Resources for a curious beginner mathematician I have a friend whose experience of math in school was horrible (each teacher, she says, either left or was fired the year after she had them).  I've been teaching her about various things: cantor's diagonal argument, irrationality of $\sqrt{2}$, etc. and she gets very excited and interested, and she picks up on mathematical concepts and thinking easily.
I am looking for some resources people would recommend for a curious but unsophisticated mathematical mind.  She is not in the target audience of most textbooks at her level; I suspect a traditional scaffolding approach would not spark her passion.  How can I help deepen her mathematical thinking without falling into the mundane algorithmics typically associated with the teaching of lower-level math?
 A: *

*There are lots of related questions already posted here that you may want to lookat for more ideas, e.g. this one and this one.  

*One specific reference I like a lot is Gowers's book Mathematics: A very short introduction.  It has a lot of interesting examples and discussions, and
while it is serious, it manages to avoid being dry.
A: Besides Martin Gardner, I would highly recommend Ian Stewart's books.
Game, Set and Math for example is an excellent (non-typical) start for various mathematical ideas.
A: There's C. Stanley Ogilvy's book Excursions in Geometry.  See my review on amazon.com.  You hardly need to know anything in order to read it.
A: She might try Combinatorics Through Guided Discovery.  It is a "textbook" but written in an interactive style, with the bulk of the book being problems for the reader to solve.  Because the subject is combinatorics, there are no prerequisites for the book other than an interest in learning how to count!  And it is also freely available for download, so there's no cost in giving it a try.
A: A good book written for college students, but also accessible to high school students and adults, is Sherman K. Stein's Mathematics: The Man-Made Universe (published by W. H. Freeman and Company).  I have the third edition (1976).  It covers topics from number theory, topology, set theory, geometry, algebra, and analysis, and it has six very useful appendices (review of arithmetic, writing mathematics, the rudiments of algebra, teaching mathematics, the geometric and harmonic series, and space of any dimension).
A: I consider myself a visual learner, so I would like to recommend giving MIT OpenCourseWare a try. There are many complete curses recorded as video lectures with very good and passionate professors.
For evaluation, you could take a look for example on Intruduction to Derivatives, or Single Variable Calculus.
I would also like to recommend a book that talks fairly enough about practical uses of Mathematics: "The World According to Wavelets", by Barbara Burke Hubbard, a journalist.
Small (out of context) fragment from the Prolog:

(...) No one claims that only geneticists doing recombinant DNA
  research should know what DNA is, or that only physicists and chemists
  should know that matter is made of atoms. In mathematics the reverse
  is often the case; (...)

This fragment is available in the book preview in Amazon, if someone thinks it should not be posted here please feel free to remove it.
A: Send her to Khan Academy and see if that  suits her. She could easily jump in just at her own level and the video lectures given there are truly great!
Khan Academy is the winner of the google 10^100 prject. It is a video library with over 2,700 videos made by Salman Khan, a graduate of MIT.
A: I think Edna Kramer's The Nature and Growth of Modern Mathematics might be a good fit for your friend (URL below), as well as Joel Reyes Noche's suggestion of Sherman K. Stein's Mathematics: The Man-Made Universe.
http://www.amazon.com/dp/0691023727
A: I always recommend "What is Mathematics" by Courant and Robbins. This is a lengthy book (around 500 pages), and it covers some very interesting and deep mathematical ideas (as compared to other "popular" books). No other book presents so strongly the motivation and intuition required to understand the deep mathematics within.
It covers very difficult ideas in a very intuitive way, and while equations frequently appear, they are not the focus. Indeed, the entire exposition reads like a novel. The revised edition by Ian Stewart contains a few corrections to the original, and also adds more recent developments (recent is a relative term, the original book was written in 1941) to address problems that were previously open (such as Fermat's Last Theorem) when the book was originally written.
This book, perhaps more than anything else, sparked an extreme motivation for me to learn more mathematics, and I often wish I had read it earlier. If you are looking for something which will do the same, I cannot recommend this book highly enough.
A: Your friend might try looking at one of Martin Gardner's many books, which are typically available at a public library. Here is a list of some of these: http://www.york.cuny.edu/~malk/biblio/martin-gardner-biblio.html
A: One possibility is The Heart of Mathematics, by Edward R. Burger & Michael Starbird. It’s a so-called liberal arts math textbook that’s rather different from most such, in that it spends some time on mathematical ideas that are more interesting in their own right than obviously of practical use. (Reading it isn’t as much fun as listening to Mike talk, but it’s a very readable book.) The liberal arts math texts Excursions in Modern Mathematics, by Peter Tannenbaum, and For All Practical Purposes, by COMAP, are more conventional, but they also have a lot of interesting and accessible mathematics.
In another direction entirely, she might find things to interest her in Martin Gardner’s old columns in Scientific American, which have been collected in various volumes; a list can be found here. Two more recent popularizers of mathematics who would probably be worth looking into are Ian Stewart and Ivars Peterson, but I’m much less familiar with their writing.
And just for fun, the kids’ story The Number Devil: A Mathematical Adventure, by Hans Magnus Enzensberger, is a cute introduction to some serious mathematical ideas.
A: Personally, if I was trying to help someone from a liberal arts background, I would start them with some of the "popular" books out there until they had some basic background in the concepts and a real desire to dive deeper.  The "popular" math books are easier to stay motivated reading, and I read a lot of them as a break from text books when I was working on my undergrad in Mathematics.
I am a fan of many of John Allen Paulos books like "Mathematics and Humor".  "From Zero to Infinity" is rigorous, yet written for the non-mathematician (or at least beginner) in an informal, easily read style.  "Beyond Numeracy" is a very approachable book with short sections on a lot of mathematical topics and targeted at the undergrad level.
For something slightly more in depth and rigorous, she may want to look at "Chapter Zero" which goes through a lot of the stuff that people should have learned by High School, but almost never did.  It is targetted at a High School level audiance and is easy to read, but unlike my other recommendations it is a text book and can be a bit dry and can require some motivation to get through.
Finally, I have to mention "Visual Complex Analysis."  It is one of the most approachable and entertaining books on serious mathematics I have ever used, and it is easy to just read selected chapters.  (That is what I did, I still haven't finished going through it cover to cover, and used it as a supplement to my text books).  But that one is about serious mathematics and is targeted at advanced undergrads in heavily mathematical fields.
