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Mathematics Reading List For High School Students?
Posted by
timothy
on Sun Feb 08, 2009 05:32 PM
from the flatland-to-hawking dept.
from the flatland-to-hawking dept.
Troy writes "I'm a high school math teacher who is trying to assemble an extra-credit reading list. I want to give my students (ages 16-18) the opportunity/motivation to learn about stimulating mathematical ideas that fall outside of the curriculum I'm bound to teach. I already do this somewhat with special lessons given throughout the year, but I would like my students to explore a particular concept in depth. I am looking for books that are well-written, engaging, and accessible to someone who doesn't have a lot of college-level mathematical training. I already have a handful of books on my list, but I want my students to be able to choose from a variety of topics. Many thanks for all suggestions!"
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Flatland (Score:5, Funny)
Sorry, my list is lacking some depth.
Re: (Score:3, Funny)
Re:Flatland (Score:5, Interesting)
Parent
Re:Flatland (Score:4, Informative)
You can always fill it out with Sphereland.
Good book. Everyone should get credit for reading anything Rudy Rucker has written. More high weirdness than math, though.
___
Here's a bunch of really good stuff:
Mathematics for the Million by Lancelot Hogben
http://www.amazon.com/Mathematics-Million-Lancelot-Thomas-Hogben/dp/0393063615 [amazon.com]
Review
"It makes alive the contents and elements of Mathematics" -- Albert Einstein"
http://www.amazon.com/Infinity-Beyond-Lillian-R-Lieber/dp/1589880366/ [amazon.com]
Infinity: Beyond the Beyond the Beyond (Paperback)
by Lillian R. Lieber (Author), Barry Mazur (Foreword), Hugh Gray Lieber (Illustrator)
http://www.amazon.com/Einstein-Theory-Relativity-Fourth-Dimension/dp/1589880447/ [amazon.com]
The Einstein Theory of Relativity: A Trip to the Fourth Dimension (Paperback)
by Lillian R. Lieber (Author), David Derbes (Foreword), Hugh Gray Lieber (Illustrator)
http://www.amazon.com/Quantity-Real-Imaginary-History-Algebra/dp/0452288533/ [amazon.com]
Unknown Quantity: A Real and Imaginary History of Algebra (Paperback)
by John Derbyshire
http://www.amazon.com/Fractal-Geometry-Nature-Benoit-Mandelbrot/dp/0716711869 [amazon.com] The Fractal Geometry of Nature
by Benoit B. Mandelbrot
http://www.amazon.com/Chaos-Making-Science-James-Gleick/dp/0140092501 [amazon.com]
Chaos: Making a New Science
by James Gleick
Rather than just reading a book, installing the following software and working through the following tutorials should be worth beaucoup extra credit:
Geometric Algebra (GA) is one of the most exciting developments in Mathematical education and Mathematical Physics. It presents in a unified mathematical language vectors, complex numbers, quaternions, spinors, and more.
GA handles rotations easily (because it includes the quaternion algebra) and also provides a mathematical description for projective geometry. Because of this, GA is being used more and more by Computer Science (virtual reality modeling, simulation, computer vision) and Robotic Engineers (arm/body movements). ...
Geometric Algebra is also called Clifford Algebra.
Geometric algebra software GAViewer for all major OSes: http://geometricalgebra.org/gaviewer_download.html [geometricalgebra.org]
http://www.science.uva.nl/ga/files/GABLE15plus.pdf [science.uva.nl]
In this tutorial we give an introduction to geometric algebra, using our GAViewer software. In the geometric algebra for 3-dimensional Euclidean space, we graphically demonstrate the ideas of the geometric product, the outer product, and the inner product, and the geometric operators that may be formed from them. We give several demonstrations of computations you can do using the geometric algebra, including projection and rejection, orthogonalization, interpolation of rotations, and intersection of linear o set spaces such as lines and planes. We emphasize the importance of blades as representations of subspaces, and the use of meet and join to manipulate them. We end with Euclidean geometry of 2-dimensional space as represented in the 3-dimensional homogeneous model.
http://www.science.uva.nl/ga/tutorials/CGA/ [science.uva.nl]
This tutorial introduces the conformal model of 3D Euclidean geometry, to date the most
Parent
How to Lie with Statistics (Score:5, Informative)
Re: (Score:3, Insightful)
I would go for things in other fields that are math-heavy - economics, science, business, stuff like that.
Shows the usefulness of math!
Re:How to Lie with Statistics (Score:4, Informative)
Darrell not Darren, at least by my printing
Parent
Re: (Score:3, Insightful)
Being about "how not to use math" and about math as such are pretty different things. It's like you were teaching a class on car repair and assigning a book on consumer fraud.
No. It's like you were teaching a class on car repair and telling your students how to not screw up. e.g. "Do not ever adjust the stabilizer based on popular arguments such as ___ and ___ because it will only screw with the engine and may even cause permanent damage." It's actually very relevant, especially in the early stages of learning.
Prime numbers online article thing (Score:5, Interesting)
I wrote this:
http://people.pwf.cam.ac.uk/jlnw3/maths/books/prime/ [cam.ac.uk]
It was meant as an introduction to the idea of proof. Perhaps you might like it.
High school is preparation for life (Score:3, Funny)
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 [amazon.com]
http://www.amazon.com/Math-Workbook-New-SAT-Barrons/dp/0764123653/ref=sr_1_2?ie=UTF8&s=books&qid=1234132532&sr=1-2 [amazon.com]
http://www.amazon.com/Petersons-Math-Exercises-Academic-Preparation/dp/0768908078/ref=sr_1_7?ie=UTF8&s=books&qid=1234132532&sr=1-7 [amazon.com]
Re:High school is preparation for life (Score:5, Insightful)
Parent
Start with Basics... (Score:5, Funny)
Principia Mathematica. It's all there ;^)
Re:Start with Basics... (Score:5, Funny)
No it's not. [everything2.com] Sorry.
Parent
Re: (Score:3, Interesting)
No, you want Fantasia Mathematica, by Clifton Fadiman, a bunch of stories with math themes. Like the guy who wants to paint a really long drive belt -- so, to keep him busy, somebody else unlaces it, flips one side over, and relaces it. Painting just the outside of a Möbius strip turns out to be tricky.
After we read that story, my kids and I made some Möbius strips and drew on them, cut them lengthwise, and so on.
Re: (Score:3, Funny)
How long did it take you to learn enough ancient Greek?
Any abstract algebra text (Score:5, Interesting)
It's normally taught as an upper-division college class but the only real prerequisite is 2nd-year high school algebra and a mind that can think abstractly.
Students will find it different enough from trig and calculus to be fresh and knowing they can do "college math" can be a real ego-boost.
By the way, if you know any elementary or middle school teachers, many of the concepts in abstract algebra can be taught to those age groups as well. Being able to do "adult math" can be a real point of pride and inspiration at those ages.
First grade isn't too early. Anyone who can add or subtract time already has the basics for abstract algebra addition and subtraction.
Re:Any abstract algebra text (Score:5, Interesting)
I teach high school and try to put in the some of the abstract algebra topics when I teach Algebra 2. Some of the students enjoy it but most of them get really pained looks and only stop me to ask if the material will be on the test. That's not a big deal all it means is I'm not presenting it right yet. It also has something to do with how they've been taught in the past. But to support your recommendation I want to share that I was able to get a review copy of the current edition of my abstract algebra book from the publisher. I think at the college level this is more common than in secondary schools. Teachers should consider this method to build their resources.
I also want to recommend Men of Mathematics by E. T. Bell. The calc kids were very interested to know about Newton and Riemann's lives. Considering that most of what we do in middle and high school is actually math history, it seemed fitting to bring some of the personalities in.
Parent
Re:Any abstract algebra text (Score:4, Interesting)
And I strongly object to trying to slip those things into early math classes; even concepts like commutativity, associativity and distributivity are simply counterproductive to students until they have some reason to need to use them in the abstract.
And FWIW, I was not personally put off by this stuff, so that's not my reason for saying this (I almost bailed thanks to calculus, though, thanks to the unreasonable focus on limits and all that garbage which any competent person can pick up practically by osmosis once they know how to actually use the damn techniques). I got quite far along in abstract math (and had the pleasure to learn some of it from Serge Lang himself, and the displeasure of fighting through the sadistic exercises in his textbooks - the guy was far more comprehensible in person!), and I absolutely love it now; however, I think it's the type of thing that a person needs to realize they want to learn about before they will be receptive to it.
On that note, I think number theory is a good soft intro to higher math, because the open problems are so easy to state and understand (3k+1, Goldbach, etc.), and you can at least see "evidence" for them using simple methods. It's hard to draw connections between number theory and the abstract stuff without a lot of machinery in place (I don't think high school students are quite ready for adeles!), but interest in those problems is what spurs a lot of work in abstract techniques, so I think it's worth nurturing.
Parent
Re: (Score:3, Insightful)
Re: (Score:3, Insightful)
Abstract algebra is beyond the capabilities of most adults.
True. We're talking about children though. All you need is a good teacher to fire up their imagination, and they can learn anything.
That's all it takes. But you better make sure it's a good teacher.
Flatland (Score:5, Informative)
*** Ponder
Re: (Score:3, Interesting)
"Somewhat"? In Flatland, the social status of men is proportional to their number of sides (triangles are the lowest class, and priests are nearly circles); women are even lower, being straight lines. Women are not allowed to walk in public spaces without swaying and emitting noises, so that men do not accidentally get impaled on them. They have to enter their houses by the back door. They are considered "wholly devoid of brain-power",
Re:Flatland (Score:5, Interesting)
An English teacher of mine lent me "Godel, Escher, Bach" in eighth grade (I suspect he taught English by necessity, not choice!), and I found it one of the most fascinating pieces of reading I'd come across in my life. Frankly, it still holds up, if you ask me - even though I don't agree with a lot of what Hofstadter says, almost everything he writes is worth reading because it brings up so many thoughts. After practically every page I would find myself feverishly jotting down my own notes and going on my own tangents, often to discover that Hofstadter would pursue exactly those ideas in the next few pages. Quite a fun read.
That simple act of lending probably had more of an impact on my future intellectual path than almost anything else in school. Gotta remember to send a thank you to that teacher one of these days.
Parent
This was just released (Score:5, Interesting)
How to Think like a Mathematician:
http://www.amazon.com/How-Think-Like-Mathematician-Undergraduate/dp/0521895464 [amazon.com]
Online here (for how much longer?):
http://www.maths.leeds.ac.uk/~khouston/httlam.html [leeds.ac.uk]
I bought this in the discount bin for $1 somewhere, I think it's (Playthinks) really good to develop logic and just try a little bit of every mathematical discipline:
http://www.amazon.com/Big-Book-Brain-Games-Mathematics/dp/0761134662 [amazon.com]
This isn't pure math, but lisp, but since Lisp is inspired by lambda calculus, perhaps it'll inspire more programming (shrugs):
http://www.cs.cmu.edu/~dst/LispBook/index.html [cmu.edu]
Kids are ungreatful bastards (Score:5, Interesting)
Even the dullest high school student has a memory that makes us adults seem slow. There is exactly one way to motivate teenagers: tell them they are not "ready", although telling them they are "not allowed" has a similar effect. With that in mind I recommend you give one or two of them a copy of All the Mathematics You Missed But Need to Know for Graduate School [amazon.com], and suggest they pass it onto someone else if they find it "too hard". It's a great book that gives a quick skim over all the different fields of mathematics that a graduate student in mathematics is expected to know. A typical college student will read this book, shake their head and decide that maybe graduate school isn't for them. A typical high school student, even one not interested in math, will read this book and decide that mathematics is awesome and maybe they should pay attention in class, because if they can't grasp differential linear equations then they're never going to understand Lebesgue integration and infinite Fourier series.
Telling students the material is hard is foolish (Score:5, Insightful)
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.
Parent
Re: (Score:3, Insightful)
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.
Godel Escher Bach (Score:5, Interesting)
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.
Martin Gardner's column in Scientific American (Score:5, Informative)
was full of the sort of stuff that's fascinating to inquiring minds. I read one of his collections many moons ago and was enthralled! Not common to find a math book that could be called a "page turner"
Link is to a CD-ROM of all his books
http://www.amazon.com/Martin-Gardners-Mathematical-Games-Gardner/dp/0883855453 [amazon.com]
Interesting math, without all the math (Score:5, Interesting)
I read Prisoner's Dilemma by William Poundstone when I was that age, and found it to be a very intriguing introduction to game theory. It is fairly light on math, providing only enough to show that there are calculable solutions to situations that are otherwise difficult to reason through. It also provides some real life examples which are easy to relate to, e.g. letting one child cut a piece of food in half and the other choose the half they want in order to ensure "fair" portions.
It's a good choice for showing that there's more to math than finding the length of the hypotenuse.
My math is cool (Score:5, Insightful)
moving outside of 'pure' math (Score:5, Informative)
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.
Re: (Score:3, Informative)
The Golden Ratio -- or some other book on the same constant -- which goes into things like sunflowers and nautilus shells IIRC.
I do hope (I have not read The Golden Ratio) that this isn't one of those popular mathematics books which presents a lot of very intriguing factoids as though they're actually true. There are some very good pop maths books (The Story of I comes to mind), and this may be one of them. However, I'm pretty leery of the "fact" that the golden ratio describes a lot
Bringing Down the House (Score:5, Informative)
Simon Singh (Score:5, Informative)
Re: (Score:3, Interesting)
Re:Simon Singh (Score:4, Informative)
I am shocked to see it not mentioned even once.
Parent
The Shape of Space (Score:5, Interesting)
I highly recommend The Shape of Space by Jeff Weeks. (He's a freelance geometer, something he can afford after winning a MacArthur Genius Grant.) I've used this book a couple of times -- once with bright high school kids and once with bright college freshman -- and even if they don't get everything, just a taste is enough.
It builds on Flatland (which someone mentioned above), but has the advantage of being more modern and not sexist. But very quickly you're learning about Klein bottles, connected sums, and all sorts of topology you typically don't see until you're well into your undergraduate (or grad!) program in math. All aimed at high school kids. Very cool stuff.
Oh, and the big punchline at the end: what is the shape of the universe? At least you'll get a good understanding of the possibilities...
Here's a taste for you from a page related to the book [geometrygames.org].
"e": The Story of A Number (Score:3, Interesting)
Surely You're Joking, Mr. Feynman! (Score:3, Informative)
Courant-Robbins (Score:3, Informative)
Fermat's Enigma (Score:3, Interesting)
It depends on what subject you want. (Score:3, Interesting)
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).
"What is Mathematics" by Courant and Robbins (Score:3, Interesting)
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.
Freakanomics... (Score:4, Informative)
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.
A Pathway Into Number Theory (Score:4, Interesting)
A Pathway Into Number Theory [amazon.com], by R. P. Burn.
It's the most unique math book I've ever read. There is no prose in the book per se; rather, the book is a series of small tasks and questions (usually starting by identifying patterns in tables of numbers) that, as the title suggests, gently lead the reader into Number Theory. All the major topics of a first course (the fundamental theorem of arithmetic, quadratic residues and forms, etc.) are there; the beauty of the book is that each task is such a small step from the previous one that the reader is led painlessly to a mastery of each concept. (Just don't skip steps!) This feature makes is suitable for advanced high school students looking for "stimulating mathematical ideas."
It's a wonderful book, on a wonderful subject. I have often wished for books written in this format on other mathematics subjects.
Here are several (Score:3, Insightful)
First, let me add my recommendation for GEB. It's an amazing book.
Here are some others that I think are good:
Two great books (Score:4, Interesting)
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.
Re: (Score:3, Insightful)
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
Re: (Score:3, Interesting)
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.