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Our approach follows the philosophy that intuitive understanding precedes formal structure. In Geometry, starting from explorers' preexisting spatial intuition, we prove facts, where a "proof" is as follows:
A proof is a convincing mathematical explanation of a statement, using
(1) facts most would agree are obvious
(2) facts we've already "proved"
(3) logic
and ideally giving us a satisfying explanation for why the statement is true.
By leaving the axiomatic approach until later, we allow you to give arguments that you find convincing, using methods of your choice, from traditional synthetic proofs to coordinates, transformations, constructions---whatever suits the moment. We no longer have to reject nice proofs because they don't fit our axiomatic system, and learners develop the maximum amount of geometric intuition.
For interested students, Foundations of geometry turns a critical eye to our foundations, asking: Which facts have we assumed are obvious? Are they actually true? Can we prove them using more obvious facts? We gradually reduce our list of obvious facts to a concise list of "axioms," and we find proofs of our theorems that use only these axioms. In the process, we examine the famous attempts of Euclid, Hilbert, and Birkhoff to do exactly this. (We also present an alternative modern approach to the foundations of geometry, based on real numbers and coordinates.)