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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 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
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.
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.
*** 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
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.
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.
Prime Obsession: A well-written history of the still-unproven Riemann Hypothesis. Maybe one of your students will solve it over summer break!
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...
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
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!
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.
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
I really like Fermat's Enigma by Simon Singh. Relatively easy read and I found it inspiring.
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).
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
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
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!
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.
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.
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.
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.
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).
Real analysis? Woof. I suppose if you want to make your students passionately despise math forever, that's one way to go.
High school kids need to be exposed to the fun parts of math, not the parts that make people that love math groan. Even complex analysis is far more enjoyable (not to mention useful) than real analysis. Nobody likes to sit around proving the obvious for no other reason than to prove that you can do it, and high school students will never realize that the reason for all of the rigor is to expose the edge cases where things break down.
Spivak's Calculus is probably the best calculus text for someone interested in mathematics. But it may be one of the worse for someone who finds mathematics difficult. But I'm biased, I learned some of the basics from Spivak himself and he left me with a lifelong love of mathematics.
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.
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
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
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.
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
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.