Resources for a curious beginner mathematician [closed]

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?

Solution 1:

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.

Solution 2:

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