You might enjoy one of the many books that exist for undergraduates to ease the transition into higher math classes, where there is a shift from the strong calculation-focus and rote-learning of most math teaching through calculus, to the more proof-centric and understanding-based approach that one finds in classes like abstract algebra, real and complex analysis, and other post-calculus math courses.

Here are some examples of the kinds of books I mean, and you can find others by following Amazon recommendations from those:

http://www.amazon.com/Nuts-Bolts-Proofs-Fourth-Edition/dp/01... http://www.amazon.com/Mathematical-Proofs-Transition-Advance...

With math textbooks especially, it pays to look for a previous edition, as the current edition can be ridiculously expensive, and the previous edition might be only 20% of the price, with no significant differences between the two.

Also, don't get them for the Kindle, as Amazon doesn't seem capable of publishing a math book with lots of notation that doesn't also have tons of errors where symbols get incorrectly imported. I've bought at least 20 and yet have to see one that didn't have lots of incorrect symbols.

My wife gave me this great book for Christmas:

  Love and Math: The Heart of Hidden Reality
  by Edward Frenkel

It begins with the author's struggle to learn the math behind quantum physics in spite of cold-war era soviet educational obstacles and leads bit by bit into the Langlands program, drawing connections between group theory, number theory and harmonic analysis.

It's definitely my favorite book of 2013.

http://www.amazon.com/Love-Math-Heart-Hidden-Reality/dp/0465...

Yup. My thought as I read Love and Math a couple of months ago was "this would be great to give to a high school senior who is wondering whether to continue with mathematics." If you truly love doing math, you'll feel it when you read this book.

Have you seen the coursera introduction to mathematical thinking [1]? That may be a good starting point.

[1]: https://www.coursera.org/course/maththink

I think Gödel, Escher Bach does a good job imparting the "mathematical mindset", albeit indirectly. It's a great read even if it's quite long and a bit repetitive. Definitely worth a read if you're not already familiar with abstract mathematics and formal logic.

I have read that actually. I think it was much heavier on the 'logic' than the mathematics though, if you know what I mean. Great book though... love how it doesn't even really get started on its main topic (AI) until about three quarters in, having covered a vast range of supporting topics.

Polya's How to Solve It was recommended below. I also like How to Prove it by Daniel Velleman (http://www.amazon.com/How-Prove-It-Structured-Approach/dp/05...).

"How to Solve It" by G. Polya is an excellent guide on how to approach mathematical problems.

Check out 'The Joy of x' by Steven Strogatz.

http://www.stevenstrogatz.com/the_joy_of_x.html

You may enjoy Mathematics for the Nonmathematician by Kline:

http://www.amazon.com/Mathematics-Nonmathematician-Dover-Boo...

I have not read it, but I have heard it recommended many times by knowledgeable educators.

Any artofproblemsolving.com book will be outstanding for this purpose. I suggest starting with Prealgebra by Richard Rusczyk.

How to Solve It, by Polya.