17

Flatland by Edwin Abbott Abbott It's a story about a 2-dimensional being's encounter with a three-dimensional being. (Well, there's a class(?) allegory at the beginning, but you can skip that.)


17

Gödel, Escher, Bach: an Eternal Golden Braid by Douglas Hofstadter is a book about Mathematics, Art, Music, Computer Science and Artificial Intelligence. It alternates story chapters and descriptive chapters. In my opinion the most valuable parts of the book are: a very understandable presentation of Gödel's incompleteness theorem. a good discussion of ...


11

Two good books that I liked when I read them years ago are Simon Singh: The Code Book. This is a great introductory book to cryptography. The book is not very mathematical heavy, but cryptography is very related to number theory, so I think the book works well and can function well as an inspirational book. Simon Singh: Fermat's Enigma. This is a book about ...


11

There are certainly many more. The following list is not meant to be exhaustive but meant just to give you a selection of the many more books that you have missed: Halmos: Linear Algebra Problem Book, A Hilbert Space Problem Book. George Polya and Gabor Szegö: Problems and Theorems in Analysis - I, II. Ram Murty: Problems in Analytic Number Theory. Jody ...


10

I am a very big fan of The Number Devil written by a famous German author (normally known for his books about history or politics). The subtitle in the original German version is "Ein Kopfkissenbuch für alle, die Angst vor der Mathematik haben." (=A pillow book for all fearing mathematic.") It's a (nicele illustrated) story about a boy having fear of ...


10

William Dunham's Journey Through Genius is, ultimately, about a bunch of facts, but it's written very well and can be inspiring to a budding math student. How to Lie with Statistics is just a classic and deserves to be mentioned, even thought it's not really "math-heavy". The Kaplans' The Art of the Infinite is genuinely playful and seeks to present math ...


10

One recommendation that I have also given in response to similar questions, check out Jo Boaler's edX course: How to Learn Math for Parents and Teachers. It's currently in progress, but I'm sure it will run again. (I don't know if you can join while it's going on). Also, the book Powerful Learning: What We Know About Teaching for Understanding has a chapter ...


10

If you looked at those other topics, you saw the links to the books page at my blog, Math Mama Writes. There are a number of books there that would work for a teen. One of my favorites is Carry On, Mr. Bowditch, by Jean Lee Latham, the (slightly fictionalized) biography of Nathaniel Bowditch, who modernized navigation in the 1700s. But maybe that has too ...


9

In terms of a book which inspires, I will offer Love and Math: The Heart of Hidden Reality by Edward Frenkel. It is an autobiography. It explains how a knowing mentor used physics to lure him into deeper math. I don't want to say too much and spoil it, but, he goes on to explain symmetries, groups, Lie groups, Loop groups, braid groups, Lie algebras, Kac-...


9

For a recent suggestion, check How Not to Be Wrong by Jordan Ellenberg. Lying in the "simple and profound" quadrant, the book also gives deserved attention to Condorcet, in addition to providing a very readable book for a wide audience. Rather than my saying more, let me direct you to some recent reviews/responses: LA Times Salon Scientific American WSJ ...


9

I highly recommend Paul Lockhart's Measurement


9

I strongly reccomend The Cauchy-Schwarz Master Class by J. Michael Steele. It could be read by advanced high school students who did well in calculus and have a strong interest in mathematics although it is probably better suited for first year undergraduate math majors. It reads like a novel that contains plenty of challenging exercises. Another book that ...


9

Lillian R. Lieber Author of Infinity: Beyond the Beyond the Beyond


9

Here goes nothing. I have many other suggestions that are not listed; there is always one more example of a great math book: "The enjoyment of Mathematics" H. Rademacher, O. Toeplitz. You can't beat this classic collection of serious (and beautiful) results with elementary proofs. "A mathematician's miscellany" J. Littlewood. This is much more advanced, ...


9

Math Girls! It gave me a small taste of generating functions, and I want more! Very challenging math. Surreal Numbers might be the best way to learn about Conway's system. Math Girls does not have math that is as original. But it is high-level math, and it's fun. (The storyline is more engaging than that of Surreal Numbers. I love both books, in different ...


9

I am very fond of the book by Steven G. Krantz, How to Teach Mathematics, published by the American Mathematical Society (AMS), now in its 3rd edition. The book is mostly aimed at professors and lecturers of undergraduate mathematics. However, I think that much the advice would be broadly applicable. There is a section addressed specifically to TA's, but ...


8

The books by Willian Dunham ("Journey through genius" (Penguin, 1991), "Euler, master of us all" (MAA, 1999), "The calculus gallery" (Princeton University Press, 2008) are the ones I've read) are outstanding. They show mathemathics in terms of the original work (more or less, using modern notation), and motivate the subject matter well. They do require some ...


8

I greatly enjoyed Another Fine Math You've Got Me Into by Ian Stewart. Entertaining and at a pace that any level of mathematician or non-mathematician would be comfortable with, but nevertheless discusses some very interesting and beautiful topics.


8

I have about twenty kids' books listed at the books page on my blog, Math Mama Writes. If I had to narrow it down, my absolute favorite is probably The Cat in Numberland, by Ivar Ekeland, a five-chapter picture book dealing with the story of the Hotel Infinity (with Mr. and Mrs. Hilbert running the hotel, of course). The confused cat is charming, Ekeland's ...


8

Martin Gardner's The Colossal Book of Mathematics. It contains many of his best columns from Scientific American on recreational mathematics. My favorite chapter is the April Fool's day chapter, which includes a 'counter-example' to the Four-Coloring Theorem.


8

Surreal Numbers by Donald Knuth is a story about two people discovering the surreal numbers and proving theorems about them.


8

My pick would be Volume 1 of Richard Courant and Fritz John's Introduction to Calculus and Analysis. For discussions of how Courant/John compares with Spivak, Apostol, and other books, see the math StackExchange question Difficulty level of Courant's book. Volume 1 of Courant/John does not have any solutions to the exercises, but complete solutions to all ...


8

Check out http://jiblm.org. There are lots of scripts here, some better than others. A nice book in this style is "Distilling Ideas" by Brain Katz and Michael Starbird. I also recommend the following method: Take a reputable text on a topic, and try to prove all the theorems for yourself. If you get stuck for a long time, take a quick peek to get unstuck…...


8

Two very good books, inspiring autobiographies: 1) "A mathematician grappling with his century", by Laurent Schwartz. Another great french mathematician. This book is even better, in my opinion than the one of André Weil, "The Apprenticeship of a Mathematician", which is already very interesting. 2) "I Want to Be a Mathematician: An Automathography", by ...


8

I recommend How to Solve It by George Polya. I merely glanced at it in high school and it helped me to work on math problems. More recently I bought it and now read the whole thing annually, and it now governs how I help my students to solve problems. This book definitely meets your first three criteria. Although it tries to be broader than just mathematics,...


7

I am going to copy and paste my answer from another question on this site, because I think one would be hard pressed to beat it in terms of the number of suggestions it covers, and the general quality with which it presents these suggestions: You might be interested in the expansive answers that were generated on math.stackexchange by the questions Book ...


7

This California college has a no-frills $612$-page textbook available for free PDF download: College of the Redwoods Mathematics Prealgebra Textbook. 2012-13. (weblink). A solutions manual is available at the same link. Because you can download it easily, you can browse through it to decide if it meets your needs before committing to your (admirable) ...


7

There are so many Calc 1, 2, and Multi video lectures online now. Watch a few... take notes on what you think are some good traits of the person teaching. MIT's Calc course (super fast) might be helpful if you're teaching that style and speed. Some resources This text, How to Teach Mathematics - S. Krantz, has many useful guides for the beginning person. ...


7

The Diamond Age, a science fiction novel by Neal Stephenson, includes a subplot that involves mechanical Turing machines that are described in great detail and that are used in solving key puzzles that move the subplot along. The subplot involving Turing machines is only one part of a complex and multilayered story, but I thought I'd bring it up in case you ...


7

"(Books in German or with German translation are a plus)" and "she is particularly interested in geometry." The OP's notes suggest: Ziegler, Günter M. Do I Count?: Stories from Mathematics. CRC Press, 2013. CRC link.                   Ziegler, Günter M.: Darf ich Zahlen? Geschichten aus der ...


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