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Table of contents

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"Philosophy is written in that great book which ever lies before our eyes -- I mean the universe -- but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written. This book is written in the mathematical language...without which one wanders in vain through a dark labyrinth."
- Galileo Galilei [got this quote from Sal Khan]

I can't remember myself ever being a fan of math while I was studying it in school; in fact, I can remember hating it a lot of the time when 1) I couldn't understand something, or 2) I had to do a lot of math drills. I do remember feeling good when I understood something. I also remember celebrating in college when I realized I would never have to take another math class for the rest of my life.

My opinion of math started to change as I read about more and more people who'd become rich through using math (or who'd studied math and then became rich with their math-trained mind). Warren Buffett loved math and numbers as a kid, Sergey Brin (of Google) was a cs / math major, the founder of plentyoffish was knowledgeable of math, the guys who improved the Netflix recommendation tool were all very fluent in advanced math (as far as I can tell).

 

Visualizing higher dimensions

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Movies

Algebra



Timothy Gowers: The Importance of Mathematics
http://www.youtube.com/view_play_list?p ... 2A6ADDB4B1

Links:



For Kids:
Brain Pop Jr: http://www.brainpopjr.com/ - I was watching a NOTACON 9 video about how this guy simplified the rules of Warhammer to make it accessible to his 6-year-old and he said that he was homeschooling his kid and that his kid would choose to spend hours on Brain Pop Jr. So I figured it was worth seeing what they were doing. It seems like they have some ideas that Khan Academy may want to consider, like 1) having animated pictures for links, 2) having a voice read what different links say. KA could tailor the design of the site to the individual user based on that user's age, gender, interests, etc.

British Mathematical Olympiad (BMO) Website - http://www.bmoc.maths.org/home/bmo.shtml

Differential Equations:
Khan Academy - http://www.khanacademy.org/math/differential-equations
user maths247 on YouTube seems pretty good: http://www.youtube.com/watch?v=cge4ss77 ... plpp_video

Fractals:
Fractals - The Colors Of Infinity
NOVA - Fractals


Movies / Documentaries:'
http://math.stackexchange.com/questions ... umentaries
http://faculty.etsu.edu/gardnerr/Math-V ... Videos.htm
Dangerous Knowledge
Hard Problems (re: training for the IMO)
Donald Duck in Mathmagic Land

Books:

Lists:
Path to advanced math and applications

google search: best pop math books - https://www.google.com/search?q=best+po ... e&ie=UTF-8
Math Overflow - best pop math book for an aunt - http://mathoverflow.net/questions/8609? ... 1#sort-top


History:
Mathematical Thought from Ancient to Modern Times by Morris Kline - rec'd here

Books to Get You Interested in Math:

Moneyball - prob. my favorite finance book, and a book that made me really appreciate the amazing things that math can do
In Pursuit of the Unknown: 17 Equations That Changed the World
Taking Sudoku Seriously: The Math Behind the World's Most Popular Pencil Puzzle
Technological Slavery: The Collected Writings of Theodore J. Kaczynski, a.k.a. "The Unabomber" - I remember reading the Unabomber Manifesto in high school and finding it to be the most clearly-written-out argument I had ever read up until that point. I've noticed that there are a lot more extremely clear writers among mathematicians (another good example is Timothy Gowers). You can find the Unabomber Manifesto online for free, but I like taking notes when I write so I have a hard copy. This book obviously doesn't talk about mathematics itself, but there are things you can read that will increase your motivation to study math even if they don't involve actually studying math. Reading Kaczynski's clear prose definitely increased my respect for mathematicians, even if I disagree with some of the conclusions he reached. While rereading a little of it after being exposed to philosophy / the LSAT / legal arguments, I could see how it had a lot of argumentation-related problems, but I still think his unpretentious prose is better than the stuff I was reading back in high school: Walt Whitman, Henry David Thoreau, guys who are more...painting with words, which is fine if that's what you're looking for, but I want just the core of the argument, plainly-spoken.
A Mathematician's Apology

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Mathematical Problem Solving:

Ask a Mathematician / Ask a Physicist - How does a scientist turn ideas into math?
http://www.askamathematician.com/2011/0 ... into-math/

George Polya Books:
Mathematics and Plausible Reasoning, Volume 1: Induction and Analogy in Mathematics
Mathematics and Plausible Reasoning: Volume II Patterns of Plausible Inference
The Stanford Mathematics Problem Book

Art of Problem Solving Books:
http://www.artofproblemsolving.com/index.php?mode=books

The Art and Craft of Problem Solving by Paul Zeitz (very high reviews)
The Mathematical Olympiad Handbook: An Introduction to Problem Solving Based on the First 32 British Mathematical Olympiads 1965-1996

Math Puzzles:
Mathematical Mind Benders by Peter Winkler
Mathematical Puzzles: A Connoisseur's Collection by Peter Winkler
My Best Mathematical and Logic Puzzles by Martin Gardner
The Colossal Book of Short Puzzles and Problems by Martin Gardner
Perplexing Puzzles and Tantalizing Teasers by Martin Gardner
Entertaining Mathematical Puzzles by Martin Gardner
The Moscow Puzzles by Boris Kordemsky

Book Lists:
http://www.amazon.com/Most-lucid-math-b ... 4DNJT5MRF9


Answers to Basic Questions About Math

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Q: Why should I learn math? How would you state, in as general terms as possible, what it is that math-solvable problems have in common that makes them solvable with math?

Q: How far should I go in my math studies?

To borrow an idea from economics: you should study as long as the marginal benefit is greater than the marginal cost. In other words, imagine that for every hour you study math, you're going to be just a little bit happier in the future (because you make smarter decisions, can get a better job, etc.). For the basic math (addition and subtraction), the extra happiness you gain from your studying is probably pretty high; basic math is really useful. As you study more and more obscure math, though, the time you're giving up may no longer be worth the little extra happiness you'll gain in the future. That's when you should stop.

Q: Why do I need to memorize how to do basic arithmetic by hand if I can always just use a calculator?


List of Situations Where Math Is Very Helpful:

Addition and Subtraction:


Multiplication and Division:




List of Math Tips:

- Listen to music while doing problems! I've been listening to the Foo Fighters albums while going doing a problem a day from the AoPS Vol. 1 book, and it's been amazing at keeping me focused instead of having my mind wander!

Precalculus
Limits
- Summary of the KhanAcademy videos: If you're given a function that's undefined at a particular point [eg, f(x) = (2x+2)/(x+1)] and you're asked to find the limit as x approaches that value [-1 in this example] you just need to get rid of the part of the division that's normally 1/1 and then becomes 0/0 at the undefined point [eg in the previous example, (x+1)/(x+1)]. Then just plug in the value of X that you want to find the limit for and that'll be your answer [eg in the previous example the answer would be 2, b/c the formula you're left with after simplifying is f(x) = 2].



Courses to get an undergraduate degree in mathematics:
TCNJ:
http://mathstat.pages.tcnj.edu/academic ... thematics/
MAT 12 - Calculus B
Proof Writing through Discrete Mathematics
Linear Algebra: Theory and Applications
Multivariable Calculus
Abstract Algebra
Real Analysis
Senior Capstone Course (writing intensive)
Five Mathematics Options at the 300/400 level, with at least two courses at the 400 level.




Regarding PhD-Level Math
Ben Tilly - Why I Left Math:
http://bentilly.blogspot.com/2009/11/wh ... -math.html
- basically he said it's very specialized, so you don't really understand what anyone is doing other than a small group of ~10 guys.

William P. Thurston - On proof and progress in mathematics
http://arxiv.org/abs/math.HO/9404236
- This was rec'd in the comments of the Tilly blog post. It looks good, I haven't read it yet.

Gowers - Two Cultures of Mathematics
https://www.dpmms.cam.ac.uk/~wtg10/2cultures.pdf
- I love Gowers' prose; I've only read the first 2 pages so far.


Successful People

Richard Rusczyk

2009 - Problem Solving: A 21st Century Education
http://mathprize.atfoundation.org/archive/2009/rusczyk
There are only two things you need to worry about with your elementary school kids:
1. Read
2. Play
"When I was a kid my parents would often say no when i said i wanted a toy or wanted candy; they never said no to a book. Any book I wanted--it could be baseball, for instance--I'm getting that book there were books all over my house. I learned to love to read."
- he makes the point that regular schools glorify sports / music but don't give similar attention to math.
- he makes the point that math clubs give good math students social rewards for being good, so there's friendly competition
36:00 - he relates an anecdote of how he was in a group of very successful mathematicians and they were supposed to figure out how to improve education, and the only thing they could agree on was to give students free time every day to work on whatever they're passionate about [independent study] Otherwise people get used to constantly being told what to do, and they don't know how to adjust when they get into the real world and life doesn't work like that anymore.''
42:00 - some math camps are heavily focused on preparing for competitions, others aren't; some are heavily structured, others arent
56 - he says if you're interested in going into finance, send him a message when you're in college

Grigory Perelman

2006 - New Yorker - MANIFOLD DESTINY
http://www.newyorker.com/archive/2006/0 ... 28fa_fact2

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Quote:
Grigory Perelman did not plan to become a mathematician. “There was never a decision point,” he said when we met. We were outside the apartment building where he lives, in Kupchino, a neighborhood of drab high-rises. Perelman’s father, who was an electrical engineer, encouraged his interest in math. “He gave me logical and other math problems to think about,” Perelman said. “He got a lot of books for me to read. He taught me how to play chess. He was proud of me.” Among the books his father gave him was a copy of “Physics for Entertainment,” which had been a best-seller in the Soviet Union in the nineteen-thirties. In the foreword, the book’s author describes the contents as “conundrums, brain-teasers, entertaining anecdotes, and unexpected comparisons,” adding, “I have quoted extensively from Jules Verne, H. G. Wells, Mark Twain and other writers, because, besides providing entertainment, the fantastic experiments these writers describe may well serve as instructive illustrations at physics classes.” The book’s topics included how to jump from a moving car, and why, “according to the law of buoyancy, we would never drown in the Dead Sea.”
[...]
“There are a lot of students of high ability who speak before thinking,” Burago said. “Grisha was different. He thought deeply. His answers were always correct. He always checked very, very carefully.” Burago added, “He was not fast. Speed means nothing. Math doesn’t depend on speed. It is about deep.”
[...]
The implications of the conjectures for other disciplines may not be apparent for years, but for mathematicians the problems are fundamental. “This is a kind of twentieth-century Pythagorean theorem,” Mazur added. “It changes the landscape.”
[...]
“I really wanted to ask [Richard Hamilton] something,” Perelman recalled. “He was smiling, and he was quite patient. He actually told me a couple of things that he published a few years later. He did not hesitate to tell me. Hamilton’s openness and generosity—it really attracted me. I can’t say that most mathematicians act like that. [Really?]
[...]
Yau was especially impressed by Hamilton, as much for his swagger as for his imagination. “I can have fun with Hamilton,” Yau told us during the string-theory conference in Beijing. “I can go swimming with him. I go out with him and his girlfriends and all that.”

 

 


Algebra