References for “high-school mathematics from an advanced point of view”

What are some good references for high-school (or even middle-school) mathematics, written from an advanced point of view, especially texts written for prospective teachers?

Some references were given in a comment for an earlier question, but I cannot find that one now ...

• I sympathize with this question, because unfortunately a lot of valuable content on this site is getting hidden in the comments. – Chris Cunningham May 5 '14 at 15:36
• – Benjamin Dickman May 5 '14 at 17:03
• Check this and this. – user5402 May 25 '14 at 21:58

A good (arguably excellent) overview of pre-calculus mathematics can be found in Precalculus Mathematics in a Nutshell, written by George F. Simmons. It is an excellent but terse resource. More thorough accounts for each topic can be found elsewhere if desired.

For algebra and trigonometry, you may wish to consider

For geometry,

There is no Calculus in Simmon's text mentioned above. However, many excellent treatments can be found elsewhere. These include,

Right now the best-known book of this sort is "Mathematics for High School Teachers -- An Advanced Perspective" by Zalman Usiskin. (I say "right now" because I have grandiose plans of writing such a textbook myself, but not until after I get tenure, i.e. not for at least 3 years.)

Comprehensive references on elementary mathematics from an advanced point of view are:

• the three-volume series Fundamentals of Mathematics (translated from German), edited by Behnke, Bachmann, Fladt, Süss and Kunle.

• the five-volume Entsyklopediya Elementarnoy Matematiki (Russian, Encyclopedia of Elementary Mathematics), edited by Aleksandrov, Markushevich and Khinchin.

• A Comprehensive Textbook of Classical Mathematics by Griffiths and Hilton

More specifically for geometry, restricting myself to Western languages, I would recommend one of the following.

• Géométrie euclidienne plane by Doneddu for discussion of a modern axiomatization along Euclidean lines, but based on isometries. The idea for this axiom system is originally due to Brisac and its most characteristic axiom is one stating that given any two flags (triples consisting of a point, a ray originating from that point, and one of the two half-planes determined by the support of the ray), there is a unique isometry carrying one flag to the other. The book also contains a rigorous construction of the real numbers, starting from Peano's axioms on the natural numbers.

• Lectures on the Foundations of Geometry, by Pogorelov, which develops the above axiom system but also looks at Lobachevskian and projective geometry, as well as geometry in space. (This book was written to be used in the geometry course in pedagogical faculties in the USSR - something that would be impossible for probably over 95% of prospective high school teachers in North America.)

• Geometría elemental by Pogorelov (the Spanish translation of Elementarnaya Geometriya) for an excellent, rigorous presentation of basic geometry which was intended to be directly usable in schools. In fact, a modified version of this system is used in Pogorelov's textbooks, which are widely used in Russian schools.

• For a rigorous development of geometry using vectors, Dieudonné's Algèbre linéaire et Géométrie élémentaire is excellent.