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.

1$\begingroup$ I don't think that this question is really appropriate for MO. $\endgroup$ – Andy Putman Sep 20 '13 at 19:02

25$\begingroup$ I realize that MO has changed since the old days, but still I think this question is appropriate. Insightful books on elementary mathematics are quite uncommon, and I'd like to see more of them. $\endgroup$ – John Stillwell Sep 20 '13 at 19:24

1$\begingroup$ Mathoverflow questions that guide up to the lists of great books  I adore. I always add to favorites. This is totally appropriate! $\endgroup$ – Olga Nov 11 '13 at 5:08

1$\begingroup$ Lockhart's Measurement is very good. $\endgroup$ – Marius Kempe Feb 11 '15 at 14:51



1$\begingroup$ Excellent book indeed, but I would not call this "elementary mathematics". $\endgroup$ – Alexandre Eremenko Sep 21 '13 at 4:13

$\begingroup$ Well, the book certainly starts off elementary, with discussions of conics, culminating in a beautiful argument that slices of cones have foci. I think it was meant to be understood by nonmathematicians. $\endgroup$ – Todd Trimble♦ Sep 21 '13 at 13:24
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.)

2$\begingroup$ I agree that it is a good and accessible book with significant mathematical content. One of the dangers is that you look for other books which attempt such holistic approaches in other sciences and you do not find them. $\endgroup$ – The Masked Avenger Sep 20 '13 at 22:08
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.

2$\begingroup$ Finally someone took the trouble to republish Hadamard's Lessons! And translate Perepelkin's solutions, even. Wow!! $\endgroup$ – darij grinberg Sep 21 '13 at 2:19
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:
The aim of this book is to explain, carefully but not technically, the differences between advanced, researchlevel mathematics, and the sort of mathematics we learn at school.
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.

3$\begingroup$ It's a curious choice. I cannot deny that it discusses elementary mathematics in an insightful way. And yet, it is so wry or ironic about how those insights are formulated that it basically subverts the presumptive purpose of discussing elementary mathematics in an insightful way! (When I first saw the book as an undergraduate, I wasn't really in on the "joke".) $\endgroup$ – Todd Trimble♦ Sep 21 '13 at 12:56
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.
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
In "On teaching mathematics", V. Arnold mentions Numbers and Figures by Rademacher and Töplitz, Geometry and the Imagination by Hilbert and CohnVossen, What is Mathematics? by Courant and Robbins, How to Solve It and Mathematics and Plausible Reasoning by Polya, and Development of Mathematics in the 19th Century by F. Klein.
Some of these have been mentioned already, so perhaps this is an appropriate list, but I'm not familiar with all of these, so if someone would like to comment on these books, your input would be appreciated.
It was ages ago that I read in a library Mathematics: Its Content, Methods, and Meaning by Aleksandrov, Kolmogorov, and Lavrent'ev, but I still remember enjoying it.

$\begingroup$ Also enjoyed Mathematics and Logic by Kac and Ulam, at a slightly higher level. $\endgroup$ – Tom Copeland Apr 9 '18 at 18:30
Walter Prenowitz and Meyer Jordan, Basic Concepts of Geometry.
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

1$\begingroup$ I disagree with the downvote. The mathematics discussed actually is basic and elemental, meant to connect with the experiences of ordinary people, and the book bristles with insight. $\endgroup$ – Todd Trimble♦ Apr 25 '17 at 14:13

2$\begingroup$ I agree it's a great book and very accessible, probably even for an interested highschooler. The ideas are clear and we'll motivated. $\endgroup$ – ಠ_ಠ Apr 26 '17 at 10:49
While it contains much beyond school mathematics, a lot of school mathematics is treated in a beautiful way in Mathematics, Form and Function by Saunders Mac Lane.
Two great ones are:
 Fuchs, Tabachnikov: Mathematical Omnibus and
 Arnold: Lectures and Problems: A Gift to Young Mathematicians.
I would suggest "Numbers and functions from a classicalexperimental mathematicians point of view" by Victor H.Moll. It contains very elementary but also some more sophisticated themes.
I once also wrote an elementary book "Grundideen der Mathematik", B.I. Wissenschaftsverlag 1992. But it is out of print and in German, thus probably does not count.
Stewart's book is an old favorite:
Stewart, Ian. From here to infinity. With a foreword by James Joseph Sylvester. The Clarendon Press, Oxford University Press, New York, 1996.
Stewart has many popularisation books some of which have reached bestseller status, e.g.:
Professor Stewart's Cabinet of Mathematical Curiosities
or
Professor Stewart's Hoard of Mathematical Treasures
etc.