Slashdot Mirror
Mathematics Reading List For High School Students?
Troy writes "I'm a high school math teacher who is trying to assemble an extra-credit reading list. I want to give my students (ages 16-18) the opportunity/motivation to learn about stimulating mathematical ideas that fall outside of the curriculum I'm bound to teach. I already do this somewhat with special lessons given throughout the year, but I would like my students to explore a particular concept in depth. I am looking for books that are well-written, engaging, and accessible to someone who doesn't have a lot of college-level mathematical training. I already have a handful of books on my list, but I want my students to be able to choose from a variety of topics. Many thanks for all suggestions!"
Sorry, my list is lacking some depth.
How to Lie with Statistics, Darren Huff, 1954
I'm wondering what constitutes "Stimulating Math?"
I wrote this:
http://people.pwf.cam.ac.uk/jlnw3/maths/books/prime/
It was meant as an introduction to the idea of proof. Perhaps you might like it.
- Jax
My 10th grade math teacher had me read it. It was very brain-stretching.
http://math.cowpi.com/flatland/
You should definitely expose your students to the following Math books:
http://www.amazon.com/Math-SAT-800-Toughest-Problems/dp/1439200068/ref=sr_1_1?ie=UTF8&s=books&qid=1234132532&sr=1-1
http://www.amazon.com/Math-Workbook-New-SAT-Barrons/dp/0764123653/ref=sr_1_2?ie=UTF8&s=books&qid=1234132532&sr=1-2
http://www.amazon.com/Petersons-Math-Exercises-Academic-Preparation/dp/0768908078/ref=sr_1_7?ie=UTF8&s=books&qid=1234132532&sr=1-7
Principia Mathematica. It's all there ;^)
Great minds think alike; fools seldom differ.
http://en.wikipedia.org/wiki/Flatland
It's normally taught as an upper-division college class but the only real prerequisite is 2nd-year high school algebra and a mind that can think abstractly.
Students will find it different enough from trig and calculus to be fresh and knowing they can do "college math" can be a real ego-boost.
By the way, if you know any elementary or middle school teachers, many of the concepts in abstract algebra can be taught to those age groups as well. Being able to do "adult math" can be a real point of pride and inspiration at those ages.
First grade isn't too early. Anyone who can add or subtract time already has the basics for abstract algebra addition and subtraction.
Knowledge is how to play a game, intelligence is how to win, wisdom is knowing what game to play.
If this book doesn't make them think that math is cool, nothing will.
*** Ponder
How to Think like a Mathematician:
http://www.amazon.com/How-Think-Like-Mathematician-Undergraduate/dp/0521895464
Online here (for how much longer?):
http://www.maths.leeds.ac.uk/~khouston/httlam.html
I bought this in the discount bin for $1 somewhere, I think it's (Playthinks) really good to develop logic and just try a little bit of every mathematical discipline:
http://www.amazon.com/Big-Book-Brain-Games-Mathematics/dp/0761134662
This isn't pure math, but lisp, but since Lisp is inspired by lambda calculus, perhaps it'll inspire more programming (shrugs):
http://www.cs.cmu.edu/~dst/LispBook/index.html
Accessible, and it'll teach them French into the bargain. One downside: you'll have to read it too --- and pretend you understood something.
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 ;)
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.
Even the dullest high school student has a memory that makes us adults seem slow. There is exactly one way to motivate teenagers: tell them they are not "ready", although telling them they are "not allowed" has a similar effect. With that in mind I recommend you give one or two of them a copy of All the Mathematics You Missed But Need to Know for Graduate School, and suggest they pass it onto someone else if they find it "too hard". It's a great book that gives a quick skim over all the different fields of mathematics that a graduate student in mathematics is expected to know. A typical college student will read this book, shake their head and decide that maybe graduate school isn't for them. A typical high school student, even one not interested in math, will read this book and decide that mathematics is awesome and maybe they should pay attention in class, because if they can't grasp differential linear equations then they're never going to understand Lebesgue integration and infinite Fourier series.
How we know is more important than what we know.
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
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.
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
The fact that no one understands you doesn't mean you're an artist.
"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).
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.
http://www.amazon.com/Godel-Escher-Bach-Eternal-Golden/dp/0465026567/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234132982&sr=8-1 Godel, Escher, Bach: An Eternal Golden Braid Very interesting book and should get students of that age excited about math and science IF they are predisposed to that sort of thing.
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. It's very dry and concise and I love it!
Swedish plasma phys. PhD student; MSc EE; knows maths, programming, electronics; finance interest; seeks opportunities
Prime Obsession: A well-written history of the still-unproven Riemann Hypothesis. Maybe one of your students will solve it over summer break!
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.
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.
#fuckbeta #iamslashdot #dicemustdie
I really enjoyed this book when I was at that stage... http://books.google.co.uk/books?id=wUdtVHBr-OQC 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...
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.
NinnleninnleninnleninnleninnleninnleninnleninnleBATMAN!
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.
Stephan
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.
https://app.box.com/WitthoftResume Code: https://github.com/cellocgw
Also AMS has some translated (or written anew) books from Russian mathematical tradition. The books that I read personally (and liked a lot):
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).
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.
Computers allow humans to make mistakes at the fastest speeds known, with the possible exception of tequila and handguns
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.
"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.
You zap the moderators with a wand of humor! The moderators resist!
- 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)
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.
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.
This side up.
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.
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?
Not strictly mathematics, but Richard Feynman's "autobiography" might be a good one for inspiring your kids to show what they can do with their math knowledge.
Taking guns away from the 99% gives the 1% 100% of the power.
Courant and Robbins, "What is mathematics?"
My first program:
Hell Segmentation fault
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: 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!
If what you're looking for is just readable books that bring forth a new perspective on maths, then I personally recommend Nassim Nicholas Taleb's Black Swan: The Impact of the Highly Improbable http://www.amazon.com/Black-Swan-Impact-Highly-Improbable/dp/1400063515/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234133817&sr=8-1 This book is a highly engaging, readable introduction to thinking about the limitations of statistical probabilities. Also, if anyone has not read Zero: The Biography of a Dangerous Idea by Charles Seife http://www.amazon.com/Zero-Biography-Dangerous-Charles-Seife/dp/0140296476/ref=sr_1_1?ie=UTF8&s=books&qid=1234134095&sr=1-1 you are depriving yourself of the fascinating history of a shockingly revolutionary idea.
fart=funny
"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
http://en.wikipedia.org/wiki/A_course_of_pure_mathematics:
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.
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.
I really like Fermat's Enigma by Simon Singh. Relatively easy read and I found it inspiring.
Chaos: Making a New Science 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.
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"?
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'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.
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.
The Great International Math On Keys Book
Okay, I'm joking. But what's the modern version of this book?
I got this with my TI-30 in 1976 and went through the whole thing because it was cool to have things to do with the calculator. I was in no way a gifted or dedicated student. I was just a bright and bored kid, and didn't get great grades.
What have we got like this today that uses the existing software on our computers? (Do our computer now all have good software like that great graphing calculator that was shipped with PPC Macs? I've no idea what's on XP/Vista these days.)
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!
Master of Space and Time by Rudy Rucker. It has some math in it, and it's funny to boot.
Learn something new.
Albert Einstein praised it as:
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.
We don't see the world as it is, we see it as we are.
-- Anais Nin
"1089 and all that" by David Acheson seems like a perfect book for your needs.
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
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.
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.
IMAGE VERIFICATION IS EVIL!
Should be required reading.
PI is on You tube -- a classic film.
A History of PI by Petr Beckmann is a great book for that age group. It has lots of historical information about PI and its calculation by various historical figures and cultures. The writing style is engaging and even moving. Another plus for that age group - it's less than 200 pages long.
I second a previous poster's suggestion of Simon Singh's The Code Book.
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
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.
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.)
"He who would learn astronomy, and other recondite arts, let him go elsewhere. " -- John Calvin, commenting on Genesis 1
Serge Lang's "Math Talks for Undergraduates".
Read the book, but dont' let them see the movie, which sucks.
A Pathway Into Number Theory, by R. P. Burn.
It's the most unique math book I've ever read. There is no prose in the book per se; rather, the book is a series of small tasks and questions (usually starting by identifying patterns in tables of numbers) that, as the title suggests, gently lead the reader into Number Theory. All the major topics of a first course (the fundamental theorem of arithmetic, quadratic residues and forms, etc.) are there; the beauty of the book is that each task is such a small step from the previous one that the reader is led painlessly to a mastery of each concept. (Just don't skip steps!) This feature makes is suitable for advanced high school students looking for "stimulating mathematical ideas."
It's a wonderful book, on a wonderful subject. I have often wished for books written in this format on other mathematics subjects.
I've used Continued Fraction by C. D. Olds successfully with kids as young as 14.
Alge
http://www.simonsingh.net/The_Code_Book.html
It's a well-written history of cryptology, with explanations of the algorithms.
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.
http://www.amazon.com/Book-Numbers-John-H-Conway/dp/038797993X/ref=pd_bbs_2?ie=UTF8&s=books&qid=1234135581&sr=8-2 I'm suprised this hasn't been mentioned yet. It is a full-color introduction to many areas of mathematics, perfect for the age group specified and not deep enough to get dull.
For best results, avoid doing stupid things.
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.
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.
"Relativity: The Special and General Theory" Einstein (about 200 pp.) http://search.barnesandnoble.com/used/product.asp?EAN=2692771891703&Itm=9
There is nothing to FEAR but NOTHING itself; and I fear there is a whole lot of nothing going on. --scorpivs
It's an old book, but I like it.
Lady Luck: The Theory of Probability
The author does a great job making the subject easy to understand for non-math people like myself.
FAQs are evil.
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.
Oh, yeah, it's not easy to pad these out to 120 characters.
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.
It seems likes kids only do what you tell them not to do, so this advice may seem wise. However, this is a form of confirmation bias; adults notice when kids don't listen because mainly because they usually do.
If you tell someone a student some skill is difficult, they will believe you. You have set them up to expect failure. This expectation is easy to meet, and most students will give up early.
If you tell a student something is easy, they are likely to believe you. Believing a subject is easy, they are more likely to follow through to mastery because they have been set up to expect success.
Reverse psychology is a trick. Tricking students is a way to alienate them; it may work on the few, but the many will respond better to affirmative attitudes.
As a high school maths teacher myself, I can recommend:
"The Mathematical Experience" by Philip J. Davis and Reuben Hersch
Google Books says "The authors of this book believe that it should be possible for these professional mathematicians to explain to non-professionals what they do, what they say they are doing, and why the world should support them at it."
It was partly the avid reading of this book from my own high school's library that inspired me to do a maths degree, and many of the other books I may have read at the time are likely due to references from this one. (Admittedly I think it is also as a result of this book that I know e to around 50 places, but don't share that with your pupils.)
I second earlier references to Godel, Escher, Bach... except that I gave up around two thirds of the way through, and haven't bothered to pick it up since. I've been meaning to for around 12 years, but lethargy keeps me from it.
In contrast, The Mathematical Experience is easy to read throughout, and can be flicked through with ease, revealing lots of intriguing results.
"1089 And All That" by David Acheson
My own college tutor wrote this book; it's not very difficult, and not very long, but it is very difficult to put down, jam-packed with anecdotes and extremely well-written. David is an expert at apt illustrations, and writes for the mathematician and the lay person both. Though this book doesn't really address the 'in-depth' requirement, it certainly creams some of the most fascinating theorems and conjectures from the last several thousand years and presents them in an entertaining and informative fashion.
http://home.jesus.ox.ac.uk/~dacheson/1089.html
Apologies if either has already been cited. (Did I mention lethargy?)
I read this for a math competition back in high school. The main advantage (apart from its relative cheapness) is that its a very easy introduction to more abstract math (non-euclidean geometry).
http://www.amazon.com/Taxicab-Geometry-Adventure-Non-Euclidean/dp/0486252027/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234136240&sr=8-1
First, let me add my recommendation for GEB. It's an amazing book.
Here are some others that I think are good:
Note to ACs: I usually delete AC replies without reading them. If you want to talk to me, log in.
Umm.. the material likely *is* too hard for them. You're not tricking them at all.. you're just giving them the opportunity to accept the challenge.
How we know is more important than what we know.
http://www.amazon.com/Godels-Proof-Ernest-Nagel/dp/0814758169
Tough but not impenetrable. You can easily interest a smart kid by
introducing it as, "here's the guy who demonstrated that everything
I'm teaching you is wrong."
If they are also interested in programming you could let them try Knuth's Fundamental Algorithms or its sequel. Just have them ignore the hard problems.
The books I'm aware of are all either too easy or too hard for that audience, although I'm sure there are some in the middle. There are some easy but interesting topics that you might look for chapters on:
1) Prove Fermat's little theorem, and show how it can be show a number is composite (but not prime).
2) Derive the closed form for the nth Fibonacci number. This should include a little more on geometric series than I think is usually taught in high school.
3) How to solve linear Diophantine equations, and why it works.
a,e,i,o,u and sometimes w and y (at be if of up cwm by)
Perhaps my favorite is The Nothing that Is: A Natural History of Zero (http://www.librarything.com/work/147631). I read it in high school in a week at church camp. It mixes history with math and isn't a hard ready but it's no Harry Potter either. You could also go with Biographies since they are less number heavy and often more interesting. I may also suggest The Drunkard's Walk: How Randomness Rules Our Lives (http://www.librarything.com/work/4850753). I have not read it yet (darn you public library) but the reviews suggest it's a good read.
The book How to Solve It by G. Polya is a
classic must read. While it was given to me by one of math
professors in undergraduate school it should not be over the heads of
advanced high school students.
Find topics in numerical analysis, so they know that after all that misery, plus calculus if they carry it into college, most of these problems can be solved with a scientific calculator and some reasonable assumptions.
My brother engineers can probably testify to how infuriating it was to spend those first couple miserable years mastering multi dimensional calculus, only to be shown how there were really damn good approximation algorithms in place for most of these problems. In my case it was twice infuriating, my professor was a drunk, he would verbally abuse anyone that dared walking into his classroom with a calculator that wasn't made by TI, HP or Casio. He did not care about price, so for example you couldn't dazzle him with a $200 HP (his ideal calculator was "the best TI you can find on sale for $20-$25)), but God protect you if you walked in with one of those calculators designed to balance checkbooks, because you wouldn't even get more than 10 precision digits.
Joking aside, try to see if you can find something that has a real world application. I was bored out of my mind because I was being taught calculus concepts in one year and they would not be needed in my major courses for at least two semesters.
Pedro
----
The Insomniac Coder
Not strictly a maths book, but it's probably the first book I read that got me to REALLY think about things. Onvolves a lot of interesting ideas from other fields (Physics, Computing, Psychology, Physiology and many more) as well.
http://www.amazon.co.uk/Emperors-New-Mind-Concerning-Computers/dp/0192861980/ref=sr_1_1?ie=UTF8&s=books&qid=1234137007&sr=1-1
I'm working through Fenman's lectures on Physics and Knuth's Art of computer programing. I find they're very hard, but that I learn a lot...
I'm actually trying to make my own math text book. You can find it here
You might have a devil of a time finding it (out of print) but Ascent to Orbit by Arthur C Clarke was very inspiring to me when I was in high school.
It's a collections of the scientific papers by ACC, explaining the mathematics of space flight (orbital velocities, geosynchronous orbit, space elevators, etc). Many of the papers were published before space flight was a reality, so it is historically interesting as well as mathematically approachable.
Life is like a web application. Sometime you need cookies just to get by.
How about "A Mathematician's Apology"
http://www.math.ualberta.ca/~mss/misc/A%20Mathematician%27s%20Apology.pdf
Other than that, how about introducing the students to R or Octave and having them solve just a few problems from their existing textbooks using those (or other) tools? Isn't "literate programming" close to "reading and writing" about math (at the high school level)?
Four Colors Suffice: How the Map Problem Was Solved by Robin Wilson
Easily readable at that age, and gets into the idea that math isn't all discovered already, and that seemingly simple things can be complex, and the problem it focuses on is something that can be easily described and understood with no math knowledge.
Others have mentioned "Flatland" by Edwin Abbott, which I also strongly recommend.
"Geometry, Relativity, and the Fourth Dimension" by Rudolf Rucker should be accessible to a high school student. It revisits Flatland, so that's probably a good book to read first.
"The Drunkard's Walk: How Randomness Rules Our Lives" by Leonard Mlodinow is an easy and entertaining read, and talks about how human intuition is often wrong when making probability estimates.
"Knotted Doughnuts" by Martin Gardner is a compilation of brain-teasers from Scientific American. Gardner has published several of these collections, but this is my favorite.
Ants, Bikes and Clocks is a wonderful introduction to applied mathematics via problem solving. Most of the material is calculus free. Students can learn an immense amount about how to approach problems and why they should study math at University. I use this as a supplement in a 200 level modeling class as well as the main text for a section I teach to math high school teachers. I also leave the book with high school classes I visit. It is very well written, approachable and filled with great problems and some hints on how to solve them. Enjoy!
The best introductory book for "real mathematics" (theorem-proof style) that I've seen is Calculus by Michael Spivak. It is a large book, lucidly explained in great detail. It teaches insight and intuition, and has a very "chatty" style, as one of my professors once put it. Stay away from his other books, though. They are very advanced and leave much to the reader to prove.
That being said, I think you ask the wrong question. Don't just give a reading list. As a teacher, you should be doing the reading and teaching things to your students. Most people will not take well to a giant (or even small) list of math books to just read.
Basic group theory is very nice and has many accessible results. The book I used is by Fraleigh and is called "A First Course in Abstract Algebra." The first half of the book is about groups.
If you are interested in computer applications, "Simulation" by Stephen Ross is quite good. It is reasonably basic and certainly requires little calculus. Most of the assignments involve programs that can be written in 20 lines of python--probably more for C/C++/Java. It shows a nice example of how computers can be used for nontrivial mathematical applications (i.e., more than just adding numbers and computing derivatives/integrals that are "hard").
Other topics of interest are Probability (the dice kind, not the measure theory kind), Combinatorics, and basic number theory. I always thought Linear Algebra was pretty cool--as long as you don't focus too much on the boring mechanical junk like Gaussian elimination and stick more to the abstract notions of vector spaces, bases, eigenvalues, and spectral theory. If you are feeling ambitious and your students have seen integral calculus, you can introduce the Fourier transform and show the equivalent of a basis in function (Hilbert) space. An excellent reference is Korner's Fourier Analysis. It has many examples of applications: lots of physics stuff, how you can use fourier analysis to estimate the age of the earth, and how it has applications all over mathematics.
My real recommendation is to take some books out of your local library and skim them yourself for topics to present in class. Pick interesting stuff that will engage students with the limitless possibilities of mathematics.
It's too simple for many adults with pre-conceived notions to understand and I'm pretty sure they won't encounter it in any standard curriculum: The Laws Of Form.
It's pretty cool, too. Spencer-Brown derives Boolean algebra from it and it uses fewer axioms.
Only his tendency toward a dazed stupor prevented him from screaming aloud.
If you don't have Journey through Genius by William Dunham, you should. It is a GREAT book that shows beautiful mathematics while telling interesting historical stories.
I read it in high school, and it helped me develop a love for mathematics.
It remains on my bookshelf today.
Dance like no ones looking and love like it's never going to hurt.
Any of Smullyan's books, particularly "What Is The Name Of This Book?", "The Lady Or The Tiger", "Alice In Puzzleland". Lots of fun, and not what high school students would consider math. "Disguised" as mere logic puzzles, they are great for learning formal logic and ultimately introduces Godel's Incompleteness Theorems. Much easier and more fun than Godel, Escher, Bach (which is truly a wonderfully fantastic book, if you have the students who are ready for it).
Feynman's Lectures on Physics are a little far afield from pure math but they do make math interesting by connecting it to the real world.
@de_machina
Cambridge UP, softcover (thus probably around $40-$50).
Terrific book, written on the level of highschool before college. Easy to read but with some real meaty stuff. (e.g. the best description of cracking the Enigma encryption that I have seen).
From the wiki page:
His beliefs on pure mathematics seem to be summarized in the following excerpt from the book,
"Pure mathematics, on the other hand, seems to me a rock on which all idealism founders: 317 is a prime, not because we think so, or because our minds are shaped in one way rather than another, but because it is, because mathematical reality is built that way."
Another theme is that mathematics is a "young man's game", so anyone with a talent for mathematics should develop and use that talent while they are young, before their ability to create original mathematics starts to decline in middle age. This view reflects Hardy's increasing depression at the wane of his own mathematical powers. For Hardy, real mathematics was essentially a creative activity, rather than an explanatory or expository one.
I see many recommendations. Let me suggest the tool to use and explore the math. APL. Powerful, easy to use, and very successfully taught to high school students.
The book, 'APL; An Interactive Approach' is a good starting point. There are many others. There are free for personal use versions of APL available.
Any questions, drop by:
comp.lang.apl
Heisenberg may have been here.
Why bother teaching little kids about mathematics, let them grow up first and then decide what they are going to do.
Conceptual Mathematics by Lawvere and Schanuel is the one maths book that I wish I'd been exposed to in high school.
sub f{($f)=@_;print"$f(q{$f});";}f(q{sub f{($f)=@_;print"$f(q{$f});";}f});
Wish I could mod parent up. Playing these sorts of tricks with people only works as long as they trust you, and it's essentially manipulation. I suppose you might argue that the ends justify the means, but in my experience the teachers that have had the most influence on me were ones that were genuinely interested in a topic, and honest enough to share their experiences without trying to influence me they way they thought best.
In the end, we're all just different from each other, and rather than trying to trick them into learning the way you want them to, I suspect it's better to show them your excitement about a topic and then just learn to accept it if they go another direction.
by James Newman. I read this, cover to cover, in high school (early 60's) and it had a positive impact on my education and career. I became a math major in college, before discovering computers.
Jan Gullberg's book Mathematics: From the Birth of Numbers is a great read! It covers, well, a hell of a lot: number theory, trig, fractals, matrices, calc, probability, diff eq, combinatorics, symbolic logic, etc. It includes anecdotes and historical notes, and does a very good job of explaining many different things.
Amazon has a couple of reviews.
i'd hit it so hard, if you pulled me out you'd be the king of britain [bash.org]
I'd recommend Super Crunchers by Ian Ayers. It greatly exceeds the scope of just mathematics, but in my opinion (I'm an engineer) anything that's useful has to apply math to something real. It's a pretty straightforward read and offers room for plenty of discussion. It's only the start of a conversation.
Expose them to Clifford Algebra before they start screwing up their brains with linear algrebra.
Only his tendency toward a dazed stupor prevented him from screaming aloud.
Men of Mathematics by E. T. Bell. Published in 1937, it is biographies of most major mathematicians "from Zeno to Poincare'". Instead of focusing on their mathematical discoveries, this book focused on what their lives were like and why they were even interested in math and how math influenced the rest of their lives.
Number 9, The Search for the Sigma Code by Cecil Balmond. This book is half fiction, half not, and looks into the weird ways that the number 9 keeps cropping up in number theory. Fun read, with lots of accessible arithmetic for high school.
Flatland by Edwin Abbot. You've probably heard of this one. But then...
Spaceland by Rudy Rucker... in which a man from our world explores a higher dimensional world in which our 3D space is but one slice of theirs, and the strange interactions he has with the beings there.
although it's not a book.
Only his tendency toward a dazed stupor prevented him from screaming aloud.
A Tour of the Calculus, by Berlinski, is a (mostly) non-technical exposition of the tools of the calculus and its various founders. I also recommend The Nothing That Is: A Natural History of Zero, and Mathematics is Not a Spectator Sport for a more workbook-like excursion into precalculus topics.
I can heartily recommend "The Enjoyment of Mathematics" to gifted students:
http://www.amazon.com/Enjoyment-Mathematics-Selections-Mathematical-Recreations/dp/0486262421
I loved this book when I was in high school. All it requires is algebra and plane geometry. It covers many interesting topics and is extremely readable. A few parts are out of date (4-color theorem and Fermat's Last Theorem are now solved), but the subtracts nothing from its value.
"Winning Ways for your Mathematical Plays" (Berlekamp, Conway and Guy) is both fun and serious mathematics. It is probably too much for most students, but even doing some of the stuff in the first few chapters is likely to open their imagination to what mathematicians can do. The first volume (of the first, two volume, edition) covers basic combinatorial game theory with the second volume covering such things as the "dots and boxes" game, the Rubik's cube and the Game of Life.
It has been reprinted in 4 volumes recently.
The first edition (at least) is filled with puns, odd drawing, and lots of other weirdness.
Hawking's God Created the Integers really shows off the beautify of some of the most seminal developments in mathematics over the millennia. Working through the proof for Gödel's Incompleteness Theorem was rewarding.
While in high school I read "Mazes for the Mind: Computers and the Unexpected" by Clifford A. Pickover. It ties math to a wide variety of topics, and should be entertaining and mostly accessible even if a little of the math goes over their heads. And it's full of pretty pictures!
Apparently he has written a number of books since then. I haven't read any of them, so I wouldn't know which to recommend.
(As mentioned a number of times already, I'd recommend Godel, Escher, Bach as well.)
Any book by Martin Gardner would be excellent, but maybe they are too wide ranging ?
The History of Pi by Beckman is pretty good.
e: The Story of a Number, by Eli Maor is very nice.
How to Read and Do Proofs by Daniel Solow is excellent.
A book that I liked when I was younger was To Infinity and Beyond by Eli Maor. It's a sort of advanced layman's look at infinity and the closely-related zero. It includes mathematical topics your students probably haven't seen before (and won't, unless they become math majors), in enough depth to be interesting but not overwhelming (not enough to really be useful mathematically, either just to make them interested and perhaps help them be more comfortable if/when they get to higher math). It also has a lot of history. It seems to be mostly available on google book search (a bunch of random pages missing, but mostly there) so you can check it out without leaving your computer!
There are 11 types of people in the world: those who can count in binary, and those who can't.
I'm not a math guy (I struggled with Calculus) so I don't understand a lot of the concepts. But I do like the hands-on approaches and casual reading. Save for the last chapter on Quantum Cryptography, Simon Sing's CodeBook is very well written and a pleasure to read.
Above all, I think its more important, in math, science or computer science, for people to get an idea as to how scientists think and might inspire them to study science or get a better appreciation for deriving new ideas by using what they already know.
I really found Dancing Naked in the Mind Field by Karry Mullis to be very inspirational and informational at the same time. I think no matter what area of study the kids will go on into, reading this book will help the students get an introduction and a better appreciation of the thinking differently 'idea' that university (and Apple) seem to encourage!
If you tell a student something is easy, they are likely to believe you. Believing a subject is easy, they are more likely to follow through to mastery because they have been set up to expect success.
An equally likely outcome is that the person will find the task difficult, and will then blame you (or worse, blame themselves) for not being able to accomplish the "easy" task, when, in fact, the task is much more difficult than they were told. Such a reaction is equally likely to result in the person giving up and ending up with a bad attitude towards mathematics.
Frankly, I'd appreciate the grandparent's strategy more, though not for the same reasons that he or she elucidated. If someone is honest with me and tells me that a task is hard, I'm more likely to take that into consideration and proceed more carefully when it comes time to do the task. At the same time, I'm probably going to appreciate the person being honest with me and telling me that the task was going to be difficult, as opposed to trying to trick me by telling me that the task was going to be easy.
We all know what to do, but we don't know how to get re-elected once we have done it
For high school students, I think the Gelfand books listed here are some of the best books available to really understand the subjects they set to teach (note that I have absolutely no experience with the program at Rutgers, that's one of the few pages I can find referencing this great collection though). I certainly find them better teaching guides than the typical mammoth text books for geometry and trig. I would seriously consider basing classes off of these books if I were allowed to.
"Men of Mathematics," by Eric Temple Bell, presents short biographies of over two dozen well-known, mainstream mathematicians down through history. According to Wikipedia, "To keep the interest of readers, the book typically focuses on unusual or dramatic aspects of its subjects' lives. While Men of Mathematics has inspired many young people, including a young John Forbes Nash Jr., to become mathematicians, it is not known for the accuracy of its historical scholarship." But that wouldn't keep interested students from reading it!
great theorems of mathematics. This book by William Dunham is a look at the lives and work of 10 or so great mathematicians throughout history.
The sections on Euclid and primes, Euler and infinite sums, and Cantor and the continuum are particularly good. Though it covers great and truly ingenious (even inspiring) mathematics, it should all be accessible to a high school student. Plus, the biographies are interesting in their own right and help to break up the parts where you actually have to concentrate.
I read this in my first year or so as an undergraduate, and I recommend it to anyone who wants to gain a feel for what mathematics is all about.
Hell, there are no rules here-- we're trying to accomplish something. --Thomas A. Edison
1. A Long Way From Euclid
Constance Reid
A survey of math from the ancient Greeks on.
Very accessible.
I spent months reading it in 6th grade.
2. Innumeracy: Mathematical Illiteracy and Its Consequences
John Allen Paulos
Lots of cool stuff on probability, estimation, and application of math to current events.
Innumeracy: Mathematical Illiteracy and Its Consequences
by John Allen Paulos
The Code Book by Simon Singh
Journey through Genius: The Great Theorems of Mathematics by William Dunham
You should DEFINITELY advise them the *great* book from Rozsa Péter (the hungarian mathematician who discovered - despite all common thoughts - the Ackermann function). "Playing with Infinity" is not well know but it was a big success in my case and for all the persons I advised it to. It is an incredibly pedagogic and fun book, definitely recommended for high school but to my mind advisable also to all maths enthusiasts, researchers included : if you don't learn anything on the technical side, you'll surely learn a lot about pedagogy and have a great time! It starts from the very beginning (how many sheeps) on a very practical point of view to elaborate concepts in various fields of mathematics, even the most complicated and abstract ones like topology and number theory. Please read it and spread the word, I assure you that is is worth it. http://www.amazon.com/Playing-Infinity-Mathematical-Explorations-Excursions/dp/0486232654/ref=sr_1_5?ie=UTF8&s=books&qid=1234139922&sr=8-5 PS: look also at her photo on Wikipedia. How could such a beautiful and sweet old lady not write beautiful things?
I love this book. Even if they don't understand any of it, it has enough problems and examples to make you think. http://www.amazon.com/Concrete-Mathematics-Foundation-Computer-Science/dp/0201558025/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234140945&sr=8-1
If you are interested in mathematics textbooks, you should consider the following...
There is a series of mathematics competitions in Illinois with an interesting category of competition. One kid from every high school team, rather than competing in the standard written exams, is designated as the "oralist" for the team. To compete, they receive a chapter of a college mathematics text to prepare in advance. Upon arriving at the competition, they get 10 or 15 minutes to prepare their answer to a set of 3 question, then present their answers to the judges. All the textbooks used for the competition are picked because they are readable to a high school student with no training in calculus and the topics vary from matrices to "taxicab" geometry to graph theory. They post the textbooks titles and chapters that have been used over the years at http://nsml.org/past/topics/. Not all of them are fantastic textbooks, but they are all reasonably accessible and some are really great.
I read this book a while back back, which is a biography of Paul ErdÃs, the most prolific mathematician of our time. The book itself does not contain that much math, but it gives a really fascinating look into the life of a mathematician.
These are my suggestions (I'm an apprentice mathematician U3):
Introducing Time, by Craig Callender
Introducing Mathematics, by Ziauddin Sardar
Introducing Chaos, by Ziauddin Sardar
Introducing Logic, by Dan Cryan
Introducing Fractal Geometry, by Nigel Lesmoir-Gordon
These books are great at explaining advanced concepts at their level. You should supplement these reading with some simple math problems that are pertinent to the subjects in the books. Anyways, I suggest you give these a try they're really fun to read.
Morris Kline's books are very accessible. Dantzig's book Number is another good one. And Gelfand's Algebra, also.
--Gabe
Dots and Lines (Paperback)
by Richard J. Trudeau (Author)
A very accessible introduction to Graph Theory.
But you will have to get it used on Amazon etc...
A great general book is "The Art of Mathematics" by Jerry King. I remember reading it as a first year undergraduate and thinking about switching majors to math. Accessible, yet it describes much of the "real math" that isn't apparent to high schoolers (or even most college graduates).
It's hard to beat some of the old classics, like Kasner and Newman's "Mathematics and the Imagination" or the four-volume "The World of Mathematics." Assuming, of course, that those are still in print.
Teen Angel - a Ghost Story
How To Solve It, Polya
Martin Gardner's written a lot of amazing stuff. You could pick up a copy of The Colossal Book of Short Puzzles and Problems and then work through puzzles from it in groups.
Also, this could be interesting, if a bit different: Unit Polyhedron Origami by Tomoko Fuse. Basically, unit origami is about building large shapes by making many small modules and combining them. It can be quite fascinating from a geometry perspective: given a square piece of paper and no other tools, the book will show you how to construct an equilateral triangle, or a regular pentagon, or a regular hexagon. In fact, not only construct them, but with pockets and tabs so you can join them together.
Repton.
They say that only an experienced wizard can do the tengu shuffle.
Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics
This is a great book - expertly researched and the author brings a lot of history and drama to the quest for the solution to Reinman's Hypthosis. The author has also presents very accessible explanations to many complex mathematical concepts.
Combinatorics is a field of math that's easily accessible, but runs very deep and is fundamental to many other areas of mathematics. All you need to start is a good sense of logic.
I TAed an introductory combinatorics university course, and this textbook was decent:
http://www.amazon.com/Applied-Combinatorics-Alan-Tucker/dp/0471735078/ref=pd_bbs_sr_4?ie=UTF8&s=books&qid=1234141953&sr=8-4
You can explore things like counting (how many ways to pack n balls into m boxes, considering the balls distinct or not, and similarly for the boxes), probability, graph theory, design theory, and all kinds of fun stuff. Sudoku, for example, is combinatorially very interesting, and a good way to motivate anyone into the field.
As a owner of The Princeton companion to mathematics, I have found a great deal of things to learn from it. This book covers math in almost every single field and easy to understand(after fully understand the introduction chapter and remember all those definitions.) There are history of each field, list of mathematicians and their biography, introduction to each field, what each field is concerned with. it can go deep into some topic, like what is on in the current research. but a high school student with enough mathematical skills have no problem reading it. I suggest this book if you have math students who are really interest to mathematics and does not fear lots of definition to remember.
Archimedes Revenge by Paul Hoffman
http://www.amazon.com/Archimedes-Revenge-Joys-Perils-Mathematics/dp/0393327752/
A collection of short essay on various topics. I particularly enjoyed the section on game theory and how it applies to voting systems.
Is the title of the book that did it for me. It's been out of print for too long, but available as a pdf download at http://www.archive.org/details/onetwothreeinfin000923mbp
Proofs and Refutations by Imre Lakatos takes the reader through an apparently simple and obvious mathematical theorem, and by the end has you deeply aware of how subtle mathematics really is.
Anything by Martin Gartner, specifically http://www.amazon.com/Aha-Insight-Gotcha-Spectrum/dp/0883855518
Challenging Mathematical Problems with Elementary Solutions
Vol 1
has some great stuff most high school students haven't seen.
Also,
Discrete Algorithmic Mathematics by Maurer and Ralston (http://www.amazon.com/Discrete-Algorithmic-Mathematics-Stephen-Maurer/dp/1568811667) is excellent, if priced as a college textbook.
If you want them doing drawings in addition to learning some math, may I recommend the three books: Cundy and Rollett's Mathematical Models, Yates's A Handbook on Curves and their Properties, and Lockwood's A Book of Curves. All three talk about how to draw the curves studied by the ancients and the math behind.
Anyone thinking about mechanical or electrical engineering should read this before any signals and systems class. It's very readable with emphasis on practical approaches to engineering problem and how mathematics can solve them.
Best of all, you can get it here free
A book I enjoyed as the text for Freshman seminar when I was in college was Niven and Zuckerman (now Niven, Zuckerman, and Montgomery) "An Introduction to the Theory of Numbers". This book might be a stretch for a good high school senior math student. But, it has a lot of interesting material and doesn't require any college level math as a pre-requisite. Besides, who doesn't like Number Theory.
It's a historical account of encryption and the math behind it along with the war stories that go along with it. It's really entertaining and educational from a historical and math perspective. Not so long that it becomes a chore either.
"Don't teach a man to fish, feed yourself. He's a grown man. Fishing's not that hard." - Ron Swanson
The Number Devil, by Hans Magnus Enzensberger
It's fun and doesn't has fun language that won't bore your students.
A Life of the Genious Ramanujan by Robert Kanagel
an ill wind that blows no good
Aigner-Ziegler: Proofs from the book
Atiyah-Macdonald: Commutative algebra
Munkres: Topology
For those who know calculus: Rudin: Real and complex analysis
Some might think these are too advanced for high school students. Give them these books and see if they really are. It's a pity no one tried it with me. Might have made high school tolerable.
You might also tell them about the library at the local university, if applicable. It tends to be a lot more informative than the public/high school library when it comes to the hard sciences.
- Math PhD
Here are two that I haven't heard mentioned. 1-2-3 Infinity by George Gamow (ISBN-10: 0486256642). The physics is a little dated, but the math is elegant, engaging, and accessible. I read it when I was a child and it was one of the reasons I went into physics. It is easily enjoyed by any motivated high school student. Another good one is The New Turing Omnibus: Sixty-Six Excursions in Computer Science" by A. K. Dewdney (ISBN-10: 0805071660). This is a little more advanced, but most of it is well within the abilities of a good high school student. The nice thing is that it is written as a series of vignettes of applying math to compute science. If one is a little too advanced, just skip it and go on to the next.
Is a brilliant and easy to follow introduction to interesting problems in Theoretical Computer Science. It doesn't require any background knowledge and you can read it in a single sitting.
The Tropic of Calculus
This book is very accessible and has some very interesting things and relationships about numbers. http://www.amazon.com/Mathematical-Mysteries-Beauty-Magic-Numbers/dp/0738202592/ref=sr_1_1?ie=UTF8&s=books&qid=1234144461&sr=1-1
The maths books by Ian Stewart are rather good for creating interest without getting too technical. "Does God Play Dice?" is a good example
I initially found this book when I was researching for a philosophy paper. I really find the combination of mathematics and philosophy to be exciting, mind expanding, etc etc. It provides a bit of math history beginning with pythagorous and his bafflement over the 2^(1/2) if I recall correctly. The GEB is an excellent book but as everyone is saying, you probably won't be inspiring any students that aren't already on the geeky side to begin with. Most students would probably flip through it and say "WTF?!" http://www.amazon.com/Mystery-Aleph-Mathematics-Kabbalah-Infinity/dp/0743422996/ref=sr_1_6?ie=UTF8&s=books&qid=1234143978&sr=1-6
Sing also wrote a great book on Fermat's Last Theorem that was a very interesting read.
It (obviously) told the story of Fermat and his theorem as well as the mathematicians and their strategies that have tried to prove it through history.
There should be a moderation category "Dumbest Comment EVER"
Cryptonomicon by Niel Stephenson.
I've used the example of mapping the US vs Europe by placing lights on the tops of people's heads and mapping when they go up and down (off of sidewalks) to explain statistical analysis to MBAs way more successfully than anything else.
0100001001100101011010010110111001100111 0100100001110101011011010110000101101110
Crypto is a tremendously engaging, true story which centers around math without going into too much depth. The math is described using very easy to follow analogies. It has the best aspects of Simon Singh's 'The Code Book' without going into the level of detail which would turn off most of your students. I could see it being a very motivational book for a young person.
Mathematics Made Difficult by Carl Linderholm. I read this book in high school and when I didn't get something, I took the time to look it. I just went to see if I could find a copy and for some reason their priced at over $100. Pretty good for a book written in 1972.
http://www.amazon.com/Mathematics-made-difficult-Carl-Linderholm/dp/0529045524/ref=sr_1_1?ie=UTF8&s=books&qid=1234144871&sr=8-1
I learned a lot about math reading martin gardiner's math columns and books.
Some drink at the fountain of knowledge. Others just gargle.
Paul Hoffman's biography of Paul Erdos is an inspiring portrait of someone for whom math was an everyday joy.
by Charles Seife
The Constants of Nature -- Barrow Prisoners Dilemma -- Poundstone The man who loved only numbers -- Hoffman Unknown Quantity: A Real and Imaginary History of Algebra -- Derbyshire Excursions in Number theory -- Ogilvy
This is perfect (not only) for high school students: "Solving mathematical problems: a personal perspective", Terry Tao You can even read it online: http://www.math.ucla.edu/~tao/preprints/problem.ps
Anything by John Allen Paulos makes excellent side reading. His "A mathematician reads the newspaper" books are presented very simply for non-genius people as a way of presenting mathematics to ordinary people.
Teaching students to deeply understand the applicability of their math knowledge should be just as helpful as the raw data.
- Michael T. Babcock (Yes, I blog)
If you want to show your students that math can be fun, Surely You're Joking Mr. Feynman! is an excellent selection. Your students will learn that one can derive great pleasure from understanding how math works.
.. Cryptonomicon.
There is a war going on for your mind.
I would recommend, as both accessible and highly enjoyable, "Journey Through Genius" by William Dunham. Also recommended is the fine historical account of the development of mathematics, "Mathematics and the Physical World" by Morris Kline. A good book for teaching students about problem solving is the very excellent "How to Solve It" by G. Polya.
I'm sure there are many more fine books for math students but these are my picks.
One additional comment: you should have the students read in groups and report what the read to the class ;)
Any good poker book can provide a basic foundation in statistics, probability, and combinatorics that will will be very helpful to them in many ways. Teaching them how the lottery works, or how scratch cards actually work and why they are such a ripoff, might also be helpful. Then apply their newfound knowledge of statistics in combination with their classes on birth control.
My brother-in-law is a mathematician and has spent some time compiling a http://kasmana.people.cofc.edu/MATHFICT/list of mathematical fiction, including novels, short stories, and other mediums. Some of these might be interesting to students to see math applied in new situations.
_Feynnman's Lost Lecture_ by Goodstein. As good as it gets for helping to understand Math, Math History and the World. Bad Karma and all, Charlie Former Math Teacher
Complex numbers are important in so many aspects of math and physics, and despite the name they are not so complex. This book has a lot to teach even those who think they know complex numbers well, since most of us never learn much about the geometry of these numbers. And for those new to the subject, this is an endlessly stimulating introduction. available here at amazon: http://www.amazon.com/Visual-Complex-Analysis-Tristan-Needham/dp/0198534469 also see the author's page about the book here: http://www.usfca.edu/vca/ Also, I'll throw in the Feynman Lectures on Computation, since it is a nice introduction to the physics of computing; plus it's hard to go wrong with anything by Feynman.
I highly recommend:
Zero the biography of a dangerous idea by Charles Seife ( I recall it was very interesting).
and
Surely You're Joking, Mr. Feynman! (Adventures of a Curious Character) Not all that mathematical, but great stories that occasionally involve math, if I recall correctly.
Surely you're Joking is a book about a physicist, but many of its pages involve how Mr. Feynman goes about thinking about Math. It is a fascinating read, and nothing too difficult for high schoolers.
I enthusiastically recommend the book How to prove it by Daniel J. Velleman. It is a very smooth introduction into proofs and rigorous mathematics. Please do check it out, your students will thank you.
This book covers quite a bit of game theory, including zero-sum, non-zero-sum, and voting theory, and is accessible to anyone with a good grasp of high school algebra. It's full of interesting examples and exercises for anyone who wants to use it for self-study. I wish I'd had this to read my senior year of high school, after I'd exhausted my school's other math.
More of a mechanical, physics based look at the world. Focuses a lot on biological comparison to things we've made. But there are some good chapters that focus on math in the real world without actually going into the math. (e.g., conical progressions in seashells)
Paperback's only $12 at Amazon: http://www.amazon.com/Cats-Paws-Catapults-Mechanical-Worlds/dp/0393319903
I can't remember the name of it, but I had a Diff. Eq. prof in college who had us read a 'history of Diff eq' sort of book as extra credit. lots of newton, galileo, celestial body tracking, etc. again, no hard math, but a lot of the who behind it all. Interesting read, especially if any of your HS math students are also rather good on the English lit. side.
Group theory in the bedroom (sounds dirty, but isn't) it has a lot of interesting essays involving mathematics and some historical stuff. For something with a little more meat, godel's proof - it's short, and sweeet.
I'm a mathematics professor. I first became interested in math after reading Excursions in Number Theory during my junior year of college. It is a wonderful introduction to the power of proof, and requires no more background than simple arithmetic. I second someone's earlier suggestion of "e" by Eli Maor. That is truly an outstanding book. Unfortunately what's good for a high school student may not be good for the rest of us. I found Zero: The History of a Dangerous Idea to be completely vapid, but it is full of intrigue and controversy, and is not difficult. A genuinely good book is John Derbyshire's history of algebra, though it is challenging in places. It is worth remembering that Ramanujan (according to myth) was strongly motivated by the book "A synopsis of elementary results in pure and applied mathematics", written by George S. Carr. It shows that a book of facts (such as Excursions) may be as good or better than a popular or historical book. I will go out on a limb and make a strange suggestion: Tractatus Logico Philosophicus, by Wittgenstein is completely fascinating, and no less comprehensible to a high school student than to the rest of us. It may create an appreciation for the mysterious and profound aspects of mathematics that could be powerfully motivational.
I second the suggestion Godel Echer Bach by Hofstadter. Also his columns in Scientific American as well as Martin Gardner's - available in book form. Plus, Feynman's QED - though there is almost no math in it, but nothing would so much motivate high school students to learn! On the heavier side: The road to reality, by Penrose, touches all the math as well as all the Physics.
I'd suggest the Manga Guide to Statistics.
We're almost to this article's end-of-life as it gets pushed to the bottom. I easily have 40 recommendations here, and I can't thank you all enough. Now my only problem is I have to read all of these books!
Many thanks!
-T
http://www.asimovonline.com/oldsite/asimov_catalogue.html has a list
I'm a consultant - I convert gibberish into cash-flow.
Mathematics can be fun - Yakov Perelman
Russian author, who explains concepts including permutations, randomness and other interesting concepts with excellent examples
I still remember huge swaths of this book 20 years after reading it!
It made me love mathematics (not that I hated it)
The OP "trick" panders to the student's hubris and gives him a challenge he'll find hard to refuse. By contrast, your "trick" panders to the student's trust and sets him up for a loss of confidence if he fails. Apples and oranges.
I'm a HS Math teacher myself and I once read a book called 'A History Of Zero'. It was pretty fascinating. It didn't deal with a lot of higher math, but had some really interesting stuff about the number which is zero. Check it out here: http://www.amazon.com/Nothing-that-Natural-History-Zero/dp/0195142373
by Tobias Dantzig
Get them doing and preparing for math contests and olympiads. There is a very strong culture worldwide around math contests. And IMHO, problem solving is a much better way to get a grip on what math is like than passive reading could be.
The toad can't burp - and for some reason can't fart either, so it swells up and eventually explodes. --Anonymous Coward
... and Michael Shermer.
Great read on how to quickly do complex arithmetic in your head.
by Kalish & Montague
The book is very readable and a classic, to boot!
It's not exactly a new book, so some of the unsolved problems listed in the book may now be solved. In any case, it's one of the few I know that help a younger student go into more depth in an area where there's still active research going on. It's a difficult subject where many of the theorems can be proved without resorting to higher mathematics.
I'd imagine that there are probably similar texts for some areas of number theory and game theory, but nothing springs to mind. Non-Euclidean geometry may also be an option if the students have already taken geometry, and there were some text books that I found at least partially accessible in high school.
The Mathematical Tourist is even more out-of-date by this time. Since it's really a survey of many areas, it doesn't really meet your need, but you may find it useful yourself for looking into other areas that may be accessible to your students.
Finally, contact your local mathematics and math education departments. The math education folks may have some good suggestions. Many mathematics departments also do some sort of outreach to high school students, so there may also be some faculty there who could offer ideas.
I would have to agree. GEB was the first book I though of as an answer to your question. I only read it a couple of years ago, but I wish I would have been introduced to it in high school. Not that I would have read it then, I only started reading for pleasure about ten years ago. I still have a file of about five programs I wrote while reading the book to solve problems or try to answer questions. fun fun.
My other answer would Chaos by james Glick. The only book I've ever more than once. It really inspired me to be a physicist. I missed and landed on Engineer but as soon as I'm done with this management gig I'm going to complete my Masters in Applied Physics.
Many readers have suggested survey or popular books, as well as books that involve math in a literary setting (like Flatland). Those are certainly all great ideas to consider, especially to give the students a taste of things. If you've got some REALLY thirsty students, or notice that a few of them get really interested, try letting them loose on some of the classic mathematical texts.
Examples include Euclid's Elements, selections from Plato (Meno, Theaetatus), selections from Apollonius' Conics, some of the works of Archimedes (Measurement of the Circle, Quadrature of the Parabola, the Sand Reckoner, Method, Stomachion, Cattle problem, etc.) If trigonometry is included in your desired student level, you might also throw in the beginning of Ptolemy's Almagest. Good translations in English exist of much of these works; many available from Dover. For Apollonius or Euclid I'd suggest the Green Lion editions; they're much more student friendly.
The content of these texts ranges from more elementary (books I-IX of the Elements) to more challenging (the other books of Euclid, Archimedes, Apollonius, etc.).
There are also good selections from more modern mathematicians, such as Pascal and Descartes. Pascal has a nice one about seeing the conics as projections of a circle, which is really interesting and can be easily demonstrated with props. Descartes has some good reads too; notably, you can use this as an opportunity to connect mathematics to other academic disciplines like logic, philosophy, etc.
-Colin
colin dash mckinney at uiowa dot edu
Asimov on Numbers. Such a fun and insightful read.
I agree, history and sociology of hard science, Mathematics, ideas and philosophy are __very__ important, as is understanding of intuitional, inductive and deductive reasoning in __everything__, NOT ONLY Mathematics. That is one of the reasons why Professional Teachers teach Math and Science so poorly. You have to like it and want to understand it yourself to teach it properly. I was fortunate to have two excellent teachers, an Oxford 2nd Wrangler and one of Fred Hoyles postdocs, and most of what they taught me was how to develop the skill to guess well, ie intuition.This leads to the debate as to whether we invent or discover Mathematics, and how far the answer extends to other sciences. E T Bell's book is good, and so is the History of Mathematics (3 volume opus, for the school library) the Mathematical Dictionary is good as is Wiki. Hard Math is usually of quite good quality.
... not say things three times ;-).
... that's a bit unkind, especially these days. If they can, and are bright, you will find you only have to spark the fire. Then they will read/think/learn faster than you can imagine, and come ask you difficult questions! This can happen at __really__ young ages, 15-25 is the top of the game.
...] and Applied [Cosmology, Quantum Theory, Relativity ... ] and are different cats!
... but especially true in Mathematics/Science.
The trick is to interest and stretch your students without loosing them, which like all good teaching, requires sensitivity, ruthlessness, and good judgement. Another thing is the Maturity and Ability to Think Abstractly of each individual student. Mathematical maturity can begin by in 1/2 grade and be complete by 6 grade, though it normally happend 3-4 years later; once it does normal school lessons become useless and boring, you get it and it becomes intuitive, you read ahead, for yourself, and need teaches to answer hard questions,
If they cannot think, and visualize abstractly, and do not enjoy introspective intellectual challenges they will never develop a working math/science intuition and (I nearly joke) should do Chemistry or Biology
G.H. Hardy, of Trinity College, Cambridge wrote A Mathematician's Apology, his essay from 1940 on the aesthetics of mathematics. The apology is often considered one of the best insights into the mind of a working mathematician written for the layman. He discovered and encouraged Srinivasa Ramanujan, a young brilliant Tamil student who later his collaborator.
The major problem with modern education is that it has the wrong goal and is not sufficiently differentiated. Why do I say this, well for me Mathematics and Hard Science, Cosmology, Physics, Physical Chemistry always came easily; I never went to Maths class after 11 and taught the Mathematics Scholarship class from 13-16, when I graduated. At the same time I was absolutely struggling in Modern languages. Now I live in Switzerland, and speak 5-6, in the worst case, and normally here, all at once! We say 'merci vielmal' in German (Schweizerdeutsch).
One thing you need to be aware of is that Mathematics(-ians) come in two favors Pure [logic, consistance
The key is interest, inform, challenge and convince the kid that "Yes you can understand", but sadly I feel that only works for teaches who also understand.
Finally, I must add that, if you teach, and are not yourself interested and good at the subject matter, dont waste your time. This is true for Languages, Economics
Let the Force, and the Source(FOSS) and your imagination, and commitment be with you, YES THEY CAN!, our students are our shared future.
by Ivan Niven.
Unlike most other books listed here, this one is actually a math book that contains real math, and it is definitely one of the best high school level books you'll find.
Not necessarily what you're looking for, but a good book involving math nonetheless: Uncle Petros and Goldbach's Conjecture
That would be availability bias, not confirmation bias.
Art of the Infinite
Excellent read. Introduces concepts such as cardinality that require almost no "higher maths" but are very abstract.
It was the most popular math book for over a millennium. Might be worth a look.
I'd highly recommend Tesla: Man out of Time, by Margaret Cheney. It is very approachable and engaging, and will give them the following:
a) appreciation of an underappreciated scientific genius
b) understanding and awe of the power of resonance
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
1) The Drunkard's Walk by Leonard Mlodinow is and likely will ever be one of my favorites. It's a very revealing look into popular misconceptions of statistics and probability. And it's extremely accessible (which I didn't necessarily feel was a good thing for me, as I might have liked more depth).
2) Foundations of Geometry by Gerard A. Venema is a particularly excellent textbook that teaches Euclidian Algebra as an introduction to the mathematical proof and axiomatic logic. It practically doesn't need an instructor to go with it at all. Of course, as a textbook, it's also quite pricey, so I doubt most of your students would want to lay down the cash for it.
3) Flatland is of course a classic.
4) And as long as we're moving more into the abstract and possibly less informative, I might suggest Foundation (Isaac Asimov). Psychohistory is undoubtedly something of a pipe dream, but it's still a fantastic look at the types of applications mathematics might have.
5) And finally, if any of the students in question have taken or are in the process of taking AP Calculus, I might recommend Prime Obsession by John Derbyshire. I read it while I was in AP Calculus BC, and I managed to get a good grasp of the subject. Any foundation short of some calculus might not be enough though (the book isn't nearly as accessible as Derbyshire makes it out to be).
This book shows how math concepts were 'discovered' as they were needed. Negative numbers help us solve all subtraction problems. Complex numbers solve all polynomial root problems. etc. It makes he student realize the math concepts are there for a reason and we still have plenty of room to add to them. Each chapter is a self-contained interesting story.
Specifically this is a subset of the confirmation bias, called the Pygmalion effect.
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>=)
open source modern art: laser taggi
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.
One criticism of Men of Mathematics was that Bell had tended to romanticize the biographies of his subjects. :)
But as long as we're going with math history, may I recommend Cajoli's A History of Mathematical Notations? :)
You might try to see if you can find some computer graphics primers to see if they can manipulate shapes on a computer screen.
Or mess with Mathematica.
Chances are, if they're going for extra credit they're doing it to please you. If they were genuinely interested in math there are much better and more interesting resources on the internet for them to research. Give them something interesting and hands-on to do that doesn't involve being buried in a book 24/7 or they'll never develop social skills and grow up to be social retards.
You don't need to throw books at them (they get plenty of that in college). You don't need to prepare them for college. If they're smart and driven, they'll do just fine. Most people make it through college high and with a perpetual hangover. Those that don't are just lazy or distracted or broke.
Professor Stewart's Cabinet of Mathematical Curiosities - Ian Stewart
Meta Math! The Quest for Omega - Gregory Chaitin
A Tour of the Calculus - David Berlinski
Excursions in Number Theory - Ogilvy and Anderson
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!
I read a collected works of Lewis Carroll and found his logic problems lots of fun. Wikikpedia refers to "Symbolic Logic" and I recall that was one of the sections in that volume.
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
One I enjoyed quite a bit was "The Nature and Growth of Modern Mathematics" by Edna Kramer.
www.amazon.com/Nature-Growth-Modern-Mathematics/dp/0691023727
From the Greeks to the early 20th Century and very readable. It's available as an oversized paperback and Amazon shows used copies from $3.50.
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
I highly recommend books from www.artofproblemsolving.com.
Also, Relativity Visualized by Epstein is a wonderful book.
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.
"Zero, The Biography of a Dangerous Idea" by Charles Seife. This book was engaging and does not require any special math skills. It traces the origins of the number zero through history and made me appreciate not having learn how to perform arithmetic on Roman Numerals. A History of Mathematics by Carl B. Boyer was recommended by a colleague. "Godel, Escher, Bach; An Eternal Golden Graid" by Douglas Hofstadler is excellent but takes a year to finish if you're intent on grokking any of it. The concept of incomplete formal systems still messes with my head. Skip "e, the story of a number" It was not engaging enough to finish.
Check these out.
I remember reading "The Education of T.C.Mits" as a teenager.
(T.C.Mits is sort of an acronym for "The Common Man-in-the-Street").
Lillian Lieber's other books are likely also wonderful.
http://pauldrybooks.com/mm5/merchant.mvc?Screen=PROD&Store_Code=PDB&Product_Code=190&Category_Code=
The Einstein Theory of Relativity
Infinity
Mits, Wits & Logic.
Winning Ways is a four volume set covering combinatorial game theory. It's pretty light hearted and covers a lot of games including some old stand bys such as Nim and Dots and Boxes.
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)
Infinity and the Mind by Rudy Rucker is great if you can find a copy. http://www.amazon.com/Infinity-Mind-Rudy-Rucker/dp/0691001723
Mr. Rucker also has a collection of fictional short stories all related to math - I think it's called 'Mathnauts'
These may stimulate some interest in math.
A Beautiful Mind.
Primer.
Pi
They might not be your typical 'reading' list, but watching them may help to get them interested in math.
I love this book. Calculus explained for normal people. It goes a little fast in the beginning - but it's a refreshing, down to earth book that explains the what and why of calculus.
It's by Silvanus Thompson.
I said no... but I missed and it came out yes.
This title is associated with the CBS TV show, and can be bought just about anywhere in a math section at bigger stores, or online.
I read this book recently, and although I have an Associate's degree in math, I thought it was easy to understand, and engaging.
Being a fan of the TV show helps, but it isn't really necessary to understand what the book teaches. For example, I was able to learn what a neural network does based on a chapter of reading. Of course I have no big reason to implement one yet, but I could try since I can program matrices... so it did whet my appetite for learning.
Joe
tom stoppard's arcadia is great. It doesn't teach much, but it's quite beautiful .
Ian Stewart's Flatterland. It's like flat land but goes in to non-euclidean geometry.
Obviously flatland. Such a great social commentary.
The Man Who Knew Infinity: A Life of the Genius Ramanujan by Robert Kanigel.
Any biography about Leonardo da Vinci, Gauss, and/or Euler.
And they should all read nate silver's blog, to learn how to become a statistician loved by all smart women: http://www.fivethirtyeight.com/
For and While loops etc have a sort of maths in them - I remember my son started programming at the age of 7 and it was a tremendous help with logical thought even though when he started algebra at school the teacher (not being familiar with programming) couldn't understand why he insisted a=a+1 could be valid! Now has a Physics Phd
As a freshmen in college I enjoyed Innumeracy, and Beyond Numeracy. They are both very easy reads, but at least introduce some very deep ideas. In general, I would recommend almost anything by John Allen Paulos. Another one that provides a more detailed but still very accessible introduction is Chapter 0, but that one is written like a text book.
Here's some laypersons math books that are a lot of fun while showing what can be done with math.
The Golden Ratio - Mario Livio
Chaos - James Gleick
Fermat's Enigma - Simon Singh
The Code Book - Simon Singh
Code - Charles Petzold
What do you care what people think? - Richard P. Feynman
Reverse psychology is a trick.
I'll prove to you what's a trick! No wait, that what you want me to do...
Sometimes, life itself is sarcasm...
Mathematics and the Imagination (Kasner & Newman) opened the door for me. It has chapters on non-Euclidean geometry, calculating pi, and the googol. Light on equations unless essential, it's kind of a "frontiers of math" thing.
Chaos, by James Gleick.
Designed for about a fourth-grade understanding of mathematics, but with enough depth to code your own Mandelbrot generator or Lorenz attractor, should you desire. One of the most popular math books of all time, and with good reason.
Keith Devlin is well known for his lay person (tourist) guides to mathematics. His works are highly approachable and enjoyable to read. A particular favourite of mine is "Mathematics: The Science of Patterns."
Two of my imaginary friends reproduced once
You mean James Gleick. Oddly enough, I have a proof copy of his book, which I found in my desk when I was a physics graduate student at the University of Chicago in the 90s; I have no idea how it got there (Gleick was never at Chicago as far as I know-- someone there must have proofread for him, or maybe it was a former student). I've kept it all these years but I've never gotten around to reading it; with your endorsement I'll have to get around to it.
(Anschauliche Geometrie) by Hilbert and Cohn-Vossen. Fantastic book and I think that much of it could be understood by high school students.
Tells you how much I've looked at the thing; I just found a letter in the pages saying it was an advance copy for Leo Kadanoff. That makes sense.
Easy read book about peoples perceptions of uncertainty with some history of statistics. Not deep but thought provoking nevertheless.
http://www.amazon.com/Drunkards-Walk-Randomness-Rules-Lives/dp/0375424040/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234157557&sr=8-1
I strongly recommend "Feynman's Lost Lecture", a reconstruction of a lecture that Richard Feynman once gave that was a proof of Newton's equations as applied to planetary motion. All of Kepler's laws are derived during the course of the lecture. When Feynman prepared for this lecture, he set himself the challenge of doing it all without using advanced calculus, and restricting himself to "high school" mathematics. It's brilliant and totally do-able for (bright) high school students.
...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
Although one could argue that GEB would be too much for an 18 year old. Even the first 3 chapters of GEB would be awesome reading for anyone even remotely interested in the subject.
Singh is good for HS students outlook on math.
I would suggest a book covering set theory and foundations of math. That was the course that really turned me on to mathematics, mostly because the problems were accessible but taught you fantastic logic and problem solving skills. The book I used was simply called Foundations of Higher Mathematics, by Fendel/Resek but I am sure there are many superb books on set theory you could find.
On a side note, I just picked up The Honors Class; Hilbert's Problems and Their Solvers and have found it extremely interesting. It would probably be horribly boring for a high school student who won't be familiar with any of the problems, but for any grad students like me who didn't pick up much math history in undergrad I highly recommend it.
"Six Degrees" -Duncan Watts
"The Collapse of Chaos" -Cohen and Stewart
Quite accessible to any reader. They were also good enough to inspire me in my grad school years.
I recommend you get your school library to carry The World of Mathematics edited by Newman, it was first published in the '50's.
I discovered this 4 volume anthology at my local Los Angeles Public library branch in the 70's and I wound up hunting for years for a used copy of the anthology. In the mid-80's it was reprinted in paperback.
It is a 4 volume anthology containing selected essays and articles about every important field of mathematics.
Really enjoyable selections in this book include a terrific introduction to double entry accounting, life insurance, the Seven Bridges of Konigsberg, early papers on Turing machines, completeness theorems, codes and codebreaking and information theory.
Two more inexpensive and interesting books are:
Great Ideas in Operational Research, published by Dover.
and
Formal Knot Theory by Kauffman.
Proofs and Refutations, by Imre Lakatos. It makes the reader think hard about what proof really is, and would give students some idea of how the informal actual work of a pure mathematician differs from the polished final products. It's also very accessible, being written as a sort of play, in which a teacher and class discuss Euler's Theorem about the relationship among the numbers of edges, faces, and vertices of polyhedra. It's an easily comprehensible and intriguing conclusion, but many quite different proofs have been offered for it since it first appeared, and the class reproduces this history. And every time a "really valid" proof is attained, some fresh student pipes up: "I have a counterexample!"
1) How to Solve Proofs.
2) How to Solve It, by Polya
3) A Compendium of Soviet High School Math Challenge Problems.
I can't recall if this is the exact title, but it was a collection of hard math problems (solvable with only a high school math understanding) meant to find math geniuses. This book was translated from Russian and was available at Barnes and Nobel about 5 years ago. I know that in high school, I would take something like the Putnam (ECS?) exam, and if I had done well enough, I could have received a math scholarship to a top rated university.
Godel, Escher, Bach even though it has been mentioned before. Also Hofstadter's "Metamagical Themas" is easier and is a collection of columns, so can be taken in small bites. Davis & Hersh, "The Mathematical Experience" Anything by Martin Gardner Lakatos, but one has to have some background in proofs Isaac Asimov's book on algebra, long out of print Feynman's Character of Physical Law
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.
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.
This one really inspired me in High School. Is fractint still around?
I didn't read GEB until grad school, but I think I could have appreciated it in High School.
Those two books can really change how you look at the world.
http://www.math.sjsu.edu/~swann/mcsqrd.html
Sort of hard to find, but they have an address to contact the publisher, who may be still willing to run some off for you.
I learned calculus from this book... When I was 8. It's pretty good.
My blog: http://www.seebs.net/log/ --- My iPhone/iPad app: http://www.seebs.net/seebsfrac/
http://www.amazon.com/One-Two-Three-Infinity-Speculations/dp/0486256642
Though not specifically mathematically focused, a very good read for students of that age.
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.
"Is life so dear, or peace so sweet, as to be purchased at the price of chains and slavery?" - Patrick Henry
On Beyond Euclid - Ben Jacobs
A great problem based approach to Non-Euclidean Geometry. Not at all hard to understand, the prerequisite for the class it's used in at my school is only pre-calculus. The book doesn't include solutions, but I believe they're available upon request.
http://www.amazon.com/Beyond-Euclid-Ben-Jacobs/dp/1411673352
Plus all profits support financial aid at my high school (the author is a teacher there.)
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
You might try Ian Stewart's Concepts of Modern Mathematics. Quoting from the end of the book:
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 :-)
This is a book written for kids with imaginary tales from the Middle East that revolve around solving math puzzles: http://en.wikipedia.org/wiki/The_Man_Who_Counted
Who is General Failure, and why is he reading my disk?
Obligatory mention of The Manga Guide to Statistics - see Slashdot review: http://books.slashdot.org/article.pl?sid=08%2F12%2F15%2F1432233&from=rss/ ... although perhaps that is because the book didn't get a very good review, not to mention that the discussion that followed the review article wasn't very, er, academic... and maybe the book is more suitable for 15 year olds rather than 16-18 year olds... Oh well...
Clifford Pickover writes on many topics of mathematics.
A few of my favourites:
Calculus and Pizza
The Zen of Magic Squares, Circles and Stars
The Loom of God
Wonders of Numbers
Fractal Horizons
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 can only speak for myself, but believing something is hard, and then managing to tackle the subject is what I find very motivating and rewarding. This has really been a driving force for me while learning about programming (first C and then assembly), I was very intimidated by both but once I started making some progress I got this wonderful 'hell yeah I can do this!' feeling which kept me going dispite how boring this stuff really is.
Anyway, it's the Principia mathematics philosophiae naturalis - the mathematical foundations of physics, in modern English. And it's a hard read.
From scarped cliff or quarried stone she cries "A thousand types are gone, I care for nothing, no not one."
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.
Especially his "Cabinet of mathematical curiosities".
"The music of the primes", by Marcus de Sautoy, is also worth reading.
1729 = 9^3 + 10^3 = 1^3 + 12^3
A great resource I found for teaching both programming and mathematics - projecteuler.net
Consists of 200+ mathematical problems, which require both mathematical insights and programming skills to solve, each producing a single numerical answer that can then be checked. Use whatever language you want - you can find examples in the forums for most problems in most languages.
Rudy Rucker's sci-fi books really built my interest in math and computer science when I was in high school.
Prime Obsession; Euclid's Window; The Code Book: Singh; The Higher Arithmetic: Davenport; Anything by W W Sawyer; Seconded Men of Mathematics: Bell
The Computational Beauty of Nature, by Gary William Flake. It's an interesting high-level look at algorithms that define natural systems (genetic algorithms, game theory, fractals, cellular automata, etc.)
I enjoyed it way back when, because it was high-level enough to give you a sense of the topics quickly, but had just enough detail (graphs, formulas, specific discussion of algorithms) that you could dig into it a little.
Maybe a little heavy for high-school, but I guess it depends (and what doesn't?)
OH and I second the motion for "Flatland" by Abbott!
nc
The Spivak Calculus book is ALWAYS good. It'll help them to truly understand modern math, but also give them an introduction into analysis. This will give them a head up on ANYTHING that will be thrown at them in the future.
I second a recommendation for Flatland by Edwin Abbot, sure, being written in the mid-1800's it's dated as all get-out, but still a good read. My high school trig teacher turned me onto it when I was about the age of your students.
John Allen Paulos is a good writer of lay math books, Innumeracy was the first I'd read of his.
A recent book steeped in math is Neal Stephenson's latest, Anathem. A good read, but kinda out there, and oh yeah, it runs to over 900 pages.
Good luck!
I recommend Donal O'Shea's "The Poincare Conjecture: In Search Of The Shape Of The Universe". It gives a very accessible introduction to some of the basic ideas behind topology, and of course details the drama and controversy behind the Perelman/Hamilton proof.
The Drunkard's Walk is a really good book about probability and how important it is in our every day lives.
Should definitely check it out.
I read and enjoyed "Mathematics for the Million" by Lancelot Hogben. It covers a lot of different mathematical areas, and provides some historical context. I generally don't like history, but he (and Isaac Asimov) does a great job of using it to make the math, and the process by which the math was discovered, more interesting.
Hi!
I agree with many of the other suggestions found here, and I'm not trying to repeat any of them. I'll suggest that your reading list contain a bit more information than people are generating here. Consider adding to the bibliographic information on your list details like:
* Are there exercises? With answers?
* What are the pre-requisites for attempting to read, and for best understanding. Be honest here, a student can always decide to ignore your 'best advice'.
* Is this a mainstream topic, or (currently considered) a sideline.
* Contents include statement of theorems? Includes proofs? Rigorous or intuitive?
* Is the book mostly on a specific mathematical subject, a range of subjects, or mostly not mathematical (but history, or entertaining instead)
I think if you aren't going to assign reading, but you want to make the book list enticing, you'll be more successful with this kind of information. Your students may not yet have the sophistication to know that so called math books can have a huge range of styles.
A way you might collect this information over time is to require a book-report from the student that contains the answers to those questions a requirement to collect your extra-credit. Then either edit it or include it wholesale into your bibliography.
I'm a big fan of the following books:
**********
*Geometrical Vectors, Gabriel Weinreich, 1998
Exercises, no answers
Contains statement of theorems, with intuitive proofs.
Subject matter: Specific to one area: Vector Calculus. This is a non-standard perspective on a mainstream area.
Prerequisites: For reading, some mathematical maturity. For best understanding, exposure to the standard treatment of Calculus or Vector Calculus.
ISBN 0226890481
**********
*What is the name of this book? Raymond Smullyan, 1986
One several logic puzzle books
Exercises (puzzles) with answers
No theorems
Subject matter: one area: Predicate Logic
Prerequisites: For reading: none. For best understanding: Exposure to predicate calculus, or other basic symbolic logic
ISBN 0671628321
Of course it'll be suggested elsewhere, but I haven't see it yet: Alice in Wonderland, just for fun. (And I can't help but emphasizing Godel, Escher, Bach)
My dad bought me this one... :-)
**********
*The Mathematics of Juggling by Burkard Polster, 2002
Exercises
Theorems, with proofs
Subject matter: Juggling Patterns and Bell Ringing. These are not mainstream subject areas.
Prerequisites: For reading: strong mathematical sophistication or dedicated experience juggling (real objects in your hands) For best understanding: I wouldn't know... I don't think I've got it.
ISBN 0387955135
**********
*The Trachtenberg Speed System of Basic Mathematics, Trachtenberg
I found this book for a few bucks on a sale rack somewhere, but the hardback is selling for about $80 on amazon. Ouch!
It has the sad and gripping story of Jakow Trachtenberg, who was a prisoner in a work camp during WWII. That is where he developed this particular system of mental arithmetic. There are no 'theorems' but the system is justified almost well enough to be proved in the text (after several chapters that have only an explanation of the technique).
If your students are now like I was then, then some like math, think they are good at math, but aren't the best at adding and subtracting. It always felt like I should be better, and I would have loved a way to get better.
ISBN: 0313232008
Hope this helps. I'd love a copy of your compiled list! Consider posting it back to this Slashdot topic, if you have thick enough skin to weather the inevitable criticism. :-)
Boring, boring, boring.
Great if you are an adult. A teenager? Boring.
Gosh, I was waiting for the first person to recommend this brick.
IANAL but write like a drunk one.
It helps to cite authors when posting - which I too should have done. The base assumption, i.e.that the Russell/Whitehead book is still current, and people would know that is the one referred to, is surely invalid. I was thinking at the time that Newton's book, on the other hand, is actually worth reading, in extracts, for 16-18 year olds because it is accessible once you work out the old notation. The Russell/Whitehead book is not. There is plenty of far better, more accessible material on foundations of maths. My pre-Cambridge reading list included stuff by Quine, Weyl, and Peano, and I got a lot more out of all of them.
From scarped cliff or quarried stone she cries "A thousand types are gone, I care for nothing, no not one."
"Proofs from the Book" is a set of mathematical problems chosen for the beauty of their proofs. Much combinatorics, many "visual" reasonings.
Everything is undergrad level at most. Your high school students might not understand it all, but they will understand some of these problems, and have a large and colourful spectrum of maths.
The title comes from PÃl ErdÅ's' expression for outstanding proofs. He said that God kept all beautiful mathematical proofs in a transfinite book, and that, as a mathematician, you might not believe in God, but had to believe in the Book...
A best selling biography of Paul Erdos.
This book conveys the magic of mathematics beautifully, by telling the life story of one of the twentieth centuries greatest mathematicians.
It's called an elephant's trunk whereas it is in fact, an elephant's nose, a nose by any other name would smell as sweet
+1 to Godel Escher Bach
Unknown Quantity by Derbyshire, I believe he has another book out for Riemann's Hypothesis
anything by Martin Gardner (may be more appropriate to younger students, but definately interesting!)
The Code Book (Simon Singh) is awesome, as is Beyond Numeracy (John Allen Paulos).
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
Do not taunt Happy Fun Ball
This topic is right up my alley! I love recreational math books and probably have read just about every one worth it's salt since college.
Top on my list for accessibility/fun has got to be
Coincidences, Chaos and All That Math Jazz:
http://www.amazon.com/Coincidences-Chaos-That-Math-Jazz/dp/0393059456
A close second is anything that the Kaplan's have written. Their books tend to be a bit more heady... but still just as accessible to a high school student. In particular - I would recommend The Nothing That Is: A Natural History of Zero http://www.amazon.com/Nothing-that-Natural-History-Zero/dp/0195142373/ref=sr_1_1?ie=UTF8&s=books&qid=1234180693&sr=1-1 . This one is a particularly good "liberal arts" read as it is part mathematics and part philosophy and makes one think about how the two subjects play off of each other.
Biographies of mathematicians can be good too. In a field like mathematics, it is important to know where certain major results came from. Of particular interest to me has always been Kurt Godel (Barns and Nobel has a book called Incompletelness: The Proof and Paradox of Kurt Godel that is very accessible to anyone with a pension for logical reasoning). There is also the classic on Paul Erdos - The Man Who Loved Only Numbers... this one may be a bit painful for a high school student to get through though.
Beginners introductions to Cryptography can be good too. Much of the basis of that math (which is a hot topic in today's information world) can be found in discrete mathematics - something most high schoolers have seen a lot of. The Code Book is a good start if you want to take this angel.
I hope you find these suggestions helpful in your search - good luck... I'll be interested to hear what book you eventually decide on.
The Annotated Turing by Charles Petzold is a very approachable presentation of Alan Turing's work.
It's more of a computer science than math book but introduces vital mathematical concepts. Website at http://www.theannotatedturing.com/.
My maths teacher recommended all of the above to me. I started with an arbitrary selection of their books and ended up with most of them. Raymond Smullyan's logic puzzle books are also definitely on my list. A good dictionary of mathematics is one of the best places though. Having a good top-level summary (and just knowing the words you don't know the meaning of) is a great spur to doing your own reading into the subject.
Without any doubt, Will Dunham's Journey Through Genius: The Great Theorems of Mathematics ought to top your list. If a student has any mathematics in his soul, this book will speak to it.
Technically, the book only requires only the most rudimentary geometry, algebra and trig to follow. I'd guess that students who have taken pre-algebra and are mathematically inclined should not only be able to follow it, but get a great deal from it. The strength of this book is that it strips the didactic shell from mathematics, laying bare its essential fascination: the struggle of an individual mind to find that one insight that will bring the solution to a problem within its grasp.
Post may contain irony: discontinue use if experiencing mood swings, nausea or elevated blood pressure.
math:
elements - euclid
principia mathematica - issac newton
Physics:
Relativity - einstein
six easy pieces & six not so easy pieces - Richard P. Feynman
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
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.
I am not sure about this, but i think that a Japanese method for Math teach called Kumon is a good way of improve math skills in our students. I am a undergraduate student in electrical engineering and Kumon helps me with math difficulty that a carry since high school. The method show to the student that math is a continuous and simple learning. If u're interested, check about it.
-- Fernando F. Linux User #263682 http://desconstruindo.eng.br
By William Dunham. The subtitle is, "A review of the great mathematical proofs of history." He also wrote another book, which was alphabetically organized instead of sequentially, but I forget the name.
This may not be what you're looking for, but when I was in high school I wrote a very simple ray casting game (a la Wolfenstein 3D), which makes good use of high school level math. Here's a tutorial for example: http://www.permadi.com/tutorial/raycast/ It definitely helped me get more interested in math, which is why I think showing how math is used in the real world is very useful (although writing video games probably won't interest everyone).
whoa, you're a blast from the past!
"Mathematics Made Difficult" by Karl Linderholm. Hilarious, but I wasn't sure how to count in positive integers for weeks afterwards.
Do not mock my vision of impractical footwear
It's a bit dated at this point, but still an absolutely fascinating read. It explains String Theory in a manner easy (by 10 dimensional theoretical physics standards) to read format. It will give them an idea of what you can accomplish/learn/theorize with higher level math. He also has another book out I think the title was "The fabric of space" Author is Brian Greene. I could only put it down when my head started to hurt :-)
How about:
The Lady Tasting Tea by David Salsburg
Symmetry and the Beautiful Universe by Leon M. Lederman and C. T. Hill
and also check out arXiv.org and look for the couple of papers by Lederman and others on teaching physics to high school students.
While in high school, my math teacher allowed me to borrow a book called "Heart of Mathematics: An Invitation to Effective Thinking." The book, IMHO, has been the best one I ever laid my hands on to intrigue me into mathematics. It explained concepts such as infinity, cardinality of infinities, fractional dimensions, etc. exceptionally well while introducing interesting math puzzles. For pleasure reading to help students appreciate the vastness and beauty of mathematics and mathematicians, I recommend "Fermat's Last Enigma."
The oldie-but-goodie four-volume "The World of Mathematics" edited by Newman;it contains a huge range of topics.
"What The Numbers Say" by Niederman and Boyum.
"The Codebreakers" by Kahn.
any of the books about the Antikythera Machine (the newest, but least mathematical, is "Decoding the Heavens."
"Sundials, Their Theory and Construction" by Waugh.
"Empires of Time" by Aveni.
"Chaos" by Gleick.
Have them follow John Baez at: http://math.ucr.edu/home/baez/ The can look over the past several years of his postings. He covers lots of topics in math and whatever else interests him.
I posted this interesting question to my blog - Cosmic Variance.
http://blogs.discovermagazine.com/cosmicvariance/2009/02/08/mathematics-reading-list-for-high-school-students/
I hope some of the answers there are useful.
I highly recommend Flatland By Edwin A. Abbott. An older book, it is excellent for introducing the key concepts of dimensionality. Amazon carries it, as I'm sure do others.
Planiverse is one that i found to be far more entertaining than flatland. Very accessible and was a fun read in high school.
Morris Kline's Mathematics for Nonmathmeticians. An excellent work, i have recommended the first few chapters to a lot of people.
And Morris Kline's Why Johnny Can't Add is excellent as well. More politics than math, but a great way to show that math is not this dead concrete progression of subjects, and that there is a lot of room for debate on when and where to introduce concepts.
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.
The millennium problems : the seven greatest unsolved mathematical puzzles of our time / Keith Devlin
This is a survey of the problems with history, why each is important and how it relates to other fields. Very readable with a few dips into actual math.
How about A New Kind Of Science by Wolfram?
http://en.wikipedia.org/wiki/A_New_Kind_of_Science
There are multiple levels to read the book, from some pretty pictures to pretty weird philosophy, so you're not too limited.
The book is readable online for free, thats always convenient for the budget.
At least some academics violently hate the book... truly at a level of reality TV drama, if not fiercer, which could appeal to the kids.
The pictures / artistic possibilities are interesting.
And you can work in some programming.
Whats not to like?
"Science flies us to the moon. Religion flies us into buildings." - Victor Stenger
... some Nova documentaries. Nothing like spending an hour alongside some of the greatest scientific minds in history to inspire you to learn more.
Ubuntu on primary work desktop since Dapper Drake (2006).
Amazon link
No doubt some of the ideas in here are outdated by now, but it almost certainly will fire the imagination of some of your students.
The Man Who Loved Only Numbers, the biography of Erdos, is hilarious.
http://www.amazon.com/Mathematicians-Delight-Dover-Science-Books/dp/0486462404 Some great stuff including finite differences that I use in teaching all the time. Sawyer was interested in mechanical representations of mathematical ideas, makes for an interesting slant on things. Anything you can find by Sawyer is worth a look. A Path To Modern Mathematics is also good but harder to find/out of print. http://www.amazon.com/Path-Modern-Mathematics-W-Sawyer/dp/B000GRL6ZA
Mario Livio has just come out with a new book entitled "Is God a Mathematician?" about the question of whether mathematics is created by the human mind or by the universe and is "discovered" by humans. Despite the rather provocative title, it is an excellent historical account of this question.
An oldie but goodie by George Polya. Readable by high school students, and not only applicable to mathematics alone, although it surely has the emphasis.
It was on the reading list of Intl. Math Olympiad training for at least one team.
http://en.wikipedia.org/wiki/How_to_Solve_It
A great book. A more-or-less gentle introduction to real analysis. It may not be "different enough" from calculus class to make the proper impression, however.
The Mathematical Theory of Communication by Claude E Shannon, Warren Weaver
http://www.amazon.com/exec/obidos/ISBN=0252725484
It was ground-breaking in its time and continues to be interesting today. It is also short, clearly written and introduces a way of thinking that is useful for all kinds of problem spaces. Randomly picking up this book was part of what convinced me to go back to school for a CS degree.
/...
Sphereland by Dionys Burger is an interesting follow-up to flatland.
http://en.wikipedia.org/wiki/Sphereland
Godel, Escher, Bach even though it has been mentioned before. Also Hofstadter's "Metamagical Themas" is easier and is a collection of columns, so can be taken in small bites.
I second this -- I actually think that Metamagical Themas is much better than GEB.
A good read that documents a high school girl's math project that became a hot item in cryptography for a while.
Check out _Mathematical Circles_ by Fomin, Genkin, and Itenberg, 1996, now published by the American Mathematical Society. The book is composed of problems for math clubs and teams. The problems are very engaging and creative. ("Cut a whole in an ordinary piece of paper such that an elephant could walk through it.") The chapters are organized by theme, and they build conceptually within chapter. I've been going through this with my 11-year-old homeschooling daughter, and she loves it, and has come to love math. Have a student read just one chapter at a time (the whole book would be too ambitious). Your students will discover the fun and creativity of math!
zero
Bruce Schechter's "My Brain Is Open: The Mathematical Journeys of Paul Erdos"
and
Richard Feynman's "Surely You're Joking, Mr. Feynman!"
3D stuff is all linear algebra, which I didn't learn until college, but is certainly something bright HS kids could grasp.
It's just a neat application to the math (rotation matrices, etc).
When reading the literature connected to mathematics, don't forget POETRY. The recent anthology STRANGE ATTRACTORS: POEMS OF LOVE AND MATHEMATICS (http://www.akpeters.com/product.asp?ProdCode=3417,) published in 2008 by A K Peters, Ltd, offers not only a wide variety of poetry with mathematical connections but also offers, in its introduction, a list of prior collections.
As well as reading there are some great youtube tutorials around where they can learn most of first year mathematics at their own pace.
One of my favourites was http://www.khanacademy.org/
Issac Asimov wrote several excellent math and science introduction books. I read "Realm of Algebra" before starting Algebra in the 8th grade and aced the course.
All ideas^H^H^H^H^Hprocesses in this post are Patent Pending. (as well as the process of patenting all postings)
Several people have already mentioned this, but I have to add my vote for Goedel, Escher, Bach. I was 16 when I read it for the first time, and it completely blew my mind and made me want to understand mathematics as a way of understanding life. I think I'm a computer scientist today largely because of that book. After I read it, I tried to get everyone I knew to read it, but I don't think I got any takers. The large text deals with a very complex set of ideas, including Goedel's incompleteness theorem. But with a teacher's help, I think most highschoolers could get through it, and possibly have their minds expanded. I also like Chaos: Making a New Science by James Gleick, for the accessible and personal introduction to chaos and fractals.
Albert Einstein's little book _Relativity: The Special and the General Theory_ is a popular book aimed at people with a high school math education (a German one, though, I suppose). I really enjoyed the special relativity part of it when I was in high school, and the math you need to know doesn't get beyond algebraic shuffling with square roots. No calculus is needed.
It's even online for free for readers in countries where 1920 books are public domain (U.S., say): http://www.bartleby.com/173/
I think there may be in-copyright in-print versions that are slightly more up-to-date, though.
Now that I am older, the positivist approach bothers me a lot, but the arguments are still pretty cool.
I suggest A Mathematician's Lament also known as "Lockhart's Lament", it was written by Paul Lockhart in 2002. It is a relatively short read and I consider it absolutely essential for anyone in mathematics, but especially the ones who dream of being teachers.
Turning the world a billionth of a degree, probability drive, the end of the universe, Marvin's vast level of intelligence (though self-claimed), music derived from stock market trends...all ignited (or rekindled, or simply increased) my love of mathematical constructs.
It's a simple matter of complex programming.
I don't know about the educational system of your own country, so forgive me if I am giving suggestions already included.
Several valuable things in real-life practice for me (special mathematics high school) were:
1. The Horner method for solving polynomial equations. Especially useful for ax^2+bx+c=0 and so on, since you could do it in your mind most of the time.
2. The vector method for solving planimetric and Solid geometry problems - really makes a geometry problem into trigonometric transformations and equations - much faster but needs a good understanding of space.
3. Interval method for polynomial inequalities - don't know the proper translation. Basically you solve for every x where y=0 and then define the - or + or - sign of the values in the intervals.
4. any info for calculations and equations with imaginary numbers. It was really helpful when studying Electrical engineering as part of my Computer science degree.
Code: The Hidden Language of Computer Hardware and Software
More scientific/electrical than mathematical, but the discussion of different base systems would have blown my hair back in high school.
The Pleasures of Counting by Tom KÃrner is excellent (as are all is books) riddled with anecdotes and interesting facts. I read it as a secondary school pupil (high school) and it encouraged me take a Maths degree.
http://www.dpmms.cam.ac.uk/~twk/
You can try "Proofs Without Words", which is an entertaining presentation of dozens of mathematical proofs using pictures only.
See http://www.amazon.com/Proofs-without-Words-Exercises-Classroom/dp/0883857006.
Phiwum's law: anyone that names an obvious law after himself and then puts it in his own sig is just pathetic.
A Certain Ambiguity: A Mathematical Novel by Gaurav Suri (Author), Hartosh Singh Bal (Author)
This is a wonderful book I enjoyed reading myself ( 60 years old ). The main characters are college students - I think high schoolers can identify. The math in the story is presented in a very understandable way. woven in is a discussion of the different infinities and the basics of Euclidean and non-Euclidean geometries.
amazon link: http://www.amazon.com/Certain-Ambiguity-Mathematical-Novel/dp/0691127093/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234194672&sr=8-1
Necessary reading at a specialty math & science academy I went to. Reads like a novel.
Here are some books that I would highly recommend in no particular order.
FlatLand: Creates a real understanding of dimensions. Great book for challenging your thinking. Also be sure to pick out the movie. This could truly be 2001 for math geeks.
A Mathematician reads a newspaper: Goes through a newspaper, and explains the math behind it. All topics covered, politics, business, lifestyle, and much more.
Innumeracy: What happens if you don't know math? Total societal collapse. Okay maybe that's a little extreme, but this subject is important, and everybody should understand it.
Why do buses come in threes": A personal favorite. Shows how math plays into everyday life. Touches every subject. This book is interesting, informative, and amusing. I highly recommend it.
How long is a piece of string: Sequel to the previous book. Not quite as good, but still better than most.
Conned Again, Watson: Where else can you have Sherlock Holmes explaining probability and statistics to a poor unlearned Dr. Watson.
A History of PI: This is more of a history book than a math book. But it is a history of math, or specifically pi.
The Joy of pi: Like the previous book, but less serious.
Euclid's Window: Now it is really time to bend the old mind. History, Adventure, and non-Euclidean Geometry. Great stuff.
Hyperspace: You thought 4 dimensions were bad? How about 10.
A Mathematician's Apology: Since I started with a classic. I will also end with a classic. G. H. Hardy's book is a must-read for any serious math aspirant.
How about a nice story about an obsessive mathematician. It's light in math, but can teach high school students that math is more than a high school chore.
http://www.amazon.com/Uncle-Petros-Goldbachs-Conjecture-Mathematical/dp/1582341281/ref=sr_1_1/188-1454914-7359420?ie=UTF8&s=books&qid=1234195922&sr=8-1
How about some research on the concepts they use in the NUMB3RS TV show?
Goedel, Escher, Bach - Douglas Hofstader
Unknown Quantity: A Real and Imaginary Histgory of Algebra- John Derbyshire
The Road to Reality - Roger Penrose
The Road to Reality might be a good book for an aspiring physics student. Some of the mathematics is very advanced, but it's can be read on several levels with varying understanding of the math. Some high school students might find it challenging and inspiring. He has an odd diagrammatic tensor notation that I found bizarre, but I still liked the book.
I know that Goedel, Escher, Bach has inspired more than a few mathemticians in their high school years
I'll second (or third) The Feynman Lectures...must reading.
There is a two volume set of How to Solve It by George Polya that you should look at. It has interesting problems, and you should find not only the problems but the thinking that is used to solve them. Should be required reading for those with long term desires in math, physics and engineering.
Choas: Making a New Science James Glieck
This book is a math history book. It doesn't use lots of equations - but uses pictures and prose. It does a very good job of introducing chaos theory.
I read this in undergrad as a Math major. It wasn't a challenging mathematical read, but it made me want to learn more. I eventually did my undergrad thesis on chaos theory and got my paper published.
This is an excellent book to show people that math is everywhere, and math is beautiful.
There are a lot of historic texts you can access parts or whole online. Otherwise some other books to consider might be.
The Structure of Proof: With Logic and Set Theory
Concrete Mathematics: A Foundation for Computer Science
Or if you feel like constructing your own based on something else you could couple this book with the episodes from the TV show and perhaps fit some data as examples of the wide applicability of mathematics. The Numbers Behind NUMB3RS: Solving Crime with Mathematics
This is just a small sampling of what could be. It depends on how much you want to fit to their current interests and career aspirations versus your own interest versus what they may see in college or simply later in life. You could show them higher things in statistics, computing, math itself, physics, biology. I mean the sky is the limit so more information would always be helpful. Any of the books I have put forward or many of the ones others have suggested seem to fit your criteria of being rigorous, but approachable.
A professor does some mathematical modeling of jai alai betting, and makes some money with it. The way he thinks about statistics is helpful and approachable, and kids who might otherwise have some trouble caring about it may be attracted by the jai alai or gambling aspects.
The Code Book, by Simon Singh
http://www.simonsingh.net/The_Code_Book.html
A Wrinkle in Time may not be super math oriented but it introduces ideas like dimensions and space travel that may pique the interest of some of your students. Next thing you know, they'll be asking about tesseracts and such.
Maybe this would be one of those books that ranks lower on the extra credit points but could be enough to get some kids to just read.
"How to Solve It" is one of the most definitive books on the heuristics of problem solving and IMO a must read for any Mathematician, budding or otherwise. Also for young readers "Flatland" is one of the great classics.
Anyone to give a comment on the "Head first" (Algebra, Physics, Statistics) books from O'Reilly? Quality? Likability to motivate youngsters?
It's strictly "popular", but I find the history of intellectual development always fascinating, and it introduces the concept and problems relating to computer-assisted proofs.
demi
Make your students think critically and mathematically about the world around them for the rest of their lives.
I recommend "The Sleepwalkers" by Arthur Koestler. Strictly speaking, it is a book about physics rather than mathematics. It roughly covers the history of physics from Galileo through Newton. I read it perhaps 40 years ago and still fondly recollect it as one of the best books that I have ever read on science/math.
One Two Three . . . Infinity by George Gamow
I have two recommendations.
1. `Concepts of Modern Mathematics', by Ian Stewart
Link
Some people here have recommended Courant and Robbins. This book is very similar, except that it was written about 30 years later (hence, is much more up-to-date) and is written in a lively, humorous, infomal style, with many whimsical pictures. It presents an excellent survey of all the core areas of modern mathematics (e.g. abstract algebra, topology, analysis, differential geometry, combinatorics, etc.), but pitched at the level of the intelligent highschool student.
2. `Visual Complex Analysis', by Tristan Needham
Link
A beautiful and very geometrically motivated introduction to complex analysis, full of amazing and imaginative pictures. The author tries, whenever possible, to avoid equation-mashing and instead appeals to the reader's geometric intuition. Interestingly, when Newton originally developed calculus (in the original `Principia Mathematica') he did not use modern `limit' arguments (these were developed later by Bolzano), but instead adapted methods of Euclidean geometry (think pictures of `infinitesimally thin triangles', etc.). This book returns to this approach. The result is not entirely `rigorous' (it can be frustrating to teach a university-level course in complex analysis out of this book), but is extremely geometrically intuitive. It also provides insight into how mathematicians `really' think about mathematics (which is different than the formal stuff we write when we want to make something precise and rigorous). When I first read this book, I often thought, `God I wish someone had given me a copy of this when I was in highschool'.
I really enjoyed this - at least until I put it down to get a copy of mathematica to be better able to follow along and found out that would set me back $2k since I'm no longer a student. It is co-authored by Sarah Flannery, and is about her adventures in cryptography as a high school student. I found the math was introduced at a very approachable pace, and would expect other students to be motivated by seeing what she was able to come up with while she was still in high school herself. Who says girls can't do math?
http://www.amazon.com/Code-Mathematical-Journey-Sarah-Flannery/dp/1565123778
Letters to a Young Mathematician by Ian Stewart.
Hi ... Math Undergrad here ... here are some of the books that I started looking at in high school ... none of them are too difficult for the high school level...
Excursions in Number Theory
C. Stanley Ogilvy and John T. Anderson
Dover: ISBN 0-486-25778-9
Introduction to Probability
John E. Freund
Dover: ISBN 0-486-67549-1
Q.E.D. Beauty in Mathematical Proof (This is a VERY fun book, high school students should really like it)
Burkhard Polster
Walker: ISBN 0-8027-1431-5
And finally, one that doesn't seem appropriate but only fits because of the lack of algebra and technical language...
Game Theory: A Very Short Introduction
Ken Binmore
Oxford: ISBN 0-19-921846-2
Project Euler is a set of progressively harder math/programming puzzles. It's a lot of fun.
Numbers and Symmetry: An Introduction to Algebra by Bernard L. Johnston, Fred Richman http://www.amazon.com/Numbers-Symmetry-Introduction-Bernard-Johnston/dp/084930301X Easy to read, very handy. I wish I had this book when I was a kid.
What word rhymes with buried alive?
also by Marcus du Sautoy, is the best "easy read" mathematics book i have ever found. I was truly surprised at how literate a mathematician could be. This book also introduces group theory and goes into the story of the classification of finite groups (fascinating!!) and some bio of John Conway, and other funny stories.
An assortment of problems (some more accessible to a high schooler than others, perhaps) with really, really neat proofs.
"A Mathematician Reads the Newspaper" by John Allen Paulos
Reveals impact of mathematics in many media stories.
"Proofs and Confirmations" by Bressoud doesn't seem to have gotten on the list yet, nor has "A=B" by Petkovsek and Zeilberger. These books introduce a student to the growing impact of computers in advancing theoretical mathematics, and don't require much background for most of the subject matter. If the school has a Maple or other math package, this would allow experimentation.
I was going to recommend that. Each chapter is a chatty essay, setting up a social context that shows the importance of mathematics in war, health care, or biology,...
However, each chapter also involves some real mathematics. For example the reader is invited to solve a simple differential equation to follow the story.
David Berlinski's A Tour of the Calculus. It reads like a novel. Really!
I loved the techniques and simplicity of complex math from this book, it is lot of fun and something to pick up for speed tests.
Ian Stewart has written numerous popular mathematics books that are lucid, educational, and entertaining. _Letters to a Young Mathematics_ (review) is likely a good bet.
_Chaos: Making a New Science_ by James Gleick was a book I read in high school that was a classic about chaos (dynamic non-linear systems) and one of books I can point to as and fractals that inspired me to maintain a heavy mathematical bend in additional to the trendy (profitable, and for me at least, easy) Computer Science courses in university.
The classic autobiographical _A Mathematician's Apology_ by G. H. Hardy might be worth considering.
Others have already mentioned _Flatland_ by Edwin A. Abbott, but the writing style might be off-putting for some readers who find its dated style strange. _Flatterland_ (review) by Ian Stewart might by an alternative.
Others have already mentioned Simon Singh's books, which I can endorse as well. In general anything about deciphering the Enigma crypto-machines during World War II, and Alan Turing are potential books to consider as well. Anything about Paul Erdos (_The Man Who Loved Only Numbers_), and the classic book turned into a movie about John Nash, _A Beautiful Mind_, by Sylvia Nasar.
As long as the book shows that mathematics is about critical thinking and problem solving, not about pushing around numbers in equations, any popular mathematics is likely worth considering.
For hands-on math education / experience, that's a different question, that's a problem to be left to the interested student...
This book will spark the imagination and is at least 25% comprehendable by a highschooler
http://www.dspguide.com/
http://en.wikipedia.org/wiki/Infinity_and_the_Mind Basically, the philosophical conundrums and ideas of the concept of infinity which is at the heart of many areas of mathematics.
I'm sorry I don't have a specific book or paper handy so that I might give you it's name, but based on my experience as a high school teacher, college instructor, and parent, I heartily recommend something on higher cardinalities.
Some specific topics would be:
- What does it mean for two infinite sets to have the same size?
- There are no more rational numbers than counting numbers.
- The size of the set of real numbers is larger than the size of the set of counting numbers.
- Some principles of transfinite arithmetic.
I've found that the material above, if presented fairly concretely, is well within the capabilities of high school students. Further, the concepts are odd-ball enough that they find it interesting. Well, some of them do.
If your question is any indication of your general work as a teacher, you're doing a great job. On behalf of the country, "Thanks".
-----------------
Note: tpzahm has appointed himself spokesman for the country. His views do not necessarily represent those of Slashdot, its editors, or its readers. -- Ed.
Taylor and Wheeler's Spacetime Physics is an exceptional text: http://www.amazon.com/Spacetime-Physics-Edwin-F-Taylor/dp/0716723271/ref=pd_bbs_sr_1?ie=UTF8&s=books&qid=1234213537&sr=8-1
It is extremely readable, it is to date the only math book I read cover-to-cover. It is accessible to a calculus student as long as they have a vague idea what a partial derivative is, and it has the huge bonus that it allows for a very geometric interpretation of some of the rules of the Calculus. Being a geometer who was trapped in an algebra-heavy curriculum, this book basically saved my life.
I'm also an educator and I recently picked up a book called "Over the Top Cranium Challenges" written by Ivan Moscovich and published as part of the Mensa group... which I think is some organization for people who are have genius level intelligence. Its a really great book that covers a wide variety of math concepts giving information along with puzzles and games. I suggest you look into books by Moscovich and the Mensa group. http://www.amazon.com/s/ref=nb_ss_b?url=search-alias%3Dstripbooks&field-keywords=mensa+math&x=0&y=0 -check this for more Mensa math books
i'm amazed no one mentioned martin gardner yet. he's the guy who turned me on to math when i was in high school.
pretty much anything by him is great but Aha! was especially fun.
For more fiction, how about The Curious Incident of the Dog in the Night-Time by Mark Haddon, which Publisher's Weekly summarized as "Christopher Boone, the autistic 15-year-old narrator of this revelatory novel, relaxes by groaning and doing math problems in his head, eats red-but not yellow or brown-foods and screams when he is touched." It has math problems scattered throughout.
Journey Through Genius and The Mathematical Universe both, by William Dunham
I can't recall which one it was but one of them actually explained the geometric proof of irrationals in a way I understood it (finally - I always understood the basics, but several geometry/trig teachers tried to explain the geometry used in ways that led me to believe that one of the two of us was dumber than previously supposed - {G})
Probably Journey through Genius - that is a fun book.
Pug
An Invisible Entity of Vast Power whose existence must be taken on faith alone: Liberal Media
The best math book that I've ever read is a history of math ideas by Morris Kline.
Mathematics: The Loss of Certainty.
For something more current, I would recommend The Poincare Conjecture: In Search of the Shape of the Universe, by Donal O'Shea.
Not sure what you might be after, but if you want them to do math for fun, there's "Sideways Arithmetic from Wayside School" -- Reading is easy, but if I recall correctly, the main character has just transferred to the school and is trying to grok the weird math they have, where the problems are things like EGGS + MAYO = SALAD and DOGS + CATS = FIGHT. So, for the second equation, S must be 1/2 T(and T must be even), and D + C must equal something that carries into the next column... Just pull the answers out of the back. ;)
Statistics you can't trust : a friendly guide to clear thinking about statistics in everyday life
by Campbell, Stephen Kent.
Parker, Colo. : Think Twice Publishing, 1999.
ISBN:
0966617150 (pbk.) :
The Education of T.C. Mits: What modern mathematics means to you, by Lillian R. Lieber.
This book explains non-Euclidean geometry, along with other math, in a way that just makes sense. It has a recommendation from Albert Einstein.
Check out Chad's News
I think a project is a good option, one that can be done on a number of topics. I remember thats what they did in high school. I think i did mine on number systems or something and touched a bit on computation. It made me really interested in maths and it even helped me in university.
When I was in my final year of HS, me and one of my mates in the advanced math class programmed a Mandelbrot set generator. That really made my mind race ... the realisation that simple rules can yield infinite variety.
We managed it in an afternoon, admittedly we were both already proficient programmers, but I will always remember that afternoon, and Mrs Munn our maths teacher who helped us (thanks Mrs Munn, you rule).
One the bests ways in attracting learning math is Math Olympiads. It's both competitive and fun. There are a lot of books on the subject, see big list here : http://olympiads.win.tue.nl/imo/books.html#FirstStepsForMathOlympians
A book "In Code" by one of their peers from Ireland about something she discovered and something that almost made her famous.
http://www.amazon.com/Code-Mathematical-Journey-Sarah-Flannery/dp/0761123849/
I'd also recommend the books by Richard Feynman, not for the mathematics in them, but for the idea you can look around in the world and find all kinds of interesting things without even trying very hard. And how life can be fun and funny while still including science.
I am a retired math prof. I've worked with high school honor students as a mentor. I've found Peitgen's book on Chaos to contain a very comprehensive amount of material and to be written so well that students for sophomore high school to college graduate level get a lot from the articles. Chaos encompasses so much of classical mathematics as well as new ideas which have originated in the last 30 years that it is a really fun way to enrich the math curriculum without simply trying to push students more rapidly through standard material. Peitgen has prepared a few high school guides as well but I do not have a reference. One of my students scanned leaves and computed the fractal dimensions of their edges. Using this data she developed and analyzed a classification scheme across both individual trees and species. Her efforts earned her three firsts (math, computer science and biology) in the International Science Fair. Hope this helps.
Well, I'm reading it right now. I am by no means a mathematician (a year of college calculus, a semester of physical chemistry and a fair amount of exposure to the kind of thermodynamics that describe chemicals in solution.)
I am enjoying it, and I'm noticing that so far (120 pages?) it's entirely math-free. Yes, there are some appendices containing proofs of some of the statements in the book. I was not impressed with the appendix on pythagoras' theorem; we did a much better job with that in high school and since it's in an appendix, I didn't understand why the treatment was as informal as it was. One big point of the section it is first cited in was to discuss the importance of formal, rigorous proofs.
I am concerned that beyond the biographical sketches of Math Greats, I'm really just getting mind candy. My training is in biology, and last night I ran across the rather insubstantial discussion of prime numbers and cicada lifespans. As presented, it was a rather weak just so story. My general rule of thumb is that if someone's not impressing me much in discussing the stuff I actually did spend time studying in depth, the prose may be entertaining on the other stuff, but it's unlikely to actually teach me anything much, and in the worst case may be telling me things which are wrong.
It's a fun book, but I wouldn't have any real interest in using it as a teaching aid for a math course.
Assigning a decent translation of one of the Greek math texts used as source material, and asking for an oral presentation of one of the classical geometric or arithmetic proofs, or perhaps of one of the proofs requiring the use of imaginary numbers? I could see doing that.
But reading a book report about a bunch of books (which is what Singh's book is) - even a well-written one - and using it in an actual math class seems wrong, somehow.
The Music of the Primes: Searching to Solve the Greatest Mystery in Mathematics by Marcus Du Sautoy.
We can say that this book is quite similar to Fermat Last Theorem by Singh the main differences are:
- It is about Riemann hypothesis and not Fermat Theorem
- It is not solved (yet):-)
I read it at least a couple time (and i just read once the Singh one), i really loved it, the format of the book is almost a collection of the biographies of the mathematicians that studied it but the thing that hook me is the feeling of the almost "need" of those scientist to work with it, not to mention the vastness of the implication it has (from pure math to physics to cryptology).
It is really readable even without great math knowledge, as there is almost no equation there.
I don't think most people even understand what mathematics is, so I recommend reading something about the history of mathematics. The World of Mathematics edited by James R. Newman has a very good collection of essays, fragments of ancient mathematics, and even short books. It is four volumes, but I eagerly read all of it in High School. The parts I remember clearly are: mathematics in antiquity, Gauss, the 19th century number theorists, an essay by Poincare about creativity, and lots of amusing applications.
A Source Book in Mathematics edited by David Eugene Smith has famous papers, from the ancient world to the 20th century. Some are advanced, but it's ok to look at stuff you don't understand.
Don't mess with The Phone Company. Piss them off and you'll be using two tin cans and a piece of string.
"How to Solve It" by George Po'lya needs to be on this list. It is a brief monograph on problem solving that anyone can use for real life. Innumeracy by Paulos has been mentioned, but I would not recommend his other books.
I found the following 2 books fascinating: An imaginary tale The Story of SQRT(-1) by Paul Nahin and e: The Story of A Number by Eli Maor