstadter

  • Beckyj

    I haven’t read a ton of math books, but I did like the History of Pi by Petr Beckman. It describes the development (or estimation) of the value of pi through the centuries.

  • Gavin Polhemus

    Four Colors Suffice: How the Map Problem Was Solved, by Robin Wilson

    This is one I haven’t seen among the many great suggestions. Four Colors Suffice is great because it connects an ancient problem to it recent solution, addresses a problem that may people wouldn’t even think of as math (map coloring), and actually does a great job of explaining the actual proof, in addition to giving the story of the people who did it.

    Early in the book readers learn how to prove that any map can be colored with six colors. This is pretty easy. Later they learn how to prove that any map can be colored with five, which is more difficult and includes all of the essential elements of the four color proof. The four color proof was eventually done with the assistance of a computer, which is an interesting twist to the story.

    It talks about many related problems as well as giving the stories of the many interesting people involved. Also, it’s not very long. It has all of the equations that are needed, but they are very simple formulas because it is a mapping problem. There are far more pictures than equations. I teach high school too, I know these will be a primary consideration for some kids. However, even with a Ph.D. in physics, I found this book fascinating.

  • robert61

    When I was that age, I read A History of Pi in one sitting after picking it off a friend’s shelf.

  • efp

    You might want to check out The Mathematical Experience by Hersh & Davis. I recall reading this around high school age. It consists of short, self-contained essays, so you could pick and choose.

    And I second the notion that there should be no such thing as extra credit. Ever.

  • http://scienceblogs.com/sunclipse/ Blake Stacey

    I read a fair number of math-y books when I was that age, but in thinking back, it’s easier to come up with glitz and glamour than it is to recall books which actually helped develop the mathematical skills I use as a science person. Knuth’s Surreal Numbers was fascinating, and I probably got some indirect benefits from encountering proof techniques (induction and such), but how often do physicists use surreal numbers? Moreover, books on more recent developments — like Singh and Aczel’s books on Fermat’s Last Theorem, or Keith Devlin’s various attempts to popularize the Riemann hypothesis — have too many gaps, too many places where the abstruse sophistication of the mathematical arguments are glossed over with a bit of narrative. It’s fun, yes, and it keeps the enthusiasm stoked, but you can’t actually solve problems in group theory by applying the life story of Evariste Galois.

    And, if you can’t actually use the mathematics, it’s not really part of your life, is it?

    Having issued all these caveats, then, here are some books I’ve enjoyed, ranked in increasing order of “I could do stuff after having read this”:

    Surreal Numbers, by Donald Knuth
    The Book of Numbers, by John Conway and Richard Guy
    QED: The Strange Theory of Light and Matter, by Richard Feynman
    Chaos, by James Gleick
    The Cartoon Guide to Statistics, by Larry Gonick and Woollcott Smith
    The Manga Guide to Statistics, by Shin Takahashi

    Also, I’m midway through Douglas Hofstadter’s I Am a Strange Loop and Marcus du Sautoy’s Symmetry: A Journey Into the Patterns of Nature (originally published as Finding Moonshine in the UK), and I bet I would have liked both of them when I was a ninth-grader.

  • John T. Scott

    I very much enjoyed John Derbyshire’s book on the Riemann Hypothesis. Blake Stacey, above, seems to downplay an apparently similar book by Keith Devlin, which I don’t know, but Derbyshire held my interest throughout, and I suspect the book might help a bright highschooler appreciate the power of mathematics. Another one would be Morris Kline’s Mathematical Thought from Ancient to Modern Times. Sure, it’s a history book, but Kline teaches a lot of mathematics along the way. I continue to find that the amount of mathematics known to the Babylonians, Greeks, Egyptians et al. quite staggering, and I hope that your high-school students would feel the same.

  • http://www.savory.de/blog.htm Eunoia

    Innumeracy.

    Beyond Numeracy.

    G.E.B

  • lcjohnson

    I read from a great text in a college course on the history of Math, but would be great for high school because it is split up into short stories – a sketch for each element explained.

    Berlinghoff and Gouvêa, Math Through The Ages. Published by Mathematical Association of America.

  • hegemonicon

    Another vote for Godel, Escher, Bach.

  • Tom S.

    For a more personal touch: “Men of Mathematics” by E. T. Bell. Although Bell doesn’t aways get the facts right, the book is a very entertaining look at the lives and work of famous mathematicians of the past.

  • JK

    I was the sort who read lots of the books mentioned above at that age (at least the ones that were published then…). The most outstanding were:

    Weeks, The Shape of Space
    Hilbert and Cohn-Vossen, Geometry and the Imagination
    Hoftstader, Godel Escher Bach

    You can give them Penrose, either The Road to Reality or, maybe better, The Emporer’s New Mind. If they’re like me, they will only understand a fraction, but the writing will set them on the right path.

    I don’t think anyone has mentioned

    Stewart and Tall, The Foundations of Mathematics

    Stewart’s popularisations have been mentioned, but this is his text book specially designed to make the transition between school mathematics and the rigour of university study. I leant how to appreciate ‘real’ mathematics by self-study of that book and would highly recommend it.

  • Scott E

    I am a high school math teacher as well and there are some interesting books on the list. Here a few not mentioned: Math Devil (an odd little book), Conned Again, Watson (Sherlock Holmes and Dr. Watson take on cases with mathematical solutions) and Journey Through Genious (a very well written -with lots of real math – history book).

  • casey jane

    THE UNIVERSE IN A TEACUP by K.C. Cole

  • Igor

    Another vote for “Godel, Escher, Bach” – this can be, no hyperbole, a life changing book for the mathematically inclined.

    I’d also suggest “When Least is Best” by Paul Nahin, which is a fantastic book about mathematical optimization. It’s a very accessible but also very rigorous introduction to some of the most interesting and important applied mathematics out there.

  • Nick

    One I wish I had read in high school is “How to prove it” by Velleman (sp?). Very useful for someone who intents to take maths in university but has not taken a class on how to write proofs.

  • Jimbo

    Frankly, most of these professionals have lost sight of the reality of HS, when you have hot blood flowing thru your veins, its hard to concentrate on anything, much less the king of abstraction, mathematics.
    Nonetheless, I’d wager 95% of your students are NOT going to major in mathematics. The bright ones will mostly go into engineering in college or engineering technology. Hence they need practical math skills, & not a cursory acquaintance with stuff they will rarely use.
    All the above esoteric math refs might be OK for gifted students, but the average Jane/Joe Schmoe will ultimately use math as a tool to facilitate earning a living, not an end to itself.
    Do not short change them. Equip them with practical algebra, trig, and elementary calculus so they leave with a diploma that empowers them, not retards or intimidates them.
    I have taught college physics in 4 states, at 4-yr universities, community colleges, & tech schools. Without questtion, math is the primary hangup.
    Conquer that, and all walls fall.

  • http://www.phys.uconn.edu/~yerubandi Kishan Yerubandi

    I’d second the nomination of Linear Algebra Done Right, 2nd edition, by Sheldon Axler.

    It explains proof ideas for the mathematical novice alongside the actual material.

    And it’s probably the next course the student will study in college anyway — except he’ll probably be forced to learn it the “wrong way” — through matrices, row reduction, systems of equations, etc.

  • Rien

    I agree with Simon Singh’s Fermat book and James Gleick’s Chaos. Both make math seem exciting and you will probably get at least a little bit of math with you from reading them. Courant’s book is very nice, but I doubt that anyone but the already quite interested will want to read it. If you are a math geek it will be perfect, though.

  • a grad student

    Calculus by Spivak

    get it done right

  • http://scienceblogs.com/sunclipse/ Blake Stacey

    John T. Scott,

    I was thinking of the chapter “Hard Problems About Complex Numbers” in Keith Devlin’s Mathematics: The New Golden Age. It’s a good book, and if you want an introduction to Mersenne primes or Cantorian higher orders of infinity, it’s probably great for your purposes. (I believe there’s since been a revised edition which updates the chapter on Fermat’s Last Theorem and such.) My only issue is that I’d have to rank it only middling on the “I could actually do math and science after reading this” scale.

    (Yes, there are all sorts of factors influencing how much practical competence one gains from a book or a class, including one’s self-discipline, but holding all else constant, there’s still a gradation of books in this regard, I believe.)

    It might be heresy to mention a TV show in a thread about books, but I have to plug the Caltech production Project Mathematics!, which is an all-around nifty treatment of geometry and trigonometry.

  • http://www.users.bigpond.com/pmurray Paul Murray

    I’d wager 95% of your students are NOT going to major in mathematics. The bright ones will mostly go into engineering in college or engineering technology. Hence they need practical math skills, & not a cursory acquaintance with stuff they will rarely use.

    Well in that case, boolean algebra and math that relates to computing is a must. I don’t know what gets taught in computing classes these days: probably “how to use microsoft word to prepare a job application”. If you are interested in exposing the students to joys of abstract math, then LISP and Prolog might be the go. There are free interpreters out there.

    Or: why not teach boolean algebra by having them assemble logic circuits on breadboards? The real rudiments of computing: AND, OR and NOT gates, and making the LEDs flash. The chips are reasonably cheap, I think. You can go onto groups and modular arithmetic and whatnot from there.

  • Barbara

    One, Two, Three . . . Infinity, by Gamow. Old, but as good as ever. It fascinated me in high school and decades later I startled a nephew by giving him a copy.

  • Methalos

    The Archimedes Codex
    By Reviel Netz, William Noel

  • Adan.Mike.Selene

    Who Is Fourier?: A Mathematical Adventure (Paperback)
    by Transnational College of LEX (Author)

    This book was written by members of a Japanese educational commune who devote themselves to innovative learning styles. They learn languages by immersing themselves in language — they say expose themselves to 11 languages simultaneously, and it works!

    They were interested in Fourier series, as they related it to understanding sound, which was related to their interest in language, and they set out to understand it in creatively.

    This book has cartoonish pictures, but don’t get put off. It sets out to derive the concepts it needs from the ground up. I love math books that lead me along a chain of logic that makes everything fall into place. This book starts by explaining graphs and trig functions, and goes through Fourier transforms.

    I love Courant’s calculus, because it explains everything (although Courant keeps making comments that his work is simplified and not truly rigorous!), and I love this book for the same quality, or though it is the opposite end of the spectrum from Courant, which is about as formal as one will be exposed to these days.

    This book is unique. Check it out, I cannot do it justice.

  • estraven

    anon at 11:36 said:
    “While in school, someone had gifted me Courant & Robbins ‘What is Mathematics?’ That book was a revelation.”
    Same happened to me. That book made me a mathematician.
    I also strongly second Feynman’s Physical Law. It takes away the guilt from wanting to be a mathematician :-) .

    To Jimbo, who claimed hormones make study difficult: I disagree. Mathematics is the only thing gripping enough to take your thoughts away from sex. Or so it seemd to me as a teenager.

  • coolstar

    Jimbo has it EXACTLY right. It’s hard to fathom how truly BAD most of these suggestions
    are (no, I don’t have any better ones other than seconding Mark’s very valuable and pragmatic suggestion). The fraction of HS students interested in pure mathematics is probably one or two orders of magnitude LOWER than those interested in a career in physics or astronomy. I’m very happy when I get students out of HS who are not 1) totally innumerate
    2) totally turned off by math and science (almost always by bad secondary teaching). Oh, want to change these numbers? Go volunteer at a high school or your local community college (and not just to teach the “smart” kids!). (my own teaching career mirrors Jimbo’s pretty well, by the way).

  • RDeen

    “Concise Introduction to Pure Mathematics”, Martin Liebeck – great book that I read a couple of months ago, and I’m in high school too.

  • http://home.comcast.net/~djmpark/ David Park

    John Stillwell,
    _____Numbers and Geometry
    ____ Mathematics And Its History

  • Jimbo

    CoolStar, Comrade in Arms…Merci beau coup, Monsieur !

    Paul Murray…Ask them to count in base 2 to 64.
    Me thinks you will get a sobering lesson in math reality,
    and soon forget about logic circuits…Better to focus only on soldering !
    The state of American HS math is worse than the American economy…
    And that’s sayin A-Lot.

    Estraven: Thinking math might keep your mind off hot hormonal rushes is like hoping abstinence pledges will also !

  • ts

    I have to agree with coolstar & Jimbo. I can see that most people here don’t deal with real high school students, outside of perhaps those occasional “feel good” science outreach programs. Looking at the book suggestions here, I can clearly see overenthusiastic math teachers who cannot communicate at all with students. So sad.

  • Scott

    If these students are already into calculus at all, I’d suggest “Calculus Made Easy” by Silvanus Thompson, or (even better) Martin Gardner’s annotated reprint of the same book.

    As for the concept of infinity, Rudy Rucker’s “Infinity and the Mind” is a good one – very entertaining!

    Lastly, if any of your students enjoyed Edwin Abbott’s “Flatland”, I’d also suggest “Flatterland” by Ian Stewart, and “Spaceland – A Novel of the Fourth Dimension” by Rudy Rucker.

  • Anil

    Maybe “What is mathematics” – by courant, robbins and stewart

    And “Does god play dice” by stewart.

  • Ginger Yellow

    “Another vote for “Godel, Escher, Bach” – this can be, no hyperbole, a life changing book for the mathematically inclined. ”

    Absolutely. I only have A-level maths, but it’s still the best non-fiction book I’ve read by some margin (sorry, Phil).

  • http://bhr-ett.blogspot.com Bhrett Parrish

    A History of Pi is a great book about the evolution of the value of pi and how we have came to know it today. It show cases fairly simple mathematics and can be found here:: http://www.amazon.com/History-Pi-Petr-Beckmann/dp/0312381859/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1237567463&sr=8-1

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Cosmic Variance

Random samplings from a universe of ideas.

About Mark Trodden

Mark Trodden holds the Fay R. and Eugene L. Langberg Endowed Chair in Physics and is co-director of the Center for Particle Cosmology at the University of Pennsylvania. He is a theoretical physicist working on particle physics and gravity— in particular on the roles they play in the evolution and structure of the universe. When asked for a short phrase to describe his research area, he says he is a particle cosmologist.

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