What are some books that discuss elementary mathematical topics ('school mathematics'), like arithmetic, basic nonabstract algebra, plane & solid geometry, trigonometry, etc, in an insightful way? I'm thinking of books like Klein's Elementary Mathematics from an Advanced Standpoint and the books of the Gelfand Correspondence School  schoollevel books with a university ethos.

Geometry and the Imagination by Hilbert and CohnVossen. 


If firstorder logic counts as "elementary mathematics", then I would like to suggest (the relevant chapters of) "Godel, Escher, Bach", by Douglas Hofstadter. (As an aside: Hofstadter's puzzle of encoding "n is a power of 10" as a predicate in Peano arithmetic is a wonderful one, quite tough even for professional mathematicians, especially if one is to avoid any form of the Godel numbering trick.) 


What Is Mathematics? An Elementary Approach to Ideas and Methods by Richard Courant and Herbert Robbins Lessons in Geometry by Jacques Hadamard, and its companion books: Hadamard's Plane Geometry and Hadamard: elementary geometry. solutions and notes to supplementary problems by Mark Saul. 


Mathematics: A Very Short Introduction by Timothy Gowers. It is very short and indeed very insightful. It is not a textbook, but includes some schoolmathematics topics. From the cover:



I really like Concrete Mathematics by Knuth, Graham and Patashnik, and the introductions to number theory by Rose and by Hardy&Wright: you will find there many interesting schoollike problems (but the whole books may not be suitable). In geometry, I can suggest Hartshorne's Geometry: Euclid and beyond. Books like Géométrie projective by Pierre Samuel or Artin's Geometric algebra contain a lot of algebra, but it is geometric instead of abstract, so you may judge they are on the safe side. 


I recommmend How to prove it by Daniel J. Velleman. The book introduces the basic logic and proof method to beginners and have many good examples and exercises to make students better understanding on what is a proof in the very elementary mathematics. 


Euclid's elements. i find it much more useful than Klein's books, but that may mean i misunderstand the question. indeed after many years of perusing them, i find Klein's "from an advanced standpoint" books more of a polemic than a useful text. Euclid on the other hand introduces many of the main ideas of modern mathematics. 


Mathematics Made Difficult by Carl E. Linderholm. 


Walter Prenowitz and Meyer Jordan, Basic Concepts of Geometry. 


I always enjoyed "How to Solve It: A New Aspect of Mathematical Method" by G. Pólya. It doesn't really cover all that much mathematics, it just helps you structure your thoughts in a mathematical sense. But it depends a lot on your actual needs. 


I personally enjoyed these books: How To Solve It by George Polya Geometry Revisited by H. S. M. Coxeter , Samuel L. Greitzer 


How about Lawvere&Schanuel's "Conceptual Mathematics: A First Introduction to Categories"? See, e.g., http://books.google.fr/books?id=h0zOGPlFmcQC&lpg=PP1&dq=lawvere&pg=PP1#v=onepage&q=lawvere&f=false 


protected by François G. Dorais♦ Sep 21 '13 at 20:53
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