Mathematics Reading List For High School Students? 630
Troy writes "I'm a high school math teacher who is trying to assemble an extra-credit reading list. I want to give my students (ages 16-18) the opportunity/motivation to learn about stimulating mathematical ideas that fall outside of the curriculum I'm bound to teach. I already do this somewhat with special lessons given throughout the year, but I would like my students to explore a particular concept in depth. I am looking for books that are well-written, engaging, and accessible to someone who doesn't have a lot of college-level mathematical training. I already have a handful of books on my list, but I want my students to be able to choose from a variety of topics. Many thanks for all suggestions!"
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.
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.
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]
Fractals (Score:1, Interesting)
While I can't think of a book offhand, I learnt complex numbers and matrices through playing with both IFS and standard fractals. Advantage is that you can get visual feedback of what you're doing in just a few seconds
A few lines of BASIC or equivalent and you can be playing with them in no time.
Kids are ungreatful bastards (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.
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.
"The HIgher Arithmetic" (Score:2, Interesting)
"The Higher Arithmetic" by Harold Davenport is a fantastic book on number theory. It explains the concept of proof in the first 10 pages without using any formal notation. All of the proofs are given in an intuitive, explanation style. Aside from being a fantastic book on Number Theory (and thus a great primer to understanding modern cryptography), it is a very good introduction to the style of thinking and argument involved in actually doing /mathematics/ (as opposed to arithmetic, which is what seems to be mostly taught in schools or the treatment of mathematics in most science and engineering fields, which tends to be algorithmic and problem focused).
Interesting math, without all the math (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.
Re:Godel Escher Bach (Score:1, Interesting)
I have no mod points, but I have to second this recommendation. I first read GEB when I was twelve and I don't think I fully understood the implications of what it was saying for another decade.
Strictly speaking, this book is about the Church-Turing thesis, but it's really about what we know--what we can know--and in what ways mathematics is and isn't useful. There's a full treatment of the fundamentals of math, formal logic, and a focus on the act of interpreting what math's little symbols really mean in the real world. All of this is couched in an entertaining Lewis Carroll'ian dialog referring to Escher's artwork and Bach's music.
John Allen Paulos books (Score:2, Interesting)
"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.
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)
Re:Simon Singh (Score:3, Interesting)
"The Code Book" by Simon Singh (Score:1, Interesting)
I really enjoyed
"The Code Book" by Simon Singh
From Publishers Weekly
In an enthralling tour de force of popular explication, Singh, author of the bestselling Fermat's Enigma, explores the impact of cryptographyAthe creation and cracking of coded messagesAon history and society. Some of his examples are familiar, notably the Allies' decryption of the Nazis' Enigma machine during WWII; less well-known is the crucial role of Queen Elizabeth's code breakers in deciphering Mary, Queen of Scots' incriminating missives to her fellow conspirators plotting to assassinate Elizabeth, which led to Mary's beheading in 1587. Singh celebrates a group of unsung heroes of WWII, the Navajo "code talkers," Native American Marine radio operators who, using a coded version of their native language, played a vital role in defeating the Japanese in the Pacific. He also elucidates the intimate links between codes or ciphers and the development of the telegraph, radio, computers and the Internet. As he ranges from Julius Caesar's secret military writing to coded diplomatic messages in feuding Renaissance Italy city-states, from the decipherment of the Rosetta Stone to the ingenuity of modern security experts battling cyber-criminals and cyber-terrorists, Singh clarifies the techniques and tricks of code makers and code breakers alike. He lightens the sometimes technical load with photos, political cartoons, charts, code grids and reproductions of historic documents. He closes with a fascinating look at cryptanalysts' planned and futuristic tools, including the "one-time pad," a seemingly unbreakable form of encryption. In Singh's expert hands, cryptography decodes as an awe-inspiring and mind-expanding story of scientific breakthrough and high drama. Agent, Patrick Walsh. (Oct.) FYI: The book includes a "Cipher Challenge," offering a $15,000 reward to the first person to crack that code. Copyright 1999 Reed Business Information, Inc. --This text refers to an out of print or unavailable edition of this title.
Fermat's Enigma (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.
1 2 3 ... infinity by George Gamow (Score:2, Interesting)
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
Re:the pleasures of counting (Score:1, Interesting)
Seconded. The author, Tom Korner, has recently published another book along similar lines: Naive Decision Making.
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.
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.
Re:Surely You're Joking, Mr. Feynman! (Score:1, Interesting)
Six Easy Pieces has a more mathematical/physics bent, and was required summer reading before my AP physics course.
Frankly, nearly anything Feynman wrote for the general population would be worth exposing someone to.
Re:moving outside of 'pure' math (Score:1, Interesting)
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.
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.
Re:Spivak's Calculus (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.
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:Start with Basics... (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:Flatland (Score:5, Interesting)
Re:Flatland (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", driven by emotion and instinct and lacking memory, and they receive no education.
But it's social satire, not a reflection of the author's views. He was "a firm believer in equality of educational opportunity, across social classes and in particular for women" [mathaware.org], and the book is attempting to highlight a Victorian mindset that was still prevalent at that time. The women in the book act in far more complex ways than their men give them credit for. The author even says "To my readers in Spaceland the condition of our Women may seem truly deplorable, and indeed it is" - he's not happy with how they're treated, and readers in Spaceland will hopefully see that it's caused by the absurd class system holding them back, though the narrator can't avoid falling back into the prejudices of his society.
The book makes more sense when you understand the context. The Annotated Flatland [amazon.com] is quite interesting, providing some background on the author and mathematics and the society of the time.
("more sense" doesn't mean it actually does make sense - it all still seems a bit muddled to me, with a random mixture of physical differences and social differences between people, and strange science (like Lamarckian evolution where the actions of a parent affect the number of sides (hence social status) not of themselves but of their offspring), and sections that I don't understand the point of (like the whole thing about colour being discovered and then banned - it makes sense within Flatland but is it meant to be satirising anything in real life?). Much of it is probably because the world has changed so drastically in 125 years that I just can't understand where the author was coming from. But it's an interesting book despite (or perhaps because of) that.)
Books, People, Ideas (Score:3, Interesting)
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.
Re:Any abstract algebra text (Score:3, Interesting)
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.
I was rather surprised how much Finnish math teaching had evolved since I was in elementary school when I recently read a first grade math book.
Those kids are solving equations in first grade.
The genius behind it is that the symbol layer is taught after the concepts. The kids learn to do the basics all with pictures. A simple equation like 2+X = 4 can be presented like a set of scales with four apples on one side, and two on the other. The task is to make the scales balance out.
Using this principle a lot of rather advanced math (for elementary school) can be taught without learning all the symbols for everything. Later on when the idea has been mastered the symbols are introduced and you just tell the kids what technique to apply with which combination of symbols.
This approach greatly reduces the tendency to do math by applying a "set of preprogrammed instructions" you learn mechanically and instead actually tackle the problem. Math problems with a lot of scary looking symbols tend to demotivate a lot of kids.
Re:Flatland (Score:3, Interesting)
I want to discourage you from this idea.
I remember when my Quantum Physics professor assigned reading to us over the summer months "for bonus points". One of them was The Structure of Scientific Revolutions by Thomas S. Kuhn, which is a very good book, but not one of us read anything on the prof's extra-credit list. I suspect you'll get a similar response from your math students.
Perhaps if you gave them something "easy" like listening to Teaching Company lectures while cruising in their cars, but even then I suspect compliance would be low. Teenagers don't want to study at home, anymore than we adults want to carry work home.
Mathematician's Lament (Score:2, Interesting)
Re:Flatland (Score:2, Interesting)